Production Economics, Mathematical Development and Applications [1 ed.] 0887380166

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PRODUCTION ECONOMICS Mathematical Development and Applications C.T.K. Ching and John F. Yanagida

•

Transaction Books New Brunswick (U.S.A.) and Oxford (U.K.)

Copyright© 1985 by Transaction, Inc. New Brunswick,-New Jersey 08903 All rights reserved under International and Pan-American Copyright Conventions. No part of this book may be reproduced or transmitted in any form or by any means, electrnnic or mechanical, including photocopy, recording, or any information storage and retrieval system, without prior permission in writing from the publisher. All inquiries should be addressed to Transaction Books, Rutgers-The State University, New Brunswick, New Jersey 08903. Library of Congress Catalog Number: 84-24109 ISBN: 0-88738-016-6 (cloth) Printed in the United States of America Library of Congress Cataloging in Publication Data Ching, C.T.K. (Chauncey T.K.) Production economics. Includes index. 1. Production (Economic theory)-Mathematical models. I. Yanagida, John F. II. Title . HB241.C492 1985 338.5 84-24109 ISBN 0-88738-016-6

0030'730

To Teddie, Donna, Cory, Jane, and Lisa

CONTENTS List of Tables List of Figures Preface

viii

ix xii

1.

Introduction

1

2.

Mathematical Review

9

3.

Production Economic Theory

4.

Empirical Production Functions

195

5.

Linear Algebra

235

6.

Linear Programming

293

7.

Generality of Production Economics

378

Index

85

397

LIST O:F TABLES Ta ble 2- 1. -Table 2-2 . Table 4- 1. Table 4-2. Table 4- 3 . Table 4- 4 . Table 4- 5 . Table 4- 6 .

Table 6-1. Table 6-2 . Table 6-3 . Table 6-4 . Table 6- 5 . Table 6- 6 . Table 7- 1 .

Rela tion between Postage Cost and Par cel Weight Tabular Form of Functi on , Y = 2 + 2X Production of Alfalfa Hay in Response to Water Application Regression Results for Alfalfa Yields Wheat Production and Response to Nitrogen and Water Application Regression Results for Wheat Yields Cobb-Douglas and CES Results for Grain Production Example Pooled Time Series and Cross Section Production Analyses (1949 , 1954, and .. 1959 Census) Simplex Tables for Wheat- Corn Problem First Simplex Table for "Greater Than" 1 Inequality Example Simplex Tables for Minimization Problem Simplex Tables for Barley-Peas Exampl e Input- Output Coefficients for 3 Milk Producing Activities Optimal Solution to the Dual Linear Programming Problem Simplex Table for Dynamic Linear Programming Model

viii

LIST OF FIGUltES Figure 2- 1. Figure 2-2. Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure

2- 3. 2-4. 2-5. 2-6 . 2- 7. 2-8. 2-9 . 2- 10 . 2-11 . 2- 12 . 2-13. 2-14 . 2-15 .

Figure 2-16. Figure 2-17. Figure 2-18 . Figure 3-1 . Figure 3-2. Figure 3-3.

Graphical Form of Function, Y = 2 + 2X Venn Diagram Showing Relations and Functions Illustration of the Concept of Limits An Extreme-Value Illustration Slope of a Linear Function Slope of a Nonlinear Function The Tangent Line as a Limit Graph of a Constant Function Graphing the Derivative Intuitive View of Optimizing Functions Function Illustrating Inflection Point Relative Maxima and Minima Inflection Points Concavity of Functions Graphical Depiction of Multiple Argument Fune tion Saddle Point Constrained Optima Three Dimensional Surfaces Production Function Illustrating Maximum Technical Efficiency Productivity Curves with Different Levels of Fixed Factor Productivity Curves

Figure 3-4. Figure 3-5.

Profit Maximizing Level of x , X~ 1 Stages of Production for Production Function Homogeneous of Degree One

Figure 3-6 .

Isoquant Diagram, q

Figure 3- 7. Figure 3-8.

Isoquants, Conv ex Upward and Downward that are Conditions for Isoquan ts Convex to the Origin Expansion Pa th Factor Dema nd Func t ion

Figure 3-9. Figure 3-10.

ix

0

=

f(X , 1

x2 )

Figure Figure Figure Figure Figure

3-11. 3-12 . 3-13 . 3-14. 3-15 .

Figure 3-16. Figure 3-17. Figure 3-18. Figure 3-19. Figure 4-1. Figure 4-2. Figure 4-3. Figure Figure Figure Figure Figure

5-1 . 5-2. ; 5-3. 5-4. 5-5.

Figure 5-6. Figure 5-7. Figure Figure Figure Figure Figure

6-1. 6-2. 6-3. 6- 4. 6-5 .

Figure 6-6. Figure 6-7. Figure 6-8.

Scale and Substitution Effects Isoquants and Expansion Path Total Cost Function Cost Fune tions Long-run Total Cost Function as an Envelope to the Short-run Cost Functions Joint Products Production Function for Three Resource Levels Time Period Definition Land Rent Function for Barley as a Function of Distance from Market Rent Functions for Barley and Corn Graph of q = f(X) Specific Linear Relationships between q and X Least Squares Estimators in Perspective Graphical Depiction of a Vector Vector in a Three-Dimensional Space Illustration of Scalar Multiplication Addition of Vectors Inconsistent Equation, No Solution Possible Unique Solution Infinite Number of Solutions Graph of the Equation y = 2x Graphing the Inequation y < 2 Graphing the Inequation y 4 Graphing the Inequation y -- x 2 < 4 Graphical Solution 2 Products-;- 2 Machines Linear Programming Problem Graphical Solution to Espe ran toPlumbing Linear Programming Problem Graphical Solution to Wheat-Corn Linear Programming Problem Venn Diagram Illustrating Basic Solutions and the Basis

>

X

Figure 6- 9. Figure 6- 10. Figure 6- 11. Figure 6-12. Figure 6-13. Figure 6-14.

Figure 6-15. Figure 6-16. Figure 6-17. Figure 6-18. Figure 6-19 . Figure 6-20. Figure 6-21. Figure 7-1.

Basic Solutions in a Linear Programming Problem Product Transformation Function for Barley and Slack Labor Milk Production_Function Graphical Depiction of Milk Production Activities Isoquants for a Single Linear Production Activity Graphical Depiction of the Linear Production Activities in the Output Dimension (Production Possibilities Curve) Graphical Depiction of the Linear Production Activities in the Input Dimension (Isoquants) Graphical Depiction of Additivity of Activities Convex and Nonconvex Sets of Points Case of Multiple Optimal Solutions Optimal! ty among Cornerpoint Feasible Solutions Nonconvex Feasible Region Barley-Peas Linear Programming Example Short-run and Long-run Factor Demand Functions

xi

PREFACE The idea for this book stemmed from two beliefs . First, production economic concepts form the basis for solving many problems in applied economics . These concepts apply not only to firm-level problems but also to those pertaining to the use of natural resources, and problems faced by communities and regions. It is extemely difficult to escape the notion of production processes in viewing practically any type of economic problem. Second, while there are many excellent books that cover the various elements of production economics, there is no single text which covers the mathematical, theoretical, and applied aspects of production economics. Given these beliefs , we have attempted to put together a .. relatively self-contained body of knowledge that covers the full range of production economics. We make a special effort to fully develop mathematical concepts relevant to production economics. W~ utilize the mathematics in exposing the pertinent relationships inherent in production economic theory. Finally, we discuss in detail some of the applied aspects of production economics--both empirical production function and mathematical programming. This book is intended to serve as a text in upper division undergraduate courses and in beginning level graduate courses dealing with production economics. For the undergraduate without the mathematical background, we hope that the review of the relevant mathematics is sufficient to ensure understanding of production economic concepts. For the more advanced students, we hope that the discussion of mathematical concepts and their applications to production economics serves as a convenient review. While we take full responsiblitity for this work there are many scholars, instructors, and students who have contributed to this effort. We wish to make s pee ial acknowledgement to two of our men tors who xii

imparted knowledge to us and stimulated our interest in production economics . Specifically, we refer to the late Professor Gerald W. Dean of the University of California at Davis, and Professor Earl R. Swanson at the University of Illinois. W~ also recognize the many graduate and undergraduate students who have endured through various versions of this material with us. S !nee both authors were former faculty members at the University of Nevada Reno, we wish to acknowledge the support and encouragement of administrators in the College of Agriculture and the Department of Agricultural Economics at that University. Without them, the task would have been infinitely more difficult . Special thanks is also extended to Nancy Taylor, Shou- Lin V. Lin, Jacki Ierien, Christie South and Judy Tomlin who graciously expended many hours of professional services in preparing this manuscript. Nancy Taylor's expert judgment and encouragement is gratefully appreciated. We also thank April Kam who exhibited an ability to make sense out of confusing instructions in the preparation of the diagrams found in this book. We also extend our gratitude to Irving Louis Horowitz, Scott Bramson, and Dalia Buzin of Transaction Books for their assistance in making this book possible and especially for their patience . In works of this sort, the support provided by colleagues who willingly review crude manuscripts are paramount. Specifically, we wish to recognize and thank Dr. Dale Henkhaus (University of Wyoming), Dr. Roger Conway (U.S. Department of Commerce), Dr. Richard Andrews (University of New Hampshire), Dr. Ping Tsui (University of Nevada Reno), Dr. Jyh-yih George Hsu (University of Hawaii), and Dr. John Halloran (University of Hawaii). Chauncey T. K. Ching John F . Yanagida

xiii

1

INTRODUCTION Production economics involves the study of optimal procedures for creating goods and services optimal in an economic as opposed to a physical sense. It is perfectly general in applying to traditional production processes of agriculture as well as to the provision of services in today's economy (e.g., banking services). Further, the term optimal implies a particular objective function to be maximized or minimized and this in turn reflects a production environment characterized by limited or scarce resources. The above view of production economics is very similar to the general definition of economics as the study of the allocation of scarce resources among competing uses. This is not an accident and attests to the significance of production economics in the training of economists. Some of the more fundamental concepts of economics - e.g., the marginal principle, opportunity cost, valuation of scarce resources - are explicitly developed in production economics. It is for this reason that production economic concepts are often the foundation of other sub-disciplines of economics. Specifically, in courses under the headings of managerial economics, business economics and agricultural economics, one finds a generous use of production economic principles. Production economics is perhaps best distinguished from the theory of the firm (as found in microeconomic theory) in that it explicitly encompasses theoretical and applied aspects. Because of this theoretical and applied emphasis, production economic concepts and methods form the basis for thinking about and solving many problems. Obvious problems include those of businesses but can also include communities, regions, and nations. Further, much of the basic concepts and methods of production economics apply to the seemingly unrelated fields of 1

2

Pro duction Economics

resource economi cs and marketing . For exampl e, the notions of ex ternal! ties and p r oducer surplus have their origins in production economic theory. For emphasis note that production economics encompa sses not only the development of thes e concepts in a theoretical sense but also considerations of the ways i n which they can be applied in a problem-solvi ng empirical sense .

1-1.

Production Economic Theory Because the real world is exceptionally complex, much of economics is necessarily theoretical. Att emp ting to capture all of these complexities would make prediction and explanation of phenomena difficult and perhaps so unwieldy as to be of limited use. These theoretical developments are abstractions of the real world focusing on a minimum set of variables and probing relationships between these varia bles that, will permit explanation or prediction of the phenomena in question. Production economic theory is a subset of microeconomic theory. The development is based on two types of as;5umptions. First, there are assumptions about the environment in which the firm operates. Second, there are assumptions about the behavior of the firm. An example of the former assumption is that the firm operates as one of many firms in an indus try . Hence, its actions will have only neg1 ig i ble effects on the aggregate situation. Of course, this assumption can be and is often relaxed in more advanced studies. An example of the latter assumption will be that the firm seeks to maximize profits . This is an explicit behavioral assumption. Alternatives include maximization of sales or minimization of costs. The assumptions made in the development of production economic theory are critical in that they condition the final conclusions . In the c lassical sense, these assumptions are not themselves under scrutiny or tested in any way . They are taken as given. Based on these assumptions , the basic set of economic concepts and relationships is derived.

Introduction

3

These in turn are used to predict and/or explain behavior to the firm. They are conditional predictions or explanations which are valid only if the assumptions are maintained. Of course, alternative sets of assumptions could lead to alternative concepts and relationships and, ultimately, different predictions or explanations. 1-2.

Production Econoaic Applications As suggested in the preceding paragraphs, production economic theory will allow analysts to make sense about a problem. Theory necessarily abstracts from the complexities of the real world. This abstraction is based on assumptions about the phenomenon in question. From these assumptions we are able to derive basic relationships among the relevant set of variables. As such, production economic theory as well as other forms of economic theory are extremely useful to analysts in providing a format for thinking about problems. Production economics, however, goes beyond the theoretical aspects of problem definition and resolution. There are explicit attempts at implementation of theory and empirical resolution of problems. For example, a basic concept in production economics is the physical production function. While this particular phenomenon is theoretically not in the realm of the production economist (i.e., the production function is taken as given), the prodµc tion economist must more often than not statistically estimate these functions and investigate the specific relationships among explanatory variables. The production function is often simplified so as to make production economic theory more operational. One simplified version is referred to as linear production activities (versus a classical production function). This yields a linear system of equations which usually has more advantages than disadvantages. In this particular form, the implemen ted production economic theory (application) has implications not only for firms but more aggregate units such as regions and nations.

