Workbook for Macroeconomic Theory: Fluctuations, Inflation and Growth in Closed and Open Economies [1st ed. 2021] 3030615472, 9783030615475

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Fernando de Holanda Barbosa Luiz Antônio de Lima Junior

Workbook for Macroeconomic Theory Fluctuations, Inflation and Growth in Closed and Open Economies

Workbook for Macroeconomic Theory

Fernando de Holanda Barbosa Luiz Antônio de Lima Junior

Workbook for Macroeconomic Theory Fluctuations, Inflation and Growth in Closed and Open Economies

Fernando de Holanda Barbosa Brazilian School of Economics and Finance Fundaç˜ao Getulio Vargas Rio de Janeiro Rio de Janeiro, Brazil

Luiz Antônio de Lima Junior Faculty of Economic Sciences Federal University of Juiz de Fora, campus Governador Valdares (Brazil) Governador Valadares Minas Gerais, Brazil

ISBN 978-3-030-61547-5 ISBN 978-3-030-61548-2 (eBook) https://doi.org/10.1007/978-3-030-61548-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book presents the answers to the exercises in Macroeconomic Theory, Fluctuations, Inflation and Growth in Closed and Open Economies by Fernando de Holanda Barbosa (Cham, Switzerland: Springer, 2018), hereafter referred as Macro Theory. Altogether, there are 166 exercises in eleven chapters and three appendices. Macro Theory points out that many of these exercises are based on, or inspired in, the literature listed in the bibliography, although the sources are not documented. We would like to thank the authors of these papers for shaping our understanding of the issues addressed in Macro Theory. It is a science tenet that models should be falsifiable representation of a phenomenon. The goal of a good number of exercises is to help the student to develop the skills necessary to obtain the models’ empirically testable predictions. You should try to solve each exercise by yourself, but do not be upset if you cannot. Some exercises are very hard and take time to work out. However, in order to learn, you should persevere and try again and again. We hope that this workbook will help you in the learning process of macroeconomic theory. Macro Theory presents almost all models with continuous variables because it is very easy to derive the qualitative results with phase diagrams. There are a few exceptions, like Appendix C, where we present the new Keynesian model using discrete variables. Macro Theory departs from the praxis of using the representative agent model to analyze the small open economy. Simpler models should be preferred to more complex models, according to Occam’s razor. Thus, why should one not use the very simple representative agent model to analyze the small open economy? The answer to this question is based on an empirical stylized fact, namely that some small economies are creditor countries, while others are debtor countries. The representative agent model is unable to provide a long run net foreign asset equilibrium either as a creditor or as a debtor country. Appendix C of Macro Theory takes stock of the new Keynesian model. Its IS curve turns the consumption Euler equation into a level curve. It is common knowledge (so, we do not have to provide references) that the Euler equation is a statement about the slope (smoothing) of the optimal consumption path. It does v

