Workbook for Microeconomics: Exercises and Solutions (Springer Texts in Business and Economics) 3031419464, 9783031419461

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Springer Texts in Business and Economics

Karolina Sobczak-Marcinkowska Krzysztof Malaga

Workbook for Microeconomics Exercises and Solutions

Springer Texts in Business and Economics

Springer Texts in Business and Economics (STBE) delivers high-quality instructional content for undergraduates and graduates in all areas of Business/Management Science and Economics. The series is comprised of self-contained books with a broad and comprehensive coverage that are suitable for class as well as for individual selfstudy. All texts are authored by established experts in their fields and offer a solid methodological background, often accompanied by problems and exercises.

Karolina Sobczak-Marcinkowska · Krzysztof Malaga

Workbook for Microeconomics Exercises and Solutions

Karolina Sobczak-Marcinkowska Institute of Informatics and Quantitative Economics Pozna´n University of Economics and Business Pozna´n, Poland

Krzysztof Malaga Institute of Informatics and Quantitative Economics Pozna´n University of Economics and Business Pozna´n, Poland

ISSN 2192-4333 ISSN 2192-4341 (electronic) Springer Texts in Business and Economics ISBN 978-3-031-41946-1 ISBN 978-3-031-41947-8 (eBook) https://doi.org/10.1007/978-3-031-41947-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

This book presents solutions and answers to exercises included in Microeconomics. Static and Dynamic Analysis by Krzysztof Malaga and Karolina SobczakMarcinkowska (Cham, Switzerland: Springer Nature, 2022). Altogether, there are 62 exercises in five chapters. Pozna´n, Poland

Krzysztof Malaga Karolina Sobczak-Marcinkowska

v

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Rationality of Choices Made by Individual Consumer . . . . . . . . . . . . .

3

3 Rationality of Choices Made by a Group of Consumers . . . . . . . . . . . .

59

4 Rationality of Choices Made by Individual Producers . . . . . . . . . . . . . .

85

5 Rationality of Choices Made by a Group of Producers by Exogenously Determined Function of Demand for a Product . . . . 163 6 Rationality of Choices Made by Groups of Producers and Groups of Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

vii

About the Authors

Krzysztof Malaga, Dr. habil. is a distinguished economics professor at Pozna´n University of Economics and Business (Poland). With 4 monographs, 3 textbooks, 6 scripts and over 70 articles, he specializes in microeconomics, macroeconomics, economic growth and development. He has translated 11 French-language books with 29 editions in Poland. Formerly serving as the vice-dean and dean, he currently directs the Institute of Computer Science and Quantitative Economics and holds key positions as an editor-in-chief and scientific director in prominent international economic associations AIELF. Karolina Sobczak-Marcinkowska, Ph.D. holds a dual doctorate in economic sciences from University Rennes 1 (France) and Pozna´n University of Economics and Business (Poland). As an assistant professor in the Department of Operations Research and Mathematical Economics, she delivers engaging lectures and classes on diverse subjects such as microeconomics, macroeconomics, mathematical economics and DSGE models. She has contributed as a co-author to a script and several textbooks focused on microeconomics.

ix

Index of Mathematical Symbols

∀ ∃ ∃1 ∃>1 ¬ ∧ ∨ ⇒ ⇔ a, x ∈ R a = (a1 , a2 , . . . , an ) ∈ Rn x = (x1 , x2 , . . . , xn ) ∈ Rn A, X A, X ∼ x∼y

x y  xy R R+ int R+ ⊂ R+ Rn = R × R × . . . × R Rn+ = { x ∈ Rn | x ≥ 0} ⊂ Rn

Universal quantifier: “for all” Existential quantifier: “there exists” Existential quantifier: “there exists exactly one” Existential quantifier: “there exists more than one” Negation; ¬p “not p” Conjunction: “and” Disjunction: “or” Implication; p ⇒ q “if p then q” Equivalence: “if and only if” Numbers, scalars Vector of parameters Vector of variables Sets Matrices (with dimensions m by n) with m rows and n columns Indifference relation Bundle x is indifferent with respect to bundle y Strong preference relation Bundle x is strongly preferred over bundle y (Weak) preference relation Bundle x is (weakly) preferred over bundle y Set of real numbers Set of nonnegative real numbers Set of positive real numbers n-dimensional space of real numbers, Cartesian product of set R Nonnegative orthant (subspace) of set Rn xi

xii

Index of Mathematical Symbols

int Rn+ ⊂ Rn+ f :X →Y

y = f (x)

y = f (x1 , x2 )

dy , dx

y  , f  (x)

d2 y , dx2

y  , f  (x)

∂y ∂ xi

∂ f (x1 ,x2 ) ∂ xi

,

∂ 2 f (x1 ,x2 ) ∂ 2 (x1 ,x2 ) , ∂x2 , ∂ xi ∂ x j i

i = 1, 2

 ∂ 2 f (x1 ,x2 ) ∂ x12 ∂ 2 f (x1 ,x2 ) ∂ x2 ∂ x1

H (x1 , x2 ) =  d f (x)  dx 

∂ 2 f (x1 ,x2 ) ∂ x1 ∂ x2 ∂ 2 f (x1 ,x2 ) ∂ x22

x = x¯



∂ f (x1 , x2 )   ∂ xi x = x¯

2 p, x = i=1 pi xi det(A) or |A|    2  1  1 2 2 2 d E x1 , x2 = i=1 x i − x i  

  d x1 , x2 = max x 1 − x 2  i=1,2

x E =

 2

i=1

i

(xi )2

21

i



Interior of set Rn Function, X —Domain of a function (set of arguments of a function) Y —Codomain of a function (set of values of a function) Scalar function of one variable, x—Independent variable (argument/input of a function) y—Dependent variable (value of a function) Scalar function of two variables, x1 , x2 —Independent variables (arguments/ inputs of a function) y—Dependent variable (value of a function) First-order derivative order of function y = f (x) Second-order derivative of second order of function y = f (x) Partial first-order derivatives of function y = f (x1 , x2 ) with respect to variable xi , i = 1, 2 Partial second-order derivatives of function y = f (x1 , x2 ) Hessian, symmetric matrix of partial second-order derivatives of function y = f (x1 , x2 ) Value of a derivative of an one-variable function at point x¯ Value of a derivative of a two variable function with respect to variable xi (i = 1, 2) at point x¯ = (x¯ 1 , x¯ 2 ) Scalar product of two vectors p, x ∈ R2+ Determinant of matrix A Euclidean metric (distance) Non-Euclidean metric (distance) Euclidean norm

Index of Mathematical Symbols

xiii

x = max {|xi |}

Non-Euclidean norm

t = 0, 1, 2, . . . t ∈ [0; +∞)

Time as discrete variable Time as continuous variable

i=1, 2

Chapter 1

Introduction

“It is not enough to have a fit mind, but to use it properly” Réné Descartes.

We present to the Reader the Workbook which is a valuable and useful supplement to our textbook Malaga K., Sobczak K., Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, Cham, 2022. The Workbook contains solutions to 62 exercises given in the textbook and assigned to five chapters: • • • •

Chapter 2 Rationality of choices made by individual consumer Chapter 3 Rationality of choices made by a group of consumers Chapter 4 Rationality of choices made by individual producers Chapter 5 Rationality of choices made by a group of producers by exogenously determined function of demand for a product • Chapter 6 Rationality of choices made by a group of producers and consumers. These exercises are aimed at a more in-depth reflection on the microeconomic categories defined in the textbook, decision-making problems and microeconomic models. Unlike in the textbook, in the Workbook a symbol (*) marks these of solved exercises that we think are a bit more difficult. We assume that this Workbook will be a supplement to our manual. In no case should it be treated as a substitute to the textbook or independent development. For this reason, in the solutions to the exercises, we draw the Reader’s attention to methods of solving problems and on the results obtained. We leave their more comprehensive interpretation to students and professors, who should use the textbook itself, including the glossary and the mathematical appendix. We consider additional materials enclosed in the form of twenty-seven MATLAB files and nine Excel files to be a very important supplement to the Workbook. We expect that they will be of particular interest to students and professors who are interested in dynamic analysis. The files allow to introduce own data if the form of Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-41947-8_1.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Sobczak and K. Malaga, Workbook for Microeconomics, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-031-41947-8_1

1

2

1 Introduction

functions and/or values of parameters and to control for these values to observe how they influence results. The Reader can use as well other type of software or create own files treating the enclosed files as exemplary. We would like to thank Springer Nature Switzerland Publishing House for its publication, in particular to the following representatives of Springer: Dr. Johannes Glaeser, Senior Editor, Economics and Political Science, Springer of Heidelberg, and Mr. Vijay Kumar Selvaraj for their extraordinary professionalism and kindness during the editing of this book. We would also like to thank Dr. Giancarlo Ianulardo from the University of Exeter Business School in the UK for his encouragement in the development of this Workbook for Microeconomics. The motto of the Workbook, taken from Discourse on Method by René Descartes, is intended to encourage students to use both the textbook and the Workbook in a way that will contribute to a full understanding of the issues and decision-making problems presented in these books, but above all in a way that will stimulate their imagination and creativity. Krzysztof MALAGA Karolina SOBCZAK Pozna´n, 16.06.2023

Chapter 2

Rationality of Choices Made by Individual Consumer

This Chapter presents 15 exercises that help in understanding the problem of describing rational behaviour of an individual consumer. In particular they concern: relation of consumer preferences, a utility function as a numerical characteristic of the relation of consumer preferences, the Marshallian demand function understood as an optimal solution to utility maximization problem, the Hicksian demand function as an optimal solution to consumer spending minimization problem, relations between the Marshallian and Hicksian demand functions, as well as substitution and income effects of changes in prices of goods, presented in a form of the Slutsky equation. E2.1. There is given an increasing and twice differentiable utility function u: R2+ → R of a form: A. B. C. D. E. F.

u(x) = a1 e x1 + a2 e x2 + a3 , ai > 0, i = 1, 2, 3. 1 1 u(x) = a1 e x1 + a2 e x2 + a3 , ai > 0, i = 1, 2, 3. x2 u(x) = a1 xx11+a , ai > 0, i = 1, 2. 2 x2 u(x) = a1 x1 + a2 x2 + ax1α1 x2α2 , a, ai , αi > 0, α1 + α2 < 1, i = 1, 2. u(x) = a1 (x1 + ln x1 ) +  a2 (x2 +  ln x2 ), ai , xi > 0, i = 1, 2. u(x) = a1 x1 1 + x1α−1 + a2 x2 1 + x2α−1 , α ∈ (0; 1), ai > 0, i = 1, 2.

1 Calculate a value and give economic interpretation of: (a) a marginal utility of i-th good, (b) a growth rate of consumption bundle utility with respect to quantity of i-th good, (c) an elasticity of consumption bundle utility with respect to quantity of i-th good, (d) a marginal rate of substitution of the first (second) good by the second (first) good, (e) an elasticity of substitution of the first (second) good by the second (first) good, for consumption bundles: x1 = (1, 1), x2 = (1, 2). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Sobczak and K. Malaga, Workbook for Microeconomics, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-031-41947-8_2

3

4

2 Rationality of Choices Made by Individual Consumer

2. Check if the given function satisfies first Gossen’s law. Solutions Ad A1a ∂u(x) ∂u(x) T1 (x) = = a1 e x1 T2 (x) = = a2 e x 2 ∂ x1 ∂ x2     T1 x1 = a1 e > 0 T2 x1 = a2 e > 0     T1 x2 = a1 e > 0 T2 x2 = a1 e2 > 0 Ad A1b a1 e x 1 a2 e x 2 T1 (x) T2 (x) = = s2 (x) = x x x u(x) a1 e 1 + a2 e 2 + a3 u(x) a1 e 1 + a2 e x 2 + a3  1   a1 e a2 e s1 x = > 0[%] s2 x1 = > 0 [%] (a1 + a2 )e + a3 (a1 + a2 )e + a3     a1 e a2 e 2 s1 x 2 = > 0[%] s2 x2 = > 0 [%] (a1 + a2 e)e + a3 (a1 + a2 e)e + a3 s1 (x) =

Ad A1c a1 e x 1 x 1 a1 e x 1 + a2 e x 2 + a3 a1 e x 2 x 2 ε2 (x) = s2 (x) · x2 = x a1 e 1 + a2 e x 2 + a3  1 a1 e ε1 x = > 0[%] (a1 + a2 )e+a3   a2 e ε2 x1 = > 0[%] (a1 + a2 )e + a3   a1 e ε1 x2 = > 0[%] (a1 + a2 e)e + a3   2a2 e2 ε2 x2 = > 0[%] (a1 + a2 e)e + a3 ε1 (x) = s1 (x) · x1 =

Ad A1d σ12 (x) =

∂u(x) ∂ x1 ∂u(x) ∂ x2

=

a1 e x 1 a2 e x 2

σ21 (x) =

a2 e x 2 1 = σ12 (x) a1 e x 1

  a1   a2 ε12 x1 = > 0[%] ε21 x1 = > 0[%] a2 a1     a2 e a1 σ21 x2 = σ12 x2 = a2 e a1

2 Rationality of Choices Made by Individual Consumer

5

Ad A1e a2 e x 2 x 2 x1 a1 e x 1 x 1 1 = = ε (x) = 21 x2 a2 e x 2 x 2 ε12 (x) a1 e x 1 x 1  1  a1  1  a2 ε12 x = > 0[%] ε21 x = > 0[%] a2 a1     2a2 e a1 ε12 x2 = > 0[%] > 0[%] ε21 x2 = 2a2 e a1 ε12 (x) = σ12 (x)

Ad A2 2 ∂ 2 u(x) x1 ∂ u(x) = a e = a2 e x 2 1 ∂ x12 ∂ x12   ∂ 2 u(x)  ∂ 2 u(x)  = a e > 0 = a2 e > 0 1 ∂ x12 x=x1 ∂ x22 x=x1   ∂ 2 u(x)  ∂ 2 u(x)  = a1 e > 0 = a2 e 2 > 0 ∂ x12 x=x2 ∂ x22 x=x2

Ad B1a 1 ∂u(x) 1 = − 2 a1 e x 1 < 0 ∂ x1 x1 1 ∂u(x) 1 T2 (x) = = − 2 a2 e x 2 < 0 ∂ x2 x2     T1 x1 = −a1 e < 0 T2 x1 = −a2 e < 0     1 1 T1 x2 = −a1 e < 0 T2 x2 = − a2 e 2 < 0 4

T1 (x) =

Ad B1b 1

− x12 a1 e x1 T1 (x) 1 = s1 (x) = 1 1 u(x) x1 a1 e + a2 e x 2 + a3 1

s2 (x) =   s1 x 1 =   s2 x 1 =   s1 x 2 =

− x12 a2 e x2 T2 (x) 2 = 1 1 u(x) a1 e x 1 + a2 e x 2 + a3 −a1 e < 0[%] (a1 + a2 )e + a3 −a2 e < 0[%] (a1 + a2 )e + a3 −a1 e < 0[%] (a1 + a2 )e + a3

6

2 Rationality of Choices Made by Individual Consumer

  s2 x 2 =

1

− 41 a2 e 2 1

a1 e + a2 e 2 + a3

< 0[%]

Ad B1c 1

ε1 (x) = s1 (x) · x1 =

− x11 a1 e x1 1

1

a1 e x 1 + a2 e x 2 + a3

0 σ12 (x) x22 a1 e x1   a1   a2 σ12 x1 = > 0 σ21 x1 = >0 a2 a1 1   4a1 e 2   a2 > 0 σ21 x2 = σ12 x2 = 1 > 0 a2 4a1 e 2 Ad B1e 1

ε12 (x) = σ12 (x)

x1 x 2 a1 e x 1 = 1 > 0[%] x2 x 1 a2 e x 2 1

x 1 a2 e x 2 1 ε21 (x) = = 1 > 0[%] ε12 (x) x 2 a1 e x 1 1  2  2a1 e 2   a2 ε12 x = > 0[%] ε21 x2 = 1 > 0[%]. a2 2a1 e 2

2 Rationality of Choices Made by Individual Consumer

  2a1 e 2   a2 ε12 x2 = > 0[%] ε21 x2 = 1 > 0[%] a2 2a1 e 2 1

Ad B2 1 1 ∂ 2 u(x) = 2x1−3 a1 e x1 + x1−4 a1 e x1 > 0 2 ∂ x1 1 1 ∂ 2 u(x) = 2x2−3 a2 e x2 + x2−4 a2 e x2 > 0 2 ∂ x2   2 ∂ u(x)  ∂ 2 u(x)  = 3a1 e > 0 = 3a2 e > 0 ∂ x12 x=x1 ∂ x22 x=x1   ∂ 2 u(x)  ∂ 2 u(x)  5 1 a2 e 2 > 0 = 3a e > 0 = 1 2  2  16 ∂ x1 x=x2 ∂ x2 x=x2

Ad C1a ∂u(x) a2 x22 ∂u(x) a1 x12 = T (x) = = 2 ∂ x1 ∂ x2 (a1 x1 + a2 x2 )2 (a1 x1 + a2 x2 )2  1   a2 a1 T1 x = >0 T2 x1 = >0 (a1 + a2 )2 (a1 + a2 )2     4a2 a1 T1 x2 = >0 T2 x2 = >0 2 (a1 + 2a2 ) (a1 + 2a2 )2 T1 (x) =

Ad C1b a2 x 2 a1 x 1 T1 (x) T2 (x) = = s2 (x) = u(x) x1 (a1 x1 + a2 x2 ) u(x) x2 (a1 x1 + a2 x2 )  1  1 a2 a1 > 0[%] s2 x = > 0[%] s1 x = a1 + a2 a1 + a2     2a2 2a1 s1 x 2 = > 0[%] s2 x2 = > 0[%] a1 + 2a2 2a1 + 4a2 s1 (x) =

Ad C1c a2 x 2 a1 x 1 ε2 (x) = s2 (x) · x2 = a1 x 1 + a2 x 2 a1 x 1 + a2 x 2  1  1 a2 a1 ε1 x = > 0[%] ε2 x = > 0[%] a1 + a2 a1 + a2     2a2 a1 > 0[%] ε2 x2 = > 0[%] ε1 x2 = a1 + 2a2 a1 + 2a2 ε1 (x) = s1 (x) · x1 =

7

8

2 Rationality of Choices Made by Individual Consumer

Ad C1d ∂u(x) ∂ x1 ∂u(x) ∂ x2

a1 x12 a2 x22 1 = σ = (x) 21 σ12 (x) a1 x12 a2 x22   a2   a1 σ12 x1 = > 0 σ21 x1 = >0 a1 a2   4a2   a1 σ12 x2 = > 0 σ21 x2 = >0 a1 4a2 σ12 (x) =

=

Ad C1e a1 x 1 x1 a2 x 2 1 = = ε21 (x) = x2 a1 x 1 ε12 (x) a2 x 2   a2   a1 ε12 x1 = > 0[%] ε21 x1 = > 0[%] a1 a2   2a2   a1 ε12 x2 = > 0[%] ε21 x2 = > 0[%] a1 2a2 ε12 (x) = σ12 (x)

Ad C2 ∂ 2 u(x) −2a1 a2 x22 ∂ 2 u(x) −2a1 a2 x12 = < 0 = 0 T1 (x) =

2 Rationality of Choices Made by Individual Consumer

Ad D1b s1 (x) = s2 (x) =   s1 x 1 =   s2 x 1 =   s1 x 2 =   s2 x 2 =

a1 + aα1 x1α1 −1 x2α2 T1 (x) = u(x) a1 x1 + a2 x2 + ax1α1 x2α2 a2 + aα2 x1α1 x2α2 −1 T2 (x) = u(x) a1 x1 + a2 x2 + ax1α1 x2α2 a1 + aα1 > 0[%] a1 + a2 + a a2 + aα2 > 0[%] a1 + 2a2 + a a1 + aα1 2α2 > 0[%] a1 + 2a2 + a2α2 a2 + aα2 2α2 −1 > 0[%] a1 + 2a2 + a2α2

Ad D1c a1 x1 + aα1 x1α1 x2α2 a1 x1 + a2 x2 + ax1α1 x2α2 a2 x2 + aα2 x1α1 x2α2 ε2 (x) = s2 (x) · x2 = a1 x1 + a2 x2 + ax1α1 x2α2   a1 + +aα1 > 0[%] ε1 x1 = a1 + a2 + a   a2 + aα2 ε2 x1 = > 0[%] a1 + a2 + a   a1 + aα1 2α2 ε1 x2 = > 0[%] a1 + 2a2 + a2α2   a2 + aα2 2α2 ε2 x2 = > 0[%] a1 + 2a2 + a2α2 ε1 (x) = s1 (x) · x1 =

Ad D1d ∂u(x) ∂ x1 ∂u(x) ∂ x2

a2 + aα2 x1α1 x2α2 −1 a1 + aα1 x1α1 −1 x2α2 1 = σ21 (x) = α1 α2 −1 a2 + aα2 x1 x σ12 (x) a1 + aα1 x1α1 −1 x α2   a1 + aα1   a2 + aα2 σ12 x1 = >0 σ21 x1 = >0 a2 + aα2 a1 + aα1    2  a2 + aα2 2α2 −1 a1 + aα1 2α2 σ12 x2 = x = > 0 σ >0 21 a2 + aα2 2α2 −1 a1 + aα1 2α2 σ12 (x) =

=

9

10

2 Rationality of Choices Made by Individual Consumer

Ad D1e ε12 (x) = σ12 (x) ε21 (x) =   ε12 x1 =   ε21 x1 =   ε12 x2 =   ε21 x2 =

x1 a1 + aα1 x1α1 x2α2 = x2 a2 + aα2 x1α1 x α2 −1

a2 + aα2 x1α1 x α2 −1 1 = ε12 (x) a1 + aα1 x1α1 x2α2 a1 + aα1 > 0[%] a2 + aα2 a2 + aα2 > 0[%] a1 + aα1 a1 + aα1 2α2 > 0[%] a2 + aα2 2α2 −1 a2 + aα2 2α2 −1 > 0[%] a1 + aα1 2α2

Ad D2 ∂ 2 u(x) = (α1 − 1)aα1 x1α1 −1 x2α2 < 0 ∂ x12 ∂ 2 u(x) ∂ x12  ∂ 2 u(x)  ∂ x12 x=x1  ∂ 2 u(x)  ∂ x22 x=x1  ∂ 2 u(x)  ∂ x12 x=x2  ∂ 2 u(x)  ∂ x22 x=x2

= (α2 − 1)aα2 x1α1 x2α2 −1 < 0 = (α1 − 1)aα1 < 0 = (α2 − 1)aα2 < 0 = (α1 − 1)aα1 2α2 < 0 = (α2 − 1)aα2 2α2 −1 < 0

Ad E1a   1 ∂u(x) = a1 1 + ∂ x1 x1   1 ∂u(x) T2 (x) = = a2 1 + ∂ x2 x2  1  1 T2 x = 2a2 > 0 T1 x = 2a1 > 0  2   3 T1 x = 2a1 > 0 T2 x2 = a2 > 0 2 T1 (x) =

2 Rationality of Choices Made by Individual Consumer

Ad E1b   a1 1 + x11 T1 (x) = s1 (x) = u(x) a1 (x1 + ln x1 ) + a2 (x2 + ln x2 ) + a3   1 a 1 + 2 x2 T2 (x) = s2 (x) = u(x) a1 (x1 + ln x1 ) + a2 (x2 + ln x2 ) + a3   2a1 s1 x 1 = > 0[%] a1 + a2 + a3   2a2 s2 x 1 = > 0[%] a1 + a2 + a3   2a1 s1 x 2 = > 0[%] a1 + a2 (2 + ln 2) + a3 3   a 2 2 s2 x 2 = > 0[%] a1 + a2 (2 + ln 2) + a3 Ad E1c a1 (x1 + 1) a1 (x1 + ln x1 ) + a2 (x2 + ln x2 ) + a3 a2 (x2 + 1) ε2 (x) = s2 (x)x2 a1 (x1 + ln x1 ) + a2 (x2 + ln x2 ) + a3     2a1 2a2 ε1 x1 = > 0[%] ε2 x1 = > 0[%] a1 + a2 + a3 a1 + a2 + a3   2a1 ε1 x2 = > 0[%] a1 + a2 (2 + ln 2) + a3   2a2 ε2 x2 = > 0[%] a1 + a2 (2 + ln 2) + a3 ε1 (x) = s1 (x)x1 =

Ad E1d σ12 (x) =

σ21 (x) =   σ12 x1 =   σ12 x2 =

  a1 1 + x11  >0 =  a2 1 + x12   1 a 1 + 2 x2 1  =  σ12 (x) a1 1 + x11   a1 a2 > 0 σ21 x1 = >0 a1 a2   4a1 3a2 > 0 σ21 x2 = >0 4a1 3a2 ∂u(x) ∂ x1 ∂u(x) ∂ x2

11

12

2 Rationality of Choices Made by Individual Consumer

Ad E1e a1 (x1 + 1) x1 = x2 a2 (x2 + 1) a2 (x2 + 1) 1 ε21 (x) = = ε12 (x) a1 (x1 + 1)  1  a2   a1 ε12 x = > 0[%] ε21 x1 = > 0[%] a1 a2   3a2   2a1 ε12 x2 = > 0[%] ε21 x2 = > 0[%] 2a1 3a2 ε12 (x) = σ12 (x)

Ad E2 ∂ 2 u(x) −a1 = 2 0 T2 x2 = a2 (1 + α)2α2 −1 > 0 T1 (x) =

Ad F1b s1 (x) =

a1 + αa1 x1α−1 T1 (x) = u(x) a1 x1 + a1 x1α + a2 + a2 x2α

a2 + αa2 x2α−1 T2 (x) = u(x) a1 x1 + a1 x1α + a2 + a2 x2α   a1 (1 + α)   a2 (1 + α) s1 x 1 = > 0[%] s2 x1 = > 0[%] 2a1 + 2a2 2a1 + 2a2   a1 (1 + α) > 0[%] s1 x 2 = 2a1 + a2 (1 + 2α ) s2 (x) =

2 Rationality of Choices Made by Individual Consumer

  a2 (1 + α)2α−1 > 0[%] s2 x 2 = 2a1 + a2 (1 + 2α ) Ad F1c a1 x1 + αa1 x1α a1 x1 + a1 x1α + a2 + a2 x2α a2 x2 + αa2 x2α ε2 (x) = s(x) · x2 = a1 x1 + a1 x1α + a2 + a2 x2α   a1 (1 + α) > 0[%] ε1 x1 = 2a1 + a2 (1 + 2α )   a2 (1 + α) > 0[%] ε2 x1 = 2a1 + a2 (1 + 2α )   a1 (1 + α) > 0[%] ε1 x2 = 2a1 + a2 (1 + 2α )   a2 (2 + α2α ) > 0[%] ε2 x2 = 2a1 + a2 (1 + 2α ) ε1 (x) = s(x) · x1 =

Ad F1d σ12 (x) = σ21 (x) =   σ12 x1 =   σ21 x1 =   σ12 x2 =   σ21 x2 =

∂u(x) ∂ x1 ∂u(x) ∂ x2

=

a1 + αa1 x1α−1

a2 + αa2 x2α−1

a2 + αa2 x2α−1 1 = σ12 (x) a1 + αa1 x1α−1 a1 (1 + α)   >0 a2 1 + α2α−1   a2 1 + α2α−1 >0 a1 (1 + α) a1 (1 + α)   >0 a2 1 + α2α−1   a2 1 + α2α−1 >0 a1 (1 + α)

Ad F1e a1 x1 + αa1 x1α x1 = x2 a2 x2 + αa2 x2α a2 x2 + αa2 x2α 1 = ε21 (x) = ε12 (x) a1 x1 + αa1 x1α

ε12 (x) = σ12 (x)

13

14

2 Rationality of Choices Made by Individual Consumer

  a1 (1 + α) ε12 x1 = a2 (2 + α2α )   a2 (2 + α2α ) ε21 x1 = a1 (1 + α)  2 a1 (1 + α) ε12 x = a2 (2 + α2α )  2  a2 (2 + α2α ) ε21 x = a1 (1 + α)

> 0[%] > 0[%] > 0[%] > 0[%]

Ad F2 ∂ 2 u(x) ∂ 2 u(x) α−2 = (α − 1)αa x < 0 = (α − 1)αa2 x2α−2 < 0 1 1 ∂ x12 ∂ x12   ∂ 2 u(x)  ∂ 2 u(x)  = (α − 1)αa < 0 = (α − 1)αa2 < 0 1 ∂ x12 x=x1 ∂ x22 x=x1   ∂ 2 u(x)  ∂ 2 u(x)  = (α − 1)αa1 < 0 = (α − 1)αa2 2α−2 < 0 ∂ x12 x=x2 ∂ x22 x=x2 E2.2. There are given utility functions u: R2+ → R: u 1 (x) = a1 x1 + a2 x2 , ai > 0, i = 1, 2, u 2 (x) = Aa1 x1 +a2 x2 , A ∈ (0; 1), ai > 0, i = 1, 2, u 3 (x) = Aa1 x1 +a2 x2 , A > 1, ai > 0, i = 1, 2, u 4 (x) = ln(a1 x11+a2 x2 ) , ai > 0, i = 1, 2, u 5 (x) = ln(a1 x1 + a2 x2 ), ai > 0, i = 1, 2, β u 6 (x) = ax1α x2 , a > 0, α, β ∈ (0; 1), u 7 (x) = α ln x1 + β ln x2 , α, β ∈ (0; 1), α β u 8 (x) = Aax1 x2 , a > 0, A, α, β ∈ (0; 1), α β u 9 (x) = Aax1 x2 , a > 0, A > 1, α, β ∈ (0; 1), u 10 (x) = min{a1 x1 , a2 x2 }, ai > 0, i = 1, 2, γ γ u 11 (x) = a1 x1 + a2 x2 , ai > 0, i = 1, 2, γ ∈ (0; 1), θ  γ γ (l) u 12 (x) = a1 x1 + a2 x2 γ , θ, ai > 0, i = 1, 2, γ ∈ (0; 1).

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

1. 2. 3. 4.

Which of these functions describe the same relation of consumer preference? Which of them are positively homogeneous of degree θ > 0? Which of them have the same degree of the homogeneity? Which of them are (weakly) increasing, which (weakly) decreasing?

2 Rationality of Choices Made by Individual Consumer

15

Solutions Ad 1 Pairs of utility functions that describe the same consumer preference relationships are following: • functions u 1 (x) and u 3 (x), because u 3 (x) is exponential function with the function u1 (x), • functions u 1 (x) and u 5 (x), because u 5 (x) is logarithmic function with the function u 1 (x), • functions u 6 (x) and u 7 (x), because u 7 (x) is logarithmic function with the function u 6 (x), • functions u 6 (x) and u 9 (x), because u 9 (x) is exponential function with the function u 6 (x), • functions u 11 (x) and u 12 (x), if γ → 1.

a composition of an increasing a composition of an increasing a composition of an increasing a composition of an increasing

Ad 2 Let us note that: (a) ∀λ > 0u 1 (λx) = a1 λx1 + a2 λx2 = λu 1 (x), which means that the degree of homogeneity of the linear utility function is equal to θ = 1, (b) ∀λ > 0u 2 (λx) = Aa1 λx1 +a2 λx2 = λθ u 2 (x), which means that the decreasing exponential utility function u 2 (x) is not a positively homogeneous function, (c) ∀λ > 0u 3 (λx) = Aa1 λx1 +a2 λx2 = λθ u 3 (x), which means that the increasing exponential utility function u 3 (x) is not a positively homogeneous function, (d) ∀λ > 0u 4 (λx) = ln(a1 λx11+a2 λx2 ) = λθ u 4 (x), which means that the inverse of a logarithmic function is not a positively homogeneous function, (e) ∀λ > 0u 5 (λx) = ln(a1 λx1 + a2 λx2 ) = λθ u 5 (x), which means that the logarithmic function u 5 (x) is not a positively homogeneous function, (f) ∀λ > 0u 6 (λx) = a(λx1 )α (λx2 )β = λα+β u 6 (x), which means that the power function u 6 (x) is a positively homogeneous function of the degree θ = α + β > 0, (g) ∀λ > 0u 7 (λx) = αlnλx1 + a2 lnλx2 = λθ u 7 (x), which means that the logarithmic function u 7 (x) is not a positively homogeneous function, α β (h) ∀λ > 0u 8 (λx) = Aa(λx1 ) (λx2 ) = λθ u 8 (x), which means that the decreasing exponential utility function u 8 (x) is not a positively homogeneous function, α β (i) ∀λ > 0u 9 (λx) = Aa(λx1 ) (λx2 ) = λθ u 9 (x), which means that the increasing exponential utility function u 9 (x) is not a positively homogeneous function, (j) ∀λ > 0u 10 (λx) = min{λa1 x1 , λa2 x2 } = λ min{a1 x1 , a2 x2 } = λu 10 (x), which means that the Koopmans-Leontief utility function u 10 (x) is a positively homogeneous function of the degree θ = 1, (k) ∀λ > 0u 11 (λx) = a1 (λx1 )γ + a2 (λx2 )γ = λγ u 10 (x), which means that a subadditive utility function u 11 (x) is a positively homogeneous function of the degree θ = γ ,

16

2 Rationality of Choices Made by Individual Consumer θ

(l) ∀λ > 0u 12 (λx) = (a1 (λx1 )γ + a2 (λx2 )γ ) γ = λθ u 12 (x), which means that the CES utility function u 12 (x) is a positively homogeneous function of the degree θ > 0. Ad 3 Functions u 1 (x) and u 10 (x) are positively homogeneous of the degree θ = 1. Functions u 6 (x), u 11 (x)and u 12 (x) are positively homogeneous of the degree θ > 0. Ad 4 (a) (b) (c) (d) (e) (f) (g) (h)

∂u(x) ∂ x1 ∂u(x) ∂ x1 ∂u(x) ∂ x1 ∂u(x) ∂ x1 ∂u(x) ∂ x1 ∂u(x) ∂ x1 ∂u(x) ∂ x1 ∂u(x) ∂ x1 ∂u(x) ∂ x1

∂u(x) = a1 > 0 = a2 > 0, ∂ x2 ∂u(x) = a1 Aa1 x1 +a2 x2 ln A < 0 = a2 Aa1 x1 +a2 x2 ln A < 0, ∂ x2 ∂u(x) a1 x1 +a2 x2 = a1 A ln A > 0 = a2 Aa1 x1 +a2 x2 ln A > 0, ∂ x2 ∂u(x) = − ln(a1 x1 + a2 x2 )−2 ax11 < 0 = − ln(a1 x1 + a2 x2 )−2 ax22 < 0, ∂ x2 ∂u(x) a1 a2 = a1 x1 +a2 x2 > 0 = a1 x1 +a2 x2 > 0, ∂ x2 β

= αax1α−1 x2 > 0 ∂u(x) = ax11 > 0 = ∂ x2 α β

β

∂u(x) ∂ x2 a2 > x2

= a1 Aax1 x2 αax1α−1 x2 ln A < 0 β ax1α x2

β−1

= βax1α x2 0,

β αax1α−1 x2

∂u(x) ∂ x2 ∂u(x) ∂ x2

>0

α β

β−1

ln A < 0,

β ax1α x2

β−1 βax1α x2

ln A > 0,

= a2 Aax1 x2 βax1α x2

(i) = a1 A ln A > 0 = a2 A (j) The Koopmans-Leontief utility function is not differentiable. γ −1 γ −1 ∂u(x) (k) ∂u(x) = γ a1 x 1 > 0 = γ a2 x2 > 0, ∂ x1 ∂ x2 θ   −1 γ γ γ −1 (l) ∂u(x) = γθ a1 x1 + a2 x2 γ γ a1 x1 > 0 ∂ x1  γ γ  θ −1 γ −1 ∂u(x) = γθ a1 x1 + a2 x2 γ γ a2 x2 > 0. ∂ x2 E2.3. Determine if a given utility function u: R2+ → R of a form: (a) u(x) = a1 x1 − a2 x2 + a3 , ai > 0, i = 1, 2, (b) u(x) = a1 ln x1 − a2 ln x2 + a3 , ai > 0, i = 1, 2, 1

1

(c) u(x) = −a1 x12 + a2 x22 + a3 , ai > 0, i = 1, 2, is (weakly) increasing or (weakly) decreasing. Solutions Ad a ∂u(x) = a1 > 0 ∂ x1

∂u(x) = −a2 < 0 ∂ x2

∂u(x) a1 = >0 ∂ x1 x1

∂u(x) a2 =− 0 ∂ x2 ∂ x1 2

It follows that none of these utility functions is either (weakly) increasing or (weakly) decreasing. E2.4. There are given:



• a budget set D(p, I ) = x ∈ R2+ | p1 x1 + p2 x2 ≤ I ⊂ X = R2+ , • a supply set B = {(x1 , x2 ) ∈ R2+ |x1 ≤ b1 , x2 ≤ b2 } ⊂ X = R2+ , such that: a. ∀α, β ≥ 0, α + β = 1 0 < b1 < b. 0 < pI1 ≤ b1 and 0 < b2 < PI2 , c. 0 < b1 < pI1 and 0 ≤ PI2 < b2 . d. 0
0, i = 1, 2, α ∈ (0; 1), a Koopmans-Leontief function: u(x) = min{a1 x1 , a2 x2 }, ai > 0, i = 1, 2, θ  γ γ F. a CES function: u(x) = a1 x1 + a2 x2 γ , θ, ai > 0, i = 1, 2, γ ∈ (−1; 0) ∩ (0; +∞),

A. B. C. D. E.

knowing that a consumer chooses an optimal consumption bundle in a set B∩D(p, I ). 2. For each of considered sets B ∩ D(p, I ) of feasible solutions by each of utility functions, determine relationships between properties of the set (convex, bounded, closed, compact) and properties of the utility function (monotonicity, convexity or strict convexity). Write conclusions about number of optimal consumption bundles. Solutions Case 1 The budget set is a subset of the supply set: D(p, I ) ⊆ B, which means that the supply of each good is sufficiently big in comparison to the consumer’s income. It is the case when: 0
0, p = λa then x = p1   I = (0, b2 ) = (b1 , 0) or x2 = 0, p2   B–F. ∃1 α, β > 0, α + β = 1 x = αp1I , βp2I = (αb1 , βb2 )

2 Rationality of Choices Made by Individual Consumer

19

Fig. 2.3 Supply set as subset of budget  set when b1 ∈ 0; αp1I and b2 ∈   0; βp2I (E2.4)

Case 2 The supply set is a proper subset of the budget set: B ⊂ D(p, I ), which means that the supply of each good is sufficiently small in comparison to the consumer’s income. It is the case when: ∀α, β ≥ 0, α + β = 1 0 < b1
0, α + β = 1 such that     B–F. I − αp1 b1 I − βp2 b2 , Case 3.2 x = , βb2 . Case 3.1 x = αb1 , p2 p1 If

Case 4 The budget set and the supply set are not disjoint but in the same time none of them is the proper subset of the other. It is the case when:

2 Rationality of Choices Made by Individual Consumer

21

Fig. 2.6 Supply set not being subset of budget set (and conversely) when 0 < I − p2 b2 < b1 < pI1 and 0 < p1 I − p1 b1 P2

< b2
0, p = λa then x = b1 , , b2 or x = p1     p2 B. B-F ∃1 α, β > 0, α + β = 1 x = α b1 + I −pp12 b2 , β b2 + I −pp21 b1 . If

E2.5. Justify by geometric means that a linear utility function: A. u(x) = a1 x1 + a2 x2 + a3 , ai > 0, i = 1, 2, 3 describes consumer goods which are perfect substitutes and not complementary for each other, and that a Koopmans-Leontief utility function: B. u(x) = min{a1 x1 , a2 x2 } + a3 , ai > 0, i = 1, 2, 3 describes consumer goods which are perfect complements and not substitute for each other. Solutions Ad a Notice that the indifference curve u(x) = a1 x1 + a2 x2 + a3 = u > 0 intersects 1 the horizontal axis at the point x1 = u−a , 0 , and the vertical axis at the point a2   u−a3 2 x = 0, a2 (Fig. 2.7).

22

2 Rationality of Choices Made by Individual Consumer

Fig. 2.7 Marginal rate of substitution of first good by second good in consumption bundle x0 ∈ G ⊂ R2+ for linear utility function (E2.5)

Fig. 2.8 Marginal rate of substitution of first good by second good in consumption bundle x0 ∈ G ⊂ R2+ for Koopmans-Leontief utility function (E2.5)

2 Let us notice that dx1 = x1 and dx2 = x2 . Since s12 (x) = − dx = tgα = dx1 const., we see that the marginal rate of substitution of the first (second) good by the second (first) in a consumption bundle x0 ∈ R2+ , which utility is described by the linear utility function, is constant. Thus, it does not depend on quantities of goods in the bundle. In this case, the goods are called perfect substitutes. In the same time, we can notice that in order to rise (reduce) the utility level of any consumption bundle x = (x1 , x2 ) ∈ R2+ , it is enough to increase (decrease) quantity of just one of the goods, not necessarily of both goods. This shows that the linear utility function describes goods which are not complementary for each other. Finally, the substitutability of consumer goods takes place when the utility of a consumption bundle x = (x1 , x2 ) ∈ R2+ is described by an increasing or by a decreasing utility function (Fig. 2.8).

Ad b Let usnotice that for an indifference  curve min{a1 x1 , a2 x2 } + a3 = u, we get that  u−a3 2 1 x1 = u−a , 0 and x = 0, . a2 a2

2 Rationality of Choices Made by Individual Consumer

23

∼ − dx2 , we Let us also notice that dx1 = x1 and dx2 = x2 = 0. Since s12 (x) = dx1 see that the marginal rate of substitution of the first good by the second in a consump  tion bundle x0 ∈ R2+ is equal to s12 x0 = 0, while that the marginal rate of substitution of the second good by the first good is undefined (its value is infinite). This shows that the Koopmans-Leontief utility function describes goods which are not substitute for each other. In the same time, we can notice that in order to rise (reduce) the utility level of any consumption bundle x = (x1 , x2 ) ∈ R2+ , it is necessary to increase (decrease) quantities of both goods in some fixed proportion, not just one of the goods. In this case, the goods are called perfect complements. Finally, the complementarity of consumer goods takes place when the utility of a consumption bundle x = (x1 , x2 ) ∈ R2+ is described by a weakly increasing or by a weakly decreasing utility function. E2.6. There are given a power utility function u(x) = ax1α1 x2α2 , a, αi > 0, α1 + α2 < 1, i = 1, 2 and a utility function u 2 (x) = α1 ln x1 + α2 ln x2 + ln a, xi > 0, i = 1, 2, which is a composition of the function u with an increasing logarithmic function. We know that u and u 2 describe the same relation of consumer preference. Solving consumption utility maximization problems with each of these functions check if they correspond to the same Marshallian demand function. Solution Let us notice that if u(x) = ax1α1 x2α2 , then u 2 (x) = ln[u(x)] = α1 ln x1 + α2 ln x2 + ln a. Thus, two problems of maximizing the utility of consumption are given: (P2) u(x) = ax1α1 x2α2 → max (1) p1 x1 + p2 x2 ≤ I (2) x1 , x2 ≥ 0 (P2) u 2 (x) = α1 ln x1 + α2 ln x2 + ln a3 → max (1) p1 x1 + p2 x2 ≤ I (2) x1 , x2 ≥ 0. The optimal solution to the problem (T1) is to solve the system of equations: (1) s12 (x) =

∂u(x) ∂ x1 ∂u(x) ∂ x2

=

p1 , p2

(2) p1 x1 + p2 x2 = I. From Eq. (1), it follows that: β

(3)

α1 ax1α−1 x2 β−1 α2 ax1α x2

=

From where:

α1 x2 α2 x1

=

p1 . p2

24

2 Rationality of Choices Made by Individual Consumer

(4) x2 =

α2 p1 x . α1 p2 1

Substituting (4) into (2) we get: (5) p1 x1 + p2 αα21 pp21 x1 = I   2 =I (6) p1 x1 α1α+α 1 (7) x1 =

α1 I (α1 +α2 ) p1

Substituting (7) into (4), we get: (8) x2 =

α2 p1 α1 I α1 p2 (α1 +α2 ) p1

=

α2 I (α1 +α2 ) p2

Finally, the  optimal bundle of  goods in the problem (T1) has the α1 α2 1 I I , , and the maximum utility is u x1 = form: x = (α +α ) p1 (α1 +α2 ) p2  α1  α2  1 2 α1 +α 2 α2 I a αp11 p2 α1 +α2 The optimal solution to the problem (T2) is to solve the system of equations: (1) s12 (x) =

∂u(x) ∂ x1 ∂u(x) ∂ x2

=

p1 , p2

(2) p1 x1 + p2 x2 = I From Eq. (1), it follows that: (3)

α1 x1 α2 x2

=

α1 x2 α2 x1

=

p1 p2

From where: (4) x2 =

α2 p1 x α1 p2 1

Substituting (4) into (2), we get: (5) p1 x1 + p2 αα21 pp21 x1 = I   2 =I (6) p1 x1 α1α+α 1 (7) x1 =

α1 I (α1 +α2 ) p1

Substituting (4) into (2), we get: (8) x2 =

α2 p1 α1 I α1 p2 (α1 +α2 ) p1

=

α2 I (α1 +α2 ) p2

Finally, the  optimal bundle of  goods in the problem (T2) has the α1 α2 I I , , and the maximum utility is u 2 x2 = form: x2 = (α1 +α2 ) p1 (α1 +α2 ) p2 α1 +α2  α1  α2  α2 I ln a αp11 p2 α1 +α2          It follows that: x1 = x2 and u 2 x2 = ln u x1 , but u x1 = u 2 x2 E2.7. Using the Kuhn-Tucker theorem,1 find an optimal solution to a consumption utility maximization problem with an additional constraint on the supply of both goods: 0 ≤ xi ≤ bi , i = 1, 2, when a utility function is: 1

The theorem (Theorem A.6) is presented in a Mathematical appendix of the textbook Malaga K., Sobczak K, Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, 2022.