4

Production Economics

Role of Mathematics In developing the role of production economic concepts, we could utilize descriptive (verbal) and graphical approaches. While these are workable, they are cumbersome. Further in selecting the descriptive or graphical approaches, the analysis usually breaks down when more than three dimensions of a particular probl em are considered. Even with a three dimensional problem, the graphics are not especially easy to use . Accordingly, mathematics does provide a very useful analytical tool for deriving the basic results inherent in production economic theory . Mathematics in this sense provides an extremely useful analytical tool in deriving relationships as well as manipulating the variables to show different forms of the relationships. Further, with mathematical techniques, there are no limits to the number of dimensions of the problem that can be considered. It is just as easy to consider three dimensions as well as n dimensions. Note, however, that mathematics is simply a means to an end. The end in this case is an explicit definition of the relevant variables and how these variables a~e related to one another. If mathematics only provided ease of exposition of production economic concepts, it would perhaps be less useful or powerful a tool. Although concise and explicit, it does require some development and this development does have a cost. We believe that this cost is justified not only in making the exposition easier, but a lso because the application of production economic theory becomes explicit with mathematics. The production function is again a good example. While one can describe a production function graphically it becomes very difficult to apply production function analysis without explicit specification in mathematical sense. Further once the production function is specified mathematically, one can apply statistical methods for estimating the parameters of the production function. Given the parameters, one can use calculus to derive basic relations such as factor demand functions as well as 1-3.

Introduction

5

to solve explicit profit maximization or cost minimization problems. To extend the analysis further, when the produe tion function is simplified to a linear process function, there is another branch of mathematics known as linear algebra which then comes into effect. With this branch of mathematics and the assumed linear production process, one can construct operational models of production behavior. These form the basis for many applications to firms as well as more aggregate units. In summary, we believe the use of mathematics in this book is justified on the following grounds. First, mathematics does provide a convenient way to expose the basic concepts of production economics. Second, mathematics is viewed simply as a means to an end rather than an end in itself. Third, the empirical implications of mathematically derived product ion economic models clearly outweigh any of the "overhead" required in learning the necessary ma thematics. 1-4.

Content of Subsequent Chapters

The intent of this book is to provide a relatively self-contained presentation of the theoretical and applied aspects of production economics. In an attempt to provide a self-contained text, we have included relatively detailed discussions on the mathematics required for both the theoretical and applied aspects of production economics. Thus, the mathematics presented is more than a brief review but less than a complete dissertation on the topic. The aim is to have enough detail so that the reader will be operational with respect to both differential calculus and linear algebra. The book consists of five main sections. The first is the review of the calculus with emphasis on optimization with and without constraints. The second is a thorough review of classical production economic theory including time and space. The third section involves empirical production functions emphasizing some of the problems and techniques of

6

Production Economics

estimating production function parameters. The fourth section is a thorough review of linear algebra with emphasis on solutions to linear equation systems. The fifth section is a description of linear programming with specific reference to theoretical as well as applied aspects. A slightly more detailed discussion of the subsequent chapters in this book · follows. Mathematical Review. This chapter contains a review of the mathematics necessary for studying production economic theory. The review assumes that the reader has knowledge of algebra. The major emphasis of this chapter is on differential calcul us emphasizing the problems of optimization. Both single argument and multiple argument functions are considered. In addition, problems of constrained optimization including Lagrangian multipliers are discussed. , Examples are included throughout the discussion. Chapter 2:

Production Economic Theory. Classical production economic theory is developed utilizing ; differential calculus. Each of the three basic cases (one factor of production-one product, two variable factors of production-one product, and two productsone factor) are developed in detail. Further, a generalized production function is considered and each of the basic cases are derived. The appropriate cost functions, supply functions, and factor demand functions are also derived explicitly. Further, instances in which the firm operates under constraints are developed. The concept of duality is briefly introduced. The chapter concludes with the introduction of time and space into the basic production model. Chapter 3:

Empirical Production Functions. This chapter discusses the appropriate functional forms for biological production functions as well as aggrega te production functions. Methods of estimation and related problems are also discussed. Several data sets, some using biological data and others

Chapter 4:

Introduction

7

using aggregate data, are used to illustrate the estimation of alternative production functions. Further, certain nonlinear production functions such as the CES and VES are also considered. Finally, a brief review of computer algorithms currently available is presented. Chapter 5: Linear Algebra. This chapter contains a detailed review of the basic concepts of linear algebra. The review assumes that the reader has had little training in the study of linear algebra. Thus, the chapter begins with a discussion of vectors and matrices. The emphasis of the chapter is on linear equation systems and the conditions under which these systems can be solved. An essential part of the discussion involves the concept of rank of ma tr ix. The "rank conditions" for solving linear equations systems are developed. Linear Programaing. One of the operational a spec ts of production economics involves the mathematical technique of linear programming. Relying heavily upon the development of linear algebra, linear programming is viewed as the core of economics in particular production economics since it deals with "optimizing an objective function subject to a set of constraints." In this instance, the objective function is assumed to be linear and the constraints are also linear. The constraints, however, can be either strict equalities or inequalities. Linear programming in this context is viewed as an application of economic thinking (marginal principles) and solving linear equation systems with elementary row operations. The concept of transfer equations are developed in some detail. Further, the tie between linear programming and production economic theory, especially duality, is developed. Chapter 6:

Generality of Production Economics . In this concluding chapter, the theoretical and operational aspects of production economics are combined to address various topics and problems. In parti-

Chapter 7:

8

Production Econoaics

cular, frontier production functions, dynamic linear programming and spatial equilibrium models are discussed. Also examined is the contribution of production economics to building a methodological framework for analyzing resource utilization issues such as general equilibrium analysis, benefit cost analysis, and factor demand policy analysis.

2

HA'ffl.EMATICAL REVIEW The primary use of mathematics in production economics is to find extreme values of functions. For example, production economists are often interested in finding maximum values of certain functions (e.g., profit) or minimum values of other functions (e.g., cost). In addition to using mathematics to find such extreme values, production economists are also interested in studying the conditions under which economic optima (maxima or minima) hold, The primary mathematical tools used in the study of economics come from concepts of algebra, analytical geometry, and calculus. The concepts from these areas of mathematics are especially useful in studying the theoretical aspects of production economics. The more applied aspects of production economics also require a basic foundation in matrix or linear algebra. This mathematical review concentrates primarily upon calculus . It is assumed that the reader has had sufficient background in algebra to progress to the study of calculus. While linear algebra is important and determinants are used in investigating secondorder conditions, they will not be covered in detail in this review. Rather, the chapter on linear programming will be prefaced with a review of the basic aspects of linear algebra . Specifically in that section, matrix definitions and solution procedures for linear equation systems will be developed. There a re many texts that con ta in the ma thema ti cal tools that are useful in the study of economics. Any text used in the first and second year courses in calculus would be adequate. In some cases, however, concepts used by economists (e.g., Lagrange multipliers) are topics that come from the study of advanced calculus , In addition to the standard calculus text, there is a set of texts dealing specifically with mathematical economics, written by economists or mathematicians with interest in economics. The books 9

10

Production Economics

by Chiang, Baumol, Henderson and Quandt, Brennan, and Allen are recommended to the interested reader. More specific references to these and other texts in mathematical economics are listed at the end of this chapter. This mathematical review contains nine sections. The first section deals with some elementary concepts, such as variables, relations, and functions. The second section deals with limits. The third section deals with the concept of the derivative, a specific application of limits. The fourth section reviews the basic rules of differentiation. The fifth section deals with optimizing single argument functions. The sixth section discusses partial derivatives and optimizing multiple argument functions. The seventh section deals with the problem of cons trained optima. The eighth section discusses differentials and the rationale of second-order conditions. .. The ninth section introduces total derivatives. 2-1.

Soae Prel iainary Concepts

In economics as well as in other scientific disciplines, when we attempt to quantify our study, we must define "variables." Mathematically, a variable is defined as "an unspecified number or element of a given set of elements." Traditionally, variables are symbolized by a letter such as X, Y, or Z. The variable "X", for example, might be defined as the integers between O and 10. Thus, X would be an unspecified number al though the possible range of numbers that variable X can assume has been restricted. An economic example of a variable would be "Y" defined as the number of dollars of annual U.S. disposable income. There are two main classes of variables. The first are continuous variables in the sense that these variables can take on any real number on any specified interval on the real number spectrum. For example, a continuous variable might be the weight (W) of a particular person whose weight might fluctuate between 100 and 125 pounds. If this person were

Mathematical Review

11

weighed on a highly accurate scale, his weight could be expressed with se1Teral decimal point precision. The interval over which a continuous variable is defined is often depicted with the following types of inequality: 100 < W < 125. The second major class of variables is discrete variables. An example of a discrete variable would be the number of people in families found in the city of Reno. If we let this variable be depicted by the 1Tariable "Y", it would be depicted as: Y = 1, 2, ••. , 15. I t may be convenient to think of a continuous variable as one in which there are no gaps in the arrangement of all possible values of this variable listed in order of magnitude. In contrast, with a discrete or discontinuous variable, all values of this variable cannot be arranged in order of magnitude without distinct gaps. In summary, two points regarding variables have been made. First, a variable may be defined as "an unspecified number or element of a given set of elements." Second, there are two main classes of variables, continuous and discontinuous, or discrete. Generally speaking, we are interested not so much in variables by themselves, but in the relationships which may exist between variables. For example, in economics we are interested in the relationship be tween the amounts of a commodity that a consumer will purchase and the prices that the commodity can assume. In a strict mathematical sense, the most general kind of relationship between variables is called "relation." A relation exists be tween variables X and Y when (1) for a specific value of X, there is a specific value or set of values of Y; and, (2) for a specific value of Y, there is a specific value or set of values of X. Relations may be expressed in tabular form, graphical form, and equation form. For example, a relation that nearly everyone encounters periodically may be found at the Post Office (Table 2-1). A relation exists between the variables X (weight of letter or parcel to be mailed) and Y (cost

12

Production Economics Table 2-1.

Weight, X (Ounces) 0

1 2

3 4

O, a < 0 for a relative maximum; and, H1 , H2 , H3 > 0 3 for a relative minimum. Recall the multiple argument function stated previously: 2 2 Y = 400 - 3X1 - 4Xl - 2X1X2 - sx2 + 48X2 The Hessian determinant for this function at the critical value is: -6

-2

H=

-2 -10 Further the first and second principal minors are: Hl = I -6 I= -6 H = H = (-6) (-10) - (-2) (-2) = 56. 2 Since the H is less than zero and H is greater than 2 1 zero, the function exhibits a relative maximum at the critical values . Consider one other example of a multiple argument extreme value problem:

60

Production Economics

First-order conditions:

aayx =

24X 2 + 2X 1

1

2

- 6X

1

= 0

3Y ax= 2X + 2X 2 = O. 1

2

Solving simultaneously yields the critical values for 3Y x1 and x2 (since ax

= 0 is a

quadratic

equation

1

there are two sets of critical values):

XO = 0,

XO =

XI= 1/3,

Xl = -1 /3 . 2

1

2

o,

and 1

Second-order conditions are: f 11 = 48X 1 - 6 f22 = ~ f12

=

2

f21

=

2

and H = -16. 2

so H 2

< O,

2

2

2

there is a saddle point 10

2

2

2

Since H > 0 H = 10 and H = 16 . 1 2 1 1 1 O, there is a relative minimum at (X 1 , x2 ).

tha t

>

Since H 2

-6

and

Mathematical Review 2-7.

61

Constrained Optima

Thus far we have been discussing optimizing functions without placing restrictions upon the values that the independent variables can assume. These are often referred to as free maxima and minima or free optima problems. In economics as well as in other disciplines, there are often restrictions or constraints placed upon the values of the independent variables. Generally speaking these problems are referred to as constrained maxima or minima or constrained optima. The added conditions specifying the restriction upon the values of the independent variables are often referred to as side conditions, or simply constraints. Baumol provides a geographical analogy to the constrained optima problem. He considers the problem of finding the highest point (altitude measured from sea level) on earth. If the search for this highest point or peak is unrestricted, the highest point would be defined by the lines of latitude and longitude corresponding to the peak of Mt. Everest. This is a free maxima problem. If, however, there are constraints imposed upon the values of latitude and longitude that can be chosen, we are dealing with a constrained maxima problem. For example, the free maxima problem can be constrained by the condition that "we must remain within the continental United States." With such a constraint, two possible changes occur. First, the specification of longitude and latitude corresponding to the highest point on earth subject to the restriction may change. Second, the height (value of the function) corresponding to the maximum point or highest point may decrease. These two points are illustrated best by noting that if the side condition or constraint had been to "remain on the Asian continent," there would be no difference between the constrained maxima and the free maxima problem. However, when the restriction is to "remain in the continental United States," the specification of the independent variables (lines of longitude and latitude) as well as the magnitude of the function (elevation) will change.

62

Production Econoaics

Graphically, the difference between free optima and constrained optima is contained in Figure 2-17. In this case a surface defined by the function Y = f(X , x ) has an obvious free maximum at the peak of 1 2 the surface. If a specific relationship between the independent variables (x , x ) is specified, the 1 2 search for an optimum is restricted to a "slice" of the surface. Thus, when a side condition or cons train tis imposed, the maximum value of the dependent variable as well as the values of · the independent variables can be quite different from the free maxima case. Figure 2-17.

Constrained Optlaa

Free ma,tim~

y

0

Since economics deals with the allocation of scarce resources among competing uses, the concept of restrictions upon choice is important. Practically speaking, there are two main approaches to solving constrained optima problems: substitution and the method of Lagrange multipliers. Both of these techniques as well as the second-order or sufficient

Mat:hellatical Review

63

conditions for extreme values will now be discussed. Consider a situation where a firm produces a product and makes a profit defined by the function Y = 5x x • However, the firm 1 2 is not able to operate as far out on this profit function as it desires (this particular function has no maximum) but must live with:fm a budget restriction. Let the two independent variables correspond to inputs with the price of the first input (X 1) being $2 per unit and the price of the second input (X ) being $1 per unit. And, assume that the maximum 2 amount of operating capital at the disposal of the firm is exactly $100. Thus, the budget constraint on the amount of inputs that the firm can purchase is described by the equation (constraint or side condition): The Method of Substit:ution.

2X

1

+ x2 = 100.