vi

Preface

not say anything about the level of consumption but just states that given the level of current consumption the Euler equation can be used to forecast the expected level of future consumption. The new Keynesian IS curve turns this feature on its head: the level of consumption today depends on the expected level of consumption tomorrow. This IS curve is solved forward and implies that the effect of the rate of interest on output gap is the same today, tomorrow and at any time in the future. Empirical observation rejects this hypothesis. A very clever, but flawed, idea to solve this “puzzle” is the discounted Euler equation, which transforms this equation into a statement about the level of consumption. Exercise 14 of Chap. 7 deals with this issue. The organization of this workbook is the same as that of Macro Theory. The first part deals with flexible price models and has five chapters and 41 exercises. Chapter 1 presents the representative agent model, Chap. 2 analyzes the open economy representative agent model, Chap. 3 addresses the overlapping generations model, Chap. 4 presents the Solow growth model and Chap. 5 introduces endogenous savings and endogenous growth in models of economic growth. The second part covers sticky price models, both Keynesian and new Keynesian, and has four chapters. Chapter 6 presents the IS, LM and Phillips curves and the Taylor monetary policy rule. Chapter 7 analyzes models of economic fluctuations and stabilization in closed economies as well as deals with the issue of chronic inflation. Chapter 8 introduces the basic concepts of open economy macroeconomics such as arbitrage in markets for goods and services and in asset markets. This chapter also presents the specifications of the IS curve, Phillips curve and monetary policy rules in an open economy. Chapter 9 deals with the models of fluctuations and stabilization in open economies. This part has 55 exercises. The third part has 2 chapters and 33 exercises. Chapter 10 introduces the government budget constraint and analyzes the following topics: (1) public debt sustainability, (2) hyperinflation, (3) Ricardian equivalence and (4) the fiscal theory of the price level. Chapter 11 addresses several monetary theory issues, such as price level determination, the optimum quantity of money, dynamic inconsistency, smoothing of the interest rate by central banks, inflation targeting, operational procedures of monetary policy and the term structure of interest rates. Macro Theory as well as this workbook has three appendices with 37 exercises. These appendices make the Macro Theory book self-contained. Appendix A deals with differential equations, Appendix B presents the essential of optimal control theory and Appendix C gives the basic tools of difference equation needed to understand the new Keynesian model. There are two types of exercises in Macro Theory. The first type aims to provide the student with material to practice for a full understanding of the subjects presented in the book. The second type of exercises addresses issues that were left out of the book because we chose to limit the size of Macro Theory to be less than 500 pages. Those exercises can be solved using the tools presented in the chapter, or appendix, where they belong. The exercises marked with an asterisk were prepared for this workbook. They cover the topics that are not dealt within Macro Theory, but we have decided to include them for the sake of completeness.

Preface

vii

The topics covered in the second type of exercises in each chapter are: Chapter 1: Discount rate and time inconsistency; cash-in-advance (CIA) constraint and money superneutrality; unpleasant monetarist arithmetic; incorrect specification of the Ramsey/Cass/Koopmans model. Chapter 2: Variable rate of interest; habit formation and small open economy model; intertemporal approach to the balance of payments. Chapter 3: Diamond growth model; social security system: fully funded versus payas-you-go. Chapter 4: CES production function, Inada conditions and endogenous growth; Solow growth model with money. Chapter 5: Human capital growth model with leisure in the utility function; human capital model with externalities. Chapter 6: Calvo Phillips curve in continuous time with discount. Chapter 7: Discounted Euler equation in the new Keynesian model. Chapter 8: Monetary approach to the balance of payments with fixed and flexible exchange rates; Harberger–Laursen–Metzler effect; portfolio balance approach to exchange rates. Chapter 9: Incorrect specification of a small open economy new Keynesian model; tradable and nontradable goods. Chapter 10: Tax smoothing. Chapter 11: Consumption asset pricing model. Appendix A: Present value models: fundamentals and bubbles; housing model; Tobin’s q model. Appendix B: Tobin’s q model with installation costs; dynamic inconsistency in monetary models. Appendix C: Taking stock of the new Keynesian model. We would like to thank the excellent work of LATEX expert Cristina Maria Igreja for processing and editing our original files. Rio de Janeiro, Brazil Governador Valadares, Brazil

Fernando de Holanda Barbosa Luiz Antônio de Lima Junior

Contents

Part I Flexible Price Models 1

The Representative Agent Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2

The Open Economy Representative Agent Model . . . . . . . . . . . . . . . . . . . . . .

29

3

Overlapping Generations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4

The Solow Growth Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5

Economic Growth: Endogenous Saving and Growth . . . . . . . . . . . . . . . . . . .