2 Rationality of Choices Made by Individual Consumer

25

A. a power function:u(x) = ax1α1 x2α2 , a, αi > 0, α1 + α2 < 1, i = 1, 2, B. logarithmic: u(x) = a1 ln x1 + a2 ln x2 , ai > 0, xi > 0, i = 1, 2, C. subadditive: u(x) = a1 x1α + a2 x2α , ai > 0, i = 1, 2, α ∈ (0; 1). Solutions There are given:



• a budget set D(p, I ) = x ∈ R2+ | p1 x1 + p2 x2 ≤ I ⊂ X = R2+ , • a supply set B = {(x1 , x2 ) ∈ R2+ |x1 ≤ b1 , x2 ≤ b2 } ⊂ X = R2+ , such that: (a) If 0 < b1 < I −pp12 b2 and 0 < b2 < I −Pp21 b1 then D(p, I ) ∩ B = B. (b) If 0 < pI1 < b1 and0 < PI2 < b2 then D(p, I ) ∩ B = D(p, I ). (c) If 0 < b1 < pI1 and0 < b2 < PI2 then D(p, I ) ∩ B = C. A Lagrange function has the following form: L(x1 , x2 , λ1 , λ2 , λ3 ) = u(x) + λ1 (b1 − x1 ) + λ2 (b2 − x2 ) + λ3 (I − p1 x1 − p2 x2 ). Ad a  ∂ L(x, λ)  = b1 − x 1 = 0 then x 1 = b1 . ∂λ1 λ1 =λ1  ∂ L(x, λ)  = b2 − x 2 = 0 then x 2 = b2 If  ∂λ If

2

λ2 =λ2

Ad A x = (x 1 , x 2 ) = (b1 , b2 ) > (0, 0) and u(x) = ab1α1 b2α2 > 0 Ad B x = (x 1 , x 2 ) = (b1 , b2 ) > (0, 0) and u(x) = a1 ln b1 + a2 ln b2 . Ad C x = (x 1 , x 2 ) = (b1 , b2 ) > (0, 0) and u(x) = a1 b1α + a2 b2α > 0.

26

2 Rationality of Choices Made by Individual Consumer

Ad b (1) (2) (3) (4)



∂ L (x,λ)  ∂ x1 x=x



∂u(x)  ∂ x1 x=x

=



  = ∂u(x) ∂ x2 

∂ L (x,λ)  ∂ x2 x=x



∂ L(x,λ)  = ∂λ3 λ =λ 3 3  ∂u(x)  ∂ x1 x=x  = pp21 ∂u(x)   ∂ x2

x=x

− λ3 p1 = 0 − λ3 p2 = 0

I − p1 x 1 − p2 x 2 = 0

x=x

(5) p1 x 1 + p2 x 2 = I . Ad A (6)

α −1 α2 x2 α α −1 α2 ax 1 1 x 2 2 α2 p1

α1 ax 1 1

=

α1 x 2 α2 x 1

p1 p2

=

x 2 = α1 p2 x 1 p1 x 1 + p2 αα21 pp21 x 1 = I α1 I x 1 = (α1 +α 2 ) p1 α2 I x 2 = (α1 +α 2 ) p2   α1 I α2 I (11) x = (α1 +α , (α1 +α2 ) p2 2 ) p1 (7) (8) (9) (10)

  (12) ∃1 α, β > 0, α + β = 1; x = αp1I , βp2I where α = 2 β = α1α+α >0 2 α1  α2  α1 +α2  α2 I (13) u(x) = a αp11 > 0. p2 α1 +α2

α1 α1 +α2

> 0 and

Ad B (6) (7) (8) (9) (10) (11)

a1 x1 a2 x2

a1 x 2 = pp21 a2 x 1 x 2 = aa21 pp21 x 1 p1 x 1 + p2 aa21 pp21 x 1 = I a1 I x 1 = (a1 +a 2 ) p1 a2 I x 2 = (a1 +a 2 ) p2   a1 I a2 I x = (a1 +a , ) p (a +a ) p 2 1 1 2 2

=

  (12) ∃1 α, β > 0, α + β = 1; x = αp1I , βp2I where α = a2 >0 a1 +a2

     a1 +a2  a1 a2 a1 a2 I . (13) u(x) = ln p1 p2 a1 +a2 Ad C (6)

αa1 x α−1 1 αa2 x α−1 2

(7) x 2 =



=

a1 x 1−α 2 a2 x 1−α 1

a2 p1 a1 p2

(8) p1 x 1 + p2

1  1−α



a2 p1 a1 p2

=

p1 p2

x1 1  1−α

x1 = I

a1 a1 +a2

> 0 and β =

2 Rationality of Choices Made by Individual Consumer

27

1

(9) x 1 =



(10) x 2 =



( p2 a1 ) 1−α I 1 p1 p21−α

1 a11−α

1

1



1

1



+ p2 p11−α a21−α 1

⎛

( p1 a2 ) 1−α I 1 p1 p21−α

⎜ x=⎜ ⎝

1 a11−α

+ p2 p11−α a21−α

( p2 a 1 ) 1 1−α

1 1−α

p1 p2 a 1

1 1−α

⎞

I 1 1−α

1 1−α

( p1 a 2 )

, 

+ p2 p1 a 2

1 1−α

1 1−α

p1 p2 a 1

1 1−α

I 1 1−α

1 1−α

+ p2 p1 a 2

⎛

(11)

⎞

⎜ =⎜ ⎝

( p2 a 1 ) 1 1−α

1 1−α

p2 a 1

1 1−α

I

α 1−α

1 1−α

1 a21−α 1

>0 ⎛

1

a11−α +a21−α



+ p2 p1 a 2

(12) ∃1 α, β > 0, α + β = 1; x =



( p1 a 2 )

, p1

αI βI , p1 p2

⎞α 1 1−α

α 1−α

1 1−α

p1 p2 a 1 

1 1−α

I

1 1−α

1 1−α

Ad c

(2) (3)



∂ L (x,λ)  ∂ x1 x=x



∂u(x)  ∂ x1 x=x

− λ3 p1 = 0   ∂ L (x,λ)   = ∂u(x) − λ3 p2 = 0 ∂ x2 x=x ∂ x2 x=x   ∂ L(x,λ) = I − p1 x 1 − p2 x 2 = 0 ∂λ3  

(4)

∂u(x)  ∂ x1 x=x  ∂u(x)  ∂ x2 x=x

=

λ3 =λ3

=

p1 p2

(5) p1 x 1 + p2 x 2 = I. Ad A α −1 α

(6) (7) (8) (9) (10) (11)

α1 ax 1 1 x 2 2 α1 x 2 = pp21 α α −1 = α x 2 1 α2 ax 1 1 x 2 2 x 2 = αα21 pp21 x 1 p1 x 1 + p2 αα21 pp21 x 1 = I α1 I x 1 = (α1 +α 2 ) p1 α2 I x 2 = (α1 +α 2 ) p2   α1 I α2 I x = (α1 +α , (α1 +α 2 ) p1 2 ) p2



+ p1 a 2

where α =

⎛

1 a11−α 1

1

a11−α +a21−α

⎞α 1 1−α

⎜ ⎟ ⎜ ⎟ a1 I a2 I ⎜  ⎟  ⎟ u(x) = a1 ⎜ ⎝ 1 ⎠ + a2 ⎝  1 ⎠ 1 1 (a11−α + a21−α p1 (a11−α + a21−α p2 (13) α  1   1 I α α 1−α 1−α a 1 p1 + a 2 p2 . = p1 p2 (1)

⎟ ⎟ ⎠

p2

⎟ ⎟ ⎠

> 0, β =

28

2 Rationality of Choices Made by Individual Consumer

  (12) ∃1 α, β > 0, α + β = 1 x = αp1I , βp2I where α = α2 >0 α1 +α2     (13) x 1 ∈ I −pp22 b2 ; b1 , x 2 ∈ I −pp11 b1 ; b2 α1 +α2  α1  α2  α2 I (14) u(x) = a αp11 > 0. p2 α1 +α2

α1 α1 +α2

> 0 and β =

Ad B (6) (7) (8) (9) (10) (11)

a1 x1 a2 x2

a1 x 2 = pp21 a2 x 1 x 2 = aa21 pp21 x 1 p1 x 1 + p2 aa21 pp21 x 1 = I a1 I x 1 = (a1 +a 2 ) p1 a2 I x 2 = (a1 +a 2 ) p2   a1 I a2 I x = (a1 +a , (a1 +a2 ) p2 2 ) p1

=

 αI βI where ∃1 α, β > 0, α + β = 1 x = , p1 p2 (12) a1 a2 α= > 0 and β = >0 a1 + a2   a1 + a2 (13) x 1 ∈ I −pp22 b2 ; b1 , x 2 ∈ I −pp11 b1 ; b2

     a1 +a2  a1 a2 a1 a2 I . (14) u(x) = ln p1 p2 a1 +a2 

Ad C (6)

αa1 x α−1 1 αa2 x α−1 2

(7) x 2 =



=

a1 x 1−α 2 a2 x 1−α 1

a2 p1 a1 p2

(8) p1 x 1 + p2 (9) x 1 = (10) x 2 =



a2 p1 a1 p2

p1 p2

x1 1  1−α

x1 = I

1 ( p2 a1 ) 1−α I  1 1 1 p1 p21−α a11−α + p2 p11−α

1



1



a21−α

1



⎛

( p1 a2 ) 1−α I 1 p1 p21−α

⎜ N x=⎜ ⎝ (11)



=

1 1−α

1 a11−α

1

+ p2 p11−α a21−α

( p2 a 1 ) 1 1−α

1 1−α

p1 p2 a 1

1 1−α

⎞

I 1 1−α

1 1−α

, 

+ p2 p1 a 2

( p1 a 2 ) 1 1−α

1 1−α

p1 p2 a 1

1 1−α

I 1 1−α

1 1−α

+ p2 p1 a 2

⎟ ⎟ ⎠

⎛ ⎜ =⎜ ⎝

⎞ ( p2 a 1 ) 1 1−α

1 1−α

p2 a 1

1 1−α

I

α 1−α

1 1−α

+ p2 p1 a 2



, p1

( p1 a 2 ) α 1−α

1 1−α

p1 p2 a 1

1 1−α

I

1 1−α

1 1−α

+ p1 a 2

 p2

⎟ ⎟ ⎠

2 Rationality of Choices Made by Individual Consumer

 ∃1 α, β > 0, α + β = 1 x = (12)

1 a11−α

α=

> 0, β =

αI βI , p1 p2

29

 where

1 a21−α

>0 1 1 1 1 1−α 1−α 1−α 1−α a + a a + a 2  1  2 1  (13) x 1 ∈ I −pp22 b2 ; b1 , x 2 ∈ I −pp11 b1 ; b2 ⎛ ⎞α ⎛ 1 1−α

⎞α 1 1−α

⎜ ⎟ ⎜ ⎟ a1 I a2 I ⎜  ⎟  ⎟ u(x) = a1 ⎜ ⎝ 1 ⎠ + a2 ⎝  1 ⎠ 1 1 (a11−α + a21−α p1 (a11−α + a21−α p2 (14) α  1   1 I a11−α p1α + a21−α p2α . = p1 p2 E2.8. There is given a market of two consumer goods, where: x = (x1 , x2 ) ∈ R2+ —a consumption bundle, p = ( p1 , p2 ) ∈ int R2+ —a vector of prices of goods, I > 0—a consumer’s income and a utility function describing a relation of consumer preference of a form: A. u(x) = min{a1 x1 , a2 x2 }, ai > 0, i = 1, 2, B. u(x) = a1 ln x1 + a2 ln x2 , ai > 0, xi > 0, i = 1, 2, β C. u(x) = ax1α x2 , a > 0, α, β ∈ (0; 1). 1. 2. 3. 4. 5. a. b. c. d.

Write a form of a consumption utility maximization problem. Determine a Marshallian demand function and an indirect utility function. Write a form of a consumer expenditure minimization problem. Determine a Hicksian demand function and a consumer expenditure function. Knowing that I = e(p, u) and that u = ν( p, I ), check if following Slutsky equations are true: ∂ϕ1 (p,I ) ∂ p1 ∂ϕ1 (p,I ) ∂ p2 ∂ϕ2 (p,I ) ∂ p2 ∂ϕ2 (p,I ) ∂ p1

= = = =

∂ f 1 (p,u) ∂ p1 ∂ f 1 (p,u) ∂ p2 ∂ f 2 (p,u) ∂ p2 ∂ f 2 (p,u) ∂ p1

− − − −

∂ϕ1 (p,I ) ϕ1 (p, ∂I ∂ϕ1 (p,I ) ϕ2 (p, ∂I ∂ϕ2 (p,I ) ϕ2 (p, ∂I ∂ϕ2 (p,I ) ϕ1 (p, ∂I

I ), I ), I ), I ),

6. Present graphic illustrations of the Slutsky equations of point 5. Solutions Ad A1 (P2) min{a1 x1 , a2 x2 } → max (1) p1 x1 + p2 x2 ≤ I (2) x1 , x2 ≥ 0.

30

2 Rationality of Choices Made by Individual Consumer

Ad B1 (P2) a1 ln x1 + a2 ln x2 → max (1) p1 x1 + p2 x2 ≤ I (2) x1 , x2 ≥ 0. Ad C1 β

(P2) ax1α x2 → max (1) p1 x1 + p2 x2 ≤ I (2) x1 , x2 ≥ 0. Ad A2 An optimal solution satisfies conditions: (3) min{a1 x 1 , a2 x 2 } = u. (4) a1 x 1 = a2 x 2 . To find the optimal solution, one needs to solve the equation system (3)–(4). The solution has a form:     p1 a 2 I I a2 I a1 I p2 a 1 = x= , , a 2 p1 + a 1 p2 a 2 p1 + a 1 p2 p1 a 2 + p2 a 1 p1 p1 a 2 + p2 a 1 p2   I I (5) . = ϕ(p, I ), where = α ,β p1 p2 p1 a 2 p2 a 1 α= > 0, β = >0 p1 a 2 + p2 a 1 p1 a 2 + p2 a 1 such thatα + β = 1.  a2 I a2 I a2 I = a2 pa11+a (6) u(x) = min a2 pa11+a , a2 pa11+a = ν(p, I ). 1 p2 1 p2 1 p2 Ad B2 An optimal solution satisfies conditions:  

(3) s12 (x 1 , x 2 ) =

∂u (x1 ,x2 )   ∂ x1

x   ∂u (x1 ,x2 )   ∂ x2 x

(4) p1 x 1 + p2 x 2 = I.

=x

=

a1 x1 a2 x2

=

a1 x 2 a1 x 1

=

p1 , p2

=x

To find the optimal solution one needs to solve the equation system (3)–(4). The solution has a form:     a1 I a2 I I I = α = ϕ(p, I ), (5) x = (a1 +a , , β p1 p2 (a1 +a2 ) p2 2 ) p1

2 Rationality of Choices Made by Individual Consumer

where α =

> 0, β =

a1 a1 +a2

(6) u(x) = ln (a

a1 a2 I 2 2 1 +a2 ) p1 p2

a2 a1 +a2

31

> 0, such that α + β = 1.

= ν(p, I ).

Ad C2 An optimal solution satisfies conditions:  

(3) s12 (x 1 , x 2 ) =

∂u (x1 ,x2 )   ∂ x1

x   ∂u (x1 ,x2 )   ∂ x2 x

(4) p1 x 1 + p2 x 2 = I

=x

=

β

αax α−1 x2 1 β−1 βax α1 x 2

=

αx 2 βx1

=

p1 p2

=x

To find the optimal solution. One needs to solve the equation system (3)–(4). The solution has a form:     βI αI  I  I = α = ϕ(p, I ), (5) x = (α+β) , , β p1 (α+β) p2 p1 p2 where α  = (6) u(x) = a

α α+β

> 0, β  =

 α  β  β p2

α p1

β α+β

I α+β

> 0, such that α  + β  = 1.

α+β

= ν(p, I ).

Ad A3 (P4) w(x) = p1 x1 + p2 x2 → min (7) min{a1 x1 , a2 x2 } = u. (8) x1 , x2 ≥ 0. Ad B3 (P4) w(x) = p1 x1 + p2 x2 → min (9) a1 ln x1 + a2 ln x2 = u. (10) x1 , x2 ≥ 0. Ad C3 (P4) w(x) = p1 x1 + p2 x2 → min β

(11) ax1α x2 = u. (12) x1 , x2 ≥ 0. Ad A4 An optimal solution satisfies conditions:

32

2 Rationality of Choices Made by Individual Consumer

(12) min{a1 x˜1 , a2 x˜2 } = u, (13) a1 x˜1 = a2 x˜2 . To find the optimal solution. One needs to solve the equation system (12)–(13). The solution has a form:   (14)  x = au1 , au2 = f (p, u) (15) w( x) = p1 au1 + p2 au2 =

u( p1 a2 + p2 a1 ) a1 a2

= e(p, I ).

Ad B4 An optimal solution satisfies conditions:  

∂u (x1 ,x2 )   ∂ x1

(16)

x   ∂u (x1 ,x2 )   ∂ x2 x

= x

=

p1 p2

= x (17) u(x˜1 , x˜2 ) = u. To find the optimal solution, one needs to solve the equation system (12)–(13). The solution has a form: (18)

a1 x˜1 a2 x˜2

=

a1 x˜2 a2 x˜1

=

p1 , p2

(19) a1 lnx˜1 + a2 lnx˜2 = u.   a a+a2   a a+a1  u u x = e a1 +a2 aa12 pp21 1 2 , e a1 +a2 aa21 pp21 1 2 = f (p, u) (20)  x) = (21) w(

a1 +a2 a1 a2 a1 +a2 a1 +a2 a1 a2

u

a1 a +a2

e a1 +a2 p1 1

a2 a +a2

p2 1

= e(p, u)

Ad C4 An optimal solution satisfies conditions: 

(22)

∂u (x1 ,x2 )   ∂ x1 x= x



∂u (x1 ,x2 )   ∂ x2 x= x

p1 p2

=

(23) u(x˜1 , x˜2 ) = u. To find the optimal solution, one needs to solve the equation system (22)–(23). The solution has a form: β

(24)

αa x˜1α−1 x˜2 β−1 βa x˜1α x˜2

=

α x˜2 β x˜1

=

p1 p2

(25) a1 lnx˜1 + a2 lnx˜2 = u. β α    1   α+β   1   α+β = f (p, u) x = ua α+β βαpp21 ; ua α+β βαpp21 (26)  (27) w( x) =

α+β

α α α+β

β

β α+β

1  u  α+β

a

α

β

p1α+β p2α+β = e(p, u).

2 Rationality of Choices Made by Individual Consumer

Ad 5A  a2 I a1 I , , a 2 p1 + a 1 p2 a 2 p1 + a 1 p2   u u , f (p, u) = , a1 a2 u · ( p1 a 2 + p2 a 1 ) e(p, u) = = I ⇒ u · ( p1 a 2 + p2 a 1 ) = I a 1 a 2 a1 a2 

ϕ(p, I ) =

a.

∂ϕ1 (p, I ) ∂ f 1 (p, u) ∂ϕ1 (p, I ) ϕ1 (p, I ), = − ∂ p1 ∂ p1 ∂I ∂ϕ1 (p, I ) −a22 I L= = 0, i = 1, 2, α ∈ (0; 1). Solutions The consumer’s expenditure minimization problem has a general form: w(x1 , x2 ) = { p1 x1 + p2 x2 } → min (P4) u(x1 , x2 ) = u = const. x1 , x2 ≥ 0 In this exercise, one imposes the additional constraints on quantities of consumption goods: xi ≤ bi i = 1, 2. For the conditional optimization problem with three constraints, one uses the following Lagrange function: L(x, λ) = w(x) + λ1 (b1 − x1 ) + λ2 (b2 − x2 ) + λ3 (u − u(x)), λi ≥ 0, which in case of (a)–(c) utility functions takes different forms: (a) L(x, λ) = p1 x1 + p2 x2 + λ1 (b1 − x1 ) 2

The theorem (Theorem A.6) is presented in a Mathematical appendix of the textbook Malaga K., Sobczak K, Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, 2022.

2 Rationality of Choices Made by Individual Consumer

43

  + λ2 (b2 − x2 ) + λ3 u − ax1α1 x2α2 (b) L(x, λ) = p1 x1 + p2 x2 + λ1 (b1 − x1 ) + λ2 (b2 − x2 ) + λ3 (u − a1 ln x1 − a2 ln x2 ) (c) L(x, λ) = p1 x1 + p2 x2 + λ1 (b1 − x1 ) + λ2 (b2 − x2 )   + λ3 u − a1 x1α − a2 x2α . A necessary condition3 for existence of a minimum is the following system of equations: ⎧ ∂ L(x,λ) = p1 − λ1 − λ3 aα1 x1α1 −1 x2α2 = 0 ⎪ ∂ x1 ⎪ ⎪ α1 α2 −1 ∂ L(x,λ) ⎪ ⎪ =0 ⎨ ∂ x2 = p2 − λ2 − λ3 aα2 x1 x2 ∂ L(x,λ) = b1 − x1 = 0 (a) ∂λ1 ⎪ ∂ L(x,λ) ⎪ ⎪ = b2 − x2 = 0 ⎪ ∂λ2 ⎪ ∂ L(x,λ) ⎩ = u − ax1α1 x2α2 = 0 ∂λ3 ⎧ ∂ L(x,λ) = p1 − λ1 − λ3 ax11 = 0 ⎪ ∂ x1 ⎪ ⎪ ∂ L(x,λ) a2 ⎪ ⎪ ⎨ ∂ x 2 = p 2 − λ2 − λ3 x 2 = 0 ∂ L(x,λ) = b1 − x1 = 0 (b) ∂λ1 ⎪ ∂ L(x,λ) ⎪ ⎪ = b2 − x2 = 0 ⎪ ∂λ2 ⎪ ⎩ ∂ L(x,λ) = u − a1 ln x1 − a2 ln x2 = 0 ∂λ3 ⎧ ∂ L(x,λ) = p1 − λ1 − λ3 a1 αx1α−1 = 0 ⎪ ∂ x1 ⎪ ∂ L(x,λ) ⎪ α−1 ⎪ ⎪ ⎨ ∂ x2 = p2 − λ2 − λ3 a2 αx2 = 0 ∂ L(x,λ) = b1 − x1 = 0 (c) ∂λ1 ⎪ ∂ L(x,λ) ⎪ ⎪ = b2 − x2 = 0 ⎪ ∂λ2 ⎪ ⎩ ∂ L(x,λ) = u − a1 x1α − a2 x2α = 0 ∂λ3 From the Kuhn-Tucker theorem, we know that the constraints can be considered in case they are binding and in case they are not binding. Let us first consider the case when constraints on quantities of goods are not binding, that is when λ1 = λ2 = 0. Then the necessary condition consists of only three equations: ⎧ ⎧ α −1 α ⎨ p1 x1 = λ3 α1 u ⎨ p1 − λ3 aα1 x1 1 x2 2 = 0 α1 α2 −1 (a) = 0 ⇔ p2 x2 = λ3 α2 u , p − λ2 aα2 x1 x2 ⎩ ⎩ 2 α1 α2 x = 0 u − ax u = ax1α1 x2α2 1 2 ⎧ ⎧ a 1 ⎪ ⎨ p 1 x 1 = λ3 a 1 ⎨ p 1 − λ3 x 1 = 0 a2 ⇔ p 2 x 2 = λ3 a 2 (b) , p 2 − λ3 x 2 = 0 ⎩ ⎪ ⎩ u − a ln x − a ln x = 0 u = a ln x + a ln x 1 1 2 2 1 1 2 2

3

A sufficient condition takes a more complex form and involves checking properties of a Hessian of the Lagrange function. But we can notice that the expenditure function w(x) is convex and the utility function u(x) is strictly concave in each (a)-(c) case; hence, the Lagrange function is strictly convex and the sufficient condition for existence of a minimum is satisfied.

44

2 Rationality of Choices Made by Individual Consumer

⎧ ⎧ α−1 ⎨ p1 x1 = λ3 a1 αx1α ⎨ p1 − λ3 a1 αx1 = 0 α−1 (c) p − λ3 a2 αx2 = 0 ⇔ p2 x2 = λ3 a2 αx2α , ⎩ ⎩ 2 u = a1 x1α + a2 x2α u − a1 x1α − a2 x2α = 0 which, after a division of first two equations one by the other, gives a relationship between quantities of both goods: (a) x2 = (b) x2 = (c) x2 =

p1 α2 x p2 α1 1 p1 a2 x p2 a1 1



p1 a p2 a1

1  α−1

x1 .

Substituting this relationship into the third equation gives the global solution, that is a solution obtained when the constraints are not binding:    1 1 α2  α +α α1  α +α     1 2 1 2 u p2 α1 u p1 α2 (a) xG = x 1G , x 2G = , a p1 α2 a p2 α1    1 1 a2  a +a a1  a +a     1 2 1 2 (b) xG = x 1G , x 2G = eu pp21 aa21 , eu pp12 aa21  − α1  α   α − α1 1   α−1  G G 1 p2 a1 α−1 p a G a1 + a2 p1 a2 (c) x = x 1 , x 2 = u α , a2 p21 aa2 + a2 uα Now, we can consider a few cases depending on whether the constraints are binding or not: (1) None of the constraints is binding: if x iG ≤ bi , i = 1, 2 then x i = x iG . (2) The constraint on the quantity of the first good is binding if x 1G > b1 and x 2G ≤ b2 then x 1 = b1 while 1 1 1 1    (a) x 2L = ua b1−α1 α2 (b) x 2L = eu b1−a1 a2 (c) x 2L = u − a1 b1α α a2α If now x 2L ≤b2 then the solution is:   1  u −α1  α1  2 (b) x = b1 , eu b1−a1 a2 (c) x = (a) x = b1 , a b1   1 1  b1 , u − a1 b1α α a2α Because of the constraint on the quantity of the first good, a consumer chooses the bigger quantity of the second good than it results from the optimization problem with no constraints on quantities of goods. If x 2L > b2 then there is no solution satisfying u(x) = u. (3) The constraint on the quantity of the second good is binding: if x 2G > b2 and x 1G ≤ b1 then x 2 = b2 while 1 1 1 1    (a) x 1L = ua b2−α2 α1 (b) x 1L = eu b2−a2 a1 (c) x 1L = u − a2 b2α α a1α . If now x 1L ≤ b1 then the solution is:

2 Rationality of Choices Made by Individual Consumer

45

    1 1 u −α2 α1 u −a2 a1 e (a) x = b , b (b) x = b , b (c) x = 2 2 2 2 a   1 1  u − a2 b2α α a1α , b2 . Because of the constraint on thequantity of the second good. A consumer chooses the bigger quantity of the first good than it results from the optimization problem with no constthe raints on quantities of goods. If x 1L > b1 then there is no solution satisfying u(x) = u. (4) The constraints on quantities of both goods are binding: if x 1G > b1 and x 2G > b2 , then there is no solution satisfying u(x) = u. E2.11. Check properties of Hicksian demand functions which are optimal solutions to consumer expenditure minimization problems of Exercise 2.8. Check properties of the corresponding consumer’s expenditure functions. Solutions The Hicksian demand functions and the corresponding consumer’s expenditure functions are obtained in Exercise 2.8 for the following utility functions: (a) a Koopmans-Leontief function: u(x) = min{a1 x1 , a2 x2 }, ai > 0, i = 1, 2, (b) logarithmic: u(x) = a1 ln x1 + a2 ln x2 , ai > 0, xi > 0, i = 1, 2, β (c) a power function: u(x) = ax1α x2 , a > 0, α, β ∈ (0; 1). The Hicksian (compensated) demand functions are:   (a) f (p, u) = au1 , au2    a2  a1  u u p2 a1 a1 +a2 a1 +a p1 a2 a1 +a2 a1 +a 2 , 2 (b) f (p, u) = e e p1 a2 p2 a1    β  α  1 1 αp2 α+β  u  α+β β p1 α+β  u  α+β (c) f (p, u) = , αp2 . β p1 a a The corresponding consumer’s expenditure functions are:   (a) e(p, u) = u ap11 + ap22   a a+a1   a a+a2 u (b) e(p, u) = e a1 +a2 (a1 + a2 ) ap11 1 2 ap22 1 2 β   1   α   α+β (c) e(p, u) = ua α+β (α + β) pα1 α+β pβ2 . The properties are stated in Theorem 2.4. We are interested in properties4 (3)–(7). (3) Homogeneity of degree 1 of the consumer’s expenditure function with respect to prices of goods:   λp2 1 = λe(p, u) (a) ∀λ > 0 e(λp, u) = u λp + a1 a2 4

The theorem (Theorem 2.4) is presented in subchapter 2.5 of the textboook Malaga K., Sobczak K., Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, 2022. In properties (1) and (2) one states that a Hicksian demand function and a consumer’s expenditure function are differentiable in their domains.

46

(b)

2 Rationality of Choices Made by Individual Consumer

∀λ > 0 a1

= λ a1 +a2 (c)

∀λ > 0

e(λp, u) = e +a

u a1 +a2



λp1 (a1 + a2 ) a1

 a a+a1  1

2

λp2 a2

 a a+a2 1

2

a2 1 +a2

e(p, u) = λ1 e(p, u) = λe(p, u)   α   β 1  u  α+β λp1 α+β λp2 α+β e(λp, u) = (α + β) a α β β

α

= λ α+β + α+β e(p, u) = λ1 e(p, u) = λe(p, u). This property shows us that when prices of both goods increase (λ > 1)/decrease (λ < 1) λ times, then the consumer’s expenditure on an optimal consumption bundle also increases/decreases λ times. (4) Homogeneity of degree 1 of the Hicksian (compensated) demand function with respect to prices of goods:  (a) ∀λ > 0

f (λp, u) =

u u , a1 a2

 = f (p, u)

Values of this function do not depend on prices. 

(b) ∀λ > 0 (c) ∀λ > 0

  a2  a1  u u λp2 a1 a1 +a2 a +a λp1 a2 a1 +a2 a +a f (λp, u) = e 1 2, e 1 2 = f (p, u) λp1 a2 λp2 a1    β  α  1 1 αλp2 α+β  u  α+β βλp1 α+β  u  α+β , f (λp, u) = = f (p, u). βλp1 a αλp2 a

This property shows us that when prices of both goods increase (λ > 1)/decrease (λ < 1) λ times. then the compensated demand of a consumer does not change. (5) Relationship between a compensated demand for a good and a change in the consumer’s expenditure when a price of this good increases: ∂e(p, u) = f i (p, u), i = 1, 2, ∂ pi (a)

∂e(p,u) ∂ p1

=

u a1

= f 1 (p, u)

∂e(p,u) ∂ p2

=

u a2

= f 2 (p, u)

2 Rationality of Choices Made by Individual Consumer

(b)

(c)

47

  a a+a2 a1 a2 u − a +a − a +a p2 1 2 ∂e(p, u) a1 1 2 1 2 a1 +a2 =e p1 (a1 + a2 )a1 ∂ p1 a1 + a2 a2  a a+a2  u p2 a1 1 2 a +a = e 1 2 = f 1 (p, u) p1 a 2 1   a +a a1 u − a +a p1 1 2 − a1a+a2 2 a2 ∂e(p, u) a1 +a2 =e a2 p2 1 2 (a1 + a2 ) ∂ p2 a1 a1 + a2  a a+a1  u p1 a2 1 2 a +a = e 1 2 = f 2 (p, u) p2 a 1 β   α+β 1 β α − α+β p2 ∂e(p, u)  u  α+β α − α+β = p (α + β)α ∂ p1 a α+β 1 β β  α+β  1  u  α+β αp2 = = f 1 (p, u) βp1 a 1 α  p  α+β β − α ∂e(p, u)  u  α+β β 1 p2 α+β = β − α+β (α + β) ∂ p2 a α α+β  α  1 βp1 α+β  u  α+β = = f 2 (p, u) αp2 a

The value of the compensated demand for a given good shows how the consumer’s expenditure will change (increase) if a price of this good increases by one unit and the consumer is interested in keeping a given level of utility from a consumption bundle. (6) An increase in a price of a good results in a decrease of the compensated demand for this good: ∂ f i (p, u) < 0, i = 1, 2 ∂ pi (a)

∂ f 1 (p,u) ∂ p1

=0

∂ f 2 (p,u) ∂ p2

= 0.

In case of the Koopmans-Leontief function. this property is not satisfied, because a consumer perceives both goods as perfect complementary goods. As a result. he/ she wants to have them only in a specific proportion (a2 units of the first good per a1 units of the second good) which does not depend on prices when it comes to the compensated demand.  a2    a2 u − a +a −1 a2 ∂ f 1 (p, u) p2 a1 a1 +a2 − p1 1 2 e a1 +a2 < 0 = ∂ p1 a2 a1 + a2 . (b)  a1    a1 u − a +a −1 a1 p1 a2 a1 +a2 ∂ f 2 (p, u) 1 2 − p = e a1 +a2 < 0 ∂ p2 a1 a1 + a2 2

48

(c)

2 Rationality of Choices Made by Individual Consumer

   1 − β −1 u α+β β p1 α+β 0, α1 + α2 < 1, i = 1, 2 or a logarithmic function u 2 (x) = α1 ln x1 + α2 ln x2 + ln a, xi > 0, i = 1, 2 then, according to the solution to Exercise 2.6, the Marshallian demand function is the same and given as:  ϕ(p, I ) =

 α1 I α2 I . , (α1 + α2 ) p1 (α1 + α2 ) p2

The corresponding Hicksian (compensated) demand functions are:    α2  α1  1 1 p2 α1 α1 +α2  u  α1 +α2 p1 α2 α1 +α2  u  α1 +α2 f (p, u) = , p2 α1 in case of the power p1 α2 a a utility function.    α2  α1  1 1 p2 α1 α1 +α2  eu  α1 +α2 p1 α2 α1 +α2  eu  α1 +α2 f (p, u) = , in case of the logap1 α2 a p2 α1 a rithmic utility function. Let us first analyse the price effect: PE1 =

∂ϕ1 (p, I ) α1 I =− < 0, ∂ p1 (α1 + α2 ) p12

PE2 =

∂ϕ2 (p, I ) α2 I = < 0, ∂ p2 (α1 + α2 ) p22

which shows us that both goods are ordinary goods. Let us now analyse the substitution effect: • in case of the solution obtained for the power utility function  1 α2 − α +α −1  u  α1 +α2 α2 − p1 1 2 0, therefore, the Walrasian equilibrium price vector, determined to an accuracy of a structure, has the following form: (12) p = λ(1, 1). Having this, after substituting (12) to (2) and (3), we can determine the values of the demand functions of both traders when prices are given by the Walrasian equilibrium price vector: (13) x1 (p) = (15, 15) (14) x2 (p) = (15, 15)

66

3 Rationality of Choices Made by a Group of Consumers

Fig. 3.4 Edgeworth box in case of logarithmic utility functions (E3.2)

and the Walrasian equilibrium allocation:   (15) x(p) = x1 (p), x2 (p) = (15, 15, 15, 15), while the initial allocation is:   (16) a = a1 , a2 = (10, 20, 20, 10). Let us notice that in the Walrasian equilibrium state, the relation between equilibrium prices is 1 : 1. This means that traders exchange goods in the relation 1 unit of the first good for 1 unit of the second good. Since the utility functions of both traders are the same, their preferences to own each of the goods are also the same. Thus, in the Walrasian equilibrium state, as a result of the exchange made by the Walrasian equilibrium prices, both traders will have identical bundles of goods. It is also worth noticing that in the only state of the Walrasian equilibrium, an increase in utility of bundles of goods purchased by traders with respect to utility of the initial consumption bundles will be the same for both traders. Figure 3.4 presents geometrical illustrations of sets of allocations: of the Walrasian equilibrium W (a) (indicated by a single point x(p)), optimal in the Pareto sense and accepted in the same time C(a), as well as a set F(a) of the allocations feasible with regard to the initial allocation: a = (10, 20, 20, 10). Since the optimal solutions to both consumption utility maximization problems must lie on the budget lines respective to each of the traders, having the Walrasian equilibrium price vector, we get: p 1 x 11 + p 2 x 12 = 10 p 1 + 20 p 2 ⇐⇒ x 11 + x 12 = 30,

(3.1)

3 Rationality of Choices Made by a Group of Consumers

67

and p 1 x 21 + p 2 x 22 = 10 p 1 + 20 p 2 ⇐⇒ x 21 + x 2 = 30.

(3.2)

This means that the budget lines of traders are not only parallel but also coincide, because each of them includes the initial bundle of goods each of the traders came to the market with. It is not difficult to notice that an angle of incline of both budget lines with respect to the horizontal axes is 45o in the coordinate system of each of the traders. It is also easy to notice that: W (a) ⊂ C(a) ⊂ S(a) ⊂ F(a) ⊂ R4+ . This means that the only Walrasian equilibrium allocation corresponding to the only Walrasian equilibrium price vector, determined to an accuracy of a structure, is a Pareto optimal allocation, accepted and feasible with regard to an initial allocation. Ad B The consumption utility maximization problems for two traders are given: 1

1

4 4 (1) u 1 (x11 , x12 ) = x11 + x12 → max (2) p1 x11 + p2 x12 ≤ 10 p1 + 20 p2 , (3) x11 , x12 ≥ 0,

and 1

1

4 4 (4) u 2 (x21 , x22 ) = x21 +x 22 → max (5) p1 x21 + p2 x22 ≤ 20 p1 + 10 p2 , (6) x21 , x22 ≥ 0.

We know that the optimal solution to the consumption utility maximization problem in case of the subadditive utility function u(x1 , x2 ) = a1 x α1 + a2 x α2 has a form:   (7) x(p) =

1

(a1 p2 ) 1−α I 1 a11−α

2−α p21−α

1 +a21−α

2−α p11−α

,

1

(a2 p1 ) 1−α I 1 a11−α

2−α p21−α

1

2−α

.

+a21−α p11−α

Substituting the parameters of the utility function of both traders into (7), we get the demand function of the first trader:  4 4 p23 (10 p1 +20 p2 ) p13 (10 p1 +20 p2 ) (8) x1 (p) = , , 7 7 7 7 p13 + p23

and of the second trader:  4 p23 (20 p1 +10 p2 ) 2 (9) x (p) = , 7 7 p13 + p23

p13 + p23

4

p13 (20 p1 +10 p2 ) 7

7

p13 + p23

.

Adding the demand functions of both traders, we get the global demand function for both consumer goods:  4 4 p23 (30 p1 +30 p2 ) p13 (30 p1 +30 p2 ) (10) x(p) = x1 (p) + x2 (p) = , . 7 7 7 7 p13 + p23

p13 + p23

68

3 Rationality of Choices Made by a Group of Consumers

The global supply function (vector) has a form: (11) a = a1 + a2 = (30, 30). The excess demand function then has a form: ⎛ 4 ⎞ 4 3 3 p + 30 p + 30 p p p p (30 ) (30 ) 1 2 1 2 ⎠ z(p) = x(p) − a = ⎝ 2 , 1 − (30, 30) 7 7 7 7 3 3 3 3 p1 + p2 p1 + p2 ⎛ ⎞  4  4 (12) 4 4 4 4 30 p1 p23 − p13 p23 30 p2 p13 − p23 p13 ⎜ ⎟ =⎝ , ⎠ = z(p) 7 7 7 7 p13 + p23 p13 + p23 Let us notice that the excess demand function is homogeneous of degree 0 because: ⎛  4 7 7 7 λ 3 p23 (30 p1 + 30 p2 ) − 30 p13 − 30 p23 ⎜ ∀λ > 0 z(λp) = ⎝ ,  7 7 7 λ 3 p13 + p23 ⎞  4 7 7 7 λ 3 p23 (30 p1 + 30 p2 ) − 30 p13 − 30 p23 ⎟ (13)  7 ⎠ 7 7 3 3 λ 3 p1 + p2 ⎞ ⎛  4  4 4 4 4 4 30 p1 p23 − p13 p23 30 p2 p13 − p23 p13 ⎟ ⎜ , =⎝ ⎠ = z(p) 7 7 7 7 p13 + p23 p13 + p23 Which means that the excess demand for any commodity does not depend on the absolute price level of both goods, but on the relationship between the prices of goods. Moreover, the function of excess demand satisfies Walras’s law, because: (14) ∀p > 0 ⟨p, z(p)⟩ = p1

 4 4 4 3 30 p1 p23 − p13 p 2 7

7

p13 + p23

+ p2

 4 4 4 3 30 p2 p13 − p23 p 1 7

7

p13 + p23

= 0,

which means that for any positive vector of consumer goods’ prices, the value of global demand is equal to the value of global supply of both goods. Let us determine the Walrasian equilibrium price vector p > 0 as a solution of a following system of equations: ⎛  4  4 ⎞ 4

⎜ 30 p1 (15) z(p) = ⎝

4

p23 − p13

2 7

7

p13 + p23

4

3

p

,

4

30 p2 p13 − p23

2 7

7

p13 + p23

which can be written in an equivalent form: 4

(17)

4

− p 13 + p 23 = 0 4

4

p 23 − p 13 = 0

3

p

⎟ ⎠ = (0, 0),

3 Rationality of Choices Made by a Group of Consumers

69

resulting in: (18) p 1 = p 2 = λ > 0, therefore, the Walrasian equilibrium price vector, determined to an accuracy of a structure, has the following form: (19) p = λ(1, 1). Having this, after substituting (19) to (8) and (9), we can determine the values of the demand functions of both traders when prices are given by the Walrasian equilibrium price vector: (20) x1 (p) = (15, 15) (21) x2 (p) = (15, 15) and the Walrasian equilibrium allocation:   (22) x(p) = x1 (p), x2 (p) = (15, 15, 15, 15), while the initial allocation is:   (23) a = a1 , a2 = (10, 20, 20, 10). Let us notice that in the Walrasian equilibrium state, the relation between equilibrium prices is 1 : 1. This means that traders exchange goods in the relation 1 unit of the first good for 1 unit of the second good. Since the utility functions of both traders are the same, their preferences to own each of the goods are also the same. Thus, in the Walrasian equilibrium state, as a result of the exchange made by the Walrasian equilibrium prices, both traders will have identical bundles of goods. It is also worth noticing that in the only state of the Walrasian equilibrium, an increase in utility of bundles of goods purchased by traders with respect to utility of the initial consumption bundles will be the same for both traders. Figure 3.5 presents geometrical illustrations of sets of allocations: of the Walrasian equilibrium W (a) (indicated by a single point x(p)), optimal in the Pareto sense and accepted in the same time C(a), as well as a set F(a) of the allocations feasible with regard to the initial allocation:a = (10, 20, 20, 10). Since the optimal solutions to both consumption utility maximization problems must lie on the budget lines respective to each of the traders, having the Walrasian equilibrium price vector we get: (24) p 1 x 11 + p 2 x 12 = 10 p 1 + 20 p 2 ⇐⇒ x 11 + x 12 = 30, and (25) p 1 x 21 + p 2 x 22 = 10 p 1 + 20 p 2 ⇐⇒ x 21 + x 2 = 30. This means that the budget lines of traders are not only parallel but also coincide, because each of them includes the initial bundle of goods each of the traders came to the market with. It is not difficult to notice that an angle of incline of both budget

70

3 Rationality of Choices Made by a Group of Consumers

Fig. 3.5 Edgeworth box in case of subadditive utility functions (E3.2)

lines with respect to the horizontal axes is 45o in the coordinate system of each of the traders. It is also easy to notice that: W (a) ⊂ C(a) ⊂ S(a) ⊂ F(a) ⊂ R4+ . This means that the only Walrasian equilibrium allocation corresponding to the only Walrasian equilibrium price vector, determined to an accuracy of a structure, is a Pareto optimal allocation, accepted and feasible with regard to an initial allocation. E3.3. Using the Edgeworth box for the static Arrow-Hurwicz model of a market with two traders and two goods present a geometric illustration of a case when there exists no Walrasian equilibrium price vector in this model. Solution There is given a market of two traders and two goods described by the static ArrowHurwicz model, in which: i = 1, 2—consumer goods, k = 1, 2—traders (consumers), X = R2+ —a goods’ space, a1 = (a11 , a12 ), a2 = (a21 , a22 ),—initial bundles the consumers come to the market with, p = ( p1 , p2 ) ∈ intR2+ —a vector of goods’ prices, xk = (xk1 , xk2 ) ∈ R2+ —a consumption bundle the k-th consumer wants to purchase, I 1 ( p1 , p2 ) = 10 p1 + 20 p2 , I 2 ( p1 , p2 ) = 20 p1 + 10 p2 —incomes of traders. Utility functions of traders have forms: u 1 (x11 , x12 ), u 1 (x11 , x12 ). The consumption utility maximization problems for two traders are given:

3 Rationality of Choices Made by a Group of Consumers

71

Fig. 3.6 Optimal baskets of goods of both traders in case of disequilibrium (E3.3)

u 1 (x11 , x12 ) → max, u 2 (x21 , x22 ) → max, p1 x11 + p2 x12 ≤ 10a11 + 20a12 , p1 x21 + p2 x22 ≤ 10a21 + 20a22 , x21 , x22 ≥ 0. x11 , x12 ≥ 0. 