Thus, the problem facing the firm is to maximize its profits subject to the budget constraint. The most obvious solution procedure is that of substitution. Specifically the budget constraint may be solved for x to yield x = 100 - 2X • This value 2 2 1 for x can be substituted into the function as 2 follows:

With these substitutions, the constrained maximum problem has been reduced to a free maximum problem in one independent variable (X ). The techniques 1 discussed earlier can simply be applied. The firstorder condition for maximizing the revised profit function is:

:i

Solving

= 500 - 20X 1

the

= O.

first-order

condition

yields x

1

= 25.

64

Production Economics

Knowing the value of

x1 , x2

to the budget constraint.

is computed by referring

In this instance, x

= 50. 2 Of course, second-order conditions should be checked:

-20 •since -20 is less than zero, the function exhibits a maximum at the critical values. The method of substitution is one way to solve constrained optima problems. In many instances, it is an acceptable method. However in some cases, the side condition or constraint may be so complicated that the method of substitution can be extremely laborious. Method of Lagrange Multipliers. The constrained optima problem may be stated generally as finding the extreme valu~ of Y = f(X , x ) subject to g(X , x ) = 1 2 1 2 O. The method credited to Lagrange involves forming an augmented function:

YA= f(~l,X2) + A[g(Xl, X2)]. Note the similarity between the original function Y and the augmented function YA. The augmented function is exactly the original function if the side condition or constraint is satisfied. In other words, YA will behave like Y if the constraint is satisfied . Recognizing this correspondence between the original and augmented function, Lagrange argued and proved that we could solve the constrained extreme value problem by: (1) forming the augmented function (called the Lagrangian function); (2) treating the unknown coefficient to the constraint as a variable; and (3) utilizing the calculus to solve the extreme value problem. In so doing, the constraint becomes part of the first-order conditions for optimization and the behavior of the original function [Y =

Ka theaa tical Review

65

f(X , x )] is not lost. Note, that in the specifica1 2 tion of the Lagrangian function, the constraint or side condition must be expressed in implicit form (i . e . , the cons training equation is rearranged so that one side is equal to zero) . Given the augmented function, the first-order conditions for optimizing the augmented function (where the independent variables- are x , x , and\) 2 1 is identical to that of optimizing a free function. More specifically, the first-order conditions are: 3Y\

-ax =

fl + Agl

=0

1

Referring to the example of the firm's profit function subject to a budget constraint, the Lagrangian function is:

YA= 5X X + \(2X + x2 - 100) 1 2 1 The first-order condition Lagrangian function is: aYA

for

optimizing

the

66

Production Econoaics

Solving the three ously yields:

first-order equations simultane-

x1 = 25 x2 = so A= -125 •

.This solution is consistent with the solution found by the method of substitution. In part, the method of Lagrange multipiiers is preferred to the method of substitution primarily because of the additional information gained from the solution. Specifically, the value of the Lagrange multiplier , has an economic interpretation. There is no counterpart to this variable in the method of substitution. Perhaps the best way to see how the Lagrange multiplier is interpreted is to reformulate the Lagrangian function as follows:

Note that in this specification, the constant term in the side condition or constraint has been expressed I as a variable. Thus in considering the first-order conditions for maximizing this Lagrangian function, there would be an additional first-order partial derivative:

This specification suggests that the negative of the calculated value of A corresponds to the change in the value of the dependent variable (from the constrained optima) as a result of a small (infinitesimal) change in the magnitude of the budget constraint. In other words, the negative of the magnitude of the calculated Lagrange multiplier represents the change in the profit function due to a, say $1.00, change in the budget constraint. Since the value of the Lagrange multipl ier calculated in this

Mathematical Review

67

problem is -125, this means that if the budget constraint were to increase by $1, the value of the profit function would increase by $125. This can be seen explicitly by recognizing that at the critical values for this cons trained optima problem, a $1 increase in budget permits an increase purchase of one unit x or one-half unit of x • 1 2 Consider the case of keeping x constant and evaluate 1 the impact of increasing x by one unit. The profit 2 function would be:

Accordingly, it follows that: Y + b. Y = 125 (X + t--X ) = 12SX + 125b.X2" 2 2 2 Finally, note that: 6Y If t--X

= 12SX2 +

125b.X - Y 2

= 125t--X 2 •

= 1, then b.Y = 125. 2 This verification of the interpretation of the Lagrangian multi plier could also have been carried out by assuming x constant and evaluating the impact 2 on the profit function of an increase (from its critical value) of x by one-half unit. 1 Thus far in our discussion of the method of Lagrangian multipliers, there has been no mention of second-order or sufficient conditions for an extreme value. These conditions are simply stated here without discussion of their rationale. The rationale for these conditions will be discussed after the concept of total differentials has been introduced in the next section. Given: YA= f(X , x ) +A[g(X , x2)J, the first2 1 1 order conditions for an extreme value are:

68

Production Economics ay A ax = f2 + )..g2 = 0 2 clY A

aI°

x2 ) =

= g(Xl'

O.

Second-order conditions, as in the free extreme value case, involve second-order partial derivatives expressed in the form of a determinant. In the constrained optima value case, this determinant is called the "bordered Hessian"--which is the Hessian surrounded by partial derivatives of the appropriate first-order condition with respect to the Lagrange multiplier. The second-order partial derivatives are:

a2Y A

- - = fll 2

ax

1 a2y

+ )..gll

= vll

.. A

=

ax ax 1 2

f12 + ).g12

= v12

a2Y

A axl dA

=

gl

=

v13

2 a Y A = f21 + ).g21 = v21 a x ax 2 1 2 a Y A 2 ax 2

=

f22 + )..g22

2 a Y A = g2 ax a). 2

=

v23

= v22

Ha thema tical Review

a 2y

A

nax

69

= gl = v31

1

2 cl y A = g2 = v32

clAclX 2

•

2 cl YA - - = 0 = v33 clA2 These second-order partials are arranged into the bordered Hessian as follows: H

vll

v12

v13

= v21

v22

v23

v31

v32

V33

Note that there are three rows and columns in H in this example. If there were three independent variables and one constraint, H would contain four rows and columns. Thus, H contains a row and column for each variable including the Lagrangian multiplier. If H > 0 and alternate in sign for higher order determinants (i.e., when there are more independent variables or constraints in the problem) the critical values exhibit a relative maximum. If H < 0 and less than zero for all higher order determinants, the critical values exhibit a relative minimum. Reconsider the constrained optima problem: YA= sxlx2 + A(2Xl + x2 - 100). The bordered Hessian is:

H=

0

5

5 2

0

2 1

1

0

= 20 > 0

= 25, x = 50, and 1 2 A= -125) exhibit a relative maximum.

Therefore, the critical values (X

70

Production Economics

Consider a final example of a constrained optima problem solved through the use of Lagrange multipliers:

YA= xlx2 + 2Xl + \(4Xl + 2X2 - 60) . First-order conditions are:

~YA

a'r

4Xl + 2X2 - 60

==

= o.

The critical values are:

=

Xl

8

,

x2 = 14 A

= -4

The bordered Hessian is: H=

0

1

1 4

0 2

4 2 0

= 16 > 0

Therefore, there is a relative maximum at the critical values. The discussion of Lagrangian multipliers has been in terms of a single restraint or side condition and two independent variables. The technique can be generalized to more independent variables as well as additional side conditions . For example, consider the three independent variable function:

Y

==

f(X , 1

x2 , x3 ).

The side conditions specifying the values that the independent variables must satisfy may be stated as:

Matheaatical Review

Given these, the Lagrangian equation is:

y

71

=

"

f(Xl, x2, X3) + "1[g(Xl, x2, X3)] + "2[h(Xl, x2,X3)]. The procedures required to solve this problem are comparable to those discussed for the two independent variables with one constraint case. In this particular example, there would be five first-order conditions--one for each of the three independent variables and one for each of the two constraints. 2-8. Differentials and Second-Order Conditions for Extreme Value Probleas

In

studying single argument functions of the dY form Y = f(X), we used the symbol dX to stand for the derivative of Y with

respect to

X, f'(X). While we dY were not explicit in referring to dX as a symbol, it

was interpreted as a total symbol and not dY ¾ dX. We will now relax this interpretation and consider dY and dX as separate en ti ties. Using the symbol "6." to represent small finite changes and the symbol "d" to represent infinitely small changes in variables, we can define the following identity: 6.Y

=

6.Y t.X t.X

If 6.Y and 6.X both approach zero, i.e., they become infinitely small, then:

~

i :! and +

6. Y "' dY, 6. X "' dX.

If this is the case, then: dY "' (

~i) dX = f

I (

X) dX.

The differential of Y is symbolized dY and the differential of Xis symbolized by dX. For example,

72

Production Econoaics

if Y

= dY

2 f(X) = x , the differential of Y is:

=

f'(X)dX

= 2XdX.

The concept of differentials applies to multiple argument as well as single argument functions. Specifically if Y = f(X , x ), then the differential of 2 1 Y would be defined as dY = f dx + f 2dx 2 • As in the 1 1 .concept of derivatives, there are higher order differentials. For example given the above function, the second-order differential of Y would be defined as d 2Y = d(dY) = d(f 1dx 1 + f 2dX 2 ). Keeping in mind that the differential simply represents a change in a particular variable, this concept plays an important role in extreme value problems--especially in providing a rationale for second-order conditions. In Figure 2-18A and 2-18B there are two three-dimensional surfaces. The first exhibits a maximum at point A and the second a minimum at point B. This means at point A everywhere in the immediate neighborhood, Y is smaller; and, at point B everywhere in the immediate neighborhood Y is larger. Being at a peak of a hill or at the bottom of a v~lley, the tangent lines TX drawn paral1

lel to the

OYX

plane have zero slope and go through

1 point A and through point B. Similarly tangent lines TX drawn parallel to the plane OYX2 have zero slope 2

and go through points A and B (Chiang, pp. 335-336). Referring to Figure 2-18A, for A to represent a point where Y reaches a maximum it is necessary that at least momentarily, Y be neither rising nor falling at that point. That is, for any changes from point A in either direction (the x or · x direction--dX 1 , 2 1 dX ), dY = f dx + f dx = O. Since dX and dX are 2 1 2 2 1 1 2 not necessarily equal to zero, the only certain way for dY = 0 is for f and f to be equal to zero. 2 1

Mathematical Review

Figure 2-18.

y

73

Three Dimensional Surfaces

(A)

y

(B)

This of course is consistent with the previous discussion of first-order conditions for an extreme value. One might also note that in Figure 2-18A, the tangent line TX is drawn parallel to the OYX 1 plane. 1

This implies that

we are

holding

x2 fixed at some

values . This is also consistent with the previous definition of the partial derivative of Y with respect to x . Similar comments hold for the tangent 1 line TX and the partial derivative of Y with respect 2

to point A. Again refer to Figure 2-18A and the maximum point A. Moving in either direction from this maximum point, Y must decrease. This is e·quivalent to saying dY < 0 since it has been previously established that dY = 0 at A, the condition dY < 0 means that dY is decreasing. An equivalent way of 2 stating this condition is d(dY) d Y < 0 . Similar conditions would hold when dealing· with a minimum. For example, at point Bin Figure 2-18B, the condi2 tion for a minimum is d Y > O. 2

and d Y

Production Econoaics

74

a(f dx

1

1

+ f dX ) 2 2

a x2

dX2 = ( f lldXl + f21dX2)dX1 + 2

. (f dx + f d~ )dX = f dX + f dX dX + 12 1 22 2 11 1 21 2 1 If f

Since

12

= f 21 , then d2Y =

dX

and dX2 can take on non-zero values, 1 Y will be greater or less than zero only if the second-order partial derivatives take on certain signs at the critical values of the independent variables. The logic underlying the signs and magnitudes of the second-order partials can be developed through the use of ordinary and linear algebra . Specifically, the concepts of quadratic forms and solving quadratic eq1,1at1ons can be used to isolate the sign and relative magnitudes of the second-order partials

i

in a2Y. A form may be defined as where each term is of uniform 3X + 2X is a linear form . 1 2 degree or quadratic would be

a polynomial expression degree. For example m = An example of a second q = 3X 2 + 4X x + x 2 • 1 1 2 2 Note that this quadratic form may be expressed in matrix form as: (3 2

In matrix notation, the quadratic form would be q X'AX, where: X

X = [

1

x2

]

A= (3 2

=

Mathellatical Review

75

Returning to the problem of second-order conditions for extreme values, note that the type of extreme value depends upon the sign of the second-order differential which can be expressed as a quadratic form:

The quadratic form is said to be positive definite if it is greater than zero and negative definite if it is less than zero. -Thus, second-order conditions for a maximum require that the second-order differential at the critical values be negative definite. And, second-order conditions for a minimum require that the second-order differential be positive definite. To investigate the definiteness of dY, a quadratic 2

form, redefined Y as follows: iY where, a

2 = q = au + 2buv +

CV

2

= fll

b

= f12 = f21

C

= f22

u

= dX 1

V

= dX 2

The investigation involves rearranging terms so that the magnitude of q is apparent (i.e., solving a quadratic equation): au au

2 2

+ 2buv + + 2buv

u2 + 2buv a

CV

2

=q

= q - CV

=

g - CV

a

2

2

76

Production Economics

Completing the square yields: 2 u

+

2buv a

+

2 2

~ = q - CV 2 a a

=q-

2 CV

a

2

2 2

b +--va

2

2 2

+ ~• a

2

·Multiplying each side of the equation by "a" yields : 2 2

bv 2 = q - cv2 + __ b v_. au+-) ( a a

Rearranging terms yields:

+ cv2 q = a(u + bv)2 a = a(u

a

2 b 2 b 2 + ~) + (c - -)v

a

a

These results show that the sign of q is independent of the sign and magnitude of u and v: q is greater than zero (positive definite--condition for 2 minimum) if a is greater than zero and ca - b is greater than zero; q is less than zero (negative definite--condition for a maximum) if a is less than 2 zero and ca - b is greater than zero. Translated into the original terms involving second-order partials, the condition for a minimum (positive definite quadratic form) is:

2 d Y

>0

if fll

>0

and f 11 f 22 - f 12

2

> 0.