81

Part II Sticky Price Models 6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

7

Economic Fluctuation and Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8

Open Economy Macroeconomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9

Economic Fluctuation and Stabilization in an Open Economy. . . . . . . . 161

Part III Monetary and Fiscal Policy Models 10

Government Budget Constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

11

Monetary Theory and Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Appendices A

Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

B

Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

C

Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

ix

x

Contents

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Macro Theory: Errata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Author Biographies

Dr. Fernando de Holanda Barbosa is Professor of Economics at the FGV EPGE Brazilian School of Economics and Finance. He has a Ph.D. in economics from the University of Chicago (US). He is the author of many academic articles in monetary economics and some are collected in the book Exploring the Mechanics of Chronic Inflation and Hyperinflation (Springer, 2017). Dr. Luiz Antônio de Lima Junior is Adjunct Professor of Economics, Federal University of Juiz de Fora, campus Governador Valdares (Brazil).

xi

Part I

Flexible Price Models

Chapter 1

The Representative Agent Model

(1) The representative agent maximizes the objective function: 

∞

β(t)u(c)dt 0

subject to the constraints a˙ = ra + y − c a(0) = a0 given The Hamiltonian is: H = β(t)u(c) + λ (ra + y − c) The first-order conditions are: ∂H = β(t)u (c) − λ = 0 ∂c

(1.1)

∂H = λr = −λ˙ ∂a

(1.2)

∂H = ra + y − c = a˙ ∂λ

(1.3)

The derivative of (1.1) with respect to time is: ˙  (c) + βu (c)c˙ = λ˙ βu

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_1

3

4

1 The Representative Agent Model

By taking into account (1.2) we get: −

β˙ u (c) c ˙ = r + u (c) β

Now let’s think about a second agent that maximizes, for s > 0: 

∞

β (t − s) u(c)dt

s

subject to the constraints a˙ = ra + y − c a(s) = as given The Hamiltonian is: H = β (t − s) u(c) + λ (ra + y − c) The first-order conditions are: ∂H = β (t − s) u (c) − λ = 0 ∂c

(1.4)

∂H = λr = −λ˙ ∂a

(1.5)

∂H = ra + y − c = a˙ ∂λ

(1.6)

The derivative of (1.4) with respect to time is: β˙ (t − s) u (c) + β (t − s) u (c)c˙ = λ˙ By taking into account (1.2) and rearranging the terms of the equation we get: −

β˙ (t − s) u (c) c˙ = r +  u (c) β (t − s)

The solution for the two agents is the same when: ˙ β˙ (t − s) β(t) = β(t) β (t − s) Thus:

1 The Representative Agent Model

5

β(t) = e−ρt β (t − s) = e−ρ(t−s) ρ = constant (2) The representative agent maximizes the objective function: 

∞

e−ρt [u(c) + υ(m)] dt

0

subject to the constraints y+τ =c+

M˙ P

M(0) given Since m = is:

M P , the derivative with respect of time of real per capita money stock

m ˙ = π=

P˙ P (inflation

M˙ − πm P

rate). The constraint can be rewritten: m ˙ = y − c + τ − mπ

(a) The current value Hamiltonian is: H = u(c) + υ(m) + λ (y − c + τ − π m) The first-order conditions are: ∂H = u (c) − λ = 0 ∂c ρλ −

∂H = ρλ − υ  (m) + ρπ = λ˙ ∂m

∂H = y − c + τ − πm = m ˙ ∂λ (b) From u (c) = λ we may write c as a function of λ : c = c(λ), since u (c) < 0. The dynamical system is:

∂c ∂λ

< 0,

6

1 The Representative Agent Model

Fig. 1.1 Phase diagram for the λ and m system



λ˙ = λ (ρ + π ) − υ  (m) m ˙ = y − c(λ) + τ − π m

The Jacobian of the system is:  J =

∂ λ˙ ∂λ ∂m ˙ ∂λ

∂ λ˙ ∂m ∂m ˙ ∂m



  ρ + π −υ  (m) = ∂c −π − ∂λ

The determinant of this Jacobian is: |J | = − (ρ + π ) π −

∂c  υ (m) < 0 ∂λ

∂c < 0 and υ  (m) < 0. Thus, the steady-state equilibrium is a because ∂λ saddle point. The m ˙ = 0 curve slopes upward and λ˙ = 0 slopes downward, as shown in Fig. 1.1. The arrows show the dynamics of the system and SS is a saddle path. (c) The marginal utility of consumption is equal to the costate u (c) = λ. By taking derivatives with respect to time we get:

u (c)c˙ = λ˙ Therefore, the equation for λ˙ can be written as: u (c)c˙ = (ρ + π ) u (c) − υ  (m) The dynamical system is given by: 