2 1 ∼



Let us denote by x , x an allocation feasible with regard to the initial allocation  1 2 , in which the first trader receives an optimal bundle of goods x1 and by a , a 1   ∼ x , x2 an allocation feasible with regard to the initial allocation a1 , a2 , in which the second trader receives an optimal bundle of goods x2 . If the price  vector p = ∼2 ( p1 , p2 ) is not a Walrasian equilibrium price vector, then allocations x1 , x and  ∼1 2 x , x are not Walrasian equilibrium allocations. In Fig. 3.6, it has been shown that although both allocations are feasibly located on the overlapping budget lines ∼1

∼2

of both traders, the bundles x and x are not the optimal bundles of both traders, as they are located on lower indifference curves than the optimal bundles x1 and x2 . Let us notice that: p1 x 11 + p2 x 12 < p1 a11 + p2 a 12 and that p1 x 21 + p2 x 22 > p1 a21 + + p 2 a 22 . This means that in the case of the first good, there is a surplus of the global supply over the global demand, and in the case of the second good there is an excess of the global demand over the global supply.

72

3 Rationality of Choices Made by a Group of Consumers

E3.4. Define the dynamic Arrow-Hurwicz model in a discrete-time and in a continuous-time versions, taking as the basis the static Arrow-Hurwicz model from Exercise 2 with: A. logarithmic utility functions of traders, B. subadditive utility functions of traders. Solutions Ad A The dynamic Arrow-Hurwicz model in a discrete-time version has the following form: (1) t ∈ [0, 1, . . . , T )   p2 (t) (2) p1 (t + 1) − p1 (t) = σ −30 p12(t)+30 p1 (t)   30 p1 (t)−30 p2 (t) (3) p2 (t + 1) − p2 (t) = σ 2 p2 (t) (4) p1 (0) = p10 > 0 (5) p2 (0) = p20 > 0. The dynamic Arrow-Hurwicz model in a continuous-time version takes a form: (6) t ∈ [0, T )   p2 (t) (7) dpdt1 (t) = σ −30 p12(t)+30 p1 (t)   p2 (t) (8) dpdt2 (t) = σ 30 p1 (t)−30 2 p2 (t) (9) p1 (0) = p10 > 0 (10) p2 (0) = p20 > 0. Ad B The dynamic Arrow-Hurwicz model in a discrete-time version takes a form: (11) t ∈ [0, 1, . . . , T )

⎛

⎜ (12) p1 (t + 1) − p1 (t) = σ ⎝ ⎛ ⎜ (13) p2 (t + 1) − p2 (t) = σ ⎝

⎞  4 4 4 3 30 p1 (t) p23 (t)− p13 (t) p (t) 2 7 p13

7 (t)+ p23

(t)

⎟ ⎠

⎞ 4 4 4 3 30 p2 (t) p13 (t)− p23 (t) p (t) 

1 7

7

p13 (t)+ p23 (t)

⎟ ⎠

(14) p1 (0) = p10 > 0 (15) p2 (0) = p20 > 0. The dynamic Arrow-Hurwicz model in a continuous-time version has the following form: (16) t ∈ [0, T )

3 Rationality of Choices Made by a Group of Consumers

⎛ (17)

dp1 (t) dt

⎜ = σ⎝ ⎛

(18)

dp2 (t) dt

⎜ = σ⎝

(19) p1 (0) = (20) p2 (0) =

p10 p20

73

⎞  4 4 4 3 30 p1 (t) p23 (t)− p13 (t) p (t) 2 7 p13

7 (t)+ p23

(t)

⎟ ⎠

⎞ 4 4 4 3 30 p2 (t) p13 (t)− p23 (t) p (t) 

1 7 p13

7 (t)+ p23

(t)

⎟ ⎠

>0 > 0.

E3.5. For the dynamic Arrow-Hurwicz model in a discrete-time version from Exercise 3.4 with logarithmic utility functions of traders and the Walrasian equilibrium price vector known from Exercise 3.2 determine a feasible price trajectory for a few subsequent periods taking some value of parameter σi = σ > 0. Present a geometric illustration of this trajectory in the state space and in the phase space. Solution Utility functions of traders are given in the following form: u 1 (x11 , x12 ) = lnx11 + lnx12 , u 2 (x21 , x22 ) = lnx21 + lnx22 . Their initial bundles and incomes are of the following form: a1 = (10, 20), a2 = (20, 10),

I 1 (p) = 10 p1 + 20 p2 , I 2 (p) = 20 p1 + 10 p2

From the solutions to the utility maximization problems in Exercise 3.2, we know that demand functions of traders are given as:   ϕ 1 p, I 1 (p) =



I 1 (p) , 2 p1  2   I (p) ϕ 2 p, I 2 (p) = , 2 p1

 p2 p1 I 1 (p) = 5 + 10 , 5 + 10 = ϕ 1 (p) = x1 (p) 2 p2 p1 p2  2 p2 I (p) p1 = 10 + 5 , 10 + 5 = ϕ 2 (p) = x2 (p). 2 p2 p1 p2

From Exercise 3.2, we also know that a function of global demand has a form:  p1 p2 x(p) = x1 (p) + x2 (p) = 15 + 15 , 15 + 15 . p1 p2 Hence, a function of excess demand has a form:  p2 p1 z(p) = x(p) − a = 15 + 15 , 15 + 15 p1 p2  p1 p2 − (30, 30) = 15 − 15, 15 − 15 p1 p2 Thus, the resulting Walrasian equilibrium price vector is given as: p = λ(1, 1), λ > 0.

74

3 Rationality of Choices Made by a Group of Consumers

Table 3.1 Price trajectories by different values of σ (E3.5)

A dynamic discrete-time Arrow-Hurwicz model for the data given in the exercise takes a form: ⎧   ⎨ p1 (t + 1) = p1 (t) + σ 15 p2 − 15   p1 ⎩ p2 (t + 1) = p2 (t) + σ 15 p1 − 15 p2 where initial prices are given: 

p1 (0) = p10 > 0 p2 (0) = p20 > 0

and prices change over time given as a discrete variable t = 0, 1, 2, . . . T . Let us take the initial prices, for example p1 (0) = 6, p2 (0) = 3 and consider a few values for σ , for example 0.05, 0.35, 1.25. The price trajectories are obtained in an Excel file named Exercise 3.5.xlsx and presented in Table 3.1. When σ = 0.05, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure in 23-rd period with prices’ levels p 1 = p 2 ≈ 4.8. When σ = 0.35, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure in seventh period with levels p 1 = p 2 ≈ 9.87. When σ = 1.25, the price trajectories are infeasible. The price trajectories in the state space and in the phase space are presented in Figs. 3.7 and 3.8.3 E3.6. For the dynamic Arrow-Hurwicz model in a discrete-time version from Exercise 3.4 with subadditive utility functions of traders and the Walrasian equilibrium price vector known from Exercise 3.2 determine a feasible price trajectory for a few 3

The price trajectories in the state space are obtained using an Excel file named Exercise 3.5.xlsx and a MATLAB file named Exercise_3_5_1.m. In the phase space, they are obtained using a MATLAB file named Exercise_3_5_2.m.

3 Rationality of Choices Made by a Group of Consumers

75

Fig. 3.7 Price trajectories in state space (E3.5)

Fig. 3.8 Price trajectories in phase space (E3.5)

subsequent periods taking some value of parameter σi = σ > 0. Present a geometric illustration of this trajectory in the state space and in the phase space. Solution Utility functions of traders are given in the following form: 1

1

1

1

4 4 4 4 u 1 (x11 , x12 ) = x11 + x11 , u 2 (x21 , x22 ) = x21 + x21 .

76

3 Rationality of Choices Made by a Group of Consumers

Their initial bundles and incomes are of the following form: a1 = (10, 20), a2 = (20, 10) I 1 (p) = 10 p1 + 20 p2 , I 2 (p) = 20 p1 + 10 p2 From the solutions to the utility maximization problems in Exercise 3.2, we know that demand functions of traders are given as: ⎞

⎛   ⎜ ϕ 1 p, I 1 (p) = ⎝

⎛

⎞

I (p) I (p) ⎟ ⎜ 10 p1 + 20 p2 10 p1 + 20 p2 ⎟ ⎠=⎝ ⎠  4  13 ,  4  13  4  13 ,  4  13 p p2 p1 p2 p1 + p12 + p p + + p 2 1 2 p1 p2 p1 ⎛ ⎞ p2 p1 ⎜ 10 + 20 p1 10 p2 + 20 ⎟ 1 =⎝ ⎠ = ϕ (p) = x1 (p)   13 ,   13 p1 p2 1 + p2 +1 p1 ⎞ ⎛ ⎞ ⎛ 1

1

  ⎜ ϕ 2 p, I 2 (p) = ⎝

I 2 (p) ⎟ ⎜ 20 p1 + 10 p2 20 p1 + 10 p2 ⎟ I 2 (p) ⎠=⎝ ⎠  4  13 ,  4  13  4  13 ,  4  13 p p2 p1 p2 p1 + p12 + p p + + p 2 1 2 p1 p2 p1 ⎞ ⎛ p2 p1 ⎜ 20 + 10 p1 20 p2 + 10 ⎟ 2 =⎝ ⎠ = ϕ (p) = x2 (p).   13 ,   13 p1 p2 1 + p2 +1 p1

From Exercise 3.2, we also know that a function of global demand has a form: ⎞ p2 p1 30 + 30 30 + 30 p1 p2 ⎟ ⎜ x(p) = x1 (p) + x2 (p) = ⎝ ⎠.   13 ,   13 p1 p2 1 + p2 +1 p1 ⎛

Hence, a function of excess demand has a form: ⎞ ⎛ p2 p1 30 + 30 30 + 30 p1 p2 ⎟ ⎜ z(p) = x(p) − a = ⎝ ⎠ − (30, 30).   13 ,   13 p2 1 + pp21 + 1 p1 Thus, the resulting Walrasian equilibrium price vector is given as: p = λ(1, 1), λ > 0. A dynamic discrete-time Arrow-Hurwicz model for the data given in the exercise takes a form:

3 Rationality of Choices Made by a Group of Consumers

77

Table 3.2 Price trajectories by different values of σ (E3.6)

⎧ ⎞ ⎛ ⎪ p2 ⎪ 30+30 ⎪ ⎪ p (t + 1) = p1 (t) + σ ⎝  p11 − 30⎠ ⎪ ⎪ 3 p ⎨ 1 1+ p1 2 ⎞ ⎛ ⎪ p1 ⎪ 30 +30 ⎪ ⎪ p (t + 1) = p2 (t) + σ ⎝  p2 1 − 30⎠ ⎪ ⎪ p2 3 ⎩ 2 +1 p1

where initial prices are given: 

p1 (0) = p10 > 0 p2 (0) = p20 > 0

and prices change over time given as a discrete variable t = 0, 1, 2, . . . T . Let us take the initial prices, for example p1 (0) = 6, p2 (0) = 3 and consider a few values for σ , for example 0.05, 0.35, 1.25. The price trajectories are obtained in an Excel file named Exercise 3.6.xlsx and presented in Table 3.2. When σ = 0.05, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure in 16-th period with prices’ levels p 1 = p 2 ≈ 4.83. When σ = 0.35, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure in 17-th period with levels p 1 = p 2 ≈ 29.76. When σ = 1.25, the price trajectories are infeasible. The price trajectories in the state space and in the phase space are presented in Figs. 3.9 and 3.10.4 E3.7. Two traders come to a market with bundles of goods: a1 = (10, 20), a2 = (20, 10). Their utility functions are following: (a) u 1 (x11 , x12 ) = ln x11 + ln x12 , u 2 (x21 , x22 ) = ln x21 + ln x22 , 1/4 1/4 1/4 1/4 (b) u 1 (x11 , x12 ) = x11 + x12 , u 2 (x21 , x22 ) = x21 + x22 , 4

The price trajectories in the state space are obtained using an Excel file named Exercise 3.6.xlsx and a MATLAB file named Exercise_3_6_1.m. In the phase space, they are obtained using a MATLAB file named Exercise_3_6_2.m.

78

3 Rationality of Choices Made by a Group of Consumers

Fig. 3.9 Price trajectories in state space (E3.6)

Fig. 3.10 Price trajectories in phase space (E3.6)

Consider a discrete-time version of dynamic discrete-time Arrow-Hurwicz model. A broker announces initial prices: p(0) = (2, 4). Using a form of the excess demand function and a structure of the Walrasian equilibrium price vector found in Exercise 2 for the static Arrow-Hurwicz model:

3 Rationality of Choices Made by a Group of Consumers

Fig. 3.11 Price trajectories in state space (E3.7 a)

Fig. 3.12 Price trajectories as functions of time (E3.7 a)

Fig. 3.13 Price trajectories in state space (E3.7 b)

79

80

3 Rationality of Choices Made by a Group of Consumers

Fig. 3.14 Price trajectories as functions of time (E3.7 b)

1. Determine trajectories of a price vector satisfying a system of equations of the dynamic discrete-time Arrow-Hurwicz model, taking a proportionality coeffi(t) and compare cient σ equal to 0.25, 0.35 and 1.25. Calculate price ratios pp21 (t) 2. 3.

4. 5.

them with the equilibrium price ratio pp2 . 1 State which trajectories determined in point 1 are feasible. State if and when (in which period) a structure of prices stabilizes around the equilibrium price structure and whether it reaches this structure in time horizon T = 15. Present graphs of the price trajectories in the state space. Present graphs of the price trajectories as functions of time.

Solutions In this exercise, we can use the results of the previous exercises: 3.2, 3.5 and 3.6. The main results are the Walrasian equilibrium price vector: p = λ(1, 1), λ > 0 and a form of the dynamic discrete-time Arrow-Hurwicz model: ⎧   ⎨ p1 (t + 1) = p1 (t) + σ 15 p2 − 15 p 1   (a) ⎩ p2 (t + 1) = p2 (t) + σ 15 p1 − 15 ⎧ ⎞ ⎛ p2 ⎪ p ⎪ 30+30 p2 ⎪ ⎪ p1 (t + 1) = p1 (t) + σ ⎝  11 − 30⎠ ⎪ ⎪ 3 p ⎨ 1+ p1 2 ⎞ ⎛ (b) ⎪ p ⎪ 30 p1 +30 ⎪ ⎪ p2 (t + 1) = p2 (t) + σ ⎝  2 1 − 30⎠ ⎪ ⎪ p2 3 ⎩ +1 p1

where prices change over time given as a discrete variable t = 0, 1, 2, . . . T . Ad 1 Initial prices announced by the broker are p1 (0) = 2, p2 (0) = 4. The price trajectories are obtained in Excel files named: Exercise 3.7 a.xlsx, Exercise 3.7 b.xlsx and presented in Tables 3.3 and 3.4.

3 Rationality of Choices Made by a Group of Consumers

81

Table 3.3 Price trajectories by different values of σ (E3.7 a)

Table 3.4 Price trajectories by different values of σ (E3.7 b)

Ad 2 and 3 (a) When σ = 0.25, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure in sixth period with prices’ levels p 1 = p 2 ≈ 7.93. When σ = 0.35, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure in 25-th period with levels p 1 = p 2 ≈ 31.47. When σ = 1.25, the price trajectories are infeasible. (b) When σ = 0.25, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure in 23-rd period with prices’ levels p 1 = p 2 ≈ 28.51. When σ = 0.35, the price trajectories are feasible but a structure of prices approaches the equilibrium price structure very slowly, stabilizing around it in 465-th period (taking an accuracy to two decimal places in prices) with levels p 1 = p 2 ≈ 576.89. When σ = 1.25, the price trajectories are infeasible.

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3 Rationality of Choices Made by a Group of Consumers

Ad 4 and 5 a. The price trajectories in the state space and as functions of time are presented in Figs. 3.11 and 3.12.5 b. The price trajectories in the state space and as functions of time are presented in Figs. 3.13 and 3.14.6 E3.8. Consider a continuous-time version of the dynamic Arrow-Hurwicz model for the same data given as in Exercise 3.8. 1. Determine trajectories of a price vector satisfying a system of equations of the dynamic continuous-time Arrow-Hurwicz model taking a proportionality coefficient σ equal to 0.25, 0.35, 1.25 and determine whether these trajectories are feasible. 2. Determine if and when (at what moment) a structure of prices stabilizes around the equilibrium price structure. 3. Present graphs of price trajectories as functions of time. Solution In this exercise, we can use the results of the previous exercises: 3.2, 3.5 and 3.6. The main results are the Walrasian equilibrium price vector: p = λ(1, 1), λ > 0 and a form of the dynamic discrete-time Arrow-Hurwicz model which we can now transform to a form of the continuous-time model. ⎧   ⎨ d p1 (t) = σ 15 p2 − 15 dt p 1   (a) ⎩ d p2 (t) = σ 15 p1 − 15 ⎧ dt ⎛ p2 ⎞ ⎪ p2 ⎪ 30+30 ⎪ d p1 (t) ⎪ = σ ⎝  p11 − 30⎠ ⎪ ⎪ 3 p ⎨ dt 1+ p1 2 ⎛ ⎞ (b) ⎪ p1 ⎪ 30 +30 ⎪ d p2 (t) ⎪ = σ ⎝  p2 1 − 30⎠ ⎪ ⎪ p2 3 ⎩ dt +1 p1

where prices change over time given as a discrete variable t ∈ [0; T ]. To determine approximate price trajectories, we can use a Euler method in which one approximates differential equations with difference equations. The system of the continuous-time model: 5

The price trajectories in the state space are obtained using an Excel file named Exercise 3.7 a.xlsx and MATLAB file named Exercise_3_7_a.m. As functions of time, they are obtained using the Excel file Exercise 3.7 a.xlsx. 6 The price trajectories in the state space are obtained using an Excel file named Exercise 3.7 b.xlsx and MATLAB file named Exercise_3_7_b.m. As functions of time, they are obtained using the Excel file Exercise 3.7 b.xlsx.

3 Rationality of Choices Made by a Group of Consumers

83

Table 3.5 Price trajectories by different values of σ (E3.8 a)

Table 3.6 Price trajectories by different values of σ (E3.8 b)

 d p1 (t) dt d p2 (t) dt

= σ z 1 ( p(t)) = σ z 2 ( p(t))

is transformed to a system of a discrete-time model in the following way: 

p1 (t + 1) − p1 (t) = σ z 1 ( p(t)) · Δt p2 (t + 1) − p2 (t) = σ z 2 ( p(t)) · Δt

where Δt denotes a time increment.7 Let us assume that a) Δt = 0.1, b) Δt = 0.35. Ad 1 Initial prices announced by the broker are p1 (0) = 2, p2 (0) = 4. The price trajectories are obtained in an Excel files named Exercise 3.8.xlsx and presented in Tables 3.5 and 3.6. Ad 2 (a) We assume that Δt = 0.1. When σ = 0.25, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure with prices’ Depending on a value of parameter Δt meaning the time increment, we observe a lack of convergence, faster or slower convergence to the equilibrium state determined as a structure of prices

7

84

3 Rationality of Choices Made by a Group of Consumers

Fig. 3.15 Price trajectories as functions of time (E3.8 a)

Fig. 3.16 Price trajectories as functions of time (E3.8 b)

levels p 1 = p 2 ≈ 3.19, at 33-rd step which means a moment 33 · 0.1 = 3.3. When σ = 0.35, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure with levels p 1 = p 2 ≈ 3.2, at 22-nd step, which means a moment 22 · 0.1 = 2.3. When σ = 1.25, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure with levels p 1 = p 2 ≈ 3.52, at fourth step, which means a moment 4 · 0.1 = 0.4. Let us notice that equilibrium price levels obtained in different cases of σ value are close to each other and not far from initial price levels p1 (0) = 2, p2 (0) = 4. (b) We assume that Δt = 0.35. When σ = 0.25, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure with prices’ levels p 1 = p 2 ≈ 3.47, at third step which means a moment 3 · 0.35 = 1.05. When σ = 0.35, the price trajectories are feasible and a structure of prices stabilizes around the equilibrium price structure with levels p 1 = p 2 ≈ 3.96, at eighth step, which means a moment 8 · 0.35 = 2.8. When σ = 1.25, the price trajectories are infeasible. Let us notice that equilibrium price levels obtained in different cases of σ value (when the price trajectories are feasible) are close to each other and not far from initial price levels p1 (0) = 2, p2 (0) = 4. Ad 3 The price trajectories as functions of time are presented in Figs. 3.15 and 3.16.8

8

The price trajectories as functions of time are obtained using an Excel file named Exercise 3.8.xlsx.

Chapter 4

Rationality of Choices Made by Individual Producers

This Chapter presents 14 exercises that help in understanding the problem of describing rational behaviour of a single producer. In particular they concern: production space, a production function as a set of technologically effective processes, profit maximization problem, production total cost minimization problem under conditions of either perfect competition or monopoly, with or without constraints on resources of production factors at the disposal of an enterprise, sensitivity of supplies and prices set by a market or by a producer to changes in values of relevant parameters. E4.1. There is a given CES production function of a form: f (x1 , x2 ) = θ  γ γ a1 x1 + a2 x2 γ , θ, ai > 0, i = 1, 2, γ ∈ (−∞; 0) ∪ (0; 1). Let us assume that a variable u = xx21 describes a quantity of the second production factor per one unit of the first production factor. 1. Justify that: (a) an elasticity of marginal rate of substitution of the first production factor by the second production factor is constant and equal to E σ12 (u) = 1 − γ , (b) an elasticity of marginal rate of substitution of the second production factor by the first production factor is constant and equal to E σ21 (u) = γ − 1. 2. Determine a range of variability of constant elasticities E σ12 (u) and E σ21 (u) of marginal rates of substitution. 3. Determine an elasticity of production with respect to scale of inputs and justify that it is equal to a degree E λ (x) = θ of positive homogeneity of the production function.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Sobczak and K. Malaga, Workbook for Microeconomics, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-031-41947-8_4

85

86

4 Rationality of Choices Made by Individual Producers

Solutions Ad 1a θ  γ γ  −1 γ −1   a1 x 1 + a2 x 2 γ γ a1 x 1 , a1 x1 γ −1 σ12 (x) =  = θ γ γ  −1 γ −1 θ a2 x 2 a1 x 1 + a2 x 2 γ γ a2 x 2 γ   a1 x2 1−γ a1 1−γ = = ·u = σ12 (u) a2 x 1 a2 a1 ∂σ12 (u) u u = (1 − γ ) · u −γ · a1 1−γ = 1 − γ E σ12 (u) = ∂u σ12 (u) a2 ·u a2

θ γ

Ad 1b θ  γ γ  −1 γ −1   a1 x 1 + a2 x 2 γ γ a2 x 2 , a2 x2 γ −1 σ21 (x) =  = θ γ γ  −1 γ −1 θ a1 x 1 a1 x 1 + a2 x 2 γ γ a1 x 1 γ a2 γ −1 = ·u = σ21 (u) a1 ∂σ21 (u) u E σ21 (u) = ∂u σ21 (u) u a2 = (γ − 1) · u −γ · a2 γ −1 = γ − 1 a1 · u a1

θ γ

Ad 2a γ ∈ (−∞; 0) ∪ (0; 1) − γ ∈ (−1; 0) ∪ (0; +∞) (1 − γ ) ∈ (0; 1) ∪ (1; +∞) Ad 2b γ ∈ (−∞; 0) ∪ (0; 1) (γ − 1) ∈ (−∞; −1) ∪ (−1; 0) Ad 3 θ  γ γ f (x) = a1 x1 + a2 x2 γ θ

f (λx) = (a1 (λx1 )γ + a2 (λx2 )γ ) γ θ  γ γ = λθ a1 x1 + a2 x2 γ = λθ f (x)

4 Rationality of Choices Made by Individual Producers

87

∂ f (λx) λ λ→1 ∂λ f (λx) λ = lim θ λθ −1 f (x) =θ λ→1 f (x)

E λ (x) = lim

E4.2. For solutions of optimization problems from examples 4.3, 4.5 and 4.7 analyze sensitivity of: 1. the demand for a production factor and the maximum profit to changes in a product price and to changes in values of parameters of a production function and of a production cost function, 2. the conditional demand for a production factor and the minimum cost of producing y output units to changes in an output level and to changes in values of parameters of a production function and of a production cost function, 3. the product supply and the maximum profit to changes in a product price and to changes in values of parameters of a production function and of a production cost function. Solutions Example 4.3 Ad 1 See Table 4.1 and 4.2.

Table 4.1 Measures of sensitivity of demand function for production factor (E4.2)

ψ( p, a, c1 ) = x G =



ap 2c1

2

xL = b

∂ψ( p,a,c1 ) ∂a

∂ψ( p,a,c1 ) ∂p

∂ψ( p,a,c1 ) ∂c1

ap 2 2c12

a2 p 2c12

− a2cp3 < 0

>0

0

2 2

>0

1

0

0

Table 4.2 a, b Measures of sensitivity of profit function (E4.2) (a)

  ∂π x G ∂a

  π xG =

a 2 p2 4c1

−d

(b) 

π x

L



ap 2 2c1

>0

  ∂π x G ∂d 1 2

= pab − c1 b − d

1 2

pb > 0

  ∂π x G ∂p a2 p 2c1

>0

−1 1 2

ab > 0

  ∂π x G ∂c1 2 2

− a4cp2 < 0

  ∂π x G ∂b

0

1

−1 −b < 0

1 2

1

pb− 2 − c1

88

4 Rationality of Choices Made by Individual Producers

Table 4.3 Measures of sensitivity of function of conditional demand for production factor (E4.2)

ξ (y) =

 y 2 a

∂ξ(y,a) ∂a

∂ξ (y,a) ∂y

2

− 2y 0

Table 4.4 Measures of sensitivity of function of minimum production cost of y units of product (E4.2)

ctot (y) = c1

 y 2 a

∂ctot (y) ∂a

∂ctot (y) ∂c1

 y 2

2

− 2ca13y < 0

+d

a

>0

∂ctot (y) ∂y

∂ctot (y) ∂d

2c1 y a2

1

>0

Table 4.5 Measures of sensitivity of product supply function (E4.2)

η( p, a, c1 ) = y G =

pa 2 2c1 1

η( p, a, c1 ) = y L = ab 2

∂η( p,a,c1 ) ∂a

∂η( p,a,c1 ) ∂p

∂η( p,a,c1 ) ∂c1

pa c1

a2 2c1

− pa 0

0

>0

2

∂η( p,a,c1 ) ∂b

1

0

>0

Table 4.6 a, b Measures of sensitivity of profit function (E4.2) (a)

  π yG =

  ∂π y G ∂a a 2 p2 2c1

−d

(b)

  1 π y L = pab 2 − bc1 − d

Ad 2 See Tables 4.3 and 4.4. Ad 3 See Tables 4.5 and 4.6. Example 4.5 Ad 1–3 See Tables 4.7 and 4.8.

ap 2 c1

>0

  ∂π y L ∂a 1

pb 2 > 0

  ∂π y G ∂p a2 p c1

>0

  ∂π y L ∂p 1

ab 2 > 0

  ∂π y G ∂c1

  ∂π y G ∂d

− a c2p < 0

−1

  ∂π y L ∂c1

  ∂π y L ∂d

−b < 0

−1

2 2 1

4 Rationality of Choices Made by Individual Producers

89

Table 4.7 Optimal solutions to problems P1m–P3m (E4.2)  2 12 6 x G = 2− 7 > 0 x˜ = ζ (y) = ay > 0 y = a2− 7 > 0   4 24 24 π x G = 7a2− 7 − d ctot (y) = ay 3 π(y) = 7a2− 7 − d

Table 4.8 Measures of sensitivity of profit function and of function of minimum production cost of y units of product (E4.2)

  ∂π x G ∂a ∂ctot (y) ∂a

  ∂π x G ∂d

24

= 7 · 2− 7  4 = −3 ay < 0

∂ctot (y) ∂y

3

p(y) = 2 7 > 0 x˜ = ζ (y) = x G  1 y = a xG 2

= −1  3 = 4 ay > 0

Table 4.9 Optimal solutions to problems P1m–P3m (E4.2)  2 12 6 x G = 2− 7 > 0 x˜ = ζ (y) = ay > 0 y = a2− 7 > 0   4 24 24 π x G = 7a2− 7 − d ctot (y) = ay 3 π(y) = 7a2− 7 − d

xL = b   1 π x L = ab 4 − ab2 − d

 y 2

x˜ = ζ (y) = ctot (y) =

a

>0

y4 a3

3

p(y) = 2 7 > 0 x˜ = ζ (y) = x G  1 y = a xG 2

1

3

y = ab 2 > 0

p(y) = 2 7 > 0

1

π(y) = ab 4 − ab2 − d

x˜ = ζ (y) = x L 1

y = ab 2

Table 4.10 Measures of sensitivity of profit function and of function of minimum production cost of y units of product (E4.2)

  ∂π x G ∂a   ∂π x L ∂a

∂ctot (y) ∂a

24

= 7 · 2− 7 1

= b 4 − b2  4 = −3 ay < 0

  ∂π x G ∂d   ∂π x L ∂b   ∂π x L ∂d

= 41 ab− 4 − 2ab

∂ctot (y) ∂y

=

= −1 3

=

  ∂π y G = −1 ∂d  y 3 4 a >0

Example 4.7 Ad 1–3 See Tables 4.9 and 4.10. E4.3. There are given: p>0 c(x) = (c1 (x1 ), c2 (x2 )) > (0, 0) ci (x i ) = ax i x = (x1 , x2 ) ≥ (0, 0)

a price of product manufactured by a firm, a vector of prices of production factors, a price of i-th production factor is proportional to the demand for i-th factor, a vector of inputs of production factors,

90

4 Rationality of Choices Made by Individual Producers

y = f (x) r (y) = py r (x) = p f (x) ctot (x) = c1 (x 1 )x 1 + c2 (x 2 )x 2 + d cv (x) = c1 (x 1 )x 1 + c2 (x 2 )x 2 cf (x) = d c(y) π (y) = r(y) – c(y) = py – c(y) π (x) = r(x) – ctot (x)

an output level described by an increasing, strictly concave and twice differentiable production function, revenue (turnover) from sales of a manufactured product as a function of output level, revenue (turnover) from sales of a manufactured product as a function of inputs of production factors, total cost of production, variable cost of production, fixed cost of production, minimum cost of producing y output units, derived as an objective function corresponding to an optimal solution to problem (P2c), firm profit as a function of output level, firm profit as a function of inputs of production factors.

For a production function: (a) power: y = f (x) = ax1α1 x2α2 , a > 0, αi ∈ (0, 1), α1 + α2 < 1, i = 1, 2, (b) logarithmic: y = f (x) = a1 lnx1 + a2 lnx2 , ai > 0, i = 1, 2, (c) subadditive: y = f (x) = a1 x1α + a2 x2α , ai > 0, α ∈ (0, 1), i = 1, 2. 1. Solve the profit maximization problem (P1c). 2. Give an economic interpretation of necessary and sufficient conditions of existence of optimal solution to problem (P1c). 3. Analyze sensitivity of the demand for production factors and of the firm maximum profit to changes in a price of a product and changes in values of parameters of the cost function and of the production function. 4. Solve the cost minimization problem (P2c). 5. Give an economic interpretation of necessary and sufficient conditions of existence of optimal solution to problem (P2c). 6. Analyze sensitivity of the conditional demand for production factors and of the firm minimum cost to changes in a price of a product and changes in values of parameters of the cost function and of the production function. 7. Solve the profit maximization problem (P3c). 8. Give an economic interpretation of necessary and sufficient conditions of existence of optimal solution to problem (P3c). 9. Analyze sensitivity of the product supply and of the firm maximum profit to changes in a price of a product and changes in values of parameters of the cost function and of the production function. 10. Justify that the profit maximizations problems (P1c) and (P3c) are equivalent.

4 Rationality of Choices Made by Individual Producers

91

Solutions Ad a P1c Let us write the problem of maximizing the firm’s profit in the case of the power production function and the nonlinear function of total cost:    π (x) = pax1α1 x2α2 − a x12 + x22 − d → max x ≥ (0, 0). Then the marginal profit functions take forms: = α1 ax1α1 −1 x2α2 − 2ax1 , = α2 ax1α1 x2α2 −1 − 2ax2 .

∂π(x) ∂ x1 ∂π(x) ∂ x2

(1) (2)

It is not difficult to check that the boundary conditions for the profit function are met: (3)

lim

∂π(x) ∂ x1

= +∞ > 0 and lim

= −∞ < 0,

lim

∂π(x) ∂ x2

= +∞ > 0 and

= −∞ < 0.

x1 →0+

(4)

x2 →0+

∂π(x) x1 →+∞ ∂ x1 lim ∂π(x) x2 →+∞ ∂ x2

On this basis, we claim that ∃1 x > (0, 0), such that:



(5) ∂π∂ x(x) = α1 pax 1α1 −1 x α2 2 − 2ax 1 = 0 ,

1

x=x

(6) ∂π∂ x(x) = α2 pax α1 1 x 2α2 −1 − 2ax 2 = 0.

2 x=x

Let us solve the system of Eqs. (5)–(6): (7) (8)

α −1 α2 x2 α α −1

α1 pax 1 1

(11)

2ax 1 , 2ax 2

  21

α2 x 1, α1 α1 −1 α2 α1 pax 1 x 2 = α

(9) x 2 = (10)

=

α2 pax 1 1 x 2 2 α1 x 2 = xx 21 , α2 x 1

α1 px 1α1 −1

(12) x 1



  22

2ax 1 ,

x α1 2 = 2x 1 ,    α2 α1 +α2 −2 α2 2 α1 px 1 − 2 = 0, α1 α2 α1

(13) x α1 1 +α2 −2 =

2 2−α2 2

pα1

α2

α2 2

2−α2 2

α

,

α2 2

1 +α2 ) = 1 2 2 , (14) x 2−(α 1 2−α2 α2   1 2(2− α +α ) 2(2− α +α ) (15) x 1 = 2p 2−(α1 +α2 ) α1 ( 1 2 ) α2 ( 1 2 ) ,

pα

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4 Rationality of Choices Made by Individual Producers

Table 4.11 Measures of sensitivity of function of demand for production factors (E4.3a)

∂x1 ∂a ∂x2 ∂a ∂x1 ∂p

0 0

x2 =

 p

1 2−(α1 +α2 )



2−α2

α +α −1

∂x2 ∂p

Table 4.12 Measures of sensitivity of profit function (E4.3a)

α1

 p  2−(α +α ) 2(2−(α1 +α2 )) 2(2−(α1 +α2 )) 1 1 2 α α2 1 2(2−(α1 +α2 ) 2 α1 +α2 −1

∂π (x) ∂a ∂π (x) ∂p

>0

∂π (x) ∂d

−1

α2 α1

 21

α2

 p 1 2 1 2−(α1 +α2 ) α 2(2−(α;1 +α2 )) α 2(2−(α1 +α2 )) 1 2 2(2−(α1 +α2 ) 2

>0

2−α1

>0

>0

2−α2 2 2−(α1 +α2 ))

α1 (

α2 2 2−(α1 +α2 ))

α2 (

= 2 α1 2−α1  p  2−(α 1+α ) 2(2− α +α ) 2(2− α +α ) 1 2 = α1 ( 1 2 ) α2 ( 1 2 ) ,  2  α2 2 1  p  2−(α 1+α ) 2(2−(2−α  p  2−(α 1+α ) 2(2−(αα11+α2 )) 2(2−2−α α;1 +α2 )) 2(2−(α1 +α2 )) (α1 +α2 )) 1 2 α 1 2 α (17) x = α , α . 1 2 1 2 2 2 (16)

Let us notice that the vector of the optimal demand for production factors is determined with an accuracy of a structure or in other words with an accuracy of a multiplication by a number λ > 0. In Tables 4.11 and 4.12 the sensitivity analysis of the demand for production factors and of the firm maximum profit is presented.   1    1  2 2 = λ 1, αα21 (18) x = (x 1 , x 2 ) = x 1 1, αα21 , ∀λ > 0,    1  1   21   2 2 p − αα21 − a − d, (19) π (x) = a p αα21 − αα21 − 1 − d = a αα21 α2 α1 α1 α2 2−α1 (2−α2 )α1  p  α1  p  α2 2(2−(α1 +α2 )) 2(2−(α1 +α2 )) 2−(α1 +α2 ) 2−(α1 +α2 ) 2−(α1 +α2 ) 2−(α1 +α2 ) π (x) = pa α α + α α 1 2 1 2 2  2 α2 α1 α1 α2 (2−α2 )α1 (2−α1 )α2 (20)  p  2α1  p  2α2 2−(α1 +α2 ) 2−(α1 +α2 ) 2−(α1 +α2 ) 2−(α1 +α2 ) −a 2 2−(α1 +α2 ) α1 α2 + 2 2−(α1 +α2 ) α1 α2 − d.

P2c Let us write down the problem of minimizing the costs of production of y units of a product:   ctot (x) = a x12 + x22 + d → min, ax1α1 x2α2 = y, x ≥ (0, 0)

4 Rationality of Choices Made by Individual Producers

93

using a the Lagrangian function:   (1) F(x, λ) = a x12 + x22 + d + λ(y − ax1α1 x2α2 ). From the Kuhn-Tucker theorem it follows that: ∂ F (x,λ˜ )

˜ 1 a x˜1α1 −1 x˜2α2 = 0 (2) = 2a x˜1 − λα ∂ x1 x= x

∂ F (x,λ˜ )

˜ 2 a x˜1α1 x˜2α2 −1 = 0 (3) = 2a x˜2 − λα ∂ x2 x=

x x,λ)

(4) ∂ F( = y − a x˜1α1 x˜2α2 = 0 ∂λ

2a x˜1 2a x˜2

λ=λ˜ α ˜ 1 a x˜1 1−1 x˜2α2 λα ˜ 2 a x˜1α1 x˜2α2 −1 λα

=   (6) x˜22 = αα21 x˜12   21 (7) x˜2 = αα21 x˜1 (5)

  α22 (8) a x˜1α1 x˜2α2 = a x˜1α1 αα21 x˜1α2 = y   α22 (9) x˜1α1 +α2 = ay αα21 2   1   2(α1α+α 2) (10) x˜1 = ay α1 +α2 αα21   2(α α+α   21   1   2(α α+α 2 1 1  y  α +α 1 2) 1 2) y α1 +α2 α1 α2 1 2 (11) x˜2 = αα21 = a α2 a α1     2(α α+α   2(α α+α 2 1 1 1  y  α +α ) y 1 2 1 2) α α (12) x = a 1 2 α21 , a α1 +α2 α21   1    1  2 α2 2 = λ 1, αα21 = x. (13) x = (x˜1 , x˜2 ) = x˜1 1, α1 Equation (13) shows that the function of demand for production factors and the function of conditional demand for production factors are identical in terms of a structure.    α2   α1 2 2  y  α +α α1 (α1 +α2 )  y  α1 +α α2 (α1 +α2 ) 1 2 2 tot c ( + x) = a +d a α2 a α1   α2   2  y  α +α α1 (α1 +α2 ) α1 1 2 (14) . 1+ +d =a a α2 α2   α2 2 α1 (α1 +α2 ) a(α1 + α2 )  y  α1 +α 2 = + d = k(y) α2 a α2 Equation (14) describes the function of the minimum cost of production of y units of a product. In Tables 4.13 and 4.14 the sensitivity analysis of the demand for production factors and of the firm maximum profit is presented. P3c Let us formulate the problem of maximizing the company’s profit as a function of the product supply:

94

4 Rationality of Choices Made by Individual Producers

Table 4.13 Measures of sensitivity of function of conditional demand for production factors (E4.3a)

Table 4.14 Measures of sensitivity of function of minimum cost of production of y units of product (E4.3a)

∂ x˜1 ∂a ∂ x˜2 ∂a

− ay2

y 1 (α 1 +α2) a

1−α1 −α2 α1 +α2

− ay2

y 1 (α 1 +α2) a

1−α1 −α2 α1 +α2

∂ x˜1 ∂y

y 1 a(α 1 +α2 ) a

1−α1 −α2 α1 +α2

∂ x˜2 ∂y

y 1 a(α 1 +α2) a

1−α1 −α2 α1 +α2

 

 

α1 α2 α2 α1

α1 α2 α2 α1



α2 2(α1 +α2 )



α1 2(α1 +α2 )



α2 2(α1 +α2 )



α1 2(α1 +α2 )

∼ ∂ctot x

>0

∂a ∼ ∂ctot x

>0

∂y ∼ ∂ctot x

1

0

∂d

π (y) = py − k(y)     α2 2 a(α1 + α2 )  y  α1 +α α1 (α1 +α2 ) 2 = py − − d → max, α2 a α2 y > 0. Then a function of the marginal profit takes a form:   α2   2−(α 1 +α2 ) (α1 +α2 ) y α1 (α1 +α2 ) 2 α1 +α2 (1) dπdy(y) = p − α1 +α · . α2 a α2 2 It is not difficult to check that the boundary conditions for the profit function are met: (2)

lim

y→0+

∂π(y) ∂y

= p > 0 and lim

y→+∞

∂π(y) ∂y

= −∞ < 0.

On this basis, we claim that ∃1 y > 0 such that:

  2−(α  α2 1 +α2 )  α1 +α2

(α1 +α2 ) y α1 (α1 +α2 ) 2 (3) dπ∂(y) = p − =0

y y=y α1 +α2 α2 a α2 2−(α +α ) α   α +α   2 1 2 (α +α ) 1 2 = 2p αα21 1 2 (4) ay   (α1 +α2 )   2−(αα12+α2 ) (5) ay = 2p 2−(α1 +α2 ) αα21 =  p  (α1 +α2 )  α2  2−(αα12+α2 ) (6) y = a 2 2−(α1 +α2 ) α1

4 Rationality of Choices Made by Individual Producers Table 4.15 Measures of sensitivity of product supply function (E4.3a)

Table 4.16 Measures of sensitivity of profit function (E4.3a)

95

∂y ∂p

>0

∂y ∂a ∂y ∂d

>0 0

∂π (y) ∂a ∂π (y) ∂p

>0

∂π (y) ∂d

0

  (α1 +α2 )   2−(αα12+α2 ) π (y) = pa 2p 2−(α1 +α2 ) αα21 2 ⎛ ⎞ α +α α2 (α1 +α2 )   1 2 (7)   α2 p 2−(α1 +α2 ) α2 2−(α1 +α2 ) a ( ) 2 α1 α1 (α1 +α2 ) 2) ⎝ ⎠ − a(α1α+α − d. a α2 2 Equation (6) describes the optimal supply of the product, and Eq. (7) the maximum profit of the company. In Tables 4.15 and 4.16 the sensitivity analysis of the product supply and of the firm maximum profit is presented. Ad b P1c Let us write the problem of maximizing the firm’s profit in the case of the logarithmic production function and the nonlinear function of total cost:    π (x) = p(a1 ln x1 + a2 ln x2 ) − a x12 + x22 − d → max, x ≥ (0, 0). Then functions of the marginal profit take forms: (1) (2)

∂π(x) ∂ x1 ∂π(x) ∂ x2

= p ax11 − 2ax1 , = p ax22 − 2ax2 .

It is not difficult to check that the boundary conditions for the firm’s profit function are met: (3) (4)

lim ∂π(x) x1 →0+ ∂ x1 lim ∂π(x) x →0+ ∂ x2 2

∂π(x) x1 →+∞ ∂ x1 lim ∂π(x) x2 →+∞ ∂ x2

= +∞ > 0 and lim

= −∞ < 0,

= +∞ > 0 and

= −∞ < 0.

On this basis, we claim that ∃1 x > (0, 0) such that:



(5) ∂π∂ x(x) = p ax 11 − 2ax 1 = 0,

1 x=x

96

4 Rationality of Choices Made by Individual Producers



∂π (x)

∂ x2 x=x

(6)

= p ax 22 − 2ax 2 = 0.

Let us solve the system of Eqs. (5)–(6). (7) (8)

a1 x1 a2 x2

=

a1 x 2 a2 x 1

2ax 1 2ax 2 = xx 21

  21 (9) x 2 = aa21 x 1 (10) 2ax 1 = p ax 11 1 (11) x 21 = pa 2apa1 2 (12) x 1 = 2a (13) 2ax 2 = p ax 22 2 (14) x 22 = pa 2apa2 2 (15) x 2 = 2a   1    1       2 pa2 2 a2 2 1 2 1, = λ 1, aa21 = (16) x = pa , x 1 2a 2a a1     pa 2  pa 2  pa1 4  pa2 4 1 2 −a −d π (x) = p a1 ln + a2 ln + 2a 2a 2a 2a (17)     p 2(a1 +a2 ) 2a1 2a2 p4  = p · ln − 3 a14 + a24 − d. a1 a2 2a a P2c In Table 4.17 the sensitivity analysis of the demand for production factors is presented. Let us write down the problem of minimizing the costs of production of y units of the product:     ctot (x) = a x12 + x22 + d → min, a1 ln x1 + a2 ln x2 = y, x ≥ (0, 0),

Table 4.17 Measures of sensitivity of function of demand for production factors (E4.3b)

∂x1 ∂a1

a1 p 2 2a 2

∂x2 ∂a

−

∂x1 ∂p

pa12 2a 2

∂x2 ∂a2

>0

p 2 a12 4a 3

p2

a2 2a 2

>0 >0

p 2 a22 4a 3

∂x1 ∂a

−

∂x2 ∂p

pa22 2a 2

0

∂ x˜2 ∂y

>0

Table 4.19 Measures of sensitivity of function of minimum cost of production of y units of product (E4.3b)

∼ ∂ctot x

>0

∂a ∼ tot x ∂c

>0

∂y ∼ tot x ∂c

1

∂d

P3c Let us formulate the problem of maximizing the company’s profit as a function of the product supply: π (y) = py − k(y)     a(a1 + a2 ) a 2y+a a1 a2 1 2 = py − e − d → max, a2 a2 y > 0. Then a function of the marginal profit takes a form:  a2 2y a1 2 2 ) a1 +a2 (1) dπdy(y) = p − a(a1a+a e = 0. a2 a1 +a2 2 It is not difficult to check that the boundary conditions for the profit function are met: (2)

lim+

y→0

∂π(y) ∂y

= p−

a

2aa1 2

a +1

a2 2

> 0 and lim

y→+∞

∂π(y) ∂y

= −∞ < 0.