And, the conditions for a maximum (negative definite quadratic form) is: , 2 d Y

o.

increasing,

we mean

1

slope of APX

is positive. 1

{f:;1']

dAPX 1

(3-18) Since x

1

dX

>0

=

dX

1

=

1

it follows that

( 3-19)

>

f' (X )

Thus MPX

is 1 increasing.

5.

Specifically,

greater

f(X ) 1

XI than

long as APX

1

1

1

When we say that APX

is negative. 1

1

is decreasing, we mean that the 1

slope of APX

is 1

< APX:

is decreasing, MPX

I f APX

as

APX

96

Production Economics

dAPX (3-20)

>

Since \

{ f::,']

1

=

dX 1 0 then

(3-21) Thus, MPX

dX

f' (X )

is


0

is

homogeneous of

2 ;1. x 2 = A2q.

=

instance,

for

if Xis

doubled (A= 2),

q in-

2

creased by A or 4. 2 Consider the function q = X + 2 for This is not a homogeneous function since,

(3-44)

(;1..X) 2 + 2

=

;1. 2x 2 + 2

=

by some constant A •

increase q

If A = 2, note that,

(3-45)' 22(X2 + _2)

2

< 22

(X + 2).

22

Consider other examples.

The function q

X 2 is homogeneous of degree 2 since, 1

(3-46) f(AX , ;1..x ) = (;\Xl >2 + o,x >2 1 2 2

== ;\2(X

x

Similarly,

the function q

of degree zero since,

> O.

;1. 2 (x2 + ~ ). A2

In this case, increasing X by A does not k

X

1

2 + X 2) 2

= X1 2 +

= ;\2q

2X

3 homogeneous = -x1 + 3X- is 2

1

(3-47) 0

A q.

Finally, note that a function homogeneous of degree one is said to be linearly homogeneous. However, it is not necessary that such functions be linear. For example, the following function is homogeneous of

Production F.coomaic Theory

103

degree 1 but it is not a linear function: 2 2 2X = _l_ + _3_ X2 Xl • X

(3-48)

The Stages of Production. In most discussions of the Factor-Product case, it is noted that the domain (input dimension) of the production function can be divided into three stages. The definition of these stages is based solely on technological grounds and can be defined prior to the addition of economic information (product or factor prices). Consider the single variable production function, ·

(3-49) where,

q

= F(V, F)

V = Variable factor of production F = Fixed factor of production

It will be noted in the next section that the restriction to functions homogeneous of degree one only affects the definition of boundaries between stages of production (Mundlak). Generally, for all homogeneous production functions, the stages of production may be defined as follows. In Stage I, the average products of the var-i able and the fixed factor are increasing; and, the marginal product of the variable factor is positive while the marginal product of the fixed factor is negative. In Stage II, the average product of the variable factor is decreasing but positive while the average product of the fixed factor is increasing and positive; and, the - marginal product of the variable and the fixed factors are positive. In Stage III, the average products of the variable and the fixed factors are decreasing; and, the marginal product of the variable factor is negative while that of the fixed factor is positive. Specifically, these stages of production for a production function homogeneous of degree one is illustrated in Figure 3-5. In defining these stages of production, it is useful to think of the ratio of the variable resource to the fixed resource. While in

104

Production Econoaics

any production situation, the fixed resource is a specific quantity, in our discussion it is most useful to think in terms of product or output obtainable from one unit of the fixed factor and variable amounts of the variable factor. It is in this sense that we are concerned with the ratio of the variable resource to the fixed resource. Leftwich has an excellent discussion of the stages of production on pages 144-154. Figure 3-5. Stages of Production for Production Function Hoaogeneous of Degree One Stage· I

Stage

II

Stage fil

Once the stages of production are defined, it is usually concluded that Stage I and Stage III are irrational areas of production. Stage I is irrational because both the average products of the variable

Production F.conoaic Theory

105

and fixed factors are increasing. This means that the technological efficiency of both factors is increasing and if production is to be undertaken at all, one should move through Stage I and at least to the boundary of Stage I and Stage II. In Stage I the ratio of variable to the fixed factor is too small-i.e., there is too much of the fixed factor relative to the variable factor. In S~ge III, the average product of the variable factor and the fixed factor are decreasing. This means for both factors additional use of the variable factor relative to fixed factor represents decreasing technological efficiency. In this stage, the ratio of variable to fixed factor is too large--i.e., there is too much of the variable factor relative to the fixed factor. Thus, the entrepreneur should not produce beyond the boundary of Stages I I and III. Thus, the rational stage of production is Stage II. If the variable factor has a positive price while the fixed factor has a zero price, the entrepreneur would produce at the boundary of Stage I and Stage II. In this way, the entrepreneur makes maximum use of the scarcer resource and produces where the variable resource ( the scarcer one) is ~t peak technological efficiency. On the other hand, if the variable resource had a zero price and the fixed resource had a positive price, the entrepreneur would produce at the boundary of Stages II and III. By so doing , the entrepreneur makes maximum use of the scarcer fixed resource and produces where the resource is at peak technological efficiency. Euler's Theorea. A useful theorem to study concerning homogeneous functions and production economics is credited to Euler. If, q = f(X , x2 ) is homogeneous 1 of degree k, Euler's Theorem states:

(3-50)

¾4x xl o 1

The proof of

+ ~ ax 2

Euler's

X2

=

kq.

Theorem is as

follows.

Since

106

Production Econoaics

the scalar, A , can be any real

number,

let A =

1 X . 2

Then,

Xl

(3-51)

F 0 11 2 12 1 2 22 1 Thus , H > 0 if and only i f: (3-103)

[f

f 2 - 2f f f + f f 2 ] 11 2 12 1 2 22 1

< O.

This is exactly the conclusion drawn when the method of substitution was discussed. Thus, in considering the optimal combination of variable factors or least-cost combination of variable factors problem both the substitution and Lagrange multiplier methods yield identical results.

Production Econmic Tbeoey

119

Specifically, the first-order or necessary condition for the optimal combination of variable factors is that the ratio of marginal products (RTS) is equal to the ratio of factor prices . The second-order or sufficient conditions are stated in equation (3-103). Convex Isoquants. It is logical for isoquants to be convex to the origin rather than f Onvex upward . If isoquants were convex upward (Figure 3-7A), this would mean that as additional units of x are substi1 tuted for x , larger and larger amounts of x are re2 2 placed by the additional units of x • The slope of 1 such an isoquant decreases and the negative of the slope, RTS, becomes larger as X is substituted for 1 x2 • In other words, it becomes easier and easier to

substitute x for x as increasing amounts of x are 1 2 1 used. In contrast, if isoquants were convex to the origin (Figure 3-7B), additional units of x substi1 tuted for x2 result in smaller and smaller reduction

in x2 . It becomes increasingly more difficult to substitute x for x2 • The slope of the isoquants in 1 this case increases and the negative of the slope (RTS) becomes smaller as x is substituted for x • 1 2 Since we logically expect the RTS to decline with increased substitution of x for x , we expect iso1 2 quants to be convex to the origin. For an isoquant to be convex to the origin or concave upward, its first derivative must be positively sloped (Figure 3-8). Mathematically, the isoquant is convex to the origin if: (3-104)

ct2x2

--2 dXl

> o.

120

Production Economics

Figure 3-7.

lsoquants, Convex Upward and Downward

(A)

Production F.cooomic Theory Figure 3-8.

d2X2 dX i2

121

Conditions for Isoquants that are Convex to the Origin

1 - - - - - - - - . . . ; . . .f" . ;(X) . . . ._ _ _ _ _ __

122

Production Economics

Given the production function in equation (3-82) and the total differential in equation (3-85), we know that

To determine the conditions under which convexity to the origin holds, we will initially ignore the negative sign in equation (3-105), take the differential ax,2 of dX, divide by ax , and multiply through by -1 : 1 1

f

(3-106)

= [

.,

dX d(-2) dXl

f

a

a c.-!.>

f2

= ax 1

f2fll - fl f21

2

f2

dXl +

ax2

] dXl + [

f2

dX2

f2f12 - flf22 2 ]dX2 . f2

(3-1 07}

Subs ti tu ting yields: (3-108)

ix2 dX 2

1

Multiply the

first term f2

equation (3-108) by (T): 2

on the

right hand

side of

Production F.coocaic Theory

123

(3-109)

2

2

- f21flf2 ....,...__;~---__;,~..;;;..__ - f12flf2 + f22fl = _fllf2 _;;;__;;;______

f 3 2

Recalling the negative sign in equation (3~105) and collecting terms: (3-110)

ix 2 dX 2 1

=

To be convex to the origin, equation (3-110) must be greater than zero. This will only occur if: (3-111)

[f

2 2 f - 2f f f + f f ] 11 2 12 1 2 22 1

< 0.

The condition expressed in equation (3-111) is exactly the second-order condition for the constrained output maximization problem discussed earlier. Thus, we conclude that if the second-order conditions for the constrained optima problem is satisfied, isoquants are convex to the origin. Summary of Optimal Combination of Factors Condition. First-order conditions for the constrained optima problem may be expressed in two equivalent ways. Alternative statements permit alternative economic interpretation of these conditions. First, firstorder conditions may be expressed as:

(3-112)

124

Production Economics

This means that resources or factors of production are combined in an optimal or least-cost manner if the ratio of marginal products equals the ratio of factor prices. Since these ratios correspond to slopes of isoquants and isocost (cost constraint) lines, resources are in an optimal combination when the slope of the isoquant is equal the slope of the isocost line--i.e., they are tangent to one another. Second, the first-order conditions may be written as: (3-113)

This condition states that resources are combined in a least cost manner if the contributions to output of the last dollar expended upon each input are equal and that these contributions equal 11.. Thus, " represents a change in output due to change in C, total cost: (3-114)-

dC =

ac ax d.Xl 1

+ ~ dX ax2 2

= rldXl + r2dX2.

Since, (3-115~

fl rl = - and

(3-116)

f2 r2 = -

"'

",

then

(3-117)

It has been previously established that:

Thus, (3-119)

Production Econoaf.c Theory

125

An isocline is defined as the locus of points on a production surface having equal slopes:

Isoclines and the Expansion Path.

(3-120)

dX2 -=L dXl

Where, L = a constant. Thus, for any value of L, there is an isocline. If L is defined as the ratio of factor prices, this is a special case of an isocline called an expansion path. Specifically, an expansion path is defined as the locus of points on the production surface with slopes equal to the ratio of factor prices: (3-121)

dX2 -=L dXl

=

rl

Graphically, an expansion path consists of the locus of tangency points between the isoquants and isocost lines (Figure 3-9). Figure 3-9.

Expansion Path

/

x,

/ Expansion path

126

Production Econoaics

Mathematically, we express the expansion path as: (3-122)

g(X , x ) = O; or, 1 2

(3-123)

X2 = g(Xl, rl, r2).

An alternative forniulation of the optimal combination of factors of production problem is to minimize the cost of production subject to the condition that the firm produces a specific level of output, q:

Constrained Cost Min:hdzatlon.

0

(3-124)

Minimize CA

= r 1x1 + r 2x2 + FC + A[q 0 First-order are:

f(X , X2 )] 1 minimizing this function

conditions for

a c, = ax1

____a

(3-125)

a CA

-

' ax2

= r

r

1

- 11.f

1

-

= 0

2 -Af 2 = 0

Second-order conditions are: - f (3-126)

H

=

1

- f2

< o.

0

The basic conclusions are the same as those drawn in the constrained output maximization case. Specifically, resources or factor of production are combined in an optimal manner (least-cost) when,

Production Ecoomic Theoey

(3-127)

127

or,

(3-128) The second part of the two variable factor analysis is to determine the optimal level of use (as opposed to combination) of the variable factors of production. It has previously been shown that resources are combined in an optimal manner if the resource combination is on the expansion path. The question now asked is "how far out on the expansion path should production occur?" To solve this problem it is assumed that the producer behaves so as to maximize profits:

Profit Maximization.

(3-129)

Max n

=

pq - rlxl - r2X2 - FC

= pf(Xl' X2) - rlxl - r2X2 - FC,

where p

=

price of product q

First-order conditions for profit maximization are:

-21!... clX 1

= pf

1

- r

1

= 0

(3-130)

Second-order conditions require; (3-131)

> o.

pf< 0 and H = p f21

f22

The two equations in (3-130) must be solved simultaneously to yield the levels of x1 and x 2 which will

maximize

profits.

Second-order

conditions

128

Production Econoaics

evaluated at these levels of factors of production must satisfy the signs specified in (3-131). Generally, the first-order conditions imply that resources or factors of production are used at an optimal level if each resource is used to the point where the value of the marginal product of each factor is exactly equal to the price of that factor:

(3-132)

Where, VMPX th i i:- factor.

=

Value of the

marginal product of the

Note that the optimal combination of factors problem is also satisfied if the equations in (3-132) are satisfied since solving simultaneously yields: (3-133) Example:

'fwd Variable Factors (Factor-Factor}.

Con-

sider the production function!/ (3-134)

q = 2(ln x ) + 4(1n X ). 1 2

Assume conditions of perfect competition so that prices of the product and factors of production are given: p = price of product (q) = $12 / unit r = price of factor x = $4/unit 1 1 r = price of factor x = $12/unit 2 2 As previously noted, the optimal or least-cost combination of factors of production can be handled in two major ways--substitution and use of Lagrange multipliers. In either case, assume that fixed costs are $20 and that the entrepreneur wishes to spend

Production F.coaaaic Theory

exactly $80 for production. (3-135)

129

Thus,

co= 80 = rlxl + r2X2 + FC.

The method of substitution involves solving equation (3-135) for x (or x ) and substituting the 1 2 results into equation (3-134): (3-136)

X = 80 - FC _ r2 1 r1 r 1

i

2

(3-137) Output (q) is maximized by (3-137) with respect to x 2 tive equal to zero: (3-138)

s= dX2

(3-139)

x2

differentiating equation and setting this deriva2

(80-FC - -r2 X) r 2 rl 1

r2

4 x2

(- - ) + - =

rl

o.