υ (m) c˙ = uu(c) (c) (ρ + π ) − u (c) m ˙ = y − c + τ − πm

1 The Representative Agent Model

7

The Jacobian of the system is: J =

 ∂ c˙

∂ c˙ ∂c ∂m ˙ ∂m ˙ ∂m ∂c ∂m





∂ c˙ ∂ c˙



− υu(m) (c) = −1 −π



The determinant of this Jacobian is: |J | = −π

∂ c˙ υ  (m) −  ∂c u (c)

The derivative of c˙ with respect to c is: ∂ c˙ = (ρ + π ) ∂c



2

 u (c) − u (c) u (c) (u (c))2

+

υ  (m)u (c) (u (c))2

which can be written as:

2  (ρ + π ) u (c) − u (c) (ρ + π ) u (c) − υ  (m) ∂ c˙ = ∂c (u (c))2 In steady-state: (ρ + π ) u (c) = υ  (m) Thus

∂ c˙ = (ρ + π ) ≥ 0 if π ≥ −ρ ∂c

It follows that the determinant of the Jacobian, evaluated at the steady-state point is negative. Thus, the system has a saddle point. The c˙ = 0 curve slopes upward and m ˙ = 0 slopes downward, as depicted in Fig. 1.2. The arrows show the dynamics of the system and SS is the saddle path. Fig. 1.2 Phase diagram for the λ and m system

8

1 The Representative Agent Model

(3) Since c = y, u (c) = λ = constant. Thus λ˙ = 0 and

υ  (m) u (c)

= (ρ + π ).

(a) The derivative with respect to time of the real quantity of money m = is:

M P

m ˙ M˙ = =μ−π m P It follows that: m ˙ = (μ − π ) m = [(μ + ρ) − (ρ − π )] m By taking into account the previous equilibrium conditions we get: m ˙ = (μ + ρ) m −

mυ  (m) u (c)

¯ as If lim mυ  (m) = 0 we get two equilibria, m = 0 and m = m, shown in Fig. 1.3. The path EA is a hyperdeflation bubble. The path E0 is a hyperinflation path. If limm→0 mυ  (m) > 0 there is one equilibrium, as shown in Fig. 1.4. The path EA is a hyperdeflation bubble and the path EB is a hyperinflation bubble. ˙ (b) When M P = constant = f , the previous real quantity of money equation can be written: m ˙ = f − mπ − ρm + ρm which is equivalent to: m ˙ = ρm + f − m (ρ + π ) Fig. 1.3 limm→0 mυ  (m) = 0

1 The Representative Agent Model Fig. 1.4 limm→0 mυ  (m) > 0

Fig. 1.5 limm→0 mυ  (m) = 0

Fig. 1.6 limm→0 mυ  (m) > 0

9

10

1 The Representative Agent Model

By taking into account the equilibrium condition we get: m ˙ = ρm + f −

mυ  (m) u (c)

If limm→0 mυ  (m) = 0 we may get two equilibrium points as shown in Fig. 1.5. There is no hyperinflation (bubble) but there is a hyperdeflation bubble path, as shown by the arrows. There is a stable low equilibrium (point m1 ) and an unstable high equilibrium (point m2 ). If limm→0 mυ  (m) > 0 there is just one equilibrium, which is unstable (Fig. 1.6). There are a hyperinflation and a hyperdeflation, both are bubbles. (4) The Leibnitz rule (Macro Theory, p. 337) for the particular case where 

β(r)

V (r) =

f (x)dx α

is given by: dV (r) dβ(r) = f (β(r), r) dr dr Applying this rule to:  F (θ ) =

t+θ

c(s)ds t

we obtain: dF (θ ) d (t + θ ) = c (t + θ ) = c (t + θ ) dθ dθ It follows that d 2 F (θ ) dc (t + θ ) = = c˙ (t + θ ) dθ dθ The Taylor Expansion of F (θ ) around the point θ = 0 is: F (θ ) = F (0) + F  (0)θ +