On this basis, we claim that ∃1 y > 0 such that:

 a2 2y

a(a1 +a2 ) a1 +a2 a1 2 (3) dπ∂ (y) = p − e =0

y y=y a2 a2 a1 +a2   a2 2y a2 2 (4) e a1 +a2 = pa 2a a1   a2  a2 2 = ln pa (5) a12y +a2 2a a1 a2  a2 2 2 ln pa (6) y = a1 +a 2 2a a1 a pa a +a a a2   a2  2 1 2 2 ln 2a2 ( a2 ) 2  1 pa2 a2 a(a1 +a2 ) a1 +a2 a1 a1 +a2 − a2 e (7) π (y) = p 2 ln 2a a1 − d. a2 Equation (7) describes the optimal supply of a product, and Eq. (8) the maximum profit of a company.

4 Rationality of Choices Made by Individual Producers Table 4.20 Measures of sensitivity of product supply function (E4.3b)

Table 4.21 Measures of sensitivity of profit function (E4.3b)

99

∂y ∂p

>0

∂y ∂a ∂y ∂d

>0 0

∂π (y) ∂a ∂π (y) ∂p

>0

∂π (y) ∂d

−1

>0

In Tables 4.20 and 4.21 the sensitivity analysis of the product supply and of the firm maximum profit is presented. Ad c P1c See Tables 4.22 and 4.23.      π (x) = pa a1 x1α + a2 x2α − a x12 + x22 − d → max

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(10)

∂π (x)

∂ x1 x=x

= αpa1 x α−1 − 2ax 1 = 0 1

∂π (x)

α−1 = αpa2 x 2 − 2ax 2 = 0 ∂ x2 x=x   α−2 x 1 αpa1 x 1 − 2a = 0   x 2 αpa2x α−2 2 − 2a = 0 2a x α−2 = αpa 1  1 2a x α−2 = αpa 2 2 1  αpa1  2−α x 1 = 2a 1  2  2−α x 2 = αpa 2a    1     1   1 1  2−α αpa1  2−α  αpa2  2−α a2 2−α = λ 1, aa21 = x 1 1, a1 x= , 2a 2a   α α   αpa  2−α αpa1  2−α 2 π (x) = pa a1 + a2 2a 2a  2 2   αpa  2−α αpa1  2−α 2 − d. + −a 2a 2a

In Tables 4.22 and 4.23 the sensitivity analysis of the demand for production factors and of the firm maximum profit is presented.

100

4 Rationality of Choices Made by Individual Producers

Table 4.22 Measures of sensitivity of function of demand for production factors (E4.3c)

Table 4.23 Measures of sensitivity of profit function (E4.3c)

∂x1 ∂a ∂x2 ∂a ∂x1 ∂p

0

0

∂π (x) ∂a ∂π (x) ∂p

>0

∂π (x) ∂d

−1

>0

P2c Let us write down the problem of minimizing the costs of production of y units of the product:     ctot (x) = a x12 + x22 + d → min, a1 x1α + a2 x2α = y, x ≥ (0, 0), using a Lagrange function:   (2) F(x, λ) = a x12 + x22 + d + λ(y − a1 x1α − a2 x2α ). From the Kuhn-Tucker theorem it follows that:

∂ F (x,λ˜ )

(16) = 2a x˜1 − λ˜ αa1 x˜1α−1 = 0 ∂ x1 x= x

∂ F (x,λ˜ )

˜ 2 x˜2α−1 = 0 = 2a x˜2 − λαa (17) ∂ x2 x=

x x,λ)

(18) ∂ F( = y − a1 x˜1α − a2 x˜2α = 0 ∂λ ˜ (19)

2a x˜1 2a x˜2

=

λ=λ ˜ 1 x˜1α−1 λαa ˜ 2 x˜2α−1 λαa

  (20) x˜22−α = aa21 x˜12−α 1   2−α (21) x˜2 = aa21 x˜1 1   2−α x˜1α = y (22) a1 x˜1α + a2 aa21  3−α 3−α (23) x˜1α

a12−α +a22−α

(24) x˜1α =

1

a12−α 1 a12−α y 3−α 3−α a12−α +a22−α

=y

4 Rationality of Choices Made by Individual Producers

101

Table 4.24 Measures of sensitivity of function of conditional demand for production factors (E4.3c)

∂ x˜1 ∂y

>0

∂ x˜2 ∂y

>0

Table 4.25 Measures of sensitivity of function of minimum cost of production of y units of the product (E4.3c)

∼ ∂ctot x

>0

∂a ∼ tot x ∂c

>0

∂y ∼ tot x ∂c

1

∂d

1

(25) x˜1 = (26) x˜2 =

a1(2−α)α y α 1

1  3−α 3−α  α a12−α +a22−α

  21 a2 a1

1

a1(2−α)α y α 1

1  3−α 3−α  α a12−α +a22−α

  1    1  2 2 = λ 1, aa21 = x. (27) x = (x˜1 , x˜2 ) = x˜1 1, aa21 Equation (27) shows that the function of demand for production factors and the function of conditional demand for production factorsare identical in terms of a structure. ⎛ ⎞ 2 2   2 2 ⎜ ⎟ a2 a (2−α)α y α a1(2−α)α y α ⎜ ⎟ +d ctot ( + x) = a ⎜  1 2 2 α α ⎟ ⎝ 3−α ⎠ a1  3−α 3−α 3−α 2−α 2−α 2−α 2−α a1 + a2 a1 + a2 2

(28)

2 α

= y a

  a2  α2 1 + a + d 3−α 1

a1(2−α)α 3−α

a12−α + a22−α

2

a(a1 + a2 ) 2 a1(2−α)α = yα   α2 + d = k(y). a1 3−α 3−α 2−α 2−α a1 + a2 Equation (28) describes the function of the minimum cost of production of y units of a product. In Tables 4.24 and 4.25 the sensitivity analysis of the conditional demand for production factors and of the minimum cost of production is presented.

102

4 Rationality of Choices Made by Individual Producers

P3c Let us formulate the problem of maximizing the firm’s profit as a function of the product supply: π (y) = py − k(y) ⎡

⎤

⎢ ⎥ a(a1 + a2 ) 2 a1 ⎢ ⎥ = ⎢ py − yα  − d ⎥ → max, 2 α ⎣ ⎦ a1 3−α 3−α 2−α 2−α a1 + a2 2 (2−α)α

y > 0. Then a function of the marginal profit takes a form: 2

(1)

dπ (y) dy

= p−

a (2−α)α 2a(a1 +a2 ) 2−α y α  3−α1 3−α  2 αa1 α a12−α +a22−α

= 0.

It is not difficult to check that the boundary conditions are met for the profit function in each case of three considered production functions: (2)

lim

y→0+

dπ(y) dy

= p > 0 and lim

y→+∞

∂π(y) ∂y

= −∞ < 0,

on this basis, we claim that ∃1 y > 0 such that: 2

a1(2−α)α dπ (y)

2a(a1 +a2 ) 2−α α = p − αa1 y  3−α 3−α  2 = 0, (3) ∂y

y=y

(4) y

(5)

(6)

2−α α

=

a12−α +a22−α

2  3−α 3−α  α pαa1 a12−α +a22−α 2

α

,

2a(a1 +a2 )a1(2−α)α 2  3−α 3−α  2−α α  (2−α)  a12−α +a22−α 1 y = 2a(apa1 +a , 2 2) 2 a1(2−α) 2  3−α 3−α  2−α α   (2−α) a12−α +a22−α 2 π (y) = p (2−α) 2a(aa11+a2 ) 2 2 a1(2−α) 2 ⎛ 2 ⎞α  3−α  3−α 2−α 2 2−α 2−α α  (2−α)  a1 +a2 ⎟ a1(2−α)α pa1 2) ⎜ − a(a1a+a ⎝ ⎠ 2 2  3−α 2a(a1 +a2 ) 1 3−α  α 2 a1(2−α) a12−α +a22−α

− d.

Equation (7) describes the optimal supply of the product and Eq. (8) the maximum profit of the company. In Tables 4.26 and 4.27 the sensitivity analysis of the product supply and of the firm maximum profit is presented.

4 Rationality of Choices Made by Individual Producers Table 4.26 Measures of sensitivity of function of product supply (E4.3c)

Table 4.27 Measures of sensitivity of profit function (E4.3c)

103

∂y ∂p

>0

∂y ∂a ∂y ∂d

0. Solutions Let us notice that the production function y = f (x1 , x2 ) has the same properties as the functions: power: y = f (x) = ax1α1 x2α2 , a > 0, αi ∈ (0, 1), α1 + α2 < 1, i = 1, 2, logarithmic: y = f (x) = a1 lnx1 + a2 lnx2 , ai > 0, i = 1, 2, subadditive: y = f (x) = a1 x1α + a2 x2α , ai > 0, α ∈ (0, 1), i = 1, 2. (P1c) Let us write the problem of maximizing the firm’s profit in the general form: π (x) = p f (x) − α( f (x))2 − β f (x) − γ  = −α( f (x))2 + ( p − β)( f (x)) − γ → max, x ≥ (0, 0). Then the marginal profit functions take forms: (1) (2)

∂π(x) ∂ x1 ∂π(x) ∂ x2

= −2α f (x) ∂∂f x(x) + ( p − β) ∂∂f x(x) , 1 1 ∂ f (x) ∂ f (x) = −2α f (x) ∂ x2 + ( p − β) ∂ x2 .

It is not difficult to check that the boundary conditions are met for the profit function in each case of three considered production functions:

104

(3) (4)

4 Rationality of Choices Made by Individual Producers

lim ∂π(x) x1 →0+ ∂ x1 lim ∂π(x) x →0+ ∂ x2

∂π(x) x1 →+∞ ∂ x1 lim ∂π(x) x2 →+∞ ∂ x2

= +∞ > 0 and lim

= −∞ < 0,

= +∞ > 0 and

= −∞ < 0.

2

On this basis, we claim that ∃1 x > (0, 0) such that:





∂ f (x)

∂ f (x)

(5) ∂π∂ x(x) = −2α f + ( p − β) = 0, (x)



∂ x1 x=x ∂ x1 x=x 1

x=x



∂ f (x)

∂ f (x)

∂π (x)

(6) ∂ x2

= −2α f (x) ∂ x2

+ ( p − β) ∂ x2

= 0. x=x

x=x

x=x

Let us write the system of Eqs. (5)–(6) in the equivalent form:



(7) (−2α f (x) + ( p − β)) ∂∂f x(x) = 0,

1

x=x ∂ f (x)

(8) (−2α f (x) + ( p − β)) ∂ x2

= 0. x=x

It follows that: (9) 2α f (x) + ( p − β) = 0, p (10) f (x) = β− . 2α This means that for none of the considered production functions we can give an analytical form of the demand function for both factors of production. (P2c) Let us write down the task of minimizing the cost of producing y units of a product:  ctot ( f (x)) = α( f (x))2 + β f (x) + γ → min f (x) = y, x ≥ (0, 0), using the Lagrange function: (1) F(x, λ) = α( f (x))2 + β f (x) + γ + λ(y − f (x)). From the Kuhn-Tucker theorem, it follows that:



∂ F (x,λ)

∂ f (x)

∂ f (x)

(2) = −2α f + (β − λ) = 0, (x)



∂ x1 ∂ x1 x=x ∂ x1 x=x

x=x



∂ F (x,λ)



(3) = −2α f (x) ∂∂f x(x) + (β − λ) ∂∂f x(x) = 0,



∂ x2 x=x 2 2 x=x x=x



= y − f (x) = 0. (4) ∂ F(x,λ) ∂λ

λ=λ

From the system of Eqs. (2)–(4) it follows that: (5) f (x) = (6) f (x) =

β− p , 2α β− p = 2α

y.

This means that for none of the considered production functions we can give an analytical form, neither for the conditional demand function nor for the function of the minimum cost of producing y units of a product.

4 Rationality of Choices Made by Individual Producers

105

(P3c) Since we do not know the analytical form of the function of the minimum cost of production of y units of a product, we cannot find a solution to the profit maximization problem (P3c), and thus the analytical form of the product supply function and the maximum profit function. E4.5. Solve exercise E4.3 assuming additionally that resources of production factors are limited: ∀i = 1, 2 0 ≤ xi ≤ bi , where ∀i = 1, 2 bi > 0 means the constrained resource of i-th production factor. Solution Based on the conclusions resulting from the solutions of the profit maximization problems in exercise E4.3, we can conclude that:  (1) If xG ∈ B = x ∈ R2+ |0 ≤ x 1 ≤ b1 , 0 ≤ x2 ≤ b2 , then we know the analytical forms of the functions of optimal demand for factors of production, conditional demand for factors of production or optimal supply of the product. Then the optimal solutions E4.3 and E.4.5 are the same.  to problems P1c, P2c and P3c in (2) If xG ∈ / B = x ∈ R2+ |0 ≤ x 1 ≤ b1 , 0 ≤ x2 ≤ b2 , then the function of optimal demand for factors of production has the form: x = x L = (b1 , b2 ), while the optimal product supply function is given as y = f (b1 , b2 ). E4.6. Solve exercise E4.3 taking simultaneously into account the data from exercises E4.4. and E4.5. Solution Based on the conclusions resulting from the solutions of the profit maximization tasks in the problem E4.4. and E4.5, we can conclude that:  (1) If xG ∈ B = x ∈ R2+ |0 ≤ x 1 ≤ b1 , 0 ≤ x2 ≤ b2 , then we do not know the analytical forms of the functions of optimal demand for factors of production, conditional demand for factors of production or optimal supply of the product.  (2) If xG ∈ / B = x ∈ R2+ |0 ≤ x 1 ≤ b1 , 0 ≤ x2 ≤ b2 , then the function of optimal demand for factors of production has the form: x = x L = (b1 , b2 ), while the optimal product supply function is given as y = f (b1 , b2 ). E4.7. There are given: x = (x1 , x2 ) ≥ (0, 0)—a vector of inputs of production factors, y = f (x)—an output level described by an increasing, strictly concave and twice differentiable   production function, α

p(y) = ay > 0—a price of product manufactured by a monopoly as a function of product supply, set by a monopoly,  α

a p( f (x)) = f (x) > 0—a price of product manufactured by a monopoly as a function of production factors’ inputs,

106

4 Rationality of Choices Made by Individual Producers

c(x) = (c1 (x1 ), c2 (x2 )) > (0, 0)—a vector of prices of production factors, each of whom is a function of demand reported by a monopoly for a given production factor, ci (x i ) = ax i —a price of i-th production factor is proportional to the demand for i-th factor, r (y) = p(y)y—revenue (turnover) from sales of a manufactured product as a function of product supply, r (x) = p( f (x)) f (x)—revenue (turnover) from sales of a manufactured product as a function of inputs of production factors,   ctot (x) = c1 (x1 )x1 + c2 (x2 )x2 + d = a x12 + x22 + d—total cost of production, cv (x) = c1 (x1 )x1 + c2 (x2 )x2 = a x12 + x22 —variable cost of production, cf (x) = d—fixed cost of production, k(y)—minimum cost of producing y output units, derived as an objective function corresponding to an optimal solution to problem (P2m), π(y) = r (y) − k(y) = p(y)y − c(y)—firm profit as a function of output level, π (x) = r (x) − ctot (x)—firm profit as a function of inputs of production factors. For a production function: (a) power: y = f (x) = ax1α1 x2α2 , a > 0, αi ∈ (0, 1), α1 + α2 < 1, i = 1, 2, (b) logarithmic: y = f (x) = a1 lnx1 + a2 lnx2 , ai > 0, i = 1, 2, (c) subadditive: y = f (x) = a1 x1α + a2 x2α , ai > 0, α ∈ (0, 1), i = 1, 2. 1. 2. 3. 4. 5.

Solve the profit maximization problem (P1m). Solve the cost minimization problem (P2m). Solve the profit maximization problem (P3m). Determine the optimal price by which a monopoly obtains the maximum profit. Justify that the profit maximizations problems (P1m) and (P3m) are equivalent.

Solutions From the data given we get the revenue function in the following form: r (y) = p(y)y =

 α a · y = a α y 1−α = a α y = a α f (x)1−α = r (x). y

Then: (a) For the power production function: y = f (x) = ax1α1 x2α2 , a > 0, αi ∈ (0, 1), α1 + α2 < 1, i = 1, 2, β

β

r (x) = a α a 1−α x1α1 (1−α) x2α2 (1−α) = ax1 1 x2 2 , β1 = α1 (1 − α) < 1, β2 = α2 (1 − α) < 1. (b) For the logarithmic production function: y = f (x) = a1 lnx1 +a2 lnx2 , a1 , a2 > 0,

4 Rationality of Choices Made by Individual Producers

107

r (x) = a α (a1 ln x1 + a2 ln x2 )1−α . (c) For the subadditive production function: y = f (x) = a1 x1α + a2 x2α , a1 , a2 > 0, 0 < α < 1,  1−α r (x) = a α a1 x1α + a2 x2α . Ad a P1m Let us write the problem (P1m) of maximizing the firm’s profit in the case of the power production function and the nonlinear function of total cost: !   β β π (x) = ax1 1 x2 2 − a x12 + x22 − d → max, x ≥ (0, 0). Then the marginal profit functions take forms: (1)

∂π(x) ∂ x1 ∂π(x) ∂ x2

β −1 β2 x2 β1 β2 −1 β2 ax1 x2

= β1 ax1 1

− 2ax1 ,

= − 2ax2 . It is not difficult to check that the boundary conditions for the profit function are met: (3) lim+ ∂π(x) = +∞ > 0 and lim ∂π(x) = −∞ < 0, ∂ x1 ∂ x1 (2)

x1 →0

(4)

lim+

x2 →0

∂π(x) ∂ x2

x1 →∞

∂π(x) x2 →∞ ∂ x2

= +∞ > 0 and lim

= −∞ < 0.

On this basis, we claim that ∃1 x > (0, 0) such that: β −1 β = β1 ax 1 1 x 2 2 − 2ax 1 = 0,



β β −1 (6) ∂π∂ x(x) = β2 ax 1 1 x 2 2 − 2ax 2 = 0.

2 x=x Let us solve the system of Eqs. (5)–(6): (7) ββ21 xx 21 = xx 21 (5)

∂π (x)

∂ x1 x=x

(8) x 22 =

β2 2 x β1 1

  21 (9) x 2 = ββ21 x 1     β2 β −2 β2 2 β2 (10) ax 1 β1 x 1 1 x − 2 =0 1 β1 2−β2 2

β2

β +β2 −2

β1

β22 = 2

β +β1 −2

β1

β22 =

(11) x 1 1 (12) x 1 1

2−β1 −β2

(13) x 1

2−β2 2

=

β2

2−β2 β1 2

2

β2 β2 2

2 2−β2 β1 2

β2

β2 2

108

4 Rationality of Choices Made by Individual Producers

(14) x 1 =

2−β2 2(2−β1 −β2 )

β1

1

2 2−β1 −β2

  21

(15) x 2 = ⎛ x=

β2 2(2−β1 −β2 )

β2

β2 β1

2−β2 2(2−β1 −β2 )

β1

β2 2(2−β1 −β2 )

β2

=

1

2−β2 2(2−β1 −β2 )

⎝ β1

2 2−β1 −β2

β2 2(2−β1 −β2 )

β2

β1 2(2−β1 −β2 )

β1

β1 2(2−β1 −β2 )

β1

2−β1 2(2−β1 −β2 )

β2 1

2 2−β1 −β2

2−β1 2(2−β1 −β2 )

β2

, 1 1 2 2−β1 −β2 2 2−β1 −β2 (16)   1 β2 2 , ∀λ > 0 = λ 1, β1    β2 β2 2 β2 (17) π (x) = a β1 − β1 − (a + d).

⎞

  1 2 ⎠ = x 1 1, β2 β1

P2m We have the problem of minimizing the cost of production of y units of a product:    ctot (x) = a x12 + x22 + d → min,

ax1α1 x2α2 = y > 0, x1 , x2 ≥ 0.

Let us write it down using Lagrange function:     (1) F(x1 , x2 , λ) = a x12 + x22 + d + λ y − ax1α1 x2α2 . From the Kuhn-Tucker theorem it follows that: ∂ F (x,λ˜ )

˜ 1 a x˜1α1 −1 x˜2α2 = 0 (2) = 2a x˜1 − λα ∂ x1 x= x

∂ F (x,λ˜ )

˜ 2 a x˜1α1 x˜2α2 −1 α = 0 (3) = 2a x˜2 − λα ∂ x2 x=

x x,λ)

(4) ∂ F( = y − a x˜1α1 x˜2α2 = 0 ∂λ

(5) (6) (7) (8) (9) (10) (11) (12)

2a x˜1 2a x˜2

λ=λ˜ α α λ˜ α a x˜ 1−1 x˜ 2

2 = ˜ 1 1α1 α2 −1  λα2 a x˜1 x˜2 x˜22 = αα21 x˜12   21 x˜2 = αα21 x˜1   α22 a x˜1α1 x˜2α2 = a x˜1α1 αα21 x˜1α2 = y   α22 x˜1α1 +α2 = ay αα21 2   1   2(α1α+α 2) x˜1 = ay α1 +α2 αα21   2(α α+α   21   1   2(α α+α 2 1 1  y  α +α 1 2) 1 2) y α1 +α2 α1 α2 1 2 x˜2 = αα21 = a α2 a α1     2(α α+α   2(α α+α 2 1 1 1  y  α +α ) y 1 2 1 2) α α α1 +α2 1 2 1 2

x= a , a α2 α1

4 Rationality of Choices Made by Individual Producers

109

  1    1  2 2 = λ 1, αα21 = x. (13) x = (x˜1 x˜2 ) = x˜1 1, αα21 Equation (13) shows that the function of demand for production factors and the function of conditional demand for production factors are identical in terms of a structure. The function of the minimum cost of production of y units of a product is:    α2   α1 2 2  y  α +α α1 (α1 +α2 )  y  α1 +α α2 (α1 +α2 ) 1 2 2 tot c ( + +d x) = a a α2 a α1   α2   2  y  α +α α1 (α1 +α2 ) α1 1 2 (14) . 1+ +d =a a α2 α2   α2 2 α1 (α1 +α2 ) a(α1 + α2 )  y  α1 +α 2 = + d = k(y) α2 a α2 P3m The problem of maximizing the profit with regard to the supply of the product has the following form: " α 1−α

π (y) = a y

2 a(α1 + α2 )  y  α1 +α 2 − α2 a



α1 α2



#

α2

(α1 +α2 )

−d

→ max

y ≥ 0. Then a function of the marginal profit takes a form:

(1)

 α2 2−α1 −α2  α1 (α1 +α2 ) dπ (y) a(α1 + α2 )  y  α1 +α 2 2 α −α = (1 − α)a y − dy α1 + α2 aα2 a α2  α2 2−α1 −α2    2 y α1 +α2 α1 (α1 +α2 ) = (1 − α)a α y −α − . α2 a α2

It is not difficult to check that the boundary conditions for the profit function are met: (2)

lim

y→0+

dπ(y) dy

= +∞ > 0 and lim

y→+∞

∂π(y) ∂y

= −∞ < 0.

On this basis, we claim that ∃1 y > 0 such that:

  2−α  α2 1 −α2  α1 +α2

α1 (α1 +α2 ) 2 y α −α (3) dπ∂(y) = − α)a y − =0 (1

y y=y α2 a α2   α2   2−α −α   1 2 2+(α−1)(α1 +α2 ) α1 (α1 +α2 ) =0 (4) y −α (1 − α)a α − α22 a1 α1 +α2 y α1 +α2 α2 2+(α−1)(α1 +α2 ) α1 +α2

2+ α1 +α2

α2 α +α

= (1−α)a2 α1( 1 2 ) α1 +α2 α2  2+(α−1)(α  2+(α−1)(α1 +α2 ) 1 +α2 ) (6) y = a α2 (1−α) α 1 2 (5) y

110

4 Rationality of Choices Made by Individual Producers

(7) y = a

  α22 α2 α1

(8) p(y) =

 α a y

=

  αα22 α1 α2

=

  β22 α1 α2

> 0.

Problems and P3m are equivalent because:  P1m   21  ∼   α22 =x and y = a αα21 x = λ 1, αα21 π (x) = π (y). Ad b P1m Let us write the problem (P1m) of maximizing the firm’s profit in case of the logarithmic production function and the nonlinear function of total cost:    π (x) = a α (a1 ln x1 + a2 ln x2 )1−α − a x12 + x22 − d → max x ≥ (0, 0). Then functions of the marginal profit take forms: (1) (2)

∂π(x) ∂ x1 ∂π(x) ∂ x2

= (1 − α)a α (a1 ln x1 + a2 ln x2 )−α ax11 − 2ax1 , = (1 − α)a α (a1 ln x1 + a2 ln x2 )−α ax22 − 2ax2 .

It is not difficult to check that the boundary conditions for the profit function are met: (3) (4)

lim ∂π(x) x1 →0+ ∂ x1 lim ∂π(x) x →0+ ∂ x2 2

∂π(x) x1 →+∞ ∂ x1 lim ∂π(x) x2 →+∞ ∂ x2

= +∞ > 0 and lim

= −∞ < 0,

= +∞ > 0 and

= −∞ < 0.

On this basis, we claim that ∃1 x > (0, 0) such that:



(5) ∂π∂ x(x) = (1 − α)a α (a1 ln x 1 + a2 ln x 2 )−α xa11 − 2ax 1 = 0,

1

x=x

(6) ∂π∂ x(x) = (1 − α)a α (a1 ln x 1 + a2 ln x 2 )−α xa22 − 2ax 2 = 0.

2 x=x

Let us solve the system of Eqs. (5)–(6): (7) aa21 xx 21 = xx 21 (8) x 22 = aa21 x 21   21 (9) x 2 = aa21 x 1   1    1    1  2 a2 2 a2 2 = λ 1, aa21 , ∀λ > 0 (10) x = x 1 , a1 x 1 = x 1 1, a1   1−α    (11) π (x) = a α 21 a2 ln aa21 − a 1 + aa21 − d.

4 Rationality of Choices Made by Individual Producers

111

P2m The problem of minimizing the cost of production of y units of the product takes a form:    ctot (x) = a x12 + x22 + d → min, a1 ln x1 + a2 ln x2 = y > 0, x1 , x2 ≥ 0. Let us write it down using the Lagrange function:   (1) F(x1 , x2 , λ) = a x12 + x22 + d + λ(y − a1 ln x1 + a2 ln x2 ). From the Kuhn-Tucker theorem it follows that:

∂ F (x,λ˜ )

(2) = 2a x˜1 − λ˜ ax˜11 = 0 ∂ x1 x= x

∂ F (x,λ˜ )

(3) = 2a x˜2 − λ˜ ax˜22 = 0 ∂ x2 x=

x x,λ)

(4) ∂ F( = y − a1 ln x˜1 − a2 ln x˜2 = 0 ∂λ

2a x˜1 2a x˜2

λ=λ˜ a λ˜ x˜1 1 a λ˜ 2

=  x˜2 (6) x˜22 = aa21 x˜12   21 (7) x˜2 = aa21 x˜1 (5)

  21 (8) a1 ln x˜1 + a2 ln aa21 x˜1 = y   a22 (9) ln x˜1a1 aa21 x˜1a2 = y a   22 (10) x˜1a1 +a2 aa21 = ey a2   y 2(α1+ a2 ) (11) = e a1 +a2 aa21   21 y   a2 2(α1+ a2 ) (12) x˜2 = aa21 e a1 +a2 aa21    a2   21 y   a2  y 2(α1+ a2 ) 2(α1+ a2 ) x = e a1 +a2 aa21 , aa21 e a1 +a2 aa21 (12)   1    1  2 α2 2 = λ 1, αα21 = x, ∀λ > 0. x = (x˜1 , x˜2 ) = x˜1 1, α1 (13) Equation (13) shows that the function of demand for production factors and the function of conditional demand for production factors are identical in terms of a structure. The function of the minimum cost of production of y units of a product is:

112

4 Rationality of Choices Made by Individual Producers



   a2   a2 a2 a 2y+a a1 (α1+ a2 ) a1 (α1+ a2 ) c ( + x) = a e +d e1 2 a2 a1 a2   a2   2y a1 (α1+ a2 ) a2 (14) . a1 +a2 = ae 1+ +d a2 a1   a2 a(a1 + a2 ) a 2y+a a1 (α1+ a2 ) ae 1 2 + d = k(y) = a1 a2 2y a1 +a2

tot



P3m The problem of maximizing the profit with regard to the supply of a product has the following form: " α 1−α

π (y) = a y

a(a1 + a2 ) a 2y+a − e1 2 a1



a1 a2



#

a2

(α1+ a2 )

+d

→ max

y ≥ 0. Then a function of the marginal profit takes a form:

(1)

dπ (y) a(a1 + a2 ) a 2y+a 2 = (1 − α)a α y −α − e1 2 dy a1 + a2 aa2   α2 2 a 2y+a α1 (α1 +α2 ) α −α = (1 − α)a y − e 1 2 . a α2



a1 a2



a2

(α1+ a2 )

It is not difficult to check that the boundary conditions for the profit function are met: (2)

lim

y→0+

dπ(y) dy

= +∞ > 0 and lim

y→+∞

∂π(y) ∂y

= −∞ < 0.

On this basis, we claim that ∃1 y > 0 such that:

  a2 2→y

(a +a ) (3) dπdy(y)

= (1 − α)a α y −α − a2 e a1 +a2 αα21 1 2 = 0 y=y   21 (4) y = a2 ln αα21 ⎛ ⎞α (5) p(y) = ⎝

a  1 a2 ln

a2 a1

2

⎠ .

Problems and P3m are equivalent because:  P1m   21  ∼   21 =x and y = a2 ln αα21 x = λ 1, αα21 π (x) = π (y).

4 Rationality of Choices Made by Individual Producers

113

Ad c P1m Let us write the problem (P1m) of maximizing the firm’s profit in case of the subadditive production function and the nonlinear function of total cost: !  1−α   π (x) = a α a1 x1α + a2 x2α − a x12 + x22 − d → max, x ≥ (0, 0). Then the marginal profit functions take forms:  −α (1) ∂π(x) = (1 − α)a α a1 x1α + a2 x2α αa1 x1α−1 − 2ax1 , ∂ x1   −α ∂π(x) (2) ∂ x2 = (1 − α)a α a1 x1α + a2 x2α αa2 x2α−1 − 2ax2 . It is not difficult to check that the boundary conditions for the profit function are met: lim ∂π(x) x1 →0+ ∂ x1 lim ∂π(x) x →0+ ∂ x2

(3) (4)

∂π(x) x1 →+∞ ∂ x1 lim ∂π(x) x2 →+∞ ∂ x2

= +∞ > 0 and lim

= −∞ < 0,

= +∞ > 0 and

= −∞ < 0.

2

On this basis, we claim that ∃1 x > (0, 0) such that:

 −α

= (1 − α)a α a1 x α1 + a2 x α2 αa1 x α−1 − 2ax 1 = 0, (5) ∂π∂ x(x)

1 1

x=x   −α

(6) ∂π∂ x(x) = (1 − α)a α a1 x α1 + a2 x α2 αa2 x α−1 − 2ax 2 = 0.

2 2 x=x

Letus solve the system of Eqs. (5)–(6): (7) (8)

a1 x α−1 1 = xx 21 a2 x α−1 2 x α−2 = aa21 x α−2 2 1

(9) x 2 =

1   2−α

a2 x1 a1     1  1    2−α 2−α a2 = λ 1, aa21 , ∀λ > 0 (10) x = x 1 1, a1    1−α 1 2    2−α   2−α a2 a2 α − d. (11) π (x) = a a1 + a2 a1 − a 1 + a1

P2m The problem of minimizing the cost of production of y units of a product takes a form:    ctot (x) = a a1 x1α + a2 x2α + d → min a1 x1α + a2 x2α = y > 0, x1 , x2 ≥ 0.

114

4 Rationality of Choices Made by Individual Producers

Let us write it down using a Lagrange function:     (1) F(x1 , x2 , λ) = a x12 + x22 + d + λ y − a1 x1α + a2 x2α . From the Kuhn-Tucker theorem it follows that:

∂ F (x,λ˜ )

(2) = 2a x˜1 − λ˜ αa1 x˜1α−1 = 0 ∂ x1 x= x

∂ F (x,λ˜ )

(3) = 2a x˜2 − λ˜ αa2 x˜2α−1 = 0 ∂ x2 x=

x x,λ)

(4) ∂ F( = y − a1 x α1 − a2 x α2 = 0 ∂λ

(5)

2a x˜1 2a x˜2

(6)

x˜22−α

λ=λ˜ ˜ x˜2α−1 λα1 ˜ 2 x˜2α−1 λαa

=

  = aa21 x˜12−α 1   2−α (7) x˜2 = aa21 x˜1  2  2 α α    2−α   2−α 2−α 2−α a2 a2 α α α α a1 +a2 = x˜1 (8) a1 x˜1 + a2 a1 x˜1 = x˜1 a1 + a2 a1 =y α a12−α  α a12−α

(9) x˜1α = y

2 a12−α



2

+a22−α

α1

α

ya12−α

(10) x˜1 =

2

2

a12−α +a22−α 1   2−α

(11) x˜2 = ⎛



x=⎝ (12)

α1

α

ya12−α

a2 a1

2

2

a12−α +a22−α

α1

α

ya12−α 2

2

a12−α +a22−α

,

1   2−α

a2 a1



α

ya12−α 2

2

α1 ⎞ ⎠

a12−α +a22−α

   1     1  2−α 2−α = λ 1, aa21 = x. (13) x = (x˜1 , x˜2 ) = x˜1 1, aa21 Equation (13) shows that the function of demand for production factors and the function of conditional demand for production factors are identical in terms of a structure. The function of the minimum cost of production of y units of a product is:

4 Rationality of Choices Made by Individual Producers

⎛⎛ ⎜ ctot ( x) = a ⎝⎝

ya1 2 2−α

2 2−α

+ a2

a1 ⎛ = a⎝ (14)

⎛

⎞ α2

α 2−α

⎞ α2 

α

ya12−α 2

⎠ +



2

⎠

a12−α + a22−α

a2 a1

 1+

2  2−α

a2 a1

115

⎛

α 2−α

ya1

⎝

2 2−α

a1 2  2−α

2

+ a22−α

⎞ α2 ⎞ ⎠ ⎟ ⎠+d

+d

⎞ 2 2 2−α 2−α a + a 2 ⎠ ⎝ 1 ⎠+d = a⎝ 2 2 2 2−α 2−α 2−α a1 + a2 a1  2  α−2 2 α 2 = ay α a12−α + a22−α + d = k(y). α

ya12−α

⎞ α2 ⎛

P3m The problem of maximizing the profit with regard to the supply of a product has the following form: " α 1−α

π (y) = a y

#  2  α−2 2 α 2−α 2−α − ay a1 + a2 − d → max 2 α

y ≥ 0. Then a function of the marginal profit takes a form:  2  α−2 2 α 2−α (1) dπdy(y) = (1 − α)a α y −α − α2 ay α a12−α + a22−α . It is not difficult to check that the boundary conditions for the profit function are met: (2)

lim

y→0+

dπ (y) dy

= +∞ > 0 and lim

y→+∞

dπ (y) dy

= −∞ < 0.

On this basis, we claim that ∃1 y > 0 such that:  2  α−2

2 α 2−α dπ (y)

2−α 2−α 2 α −α (3) = (1 − α)a y − α a y α a1 + a2 =0 ∂y

y=y 2

(4) y =

2

a12−α +a22−α

(5) p(y) =

α

a12−α

 α a y

 =

α

α

aa12−α 2

2

a12−α +a22−α

> 0.

Problems and P3m are equivalent because:  P1m   21  ∼   α22 =x and y = a αα21 x = λ 1, αα21 π (x) = π (y).

116

4 Rationality of Choices Made by Individual Producers

E4.8.*1 Solve exercise E4.7, when the production total cost function has a form: ctot (x1 , x2 ) = α( f (x1 , x2 ))2 + β f (x1 , x2 ) + γ , or equivalently: ctot (y) = αy 2 + βy + γ , where y = f (x1 , x2 ) is an increasing, strictly concave and twice differentiable production function and Δ = β 2 − 4αγ = 0, α, β, γ > 0. Solutions The data given are: a production function: (a) power function: y = f (x) = ax1α1 x2α2 , a > 0, αi ∈ (0, 1), α1 + α2 < 1, i = 1, 2, (b) logarithmic: y = f (x) = a1 lnx1 + a2 lnx2 , ai > 0, i = 1, 2, (c) subadditive: y = f (x) = a1 x1α + a2 x2α , ai > 0, α ∈ (0, 1), i = 1, 2, and p(y) =

 α a y

>0

a price of product manufactured by a monopoly as a function of product supply, set by a monopoly.

Ad 1 Solve the profit maximization problem (P1m). The profit function with inputs of production factors as arguments has a form:  π (x) = p( f (x)) · f (x) − c (x) =   − α( f (x))2 + β f (x) + γ tot

a f (x)

α

f (x)

= a α f (x)1−α − α( f (x))2 − β f (x) − γ . A necessary condition for existence of a maximum of this function takes the following form: ∂π(x) ∂ f (x) ∂ f (x) ∂ f (x) = a α (1 − α) f (x)−α · − 2α f (x) −β = 0 i = 1, 2, ∂ xi ∂ xi ∂ xi ∂ xi 1

This exercise is not a standard one due to a form of the production total cost function resulting in a not standard necessary condition for existence of a profit function’s maximum. Thus, one needs to take into account that solutions and answers to the subsequent tasks neither do have standard forms.

4 Rationality of Choices Made by Individual Producers

117

where ∂∂f x(x) > 0 since function f (x) is increasing. i Simplifying the necessary condition to: a α (1 − α) f (x)−α − 2α f (x) − β = 0 we can notice that in fact we are left with only one equation while there are two variables x1 , x2 for which we would like to find their optimal values. Having one equation it is not possible. Instead we can determine an optimal level of output and then a relationship between inputs of production factors for a given level of output. Using the fact that y = f (x) the necessary condition can be written as: a α (1 − α)y −α − 2αy − β = 0. This is a nonlinear equation and n be solved by numerical methods.2 For a given optimal output level y the relationship between inputs of production factors results from the form of production function:    α1  α1   2 α1 α2 y y (a) ax1 x2 = y x2 = ax α1 x = x1 , ax α1 2 , x1 > 0 1 1   1  y  a1 y a 2 e (b) a1 ln x1 + a2 ln x2 = y x2 = x a1 x = x1 , xea1 2 , x1 > 0 1 1   1 1   y−a1 x1α α y−a1 x1α α (c) a1 x1α + a2 x2α = y x2 = x = x , , 0 ≤ x1 ≤ 1 a a   α1 y . a1 Ad 2 Present a geometric illustration of the profit maximization problem (P1m). Taking for example a = 10, α = 0.75, γ = 1 (which gives β ≈ 1.73) we get the optimal output y equal to about 0.4773. The geometric illustration of the problem is presented in Fig. 4.1. We can notice that the necessary condition determines the optimal output level for which the profit reaches its maximum. On the right-hand side of Fig. 4.1, we present graphs of the function resulting from the relationship f (x1 , x2 ) = y in cases (a)–(c), taking y ≈ 0.4773. The green curve is a graph in the case of the power production function (α1 = α2 = 0.75), the pink one—in the case of the logarithmic function (a1 = a2 = 0.75) and the black one—in the case of the subadditive production function (a1 = a2 = 0.75). One can notice that taking specified values for parameters we bound a set of solutions, that is the set combinations of inputs that satisfy the condition f (x1 , x2 ) = y. But in every case (a)–(c) the number of solutions is uncountably infinite. There is no uniquely determined solution. Here the solution is determined with an accuracy of a relationship between inputs of production factors. 2

MATLAB files to find the optimal value of output using numerical methods are called myfun_ E4_8.m and solvery_E4_8.m. The latter is the file to be run. Taking for example a = 10, α = 0.75, γ = 1 (which gives β ≈ 1.73) we get the optimal output y equal to about 0.4773.

118

4 Rationality of Choices Made by Individual Producers π(y) r (y) ctot (y) necessary condition Profit function

2

Relationship of inputs f (x1, x 2) =y a) power b) logarithmic c) subadditive

1.5

x2

6 4 2

1

0.5

0 0

0.5

1

y

1.5

2

2.5

0 0

0.5

1 x1

1.5

2

Fig. 4.1 Profit maximization problem (P1m, P3m) (E4.8)

Ad 3 Give an economic interpretation of the necessary and sufficient conditions of the existence of an optimal solution to problem (P1m). • For the necessary condition If by given values of parameters the profit function π(x) has a maximum value obtained for an optimal level y of output, then the necessary condition is satisfied for this level of output. Inputs of production factors should then satisfy a condition f (x1 , x2 ) = y. • For the sufficient condition If by some values of parameters the profit function is strictly concave3 then there exists the maximum of this function.

3

Let us recall that strict concavity in the case of a function of many variables requires that the Hessian matrix of the profit function is negative definite. Here, it depends on values of parameters of the production function and of the production total cost function. Let us assume that these values are such that the profit function is strictly concave, which requires, among others that α ∈ (0; 1).

4 Rationality of Choices Made by Individual Producers

119

Ad 4 Solve the cost minimization problem (P2m). The problem has the following form:  ctot (x) = α( f (x))2 + β f (x) + γ → min y = f (x) = const. > 0, x1 , x2 ≥ 0. The problem can be expressed using a Lagrange function: L(x, λ) = ctot (x) + λ(y − f (x)) = α( f (x))2 + β f (x) + γ + λ(y − f (x)) for which a necessary condition for existence of a minimum takes a form of the following system: "

where

∂ f (x) ∂ xi

∂ L(x,λ) ∂ xi ∂ L(x,λ) ∂λ

= 2α f (x) · ∂∂f x(x) + β ∂∂f x(x) − λ ∂∂f x(x) = 0 i = 1, 2 i i i , = y − f (x) = 0

> 0 since function f (x) is increasing. The system can be simplified to: $

2α f (x) + β − λ = 0 y = f (x)

and then to: $

2αy + β − λ = 0 . y = f (x)

We can notice that by a given level y of output and given values of parameters using this system one can find a value of λ and only a relationship between inputs of production factors. But determining optimal inputs is not possible since there is only one equation for two variables x1 , x2 . For a given output level y the relationship between inputs of production factors results from the form of a production function:    α1  α1   (a) ax1α1 x2α2 = y x2 = axyα1 2 x = x1 , axyα1 2 , x1 > 0 1 1   1  y  a1 y a 2 e (b) a1 ln x1 + a2 ln x2 = y x2 = x a1

x = x1 , xea1 2 , x1 > 0 1 1   1 1   y−a1 x1α α y−a1 x1α α α α (c) a1 x1 + a2 x2 = y x2 =

x = x1 , , 0 ≤ x1 ≤ a a   α1 y . a1

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4 Rationality of Choices Made by Individual Producers

Fig. 4.2 Firm’s minimal cost function of producing y output units (E4.8)

7 c(y) 6 5 4 3 2 1 0

0

1

2

3

4

5

6

7

y

Ad 5 Present a geometric illustration of the cost minimization problem (P2m). Taking for example α = 0.75, γ = 1 (which gives β ≈ 1.73) we get a monopoly’s minimal cost function of producing y output units in the following form (Fig. 4.2). ctot (y) = αy 2 + βy + γ = c(y), ctot (y) = 0.75y 2 + 1.73y + 1 = c(y). Ad 6 Give an economic interpretation of the necessary and sufficient conditions of the existence of an optimal solution to problem (P2m). Any inputs of production factors satisfying a condition resulting from the production function f (x1 , x2 ) = y are the solution to the problem. The solution is determined with an accuracy of a relationship between the inputs. Ad 7 Solve the profit maximization problem (P3m). The profit function with output level as an argument has a form:  α   a π (y) = p(y) · y − c(y) = y − c(y) = a α y 1−α − αy 2 + βy + γ . y A necessary condition for existence of a maximum of this function takes the following form:

4 Rationality of Choices Made by Individual Producers

121

∂π(y) = a α (1 − α)y −α − 2αy − β = 0. ∂y This is a nonlinear equation and can be solved by numerical methods.4 Let us check if and when the profit function π (y) is strictly concave: ∂ 2 π(y) = −a α (1 − α)αy −α−1 − 2α < 0. ∂ y2 One can notice that the inequality above is satisfied for example when α ∈ (0; 1). Ad 8 Present a geometric illustration of the profit maximization problem (P3m). The geometric illustration of the problem is presented in Fig. 4.1 on its left-hand side. This is the same figure as for the problem (P1m). Taking for example a = 10, α = 0.75, γ = 1 (which gives β ≈ 1.73) we get the optimal output y equal to about 0.4773. Ad 9 Give an economic interpretation of the necessary and sufficient conditions of the existence of an optimal solution to problem (P3m). • For the necessary condition If by given values of parameters the profit function π(y) has a maximum value obtained for an optimal level y of output, then the necessary condition is satisfied for this level of output. • For the sufficient condition If by some values of parameters the profit function is strictly concave, then there exists the maximum of this function. Ad 10 Justify that the profit maximizations problems (P1m) and (P3m) are equivalent. The solutions to problems (P1m) and (P3m) indicate the same optimal level y of output (found using numerical methods from the nonlinear equation). To have this level of output, a producer has to combine production factors in the amounts resulting from the production function formula. This means that he/she is able to determine the optimal level of output and then to use the proper inputs of production factors. Ad 11 Determine the optimal price by which a monopoly obtains the maximum profit. 4

MATLAB files to find the optimal value of output using numerical methods are called myfun_ E4_8.m and solvery_E4_8.m. The latter is the file to be run. Taking for example, a = 10, α = 0.75, γ = 1 (which gives β ≈ 1.73) we get the optimal output y equal to about 0.4773.