= 4(80 - FC) 6r2

Second-order conditions require: (3-140)

£9.= dX 2 2

Substituting factor prices and fixed costs into equation (3-139) and (3-140) yields:

x2 = 3.33

a2

~ dX

2

2

= -1.08 < o.

130

Production Economics

Utilizing equation (3-136) yields:

x1 = 5. Thus, output is maximized subject to a total cost (outlay) constraint of $80 when x = 5 and x2 = 3.33. 1 The level of output corresponding to this combination of inputs is 8.03 units. An alternative way to solve the same problem is to maximize the Lagrangian function:

(3-141) qA = 2 ln

x1+

4 ln

x2+

A(C 0

r 1x1 - r 2 x2 - FC).

-

First-order conditions are:

(3-1425

-=C

en

o

- r X - r X - FC 1 1

2 2

=0

Subs ti tu ting prices and cos ts and solving the equations in (3-142) simultaneously yields: x1 = 5, x2 = 1

3.33, and A= To"• . Second-order conditions require: 2 - ~2

0

-

rl

4 - ~2

-

r2

1 H

0

=

2

-

rl

- r2

0

Production Ecoooaic Theory

2 25 =

- 4

0

4 - -11.1 - 12

0

- 4

131

12

= 17.28

> o.

0

Thus, output is maximized subj~ct to the cost constraint when x = 5 and x = 3.33. This is consis1 2 tent with case solved by substitution. Note that with the Lagrange multipliers method there is an additional variable, A, and it has an economic interpretation. In this instance, X may be interpreted as the change in output if the level of the cost constraint were to change by one unit. Specifically, (3-143)

dq = fl dXl + f 2 dX 2

(3-144)

dC = r dX + r dX • 2 2 1 1

From the first two equations shown in (3-95) or (3-142) we know that

t

(3-145)

1

= r X

1

(3-146) Substituting equations (3-143) yields: (3-147) dq

=

(3-145)

r XdX + r AdX 2 2 1 1

and

(3-146)

into

= A(r1dX1 + r 2 dX 2 ).

Dividing equation (3-147) by equation (3-144) yields: (3-148) In

this

d

2.9.. dC --

\(rldXl + r2dX2)

-----~=A. ( r dX + r dX )

instance, a

1

$1

1

2

2

increase

results in an i ncrease of output of

in expenditures 1

10 of a unit.

The Lagrange multiplier approach permits a clear derivation of the expansion pa~h. S~lving the first

132

Production Econoaics

two equations of (3-142) yields:

(3-149)

2 xl rl -4- = r 2

x2

Substituting terms of

factor

x yields 1

prices the

and

for x in 2 the expansion

solving

equation for

pa th: (3-150) This means that given the factor prices, the variable factors of production must be combined in such a manner so as to satisfy this equation. The solution to the constrained output maximization problem (X = 1 5 and x = 3.33) satisfies this equation. 2

alternative way of viewing the optimal combination of ~esources problem is to minimize costs subject to the condition that the output level be constant, say, 9 units. More specifically, An

(3-151) 1 Minimize CA=

4X + 12X + 20 + A[9 - 2(ln 2 1

x1 )

- 4(ln

x2)] .

First-order conditions are:

(3-152)

acA

aT°

= 9 - 2(ln

x1 )

- 4(ln

x2 ) = 0

Solving the first two equations of (3-152) yields:

Production Kcooaaic Theoey

133

(3-153) (3-154) These are alternative statements of the expansion path equation. • Solving all three equations in (3-152) simultaneously yields:

= 5.9 X2 = 3.9 A = 11. 7 Xl

(3-155)

Second-order condition require: 2A ~2 1

(3-156)

H

=

4A ~2 2

0

4 x2

4 x2

2 Xl

=

2 Xl

0

0

0.68

0

- 0.34

0

3.07

- 1.02

- 0.34

- 1.02

=

- 1.06

< o.

0

Thus, the critical values in (3-155) indicate the levels of x , x2 , and A which will minimize cost sub1 ject to the output constraint. The value for A = 11 • 7 reflects the change in cost should output ( the constraint) be increased by one unit. As noted previously, the second part to the two variable factor problem is determining levels of the factors which will maximize profits. Alternatively, the problem is one of determining how far out on the expansion path an entrepreneur should be operating.

134

Production Econoaics

In addition to the information previously given, we specify the price of the product at $12 per unit. The profit function is:

(3-157) n = 12[2(ln X ) + 4(1n X )] - 4X

1

2

1

- 12X - 20. 2

First-order conditions for profit maximization are:

an ax1

12 0.

With the above equations, we can define the long-run cost function as the minimum cost of producing each output level if size of plant is variable. More specifically, we say that the long-run total cost function (LRTC) is the envelope of the short-run total cost curve (Figure 3-15) . Note that in this figure, k(l) < k(2) < k(3). The long-run average cost function (LRAC) is defined as the long-run total cost function divided by output. Alternatively, we say that the long-run average cost function is simply that envelope of the short-run average cost functions . The long-run marginal cost function (LRMC) is defined as the derivative of the long-run total cost function with respect to output (q) . The long-run marginal cost function, however, is not the envelope of the short-run marginal cost functions. Rather, we

156

Production Economics

say that the long-run marginal cost function is the locus of points on the short-run marginal cost functions corresponding to optimum plant sizes. This is apparent by inspecting Figure 3-15. Here the envelope to the short-run total cost functions corresponds to the long-run total cost function. And, the slope of the long-run total cost function is the long-run marginal cost function. Since the envelope is tangent to (i.e., equal slope) the short-run total cost functions , the long-run marginal cost function corresponds to those points on the short-run total cost functions that just touch the long-run marginal cost function. Figure 3-15. Long-run Total Cost Function as an Envelope to the Shor~run Cost Functions

$ 7

,_. ::;.,- 4 - _ En v elope LRTC

q

And since the long-run total cost function describes the minimum cost of production, each point on the long-run total cost function corresponds to an

Production F.coomic Theory

157

optimum plant size. Mathematically, we derive the long-run total cost function by starting with equations (3-233), (3234), and (3-235). Generally, two of these equations can be used to eliminate x and x to yield: 1 2 (3-237)

C = • (q, k) + '1-'(k).

This is an equation which descr ibes a family of short-run total cost functions. Thus, to define the long-run total cost function, we need simply define the envelope to this family of short-run total cost curves. The envelope of this family of curves is a function with the property that it is tangent to each member of the family. The equation for the envelope curve is secured by taking the partial derivative of equation (3-237), with respect to k, and eliminating k from the two equations. Specifically, we write the family of short-run total cost curves in implicit form: (3-238)

C - .(q, k) - '1-'(k)

= 0.

More generally, we can write the equation as: (3-239)

G(C, q, k) = 0

To find the envelope or long-run total cost curve, we set the partial derivative with respect to k of equation (3-238) or (3-239) equal to zero; (3-240) Since there are three variables (C, q,_ k) in equations ( 3-239) and (3-240), we can usually eliminate k from these equations and express C as some function of q: (3-241)

C

=

(q) .

This is the general designation of the long-run total cost function. It is useful to consider a specific example of a long-run total cost function derivation. This example is taken from Henderson and Quandt ( 1980, pp. 91-92). The equation for the family of short-run

158

Production Economics

total cost curves is: (3-242) C - 0.04q

3

2 2 - 0.9q + (11 - k)q + Sk = O.

The partial derivative with respect to k of this equation is: (3-243) Setting this partial derivative equal to zero and solving fork yields: (3-244)

k = O.lq.

Substituting equation (3-244) into equation (3-242) yields the long-run total cost function: 2 3 (3-245) C = 0.04q - 0.9Sq + llq. Given the long-run total cost function derived above, the entrepreneur is assumed to maximize profits in the long-run. The analysis is very similar to that of the short-run optimization. Specifically, in the e:>Eample in question, the profit function is defined as: (3-246)

rr

= 6q - 0.04q 3 + 0.9Sq 2 - llq.

In this ca;se, the price of the product or output is $6. To maximize this function, we take the first derivative with respect to q and equate it zero: (3-247)

drr

2

dq = 6 - .12q + 1.9q - 11 = O.

Second-order conditions require that: (3-248)

a27T = -.24q + 1.9 < o.

-2 dq

Solving the first-order condition, which is a quadratic equation, yields q = 3.3 and q = 12.S. Substituting these critical values into the secondorder condition shows that when q is equal to 3.3 the second derivative is equal to 1.108 which is greater than zero; and, substituting q = 12.5 yields a value of the second derivative equal to -1.1, which is less

Production F.conanic Theoey

159

than zero . Thus, profits maximized when q = 12.s . Given q and equation (3-244), scale of plant is equal to 1.25. 3-6.

The Supply Function Knowledge of the cost function permits an alternative determination of profit maximization , one in which the decision variable 1$ output rather than input. An alternative and closely related use of cost functions is the derivation of the firm's supply function . Such a function is defined as the relationship between price of the produce (output) and the quantity of that output that the producer will offer for sale. However, it is not just any pricequantity relationship but is one which describes the level of output which will maximize profits, once price is given. Clearly, such a relationship must be derived from conditions of profit maximization. Specifically,

(3-249)

TI

= pq - h(q) - FC

First- and second-order conditions for profit maximization are: (3-250) (3-251)

dTI

dq a2TI

di

= p -

h' (q) = 0

= - h"(q)

< O;

or, h"(q)

> O.

The supply function is derived from equation (3- 250) . Since h'(q) is a function of q, ·it can be solved for q as a function p: (3-252)

q = f(p).

Since equation (3-252) was derived from the firstorder condition for profit maximization it necessarily depicts the level of output which will maximize profits--given price. This assumes second-order conditions are satisfied. If the first-order condition for profit maximization is plotted for specific values of p, we

160

Production Economics

would simply plot the output:--price combinations that coincide exactly with the marginal cost function . Thus, the supply function is nothing more than the marginal cost function. However, the entire mar ginal cost function is not the supply function for two reasons. First, second-order conditions require that the entrepreneur operate on the upward sloping portion of the marginal cost function. Thus , the supply function is restricted to the upward sloping portion of the marginal cost function. Second, the entrepreneur need not lose more than his fixed costs. The entrepreneur will not operate at a level of output (even if it is on the upward sloping part of the marginal cost function) if total returns are such that he will not at least cover his variable costs . If he cannot generate enough revenue to pay his variable costs, this would mean that he has lost some portion of variable costs as well as fixed costs. If he did nothing (output equal zero), all he would lose is his fixed costs. Thus , the relevant portion of the supply function is the upward sloping portion of the marginal cost curve above the average variable cost curve. The entrepreneur would be indifferent as to production at the point where marginal cost equals average variable cost. As an illustration of the supply function , consider the cost function derived for the two variable factors of production example. In this instance, the profit function is: (3-253)

rr

= pq - 12 exp[q - 4 ~n( 2 / 3 )] - FC,

The first-order condition for profit maximization is: (3-254)

dn = p _ (g) exp[q - 4 ln(2/3)] = O. dq 6 6

Solving for q as a supply function: (3-255)

function of

price

yields the

q = 6 ln(}) + 4 ln(~) •

Finally, note that the supply function as de-

Production Econanic Theory

161

picted is a simplification in terms of variables. A number of factors have been assumed to be given. For instance, the production function (technology) and factor prices are assumed constant in the specification of the supply function. Thus, if these change, there will be a change ( shift) in the supply function. If some of these other variables had been left in their general form (e.g., instead of specific prices for the variable factors 'they could have been specified as r 1 and r ), the supply function deriva2 tion would have been more general: (3-256)

With such a specification, the effect of changes in factor prices on supply can be noted explicitly. 3-7. Joint Products (Two Prodocts--One Resource or Factor of Production)

In the case of joint products, or as it is sometimes called, the product-product case, we are concerned with production processes that yield more than a single output or product. An example would be wool and mutton from the production of sheep. Another example would be two crops such as barley and corn that could be produced from a single acre of land. In other words, we are concerned with products that are technically interdependent. If these products are produced only in fixed proportions so there is no choice in varying the combination of the two outputs or products, we simply treat the two prod~c ts as a single product in a specific combination and apply the single product analysis to the situation. In other words, in the joint products case, we ignore all production situations in which the products are technically independent. More specifically, consider production function satisfying the joint products requirement. In implicit form, the production function is:

The Production Function. a

162

Production Economics (3- 257)

H(q , q , X) = 0 1 2

In this instance,

q and q are joint or technically 1 2 interdependent products and Xis a resource or factor of production. If a specific level of resource were considered (i.e ., X would be set at a constant value), the production of the joint products could be shown graphically as in Figure 3-16. In this instance, q might correspond to beef cattle on the 1 range and q to wildlife such as deer with X being a 2 fixed quantity of grazing land. The graphical depiction of the joint products production function is very similar to the concept of isoquants described earlier. In the case of joint products, the function shown in Figure 3-16 is referred to as the production possibilities curve, the product transformation curve, or sometimes the isocost curve where cost of production is measured in terms of the physical resource, X:. Figure 3-16. Joint Products Production Function for Three Resource Levels

Production F.coooaic Theory

163

The graphical depiction of the production function is one way of reducing a three-dimensional space to two dimensions. It should be noted that in the case depicted graphically, x< 3 ) is larger than x< 2 ); 2 and x< > is greater than X(l). The equation for the product transformation curve is an explicit function of q 1 and q where the resource ~ level is held con2 o

stant at X :

(3-258) The Rate of Production Transfomation {RPT). The rate at which q must be sacrificed to obtain more of q 2 1 (or vice versa) without varying the amount of the resource, X, used is called the rate of product transformation (RPT). The negative of the slope of the line tangent to a point on the product transformation curve is the rate of product transformation:

(3-259 )

dq2 RPT=--.

dql

(3-260) 0

For a specific level of X, x

,

dX0

= O. Thus,

(3-261) Solving equation (3-261) yields: (3-262)

dq2 - dql

=

bl h2 •

Equation (3-262) indicates that the negative of the slope of the production transformation curve is equal to the ratio of first partial derivatives of the explicit form of the production function. Since,

164

Production

Econo■ics

ax

(3-263)

and bl =clq1 '

(3-264)

h2 =a' 42

cl X

hl is often referred to as the marginal cost of q1 measured in terms of X; and, is the marginal cost

~

measured in terms o°f X. In other words, these 2 "marginal cost" interpretations of these first partial derivatives are valid if costs are measured in physical quantities of the resource in question. For this reason, it is sometimes stated that the t:ate of product transformation is equal to the ratio of marginal costs (measured in physical terms) of proproducing q and q , respectively. 1 2 An alternative way of viewing the rate of product transformation is to note that: of q

(3-265) Here, MPX*

1 is the marginal product of the resource, ql X, used in the production of q • Using the defini1 tion of h 1 in equation (3-263), we note that:

(3-266)

Thus, the rate of product transformation may be rewritten in terms of the marginal products:

(3-267)

dq2

hl

RPT = - - - = - = dql ~

1 h2 -1 hl

42 X =-

ql

X

MPX* =

MPX*

42 ql

Production F.coooalc Theory

165

Thus, we say that the rate of product transformation is also equal to the ratio of the marginal product of X in producing q 2 to the marginal product of X in producing q • 1 In analyzing the joint products case, we proceed as in the case of two variable factors. The overall problem may 'be decomposed into two subproblems that are very much related. First, we investigate the optimal combination of products where we define the way in which the two products are combined in a revenue maximizing manner. Second, we investigate the optimal · levels of output of. the two products in question. Again, as in the two variable factor case, both problems are solved simultaneously when profits are maximized.