F  (0) 2 θ + ··· 2

Thus ˙ 2 (a) F (θ ) = c(t)θ + c(t) 2 θ + ... (b) M(t) ≥ F (θ ) From (a) we can write the second-order approximation:

1 The Representative Agent Model

11

M(t) ≥ c(t)θ +

c(t)θ 2 θ + ··· 2

It follows that: M(t) ≥ c(t)θ (5) Total assets are given by: a =k+m Thus the dynamic equation can be written as: a˙ = f (a − m) + τ − c − δ (a − m) − π m The cash-in-advance constraint (CIA) is m = c. The a˙ equation becomes: a˙ = f (a − c) + τ − c − δ (a − c) − π c The Hamiltonian of this problem is: H = u(c) + λ [f (a − c) + τ − (1 − δ + π ) c − δa] (a) The first-order conditions are:  ∂H = u (c) + λ f  (a − c) (−1) − (1 − δ + π ) = 0 ∂c ρλ −

 ∂H = ρλ − λ f  (a − c) − δ = λ˙ ∂a

∂H = f (a − c) + τ − (1 − δ + π ) c − δa = a˙ ∂λ (b) In the steady-state equilibrium:  ¯ +δ λ=0 λ˙ = ρ − f  (k) Thus: ¯ −δ ρ = f  (k) Therefore, money is superneutral in this model because k¯ does not depend on the rate of growth of money.

12

1 The Representative Agent Model

(6) By taking into account the CIA and the fact that: M˙ =m ˙ + mπ P we can write: f (k) + τ = c + k˙ + δk +

M˙ =m+m ˙ + mπ P

and m ˙ = f (k) + τ − (1 + π ) m The CIA can be written as: k˙ = m − c − δk The representative agent maximizes the objective function: 

∞

e−ρt u(c)dt

0

subject to the constraints m ˙ = f (k) + τ − (1 + π ) m k˙ = m − c − δk k(0) and M(0) given (a) The Hamiltonian of this problem is: H = u(c) + λ (m − c − δk) + μ [f (k) + τ − (1 + π ) m] The first-order conditions are: ∂H = u (c) − λ = 0 ∂c λ˙ = ρλ −

 ∂H = ρλ − −λδ + μf  (k) ∂k

μ˙ = ρμ −

∂H = ρμ − [λ − μ (1 + π )] ∂m

In steady-state equilibrium

1 The Representative Agent Model

13

λ˙ = 0 μ˙ = 0 Thus:

 λ˙ = (ρ + δ) λ − μf  (k) = 0 μ˙ = μ (ρ + 1 + π ) − λ = 0

From the first expression we get: ρ+δ =

μ  f (k) λ

and from the second expression, we obtain: (1 + ρ + π ) =

λ μ

The marginal product of capital is: f  (k) = (ρ + δ) (1 + i) where i = ρ + π is the nominal interest rate. The real rate of interest is: f  (k) − δ = ρ + i (ρ + δ) (b) Money is neutral since a change of its level does not change the nominal interest rate. (c) Money is not superneutral because the rate of growth of the stock of money changes the nominal interest rate, and a change of the nominal interest rate affects the quantity of capital in steady-state equilibrium. (7) First, let us change notation. We denote the nominal interest rate by i instead of r. Thus, the representative agent budget is: (1 − τ ) (ib + y) = c +

M˙ B˙ + P P

and the government budget constraint is: g + ib − τ (ib + y) = Total assets are defined by:

B˙ M˙ + P P

14

1 The Representative Agent Model

a =m+b By using the a equation we can rewrite the agent budget constraint as: ˙ + π m = c + a˙ + π a (1 − τ ) (ib + y) = c + b˙ + π b + m Taking into account that b = a − m, we may write: a˙ = (1 − τ ) [i (a − m) + y] − π a − c or: a˙ = [(1 − τ ) i − π ] a − (1 − τ ) im + (1 − τ ) y − c The representative agent maximizes the objective function: 