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4 Rationality of Choices Made by Individual Producers

The optimal level of a price results from a relation between the price and the supply:  α a p(y) = . y We can notice that here the optimal price does not depend on the form of the production function. It depends on the form of the production total cost function and its parameters since the optimal level y of output is determined from an equation: a α (1 − α)y −α = 2αy + β, where the right-hand side of the equation (the necessary condition in problems (P1m) and (P3m) results from the total cost function. E4.9. Solve exercise E4.7 assuming additionally that resources of production factors are limited: ∀i = 1, 2 0 ≤ xi ≤ bi , where ∀i = 1, 2 bi > 0 means the constrained resource of i-th production factor. Solutions The data given are: a production function: (a) power function: y = f (x) = ax1α1 x2α2 , a > 0, αi ∈ (0, 1), α1 + α2 < 1, i = 1, 2, (b) logarithmic: y = f (x) = a1 lnx1 + a2 lnx2 , ai > 0, i = 1, 2, (c) subadditive: y = f (x) = a1 x1α + a2 x2α , ai > 0, α ∈ (0, 1), i = 1, 2, and p(y) =

 α a y

>0

a price of product manufactured by a monopoly as a function of product supply, set by a monopoly.

Ad 1 Solve the profit maximization problem (P1m). The profit function with inputs of production factors as arguments has a general form: α  a tot π (x) = p( f (x)) · f (x) − c (x) = f (x) f (x)   2     − a x1 + x22 + d = a α f (x)1−α − a x12 + x22 − d. In this exercise one imposes the additional constraints on inputs of production factors:

4 Rationality of Choices Made by Individual Producers

123

xi ≤ bi i = 1, 2. necessary condition for existence of a maximum of the profit function takes the following form: "

∂π (x) ∂ x1 ∂π(x) ∂ x2

= a α (1 − α) f (x)−α · = a α (1 − α) f (x)−α ·

∂ f (x) ∂ x1 ∂ f (x) ∂ x2

− 2ax1 = 0 − 2ax2 = 0.

Transforming the necessary condition leads to: ∂ f (x) ∂ x2 ∂ f (x) ∂ x1

x2 =

x1

which gives the following relationships between inputs of production factors: 1   2−α / / (a) x2 = αα21 x1 (b) x2 = aa21 x1 (c) x2 = αα21 x1 . Substituting this relationship into one of the equations of the necessary condition gives the global solution that is a solution obtained when the constraints on inputs of production factors are not binding: 1   1 1 (α1 +α2 )(1−α)−2 2 G 2 α2 (1−α)−1 2 α2 (α−1) α x1 = α2 1−α 1 (a) 1   1 2 α1 (α−1) 21 α1 (1−α)−1 (α1 +α2 )(1−α)−2 α12 x 2G = α2  1 − α (b) xG = x 1G , x 2G can be found using numerical methods from a system: " α−1  a1 a2 α 2 a (1 /− α)a1 = 2x1 ln x1 x2 , x2 = aa21 x1 if values of parameters are given  x 1G = (c)

 x 2G =

1 α−1 a α(1 − α)a1 2 1 α−1 a α(1 − α)a2 2

 

1 α 2 −α+2

1 α 2 −α+2





a1 + a2  a2 + a1



a2 a1 a1 a2

−α α 2  2−α α −α+2

−α α 2  2−α α −α+2

,

Let us assume that a = 1, α = 0.75 and that parameters of the production function have the following values: (a) α1 = α2 = 0.75 (b) a1 = a2 = 5 (c) a1 = a2 = 3.

124

4 Rationality of Choices Made by Individual Producers

Then the optimal inputs as the global solution are:   (a) x G = x 1G , x 2G ≈ (0.233, 0.233)   (b) x G = x 1G , x 2G ≈ (1.0483, 1.0483)—found using numerical methods5   (c) x G = x 1G , x 2G ≈ (0.2366, 0.2366), which gives the optimal output as the global solution6 :   (a) y G = f x 1G , x 2G ≈ 0.1125   (b) y G = f x 1G , x 2G ≈ 0.4713   (c) y G = f x 1G , x 2G ≈ 2.0356. Now we can consider a few cases depending on whether the constraints are binding or not: (1) None of the constraints is binding if x iG ≤ bi , i = 1, 2 then x i = x iG . The solution takes the following form:   x = x 1G , x 2G , which means that the firm can use the production factors in quantities resulting from the global solution. Using less than available resources is optimal. (2) The constraint on resource of the first production factor is binding if x 1G > b1 and x 2G ≤ b2 then x 1 = b1 while x 1 = x 2G . The solution takes the following form:   x = b1 , x 2G . (3) The constraint on resource of the second production factor is binding if x 1G ≤ b1 and x 2G > b2 then x 1 = x 1G while x 2 = b2 . The solution takes the following form:   x = x 1G , b2 . (4) The constraints on quantities of both goods are binding if x 1G > b1 and x 2G > b2 following form:

then x 1 = b1 and x 2 = b2 . The solution takes the x = (b1 , b2 ),

5

MATLAB files to find the optimal value of output using numerical methods are called myfun_ E4_9_1.m and solvery_E4_9_1.m. The latter is the file to be run. 6 These are the same values as obtained in task 7 (ad 7).

4 Rationality of Choices Made by Individual Producers

125

Fig. 4.3 Profit maximization problem (P1m) (E4.9)

which means that the firm can use the production factors only in quantities constrained by the resources. Using all available resources is optimal. Ad 2 Present a geometric illustration of the profit maximization problem (P1m). Taking for example α = 0.75, α1 = α2 = 0.75 in the example (a) we get the global solution xG = x 1G , x 2G ≈ (0.23, 0.23). The global solution has to be compared with the resources of production factors b1 and b2 (Fig. 4.3). Let us assume that the constrained resources of production factors are (b1 , b2 ) = (0.25, 0.2). Because of these constraints the global solution (0.23, 0.23) can’t be used by the firm—the firm has to use inputs (0.23, 0.2). The constraint on the first production factor is not binding, the constraint on the second factor is binding. Ad 3 Give an economic interpretation of the necessary and sufficient conditions of the existence of an optimal solution to problem (P1m). • For the necessary condition If by given values of parameters the profit  function π(x) has a maximum value obtained for optimal inputs xG = x 1G , x 2G of production factors, then the necessary condition is satisfied for these inputs. From the necessary condition it results that when the inputs are optimal as the global solution to the problem, then the marginal revenue equals the marginal cost of production. • For the sufficient condition If by some values of parameters the profit function is strictly concave7 then   there exists the maximum of this function obtained for the global solution xG = x 1G , x 2G . 7

Let us recall that strict concavity in case of a function of many variables requires that the Hessian matrix of the profit function is negative definite. Here, it depends on values of parameters of the production function. Let us assume that these values are such that the profit function is strictly concave, which requires, among others that α ∈ (0; 1).

126

4 Rationality of Choices Made by Individual Producers

Ad 4 Solve the cost minimization problem (P2m). The problem has a general form:    ctot (x) = a x12 + x22 + d → min y = f (x) = const. > 0, x1 , x2 ≥ 0. There are additional constraints on inputs of production factors: xi ≤ bi i = 1, 2. The problem can be expressed using a Lagrange function:   L(x, λ) = ctot (x) + λ(y − f (x)) = a x12 + x22 + d + λ(y − f (x)) for which a necessary condition for existence of a minimum takes a form of the following system: "

∂ L(x,λ) ∂ xi ∂ L(x,λ) ∂λ

= 2axi − λ ∂∂f x(x) = 0 i = 1, 2 i = y − f (x) = 0.

Transforming the necessary condition leads to: ⎧ ∂ f (x) ⎨ x = ∂ x2 x 2 ∂ f (x) 1 ∂ x1 ⎩ y = f (x), which gives the following relationships between inputs of production factors: 1   2−α / / (a) x2 = αα21 x1 (b) x2 = aa21 x1 (c) x2 = αα21 x1 . Substituting this relationship into the third equation of the necessary condition gives the global solution that is a solution obtained when the constraints on inputs of production factors are not binding:  1 1    1  α +α   1 α  α1 +α α 2 1 2  G G y α1 2 2 y α2 2 1 G (a) x = x1 , x2 = , a α1 a α2  1 1    1  a +a   21 a2  a1 +a a1 1 2 2   2 e y aa21 (b) xG = x1G , x2G = , e y aa21  − α1    α 1 ) α *− α     1 1 2−α a a (c) xG = x1G , x2G = a1 + a2 a2 2−α y α , a1 a21 + a2 yα .

4 Rationality of Choices Made by Individual Producers

127

Now we can consider a few cases depending on whether the constraints on production factors are binding or not: (1) None of the constraints is binding if xiG ≤ bi , i = 1, 2 then xi = xiG . (2) The constraint on quantity of the first good is binding if x1G > b1 and x2G ≤ b2 then x˜1 = b1 while 1 1 1 −1    (a) x2L = ay b1−α1 α2 (b) x2L = e y b1−a1 a2 (c) x2L = y − a1 b1α α a2 α .   ∼ If now x 2L ≤ b2 then the solution is x= b1 , x˜2L . Because of the constraint on resource of the first production factor a producer chooses bigger input of the second factor than it results from the optimization problem with no constraints on inputs of production factors. If x 2L > b2 then there is no solution satisfying f (x) = y. (3) The constraint on quantity of the second good is binding if x2G > b2 and x1G ≤ b1 then x˜2 = b2 while 1 1 1 −1    (a) x1L = ay b2−α2 α1 (b) x1L = e y b2−a2 a1 (c) x1L = y − a2 b2α α a1 α .   L If now x1L ≤ b1 then the solution is x= x1 , b2 . Because of the constraint on resource of the second production factor a producer chooses bigger input of the first factor than it results from the optimization problem with no constraints on inputs of production factors. If x 1L > b1 then there is no solution satisfying f (x) = y. (4) The constraints on quantities of both goods are binding if x1G > b1 and x2G > b2

then there is no solution satisfying f (x) = y.

Ad 5 Present a geometric illustration of the cost minimization problem (P2m). Taking for example: (a) a = 1, α1 = α2 = 0.75 (b) a1 = a2 = 5 (c) a1 = a2 = 3, α = 0.75, we get the global solution:  G G   y  23  y  23  (a) xG = x1 , x2 = 10 , 10   G G   0.1y G x = x1 , x2 = e , e0.1y (b)  G G   y  43  y  43  (c) xG = x1 , x2 = 6 , 6 .

128

4 Rationality of Choices Made by Individual Producers Inputs and resources

4 3.5

30 a) power b) logarithmic c) subadditive

25

Minimal cost function a) power b) logarithmic c) subadditive

3 20 c ( y)

~ ~G xG 1 =x 2

2.5 2

1.5

15 10

1 5

0.5 0 0

2

y

4

6

0 0

2

y

4

6

Fig. 4.4 Cost minimization problem (P2m) (E4.9)

Taking for example a = 1, α = 0.75, d = 0.25 we get a monopoly’s minimal cost function of producing y output units in the following form: ) 2  2 *  G G + d = c(y), x1 , x1G + ctot x2 = a x2G  G G   G 2  G 2 ctot x1 , x1 + x2 = x2 + 0.25 = c(y). The global solution has to be compared with the resources of production factors b1 and b2 (Fig. 4.4). Let us assume that b1 = b2 = 2.5 and that the producer wants to have an output level y = 6. In the case of the logarithmic and the subadditive production functions the resources are sufficient to obtain this output and the producer can use the optimal   ∼G inputs x = x˜1G , x˜2G that are smaller than resources and guarantee the minimal cost of production. In the case of the power production function, the resources are not sufficient to obtain an output level y = 6. We can state that there is no optimal solution in such case and that the producer has to change an output level he/she wants to obtain choosing a smaller one. Ad 6 Give an economic interpretation of the necessary and sufficient conditions of the existence of an optimal solution to problem (P2m).   ∼ ∼G An optimal Lagrange multiplier λ and optimal inputs x = x˜1G , x˜2G of production factors satisfying the necessary condition are the global solution to the problem. The inputs which satisfy the necessary condition guarantee the desired output level and in the same time minimal cost of production.

4 Rationality of Choices Made by Individual Producers

129

Ad 7 Solve the profit maximization problem (P3m). From point 5 we know that the minimal cost function has the following form: ) 2  2 * + d. c(y) = a x˜1G + x˜2G The profit function with output level as an argument has a form: π (y) = r (y) − c(y) = p(y) · y − c(y) =

 α a y − c(y) = a α y 1−α − c(y). y

There is an additional constraint on output level resulting from the constraints on inputs of production factors: y ≤ f (b1 , b2 ) A necessary condition for existence of a maximum of the profit function takes the following form: ∂r (y) ∂c(y) ∂π(y) ∂c(y) = − = a α (1 − α)y −α − = 0. ∂y ∂y ∂y ∂y Let us assume that a = 1, α = 0.75, d = 0.25 and that parameters of the production function have the following values: (a) α1 = α2 = 0.75 (b) a1 = a2 = 5 (c) a1 = a2 = 3. Then the profit function takes a form: 4

(a) π (y) = y 0.25 − 2y 3 − 0.25 (b) π (y) = y 0.25 − 2e0.2y − 0.25  8 (c) π (y) = y 0.25 − 2 6y 3 − 0.25. The necessary condition gives the global solution,8 that is a solution obtained when the constraints on inputs of production factors are not binding:  3  1213 (a) y G = 32 ≈ 0.1125 (b) y G ≈ 0.4713—found using numerical methods9 from an equation: 0.25y −0.75 − 0.4e0.2y = 0 12   8 3 29 (c) y G = 6 3 · 64 ≈ 2.0356. 8

These are the same values as obtained in task 1 (ad 1). MATLAB files to find the optimal value of output using numerical methods are called myfun_ E4_9_2.m and solvery_E4_9_2.m. The latter is the file to be run.

9

130

4 Rationality of Choices Made by Individual Producers

Fig. 4.5 Profit maximization problem (P3m) (E4.9)

π(y) r (y) c(y) necessary condition Profit function

1 0.5 0 -0.5 -1

0

0.2

0.4

y

0.6

0.8

1

Now we can consider a few cases depending on whether the constraint y ≤ f (b1 , b2 ) is binding or not: (1) The constraint is not binding: if y G ≤ f (b1 , b2 ) then the solution is y = y G , which means that the firm can use the production factors in quantities resulting from the global solution. Using less than available resources is optimal. (2) The constraint is binding which means that the firm can use the production factors only in quantities constrained by the resources: • if y G > f (b1 , b2) and x 1G ≤ b1 while x 2G > b2 then the solution is y = f x 1G , b2 , • if y G > f (b1 , b2) and x 1G > b1 while x 2G ≤ b2 then the solution is y = f b1 , x 2G , • if y G > f (b1 , b2 ) and x 1G > b1 , x 2G > b2 then the solution is y = f (b1 , b2 ). Ad 8 Present a geometric illustration of the profit maximization problem (P3m). In Fig. 4.5, the problem is presented for point (a) assuming a = 1, α = 0.75, d = 0.25, α1 = α2 = 0.75. When the constraint y ≤ f (b1 , b2 ) on the output resulting from the constraints on the production factors is binding, for example having f (b1 , b2 ) = 0.05, then the optimal supply equal to y G = 0.11 can’t be used by the firm. The firm has to use the supply 0.05, so the maximum supply can be obtained from the resources of production factors. When the constraint on the output is not binding, for example having f (b1 , b2 ) = 0.15, then the firm uses the supply resulting from the global

4 Rationality of Choices Made by Individual Producers

131

solution that is y = y G = 0.11 which is less than the output resulting from the usage of all available resources. Ad 9 Give an economic interpretation of the necessary and sufficient conditions of the existence of an optimal solution to problem (P3m). • For the necessary condition If by given values of parameters the profit function π(y) has a maximum value obtained for an optimal level y of output, then the necessary condition is satisfied for this level of output. From the necessary condition it results that when the output is optimal as the global solution to the problem, then the marginal revenue equals the marginal minimal cost of production. • For the sufficient condition If by some values of parameters the profit function is strictly concave, then there exists the maximum of this function obtained for the global solution y G . Ad 10 Justify that the profit maximizations problems (P1m) and (P3m) are equivalent. The solutions to problems (P1m) and (P3m) indicate the same optimal level y G of output.10 If we substitute the optimal inputs of production factors obtained as the global solution to the problem (P1m) into the production function:     f xG = f x 1G , x 2G = y G , we get the same value of the optimal supply obtained as the global solution to the problem (P3m). Moreover the maximum profit is the same in both problems, (P1m) and (P3m). Ad 11 Determine the optimal price by which a monopoly obtains the maximum profit. The optimal price in a general form is given as: α  α  a a p(y) = or p(x) = , y f (x) • • • • 10

if if if if

x iG x 1G x 1G x 1G

≤ bi , > b1 ≤ b1 > b1

i = 1, 2 then and x 2G ≤ b2 and x 2G > b2 and x 2G > b2

  p(x) = p xG   then p(x) = p b1 , x 2G  G  then p(x) = p x 1 , b2 then p(x) = p(b1 , b2 ) .

One can compare values obtained in tasks 1 and 7 (ad 1 and ad 7).

132

4 Rationality of Choices Made by Individual Producers

The producer determines the price of her/his product on the basis of inputs that she/he uses in the production. E4.10.*11 Solve exercise E4.7 taking simultaneously into account the data from exercises E4.8 and E4.9. Solutions The data given are: • a production function: (a) power function: y = f (x) = ax1α1 x2α2 , a > 0, αi ∈ (0, 1), α1 + α2 < 1, i = 1, 2, (b) logarithmic: y = f (x) = a1 lnx1 + a2 lnx2 , ai > 0, i = 1, 2, (c) subadditive: y = f (x) = a1 x1α + a2 x2α , ai > 0, α ∈ (0, 1), i = 1, 2, • a price of product manufactured by a monopoly as a function of product supply, set by a monopoly:  α a p(y) = > 0, y • a production total cost function of a form ctot (x1 , x2 ) = α( f (x1 , x2 ))2 + β f (x1 , x2 ) + γ Δ = β 2 − 4αγ = 0, α, β, γ > 0, • and constraints on inputs of production factors xi ≤ bi i = 1, 2. Ad 1 Solve the profit maximization problem (P1m). In exercise E4.8 we found that there exists the optimal output guaranteeing the maximum profit. But this optimal output can be obtained by infinitely many combinations of inputs of production factors. In other words, the optimal output is uniquely determined but the inputs are not uniquely determined. The production factors have to be used in amount that result from the optimal output and the production function. The relationship between the inputs obtained in exercise E4.8 can be treated now as 11

This exercise is not a standard one (see exercise E4.8) due to a form of the production total cost function resulting in a not standard necessary condition for existence of a profit function’s maximum. Thus, one needs to take into account that solutions and answers to the subsequent tasks neither do have standard forms.

4 Rationality of Choices Made by Individual Producers Fig. 4.6 Profit maximization problem (P1m) (E4.10)

133 Inputs and resources

2

a) power b) logarithmic c) subadditive

x2

1.5

1

0.5

0 0

0.5

1 x1

1.5

2

the global solution that is a solution that would be if there were no constraints on inputs of production factors:    α1  α1   (a) ax1α1 x2α2 = y x2G = axyα1 2 xG = x1 , axyα1 2 , x1 > 0 1 1   1  y  a1 y a 2 G e G (b) a1 ln x1 + a2 ln x2 = y x2 = x a1 x = x1 , xea1 2 , x1 > 0 1 1   1 1   y−a1 x1α α y−a1 x1α α G , 0 ≤ x1 ≤ (c) a1 x1α + a2 x2α = y x2G = x = x , 1 a a   α1 y . a1 Now there are additional constraints on inputs of production factors: xi ≤ bi i = 1, 2. Among combinations of inputs of production factors that give the maximum profit, a producer has to choose such amounts that do not exceed available resources. Ad 2 Present a geometric illustration of the profit maximization problem (P1m). Taking for example a = 10, α = 0.75, γ = 1 (which gives β ≈ 1.73) we get the optimal output y equal to about 0.4773. Let us assume that the resources of production factors are b1 = 0.5 and b2 = 0.25. The problem is presented in Fig. 4.6. It can be noticed that among all combinations guaranteeing the optimal output y only part of them satisfy the conditions: xi ≤ bi i = 1, 2.

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4 Rationality of Choices Made by Individual Producers

Hence, the solution to the problem in which one accounts for the constraints on inputs of production factors can be expressed as:    α1  (a) x = x1 , axyα1 2 , 0 < x1 ≤ b1 ∧ 0 < x 2 ≤ b2   1 1  y a (b) x = x1 , xea1 2 , 0 < x1 ≤ b1 ∧ 0 < x 2 ≤ b2 1     1  1  α y−a1 x1α α ∧ 0 < x 2 ≤ b2 , (c) x = x1 , , 0 ≤ x1 ≤ min b1 , ay1 a By the given values of parameters the constraints are binding in case of all production functions (a)–(c). The producer can choose only these combinations of inputs that do not exceed the resources. Ad 3 Give an economic interpretation of the necessary and sufficient conditions of the existence of an optimal solution to problem (P1m). • For the necessary condition If by given values of parameters the profit function π(x) has a maximum value obtained for an optimal level y of output, then the necessary condition is satisfied for this level of output. Inputs of production factors should then satisfy a condition f (x1 , x2 ) = y. Because of the constraints on inputs of production factors, the set of inputs treated as the global solution is now bounded by b1 and b2 . • For the sufficient condition If by some values of parameters the profit function is strictly concave,12 then there exists the maximum of this function obtained for the global solution xG . Ad 4 Solve the cost minimization problem (P2m). The relationship between the inputs obtained in exercise E4.8 can be treated now as the global solution that is a solution that would be if there were no constraints on inputs of production factors:    α1  α1   (a) ax1α1 x2α2 = y x2 = axyα1 2 xG = x1 , axyα1 2 , x1 > 0 1 1   1  y  a1 y α 2 e G (b) a1 ln x1 + a2 ln x2 = y x2 = x a1

x = x1 , xeα1 2 , x1 > 0 1

12

1

Let us recall that strict concavity in case of a function of many variables requires that the Hessian matrix of the profit function is negative definite. Here it depends on values of parameters of the production function and of the production total cost function. Let us assume that these values are such that the profit function is strictly concave, which requires, among others that α ∈ (0; 1).

4 Rationality of Choices Made by Individual Producers

(c) a1 x1α + a2 x2α = y x2 =   α1 y . a1



y−a1 x1α a

 α1

135



xG =

x1 ,



y−a1 x1α a

 α1 

, 0 ≤ x1 ≤

Now there are additional constraints on inputs of production factors: xi ≤ bi i = 1, 2. Among combinations of inputs of production factors that allow to obtain a given output level y, a producer has to choose such amounts that do not exceed available resources. The solution to the problem in which one accounts for the constraints on inputs of production factors can be expressed as:    α1  y (a) x = x1 , ax α1 2 , 0 < x1 ≤ b1 ∧ 0 < x2 ≤ b2   1 1  y a (b) x = x1 , xea1 2 , 0 < x1 ≤ b1 ∧ 0 < x2 ≤ b2   1   1    α1  y−a1 x1α α y (c) x = x1 , , 0 ≤ x ≤ min b , ∧ 0 < x2 ≤ b2 . 1 1 a a1 Ad 5 Present a geometric illustration of the cost minimization problem (P2m). Taking for example α = 0.75, γ = 1 (which gives β ≈ 1.73) we get a monopoly’s minimal cost function of producing y output units in the following form: ctot (y) = αy 2 + βy + γ = c(y), ctot (y) = 0.75y 2 + 1.73y + 1 = c(y). Let us assume that the resources of production factors are b1 = 0.5 and b2 = 0.25 which gives a constraint on the output level y ≤ f (b1 , b2 ) = f (0.5, 0.25). In the case of the power production function from point (a), taking a = 10, α1 = α2 = 0.75, the constraint is y ≤ 2.1. In Fig. 4.7, a firm’s minimal cost function is presented in comparison with this constraint. Ad 6 Give an economic interpretation of the necessary and sufficient conditions of the existence of an optimal solution to problem (P2m). Any inputs of production factors satisfying a condition resulting from the production function f (x1 , x2 ) = y are the global solution to the problem. The solution is determined with an accuracy of a relationship between the inputs. Ad 7 Solve the profit maximization problem (P3m).

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4 Rationality of Choices Made by Individual Producers

Fig. 4.7 Firm’s minimal cost function of producing y output units (E4.10)

9 c(y) 8 7 6 5 4 3 2 1 0

0

0.5

1

1.5 y

2

2.5

3

In exercise E4.8 we found that there exists the optimal output guaranteeing the maximum profit. It is derived from a necessary condition for existence of a maximum of the profit function: ∂π(y) = a α (1 − α)y −α − 2αy − β = 0 ∂y and treated as the global solution y G . When one accounts for the constraint on the output level y ≤ f (b1 , b2 ) resulting from the constraints on inputs of production factors, then the solution takes a form: y = min( f (b1 , b2 ), y G ). Ad 8 Present a geometric illustration of the profit maximization problem (P3m). Taking for example a = 10, α = 0.75, γ = 1 (which gives β ≈ 1.73), we get the optimal output y equal to about 0.4773. Let us first assume that the resources of production factors are b1 = 0.5 and b2 = 0.25 which gives a constraint on the output level y ≤ f (b1 , b2 ) = f (0.5, 0.25). In the case of the power production function from point a), taking a = 10, α1 = α2 = 0.75, the constraint is y ≤ 2.1 and is not 1 , the binding. When the resources of production factors are b1 = 18 and b2 = 16 constraint is y ≤ 0.26 and is binding. In this case, the profit corresponding to the output level f (0.5, 0.25) is smaller than the profit corresponding to the optimal output as the global solution. In Fig. 4.8, the optimal output as the global solution is presented in comparison with the constraints on output level.

4 Rationality of Choices Made by Individual Producers Fig. 4.8 Profit maximization problem (P3m) (E4.10)

137

7 6 5

π (y)

4 3 2 1 0 -1 0

0.5

1

1.5

2

2.5

y

Ad 9 Give an economic interpretation of the necessary and sufficient conditions of the existence of an optimal solution to problem (P3m). • For the necessary condition If by given values of parameters, the profit function π(y) has a maximum value obtained for an optimal level y G of output treated as the global solution, then the necessary condition is satisfied for this level of output. • For the sufficient condition If by some values of parameters the profit function is strictly concave, then there exists the maximum of this function obtained for the global solution y G . Ad 10 The solutions to problems (P1m) and (P3m) indicate the same optimal level y of output (found using numerical methods from the nonlinear equation). To have this level of output a producer has to combine production factors in the amounts resulting from the production function formula. This means that he/she is able to determine the optimal level of output and then to use the proper inputs of production factors. If the producer has limited resources of production factors he/she has to compare the optimal inputs obtained as the global solution with the maximization problems with the resources. If the constraints on inputs are not binding then using less than resources is optimal. If the constraints are binding then using all available resources in the proper relationship resulting from the production function is optimal. Ad 11 The optimal level of a price results from a relation between the price and the supply:

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4 Rationality of Choices Made by Individual Producers

p(y) =

 α a . y

We can notice that here the optimal price does not depend on the form of the production function. It depends on the form of the production total cost function and its parameters, since the optimal level y of output is determined from an equation: a α (1 − α)y −α = 2αy + β, where the right-hand side of the equation (the necessary condition in problems (P1m) and (P3m)) results from the total cost function. Accounting for the constraints on inputs of production factors gives the optimal price as:   • if y G ≤ f (b1 , b2 ), then p(y) = p y G , • if y G > f (b1 , b2 ), then p(y) = p( f (b1 , b2 )). E4.11. A production process in some firm acting in the perfect competition is described by a one-variable production function of a form: f (x(t)) = x(t)0.25 . In periods t = 0, 1, 2, . . . , 20 the price of a production factor, the product price and the production fixed cost evolve according to the following equations: C(t) = 4 · 0.98−t , p(t) = −0.006t 2 + 0.1t + 3, d(t) =

(−0.006t 2 + 0.1t + 3)2 t t + 1. + 480 · 0.98−t 30

Using the dynamic approach: 1. Solve the profit maximization problem with regard to an input of a production factor. 2. Present a trajectory of the demand for a production factor and a trajectory of the firm’s maximum profit. 3. Solve the production cost minimization problem assuming that the fixed output level y(t) that the firm wants to achieve in subsequent periods is given by a formula: y(t) = 0.00035(t + 15)2 + 1.25. 4. Present a trajectory of the conditional demand for a production factor and a trajectory of the production minimum cost.

4 Rationality of Choices Made by Individual Producers

139

5. Solve the profit maximization problem with regard to output level. 6. Present a trajectory of the product optimal supply and a trajectory of the firm’s maximum profit. Solution Ad 1 Solve the profit maximization problem with regard to an input of a production factor. The profit function with inputs of production factors as arguments has a form: π (x(t)) = r (x(t)) − ctot (x(t)) = p(t) f (x(t)) − (C(t)x(t) + d(t)) = p(t)x(t)0.25 − c(t)x(t) − d(t). A necessary condition for existence of a maximum of this function takes the following form: dπ(x(t)) = 0.25 p(t)x(t)−0.75 − C(t) = 0 dx(t) and gives an optimal input of the production factor:  x(t) =

p(t) 4C(t)

 43

and the optimal supply:  y(t) =

p(t) 4C(t)

 13

.

The optimal input and the optimal supply depend positively on the price p(t) of the product and negatively on the price C(t) of the production factor. A sufficient condition for existence of the maximum of the profit function has a form: d2 π(x(t)) = −0.25 · 0.75 p(t)x(t)−1.75 < 0 dx(t)2 and is satisfied since the price of the product and an input of the production factor have positive values. The maximum profit function then takes the following form:

140

4 Rationality of Choices Made by Individual Producers

Fig. 4.9 Demand for production factor (E4.11)

0.2 x(t)

0.15

0.1

0.05

0 0

5

10

15

20

⊓( p(t), C(t), d(t)) = max π (x(t)) = π (x(t)) = p(t)x(t)0.25 − C(t)x(t) − d(t)   1 4 p(t) 3 p(t) 3 = p(t) − C(t) − d(t) 4C(t) 4C(t)  1 p(t) 3 =3 − d(t) 4c(t) The maximum profit depends positively on the price p(t) of the product and negatively on the price C(t) of the production factor and on the production fixed cost d(t). Whether it is positive, equal to 0 or negative depends ultimately on the production fixed cost. Ad 2 Present a trajectory of the demand for a production factor (Fig. 4.9) and a trajectory of the firm’s maximum profit (Fig. 4.10). It can be seen in Fig. 4.10 that the maximum profit is negative from 12th period to the last 20th period. The firm should decide whether to continue the production activity or to leave the market. It can try to lower the production fixed cost. When it is not possible, the firms should quit its production activity. Ad 3 Solve the production cost minimization problem assuming that the fixed output level y(t) that the firm wants to achieve in subsequent periods is given by a formula: y(t) = 0.00035(t + 15)2 + 1.25.

4 Rationality of Choices Made by Individual Producers Fig. 4.10 Maximum profit of firm (P1c) (E4.11)

0.6

141

π (p(t),C(t),d(t))

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

5

10

15

20

The problem has the following form: ctot (x(t)) = {C(t)x(t) + d(t)} → min y(t) = f (x(t)) = const. > 0, x ≥ 0. Because there is only one production factor its optimal input x(t) ˜ can be found directly from an equation: y(t) = x(t)0.25 , which gives x(t) ˜ = y(t)4 . Hence the minimal cost function takes a form: ctot (x(t)) ˜ = C(t)x(t) ˜ + d(t) = C(t)y(t)4 + d(t) = c(y(t)). Ad 4 Present a trajectory of the conditional demand for a production factor (Fig. 4.11) and a trajectory of the production minimum cost (Fig. 4.12). Ad 5 Solve the profit maximization problem with regard to output level. The profit function with output level as an argument has a form: π (y(t)) = r (t) − c(y(t)) = p(t)y(t) − c(y(t)) = p(t)y(t) − C(t)y(t)4 − d(t).

142

4 Rationality of Choices Made by Individual Producers

Fig. 4.11 Conditional demand for production factor (E4.11)

8 x (t ) 7

6

5

4

3 0

Fig. 4.12 Production minimum cost (E4.11)

5

10

15

20

10

15

20

50 c(y(t)) 45 40 35 30 25 20 15 10 0

5

A necessary condition for existence of a maximum of this function takes the following form: dπ(y(t)) = p(t) − 4C(t)y(t)3 = 0 dy(t) and gives the optimal output level:  y(t) =

p(t) 4C(t)

 13

.

The optimal supply depends positively on the price p(t) of the product and negatively on the price C(t) of the production factor. We can notice that the solution to the profit maximization problem with regard to output level is the same as optimal

4 Rationality of Choices Made by Individual Producers

143

output level that results from the solution to the profit maximization problem with regard to an input of a production factor. A sufficient condition for existence of the maximum of the profit function has a form: d2 π(y(t)) = −12C(t)y(t)2 < 0, dy(t)2 and is satisfied since the price of the production factor and an output have positive values. The maximum profit function then takes the following form: ⊓( p(t), C(t), d(t)) = max π (y(t)) = π (y(t)) = p(t)y(t) − C(t)y(t)4 − d(t)   1 4 p(t) 3 p(t) 3 = p(t) − C(t) − d(t) 4C(t) 4C(t)  1 p(t) 3 =3 − d(t) 4c(t) The maximum profit depends positively on the price p(t) of the product and negatively on the price C(t) of the production factor and on the production fixed cost d(t). Whether it is positive, equal to 0 or negative depends ultimately on the production fixed cost. We can notice that the maximum profit obtained from the solution to the profit maximization problem with regard to output level is the same as the maximum profit obtained from the solution to the profit maximization problem with regard to an input of a production factor. This shows that these two problems are equivalent. Ad 6 Present a trajectory of the product optimal supply (Fig. 4.13) and a trajectory of the firm’s maximum profit (Fig. 4.14). It can be seen in Fig. 4.14 that the maximum profit is negative from 12th period to the last 20th period. The firm should decide whether to continue the production activity or to leave the market. It can try to lower the production fixed cost. When it is not possible the firms should quit its production activity. E4.12. At any moment t ∈ [0; 20] a firm and conditions in which it acts are described as in exercise E4.11, except additional constraint in a form of a production factor resource: b(t) = 0.01(t + 1).

144

4 Rationality of Choices Made by Individual Producers

Fig. 4.13 Optimal supply of product (E4.11)

0.6 y(t) 0.58 0.56 0.54 0.52 0.5 0.48 0.46

Fig. 4.14 Maximum profit of firm (P3c) (E4.11)

0

0.6

5

10

15

20

10

15

20

Π(p(t), C(t), d(t))

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

5

Using the dynamic approach: 1. Solve the profit maximization problem with regard to an input of a production factor and determine time intervals in which the constraint on the production factor resource is binding. 2. Present a trajectory of the demand for a production factor and a trajectory of the firm’s maximum profit. 3. Solve the production cost minimization problem assuming that the fixed output level y(t) that the firm wants to achieve in subsequent periods is given by a formula: y(t) = 0.00035(t + 15)2 + 1.25. Determine time intervals in which the constraint on the production factor resource is binding.

4 Rationality of Choices Made by Individual Producers

145

4. Present a trajectory of the conditional demand for a production factor and a trajectory of the production minimum cost. 5. Solve the profit maximization problem with regard to output level and determine time intervals in which a constraint on output level resulting from the constraint on the production factor resource is binding. 6. Present a trajectory of the product optimal supply and a trajectory of the firm’s maximum profit. Solution Ad 1 Solve the profit maximization problem with regard to an input of a production factor and determine time intervals in which the constraint on the production factor resource is binding. The global solution obtained in exercise E4.11:  x G (t) =

p(t) 4C(t)

 43

needs to be compared with the constraint on input of the production factor: b(t) = 0.01(t + 1). When x G (t) ≤ b(t) then the constraint is not binding and the solution is x(t) = x (t). When x G (t) > b(t) then the constraint is binding and the solution is x(t) = b(t). Summing up x(t) = min(x G (t), b(t)). Time intervals in which the constraint on the production factor resource is binding can be found from an equation13 : G

x G (t) = b(t) that gives t ∈ [0; 9). Ad 2 Present a trajectory of the demand for a production factor (Fig. 4.15) and a trajectory of the firm’s maximum profit (Fig. 4.16).

13

In a MATLAB file called Exercise_4_12.m the intervals are found using commands: T1=find(xG>b,1,’first’)/100-0.01 and T2=find(xG>b,1,’last’)/1000.01.

146 0.25

0.2

4 Rationality of Choices Made by Individual Producers 0.2

x G (t) b(t) x(t)

x(t)

0.15

0.15 0.1

0.1 0.05

0.05

0 0

5

10

15

0 0

20

5

10

15

20

Fig. 4.15 Demand for production factor (E4.12)

0.6

π(x(t)) π(x G (t))

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

0

5

10

15

20

Fig. 4.16 Maximum profit of firm (P1c) (E4.12)

It can be seen in Fig. 4.16 that the maximum profit is negative at the beginning of the time horizon and from moment t ≈ 11.4 to the end of the time horizon T = 20. The firm should decide whether to continue the production activity or to leave the market. It can try to lower the production fixed cost. When it is not possible the firms should quit its production activity. Ad 3 Solve the production cost minimization problem assuming that the fixed output level y(t) that the firm wants to achieve in subsequent periods is given by a formula: y(t) = 0.00035(t + 15)2 + 1.25. Determine time intervals in which the constraint on the production factor resource is binding. The global solution obtained in exercise E4.11:

4 Rationality of Choices Made by Individual Producers

147

x˜ G (t) = y(t)4 needs to be compared with the constraint on input of the production factor: b(t) = 0.01(t + 1). When x˜ G (t) ≤ b(t) then the constraint is not binding and the solution is x(t) ˜ = x˜ (t). When x˜ G (t) > b(t) then the constraint is binding and the solution is x(t) ˜ = b(t). Summing up x(t) ˜ = min(x˜ G (t), b(t)). Time intervals in which the constraint on the production factor resource is binding can be found from an equation14 : G

x˜ G (t) = b(t) that gives t ∈ [0; 20]. It means that in the whole time interval, the constraint on input of the production factor is binding when one regards the cost minimization problem and the conditional demand for the production factor. The firm can’t produce the desired amount of its product because it has the limited resource of the production factor. The firm has to decide whether to continue the production activity taking into account the maximum profit it can achieve over time. Let us recall, from exercise E4.11 that the minimal cost function is: ctot (x(t)) ˜ = C(t)x(t) ˜ + d(t) = C(t)y(t)4 + d(t) = c(y(t)), where C(t) is an increasing function given as: C(t) = 4 · 0.98−t . Ad 4 Present a trajectory of the conditional demand for a production factor and a trajectory of the production minimum cost (Figs. 4.17 and 4.18). Ad 5 Solve the profit maximization problem with regard to output level and determine time intervals in which a constraint on output level resulting from the constraint on the production factor resource is binding. The global solution obtained in exercise E4.11:  y (t) = G

14

p(t) 4C(t)

 13

In a MATLAB file called Exercise_4_12.m the intervals are found using commands: T1=find(xGG>b,1,’first’)/100-0.01 and T2=find(xGG>b,1,’last’)/100-0.01.

148

4 Rationality of Choices Made by Individual Producers

Fig. 4.17 Conditional demand for production factor (E4.12)

Fig. 4.18 Minimum cost of production (E4.12)

needs to be compared with a constraint resulting from the constraint on input of the production factor: f (b(t)) = b(t)0.25 = (0.01(t + 1))0.25 . When y G (t) ≤ f (b(t)) then the constraint is not binding and the solution is y(t) = y G (t). When y G (t) > f (b(t)) then the constraint is binding and the solution is y(t) = f (b(t)). Summing up y(t) = min(y G (t), f (b(t))). Time intervals in which the constraint on the production factor resource is binding can be found from an equation15 : y G (t) = f (b(t))

15

In a MATLAB file called Exercise_4_12.m the intervals are found using commands: T1=find(yG>yL,1,’first’)/100-0.01 and T2=find(yG>yL,1,’last’)/100-0.01.

4 Rationality of Choices Made by Individual Producers

149 y(t)

0.65

0.65 0.6

0.6

0.55

0.55

0.5

0.5

0.45

0.45

0.4

0.4

yG (t) b(t) 0:25 y(t)

0.35 0.3 0

5

10

15

0.35 0

20

5

10

15

20

Fig. 4.19 Optimal supply of product (E4.12)

0.6

π (y(t)) π (y G(t))

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

10

5

15

20

Fig. 4.20 Maximum profit of firm (P3c) (E4.12)

that gives t ∈ [0; 9). Ad 6 Present a trajectory of the product optimal supply (Fig. 4.19) and a trajectory of the firm’s maximum profit (Fig. 4.20). It can be seen in Fig. 4.20 that the maximum profit is negative at the beginning of the time horizon and from moment t ≈ 11.4 to the end of the time horizon T = 20. The firm should decide whether to continue the production activity or to leave the market. It can try to lower the production fixed cost. When it is not possible the firms should quit its production activity. E4.13. A production process in a firm acting as a monopoly is described by a onevariable production function of a form: f (x(t)) = x(t)0.25 . The price of a product manufactured by this monopoly changes according to a function of a form:

150

4 Rationality of Choices Made by Individual Producers

 p(y(t)) =

a(t) y(t)

0.5 , where a(t) > 0 ∀t,

and a production factor price changes in the following way: c(x(t)) = C(t)x(t), where C(t) > 0 ∀t. In periods t = 0, 1, 2, ..., 20 a value of a(t), a value of C(t) and the fixed production cost change according to equations: a(t) = 30.1t + 3, C(t) = 3−0.1t , d(t) =

(−0.006t 2 + 0.1t + 3)2 t t + 1. + −t 480 · 0.98 30

Using the dynamic approach: 1. Solve the monopoly profit maximization problem with regard to an input of a production factor. 2. Present a trajectory of the demand for a production factor and a trajectory of the monopoly maximum profit. 3. Solve the production cost minimization problem assuming that the fixed output level y(t) that the monopoly wants to achieve in subsequent periods is given by a formula: y(t) = 0.00035(t + 15)2 + 1.25. 4. Present a trajectory of the conditional demand for a production factor and a trajectory of the production minimum cost. 5. Solve the monopoly profit maximization problem with regard to output level. 6. Present a trajectory of the product optimal supply and a trajectory of the monopoly maximum profit. 7. Determine the product optimal price and present its trajectory. Solution Ad 1 Solve the monopoly profit maximization problem with regard to an input of a production factor. The profit function with inputs of production factors as arguments has a form:

4 Rationality of Choices Made by Individual Producers

151

π (x(t)) = r (x(t)) − ctot (x(t)) = p(t) f (x(t)) − (c(x(t))x(t) + d(t)) = a(t)0.5 f (x(t))−0.5 f (x(t)) − C(t)x(t)x(t) − d(t) = a(t)0.5 f (x(t))0.5 − C(t)x(t)2 − d(t) = a(t)0.5 (x(t)0.25 )0.5 − C(t)x(t)2 − d(t) = a(t)0.5 x(t)0.125 − C(t)x(t)2 − d(t) A necessary condition for existence of a maximum of this function takes the following form: dπ(x(t)) = 0.125a(t)0.5 x(t)−0.875 − 2C(t)x(t) = 0 dx(t) and gives an optimal input of the production factor:  x(t) =

a(t)0.5 16C(t)

158

and the optimal supply:  y(t) =

a(t)0.5 16C(t)

152 .

A sufficient condition for existence of the maximum of the profit function has a form: d2 π(x(t)) = −0.875 · 0.125a(t)0.5 x(t)−1.875 − 2C(t) < 0 dx(t)2 and is satisfied. The maximum profit is: maxπ (x(t)) = π (x(t)) = a(t)0.5 x(t)0.125 − C(t)x(t)2 − d(t) and whether it is positive, equal to 0 or negative depends ultimately on the production fixed cost d(t). Ad 2 Present a trajectory of the demand for a production factor (Fig. 4.21) and a trajectory of the monopoly maximum profit (Fig. 4.22).

152

4 Rationality of Choices Made by Individual Producers

Fig. 4.21 Demand for production factor (E4.13)

1.6

x(t)

1.4 1.2 1 0.8 0.6 0.4 0.2 0

Fig. 4.22 Maximum profit of monopoly (P1m) (E4.13)

1.8

5

10

15

20

5

10

15

20

Π(x(t))

1.6 1.4 1.2 1 0.8 0.6 0.4 0

It can be seen in Fig. 4.22 that the maximum profit of the monopoly is positive in all considered periods. It results from sufficiently low production fixed cost d(t) and the fact that a monopoly sets the highest price among all forms of market that firm can operate on. Generally, it can happen that the maximum profit of a monopoly is negative, but only due to very high production fixed cost. Ad 3 Solve the production cost minimization problem assuming that the fixed output level y(t) that the monopoly wants to achieve in subsequent periods is given by a formula: y(t) = 0.00035(t + 15)2 + 1.25. The problem has the following form:  ctot (x(t)) = C(t)x(t)2 + d(t) → min y(t) = f (x(t)) = const. > 0, x ≥ 0.

4 Rationality of Choices Made by Individual Producers

153

Because there is only one production factor its optimal input x(t) ˜ can be found directly from an equation: y(t) = x(t)0.25 , which gives x(t) ˜ = y(t)4 . Hence, the minimal cost function takes a form: ctot (x(t)) ˜ = C(t)x(t) ˜ 2 + d(t) = C(t)y(t)8 + d(t) = c(y(t)). Ad 4 Present a trajectory of the conditional demand for a production factor and a trajectory of the production minimum cost (Figs. 4.23 and 4.24).