Analytical Procedure.

Since there are two products, the revenue function is:

Cons trained Revenue Haxiaiza tion.

(3-268)

where, R in this case corresponds to revenue; and, p1 and Pz are prices for the two products, respectively. It is often desirable to depict the revenue function on the production possibilities curve diagram. To do so, we must hold revenue constant (R and express q as a function of prices and revenue: 2

0 )

(3-269)

Since it is possible to hold revenue at any level, there is a family of functions corresponding to equation (3-268), Each of these functions exhibits the various combinations of q and q that will yield a 1 2 Thus, they are called specific level of revenue. isorevenue lines. Three such isorevenue lines R(l), (2) (3) R , and R are depicted in Figure 3-16.

166

Production Econoaics

One way of defining the optimal combination of products is to maximize revenue subject to the condition that the producer completely exhaust the resource, when the resource is held at a fixed level. This is known as the constrained revenue maximization problem. Specifically, maximize: (3-270) First-order conditions for revenue subject to the constraint are: 3R

11

a ql =

P1 - "hl

maximization

=0

(3-271)

Solving the first two equations of the system (3-271) yields: (3-272) This means that the products are combined in an optimal manner when the ratio of the marginal cos ts (measured in terms of the physical resource, X) is equal to the ratio of product prices. Or, products are combined in an optimal manner when the slope of the product transformation curve is exactly equal to that of the isorevenue line. Utilizing the relation that the inverse of the first - partial derivatives of the production function, h and h , are marginal pro1 2 ducts, we can write the condition for an optimal combination of products as:

Production F.conanic Theory

(3-273)

1 h2 -1-

aq2

hl

=

h2

ii

=

ax aql ax

MPX*

=

MPX*

q2

167

=

ql

. -P1 P2

Note that when we describe the optimal combination of outputs, we simply mean the manner in which the two outputs are combined sol' that when the resource is completely exhausted, revenue will be at a maximum. This is not a very general statement of the product-product problem, since the level of output of q and q is defined for only one level of resource 1 2 use. If there are other levels of resource ·use to be considered, there would be different levels of output produced . However, first-order conditions for this constrained revenue maximization problem define the relationship that must exist when outputs are combined in an optimal manner. If the first two equations of equation system (3-271) are solved for A and equated, we have: a 41

(3- 274)

A

aq2

= ax P1 = ax P2•

Thus, A corresponds to the value of the marginal products of X in producing q and q when products are 2 1 combined in an optimal manner, i.e., when first-order conditions are satisfied , An alternative interpretation of A is that it corresponds to a change in revenue if there were an additional unit of resource X available, · holding prices constant. This can be seen more specifically as follows. The differential of the revenue function is: (3-275)

dR =

p

1

dq + p dq • 2 2 1

The differential of the production function is: (3-276)

168

Production Econoaics

Since,

(3-277)

pl bl =>..-

(3-278)

P2 h2 = >..-

'

The differential of the production function may be written as :

Dividing the differential of the revenue function by the differential of the production function yields:

(3-280)

dR

ax =

(pldql + p2dq2) = >.. •

1(pldql + p2dq2)

Thus, the Lagrange multiplier,>.., is interpreted as the change tn revenue if the resource level, X, were to change. Second-order conditions for constrained revenue maximization require that:

-h (3-281)

H =

1

-h

2

> o.

0

In equation form, this second-order condition translates to:

( 3-282) Since>.. corresponds to the value of the marginal products of X in producing q 1 and q 2 , >.. must be positive for relevant production levels . Thus, the second-order conditions for revenue maximization subject to the constraint that the entrepreneur operates on the production function with a specific level

Product.loo. Ecooalic 'l'beory

169

of resource Xis satisfied if the term in brackets in equation (3-282) is positive. The importance of second-order conditions is apparent when one considers the shape of the production transformation or production possibilities curve. As depicted in Figure 3-16, it is convex upward or concave to the origin. Mathematically, this requires that: 2

(3-283)

d q2 --2

< o.

dql It can be shown that the sign in (3-283) depends upon the second-order condition for revenue maximization subject to the resource restraint in the production function, the term in brackets in equation (3-282). Given the differential of the production function, equation (3-261), we can rearrange terms to yield: (3-284) Since both q and q are variables in the equation 1 2 being considered, we must take the total derivative of equation (3-284):

( 3-285)

2 d q2 --2 = dql

dq d(-2) dql dql

=

hl d(- - ) h2 dql

More specifically, the total derivative is:

(3-286)

170

Production Economics

=

=

hl ~1 - 1½hu

+

hz2 hl h21 - hzhu

[

hz

2 -bl h22 + hl h2h12

+

h 2 2

hl h1~2 - h2hl2 ] [- - ] 2 h2

[

3

]

h2

h 3 2

= -

1 h 3

[ hll h2

2

2 - 2bl 2 hl h2 + h2 2hl ] •

2

Since , h_ = dX -L dq

= -dq1- '

2

1½ must be greater than O for

2

dX

any reasonabte

production

situation.

Therefore, we

2

d 42

2

hz

2

conclude that - + h22 h 1 ] 2 < 0 if [hll1½ - 2h12h1 dql is positive. And, it is positive if the second-order conditions for the constrained revenue maximization problem are satisfied. Profit Maxiaization. The second phase of the joint products problem is to determine the level of output when there is no constraint on the resource in question. Although the optimal combination of products problem is still important, it is subsumed under the general analysis to follow. Specifically, assume that the entrepreneur wishes to maximize profits where profits are defined as:

Production F..conmic Theory

(3-287)

171

= pl ql + p2q2 - rX

1T

= plql + p2q2 - r[h(ql' ql)]. The first-order conditions for profit maximization are: 81T

aq1

= p

1

-

rh = 0 1

•

(3-288)

Second-order conditions require that: ,i21T

(3-289)

H1 = aql 2 = -rh11

o.

(3-290)

Of interest are the first-order conditions expressed in equation system (3-288). Dividing the first equation by the second equation yields the previously derived condition for the optimal combination of outputs: (3-291) The optimal level of output is determined by solving the first-order condition equations simultaneously: P2

(3-292) dql

1

Since, h

=h 1

= - - and -

.

dq2 = dX.,

dX. h2 optimal level of output is: 1

2

the conditions for the

172

Production Economics

(3-293) This condition, also derived in the one variable factor one product case, states that profits are maximized when the value of the marginal product is equal to factor price. More specifically, the value of the marginal product of X in producing q must be 1 equal to the value of the marginal product of X in producing q2 ; and, each must be equal to the price of the factor. For a specific example, consider the production function: (3-294)

3q 1

2

+ 9q 2

2

- 3X

Let p 1 = $10, p 2= $12, and r tion is:

= 10q1 +

(3-295 ),- n

=

=

0.

$1.

The profit func-

12q - lX. 2

Solving equation (3-294) for X and substituting into equation (3-295) , yields: (3-296) n First-order conditions for profit maximization are:

..£2I. = 10 - 2q = 0 aq

(3-297)

1

1

11L = 12 - 6q2 = aq2

Solving

simultaneously

yields

o. q

1

=

5

and

Second-order conditions are satisfied since: (3-298)

H1

a2n

=-

aql

2

= -2

( 0.

q 2

= 2.

Production F.cooolic Theory

(3-299)

-2

0

0

-6

Thus, profits are maximized when q 3-8.

The General Case of

■

= 12

1

=5

173

> o. and q

2

= 2.

Inputs and n Products

In this section on the theory df the firm, we discuss a situation in which there are many inputs (m) and many products (n). We assume that there is a production function tha.t relates these m inputs and n products. This case is discussed primarily to present a summary of the main results from production theory. Specifically, the results from . the one variable-one product case, the two variable factor case, and the two products or joint products case are derived. I■plicit Function Rule. The development of the general case requires differentiation of an implicit function. Given the implicit function f(X , x , ••• , 1 2 X ) = O, the total differential is:

The

n

(3-300)

af af dX = ax dXl + ax dX2 + ••• + 1!. ax n 1 2 n

o.

Dividing by dXi yields: (3-301)

af ax

dXl dxi

af ax2

--+1

dX2 dX + i

af + ax

+ ••• i

+ ••• Setting all differentials except dXi and dXj equal to zero results in: (3-302)

·,af 1!_ dXj _ axi + ax dX - 0 · j i

Assuming all differentials other than dXi and dXj are

174

Production Econoaics

equal to zero cause

dX ,ii to

become

5_

aX •

i

Rearranging yields:

elf

=

(3-303)

i

-

axi elf

axj tells

. Thus, the implicit function rule f(X , x , ••. , Xn) = O, 2 1

us

that

if

(3-304)

Consider the example: F(ql, qz, q3, q4) = ql To

find

clq2

a''

that:

ql

(3-305)

+ q2

2

+ q3 + q4 = O

the implicit function

clq2 I

2

tells us

clF aql clF clq2

=

-

clF clql

=

2ql

clF clq2

=

clql

rule

where, and

(3-306)

(3-307)

clq2 clql

2q2' so that

2ql

=

-2q2 = clq4

Similarly, is:

to find

clq3 '

ql

4z

the implicit function rule

Production F.conanic Theory

175

(3-308) where, and

1

(3-309) clF clq4 (3-310)

=

clq4

1, so that,

= -1.

clq3

The Generalized Production Function. In the general case of m products and n variable factors, the production function in implicit form is:

(3-311) F(q1 , q 2 , ••• , qm'

x1 , x2 ,

••• , Xn)

= 0.

This function relates them products to then factors of production and is stated in implicit form so that the analyst need not identify a dependent variable. This function is primarily a conceptual one since it would be most difficult to find an empirical counterpart. Given the production function in equation (3-311), the profit function is defined as total revenues less total costs: Profit Haxiaization.

(3-312)

1r

=

TR - TC

where, (3-313)

TR

=

(3-314)

TC

=

n

L

rjxj

+ FC.

j=l

While this is a valid statement of the profit func-

176

Production Economics

tion, it is not a useful statement because it has no mention of the production function or the way in which the output and input variables are related. To integrate this aspect of the problem into the equation, we convert the profit function in equation (3-312) into a profit function with a restriction that the entrepreneur operate on the production function. In other words, we form the Lagrangian function: (3-315) n

= TR - TC+ AF(ql, 42, ••• , qm, xl, x2, ••• , Xn).

The first-order conditions for maximizing this function are: = p

(3-316) (3-317)

.lL

axj

i

= -r

+ ,a F = O; i = 1,2, ••• ,m /\aq i

j

+ A J.!._

axj

= 0;

j

=

1 , 2 , ••• , n

(3-318)

an

~

F(q1,4z,··•,4m,xl,x2, ••• ,xn)=O.

= r

In total, there are m + n + 1 equations and m + n + 1 variables or unknowns and generally speaking, we can solve this equation system. The results would indicate the levels of output and input which would maximize profits while operating on the production function. Of course, second-order conditions for maximizing the Lagrangian function must also hold. Theory of the Fira. The main advantage of discussing this general case is that it provides a convenient summary of the main results from the theory of the firm. Specifically, based on the first-order conditions in equations (3-316), (3-317), and (3-318), the relationships between a factor of production and a product, between two variable factors of production, and two products can

Generalized Results:

Production Economic Theory

177

be explicitly derived. First, the relationship between a variable factor of production and a product can be derived by rearranging equations (3-316) and (3-317) and dividing the latter by the former: -r

j

(3-319)

axj

or, •

= -). 3F

pi

(3-320)

3F = ->..aqi

~ = pi

3F _ axj 3F aqi

Util iz ing the implicit function rule yields:

(3-321)

~

(3-322)

rj

pi

aqi =

or,

axj aqi

=

Pi

axj

Equation (3-322) is simply the relationship that must exist between a variable factor and a product when profits are maximized. Specifically, when profits are maximized, the value of the marginal product of Xj in producing qi must equal the factor price, rj. The relationship between two factors of production when profits are maximized may be obtained by expanding the equations in (3-317) for two variable factors, say, j and j + 1. Rearranging these two equations and dividing the former by the latter yields: -r

j

=

(3-323) -rj+l

178

Production Econ11ics

. .2!.... = 0 a q22

=0

11

+;\ 2!_

ax21

=0

.

0 f course, second-order conditions (not shown here) must also hold. As in the case of static analysis, the most important aspect of these first-order conditions is that they illustrate relationships between and among variables in the system. The important rela tlonships involved are those between a factor of production and a product, two factors of production, and two products. These are shown in order. There is a relationship between a factor of production purchased on the first marketing day (e.g., x ) and a product sold on the second market10 ing day (e.g., q ). For x and q , the relation11 10 10 ship is derived by dividing equation (3-340) by (3-338): aF aqll . 3X10 rlO (3-346) = - -aF- = pll ax10 clqll (1 + 11)

Relationships aaong Variab1es.