∞

e−ρt [u(c) + υ(m)] dt

0

subject to the previous transition equation. The Hamiltonian of this problem is: H = u(c) + υ(m) + λ [((1 − τ ) i − π ) a − (1 − τ ) im + (1 − τ ) y − c] The first-order conditions are: ∂H = u (c) − λ = 0 ∂c ∂H = υ  (m) − λ (1 − τ ) i = 0 ∂m ρλ −

∂H = ρλ − λ [(1 − τ ) i − π ] = λ˙ ∂a

The goods and services market is in equilibrium. Thus, c is constant and the costate variable is constant, so λ˙ = 0. Therefore: (ρ + π ) = (1 − τ ) i From the second first-order condition equation we get: υ  (m) = λ (1 − τ ) i Since u (c) = λ, we obtain:

1 The Representative Agent Model

15

υ  (m) = (1 − τ ) i u (c) From the differential equation of the real stock of money, we get: m ˙ = μm ¯ − mπ − ρm + ρm or: m ˙ = μm ¯ − (π + ρ) m + ρm = (μ¯ + ρ) m −

υ  (m)m u (c)

The government budget constraint can be written as: g + ib − τ (ib + y) = m ˙ + mπ + b˙ + bπ From the monetary policy rule: m ˙ = m (μ − π ) Thus, we get the following equation for the government budget constraint: b˙ = g − τy + [(1 − τ ) i − π ] b − μm ¯ or b˙ = ρb − μm ¯ + g − τy The dynamical system has two equations, one for m and another for b, that is: 



m ˙ = (μ¯ + ρ) m − m υu(m) (c) b˙ = ρb − μm ¯ + g − τy

The Jacobian of the system is:  J =

∂m ˙ ∂m ∂ b˙ ∂m

∂m ˙ ∂b ∂ b˙ ∂b

 =

 ∂ m˙

0 −μ¯ ρ



∂m

where:    υ (m) + mυ  (m) υ  (m) mυ  (m) ∂m ˙ = (μ¯ + ρ) − = μ ¯ + ρ − − ∂m u (c) u (c) u (c) Since:

16

1 The Representative Agent Model

υ  (m) = (1 − τ ) i = ρ + π u (c) we get ∂m ˙ mυ  (m) = μ¯ + ρ − (ρ + π ) − ∂m u (c) In equilibrium μ¯ = π , therefore ∂m ˙ mυ  (m) =−  ∂m u (c) (a) The determinant of this Jacobian is positive: |J | =

∂m ˙ ρ>0 ∂m

The trace of J is positive: trJ =

∂m ˙ +ρ >0 ∂m

We may conclude that the steady-state equilibrium is unstable. Figure 1.7 shows the phase diagram for the dynamical system of two differential equations, one for the real quantity of public debt (b) and the other for the real quantity of money (m). The arrows in this figure show the dynamics of the system. (b) Figure 1.8 shows the monetary experiment whereby the central bank reduces the monetary expansion rate from μ0 to μ1 at instant zero. The adjustment of the economy is shown in Fig. 1.9. The path E0 ET shows the system adjustment to the tight monetary policy with a ceiling of the Fig. 1.7 The phase diagram for the b and m system

1 The Representative Agent Model

17

Fig. 1.8 Experiment of a tight monetary policy

Fig. 1.9 Dynamic adjustment of the system

real stock of debt bs . The real quantity of money decreases and inflation increases since the announcement of the tight monetary policy. (8) When b˙ = f + ρb we may write: d −ρs be = f e−ρs ds or: dbe−ρs = f e−ρs ds (a) By integrating this equation, we obtain:  t