Fig. 4.23 Conditional demand for production factor (E4.13)

8

x (t )

7

6

5

4

3 0

Fig. 4.24 Minimum cost of production (E4.13)

5

11

10

15

20

15

20

ctot ( x ( t ) )

10.5

10

9.5

9

8.5 0

5

10

154

4 Rationality of Choices Made by Individual Producers

Ad 5 Solve the monopoly profit maximization problem with regard to output level. The profit function with output level as an argument has a form: π (y(t)) = r (y(t)) − c(y(t)) = p(y(t))y(t) − C(t)y(t)8 − d(t) = a(t)0.5 y(t)−0.5 y(t) − C(t)y(t)8 − d(t) = a(t)0.5 y(t)0.5 − C(t)y(t)8 − d(t). A necessary condition for existence of a maximum of this function takes the following form: dπ(y(t)) = 0.5a(t)0.5 y(t)−0.5 − 8C(t)y(t)7 = 0 dy(t) and gives the optimal output level:  y(t) =

a(t)0.5 16C(t)

152 .

We can notice that the solution to the profit maximization problem with regard to output level is the same as optimal output level that results from the solution to the profit maximization problem with regard to an input of a production factor. A sufficient condition for existence of the maximum of the profit function has a form: d2 π(y(t)) = −0.25a(t)y(t)−1.5 − 56C(t)y(t)6 < 0 dy(t)2 and is satisfied. The maximum profit is: maxπ (y(t)) = π (y(t)) = a(t)0.5 y(t)0.5 − C(t)y(t)8 − d(t) and whether it is positive, equal to 0 or negative depends ultimately on the production fixed cost d(t). We can notice that the maximum profit obtained from the solution to the profit maximization problem with regard to output level is the same as the maximum profit obtained from the solution to the profit maximization problem with regard to an input of a production factor. This shows that these two problems are equivalent. Ad 6 Present a trajectory of the product optimal supply and a trajectory of the monopoly maximum profit (Figs. 4.25 and 4.26).

4 Rationality of Choices Made by Individual Producers Fig. 4.25 Optimal supply of product (E4.13)

155

1.15 y(t)

1.1 1.05 1 0.95 0.9 0.85 0.8 0.75

Fig. 4.26 Maximum profit of monopoly (P3m) (E4.13)

1.8

0

5

10

15

20

15

20

Π(y(t))

1.6 1.4 1.2 1 0.8 0.6 0.4 0

5

10

Ad 7 Determine the product optimal price and present its trajectory (Fig. 4.27). The optimal price of the product is  p(y(t)) =

a(t) y(t)

0.5

⎛



= ⎝a(t) ·

a(t)0.5 16C(t)

− 152 ⎞0.5 ⎠

 1 = 16a(t)7 C(t) 15 .

E4.14. At any moment t ∈ [0; 20] a monopoly and conditions in which it acts are described as in exercise E4.13, except for an additional constraint in a form of a production factor resource: b(t) = 0.01t + 1. Using the dynamic approach:

156

4 Rationality of Choices Made by Individual Producers

Fig. 4.27 Optimal price of product (E4.13)

3.6 p(y(t)) 3.4 3.2 3 2.8 2.6 2.4 2.2 0

5

10

15

20

1. Solve the monopoly profit maximization problem with regard to an input of a production factor and determine time intervals in which the constraint on the production factor resource is binding. 2. Present a trajectory of the demand for a production factor and a trajectory of the monopoly maximum profit. 3. Solve the production cost minimization problem assuming that the fixed output level y(t) that the monopoly wants to achieve in subsequent periods is given by a formula: y(t) = 0.00035(t + 15)2 + 1.25.

4. 5.

6. 7.

Determine time intervals in which the constraint on the production factor resource is binding. Present a trajectory of the conditional demand for a production factor and a trajectory of the production minimum cost. Solve the monopoly profit maximization problem with regard to output level and determine time intervals in which a constraint on output level resulting from the constraint on the production factor resource is binding. Present a trajectory of the product optimal supply and a trajectory of the monopoly maximum profit. Determine the product optimal price and present its trajectory in comparison with a trajectory of a price which would be set by the monopoly if there was no constraint on the production factor resource.

Solution Ad 1 Solve the monopoly profit maximization problem with regard to an input of a production factor and determine time intervals in which the constraint on the production factor resource is binding. The global solution obtained in exercise E4.13:

4 Rationality of Choices Made by Individual Producers 1.6

1.6

x G (t) b(t) x(t)

1.4

x(t)

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

157

0

5

10

15

0.2

20

5

0

10

15

20

Fig. 4.28 Demand for production factor (E4.14)

 x G (t) =

a(t)0.5 16C(t)

158

needs to be compared with the constraint on input of the production factor: b(t) = 0.01t + 1. When x G (t) ≤ b(t) then the constraint is not binding and the solution is x(t) = x (t). When x G (t) > b(t) then the constraint is binding and the solution is x(t) = b(t). Summing up x(t) = min(x G (t), b(t)). Time intervals in which the constraint on the production factor resource is binding can be found from an equation16 : G

x G (t) = b(t) that gives t ∈ (17.6; 20]. Ad 2 Present a trajectory of the demand for a production factor and a trajectory of the monopoly maximum profit (Figs. 4.28 and 4.29). Ad 3 Solve the production cost minimization problem assuming that the fixed output level y(t) that the monopoly wants to achieve in subsequent periods is given by a formula: y(t) = 0.00035(t + 15)2 + 1.25.

16

In a MATLAB file called Exercise_4_14.m the intervals are found using commands: T1=find(xG>b,1,’first’)/100-0.01 and T2=find(xG>b,1,’last’)/100-0.01.

158

4 Rationality of Choices Made by Individual Producers

Fig. 4.29 Maximum profit of monopoly (P1m) (E4.14)

1.6

π(x(t)) π(x G (t))

1.55 1.5 1.45 1.4 1.35 1.3 1.25 1.2 17.5

18

18.5

19

19.5

20

Determine time intervals in which the constraint on the production factor resource is binding. The global solution obtained in exercise E4.13: x˜ G (t) = y(t)4 needs to be compared with the constraint on input of the production factor: b(t) = 0.01t + 1. When x˜ G (t) ≤ b(t) then the constraint is not binding and the solution is x(t) ˜ = x˜ (t). When x˜ G (t) > b(t) then the constraint is binding and the solution is x(t) ˜ = b(t). Summing up x(t) ˜ = min(x˜ G (t), b(t)). Time intervals in which the constraint on the production factor resource is binding can be found from an equation17 : G

x˜ G (t) = b(t) that gives t ∈ [0; 20]. It means that in the whole time interval the constraint on input of the production factor is binding when one regards the cost minimization problem and the conditional demand for the production factor. The firm can’t produce the desired amount of its product because it has the limited resource of the production factor. Let us recall, from exercise E4.13 that the minimal cost function is: ctot (x(t)) ˜ = C(t)x(t) ˜ 2 + d(t) = C(t)y(t)8 + d(t) = c(y(t)), where C(t) is a decreasing function given as: 17

In a MATLAB file called Exercise_4_14.m the intervals are found using commands: T1=find(xGG>b,1,’first’)/100-0.01 and T2=find(xGG>b,1,’last’)/100-0.01.

4 Rationality of Choices Made by Individual Producers

159

Fig. 4.30 Conditional demand for production factor (E4.14)

Fig. 4.31 Minimum cost of production (E4.14)

C(t) = 3−0.1t . Ad 4 Present a trajectory of the conditional demand for a production factor and a trajectory of the production minimum cost (Figs. 4.30 and 4.31). Ad 5 Solve the monopoly profit maximization problem with regard to output level and determine time intervals in which a constraint on output level resulting from the constraint on the production factor resource is binding.

160

4 Rationality of Choices Made by Individual Producers

The global solution obtained in exercise E4.13:  y (t) = y(t) = G

a(t)0.5 16C(t)

152

needs to be compared with a constraint resulting from the constraint on input of the production factor: f (b(t)) = b(t)0.25 = (0.01t + 1)0.25 . When y G (t) ≤ f (b(t)) then the constraint is not binding and the solution is y(t) = y G (t). When y G (t) > f (b(t)) then the constraint is binding and the solution is y(t) = f (b(t)). Summing up y(t) = min(y G (t), f (b(t))). Time intervals in which the constraint on the production factor resource is binding can be found from an equation18 : y G (t) = f (b(t)) that gives t ∈ (17.6; 20]. Ad 6 Present a trajectory of the product optimal supply and a trajectory of the monopoly maximum profit (Figs. 4.32 and 4.33). Ad 7 Determine the product optimal price and present its trajectory in comparison with a trajectory of a price which would be set by the monopoly if there was no constraint on the production factor resource.

Fig. 4.32 Optimal supply of product (E4.14) 18

In a MATLAB file called Exercise_4_14.m the intervals are found using commands: T1=find(yG>yL,1,’first’)/100-0.01 and T2=find(yG>yL,1,’last’)/100-0.01.

4 Rationality of Choices Made by Individual Producers Fig. 4.33 Maximum profit of monopoly (P3m) (E4.14)

161

π π

Fig. 4.34 Optimal price of product (E4.14)

The optimal price of the product is:  p(y(t)) =

a(t) y(t)

where y(t) = min(y G (t), f (b(t))) (Fig. 4.34).

0.5 ,

Chapter 5

Rationality of Choices Made by a Group of Producers by Exogenously Determined Function of Demand for a Product

This Chapter presents 18 exercises that help in understanding the problem of describing rational decisions made by individual producers in the conditions of perfect competition, monopoly or duopoly with exogenously defined demand functions. In particular they concern following problems: determining the supply by an enterprise operating in perfect competition, setting a price of a product and the supply by a monopolist, price discrimination by a monopolist when the supply of a product is intended for two markets. Part of the exercises deals with the issues of quantitative competition, on the basis of the Cournot and Stackelberg duopoly models and with the issues of price competition, on the basis of the Bertrand duopoly model. These exercises are presented in the static and the dynamic context. The focus is to consider problems of a partial equilibrium of an enterprise and a general equilibrium of a market. E5.1. Determine an inverse function of demand and a function inverse to a given demand function: y d ( p) = −ap α + b, a, b > 0. Draw graphs of these functions in the case when: A. α ∈ (0; 1), B. α > 1. Solutions Ad A α ∈ (0; 1) y d ( p) = −ap α + b, a, b > 0 )1 ( b − y d ( p) α p(y d ) = a )1 ( b− p α y i ( p) = . a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Sobczak and K. Malaga, Workbook for Microeconomics, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-031-41947-8_5

163

164

5 Rationality of Choices Made by a Group of Producers by Exogenously …

Fig. 5.1 Function of demand and function inverse to demand function when α ∈ (0; 1) (E5.1 A)

Fig. 5.2 Function of demand and function inverse to demand function when α > 1 (E5.1 B)

See Fig. 5.1. Ad B α>1 y d ( p) = −ap α + b, a, b > 0 )1 ( b − y d ( p) α d p(y ) = a )1 ( b− p α i y ( p) = . a See Fig. 5.2.

5 Rationality of Choices Made by a Group of Producers by Exogenously …

165

Fig. 5.3 Demand function and supply function in space R2+ (E5.2 A)

E5.2. There is a market of a product with exogenously determined demand function and product supply function: A. B. 1. 2. 3.

y d ( p) = −ap 2 + b, a, b > 0, y s ( p) = cp 2 + d, c, d > 0, b > d, 1 1 y d ( p) = −ap 2 + b, a, b > 0, y s ( p) = cp 2 + d, c, d > 0, b > d. Determine ranges of: a product price, demand for the product, the product supply. Draw graphs of the demand function and the supply function in space R2+ . Determine an inverse function of demand and a function inverse to a given demand function. Draw their graphs in space R2+ .

Solutions Ad A y d ( p) = −ap 2 + b, a, b > 0, y d ( p) ∈ [0; b] 1.

y s ( p) = cp 2 + d, c, d > 0, b > d [ ( )1 ] b 2 p ∈ 0; a

y s ( p) ∈ [d; +∞). 2. Figure 5.3. b − yd , then p = p = a )1 ( b− p 2 i y = . a 2

3.

See Fig. 5.4.

(

b − yd a

) 21

166

5 Rationality of Choices Made by a Group of Producers by Exogenously …

Fig. 5.4 Function of demand and function inverse to demand function (E5.2 A)

y d ( p) = y s ( p) − a p2 + b = c p2 + d 4.

p 2 (a + c) = b − d )1 ( b−d 2 p= > 0. a+c

Ad B 1

y d ( p) = −ap 2 + b, a, b > 0, y d ( p) ∈ [0; b] 1

1.

y s ( p) = cp 2 + d, c, d > 0, b > d [ ( ) ] b 2 p ∈ 0; a

y s ( p) ∈ [d; +∞). 2. Figure 5.5. b − yd , then p = p = a ) ( b− p 2 yi = p = . a 1 2

3.

See Fig. 5.6.

(

b − yd a

)2

5 Rationality of Choices Made by a Group of Producers by Exogenously …

167

Fig. 5.5 Demand function and supply function in space R2+ (E5.2 B)

Fig. 5.6 Function of demand and function inverse to demand function (E5.2 B)

y d ( p) = y s ( p) 1

1

− ap2 + b = cp2 + d 4.

1

p 2 (a + c) = b − d ) ( b−d 2 p= > 0. a+c

E5.3. Two producers act in perfect competition on a market described in Sect. 5.11 , supplying one homogenous product. A function of demand for the product is linear: y d ( p) = −ap + b,( a,)b > 0, while functions of production total costs are nonlinear y j = cy 2j + d j , c j , d j > 0, j = 1, 2. and of a form: ctot j 1. Determine the optimal supply of a product: 1

See a textbook Malaga K., Sobczak K, Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, 2022.

168

5 Rationality of Choices Made by a Group of Producers by Exogenously …

(a) By the first producer, (b) By the second producer, (c) By both producers—the total supply. 2. Analyse sensitivity of the optimal supply by the first producer, by the second producer and the total supply by both producers to changes in values of parameters describing the market: a market capacity, strength of consumers’ reaction to changes in a product price, production marginal cost of each producer. Solutions y d ( p) = −ap + b, a, b > 0, ( ) y j = c j y 2j + d j , c j , d j > 0, j = 1, 2. ctot j p=

b − (y1 + y2 ) 1 b b − y d ( p) = = α − β(y1 + y2 ), α = , β = a a a a

Ad 1a ) ) ( ( ⊓1 (y1 , y2 ) = py1 − c1 y12 + d1 = (α − β(y1 + y2 ))y1 − c1 y12 + d1 { } = (α − c1 )y12 − βy1 y2 − d1 → max Ad 1b ) ) ( ( ⊓2 (y1 , y2 ) = py2 − c2 y22 + d2 = (α − β(y1 + y2 ))y2 − c2 y22 + d1 { } = (α − c2 )y22 − βy1 y2 − d2 → max Ad 1c

|

∂⊓1 (y1 ,y2 ) | | ∂ y1 y=y

(1)

| ∂⊓1 (y1 ,y2 ) | | ∂ y2

(2)

y=y

= 2(α − c1 )y 1 − β y 2 = 0. = 2(α − c2 )y 2 − β y 1 = 0.

Let us divide Eq. (1) by Eq. (2): (3) (4)

(α−c1 )y 1 = yy 2 . (α−c2 )y(2 )1 2 y 21 = α−c y 22 α−c1

(5) Then y 1 = or (6) y 2 =

(

(7) y = y 2 or

b−ac1 b−ac2

(

(

=

(

b−ac2 b−ac1

) 21

y1, )

b−ac2 ,1 b−ac1

b−ac2 b−ac1

) 21

=λ

)

y 22 .

y2

(

b−ac2 ,1 b−ac1

)

> (0, 0),

λ > 0, c1 < ab , c2 < ab ,

5 Rationality of Choices Made by a Group of Producers by Exogenously …

169

Table 5.1 (a) Measures of sensitivity of product supply of two producers to changes in parameters of demand function and cost function, (b) Measures of sensitivity of total supply of product of two producers to changes in parameters of demand function and cost function (a) y1 =

(

b−ac2 b−ac1

∂ y1 ∂a

= − 21

∂ y1 ∂b

=

1 2

∂ y1 ∂c1

=

1 2

∂ y1 ∂c2

= − 21

∂ y1 ∂ y2

=

( (

(

)1 2

b−ac2 b−ac1

b−ac2 b−ac1 b−ac2 b−ac1

(

)− 1 2

)− 1 2

)− 1

b−ac2 b−ac1

b−ac1 b−ac2

y2 =

y2

2

2b−a(c1 +c2 ) y2 (b−ac1 )2

0

∂ y2 ∂b

=

a(b−ac2 ) y (b−ac1 )2 2

>0

∂ y2 ∂c1

= − 21

∂ y1 ∂c2

=

1 2

∂ y2 ∂ y1

=

b−ac2 b−ac1

)− 1 2

a(b−ac1 ) y (b−ac1 )2 2

0

(

2

b−ac1 b−ac2

(

y1

b−ac1 b−ac2

a(c2 −c1 ) y (b−ac1 )2 2

1 2

)1

2

)− 1 2

b−ac1 b−ac2

b−ac1 b−ac2

)− 1

a(c1 −c2 ) y (b−ac2 )2 1

)− 1 2

)− 1 2

2b−a(c1 +c2 ) y1 (b−ac2 )2

>0

a(b−ac2 ) y (b−ac2 )2 2

a(b−ac1 ) y (b−ac1 )2 2

0

(b) 1 +c2 ) y = λ 2b−a(c b−ac1

( ) b(c −c ) c2 +c c λ 1 2 1212 (b−ac1 )

∂y ∂a

=

∂y ∂b

= λ a(c2 −c12) ∈ R

1 +c2 ) y = λ 2b−a(c b−ac2

∈R

(b−ac1 )

∂y ∂a

=λ

( ) b(c2 −c1 ) c22 +c1 c2 2 (b−ac2 )

∂ y2 ∂b

= λ a(c1 −c22) ∈ R

∈R

(b−ac2 )

∂y ∂c1

= λ a(b−ac22) > 0 (b−ac1 )

∂y ∂c1

a = −λ b−ac (0, 0), (8) y = y 1 1, b−ac b−ac 2 2

(b−ac2 )

λ > 0, c1 < ab , c2 < ab ,

1 +c2 ) (9) y = y 1 + y 2 = λ 2b−a(c > 0, b−ac1

or 1 +c2 ) > 0. (10) y = y 1 + y 2 = λ 2b−a(c b−ac2

Ad 2 In Table 5.1, the sensitivity analysis of the product supplies is presented. E5.4. Three producers act in conditions of an oligopoly on a market of one homogenous product, having equal positions on the market. A function of demand for the product is linear y d ( p) = −ap + b, a, b > 0, functions of production total costs are also linear: citot (yi ) = ci yi + di , ci , di > 0, i = 1, 2, 3. 1. Determine the optimal supply of a product by each of three producers. 2. Determine the optimal total supply by all three producers. 3. Analyse sensitivity of the optimal supply by each producer and the total supply by all three producers to changes in values of parameters describing the market:

170

5 Rationality of Choices Made by a Group of Producers by Exogenously …

a market capacity, strength of consumers’ reaction to changes in a product price, production marginal cost of each producer. 4. Generalize conclusions derived in points 1–3 to the case of r producers (r ∈ N, r ≥ 3). Solution Ad 1 We have: • Functions of the total production costs of three producers: (1) ∀i = 1, 2, 3 citot (yi ) = civ (yi ) + ci (yi ) = ci yi + di , ci , di > 0. f

being sums of variable cost functions: (2) ∀i = 1, 2, 3 civ (yi ) = ci yi , ci > 0, and the fixed cost functions: f

(3) ∀i = 1, 2, 3 ci (yi ) = di > 0. Since the total cost functions are linear functions of output, then: (1) ∀i = 1, 2, 3

dcitot (yi ) dyi

=

dciv (yi ) dyi

= ci > 0,

the marginal total and variable costs of the i-th producer are increasing functions of output. The function of demand for a product reported by consumers, depending on its price set by producers, is: (5) y d ( p) = −ap + b, a, b > 0, where: a A measure of consumer reaction to a change in the price of a product, b Measure of market capacity. Since the demand function must take non-negative values, so: [ ] (6) p ∈ 0; ab . The total supply by the three producers adjusts to the demand that is reported by consumers by the given price of the product: (7) y1 + y2 + y3 = y d ( p) = −ap + b, a, b > 0. The first producer wants to determine such a quantity of output that, given the output of the second and third producers, guarantees her/him the maximum profit: (8)

⊓1 (y1 )| y2 ,y3 = const.≥0 → max y1 ≥ 0.

.

5 Rationality of Choices Made by a Group of Producers by Exogenously …

171

The second producer wants to determine such a quantity of output that, given the output of the first and third producers, guarantees her/him the maximum profit: (9)

⊓2 (y2 )| y1 ,y3 = const.≥0 → max y2 ≥ 0.

.

The third producer wants to determine such a quantity of output that, given the production volumes of the first and second producers, guarantees her/him the maximum profit: (10)

⊓3 (y3 )| y1 ,y2 =const.≥0 → max y3 ≥ 0.

.

The profit function of i-th producer can be expressed as the difference between its revenue function from the sale of the product and the function of the total cost of production: (11) ∀i = 1, 2, 3 ⊓i (yi ) = p(y)yi − ci yi − di = ( p(y) − ci )yi − di , Substituting into the system of Eqs. (11) an inverse demand function p(y) = = α − β(y1 + y2 + y3 ) where: α = ab , β = a1 , we get the profit functions of the three producers as functions of their output: b−y a

• For the first producer: (12)

⊓1 (y1 , y2 , y3 ) = [α − β(y1 + y2 + y3 )]y1 ( ) − c1 y1 − d1 = [α − c1 ]y1 − β y12 + y1 y2 + y1 y3 − d1 ,

• For the second producer: (13)

⊓2 (y1 , y2 , y3 ) = [α − β(y1 + y2 + y3 )]y2 − c2 y2 − d2 = [α − c2 ]y2 − β(y22 + y1 y2 + y2 y3 ) − d2 ,

• For the third producer: (14)

⊓3 (y1 , y2 , y3 ) = [α − β(y1 + y2 + y3 )]y3 − c3 y3 − d3 = [α − c3 ]y3 − β(y32 + y1 y3 + y2 y3 ) − d3 .

The necessary and sufficient conditions for the profit maximization problem for the first producer, taking levels of production of the second and third producers as given, are2 : | | (15) ∂⊓1 (y∂1y,y1 2 ,y3 ) | = 0 −necessary condition, y1 =y 1 ,y2 ,y3 =const.≥0

2

The profit function of the first (second, third) producer with a given supply of a product of the other two enterprises is a one-variable function. In conditions (15)–(20), we use notations appropriate for first- and second-order partial derivatives, but the necessary and sufficient conditions for the existence of an optimum refer in fact to one-variable functions.

172

5 Rationality of Choices Made by a Group of Producers by Exogenously …

|

(16)

∂ 2 ⊓1 (y1 ,y2 ,y3 ) | | ∂ y12 y1 =y 1 ,y2 ,y3 =const.≥0

< 0 −sufficient condition.

The necessary and sufficient conditions for the profit maximization problem for the second producer, taking levels of production of the second and third producers as given, are: | | (17) ∂⊓2 (y∂1y,y2 2 ,y3 ) | = 0 − necessary condition, y2 =y 2 ,y1 ,y3 =const.≥0 | 2 | (18) ∂ ⊓2 (y∂ y1 2,y2 ,y3 ) | < 0 − sufficient condition. 2

y2 =y 2 ,y1 ,y3 =const.≥0

The necessary and sufficient conditions for the profit maximization problem for the third producer, taking levels of production of the first and second producers as given, are: | | (19) ∂⊓3 (y∂1y,y3 2 ,y3 ) | = 0 − necessary condition, |y3 =y 3 ,y1 ,y2 =const.≥0 2 | (20) ∂ ⊓3 (y∂ y1 2,y2 ,y3 ) | < 0 − sufficient condition. 3

y3 =y 3 ,y1 ,y2 =const.≥0

Considering the necessary and sufficient conditions for the profit functions described by Eqs. (12)–(14), we obtain: • For the first producer: | | (21) ∂⊓1 (y∂1y,y1 2 ,y3 ) | = α − c1 − 2β y 1 −β(y2 + y3 ) = 0, |y1 =y 1 , y2 ,y3 =const.≥0 2 | (22) ∂ ⊓1 (y∂ y1 2,y2 ,y3 ) | = −2β < 0,, 1

y1 =y 1 , y2 ,y3 =const.≥0

which means that for any (given) levels of production of the second and third producers, the first producer obtains a maximum profit for y1 = y 1 . • For the second producer: | | (23) ∂⊓2 (y∂1y,y2 2 ,y3 ) | = α − c2 − 2β y 2 −β(y1 + y3 ) = 0, y2 =y 2 , y1 ,y3 =const.≥0 | 2 | (24) ∂ ⊓2 (y∂ y1 2,y2 ,y3 ) | = −2β < 0, 2

y2 =y 2 , y1 ,y3 =const.≥0

which means that for any (given) levels of production of the first and third producers, the second producer obtains a maximum profit for y2 = y 2 : • For the third producer: | | (25) ∂⊓(y∂1y,y3 2 ,y3 ) | = α − c3 − 2β y 3 −β(y1 + y2 ) = 0, y|3 =y 3 , y1 ,y2 =const.≥0 2 | (26) ∂ ⊓3 (y∂ y1 2,y2 ,y3 ) | = −2β < 0, 3

y3 =y 3 , y1 ,y2 =const.≥0

which means that for any (given) levels of production of the first and second producers, the third producer obtains a maximum profit for y3 = y 3 . In order to find the equilibrium state in the Cournot oligopoly model, one needs to solve the following system of equations:

5 Rationality of Choices Made by a Group of Producers by Exogenously …

173

(27) α − c1 − 2β y 1 −β y 2 −β y 3 = 0, (28) α − c2 − 2β y 2 −β y 1 −β y 3 = 0, (29) α − c3 − 2β y 3 −β y 1 −β y 2 = 0, equivalent to the system of equations: (30) 2 y 1 + y 2 + y 3 = (31) 2 y 2 + y 1 + y 3 = (32) 2 y 3 + y 1 + y 2 =

α−c1 β α−c2 β α−c3 β

= b − ac1 , = b − ac2 , = b − ac3 .

Finally, after simple transformations, the equilibrium state in the Cournot oligopoly model is described by a vector3 : ) ( ) ( (C) (C) b−a(3c1 −(c2 +c3 )) b−a(3c2 −(c1 +c3 )) b−a(3c3 −(c1 +c2 ) = . , y , y , , (30) y(C) = y (C) 1 2 3 4 4 4 This is the optimal state of equilibrium in the sense of Nash. It follows that the optimal total supply of a product in the Cournot oligopoly market is: (C) (C) (31) y = y (C) 1 + y2 + y3 =

3b−a(c1 +c2 +c3 ) , 4

and the Walrasian equilibrium price takes the value: (32) p =

b+a(c1 +c2 +c3 ) . 4a

The maximum profits of enterprises are at the following levels: ( ) (b−a(3c1 −(c2 +c3 ))(b+a (c1 +c2 +c3) −4c1 ) − d1 . (33) ⊓1 y 1 , y 2 , y 3 = 16 ( ) (b−a(3c2 −(c1 +c3 ))(b+a c1 +c2 +c3) −4c2 ) ( (34) ⊓2 y 1 , y 2 , y 3 = − d2 . 16 ( ) (b−a(3c3 −(c2 +c3 ))(b+a c1 +c2 +c3) −4c3 ) ( (35) ⊓3 y 1 , y 2 , y 3 = − d3 . 16 Ad 3 See Table 5.2. Ad 4 In case of four producers: b − a(4c1 − (c2 + c3 + c4 )) b − a(4c2 − (c1 + c3 + c4 )) y2 = 5 5 b − a(4c3 − (c1 + c1 + c4 )) b − a(4c4 − (c1 + c2 + c3 )) y3 = y4 = 5 5 4b − a(c1 + c2 + c3 + c4 ) y= 5 b + a(c1 + c2 + c3 + c4 ) . p= 5a y1 =

3

The Reader interested in a specific solution by specific values of parameters can exploit an Excel file called Exercise 5.4.xlsx.

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

Table 5.2 Measures of sensitivity to changes in parameters of demand function and cost function for: (a) supply of first producer’s product, (b) supply of second producer’s product, (c) supply of third producer’s product, (d) total supply of a product, (e) Walrasian equilibrium price (a) y1

∂ y1 ∂a

∂ y1 ∂b

∂ y1 ∂c1

∂ y1 ∂c2

∂ y1 ∂c3

b−a(3c1 −(c2 +c3 )) 4

−3c1 +(c2 +c3 ) 4

1 4

− 3a 4

a 4

a 4

y2

∂ y2 ∂a

∂ y2 ∂b

∂ y2 ∂c1

∂ y2 ∂c2

∂ y2 ∂c3

b−a(3c2 −(c1 +c3 )) 4

−3c2 +(c1 +c3 ) 4

1 4

a 4

− 3a 4

a 4

y3

∂ y3 ∂a

∂ y3 ∂b

∂ y3 ∂c1

∂ y3 ∂c2

∂ y3 ∂c3

b−a(3c3 −(c1 +c2 )) 4

−3c3 +(c1 +c2 ) 4

1 4

a 4

a 4

− 3a 4

∂y ∂a −(c1 +c2 +c3 ) 4

∂y ∂b

∂y ∂c1

∂y ∂c2

∂y ∂c3

3 4

− a4

− a4

− a4

p

∂p ∂a

∂p ∂b

∂p ∂c1

∂p ∂c2

∂p ∂c3

b+a(c1 +c2 +c3 ) 4a

− 4ab 2

1 4a

1 4

1 4

1 4

(b)

(c)

(d) y 3b−a(c1 +c2 +c3 ) 4

(e)

In case of r producers (r ≥ 3): yi =

( ) ∑ b − a r · ci − rj=1, j/=i ci

r +1 r · b − a(c1 + c2 + · · · + cr ) y= r +1 b + a(c1 + c2 + · · · + cr ) . p= (r + 1)a

i = 1, 2, . . . , r

E5.5. Consider two markets of one homogenous product. The first one is a perfect competition market described in Sect. 5.14 , the second one is a monopolistic market (a monopoly) presented in Sect. 5.25 . Determine conditions by which the optimal supply of a product on the perfect competition market is: (a) bigger than, (b) equal to, 4

See a textbook Malaga K., Sobczak K, Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, 2022. 5 See a textbook Malaga K., Sobczak K, Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, 2022.

5 Rationality of Choices Made by a Group of Producers by Exogenously …

(c) smaller than the optimal supply of a product on the monopolistic market. Solutions Perfect competition market y1d ( p) = −a1 p1 + b1 b1 − y1d ( p) b1 1 = α1 − β1 y1d ( p), where α1 = , β1 = a1 a1 a1 If y1d ( p) = y1s ( p) = y1 , then p1 y1 = (α1 − β1 y1 )y1 ⊓1 (y1 ) = p1 y1 − c1tot (y1 ) = {(α1 − β1 y1 )y1 − (c1 y1 + d1 )} → max d⊓1 (y1 ) = −2β1 y1 + (α1 − c1 ) dy1 d⊓1 (y1 ) b1 lim = α1 − c1 > 0 for c1 < α1 = y1 →0+ dy1 a1 d⊓1 (y1 ) lim = −∞ y1 →+∞ dy | 1 d⊓1 (y1 ) || = −2β1 y 1 + (α1 − c1 ) = 0 | dy p1 =

1

y1 =y 1

b1 − a1 c1 α1 − c1 = >0 y1 = 2β 2

for

c1
0 for c2 < α2 = y2 →0 dy2 a2 d⊓2 (y2 ) lim = −∞ y2 →+∞ dy | 2 d⊓2 (y2 ) || = −2β2 y 2 + (α2 − c2 ) = 0 dy2 | y2 =y 2 p2 =

y2 =

α2 − c2 b2 − a2 c2 >0 = 2β2 2

for

c2
y2

⇔

b1 > b2

∧ a1 c1 > a2 c2

y1 = y2

⇔

b1 = b2

∧ a1 c1 = a2 c2

y1 < y2

⇔

b1 < b2

∧ a1 c1 < a2 c2 .

Ad b

Ad c

E5.6. Formulate and solve a problem of choice of the optimal supply and of the optimal price set by a monopolistic company considered in Example 5.3. Assume a nonlinear function of production total cost of a form: ctot (y) = cy 2 + d, c, d > 0. Analyse sensitivity of the optimal supply and of a product optimal price to changes in values of parameters describing the market: a market capacity, strength of consumers’ reaction to changes in a product price, production marginal cost. Solution Following functions are given: • A linear function of demand for a product: y d = −ap + b, a, b > 0, • A linear inverse function of demand for a product: ( ) b − yd 1 b = α − βy d , a, b > 0, α = > 0, β = > 0, p yd = a a a • A nonlinear function of total cost of production: ( ) ctot y s ( p) = cy s2 + d, c, d > 0 • r (y s ) = p(y s )y s —revenue (turnover) from sales of a product. We assume that the supply of a product is equal to the demand reported by consumers: y s = y d = y.

5 Rationality of Choices Made by a Group of Producers by Exogenously …

177

Then the profit maximization problem for a monopolist takes the form: ⊓(y) = r (y) − ctot (y) = p(y)y − ctot (y) = (α − βy)y − cy 2 − d { } = −(β + c)y 2 + αy − d → max y ≥ 0.

| d⊓(y) || = −2(β + c)y + α = 0 ⇔ dy | y=y ⇔y=

b α b a = 1+ac > 0, = 2(β + c) 2(1 + ac) 2 a

which means that an output level guaranteeing the maximum profit for a monopolist is positive. Then a level of a product price set by a monopolist is: p(y) =

b b − 2(1+ac) b(1 + 2ac) b−y = = > 0. a a 2a(1 + ac)

The monopolist’s maximum profit at the equilibrium price is: ⊓(y) = p(y)y − cy 2 − d =

b2 − d. 4a(1 + ac)

If in addition: b2 > d, 4a(1 + ac) then the monopolist’s maximum profit is positive. Let us notice that the product price and the product supply set by a monopolist, which by given demand for a product guarantee the maximum profit, depend on: a market capacity b, the strength a of consumers’ reaction to changes in a product price, the production marginal total (variable) cost c. The maximum profit of a monopolist depends also on the fixed cost of production d. Let us analyse sensitivity of the product optimal supply by a monopolist and of the product price to changes of values of parameters that describe the considered model of a monopolistic market. Values of adequate measures are presented in Table 5.3a, b. From Table 5.3a and b, it follows that a unit increase in market capacity b results in an increase in the optimal supply and in the product price. A unit increase in the marginal variable cost of production c results in a decrease in the product optimal supply and an increase in the product optimal price. A unit increase in the strength a of consumers’ reaction to changes in the price of a product set by a monopolist results in a decrease in the product price and in the product optimal supply.

178

5 Rationality of Choices Made by a Group of Producers by Exogenously …

Table 5.3 Measures of response to changes in parameters’ values for: (a) product supply by monopolist, (b) product price set by monopolist (a) Characteristic b y = 2(1+ac) Value

∂y ∂a −bc 2(1+ac)2

∂y ∂b

0

−ab 2(1+ac)2

0

E5.7. Formulate and solve a problem of choice of the optimal supply and of the optimal price set by a monopolistic company considered in Example 5.46 . Assume that the monopolist can supply her/his product to two different markets, thus regards discriminatory pricing. Assume a nonlinear function of production total cost: ctot (y) = cy 2 + d, c, d > 0. Analyse how the optimal supply and a product optimal price react to changes in values of parameters describing the markets: market capacities, consumers’ sensitivities to changes in a product price, production marginal cost. Solution Let us take following notation: y1 —an output level intended for the first market, y2 —an output level intended for the second market, y = ξ (y1 , y2 ) = y1 + y2 —total quantity of the product manufactured by the monopolist supplied to both markets, ctot (y) = cv (y) + c f (y)—production total cost, cv (y)—production variable cost, c f (y)—production fixed cost, p1 (y1 ) > 0—a product price set by the monopolist on the first market, p2 (y2 ) > 0—a product price set by the monopolist on the second market, y1d = y1d ( p1 )—the demand reported by consumers for a product on the first market, y2d = y2d ( p2 )—the demand reported by consumers for a product on the second market, y1s = y1s ( p1 )—the supply of the product intended for the first market, y2s = y2s ( p2 )—the supply of the product intended for the second market, r1 (y1 ) = p1 (y1 )y1 —revenue from sales of the product on the first market, r2 (y2 ) = p2 (y2 )y2 —revenue from sales of the product on the second market. 6

See a textbook Malaga K., Sobczak K, Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, 2022.

5 Rationality of Choices Made by a Group of Producers by Exogenously …

179

From now on it is assumed that: (1) ∀i = 1, 2 yid = yid ( pi ) = yis ( pi ) = yi , which means that the output level on each market matches the demand reported for the product on this market. We assume also that is not possible to resell the product between the markets. There are given: A production total cost function: (2) ctot (y) = cv (y) + c f (y) = cy 2 + d = c(y1 + y2 )2 + d, c, d > 0, according to which the production total cost depends on the total output, regardless it is intended for the first or for the second market. A function of demand reported for a product on the first market: (3) y1d ( p1 ) = −a1 p1 + b1 , such that: dy1d ( p1 ) d p1

(4)

= −a1 < 0,

and is decreasing in a price of a product price set on the first market. An inverse function of consumer demand reported for a product on the first market: ( ) yd (5) p1 y1d = ab11 − a11 = α1 − β1 y1d , a1 , b1 > 0 ⇒ α1 , β1 > 0. A function of revenue from sales of a product one the first market: (6) r1 (y1 ) = p1 (y1 )y1 = (α1 − β1 y1 )y1 = α1 y1 − β1 y12 . A function of demand reported for a product on the second market: (7) y2d ( p2 ) = −a2 p2 + b2 , such that: dy2d ( p2 ) d p2

(7)

= −a2 < 0.

and is decreasing in a price of a product price set on the second market. An inverse function of consumer demand reported for a product on the second market: ( ) yd (8) p2 y2d = ab22 − a22 = α2 − β2 y2d , a2 , b2 > 0 ⇒ α2 , β2 > 0. A function of revenue from sales of a product one the second market: (9) r2 (y2 ) = p2 (y2 )y2 = (α2 − β2 y2 )y2 = α2 y2 − β2 y22 . A monopolist’s profit function of a form: ⊓(y1 , y2 ) = r1 (y1 ) + r2 (y2 ) − ctot (y1 , y2 ) (10)

= α1 y1 − β1 y12 + α2 y2 − β2 y22 − c(y1 + y2 )2 − d y1 , y2 ≥ 0.

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

Formulate and solve a problem of choice of the optimal supply and of the optimal price set by a monopolistic company. The necessary condition for existence of an optimal solution to the monopolist’s profit maximization problem has a form of two equations: | 1 ,y2 ) | = α1 − (2β1 + c)y 1 = 0, (11) ∂⊓(y | ∂ y1 | y=y 1 ,y2 ) | (12) ∂⊓(y = α2 − (2β2 + c)y 2 = 0, | ∂ y2 y=y

which gives: • For the first market: α1 2β1 +c

(13) y 1 =

=

b1 2(1+a1 c)

> 0, where α1 =

b1 , β1 a1

=

1 , a1

• For the second market: α2 2β2 +c2

(14) y 2 =

=

b2 2(1+a2 c)

> 0, where α2 =

b2 , β2 a2

=

1 , a2

The sufficient condition for existence of an optimal solution to the monopolist’s profit maximization problem has a form of two inequalities: | 2 1 ,y2 ) | (15) ∂ ⊓(y = −(2β1 + c) < 0, | 2 ∂ y1 | y=y 2 1 ,y2 ) | (16) ∂ ⊓(y = −(2β2 + c) < 0, | ∂ y22 y=y | | | | 2 ∂ 2 ⊓(y1 ,y2 ) | ∂ 2 ⊓(y1 ,y2 ) | ∂ 2 ⊓(y1 ,y2 ) | 1 ,y2 ) | (17) ∂ ⊓(y · − · > 0. | | | ∂ y1 ∂ y2 ∂ y2 ∂ y1 | ∂ y2 ∂ y2 1

y=y

2

y=y

y=y

y=y

The Hessian of the profit function has a form: ] [ 0 −2β1 − c (18) H (y1 , y2 ) = . 0 −2β2 − c Since a condition det H (y1 , y2 ) = 4β1 β2 + 2c(β1 + β2 ) + c2 > 0 is satisfied, then for the output levels, the monopolist obtains the maximum profit. Let us notice that the optimal supply of a product intended for each of the markets depends on: i-th’s market capacity bi > 0 (the demand for a product by zero price), the strength ai > 0 of consumers’ reaction to changes in a product price on i-th market and on the production marginal total (variable) cost c > 0. The optimal total supply of a product supplied for both market is: (19) y = y 1 + y 2 =

b1 2(1+a1 c)

+

b2 2(1+a2 c)

=

b1 (1+a2 c)+b2 (1+a1 c) 2(1+a1 c)(1+a2 c)

> 0.

Let us notice that the optimal total supply of a product intended for both markets depends on: market capacities b1 , b2 > 0, the strength a1 , a2 > 0 of consumers’ reaction to changes in product prices on both markets, the production marginal total cost c > 0. From the assumption, it is known that the output level on i-th market (i = 1, 2), including the optimal level, matches the demand reported for the product on this

5 Rationality of Choices Made by a Group of Producers by Exogenously …

181

market. On the basis of conditions describing, the inverse demand functions one can determine the optimal levels of a product price on two markets that a monopolist wants to set to maximize her/his profit. The product price on the first market is: ( ) b1 y b1 (1 + 2a1 c) , p1 y 1 = − 1 = a1 a1 2a1 (1 + a1 c) and on the second market: ( ) b2 y b2 (1 + 2a2 c) . p2 y 2 = − 2 = a2 a2 2a2 (1 + a2 c) Let us notice that the optimal price of a product on i-th market (i = 1, 2) depends on: i-th’s market capacity bi > 0, the strength ai > 0 of consumers’ reaction to changes in a product price on i-th market and on the production marginal total (variable) cost c > 0. Analyse how the optimal supply and a product optimal price react to changes in values of parameters describing the markets: market capacities, consumers’ sensitivities to changes in a product price, production marginal cost. Table 5.4a–d present measures of sensitivity of the optimal supply and optimal prices on each market to changes of values of parameters. E5.8. Consider the Cournot duopoly model when a production total cost function for i-th producer (i = 1, 2) is nonlinear and of a form: citot (y) = ci yi2 + di , ci , di > 0. Determine optimal levels of: the product supply by each producer, the total supply by both producers, an equilibrium price. Analyse how these levels react to changes in values of parameters describing the market: a market capacity, strength of consumers’ reaction to changes in a product price, production marginal costs. Solution Let us apply a following set of assumptions: Two producers (i = 1, 2) act on a market of a homogeneous (undifferentiated) product. Functions of their production total cost are as follows: (1) ∀i = 1, 2 citot (yi ) = civ (yi ) + ci (yi ) = ci yi2 + di , ci , di > 0, f

being the sum of variable cost functions: (2) ∀i = 1, 2 civ (yi ) = ci yi2 , ci > 0. and the fixed costs: f

(3) ∀i = 1, 2 ci (yi ) = di > 0. Since the total cost functions are nonlinear functions of output levels, then: (4) ∀i = 1, 2

dcitot (yi ) dyi

=

dciv (yi ) dyi

= 2ci yi > 0,

Value

Characteristic ( ) 2 c) p2 y 2 = b2a2 (1+2a 2 (1+a2 c)

(d)

Value

Characteristic ( ) 1 c) p1 y 1 = b2a1 (1+2a 1 (1+a1 c)

(c)

Value

Characteristic b2 y 1 = 2(1+a 2 c)

(b)

Value

Characteristic b1 y 1 = 2(1+a 1 c)

(a)

a 1 b1 2(1+a1 c)2

0. The first producer wants to determine such an output level that guarantees the maximum profit for her/him taking an output level of the second producer as given: (8) ⊓1 (y1 )| y2 =const.≥0 → max y1 ≥ 0. The second producer wants to determine such an output level that guarantees the maximum profit for her/him taking an output level of the first producer as given: (9) ⊓2 (y2 )| y1 =const.≥0 → max y2 ≥ 0. A profit function of i-th producer (i = 1, 2) can be expressed as the difference between revenue from sales of a product and total cost of production: ] [ (10) ∀i = 1, 2 ⊓i (yi ) = p(y)yi − ci yi2 + di . Substituting an inverse function of demand p(y) = b−y = α−β(y1 + y2 ), where: a b 1 α = a , β = a , into Eq. (1), one obtains the profit functions of both producers as functions of their output levels: • For the first producer:

] [ (11) ⊓1 (y1 , y2 ) = [α − β(y1 + y2 )]y1 − c1 y12 + d1 = αy1 −(β +c1 )y12 −βy1 y2 − d1 , • For the second producer:

] [ (12) ⊓2 (y1 , y2 ) = [α − β(y1 + y2 )]y2 − c2 y22 + d2 = αy2 − (β + c2 )y22 − βy1 y2 − d2 . When the output level of the second producer is taken as given, thus treated as a parameter, the necessary condition and the sufficient condition for the profit maximization problem of the first producer are following7 : 7

The profit function of the first (second) producer is an one-variable function when the supply of a product by the second (first) producer is set. In conditions, we do use notions appropriate for the first- and second-order partial derivatives, but the necessary and sufficient conditions of the optimum existence refer actually to one-variable functions.