This utilizes the implicit function rule described earlier. This relationship may be rewritten as follows: (3-347)

186

Production Economics

This relationship means that when profits are maximized over time, each input is used up to the point where the discounted value of the marginal product is equal to the discounted factor price. The same type of relationship could have been derived for other factors of production and products. The relationship be tween factors of production may be illustrated by dividing equation (3-340) by equation (3-341): aF ax20 ax10 rlO (3-348) = RTSX X • = aF = --ax10 r20 10 20 axzo This shows that the relationship between two variable factors purchased at the same time will be used up to that point where the rate of technical substitution between these factors is equal to the ratio of factor prices. It does not matter which time period is being considered. For example, the seventh firstorder condition (3-344) divided by the eighth firstorder condition (3-345) would yield the same result. The relationship of factors between time periods, can be shown by dividing equation (3-340) by equation (3~344): aF ax ax10 rlO 11 = RTS = (3-349) = aF ax10 xlOxll rll ax 0). 1

1

1

Variations of the quadratic and square root forms can be used where the exponents have values other than 1/2 or 2, e.g., 3/2. The linear production function, unlike the quadratic and square root forms, lacks the property of diminishing marginal productivity. The marginal physical products of each input is constant over the entire range of factor utilization. The generalized power production functions (GPPF) include as special cases the Cobb-Douglas and Transcendental forms. The GPPF can be written as : m f (X) g(X)

Generalized Power Functions.

(4-4)

q = A

IT

i=l

x1

1

e

202

Production Economics

where IT is a mathematical symbol denoting product multiplication, A is an efficiency parameter, fi(X) and g(X) are polynomials of any degree in the arguments of them-dimensional input vector X, and e denotes an eKponential function (see de Janvry). Four special cases can be derived from equation (4-4). First, if fi(X) = ai and g(X) = O, this is the Cobb-Douglas production function.

Second,

if g(X)

=

m

E biXi and fi(X) = ai for all 1, this is the Trani=l scendental production function (for more information on this form see Halter et al.). Third, if g(X) = 0 and fi(X) is homogeneous of degree zero in X for all

i, this is the Cobb-Douglas production function with variable returns to scale. Fourth, if g(X) = O, this is the Cobb-Douglas production function with variable elasticities of production. The commonly used Cobb-Douglas and Transcendental forms are: al a2 (4-5) Cobb-Douglas q = AXl X2 (4-6)

Transcendental

q

= AX1

al b?1 e

a2 b2X2

x2 e

CES Production Functions. Ever since 1928 when Cobb and Douglas formalized the multiplicative homogeneous production function depicted in equation (4-5), economists have relied upon it for investigating production processes of economic systems. The ease by which parameters can be estimated and the simplicity of its economic interpretation coupled with its apparent satisfactory fit in a variety of situations have led to its popularity with economists. Of more recent vintage have been the Transcendental form illustrated in the previous section and the Constant Elasticity of Substitution (CES) production function. The latter form is an outgrowth of the Cobb-Douglas function in that the Cobb-Douglas function is a

Dlpirical Production Functions

203

special case of the CES production function. The CES function was first presented by Arrow, Chenery, Minhas, and Solow in 1961. The CES production function has the following form: -r q

(4-7)

= A(b X -p + b X -p) 1 1

2 2

p

~

where pis the substitution parameter, r is the returns to scale parameter, and other variables are as previously defined. · 4-5.

Properties of Various Functional Foras

The following characteristics are determined and compared for some of the algebraic forms discussed in the previous sections: (i) marginal and average physical products (ii) output elasticity (iii) homogeneity Output q is assumed to be produced with inputs x and 1 x • The above three characteristics are determined 2 only for factor x • Similar procedures can be fol1 lowed to derive these characteristics for factor x2 • The forms discussed are the quadratic, CobbDouglas, and CES. The Cobb-Douglas form is representative of the generalized power design and the quadratic and square root forms are similar, differing primarily in the exponents used. Quadratic Production Function (4-8) ( i)

q =

bO + blXl + b2X2 + b3Xl

2

+ b4X2

2

+ bSX1X2

marginal and average physical products MPPXl

=

~il =bl+ 2b3Xl + bSX2

=_g__= Xl

204

Production

Econo■lce

Xl output elastic! ty

( ii)

X MPPX =~*_!_= 1 axl q APPX 1

bl+ 2b X + b x 3 1 5 2

=

b

= (iii)

x

1 1

2

+ 2b x + b x x 3 1 5 1 2 q

homogeneity k

Aq

~

= b0 + bl (AX1 ) + b2 (AX2 ) + b3 (AX1 )

2

2

+ b

(AX ) + b (AX )(AX ) 2 5 1 2 = b + A(b x + b X ) 4 r 0

2

1 1

+ A (b

x2

3 1

2 2

+ b X 4 2

2

+ b

xx)

5 1 2

.

This form of the production function is not homogeneous. Cobb-Douglas Production Function

(4-9) (i)

q = b

X o 1

bl

x2

b2

•

marginal and average physical products

Faplrical. Production FunctiODB b

b

1

x2

b X

o 1

205

2

XI ( ii)

output elasticity MPPX

E

a XI 1 =~*-= = ax q p,Xl APPX 1 1

( iii)

bl L . XI

.L

= bl

Xl

homogeneity b

Akq

b

= bo(AXl) 1 (AX2) 2

ht

= b0 A

This production bl+ b2.

b2

b (X

1

b

IX

bl+ b2 2) = A q 2

function is

CES Production Function

homogeneous of

degree

-r

(4- 10)

(i) marginal and average physical products -r

MPPxl

=

aq -r -p -pp-I -p-1 axl = p bo (blXl + b2X2 ) (-pblXl ) -r

bO (blXl-p + b2X2-p) p APP XI

= L = ---------Xl

XI

(ii) output elasticity X

=~*-1 = axl

q

MPPX APPX

1 1

206

Production Economics

Xl -r = - (blXl p

-p

+ b2X2

-p -1 -p ) (-pblXl )

=

\kq =

Po

-p -p -p (A (b1x1 + b2x2 >

-r ) p

-r \kq = \rb (b X -p + b X -p) p 0 1 1 2 2 r

\kq

r = \ q.

This function is homogeneous of degree r.

4-6.

Estf.llatlon Procedures Following World War II, agricultural economists made a conscious effort to improve the analytical methodology for quantifying production response. Ensuing studies incorporated mathematics and statistics to develop more efficient experimental designs and validation tests. The development and rapid technological advancement of electronic data processing increased the speed and efficiency of estimation procedures, Improved computational capabilities also made possible estimating techniques that were previously too tedious to be practical. Major advancements in estimating and computational procedures have

Fapfrical Production Functions

'207

been in the areas of linear regression, nonlinear regression, and mathematical programming. The most frequently used method for estimating production functions has been linear regression or ordinary least squares (OLS). Given the following functional relationship: q = f(X), the data for X and q depicting this relationship can be expressed graphically (see Figure 4-1).

Linear Regression.

Figure 4-1.

Graph of q = f(X)

•

• •

q

•

•

•

• •

• X

A more specific relationship between q and X can be obtained by drawing a line through these data points that best describes the relationship between q and X ( see Figure 4-2). However, one can draw several lines through this data set. Which line best represents the relationship between q and X? Through regression analysis, one can choose a specific line using the criterion that the line should minimize the sum of squared errors or dispersion. The term dispersion refers to the difference between the actual q value and the q corresponding to the value on the estimated regression line.

208

Production Econ011ics Figure 4-2.

Specific Linear Relationships between q and X

q

•

X

Thus, linear regression or the ordinary least squares (O~S) procedure involves fitting a line in the case of a single explanatory variable or a hyperplane in the case of multiple regression ( for more than one explanatory variable) through a set of observed data. The objective is to find that line or 1 hyperplane that minimizes the sum of squared errors or residuals. More elaborate discussion of the OLS procedure is found in econometric textbooks (e.g., Johnston, Kmenta, Wonnacott and Wonnacott, Gujarati, Draper and Smith, Theil, etc.). Simple and multiple regressions are usually expressed as follows: (4-11)

q

=

(simple linear regression) (4-12)

q

= b 0 + b 1X1 + b 2X2 +

(multiple regression)

+ b n Xn+ e

Eapirlcal Production Functions

209

where: q = dependent variable Xi = independent or explanatory variables i = 1,2, ••• ,n e = error terms a ,b = intercept terms 0 0 a , b ,b , ••• , bn = coefficients of explanatory 1 1 2 variables. According to the Gauss-Markov Theorem, within the class of 1 inear unbiased estimators, the least squares es tima tor has minimum variance. Therefore, it is often referred to as BLUE or the best linear unbiased estimator. Figure 4-3 illustrates the relationship of least squares estimator (BLUE) to all estimators.

Assuaptlons Underlying the OLS Procedure.

Figure 4-3.

Least Squares Estf.llators in Perspective Al l estimators

Li n e a r - - + - - - - ~ estimators UnbiaHd estimators

Least squares estimators

There are five major assumptions enabling OLS estimators to be BLUE. These assumptions are illustrated by using the general linear model written in matrix notation (refer to Chapter S for a review of matrix algebra and notation) .

Production Econ011ics

210

General 1 inear model q = X(3 + µ where:

=

(nxl) matrix of observations for the dependent variable. X = (n x k + I) matrix of observation for the k explanatory variable and constant term. (3 = (k +Ix 1) matrix of regression parameters or coef fie ien ts. µ = (nxl) matrix of errors or residuals.

q

( i) The first assumption is tqa t the observations on q are a linear function of the X observations and µ' s. 2

= 0 and

N(O,o ). This assumption states that the expected value of the disturbance term or its mean value is zero. This implies that external influences not captured by the set of explanatory variables are expected to cancel out and have no influence on the explanatory variables. Also, the error term is normally distributed with zero mean and variance equal

(ii) E(µ)

2

to o • (iii) E(µi~µj)

= 0 21 for i

=

j

or

= 0 for

i # j.

The assumption for i = j states that the error term has constant variance or is homoscedastic. For i -=I- j, the assumption is that the error terms are not correlated, (iv)

is a matrix of fixed numbers. This assumption states that the explanatory variables are nons toe has tic. X

(v) Rank of X = k +I< n. This assumption states that there are no exact linear relationships among the explanatory variables or that the rank of Xis k + I.

Eapirical Production Functions

211

Coaputational Procedures. With the exception of the CES production function, all other algebraic forms discussed above can be estimated by OLS. The CobbDouglas and Transcendental forms are nonlinear equations. They can be estimated by converting all variables to natural logarithms and applying OLS to estimate the parameters. There are many software packages containing least squares procedures. For IBM main frame computers, commonly used packages are SAS (Statistical Analysis System), ESP (Econometric Software Package), SPSS (Statistical Package for the Social Sciences), SHAZAM, and SPEAKEASY. For CDC or Cyber main frame computers, SPSS, SHAZAM, and TSP (Time Series Processor) are available. Further, similar software for popular microcomputers are beginning to appear. Thus, the computational aspects for OLS estimated production functions are easily handled. Unlike the other forms, the parameters of the CES production function cannot be estimated by linear regression techniques. There are several estimation techniques for this nonlinear production function (see Miller et al. and Judge et al.). It should be pointed out that ranking of nonlinear algorithms may not be useful to the researcher since successful evaluation depends upon the problem considered. Miller et al. describe three estimation procedures. First, there is the truncated Taylor series approximation. Second, often used, is the method of steepest descent or gradient method. Both are iterative techniques in which initial guesses of the unknown parameters (bi' r, and p) are used in the computer

algorithm to minimize the sum of squared error. The drawback to the Taylor series approach is the tendency for the sum of squared errors to diverge rather than converge toward a minimum. The gradient technique, on the other hand, is often slow to converge. The third method utilizes the Marquardt algorithm which combines the properties of the gradient technique and the Taylor series approach. The Marquardt algorithm is also an iterative process that utilizes

212

Production Economics

not only initial guesses of the unknown parameters but also the angle to which convergence takes place to minimize the sum of squared errors. Judge et al. describe several more nonlinear procedures (e.g., Gauss, Quasi-Newton, Newton Raphson, etc.). Different software packages generally utilize different nonlinear estimation techniques. For instance, SHAZAM employs the Quasi-Newton approach. SAS, on the other hand, uses three methods; Marquardt, modified Gauss-Newton, and gradient or steepest descent. Likewise, TSP offers both the Gauss and gradient methods. In summary, the method chosen will depend on the given problem, available computer programs, and the analyst's computer budget.

4-7.

Kapirical Exaaples Four empirical production examples are presented 1n this section. The first two examples deal with biological production functions. The third example illustrates a production function estimated with data from a sample of grain farms. The last example involves estimates of an aggregate production function obtained by pooling time series and cross section data. Several economic characteristics are examined fo,: these functions, e.g., output elasticity, optimal input combinations, derivation of factor demands, and determination of elasticities of substitution.

Example A: Production with One Variable Input. The following data describe the production of Narragansett alfalfa hay in Northern Nevada during the period 1972-1975. The variable input used in the production process is irrigation water. The following functional forms of the production function are estimated:

x + a 2x 2

(4-13)

q

= a0

+ a

(4-14)

q

= bo

+ blX + bzXl/2

1

Eapirical Production Functions

213

where: q X

= alfalfa hay = irrigation

yield (tons/acre) water application (acre-inches).

Data for alfalfa hay yield and irrigation water applied are shown in Table 4-1. • The estimated results (0LS) are shown in Table 4-2. All explanatory variables are statistically significant at the 5% level.