T

dbe−ρs =



T

f e−ρs ds

t

or: b(T )e−ρT − b(t)e−ρt =

 t

T

f e−ρs ds

18

1 The Representative Agent Model

It follows that: b(T ) = b(t)eρT e−ρT +



T

 f e−ρs ds eρT

t

(b) From this expression we get: lim λb(T )e

−ρT

T →∞

Since

= lim λb(t)e

−ρt

T →∞



∞

 + lim λ T →∞

e−ρs ds =

t

T

fe

−ρs

 ds

t

e−ρt ρ

the former becomes: lim λb(T )e−ρT = lim λb(t)e−ρt + lim λ

T →∞

T →∞

T →∞

e−ρt ρ

which is equal to:

 e−ρt = 0 lim λb(T )e−ρT = λe−ρt b(t) + T →∞ ρ (c) The real deficit is constant: b˙ = f Thus, 

T

t

 db =

T

f ds t

and b(T ) − b(t) = f (T − t) or b(T ) = b(t) + f (T − t) The transversality condition is: lim λb(T )e−ρT = lim λb(t)e−ρT + lim λf (T − t) e−ρT

T →∞

T →∞

T →∞

1 The Representative Agent Model

19

It is easy to verify that: lim λb(t)e−ρT = 0

T →∞

and f (T − t) lim λ ρT = 0 e

T →∞

We conclude that lim λb(T )e−ρT = 0

T →∞

(d) The conclusions drawn from items (b) and (c) are that a constant primary deficit does not obey the transversality condition, but a constant real deficit satisfies the transversality condition. (9) The real business cycle model with a government is given by the dynamical system: 

k˙ = Ak α − ρ+δ α k k α − g − δk K˙ = AKk α−1 − A(1−α) β

(a) The first experiment is an unanticipated permanent increase in government spending as described in Fig. 1.10. The k˙ = 0 curve does not shift but the K˙ = 0 curve shifts downward as Fig. 1.11 shows. The variable K is predetermined but the variable k can jump. At the instant of the increase in the government expenditure k jumps to the point I , with k(0+ ), in saddle path SS, and then converges on the new equilibrium (point Ef ). (b) The second experiment is an anticipated permanent increase in government spending as described in Fig. 1.12. The k˙ = 0 will shift at instant t. Fig. 1.10 An unanticipated permanent increase in government spending

20

1 The Representative Agent Model

Fig. 1.11 Dynamic adjustment of the economy to unanticipated change in government spending

Fig. 1.12 Anticipated permanent increase in government spending

Fig. 1.13 Dynamic adjustment of the economy to anticipated change in government spending

1 The Representative Agent Model

21

However the economy jumps at instant zero in such way that at instant T will be at the saddle path SS, as shown in Fig. 1.13. The economy converges on the point Ef , the new steady-state equilibrium. (c) The third experiment is an unanticipated transitory increase in government spending as described in Fig. 1.14. This transitory increase last until time T , when government goes back to the previous level. Figure 1.15 shows the dynamic adjustment of the economy. The variable k jumps to point I in such a way that by time T it will reach the point IT , and then converges back on the previous equilibrium E0 . (d) The fourth experiment is an anticipated transitory increase in government spending as described in Fig. 1.16. The k˙ = 0 will change at time T1 and it will be back to the original position at T2 . Figure 1.17 shows the dynamic adjustment of the economy. The variable k jumps to point I when the fiscal policy is announced. The capital

Fig. 1.14 An unanticipated transitory increase in government spending

Fig. 1.15 Dynamic adjustment of the economy to an unanticipated transitory change in government spending

22

1 The Representative Agent Model

Fig. 1.16 An anticipated transitory increase in government spending Fig. 1.17 Dynamic adjustment of the economy to an anticipated transitory change in permanent spending

stock decreases until time T1 and starts increasing in such a way to arrive at the saddle path at instant T2 . Then, it converges on the initial equilibrium point E0 . (10) The representative agent maximizes:  ∞ e−ρt u(c)dt subject to the constraints k˙ = f (k) − (η + δ) k − c k(0) given The Hamiltonian of this problem is: H = u(c) + λ [f (k) − (η + δ) k − c]