184

5 Rationality of Choices Made by a Group of Producers by Exogenously …

|

(13) (14)

∂⊓1 (y1 ,y2 ) | =0 | ∂ y1 |y1 =y 1 , y2 =const.≥0 2 | ∂ ⊓1 (y1 ,y2 ) 0. Determine the product optimal supply by: a leader, a follower, both producers and an equilibrium price. Analyse how these levels react to changes in values of parameters describing the market: a market capacity, strength of consumers’ reaction to changes in a product price, production marginal costs. Solution Let us apply a following set of assumptions: (S1)Two producers (i = 1, 2) act on a market of a homogeneous (undifferentiated) product. The first producer is a leader and the second is a follower. (S2)Functions of production total cost for producers are as follows: (1) ∀i = 1, 2 citot (yi ) = civ (yi ) + ci (yi ) = ci yi2 + di , ci , di > 0, f

being the sum of variable cost functions: (2) ∀i = 1, 2 civ (yi ) = ci yi2 , ci > 0, and the fixed costs: f

(3) ∀i = 1, 2 ci (yi ) = di > 0. Since the total cost functions are nonlinear functions of output levels, then: (4) ∀i = 1, 2

dcitot (yi ) dyi

=

dciv (yi ) dyi

= 2ci yi > 0,

that is: the marginal total cost and the marginal variable cost for the i-th producer are equal and increasing in an output level. (S3) A function of demand reported for a product by consumers, depending on its price set by producers, is as follows: (5) y d ( p) = −ap + b a, b > 0, where a denotes a measure of the consumers’ reaction strength to an unit increase in a price of a product and b denotes a measure of a market capacity. Since values of the demand function have to be non-negative then: ] [ (6) p ∈ 0; ab .

5 Rationality of Choices Made by a Group of Producers by Exogenously …

187

Table 5.5 (a) Optimal quantities of producers’ supply, total supply and prices in Walrasian equilibrium; Sensitivity measures of: (b) first producer’s supply, (c) second producer’s supply, (d) of both producers’ total supply, (e) Walrasian equilibrium price Characteristic

Value

(a) y (C) 1 (C)

y2

y (C) ) ( p y (C)

b(1+2ac2 ) 3+4a(c1 +c2 )+4a 2 c1 c2 b(1+2ac1 ) 3+4a(c1 +c2 )+4a 2 c1 c2 2b[1+a(c1 +c2 )] 3+4a(c1 +c2 )+4a 2 c1 c2 [ ] b 1+2a(c1 +c2 )+4a 2 c1 c2 2 a [3+4a(c1 +c2 )+4a c1 c2 ]

=

b a

2b[1+a(c1 +c2 )] − a [3+4a(c 2 1 +c2 )+4a c1 c2 ]

(b) (C)

∂ y1 ∂a

2b[c2 −2c1 −4ac1 c2 (1+ac2 )] [3+4a(c1 +c2 )+4a 2 c1 c2 ]2

0

2ab(1+2ac1 )

[3+4a(c1 +c2 )+4a 2 c1 c2 ]2

0

(c) (C)

∂ y2 ∂a

2b[c1 −2c2 −4ac1 c2 (1+ac1 )] [3+4a(c1 +c2 )+4a 2 c1 c2 ]2

0

2ab(1+2ac2 )

[3+4a(c1 +c2 )+4a 2 c1 c2 ]2 −

>0

8ab(1+2ac1 )(1+ac1 )

[3+4a(c1 +c2 )+4a 2 c1 c2 ]2

0

[3+4a(c1 +c2 )+4a 2 c1 c2 ]2

∂c1 ( ) ∂ p y (C)

[3+4a(c1 +c2 )+4a 2 c1 c2 ]2

∂c2

0. (S5) An inverse function of consumer demand for a product manufactured by producers has a form: (8) p(y1 , y2 ) =

b a

− a1 (y1 + y2 ) = α − β(y1 + y2 ), α, β > 0, α = ab , β = a1 .

Let us notice that: (9)

∂ p(y1 ,y2 ) ∂ y1

=

∂ p(y1 ,y2 ) ∂ y2

= −β = − a1 < 0,

thus, no matter which producer increases the output by one physical unit, it leads to a decrease in a product price by β = a1 money units. (S6)The first producer (the leader) wants to determine such an output level that guarantees the maximum profit for her/him: (10) ⊓1 (y1 ) → max y1 ≥ 0. A profit function of the first producer can be expressed as the difference between her/his revenue from sales of a product and total cost of production: (11) ⊓1 (y1 ) = p(y1 , y2 )y1 − c1 y12 − d1 . Substituting the inverse function of demand (8) into Eq. (11), one obtains the profit function of the first producer as: ] [ ⊓1 (y1 , y2 ) = [α − β(y1 + y2 )]y1 − c1 y12 + d1 (12) . = αy1 − (β + c1 )y12 − βy1 y2 − d1 (S7)The second producer (the follower) wants to determine such an output level that guarantees the maximum profit for her/him taking an output level of the first producer as given: (13) ⊓2 (y2 )| y1 =const.≥0 → max y2 ≥ 0.

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189

A profit function of the second producer can be expressed as the difference between her/his revenue from sales of a product and total cost of production: (14) ⊓2 (y2 ) = p(y1 , y2 )y2 − c2 y22 − d2 . Substituting the inverse function of demand (8) into Eq. (14), one obtains the profit function of the second producer as: ] [ (15) ⊓2 (y1 , y2 ) = [α − β(y1 + y2 )]y2 − c2 y22 + d2 = αy2 − (β + c2 )y22 − βy1 y2 − d2 . Let us now derive optimal solutions to the profit maximization problems of both producers in a duopoly in the Stackelberg model. The necessary condition and the sufficient condition for the profit maximization problem of the first producer are following: | | (16) ∂⊓1∂(yy11 ,y2 ) | = 0 − the necessary condition, y1 =y 1 | 2 | (17) ∂ ⊓1∂(yy 21 ,y2 ) | < 0 − the sufficient condition. 1

y1 =y 1

When the output level of the first producer is taken as given, thus treated as a parameter, the necessary condition and the sufficient condition for the profit maximization problem of the second producer are following: | | (18) ∂⊓2∂(yy21 ,y2 ) | = 0 − the necessary condition, y2 =y 2 , y1 =const.≥0 | 2 | < 0 − the sufficient condition. (19) ∂ ⊓2∂(yy 21 ,y2 ) | 2

y2 =y 2 , y1 =const.≥0

Deriving the necessary condition and the sufficient condition for the profit function of the second producer (the follower), we get: | | (20) ∂⊓2∂(yy21 ,y2 ) | = α − 2(β + c2 )y 2 − βy1 = 0, |y2 =y 2 , y1 =const.≥0 2 | (21) ∂ ⊓2∂(yy 21 ,y2 ) | = −2(β + c2 ) < 0, 2

y2 =y 2 , y1 =const.≥0

which means that for any (given) output level y1 ≥ 0 set by the first producer, the second producer obtains the maximum profit when y2 = y 2 . From condition (20), it results that: (22) y 2 =

α 2(β+c2 )

−

βy1 2(β+c2 )

RL 2 .

Equation (22) is called a reaction line of the follower. From perspective of the follower, it describes her/his output level which, by the output level of the leader taken as given, guarantees the maximum profit for the follower. From perspective of the leader, the equation describes a share of the market which, by a given output level of the leader, is to be captured by the follower. Substituting expression (22) into the leader’s profit function, one gets:

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

Fig. 5.8 Equilibrium state in the Stackelberg duopoly model

(

) α βy1 ⊓1 (y1 , y2 ) = αy1 − (β + − βy1 − − d1 2(β + c2 ) 2(β + c2 ) [ 2 ] 2 (23) . α(β + 2c2 )y1 − β + 2β(c1 + c2 ) + 2c1 c2 y1 = − d1 . 2(β + c2 ) c1 )y12

Then the necessary condition for the leader’s profit function takes a form: | α(β+2c2 )−2[β 2 +2β(c1 +c2 )+2c1 c2 ] y 1 | = = 0, (24) ∂⊓1∂(yy11 ,y2 ) | 2(β+c2 ) y1 =y 1

and the sufficient condition for the leader’s profit function is: | 2 2 | 1 +c2 )+2c1 c2 (25) ∂ ⊓1∂(yy 21 ,y2 ) | = − β +2β(cβ+c < 0. 2 1

y1 =y 1

From conditions (24) and (25), it results that for an output level:

(26)

α(β + 2c2 ) b(1 + 2ac2 ) ]= [ ] y (S) 1 = [ 2 2 β + 2β(c1 + c2 ) + 2c1 c2 2 1 + 2a(c1 + c2 ) + 2a 2 c1 c2 . abc2 b ] [ = + , 1 + 2a(c1 + c2 ) + 2a 2 c1 c2 2 1 + 2a(c1 + c2 ) + 2a 2 c1 c2

the leader obtains the maximum profit. Substituting expression (26) into Eq. (22): α β α(β + 2c2 ) ] − · [ 2 2(β + c2 ) 2(β + c2 ) 2 β + 2β(c1 + c2 ) + 2c1 c2 . αc1 abc1 = 2 = , β + 2β(c1 + c2 ) + 2c1 c2 1 + 2a(c1 + c2 ) + 2a 2 c1 c2

y (S) 2 = (27)

one derives the optimal output level of the follower. When we compare the optimal output levels of the leader and of the follower, assuming the same value of c1 = c2 = c, we can notice that the leader’s output level is bigger. The equilibrium state in the Stackelber duopoly model is presentedin Fig. 5.8. Let us notice that in the Stackelberg duopoly model, there is only the reaction line of

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191

the follower, who has to accept unconditionally the choice made by the leader. On the other hand the leader, when deciding her/his optimal output level, does not have to take into account the decisions made by the follower. The product supply by the leader is determined by her/his profit function which depends on revenue from sales and on production total cost. The equilibrium state in the Stackelberg duopoly model exists, there is exactly one such state and it is determined uniquely by the product supplies by the leader and by the follower. Deciding her/his output level, the follower relies on the choice made by the leader as well as on the maximum profit that can be obtained when the leader has made decision to manufacture y 1(S) units of a product. As a result, in the equilibrium state, the optimal supply of the product by the leader and by the follower in the Stackelberg duopoly model is given as: ) ( ) ( b(1+2ac2 ) abc1 , (28) y(S) = y 1(S) , y 2(S) = 2[1+2a(c . +c )+2a 2 c c ] 1+2a(c1 +c2 )+2a 2 c1 c2 1

2

1 2

Then the total supply of the product equals: b[1+2a(c1 +c2 )] ; (29) y (S) = y 1(S) + y 2(S) = 2[1+2a(c 2 1 +c2 )+2a c1 c2 ]

hence, the equilibrium price in the Stackelberg duopoly model is: ( ) ( ) p (S) y 1(S) , y 2(S) = α − β y 1(S) + y 2(S) 1 b[1 + 2a(c1 + c2 )] b ] − · [ a a 2 1 + 2a(c1 + c2 ) + 2a 2 c1 c2 ] [ , b 1 + 2a(c1 + c2 ) + 4a 2 c1 c2 ] = [ 2a 1 + 2a(c1 + c2 ) + 2a 2 c1 c2 abc1 c2 b ] +[ = 2a 1 + 2a(c1 + c2 ) + 2a 2 c1 c2 =

(30)

which means that the total supply of a product by both producers and the equilibrium price set by them depend on: the market capacity b > 0, the strength of consumers’ reaction a > 0 to changes in a product price and on the marginal (variable or total) costs of production c1 , c2 > 0. Let us analyse the sensitivity of the product optimal supply and of the equilibrium price to changes in values of the parameters of the Stackelberg duopoly model (Table 5.6). E5.10. Compare conclusions drawn from solutions to exercises E5.6, E5.8 and E5.9 assuming they relate to a market of one homogenous product in cases of the: pure monopoly, Cournot duopoly, the Stackelberg duopoly. Solution Let us compare the supply of a product by a monopolist with the total supply of the product by both producers in the Cournot duopoly and the total supply by the leader and the follower in the Stackelberg duopoly. Let us also compare optimal prices set by the producers. For this purpose, let us assume that in each case the same value of the parameter c is used in the function of the production total cost:

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

Table 5.6 (a) Optimal quantities of producers’ supply, total supply and prices in Walrasian equilibrium; Sensitivity measures to changes in parameters’ values for: (a) leader’s optimal supply, (c) follower’s optimal supply, (d) total optimal supply from both producers, (e) Walrasian equilibrium price Characteristic

Value

(a) y (S) 1 (S)

y2

y (S) ( ) p y (S)

b(1+2ac2 ) 2[1+2a(c1 +c2 )+2a 2 c1 c2 ] abc1 1+2a(c1 +c2 )+2a 2 c1 c2 b[1+2a(c1 +c2 )] 2[1+2a(c1 +c2 )+2a 2 c1 c2 ] [ ] b 1+2a(c1 +c2 )+4a 2 c1 c2 2a [1+2a(c1 +c2 )+2a 2 c1 c2 ]

=

b 2a

1 c2 + [1+2a(c abc 2 1 +c2 )+2a c1 c2 ]

(b) (S)

∂ y1 ∂a

−2abc1 c2 (1+ac2 )−bc1

0

(S) ∂ y1 ∂c1

2) − ab(1+2ac2 )(1+ac [1+2a(c1 +c2 )+2a 2 c1 c2 ]2

(S)

[1+2a(c1 +c2 )+2a 2 c1 c2 ]2

(S)

∂ y1 ∂c2

a 2 bc1

[1+2a(c1 +c2 )+2a 2 c1 c2 ]2

0

(c) (S)

∂ y2 ∂a

( ) bc1 1−2a 2 c1 c2 2

[1+2a(c1 +c2 )+2a 2 c1 c2 ]

1 (S) ∂ y2

∂b

(S) ∂ y2 ∂c1 (S)

∂ y2 ∂c2

ac1 1+2a(c1 +c2 )+2a 2 c1 c2

>0

ab(1+2ac2 ) [1+2a(c1 +c2 )+2a 2 c1 c2 ]2

−

>0

2a 2 bc1 (1+ac1 )

0, f

• The supply of each product matches the demand that consumers report for this product y1s ( p1 , p2 ) = y1d ( p1 , p2 ) = y1 = b1 − a1 p1 + γ1 p2 , y2s ( p1 , p2 ) = y2d ( p1 , p2 ) = y2 = b2 − a2 p2 + γ2 p1 . By this assumptions, we get following optimal supplies of substitute products offered by the producers in the Bertrand duopoly: y (B) 1

9

[ ] a1 2a2 (b1 − a1 c1 ) + γ1 (b2 + a2 c2 + γ2 c1 ) = 4a1 a2 − γ1 γ2

See a textbook Malaga K., Sobczak K, Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, 2022. 10 For the sake of simplicity, it is assumed that functions of production total costs are linear functions of output levels. One can take as well into account some nonlinear functions of output levels, getting different results.

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Table 5.9 Measures of response to changes in parameters’ values for: (a) first product’s optimal suppply, (b) second product’s optimal supply Characteristic

Value

(a) (B)

−a1 (2a1 a2 −γ1 γ2 ) 4a1 a2 −γ1 γ2

(B)

a 1 a 2 γ1 4a1 a2 −γ1 γ2

>0

(B)

2a1 a2 4a1 a2 −γ1 γ2

>0

(B)

a 1 γ1 4a1 a2 −γ1 γ2

>0

(B)

2a1 a2 [γ2 (b1 +a1 c1 )+2a1 (b2 +a2 c2 )] (4a1 a2 −γ1 γ2 )2

(B)

a1 γ1 [2a2 (b1 +a1 c1 )+γ1 (b2 +a2 c2 )] (4a1 a2 −γ1 γ2 )2

∂ y1 ∂c1 ∂ y1 ∂c2

∂ y1 ∂b1 ∂ y1 ∂b2

∂ y1 ∂γ1 ∂ y1 ∂γ2

(B)

0 >0

∂ y1 ∂a1

−

γ1 γ2 [2a2 (b1 −2a1 c1 )+γ1 (b2 +a2 c2 +γ2 c1 )]+8a12 a22 c1 (4a1 a2 −γ1 γ2 )2

(B) ∂ y1 ∂a2

−

a1 γ1 [γ2 (2b1 +2a1 c1 +γ1 c2 )+4a1 b2 ] (4a1 a2 −γ1 γ2 )2

0

(B)

a2 γ2 [2a1 (b2 +a2 c2 )+γ2 (b1 +a1 c1 )] (4a1 a2 −γ1 γ2 )2

(B)

2a1 a2 [γ1 (b2 +a2 c2 )+2a2 (b1 +a1 c1 )] (4a1 a2 −γ1 γ2 )2

(B)

−

a2 γ2 [γ1 (2b2 +2a2 c2 +γ2 c1 )+4a2 b1 ] (4a1 a2 −γ1 γ2 )2

−

γ1 γ2 [2a1 (b2 −2a2 c2 )+γ2 (b1 +a1 c1 +γ1 c2 )]+8a12 a22 c2 (4a1 a2 −γ1 γ2 )2

∂ y2 ∂c1 ∂ y2 ∂c2

∂ y2 ∂b1 ∂ y2 ∂b2

∂ y2 ∂γ1 ∂ y2 ∂γ2

∂ y2 ∂a1

(B)

∂ y2 ∂a2

y (B) 2

>0 0 >0 0 ∂b1 4a1 a2 − γ1 γ2 and means that when this capacity increases c. p. (ceteris paribus), then the optimal supply of the first product increases. Reaction of the first product’s optimal supply to a change in the capacity of the second product’s market is expressed as: ∂ y 1(B) a 1 γ1 = >0 ∂b2 4a1 a2 − γ1 γ2 and means that when this capacity increases c. p. (ceteris paribus), then the optimal supply of the first product increases. Reaction of the first product’s optimal supply to a change in the sensitivity of the first product’s consumers to changes in the price of the second product is expressed as:

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[ ] 2a1 a2 γ2 (b1 + a1 c1 ) + 2a1 (b2 + a2 c2 ) ∂ y (B) 1 = >0 ∂γ1 (4a1 a2 − γ1 γ2 )2 and means that when this sensitivity increases c. p. (ceteris paribus), then the optimal supply of the first product increases. Reaction of the first product’s optimal supply to a change in the sensitivity of the second product’s consumers to changes in the price of the first product is expressed as: [ ] a1 γ1 2a2 (b1 + a1 c1 ) + γ1 (b2 + a2 c2 ) ∂ y 1(B) = >0 ∂γ2 (4a1 a2 − γ1 γ2 )2 and means that when this sensitivity increases c. p. (ceteris paribus), then the optimal supply of the first product increases. Reaction of the first product’s optimal supply to a change in the sensitivity of the first product’s consumers to changes in the price of the first product is expressed as: [ ] γ1 γ2 2a2 (b1 − 2a1 c1 ) + γ1 (b2 + a2 c2 + γ2 c1 ) + 8a12 a22 c1 ∂ y (B) 1 =− 0, (c) Demand for the second producer’s (follower) product: y2d ( p1 , p2 ) = b2 − a2 p2 + γ2 p1 , a2 , b2 , γ2 > 0. The supply of each producer’s product is assumed to match the demand reported for this product: y1s ( p1 , p2 ) = y1d ( p1 , p2 ) = y1 = b2 − a1 p1 , y2s ( p1 , p2 ) = y2d ( p1 , p2 ) = y2 = b2 − a2 p2 + γ2 p1 . 1. Formulate the profit maximization problem for: • The first producer: ⊓1 ( p1 ) → max p1 ≥ 0 • The second producer: ⊓2 ( p1 , p2 )| p1 =const.≥0 → max p1 , p2 ≥ 0. 2. Solve the profit maximization problems for both producers determining optimal levels of prices of both products and maximum profits of both producers. 3. Analyse how optimal prices of both products react to changes in values of parameters describing the market: market capacities, consumers’ sensitivities to changes in prices of products, production marginal costs. 4. Draw reaction lines of both producers and illustrate an equilibrium state of this modified Bertrand duopoly model. 5. Determine the optimal supply of each product manufactured by a given producer. 6. State which of the strategies of setting an optimal price level of a product is more rational and more beneficial: the one of the leader or the one of the follower? Solutions Ad 1 Formulate the profit maximization problem for: • The first producer: ⊓1 ( p1 ) → max p1 ≥ 0 • The second producer: ⊓2 ( p1 , p2 )| p1 =const.≥0 → max p1 , p2 ≥ 0. The task is to determine the profit functions of the leader and the follower. The leader’s revenue, production total cost and profit are described by functions: r1 ( p1 ) = p1 y1 = p1 (b1 − a1 p1 ) = b1 p1 − a1 p12 c1tot (y1 ) = c1 y1 + d1 = c1 (b1 − a1 p1 ) + d1 = b1 c1 − a1 c1 p1 + d1 = c1tot ( p1 ) ⊓1 ( p1 ) = r1 ( p1 ) − ctot ( p1 ) = −(b1 c1 + d1 ) + (b1 + a1 c1 ) p1 − a1 p12 .

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The follower’s revenue, production total cost and profit are described by functions: r2 ( p1 , p2 ) = p2 y2 = p2 (b2 − a2 p2 + γ2 p1 ) = b2 p2 − a2 p22 + γ2 p1 p2 c2tot (y2 ) = c2 y2 + d2 = c2 (b2 − a2 p2 + γ2 p1 ) + d1 = c2tot ( p1 , p2 ) ⊓2 ( p1 , p2 ) = r2 ( p1 , p2 ) − c2tot ( p1 , p2 ) = −(b2 c2 + d2 ) + (b2 + a2 c2 ) p2 + γ2 p1 ( p2 − c2 ) − a2 p22 . Ad 2 Solve the profit maximization problems for both producers determining optimal levels of prices of both products and maximum profits of both producers. The necessary and the sufficient conditions for existence of a maximum of the profit function in case of the follower take the following form: | ∂⊓2 ( p1 , p2 ) || = b2 + a2 c2 + γ2 p1 − 2a2 p 2 = 0 | ∂ p2 p2 = p 2 , p1 =const.≥0 | ∂ 2 ⊓2 ( p1 , p2 ) || = −2a2 p 2 < 0, | ∂ p22 p2 = p 2 , p1 =const.≥0 which gives a line of the follower’s reaction to changes in the price of a product of the leader: p2 =

b2 + a2 c2 γ2 + p1 R L 2 . 2a2 2a2

The necessary and the sufficient conditions for existence of a maximum of the profit function in case of the leader take the following form: | d⊓1 ( p1 ) || = b1 + a1 c1 − 2a1 p 1 = 0 dp1 | p1 = p1 | d 2 ⊓1 ( p1 ) || = −2a1 < 0 dp12 | p1 = p1 which gives the optimal price of the leader’s product: p1 =

b1 + a1 c1 . 2a1

Substituting this price into the follower’s reaction line equation, we get the optimal price of the follower’s product: p2 =

b2 + a2 c2 γ2 b2 + a2 c2 γ2 b1 + a1 c1 + p = + · 2a2 2a2 1 2a2 2a2 2a1

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

=

2a1 (b2 + a2 c2 ) + γ2 (b1 + a1 c1 ) b2 + a2 c2 γ2 (b1 + a1 c1 ) = + 4a1 a2 2a2 4a1 a2

We can notice that by the assumption a1 = a2 , b1 = b2 , c1 = c2 , the optimal price of the follower’s product is bigger than the optimal price of the leader’s product. Having optimal prices of both products, we can determine maximum profit of the leader: ( ) max ⊓1 ( p1 ) = ⊓1 p 1 = −(b1 c1 + d1 ) + (b1 + a1 c1 ) p 1 − a1 p 21 ( ) b1 + a1 c1 2 b1 + a1 c1 = −(b1 c1 + d1 ) + (b1 + a1 c1 ) − a1 2a1 2a1 2 1 b = c1 (a1 c1 − 2b1 ) + 1 − d1 4 4a1 and of the follower ( ) max ⊓2 ( p1 , p2 ) = ⊓2 p¯ 1 , p¯ 2

( ) = −(b2 c2 + d2 ) + (b2 + a2 c2 ) p¯ 2 + γ2 p¯ 1 p¯ 2 − c2 − a2 p¯ 22 ( ) = −(b2 c2 + d2 ) + p¯ 2 b2 + a2 c2 + γ2 p¯ 1 − a2 p¯ 2 − c2 γ2 p¯ 1 2a1 (b2 + a2 c2 ) + γ2 (b1 + a1 c1 ) c2 γ2 (b1 + a1 c1 ) = −(b2 c2 + d2 ) + p¯ 2 · − 4a1 2a1 ]2 [ 2a1 (b2 + a2 c2 ) + γ2 (b1 + a1 c1 ) c2 γ2 (b1 + a1 c1 ) = −(b2 c2 + d2 ) + − 2a1 16a12 a2 (b2 + a2 c2 )2 (b2 + a2 c2 )2 + 4a2 4a2 γ2 (b1 + a1 c1 )(b2 + a2 c2 ) γ22 (b1 + a1 c1 ) c2 γ2 (b1 + a1 c1 ) + + − . 4a1 a2 2a1 16a12 a2

= −(b2 c2 + d2 ) +

Ad 3 Analyse how optimal prices of both products react to changes in values of parameters describing the market: market capacities, consumers’ sensitivities to changes in prices of products, production marginal costs. From point 2, we know the optimal prices’ forms: b1 + a1 c1 1 1 = a1−1 b1 + c1 2a1 2 2 b2 + a2 c2 γ2 (b1 + a1 c1 ) p2 = + 2a2 4a1 a2 1 −1 1 γ2 b1 γ2 c1 = a2 b2 + c2 + + . 2 2 4a1 a2 4a2 p1 =

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Let us recall the meaning of the parameters: ci Marginal total (variable) cost of production of i-th product (i = 1, 2), bi Capacity of i-th product’s market (i = 1, 2), γ2 Sensitivity of consumers of the second product (offered by the follower) to changes in a price of the first product (offered by the leader), ai Sensitivity of i-th (i = 1, 2) product’s consumers to changes in a price of i-th product (a2 > γ2 ). Reaction of i-th product’s optimal price to a change in the marginal total cost of production of this product is expressed as: ∂ pi 1 i = 1, 2 = ∂ci 2 and means that when this marginal cost increases c. p. (ceteris paribus) by one money unit, then the optimal price of i-th product increases by 0.5 of money unit. Reaction of i-th (i = 1, 2) product’s optimal price to a change in the marginal total cost of production of j-th ( j /= i) product is expressed as: ∂ p1 = 0, ∂c2

∂ p2 γ2 = >0 ∂c1 4a2

and means that when the marginal cost of production of the follower’s product increases c. p. (ceteris paribus), then the optimal price of the leader’s product does not change while the optimal price of the follower’s product increases when the marginal cost of production of the leader’s product increases c. p. This shows that the marginal cost of production for the follower does not affect the optimal price of the leader’s product. In contrast, the marginal cost of production for the leader affects the optimal price of the follower’s product Reaction of i-th product’s optimal price to a change in the capacity of the first product’s market is expressed as: ∂ pi 1 = a1−1 > 0 i = 1, 2 ∂bi 2 and means that when this capacity increases c. p. (ceteris paribus), then the optimal price of i-th product increases. Reaction of i-th (i = 1, 2) product’s optimal price to a change in the capacity of j-th ( j /= i) product’s market is expressed as: ∂ p1 = 0, ∂b2

∂ p2 γ2 = >0 ∂b1 4a1 a2

and means that when the capacity of the market for the follower’s product increases c. p. (ceteris paribus), then the optimal price of the leader’s product does not change

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

while the optimal price of the follower’s product increases when the capacity of the market for the leader’s product increases c. p. This shows that the capacity of the market of the follower’s product does not affect the optimal price of the leader’s product. In contrast, the capacity of the market of the leader’s product affects the optimal price of the follower’s product. Reaction of i-th product’s optimal price to a change in the sensitivity of i-th product’s consumers to changes in the price of i-th product is expressed as: ∂ p1 1 = − a1−2 b1 < 0, ∂a1 2

∂ p2 1 γ2 b1 γ2 c1 = − a2−2 b2 − − ac is an assumption that has to be satisfied. Otherwise, the optimal supply by the leader would be negative which has no economic sense. Now, assuming moreover that d1 = d2 = d, we get the maximum profits for producers as follows: b − ac b + ac b − ac · −c −d 2 2 ( ) ) ( 2a b − ac γ2 (b + ac) b + ac γ2 (b + ac) max ⊓2 = p¯ 2 y¯ 2 − c y¯ 2 − d2 = · + + 2a 4a 2 2 4a ) ( b + ac b − ac b − ac b − ac γ2 (b + ac) + −d = · −c −c 2 4a 2a 2 2 γ2 (b + ac) b + ac γ2 (b + ac) γ2 (b + ac) b − ac · + −c · −d + 2a 4a 4a 2 ) 2 4a ( γ2 b 2 − a 2 c 2 γ2 (b + ac)2 γ2 (b + ac) + −c = max ⊓1 + 2 8a 2 ( 8a ) 4a( ) γ2 b2 + 2abc + a 2 c2 + b2 − a 2 c2 γ2 bc + ac2 − = max ⊓1 + 2 4a ( 2 ( ) )8a γ2 2b + 2abc γ2 2abc + 2a 2 c2 −c = max ⊓1 + 2 8a 2 ( ( 28a ) ) γ2 2b − 2a 2 c2 2γ2 b2 − a 2 c2 = max ⊓1 + = max ⊓1 + . 8a 2 8a 2 max ⊓1 = p¯ 1 y¯ 1 − c y¯ 1 − d1 =

One can notice that the maximum profit for the follower is bigger whenever b2 > a 2 c2 , thus always by the required assumption that b > ac. This shows that the follower’s strategy of setting an optimal price level of her/his product is more rational and beneficial than the leader’s strategy.

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E5.13. Compare the original and the modified (E5.12) Bertrand duopoly models.11 State if the leader position in the modified Bertrand duopoly model is more beneficial for the first producer than a market position equal to the one of the second producer when these two producers compete on prices. Solution For the sake of simplicity, let us assume that a1 = a2 = a, b1 = b2 = b, c1 = c2 = c and γ1 = γ2 = γ . Then the optimal price and optimal supply of the leader’s product in both models are: 2a(b + ac) + γ (b + ac) (b + ac)(2a + γ ) = 2 2 4a − γ 4a 2 − γ 2 b + ac b + ac (b + ac)(2a + γ ) > = p1 = = 2a − γ 2a (2a − γ )(2a + γ ) ] || [ || || : 2a 2 || a 2a(b − ac) + γ (b + ac + γ c) || || = = || 4a 2 − γ 2 : 2a 2 ||

p (B) 1 =

y (B) 1

=

b − ac 2−

γ2 2a 2

+

1 γ 2a (b

+ ac + γ c)

2−

γ2 2a 2

>

b − ac = y1. 2

One can notice that the optimal price and the optimal supply of the leader’s product are bigger in the standard Bertrand duopoly model than in the modified model. This shows again that the leader’s strategy is less rational and less beneficial than the follower’s strategy. It is also less rational and less beneficial than the one he/she would have in case of the equivalent market position. The leader would benefit if he/she decided to take market position equivalent to her/his competitor. The leader position in the market conditions described in exercise E5.12 results in a situation in which the leader does not take into account the consumers of her/his competitor’s product and the fact their products are substitutes for each other. If we again assume (for the sake of simplicity) that a1 = a2 = a, b1 = b2 = b, c1 = c2 = c and γ1 = γ2 = γ , then the optimal price and optimal supply of the follower’s product in both models are: 2a(b + ac) + γ (b + ac) 2a(b + ac) + γ (b + ac) > = p2 4a 2 − γ 2 4a 2 ] || || [ a 2a(b − ac) + γ (b + ac + γ c) || : a || = = || || 4a 2 − γ 2 :a 2a(b − ac) + γ (b + ac + γ c) 2a(b − ac) + γ (b + ac) = = y2 > γ2 4a 4a −

p 2(B) = y 2(B)

a

11

The reference for the comparison can be the comparative analysis, presented in Sect. 5.4.3 (See a textbook Malaga K., Sobczak K, Microeconomics. Static and Dynamic Analysis, Springer Nature Switzerland, 2022.), of Cournot and Stackelberg duopoly models.

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One can notice that the optimal price and the optimal supply of the follower’s product are bigger in the standard Bertrand duopoly model than in the modified model. This shows that if the leader decided to take equivalent position to her/his competitor, then not only he/she would benefit but also the follower. E5.14. The demand for a product of a monopolistic company evolves according to a linear function of a form: y d (t) = −a(t) p(t) + b(t), a(t), b(t) > 0, ∀t = 0, 1, . . . , 20. A function of production total cost is given as: ( ) ctot y s (t) = c(t)y s (t) + d(t), c(t), d(t) > 0, ∀t = 0, 1, . . . , 20. 1. Solve the profit maximization problem determining the optimal supply and an optimal price level by the following assumptions: 1 (a) a(t) = − t+1 + 2, b(t) = 10, c(t) = 2, (b) a(t) = 1, b(t) = −0.025t 2 + 0.75t + 10, c(t) = 2, 1 (c) a(t) = 1, b(t) = 10, c(t) = t+1 + 1.

2. Present trajectories of the optimal supply and of the optimal price. 3. Analyse how the optimal supply and the optimal price react to changes (ceteris paribus) in values of parameters describing the market: market capacity, consumers’ sensitivity to changes in a product price, production marginal costs. Solutions Ad 1 Solve the profit maximization problem determining the optimal supply and an optimal price level. The profit function takes a form: π (y(t)) = r (y(t)) − ctot (y(t)) = p(y(t)) · y(t) − ctot (y(t)). We need first to derive an inverted function p(y(t)) of the consumer demand: y(t) = −a(t) p(t) + b(t) a(t) p(t) = −y(t) + b(t) b(t) 1 y(t) + . p(t) = − a(t) a(t) Substituting this function into the profit function and using the form of the production total cost function we get:

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207

) ( b(t) 1 y(t) − (c(t)y(t) + d(t)) y(t) + π (y(t)) = − a(t) a(t) ) ( 1 b(t) =− y(t)2 + − c(t) y(t) − d(t). a(t) a(t) The necessary condition for existence of a maximum of this function is following: 2 b(t) dπ (y(t)) =− y(t) + − c(t) = 0 dy(t) a(t) a(t) and gives the optimal supply: y(t) =

b(t) − a(t)c(t) . 2

The sufficient condition for existence of a maximum of the profit function is following: d2 π (y(t)) 2 0 ∂b(t) 2a(t)

and mean that when the capacity increases c. p. (ceteris paribus) by one physical unit, then the optimal supply increases by 0.5 of the physical unit and the optimal price increases. Reactions of the optimal supply and of the optimal price to a change in the sensitivity of consumers to changes in a price of the product are expressed as: 12

A MATLAB file to obtain these trajectories is called Exercise_5_14.m.

5 Rationality of Choices Made by a Group of Producers by Exogenously … 10

p(t)-(a)

9

p(t)-(b)

209

p(t)-(c)

8 7 6 5 4 3 2

0

5

10

15

20

Fig. 5.11 Optimal price (E5.14)

1 ∂ y(t) = − c(t) < 0, ∂a(t) 2

b(t) ∂ p(t) =− 0, i = 1, 2, ∀t ∈ [0; 20]. A function of production total cost is given as: ( ) ctot y s (t) = c(t)y s (t) + d(t), c(t), d(t) > 0, t ∈ [0; 20], where c(t) =

1 + 1. t +1

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

1. Solve the profit maximization problem determining the optimal supply and optimal levels of prices by the following assumptions: (a) a1 (t) = 4 · 0.98t , a2 (t) = −0.006t 2 + 0.1t + 4, b1 (t) = b2 (t) = 15, (b) a1 (t) = a2 (t) = 4, b1 (t) = 0.025t 2 − 0.5t + 15, b2 (t) = −0.025t 2 + 0.5t + 15, a1 (t) = 4 · 0.98t , a2 (t) = −0.006t 2 + 0.1t + 4, (c) b1 (t) = 0.025t 2 − 0.5t + 15, b2 (t) = −0.025t 2 + 0.5t + 15. 2. Present trajectories of the product optimal supplies intended for each of the markets and trajectories of the optimal prices of the product on each of the market. 3. Analyse how the optimal supplies and the optimal prices of the product supplied to both markets react to changes (ceteris paribus) in values of parameters describing the markets: market capacities, consumers’ sensitivities to changes in a product price, production marginal cost. Analyse what the importance of differences in these values between two markets is. Solutions Ad 1 Solve the profit maximization problem determining the optimal supply and optimal levels of prices. The profit function takes a form: π (y1 (t), y2 (t)) = r1 (y1 (t)) + r2 (y2 (t)) − ctot (y1 (t), y2 (t)) = p1 (y1 (t)) · y1 (t) − p2 (y2 (t)) · y2 (t) − (c(t)y(t) + d(t)). We need first to derive inverted functions pi (yi (t)) (i = 1, 2) of the consumer demand: yi (t) = −ai (t) pi (t) + bi (t) ai (t) pi (t) = −yi (t) + bi (t) 1 bi (t) yi (t) + . pi (t) = − ai (t) ai (t) The total supply of the monopoly consists of supplies of the product intended for two markets: y(t) = y1 (t) + y2 (t). Substituting the inverted functions of the consumer demand into the profit function and using the form of the production total cost function we get:

5 Rationality of Choices Made by a Group of Producers by Exogenously …

211

) ) ( ( 1 b1 (t) 1 b2 (t) y1 (t) + − y2 (t) y1 (t) + y2 (t) + π (y1 (t), y2 (t)) = − a1 (t) a1 (t) a2 (t) a2 (t) ) ( b1 (t) 1 − c(t)(y1 (t) + y2 (t)) − d(t) = − (y1 (t))2 + − c(t) y1 (t) a1 (t) a1 (t) ) ( 1 b2 (t) − (y2 (t))2 + − c(t) y2 (t) − d(t). a2 (t) a2 (t) The necessary condition for existence of a maximum of this function takes a form of two equations: 2 ∂π (y1 (t), y2 (t)) bi (t) =− yi (t) + − c(t) = 0 i = 1, 2 ∂ yi (t) ai (t) ai (t) and gives the optimal supply intended for each market: y i (t) =

bi (t) − ai (t)c(t) i = 1, 2. 2

The sufficient condition for existence of a maximum of the profit function takes a form of two inequalities: ∂ 2 π (y1 (t), y2 (t)) 2 < 0, i = 1, 2 =− ai (t) ∂ yi (t)2 and is satisfied. The optimal total supply of the monopoly consists of the optimal supplies of the product intended for two markets: y(t) = y 1 (t) + y 2 (t) =

b1 (t) + b1 (t) − c(t)(a1 (t) + a2 (t)) . 2

The optimal price set by the monopoly on each market takes a form: bi (t) 1 y¯ (t) + ai (t) a (t) ) ( i 1 bi (t) bi (t) + ai (t)ci (t) = + c(t) i = 1, 2. = 2ai (t) 2 ai (t)

p¯ i (t) = pi (yi (t)) = −

Ad 2 Present trajectories13 of the product optimal supplies intended for each of the markets and trajectories of the optimal prices of the product on each of the market (Figs. 5.12, 5.13 and 5.14). 13

A MATLAB file to obtain these trajectories is called Exercise_5_15.m.

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Fig. 5.12 Optimal supplies intended for the first and second markets (E5.15)

Fig. 5.13 Optimal total supply (E5.15)

Ad 3 Analyse how the optimal supplies and the optimal prices of the product supplied to both markets react to changes (ceteris paribus) in values of parameters describing the markets: market capacities, consumers’ sensitivities to changes in a product price, production marginal cost. Analyse what the importance of differences in these values between two markets is. Reactions of the optimal supply intended for i-th market (i = 1, 2), of the optimal total supply and of the optimal price on i-th market to a change in i-th market capacity are expressed as:

5 Rationality of Choices Made by a Group of Producers by Exogenously …

213

Fig. 5.14 Optimal prices on the first and second markets (E5.15)

∂ y i (t) 1 = ∂bi (t) 2

∂ y(t) 1 = , ∂bi (t) 2

∂ pi (t) 1 = > 0 i = 1, 2 ∂b(t) 2ai (t)

and mean that when the capacity of i-the market increases c. p. (ceteris paribus) by one physical unit, then the optimal supply intended for i-th market increases by 0.5 of the physical unit, the optimal total supply increases the same and the optimal price on i-th market increases. Reactions of the optimal supply intended for i-th market (i = 1, 2), of the optimal total supply and of the optimal price on i-th market to a change in the sensitivity of ith market consumers to changes in a price of the product on i-th market are expressed as: ∂ y i (t) 1 = − c(t) < 0, ∂ai (t) 2

1 ∂ y(t) = − c(t) < 0, ∂ai (t) 2

bi (t) ∂ pi (t) =− < 0, ∂b(t) 2(ai (t))2

i = 1, 2

and mean that when the sensitivity of consumers on i-the market increases c. p. (ceteris paribus), then the optimal supply intended for i-th market decreases, the optimal total supply decreases the same and the optimal price on i-th market decreases as well. Reactions of the optimal supply intended for i-th market (i = 1, 2), of the optimal total supply and of the optimal price on i-th market to a change in the marginal cost of production are expressed as: ∂ y i (t) 1 = − ai (t) < 0, ∂c(t) 2

1 ∂ y(t) = − (a1 (t) + a2 (t)) < 0, ∂c(t) 2

1 ∂ pi (t) = , ∂c(t) 2

i = 1, 2

and mean that when the marginal cost of production increases c. p. (ceteris paribus) by one money unit, then the optimal supply intended for i-th market decreases, the

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optimal total supply decreases while the optimal price on each market increases by 0.5 of the money unit. Let us recall the form of the optimal supply intended for i-th market and of the optimal price on i-th market: bi (t) − ai (t)c(t) i = 1, 2 2 ) ( 1 bi (t) + c(t) i = 1, 2. pi (t) = 2 ai (t) y i (t) =

From the forms of these values, it results that the bigger is the capacity of ith market, the higher price can be set by the monopoly and the bigger supply is intended for this market. It results also that the bigger is the sensitivity of i-th market consumers to changes in a price of the product on i-th market, the lower price has to be set by the monopoly and the smaller supply is intended for this market. Finally, it results that the bigger is the production marginal cost, the higher price is set by the monopoly on each market and the smaller supply is intended for each market. E5.16. Two producers having equal positions act on a market of some homogeneous product. The demand for this product evolves according to a demand function: y d ( p) = −ap + b, a, b > 0. Production total costs for the first and for the second firm, respectively, are as follows: c1tot (y1 ) = c1 y1 + d1 , c1 , d1 > 0, c2tot (y2 ) = c2 y2 + d2 , c2 , d2 > 0. The total output by both producers matches the demand for the product reported by consumers by a given price: y1 + y2 = y d ( p). 1. Determine an equilibrium state in the Cournot duopoly model by the following assumptions: (a) a = 2, b = 20, c1 = 1, c2 = 1, (b) a = 2, b = 20, c1 = 2, c2 = 1, (c) a = 2, b = 200, c1 = 2, c2 = 1. 2. Present a mechanism of reaching the equilibrium state when: (a) the first producer decides about the level of supply as first, assuming the competitor’s supply equals 0,

5 Rationality of Choices Made by a Group of Producers by Exogenously …

215

(b) the second producer decides about the level of supply as first, assuming the competitor’s supply equals 0. State what the number of iterations in determining the levels of supply by each producer needed to reach the equilibrium state is. 3. Present trajectories of the optimal supplies by both producers. Solutions Ad 1 Determine an equilibrium state in the Cournot duopoly model. From the description of the Cournot duopoly model,14 one knows a form of reaction lines of producers: y2 b − ac1 − 2 2 y1 b − ac2 − y2 = 2 2 y1 =

R L1 R L 2,

a form of optimal supplies by the producers: ) ( b − a(2c − c ) b − a(2c − c ) ) ( 1 2 2 1 (C) , = y(C) = y (C) , y 1 2 3 3 and a form of optimal price of their product: p (C) =

(a)

(b)

14

b + a(c1 + c2 ) 3a

a = 2, b = 20, c1 = 1, c2 = 1, 20 − 2(2 · 1 − 1) (C) =6 y (C) 1 = y2 = 3 20 + 2(1 + 1) (C) (C) =4 y (C) = y (C) = 1 + y 2 = 12 p 3·2 a = 2, b = 20, c1 = 2, c2 = 1, 20 − 2(2 · 2 − 1) 20 − 2(2 · 1 − 2) ≈ 4.67 y (C) ≈ 6.67 y (C) 1 = 2 = 3 3 20 + 2(2 + 1) (C) (C) ≈ 4.33 y (C) = y (C) = 1 + y 2 ≈ 11.33 p 3·2

Having the standard assumptions that the demand function and the functions of production total cost are linear.

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

Fig. 5.15 Mechanism of reaching equilibrium state—first producer decides as first (E5.16)

(c)

a = 2, b = 200, c1 = 2, c2 = 1 200 − 2(2 · 2 − 1) 200 − 2(2 · 1 − 2) y (C) ≈ 64.67 y (C) ≈ 66.67 1 = 2 = 3 3 200 + 2(2 + 1) (C) (C) y (C) = y (C) = ≈ 34.33 1 + y 2 ≈ 131.33 p 3·2

Ad 2 Present a mechanism of reaching the equilibrium state. State what the number of iterations in determining the levels of supply by each producer needed to reach the equilibrium state is. (A) The first producer decides about the level of supply as first, assuming the competitor’s supply equals 0 (Fig. 5.15). (a) The equilibrium supply is obtained by the first producer in 17-th iteration,15 by the second producer in 18-th iteration. Thus, the equilibrium state is reached in 18-th iteration. (b) The equilibrium supply is obtained by the first producer in 17-th iteration, by the second producer in 18-th iteration. Thus, the equilibrium state is reached in 18-th iteration. (c) The equilibrium supply is obtained by the first producer in 21-st iteration, by the second producer in 20-th iteration. Thus, the equilibrium state is reached in 21-st iteration. (B) The second producer decides about the level of supply as first, assuming the competitor’s supply equals 0 (Fig. 5.16).