Table 4-1. Production of Alfalfa Hay in Response to Vater Application · Alfalfa Hay Yield ( tons/ acre) 4.12 5.08 4.52

4.67 4.15

Irrigation Water Applied (acre-inches)

27 42 57 75 24

5.63 6.04

36

5.52

72

48

4.17

24

5.94

33

5.66

42

5.89

60

3.46

21

5.49 4.79

30

6.43

39 57

Source: R. A. Young et al., "Irrigation of Alfalfa and Bromegrass." Manuscript, Nevada Experiment Station, University of Nevada, 1976.

214

Production Econoaics

Table 4- 2. q

Regression Results for Alfalfa Yields

= 0.3266 + 0.2084X - 0.0020X 2 (3.68) 2

Adjusted R F-statistic q

= 0.50 = 8.56

= -11.1132 - 0.3357X + 4 . 7604x 112 (-3.31)

2 Adjusted R F-s ta tis tic q

(-3.32)

(3.55)

= 0.59 = 9.52

= -0.9345 + 0.3893X - 0.0360x312 (3.60)

2 Adjusted R F-statistic

(-3.41)

= 0.51 = 8.90

Note: (1) Figures in parentheses denote t-statistics which show statistical significance of explanatory variables. Statistical significance at the 5% level 1requires a critical t-value of 2.16. (2)

2

The adjusted R statistic is the coefficient of determination adjusted for degrees of freedom. 2

An adjusted R of 0.50 implies that the explanatory variables have explained 50% of the variation in the dependent variable adjusting for degrees of freedom . The three regressions explain 50-59% of the variation in alfalfa hay yield. (3) The F-statistic measures the statistical significance of the entire regression equation. Statistical significance at the 5% level requires a critical value of 4.08 .

Eapirical Production Functions

215

The output elasticity of irrigation water for the quadratic production function equation (4-13), is calculated as follows: E

p,X

*! -q

a.9.

=

ax

where q and X denote means of q and X, res pee ti vely

ll ax = 0.2084

- .0040X

= 5.10 tons/acre

q

X = 42.94 acre-inches 4 Ep,X = (0.2084 - . 0040 (42.94)) ( ;:~~)

= 0.31.

This .estimate suggests that if water use were to increase by 1 percent, production would increase by 0.31 percent. The output elasticities of irrigation water for the square-root and three-halves forms are calculated and interpreted in similar fashion. The amount of water that maximizes alfalfa hay yield can be determine'd using the procedures described in the preceding chapter. The first-order and second-order conditions for the estimated quadratic production function are:

f=

. 2084 - .004X = 0

X = 52.1 acre-inches

i

!!....i dX2

= - . 004

r(A), the equation system is inconsistent-and and there is no solution. (2) If r(A!E_) = r(A) = n, the equation system is consistent and a unique solution is exists. (3) If r(AIE_) = r(A) = p, p r(A), the system of equations is inconsistent and

Linear Algebra

there is no solution possible. This case can be illustrated Consider the equation system :

=

1

2x - 2x 1 2

=

x

1

- x

2

285

graphically.

3.

In this example, r(A) = 1 and r(Alb) = 2. The equations are inconsistent and there is no solution to the equation system. Graphically, the two equations in the system may be depicted as in Figure 5-5. The two equations in the sys tern are parallel lines in the x , x space . There is no instance in which the two 1 2 equations intersect. There are no values of x and 1 x which will satisfy both equations at the same 2 time. There is no solution. Figure 5-5. Inconsistent Equation, No So1ution Possible

1.0

1.5

- 0.5

-1.0

-1.5

286

Production Econoaics

The reasoning for this case is as follows. We know that r(Alb) > r(A). Therefore, the largest nonvanishing determinant in (Al~) must contain E.· That is, if we add a vector to a set of vectors and the rank of the expanded set increases, then the added vector must be linearly independent of the original set of vectors. Conversely, if the "added vector" were linearly dependent upon the original set of vectors, then its addition to the original set would not change the rank of the augmented set. Thus, we conclude that b is linearly independent of the columns in the original A matrix. Accordingly, the linear combination of the original vectors cannot yield!_. In other words: (n)

a :/: b. + ••• + xnWe can only form b as a linear combination of the vectors in A if b is linearly dependent on the vectors in A. Because bis linearly independent of the vectors of" A, we conclude there is no solution possible. To illustrate the instance where there is a unique solution, consider the example: I xl + 2x2 + 3x3 = 6 X X

1 1

-

=

x2 x3

2

= -1

In matrix notation, expressed as:

this equation system would be

It can be shown that r(A) = r(A I~) = 3. Further, applying ERO to solve the equation system yields x = 7/6 and x = -5/6, and x 3 = 13/6. Thus, 2 1

J.fnear Algebra

'JB7

if r(A) = r(Ajb) = n, a unique solution exists. Graphically, this case can be illustrated by the following example:

=

x 1 - x2 4x

1

+ 2x

2

2

= 2.

When these two equations are plotted, they intersect at x 1 = 1 and~= -1 (Figure 5-6). This means that there are unique values of x and x which will 1 2 satisfy both equations simultaneously. Figure 5-6.

Unique So1ution

0.5

--+----+----+-_ _ _ _ _.,__ ___,. 1.0 1.5 2.5 -0.5

-1.0

-1.5

-2.0

x,

288

Production Economics

The rationale for this case can be explained as follows. We know or start with the condition that r(A) = r(Al1) = n. If m = n, we have the same number of equations as unknowns. If, m > n, there are m - n redundant equations . When we augment A by b, the rank of the augmented matrix does not change. Therefore, b is linearly dependent on the columns of A. The only way in which the rank of the augmented ma·trix could change is if bis linearly independent of the columns of A. Thus, the vectors in following set are linearly dependent: (1)

(n)

(2)

!!. '~ ' ••• , a '1. Since the vectors in this set are linearly dependent, we can form the following linear combination of the vectors of the set: (1)

(n)

(2)

xI a + x a + • •• + x a = b. 2nAlternatively, we could form the linear combination: xI~(l) + x a( 2 ) + ••• + x~(n) + k 2

E.

= O, where, k= -1.

In this las t r instance, since the scalar associated with b is defined by the equation system to be -1, there- is a unique linear combination. This 1 inear combination is the unique solution to the equation sys tern. The third case or possible type of solution to an equation system is where there is an infinite number of solutions. Specifically, there is an infinite number of solutions to an equation system when r(A) = r(Al1) = p < n. Here, p of the xj's can be solved for in terms of the remaining n-p xj's. To see this case, consider the example:

XI + x2 + X3 + x4 = 8 xl - X2 - X3 - X4 3xI +

Xz

=

6

+ X3 + X4 = 22

Linear Algebra

289

In matrix notation, this equation system is: 1

1

1

1

xl

1

-1

-1

-1

x2

3

1

1

1

X3

8

=

6

22

It can be shown that r(A) .= r(A IE.) = 2. The equations are consistent but a unique solution to the equation sys tern is not possible. With appropriate ERO on (Alb) we have: xl + x2 + X3 + X4 = 8 x2 + x3 + x4 = 1. Rearranging these equations, we have:

x2 = 1 - X3 - X4 xl = 8 - (1 Thus,

-

X

3

-

X

4

the solution values

in terms of x

)

-

X

3

-

x4

.

for x

and x are defined 1 2 Other combinations are possi-

and x • 4 ble since we can solve for any two of the unknown variables in the equation system in terms of the remaining two of the unknown variables in the system. In general, we say that when r(A) = r(Alb) = p < n, then p of the xj's can be solved for in terms of the 3

remaining n-p xj's. To see this equation sys tern: x

1

- x

2

case

graphically,

consider

the

= 1

2x - 2x = 1. 1 2 These two equations are plotted in a two-dimensional space in Figure 5-7. Note that both equations coincide exactly. Thus, there is an infinite number of solutions to the equation system since there is an

290

Production Econoaics

Figure 5-7.

Infinite Number of Solutions

1.0

--+--------r#----------1

X1

2.0

-1.0

infinite

number of combinations of

x

and x which 2 1 will satisfy both equations simultaneously. The rationale for this case can be shown as follows. We know that n > p and because of that, (A I~) necessarily con ta ins n + 1 linearly dependent vectors (then columns of A plus the vector b), Because these vectors are linearly dependent, any column of (Alb) can be expressed as a linear combination of the-remaining columns of (Alb). Since the vectors of (Alb) are linearly dependent, there is a linear combination of these vectors which will yield a null vector:

Linear Algebra

291

consistent with the original equation system, specify k = -1 so that: b

a(l) = x1xp+l

+ x

a( 2 ) + ... + x• a(p) + 2p-

a(p+l) + ..• + x

a(n)_ n -

In this case, arbitrary values for x

n

have been specified.

xp+l'

Because we

xp+2 , ••• ,

can set

these

values arbitrarily, we say that there is an infinite number of solutions to equations systems of this type.

292

Production Economics

REFERENCES Aitken, A. C. 1962. Determinants and Matrices. York: Interscience.

New

Campbell, Hugh G. 1965 . An Introduction to Matrices, Vectors, and Linear Programming. Englewood Cliffs, N.J. : Prentice-Hall. Hadley, G. 1961. Addison-Wesley.

Linear Algebra.

Reading, Mass.:

6

LINEAR PROGRAMKING Having completed a review of linear algebra, we now apply these concepts to a powerful technique used by economists--linear programming. Linear programming is a mathematical techniqu~ for minimizing or maximizing a linear objective function subject to a set of linear constraints. The constraints may be expressed as linear equations or inequations. And, the variables or activities are cons trained to be nonnegative. The technique of linear programming is not unlike the two-variable factor case found in the theory of the firm. There, output is maximized subject to cost constraint. While there may be several ways of solving such a problem, the method of Lagrangian multipliers is often used. This approach is comparable to linear programming except that t he constraint is stated as an equality. In linear programming, the ~onstraints may be equalities or inequalities, In any case, the constrained optima problem posed in the theory of the firm and linear programming are especially appealing to economists because they capture the essence of economics--optimizing an objective function subject to constraints on the function. In this chapter, we discuss ways of solving linear programming problems graphically and analytically. In the process, we formulate alternative linear programming models and interpret their results. 6-1.

Graphing Inequalities Prior to discussing ways of graphing inequalities, it is useful to review the graphing of equalities or equations. For example, given the function y. = 2x or y - 2x = O, we can graph this function by placing y on the vertical axis and x on the horizontal axis (Figure 6-1), The graph of this function means that all points on the line or making up the line satisfies the equation y = 2x or y - 2x = 0.

293

294

Production Econoaics

Alternatively, of all the points (xi yi)

=

pi in

the

two-dimensional space, only those on the line will satisfy the equation y = 2x. We might also view the graph of the equation y = 2x as a way of dividing the points in the two-dimensional space or x y plane into two groups: (1) those points that satisfy the equation and hence are on the line; and, (2) those points that do not satisfy the equation and hence are not on the line.

Figure 6-1.

Graph of the Equation y

3

=

2x

4

It is this last interpretation of graphing an equation that interests us as we consider graphing inequalities. Specifically, we utilize the following procedure for graphing an inequality or inequa tion. First, we ignore the inequality sign and graph the function as if it were an equation. Second, we identify the two sets of points in the two dimensional space that is defined by the inequation: (1) those that satisfy the inequa tion; and, ( 2) those that do not satisfy the inequation. Consider the inequa tion y < 2. To graph this inequation, we ignore the inequality and consider only the equality portion of the inequa tion. This

Linear Prograaing

295

equation is plotted in Figure 6-2. To graph the inequation, we must determine the points that satisfy the inequation and those that do not. One way of doing this is to simply choose a point that is obviously in the set of points that is obviously on one side of the line (equation) in question. In this example, the origin where x = 0 and y = 0 is clearly away from the line. We then ask, . "when x = 0 and y = O, is the inequation satisfied?" Clearly, when y = 0 and x = O, y is< 2. Thus, we conclude that all points on the line-and below the line satisfy the inequation y < 2; and, all points above the line do not satisfy the-inequation y .s_ 2. Figure 6-2.

Graphing the lnequation y

4. If we ignore the inequality, we would plot the equation y = 4 (Figure 6-3). We next identify the points that satisfy the inequation and those that do not. Again, we arbitrarily choose any point obviously away from the line and determine whether at this point the inequation is satisfied, Consider the origin where x = 0 and y = O. At these values for x and y is the inequation satisfied? When x = 0 and y = O, the value of y is clearly less than 4. Therefore, we

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Production Economics

conclude that all points on and above the line satisfies the inequation y ~ 4. Figure 6-3.

Graphing the Inequation y

~

4

y

3

2

..

--------------1

-1

2

3X

The procedures for graphing inequalities also apply to nonlinear equations. For example, consider the inequation y - x

2

< 4. -

Ignoring

the inequality,

2

we plot the equation y - x = 4 (Figure 6-4). Again, all points on the graph of the parabola satisfies the equation. All points below the graph of the parabola 2

and on the parabola satisfies the inequation y - x < 4. And, all points above the graph of the parabola would not satisfy this inequation. 6- 2.

Kxaaples of Linear Prograaaing Probleas

As in introduction to linear programming, we discuss three examples. These examples have been selected for their simplic~ ty in Ulustra ting the basic type of problem addressed by linear programming models.

Linear Progr Figure 6-4.

fng

Graphing the Inequation y - x2

297

0

number of acres of wheat produced number of acres of corn produced.

The constraints the the above problem are plotted in Figure 6-7. The points satisfying all constraints are bounded by bold lines. An arbitrary level of profit ($1,600) is also plotted. Profits are maximized by shifting the profit line away from the origin, as far as possible while still touching at least one point that satisfies all constraints. Profits are maximized when x = 130 and x = 30. In W

C

other words, profits are maximized when the farmer produces 130 acres of wheat and 30 acres of corn. In this case profits are $4,640. Figure 6-7. Graphical Solution to Wheat-Com Linear Programming Problem x,.. = 130 140

Xe=

110

100 Xe

60

180

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Production Econoaics

6-3. Analytical Solution to the Linear Prograaaing Problea-The Shlp1ex Method

Referring back to the wheat-corn problem, linear programming statement of the problem is: Maximize rr = 32 x subject to:

+

X W

X