15

In a MATLAB file called Exercise_5_16.m one can find commands to obtain the numbers of iteration in which the equilibrium supplies are reached: t1_A = find(abs(y1-yC1) < 0.00009)and t2_A = find(abs(y2-yC2) < 0.00009).

5 Rationality of Choices Made by a Group of Producers by Exogenously …

217

Fig. 5.16 Mechanism of reaching equilibrium state—second producer decides as first (E5.16)

(a) The equilibrium supply is obtained by the first producer in 18-th iteration,16 by the second producer in 17-th iteration. Thus, the equilibrium state is reached in 18-th iteration. (b) The equilibrium supply is obtained by the first producer in 16-th iteration, by the second producer in 17-th iteration. Thus, the equilibrium state is reached in 17-th iteration. (c) The equilibrium supply is obtained by the first producer in 20-th iteration, by the second producer in 21-th iteration. Thus, the equilibrium state is reached in 21-st iteration. Ad 3 Present trajectories of the optimal supplies by both producers (Figs. 5.17, 5.18 and 5.19). E5.17. Two producers act on a market of some homogeneous product. The first of them has a position of the leader and the other a position of the follower. The demand for this product evolves according to a demand function: y d ( p(t)) = −2 p(t) + 20,

a(t), b(t) > 0.

The total output by both producers matches the demand for the product reported by consumers by a given price: y1 (t) + y2 (t) = y d ( p(t)). Production total costs for the leader and for the follower, respectively, are as follows: c1tot (y1 (t)) = c1 (t)y1 + 1, 16

In a MATLAB file called Exercise_5_16.m one can find commands to obtain the numbers of iteration in which the equilibrium supplies are reached: T1_A = find(abs(Y1-yC1) < 0.00009)and T2_A = find(abs(Y2-yC2) < 0.00009).

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Fig. 5.17 Trajectories of optimal supplies by both producers—(a) (E5.16)

Fig. 5.18 Trajectories of optimal supplies by both producers—(b) (E5.16)

c2tot (y2 (t)) = 2.5y2 (t) + 1, where

5 Rationality of Choices Made by a Group of Producers by Exogenously …

219

Fig. 5.19 Trajectories of optimal supplies by both producers—(c) (E5.16)

c1 (t) = −

1 5 t + 6 , ∀t ∈ [0; 20]. 16 4

1. Determine an equation describing a line of reaction of the follower and an equilibrium state in the Stackelberg duopoly model when the production marginal cost for the leader varies in time. Present a graph of the follower’s reaction line and the equilibrium states depending on changes in the marginal cost for the leader. 2. At which moment are marginal costs for the leader and for the follower equal? What are then their shares in the market of a product? 3. Present trajectories of the optimal supplies by the leader and by the follower. Solutions Ad 1 Determine an equation describing a line of reaction of the follower and an equilibrium state in the Stackelberg duopoly model when the production marginal cost for the leader varies in time. Present a graph of the follower’s reaction line and the equilibrium states depending on changes in the marginal cost for the leader. The profit function of the follower takes a form: π2 (y2 (t)) = r2 (y2 (t)) − c2tot (y2 (t)) = p(t) · y2 (t) − c2tot (y2 (t)). We need first to derive an inverted functions p(y(t)) of the consumer demand:

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

y(t) = −2 p(t) + 20 2 p(t) = −y(t) + 20 1 p(t) = − y(t) + 10. 2 The total supply on the market consists of supplies by both producers: y(t) = y1 (t) + y2 (t). Substituting the inverted function of the consumer demand into the follower’s profit function and using the form of the production total cost function for the follower, we get: ) ( 1 π2 (y2 (t)) = − y(t) + 10 · y2 (t) − (2.5y2 (t) + 1) 2 [ ] 1 = − (y1 (t) + y2 (t)) + 10 · y2 (t) − (2.5y2 (t) + 1) 2 1 1 = (10 − 2.5)y2 (t) − (y2 (t))2 − y1 (t)y2 (t) − 1 2 2 1 1 2 = 7.5y2 (t) − (y2 (t)) − y1 (t)y2 (t) − 1 = π2 (y1 (t), y2 (t)). 2 2 Since among variables y1 (t), y2 (t), the second producer can decide only about y2 (t) the necessary condition for existence of a maximum of this function takes the form of one equation: dπ2 (y1 (t), y2 (t)) 1 = 7.5 − y2 (t) − y1 (t) = 0 dy2 (t) 2 and gives the reaction line of the follower (Fig. 5.20): y 2 (t) = 7.5 −

1 y1 (t). 2

The sufficient condition for existence of a maximum of the follower’s profit function takes a form: d2 π2 (y1 (t), y2 (t)) = −1 < 0 dy2 (t)2 and is satisfied. The profit function of the leader takes a form: π1 (y1 (t)) = r1 (y1 (t)) − c1tot (y2 (t)) = p(t) · y1 (t) − c1tot (y1 (t)).

5 Rationality of Choices Made by a Group of Producers by Exogenously …

221

Fig. 5.20 Reaction line of follower and equilibrium states depending on production marginal costs (E5.17)

Substituting the inverted function of the consumer demand and the supply by the follower found above into the leader’s profit function and using the form of the production total cost function for the leader, we get: [ ] 1 π1 (y1 (t)) = − (y1 (t) + y2 (t)) + 10 · y1 (t) − (c1 (t)y1 (t) + 1) 2 1 1 = (10 − c1 (t))y1 (t) − (y1 (t))2 − y1 (t)y2 (t) − 1 2 2 ( ) 1 1 1 2 = (10 − c1 (t))y1 (t) − (y1 (t)) − y1 (t) 7.5 − y1 (t) − 1 2 2 2 1 = (6.25 − c1 (t))y1 (t) − (y1 (t))2 − 1 4 The necessary condition for existence of a maximum of this function takes a form: dπ1 (y1 (t)) 1 = 6.25 − c1 (t) − y1 (t) = 0 dy1 (t) 2 and gives the optimal supply by the leader: y 1(S) (t) = 12.5 − 2c1 (t). The sufficient condition for existence of a maximum of the leader’s profit function takes a form: d2 π1 (y1 (t)) = −2c1 (t) < 0 dy1 (t)2 and is satisfied whenever the marginal cost of production c1 (t) is positive.

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When the optimal supply by the leader is stated, one can determine the optimal supply by the follower: y (S) 2 (t) = 7.5 −

1 (S) 1 y (t) = 7.5 − (12.5 − 2c1 (t)) = 1.25 + c1 (t). 2 1 2

The optimal total supply on the market consists of the optimal supplies by both producers: y (S) (t) = y 1(S) (t) + y 2(S) (t) = 12.5 − 2c1 (t) + 1.25 + c1 (t) = 13.75 − c1 (t). The optimal price set on the market by both producers takes a form: ( ) 1 1 p (S) (t) = p (S) y (S) (t) = − y (S) (t) + 10 = − (13.75 − c1 (t)) + 10 = 3.125 + c1 (t). 2 2

Ad 2 At which moment are marginal costs for the leader and for the follower equal? What are then their shares in the market of a product? The production marginal costs have the same value for both producers at moment (S) t( = 12: c1 (12) = ) c2 (12) = 2.5. Then the equilibrium state is y (12) = (S) (S) y 1 (12), y 1 (12) = (7.5, 3.75). Their shares in the market of the product are: 7.5 ≈ 0.67 7.5 + 3.75 3.55 (S) ≈ 0.33 U 1 (12) = 7.5 + 3.75 (S)

U 1 (12) =

for the leader for the follower.

Ad 3 Present trajectories of the optimal supplies by the leader and by the follower. In Fig. 5.21, we can notice that at moment t = 8, both producers have the same optimal supplies. At this moment, the production marginal cost for the leader is: c1 (8) = −

5 · 8 + 6.25 = 3.75. 16

Then the optimal supplies are obtained as: y (S) 1 (8) = 12.5 − 2 · 3.75 = 12.5 − 7.5 = 5 y 1(S) (8) = 1.25 + 3.75 = 5.

5 Rationality of Choices Made by a Group of Producers by Exogenously …

223

Fig. 5.21 Optimal supplies by leader and by follower (E5.17)

E5.18. Two producers having equal positions on a market offer two substitute products. The demand for these products evolves according to the following demand functions: y1d ( p1 , p2 ) = −a1 p1 + γ1 p2 + b1 , y2d ( p1 , p2 ) = −a2 p2 + γ2 p1 + b2 , ai , γi , bi > 0, ai > γi i = 1, 2. Production total costs for the first and for the second firm, respectively, are as follows: c1tot (y1 ) = c1 y1 + d1 , c1 , d1 > 0, c2tot (y2 ) = c2 y2 + d2 , c2 , d2 > 0. An output level of each product matches the demand reported by consumers for this product by its given price: yi = yid ( p1 , p2 ), i = 1, 2. 1. Determine an equilibrium state in the Bertrand duopoly model by the following assumptions for i = 1, 2: (a) b) (c) (d) (e)

ai ai ai ai ai

= 2, = 3, = 2, = 2, = 2,

bi bi bi bi bi

= 20, = 20, = 21, = 20, = 20,

γi γi γi γi γi

= 1, ci = 1, = 1, ci = 1, = 1, ci = 1, = 1.9, ci = 1, = 1, ci = 2,

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5 Rationality of Choices Made by a Group of Producers by Exogenously …

(f) ai = 3, bi = 21, γi = 1.9, ci = 2. 2. Present a mechanism of reaching the equilibrium state when: (a) the first producer decides about the level of a price as first, assuming a price of the competitor’s product equals 0, (b) the second producer decides about the level of a price as first, assuming a price of the competitor’s product equals 0. State what the number of iterations in determining the levels of a price by each producer needed to reach the equilibrium state is. 3. Present trajectories of the optimal prices of both producers’ products. Solutions Ad 1 Determine an equilibrium state in the Bertrand duopoly model. From the description of the Bertrand duopoly model,17 one knows a form of reaction lines of producers: b1 + a1 c1 γ1 + p2 2a1 2a1 b2 + a2 c2 γ1 p2 = + p2 2a2 2a2

p1 =

R L1 R L 2,

a form of optimal supplies by the producers: 2a2 (b1 + a1 c1 ) + γ1 (b2 + a2 c2 ) 4a1 a2 − γ1 γ2 2a1 (b2 + a2 c2 ) + γ2 (b1 + a1 c1 ) = 4a1 a2 − γ1 γ2

p 1(B) = p 2(B)

and a form of optimal price of their product: [ ] a1 2a2 (b1 − a1 c1 ) + γ1 (b2 + a2 c2 + γ2 c1 ) 4a1 a2 − γ1 γ2 [ ] a2 2a1 (b2 − a2 c2 ) + γ2 (b1 + a1 c1 + γ1 c2 ) = . 4a1 a2 − γ1 γ2

y (B) 1 = y (B) 2

(a) ai = 2, bi = 20, γi = 1, ci = 1, p 1(B) = p (B) 2 ≈ 7.33 17

y 1(B) = y 1(B) ≈ 12.67

Having the standard assumptions that the demand function and the functions of production total cost are linear.

5 Rationality of Choices Made by a Group of Producers by Exogenously …

225

(b) ai = 3, bi = 20, γi = 1, ci = 1, p 1(B) = p 2(B) = 4.6

y 1(B) = y 1(B) = 10.8

(c) ai = 2, bi = 21, γi = 1, ci = 1, p 1(B) = p (B) 2 ≈ 7.67

y 1(B) = y 1(B) ≈ 13.33

(d) ai = 2, bi = 20, γi = 1.9, ci = 1, p 1(B) = p 2(B) ≈ 10.48

(B) y (B) 1 = y 1 ≈ 18.95

(e) ai = 2, bi = 20, γi = 1, ci = 2, (B) p (B) 1 = p2 = 8

y 1(B) = y 1(B) = 12

(f) ai = 3, bi = 21, γi = 1.9, ci = 2 (B) p (B) 1 = p 2 ≈ 6.59

(B) y (B) 1 = y 1 ≈ 13.75.

Ad 2 Present a mechanism of reaching the equilibrium state.18 State what the number of iterations in determining the levels of a price by each producer needed to reach the equilibrium state is. (A) the first producer decides about the level of a price as first, assuming a price of the competitor’s product equals 0 (Fig. 5.22; Table 5.10). (B) the second producer decides about the level of a price as first, assuming a price of the competitor’s product equals 0(Fig. 5.23; Table 5.11).

18

We present the graphic results only for point (a). For the remaining points (b)–(f), the graphic results are similar.

226

5 Rationality of Choices Made by a Group of Producers by Exogenously …

Table 5.10 Number of iterations needed to reach the equilibrium supply—first producer decides as first19 Number of iterations needed to reach the equilibrium supply by the first producer

by the second producer

by both producers

(a)

9

10

10

(b)

7

8

8

(c)

9

10

10

(d)

17

16

17

(e)

9

10

10

(f)

11

10

11

Ad 3 Present trajectories of the optimal prices of both producers’ products.21 (Fig. 5.24)

Table 5.11 Number of iterations needed to reach the equilibrium supply—second producer decides as first20 Number of iterations needed to reach the equilibrium supply By the first producer

By the second producer

By both producers

(a)

10

9

10

(b)

8

7

8

(c)

10

9

10

(d)

16

17

17

(e)

10

9

10

(f)

10

11

11

19

In a MATLAB file called Exercise_5_18.m one can find commands to obtain the numbers of iteration in which the equilibrium supplies are reached. t1_A = find(abs(p1-pB1) < 0.00009) and t2_A = find(abs(p2-pCB2) < 0.00009). 20 In a MATLAB file called Exercise_5_18.m one can find commands to obtain the numbers of iteration in which the equilibrium supplies are reached. T1_A = find(abs(P1-pB1) < 0.00009) and T2_A = find(abs(P2-pCB2) < 0.00009). 21 We present the trajectories only for point (a). For the remaining points (b)–(f) trajectories are similar.

5 Rationality of Choices Made by a Group of Producers by Exogenously …

227

Fig. 5.22 Mechanism of reaching equilibrium state—first producer decides as first (E5.18)

Fig. 5.23 Mechanism of reaching equilibrium state—second producer decides as first (E5.18)

228

5 Rationality of Choices Made by a Group of Producers by Exogenously …

a

Fig. 5.24 Trajectories of optimal prices (E5.18)

a

Chapter 6

Rationality of Choices Made by Groups of Producers and Groups of Consumers

This Chapter presents 7 exercises that help in understanding the problem of describing behaviour of the community of consumers and producers. They concern two different types of static and dynamic two-commodity models: 1) with supply functions and functions of demand for a product that are defined exogenously, 2) with supply functions and functions of demand for a product that are defined endogenously—that is the Arrow-Debreu-McKenzie general equilibrium models. The dynamic models are considered in a discrete-time and in a continuous-time versions. In particular the focus of the exercises is to deal with a concept of the Walrasian general equilibrium and its existence, uniqueness and asymptotical global stability. E6.1. There is a market of a homogeneous product with exogenously determined demand function: y d ( p) = −ap α + b, a, b > 0 and supply function: y s ( p) = cp α + d, c, d > 0. 1. For a product price, the demand and the supply levels determine intervals of values resulting from analytical forms of the demand and supply functions. 2. Draw graphs of both functions in space R2+ (in one figure). Determine by what values of parameters b, d > 0 there exists the equilibrium price and indicate this price in the figure, when: (a) α ∈ (0; 1), (b) α = 1, (c) α > 1. 3. Determine the equilibrium price. 4. Determine elasticities of the equilibrium price with regard to parameters of the supply and the demand functions that determine this price and give their economic interpretation.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Sobczak and K. Malaga, Workbook for Microeconomics, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-031-41947-8_6

229

230

6 Rationality of Choices Made by Groups of Producers and Groups …

Fig. 6.1 Demand function and supply function, α ∈ (0; 1) (E.6.1a)

Solutions Ad 1 y d ( p) = −ap α + b, a, b > 0 and y s ( p) = cp α + d, c, d > 0. If p = 0 then y d ( p) = b and y s ( p) = d. ( )1 If y d ( p) = 0 then −ap α + b = 0 , so p = ab α . [

( ) α1 ] b p ∈ 0; a y d ( p) ∈ [0; b] y s ( p) ∈ [d; +∞). Ad 2a (Fig. 6.1). Ad 2b (Fig. 6.2). Ad 2c (Fig. 6.3). Ad 3a α ∈ (0; 1) ( ) α1 (1) ∃1 p = b−d > 0, y d ( p) = y s ( p), ∀b, d > 0, b > d a+c d (2) ∃1 p = 0, y ( p) = y s ( p), ∀b, d > 0, b = d 3) ¬∃ p ≥ 0, y d ( p) = y s ( p), ∀b, d > 0, d > b. Ad 3b α=1

6 Rationality of Choices Made by Groups of Producers and Groups … Fig. 6.2 Demand function and supply function, α = 1 (E.6.1b)

Fig. 6.3 Demand function and supply function, α > 1 (E.6.1c)

(1) ∃1 p = b−d > 0, y d ( p) = y s ( p), ∀b, d > 0, b > d a+c (2) ∃1 p = 0, y d ( p) = y s ( p), ∀b, d > 0, b = d (3) ¬∃ p ≥ 0, y d ( p) = y s ( p), ∀b, d > 0, d > b. Ad 3c α>1 ) α1 ( (1) ∃1 p = b−d > 0, y d ( p) = y s ( p), ∀b, d > 0, b > d a+c d (2) ∃1 p = 0, y ( p) = y s ( p), ∀b, d > 0, b = d (3) ¬∃ p ≥ 0, y d ( p) = y s ( p), ∀b, d > 0, d > b. Ad 4 ( ∃1 p =

b−d a+c

) α1

>0

231

232

6 Rationality of Choices Made by Groups of Producers and Groups …

E a ( p) =

( )1 1 b − d α −1 (b − d) a ∂p a a =− < 0[%] 1 = − ( ) 2 ∂a p α a+c α(a + c) (a + c) b−d α (

E c ( p) =

1 b−d ∂p c =− ∂c p α a+c (

E b ( p) =

1 b−d ∂p b = ∂b p α a+c

) α1 −1

(

E d ( p) =

) α1 −1

1 b−d ∂p d =− ∂d p α a+c

a+c

c c (b − d) < 0[%] 1 = − α(a + c) (a + c)2 ( b−d ) α a+c

b 1 b > 0[%] 1 = ( ) α(b − d) (a + c) b−d α

) α1 −1

a+c

d 1 d < 0[%]. 1 = − ( ) α(b − d) (a + c) b−d α a+c

E6.2. There is given a market of two products with exogenous demand functions: y1d (p) = −a1 p1 + γ1 p2 + b1 , y2d (p) = −a2 p2 + γ2 p1 + b2 , ai , bi , γi > 0, i = 1, 2 and exogenous supply functions: y1s (p) = c1 p1 + δ1 p2 + d1 , y2s (p) = c2 p2 + δ2 p1 + d2 , ci , di , δi > 0, i = 1, 2. 1. For price of products, the demand and the supply levels determine intervals of values resulting from analytical forms of the demand and supply functions. 2. Check if the two products are: (a) independent, (b) complementary, (c) substitute, to each other. 3. Draw graphs of these functions in space R3+ (in one figure). Determine by what values of parameters bi , di > 0 (i = 1, 2) there exists the equilibrium price vector and indicate this vector in the figure. 4. Determine the equilibrium price vector. 5. Determine elasticities of the equilibrium prices with regard to parameters of the supply and the demand functions that determine the prices and give economic interpretation of these elasticities. Solutions Ad 1- 2a Independents Products y1d ( p1 ) = −a1 p1 + b1 ,

y2d ( p2 ) = −a2 p2 + b2 , ai , bi , γi > 0, i = 1, 2

6 Rationality of Choices Made by Groups of Producers and Groups …

233

Fig. 6.4 Demand function and supply function, first product (E.6.2)

y1s ( p1 ) = c1 p1 + d1 ,

y2s ( p2 ) = c2 p2 + d2 , ci , di , δi > 0, i = 1, 2

y1d ( p1 ) ∈ [0; b1 ], y2d ( p2 ) ∈ [0; b2 ] y1s ( p1 ) ∈ [d1 ; +∞), y2s ( p2 ) ∈ [d2 ; +∞) [ [ ] ] b1 b2 p1 ∈ 0; , p2 ∈ 0; . a1 a2 Ad 3 Since we are interested only in positive prices, the equilibrium price vector exists when b1 > d1 and b2 > d2 . These two conditions have to be satisfied simultaneously. We present figures for exemplary values of the demand functions and the supply functions: y1d (p) = −a1 p1 + γ1 p2 + b1 = − p1 + y1s (p) = c1 p1 + δ1 p2 + d1 = p1 +

1 p2 + 2 2

1 p2 + 1 2

and b1 = 2 > 1 = d1 y2d (p) = −a2 p2 + γ2 p1 + b2 = −2 p2 + y2s (p) = c2 p2 + δ2 p1 + d2 = 2 p2 +

1 p1 + 2 2

1 p1 + 1 2

and b2 = 2 > 1 = d2 (Figs. 6.4 and 6.5). There exists only one point belonging to (all four planes. point indicates the ) This ( 2−1 2−1 ) ) ( ) ( b1 −d1 b2 −d2 equilibrium price vector p = p 1 , p 2 = a1 +c1 , a2 +c2 = 1+1 , 2+2 = 21 , 41 (Fig. 6.6).

234

6 Rationality of Choices Made by Groups of Producers and Groups …

Fig. 6.5 Demand function and supply function, second product (E.6.2)

Fig. 6.6 Demand functions and supply functions, both products (E.6.2)

Ad 4 ( ) ( ) b1 − d1 b1 y1d p 1 = y1s p 1 ⇒ p 1 = if b1 > d1 then 0 < p 1 < , a1 + c1 a1 ( ) ( ) − d b b2 2 2 y2d p 2 = y2s p 2 ⇒ p 2 = if b2 > d2 then 0 < p 2 < . a2 + c2 a2 Ad 5 ( ) d pi x E x pi = , i = 1, 2, d x pi ( ) ( ) a1 a2 < 0, E a2 p 2 = − < 0, E a1 p 1 = − a1 + c1 a2 + c2 ( ) ( ) 1 1 E b1 p 1 = > 0, E b2 p 2 = >0 a1 + c1 a2 + c2 ( ) ( ) c1 c2 E c1 p 1 = − < 0, E c2 p 2 = − < 0, a1 + c1 a2 + c2 ( ) ( ) 1 1 E d1 p 1 = − < 0, E c2 p 2 = − < 0. a1 + c1 a2 + c2

6 Rationality of Choices Made by Groups of Producers and Groups …

235

E6.3. Present the model of a market of a single good from Exercise E6.1 as: (a) a dynamic discrete-time model, (b) a dynamic continuous-time model. Check whether in these models, Walras’s law is satisfied for an excess demand function. Selecting proper values for parameter σ > 0 in the dynamic discretetime model determine a feasible price trajectory for 10 subsequent periods: t = 1, 2, . . . , 10. Solutions If y d ( p(t)) = −ap(t)α + b, a, b > 0, cp(t)α + d, then:

and the supply function

y s ( p(t)) =

z( p(t)) = y d ( p(t)) − y s ( p(t)) = −ap(t)α + b − cp(t)α − d = −(a + c) p(t)α + b − d Ad a Dynamic discrete-time model (1) p(t + 1) = p(t) + σ (−(a + c) p(t)α + b − d) (2) p(0) = p0 > 0 (3) t = 1, 2, . . . , 10. Ad b Dynamic continuous-time model = σ (−(a + c) p(t)a + b − d) (1) d p(t) dt (2) p(0) = p0 > 0 (3) t ∈ [0; 10]. E6.4. Present the model of a market of two goods from Exercise E6.2 as: (a) a dynamic discrete-time model, (b) a dynamic continuous-time model. Check whether in these models Walras’s law is satisfied for a vector function of the excess demand. Selecting proper values for parameter σ > 0 in the dynamic discrete-time model, determine feasible price trajectories for 10 subsequent periods: t = 1, 2, . . . , 10. Solution In exercise E6.2, the static model of a market of two goods is given. The demand for goods is described by functions: y1d (p) = −a1 p1 + γ1 p2 + b1

236

6 Rationality of Choices Made by Groups of Producers and Groups …

y2d (p) = −a2 p2 + γ2 p1 + b2 ai , bi , γi > 0 i = 1, 2. The supply of goods is described by functions: y1d (p) = c1 p1 + δ1 p2 + d1 y2d (p) = c2 p2 + δ2 p1 + d2 ci , di , δi > 0 i = 1, 2. Functions of excess demand for goods take the following forms: z 1 (p) = y1d (p) − y1s (p) = −a1 p1 + γ1 p2 + b1 − (c1 p1 + δ1 p2 + d1 ) = −(a1 + c1 ) p1 + (γ1 − δ1 ) p2 + b1 − d1 z 2 (p) = y2d (p) − y2s (p) = −(a2 + c2 ) p2 + (γ2 − δ2 ) p1 + b2 − d2 . A price vector of the Walrasian equilibrium p is a solution to the following system: {

z 1 (p) = 0 z 2 (p) = 0

that gives {

−(a1 + c1 ) p1 + (γ1 − δ1 ) p2 + b1 − d1 = 0 −(a2 + c2 ) p2 + (γ2 − δ2 ) p1 + b2 − d2 = 0 ( ) (γ1 − δ1 )(b2 − d2 ) + (b1 − d1 )(a2 + c2 ) (γ2 − δ2 )(b1 − d1 ) + (b2 − d2 )(a1 + c1 ) p= , . (a1 + c1 )(a2 + c2 ) − (γ1 − δ1 )(γ2 − δ2 ) (a1 + c1 )(a2 + c2 ) − (γ1 − δ1 )(γ2 − δ2 )

(a) The dynamic discrete-time model has a form: pi (t + 1) = pi (t) + σ z i (p(t)) i = 1, 2, t = 0, 1, 2 . . . pi (0) = const. > 0 i = 1, 2 (b) The dynamic continuous-time mode has a form: dpi (t) = σ z i (p(t)) i = 1, 2, t ∈ (0; +∞) dt pi (0) = const. > 0 i = 1, 2. Let us recall the Walras’s law: ∀p > 0 ⟨p, z(p)⟩ = p1 z 1 (p) + p2 z 2 (p) = 0. From the form of the excess demand function, one can notice tat in general, the Walras’s law is not satisfied in the models considered in this exercise. But the equation

6 Rationality of Choices Made by Groups of Producers and Groups …

237

Fig. 6.7 Price trajectories in discrete-time model (E6.4)

of the Walras’s law is satisfied for these models in a special case, when prices are the equilibrium prices: ⟨p, z(p)⟩ = p1 z 1 (p) + p2 z 2 (p) = 0 since z 1 (p) = 0 and z 2 (p) = 0. Let us now consider different values for parameter σ, to see by which values of this parameter we get feasible price trajectories. When σ = 0.05, the price trajectories are feasible but the equilibrium is not reached in time horizon T = 10. When σ = 0.35, the price trajectories are feasible and the equilibrium is reached in period t = 5. When σ = 0.7, the price trajectories are not feasible. The price trajectories1 in all three cases are presented in Fig. 6.7. E6.5. owner of a strawberry plantation hires one worker who has 24 units of time. The employee can allocate part of the time to work and part to rest. He/she owns 1

The price trajectories are obtained in a file called Exercise 6.4.xlsx.

238

6 Rationality of Choices Made by Groups of Producers and Groups …

20% of shares in profits of the plantation. The only production factor in producing strawberries is the labour of the worker. The process of production of strawberries is described by a power production function of a form: w1 = f (n 2 ) = n 0.5 2 , where w1 denotes the output in the production process and n 2 denotes the input of labour. The plantation owner wants to maximize her/his profit from production and sales of strawberries for which their market price equals p1 per unit (e.g. one kilogram, one crate, etc.). The source of production cost is hiring the labour force for which its market price equals p2 per one hour. The plantation owner, as a consumer, wants to maximize her/his utility u 1 (x11 ) described as an increasing function of consumed strawberries’ quantity. Her/his income consists of 80% of shares in the profit of the plantation. The worker wants to maximize her/his utility described as an increasing function of consumed strawberries’ quantity and of leisure: 0.5 0.3 u 2 (x21 , x22 ) = x21 x22 .

Income of the worker consists of 20% of shares in the profit of the plantation and a money value of 24 time units allocated partly to work and partly to leisure. For the static Arrow-Debreu-McKenzie model: 1. Write a space of the net output of the strawberries’ plantation. Note: according to the description of the exercise the input n 1 of strawberries equals 0. While labour is not produced, hence w2 = 0. 2. Formulate and solve the profit maximization problem for the plantation owner. 3. Formulate and solve the consumption utility maximization problems for the plantation owner and for the worker. 4. Write a form of an excess demand function. 5. Determine a system of general equilibrium conditions and a Walrasian equilibrium price vector. 6. Determine components of a Walrasian equilibrium allocation. Solutions Ad 1 Write a space of the net output of the strawberries’ plantation. Note: according to the description of the exercise, the input n 1 of strawberries equals 0. While labour is not produced, hence w2 = 0. A vector of the net output of the strawberries’ plantation is given as: y1 = (w1 − n 1 , w2 − n 2 ) = (w1 − 0, 0 − n 2 ) = (w1 , −n 2 ),

where w1 ≤ n 0.5 2 , n2 ≥ 0

6 Rationality of Choices Made by Groups of Producers and Groups …

239

Hence, the space of the net output of the strawberries’ plantation can be written in the following way: Y 1 = {y1 ∈ R2 : y1 = (w1 , −n 2 ), wherew1 ≤ n 0.5 2 , n 2 ≥ 0}. Ad 2 Formulate and solve the profit maximization problem for the plantation owner. The profit maximization problem for the plantation owner takes a form: w1 = f (n 2 ) = n 0.5 2 π (n 2 ) = p1 w1 − p2 n 2 = p1 n 0.5 2 − p2 n 2 π (n 2 ) → max n 2 ≥ 0 The decision variable for the plantation owner is an input n 2 of labour. Hence, a necessary condition for existence of a maximum of the profit function takes the following form: dπ (n 2 ) = 0.5 p1 n −0.5 − p2 = 0 2 dn 2 and gives the optimal input of labour: ( n2 =

p1 2 p2

)2 .

A sufficient condition for existence of a maximum of the profit function takes a form: d2 π (n 2 ) = −0.25 p1 n −1.5 0. The maximum profit obtained in the plantation is: max π (n 2 ) = π (n 2 ) =

p1 n 0.5 2

( ) p1 2 p1 p2 − p2 n 2 = p1 · − p2 = 1 = π (p). 2 p2 2 p2 4 p2

The optimal supply of strawberries is: w1 = n 0,5 2 =

p1 . 2 p2

240

6 Rationality of Choices Made by Groups of Producers and Groups …

Now we can also write a function of supply of the plantation taking into account all products considered (strawberries and the labour): ]

[

y (p) = y 11 (p), y 12 (p) = 1

[

) ] ( p1 p1 2 ,− . 2 p2 2 p2

Let us emphasize that the supply of labour is written with the negative sign since in fact in this vector, it expresses the demand of the plantation owner for labour needed in the production of strawberries. Ad 3 Formulate and solve the consumption utility maximization problems for the plantation owner and for the worker. The consumption utility maximization problem for the plantation owner takes a form: a1 = (0, 0)—initial endowment of the plantation owner. u 1 (x11 )— the plantation owner’s utility function depending only on consumption of strawberries u 1 (x11 ) → max x11 ≥ 0 p1 x11 ≤ I 1 (p) = 0.8π (p) u 1 (x11 ) is an increasing function, hence dudx(x1111 ) > 0. Because of the insatiability phenomenon reflected in the increasing utility function instead of the inequality above, one considers an equality which means that the plantation owner wants to spend all her/his income: 1

p1 x11 = I 1 (p). From this equation, we can find the demand of the plantation owner for strawberries: x11 (p) =

0.8π (p) I 1 (p) = . p1 p1

Now, we can also write a function of demand of the plantation owner, being a consumer, taking into account all goods considered (strawberries and the leisure): [ x1 (p) =

] 0.8π (p) ,0 . p1

Let us emphasize that the plantation owner’s demand for leisure equals zero because her/his decision how much time he/she wants to spend working and how much resting is irrelevant for other agents in the economy, here only the worker. The

6 Rationality of Choices Made by Groups of Producers and Groups …

241

plantation owner does not offer her/his hours of work to other employers; he/she is not employee in any company. The consumption utility maximization problem for the worker takes a form: a2 = (0, 24)—– initial endowment of the worker 0.5 0.3 u 2 (x21 , x22 ) = x21 x22 → max

x21 , x22 ≥ 0

p1 x21 + p2 x22 ≤ I (p) = 0.2π (p) + 24 p2 . 2

Again, because of the insatiability phenomenon reflected in the increasing utility function instead of the inequality above, one considers an equality which means that the worker wants to spend all her/his income. Hence, we have one equation of the system of two equations. The other equation results from the second Gossen’s law. The system is: {

2 K SS12 =

−0,5 0,3 0.5x21 x22 0,5 −0,7 0.3x21 x22

=

5 3

·

x22 x21

=

p1 p2

⇒ x22

=

3 p1 x 5 p2 21

p1 x21 + p2 x22 = 0.2π (p) + 24 p2 and is transformed to: p1 x21 + p2

3 p1 x21 = 0.2π (p) + 24 p2 5 p2

8 p1 x21 = 0.2π (p) + 24 p2 5 giving the demand reported by the worker: {

x 21 (p) = x 22 (p) =

5(0.2π (p)+24 p2 ) 8 p1 3(0.2π (p)+24 p2 ) 8 p2

= =

π (p) + 15 pp21 8 p1 0.6π (p) +9 8 p2

the demand for leisure.

The time the worker wants to spend working in the plantation is the labour supply: 24 − x 22 (p) = 24 −

0.6π (p) 0.6π (p) − 9 = 15 − . 8 p2 8 p2

Now, we can write a function of demand of the worker, taking into account all goods considered (strawberries and the leisure): ] p2 0.6π (p) π (p) x (p) = + 15 , +9 . 8 p1 p1 8 p2 [

2

242

6 Rationality of Choices Made by Groups of Producers and Groups …

Ad 4 Write a form of an excess demand function. A function of the excess demand is defined as: F(p) = x(p) − y(p) − aˆ = (F1 (p), F2 (p)) where the excess demand for strawberries is: F1 (p) = x 11 (p) + x 21 (p) − y 11 (p) =

p2 p1 0.8π (p) π (p) + + 15 − p1 8 p1 p1 2 p2

and the excess demand for leisure is: F2 (p) = x¯ 12 (p) + x¯ 22 (p) − y¯ 12 (p) − a22 ( ( ) ) p1 2 0.6π (p) =0+ +9− − − 24 8 p2 2 p2 ) ) ( ( p1 2 0.6π (p) . = − 15 − 2 p2 8 p2 The excess demand for leisure can be seen as well as the excess demand for labour, hence the difference between the demand for labour and the labour supply. Ad 5 Determine a system of general equilibrium conditions and a Walrasian equilibrium price vector. A system of general equilibrium conditions is derived from equalizing the excess demand function with 0: ⎧ ⎨ 0.8πp (p) + π8(p) + 15 pp21 − 2pp12 = 0 ( 1 )2 (p1 ) (p) ⎩ p1 =0 − 15 − 0.6π 2 p2 8 p2 which can be also presented in a form showing the equalization of the demand and the supply, thus partial equilibrium on market of each product, hence the general equilibrium conditions: ⎧ ⎨ 0.8πp (p) + π8(p) + 15 pp21 = p1 ( 1 )2 ⎩ p1 = 15 − 0.6π (p) . 2 p2

p1 2 p2

8 p2

From point 2, we know that the profit obtained in the plantation is:

6 Rationality of Choices Made by Groups of Producers and Groups …

π (p) =

243

p12 . 4 p2

Substituting this profit into each equation of the system, we get the same relationship prices of both products, that is p1 of strawberries and p2 of the labour: 43 p12 = 15 160 p22

⇔

p12 =

2400 2 p 43 2

which gives an equilibrium prices in a form of the following relationship: p 1 ≈ 7.47 · p 2 This relationship shows a structure of the Walrasian equilibrium price vector: pλ ≈ λ(7.47, 1), λ > 0. That is why one says that the Walrasian equilibrium price vector is defined with an accuracy of a structure. This means that any prices p1 , p2 being in such a relationship satisfy the equilibrium conditions, thus guarantee the equilibrium on all markets considered. Ad 6 Determine components of a Walrasian equilibrium allocation. By the Walrasian general equilibrium prices, the demand of the plantation owner for strawberries and the leisure is: ] [ ] [ ] [ 0.8π (p) 0.8 p 21 0.8 1 · 7.47, 0 = [1.49, 0] x (p) = ,0 = · ,0 = p1 p1 4 p2 4 and the demand of the worker for strawberries and the leisure is: [ ] π (p) p2 0.6π (p) 2 x (p) = + 15 , +9 8 p1 p1 8 p2 ] [ p2 p 0.6 p 21 1 · 1 + 15 2 , · +9 = 8 p1 4 p 2 p1 8 p2 4 p2 ] [ 15 0.6 1 · 7.47 + , · 55.81 + 9 = [2.24, 10.05]. = 32 7.47 32 Thus, the time the worker wants to spend working, that is her/his labour supply is: 24 − 10.05 = 13.95.

244

6 Rationality of Choices Made by Groups of Producers and Groups …

The supply of the plantation by the Walrasian general equilibrium prices is: [ y (p) = 1

) ] [ ( )2 ] ( 1 1 p1 p1 2 · 7.47, − · 7.47 ,− = = [3.73, −13.95]. 2 p2 2 p2 2 2

E6.6. Consider a discrete-time version of the dynamic Arrow-Debreu-McKenzie model for the same data given as in Exercise E6.5. Initial prices are: p(0) = (10, 2). Using formulas for the excess demand function and for a structure of the Walrasian equilibrium price vector derived in Exercise E6.5 for the static Arrow-DebreuMcKenzie model: 1. Determine trajectories of a price vector satisfying a system of equations of the dynamic discrete-time Arrow-Debreu-McKenzie model, taking a proportionality (t) and compare coefficient σ equal to 0.01, 0.05 and 0.1. Calculate price ratios pp21 (t) 2. 3.

4. 5.

them with the equilibrium price ratio pp2 . 1 State which of trajectories determined in point 1 are feasible. State if and when (in which period) a structure of prices stabilizes around the equilibrium price structure and whether it reaches this structure in time horizon T = 30. Present graphs of the price trajectories in the state space. Present graphs of the price trajectories as functions of time.

Solutions Ad 1 Determine trajectories2 of a price vector satisfying a system of equations of the dynamic discrete-time Arrow-Debreu-McKenzie model, taking a proportion(t) and ality coefficient σ equal to 0.01, 0.05 and 0.1. Calculate price ratios pp21 (t)

compare them with the equilibrium price ratio pp2 . 1 The dynamic discrete-time Arrow-Debreu-McKenzie model in case of two goods has a general form: pi (t + 1) = pi (t) + σ Fi (p(t)) i = 1, 2, t = 0, 1, 2 . . . pi (0) = const. > 0 i = 1, 2. On the basis of the results of exercise E6.5, we have the excess demand function given as:

2

The trajectories are obtained in a file called Exercise 6.6.xlsx.

6 Rationality of Choices Made by Groups of Producers and Groups …

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p2 (t) p1 (t) 0.8π (p(t)) π (p(t)) + + 15 − p1 (t) 8 p1 (t) p1 (t) 2 p2 (t) )2 ( ) ( 0.6π (p(t)) p1 (t) . − 15 − F2 (p(t)) = 2 p2 (t) 8 p2 (t)

F1 (p(t)) =

Starting from the initial condition p(0) = (10, 2), one can find levels of prices of both goods in all subsequent periods. Ad 2 State which of trajectories determined in point 1 are feasible. It turns out that all trajectories are feasible. However, when σ = 0.1 the trajectories are oscillatory (Fig. 6.9). Ad 3 State if and when (in which period) a structure of prices stabilizes around the equilibrium price structure and whether it reaches this structure in time horizon T = 30. This can be stated on the basis of calculations provided in a file (Exercise 6.6.xlsx). From the static Arrow-Debreu-McKenzie model, we know that the Walrasian equi≈ 7.47 and librium price vector is pλ ≈ λ(7.47, 1); thus, we can focus on pp21 (t) (t) when this equation is satisfied for the first time. It turns out that when σ = 0.01 the equilibrium state is reached in period t = 51, when σ = 0.05 it is reached very quickly because already in period t = 7, when σ = 0.1, it is reached very slowly because not earlier than in period t = 822. In this last case, the price trajectories are oscillatory but one can notice that the oscillations are smaller and smaller in time. Ad 4 Present graphs of the price trajectories in the state space (Fig. 6.8).3 Ad 5 Present graphs of the price trajectories as functions of time (Fig. 6.9).4 E6.7. Consider a continuous-time version of the dynamic Arrow-Debreu-McKenzie model for the same data given as in Exercise E5. 1. Determine trajectories of a price vector satisfying a system of equations of the dynamic continuous-time Arrow-Debreu-McKenzie model taking a proportionality coefficient σ equal to 0.01, 0.05, 0.1 and determine whether these trajectories are feasible. 2. Determine if and when (at what moment) a structure of prices stabilizes around the equilibrium price structure. 3 4

The trajectories are obtained in an Excel file called Exercise 6.6.xlsx. The trajectories are obtained in an Excel file called Exercise 6.6.xlsx.

246

6 Rationality of Choices Made by Groups of Producers and Groups …

Fig. 6.8 Price trajectories in state space (E6.6)

6 Rationality of Choices Made by Groups of Producers and Groups …

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Fig. 6.9 Price trajectories as functions of time (E6.6)

3. Present graphs of price trajectories as functions of time. Solutions Ad 1 Determine trajectories of a price vector satisfying a system of equations of the dynamic continuous-time Arrow-Debreu-McKenzie model taking a proportionality coefficient σ equal to 0.01, 0.05, 0.1 and determine whether these trajectories are feasible. The dynamic continuous-time Arrow-Debreu-McKenzie model in case of two goods has a general form: d pi (t) = σ z i (p(t)) i = 1, 2, t ∈ (0; +∞) dt pi (0) = const. > 0 i = 1, 2.

248

6 Rationality of Choices Made by Groups of Producers and Groups …

On the basis of the results of exercise E6.5, we have the excess demand function given as: p2 (t) p1 (t) 0.8π (p(t)) π (p(t)) + + 15 − p1 (t) 8 p1 (t) p1 (t) 2 p2 (t) ) ) ( ( 0.6π (p(t)) p1 (t) 2 . − 15 − F2 (p(t)) = 2 p2 (t) 8 p2 (t)

F1 (p(t)) =

To find levels of prices of goods in subsequent moments, one can approximate the original continuous-time model with a discrete-time model in the following way: pi (t + Δt) = pi (t) + σ Fi (p(t))Δt i = 1, 2, t ∈ [0; +∞) pi (0) = const. > 0 i = 1, 2, where Δt means a time increment. Having this discrete-time approximation of the continuous-time model and starting from the initial condition p(0) = (10, 2), one can find levels of prices of both goods in all subsequent moments. To find feasible price trajectories, one can use various values for σ and for the time increment Δt. We use following sets of these parameters: σ = 0.01, Δt = 1 σ = 0.05, Δt = 0.5 σ = 0.1, Δt = 2. It turns out that all trajectories are feasible. Ad 2 Determine if and when (at what moment) a structure of prices stabilizes around the equilibrium price structure. The structure of prices in a sense pp21 (t) stabilizes around the equilibrium price (t) structure, that is pp2 at following moments: 1 when σ = 0.01, Δt = 1 − in 51-th step, thus at moment t = 51 · 1 = 51, when σ = 0.05, Δt = 0.5 − in 17-th step, thus at moment t = 17 · 0.5 = 8.5, when σ = 0.1, Δt = 2 − in 22-nd step, thus at moment t = 22 · 2 = 44.

Ad 3 Present graphs of price trajectories as functions of time (Fig. 6.10).5

5

The trajectories are obtained in an Excel file called Exercise 6.7.xlsx.

6 Rationality of Choices Made by Groups of Producers and Groups … Fig. 6.10 Price trajectories as functions of time (E6.7)

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References

Anholcer, M. (2015). Mathematics and management examples and exercises. MD 290. Wydawnictwo Uniwersytetu Ekonomiczego w Poznaniu. Chiang, A. C., & Wainwright, K. (2005). Fundamental methods of mathematical economics. Mcgraw-Hill Education. Jacques, I. (2006). Mathematics for economics and business (5th ed.). Prentice Hall. Malaga, K., & Sobczak, K. (2021). Advanced microeconomics, learning materials for Ph.D. students. MD 367. PUEB Press. Malaga, K., & Sobczak, K. (2022). Microeconomics. Static and dynamic analysis. Springer Nature. O’Brien, R. J., & Garcia, G. G. (1971). Mathematics for economists and social scientists. Pitman Press. Ostoja-Ostaszewski, A. (1996a). Matematyka w ekonomii. Modele i metody (Vol. 1). Algebra elementarna. PWN. Ostoja-Ostaszewski, A. (1996b). Matematyka w ekonomii. Modele i metody (Vol. 2). Elementarny rachunek ró˙zniczkowy. PWN. Sydsaeter, K., & Hammond, P. (2008). Essential mathematics for economic analysis. Prentice Hall.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Sobczak and K. Malaga, Workbook for Microeconomics, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-031-41947-8

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