1 The Representative Agent Model

23

The first-order conditions are: ∂H = u (c) − λ = 0 ∂c ρλ −

 ∂H = ρλ − λ f  (k) − (η + δ) = λ˙ ∂k

∂H = f  (k) − (η + δ) k − c = k˙ ∂λ In equilibrium λ˙ = 0 we get:  ρλ = λ f  (k) − (η + δ) This is equivalent to: f  (k) − δ = ρ + n (a) The net marginal product is not equal to the rate of time preference. (b) The previous items result was obtained because the objective function does not take into account the fact that population is growing at a rate n. The objective function should be: 

∞

ent e−ρt u(c)dt

0

(11) *(Real Business Cycle (RBC)). The dynamical RBC system is given from equations (1.38) and (1.39). [Macro Theory, p. 24] by: ⎧ ⎨ k˙ = Ak α −

ρ+δk α

⎩ K˙ = AKk α−1 − where k =

K L

(1.7) A(1−α)k α β

− δK

(1.8)

and the steady-state values for k, K and L are: k¯ = K¯ =

αA ρ+δ



1 1−α

;

α (1 − α) A β [ρ + (1 − α) δ]

αA ρ+δ



α 1−α

;

(1 − α) (ρ + δ) L¯ = β [ρ + (1 − α) δ] ¯ does not depend on the technology index The steady-state value for labor (L) (A).

24

1 The Representative Agent Model

(a) Solve the RBC model using a dynamical system for k and L such as: 

k˙ = G (k, L, · · · )

(1.9)

L˙ = H (k, L, · · · )

(1.10)

(b) Linearizes the dynamical system of item (a) taking a first-order Taylor expansion around the steady-state values. (c) Analyze an unanticipated permanent increase in the technological index A. (d) Analyze an unanticipated permanent decrease in the technological index A. (a) The first equation of the dynamical system is Eq. (1.7). To obtain the second equation we use the definition: L˙ K˙ k˙ = − L K k By substituting (1.7) and (1.8) and rearranging terms results: ρ + δ (1 − α) L A (1 − α) k α−1 L˙ = − α β

(1.11)

Therefore, the RBC dynamical system is given by: 

k˙ = Ak α − ρ+δ α k L˙ = ρ+δ(1−α)L − α

A(1−α)k α−1 β



¯ L¯ as follows. (b) We linearize this system around the steady-state point k, When k˙ = 0: ρ+δ¯ 0 = Ak¯ α − k α Subtracting from the differential equation for k this expression yields:

ρ+δ k − k¯ k˙ = A k α − k¯ α − α A first-order Taylor approximation of the first term in the right-hand side gives:

k α = k¯ α + α k¯ α−1 k − k¯ By combining the two last expressions we obtain:

1 The Representative Agent Model

25

(1 − α) (ρ + δ) k − k¯ k˙ = − α

(1.12)

By the same token, when L˙ = 0 0=

ρ + δ (1 − α) L¯ A (1 − α) k¯ α−1 − α β

Subtracting this from the L˙ equation yields: 

A (1 − α)  α−1 ρ + δ (1 − α) L − L¯ − k − k¯ α−1 L˙ = α β By using the first-order Taylor approximation:

k α−1 = k¯ α−1 + (α − 1) k¯ α−2 k − k¯ we obtain the linear differential equation for labor:



A (1 − α)2 k¯ α−2 k − k¯ ρ + δ − α) (1 L − L¯ + L˙ = α β Therefore, the linearized dynamical RBC system of differential equations is given by: 



k˙ = − (1−α)(ρ+δ) k − k¯ α

L˙ = ρ+δ(1−α) L − L¯ + α

A(1−α)2 k¯ α−2 β



k − k¯

It should be noticed that both variables, k and L, are not predetermined. They are jump variables related by: kL = K and K is a predetermined variable. The Jacobian of this system is:  J =

∂ k˙ ∂k ∂ L˙ ∂k

∂ k˙ ∂L ∂ L˙ ∂L

 =

 (1−α)(ρ+δ) − α



0

A(1−α)2 k¯ α−2 ρ+δ(1−α) β α

The determinant of this matrix is negative: |J | = −

(1 − α) (ρ + δ) α

ρ + δ (1 − α) α

 0 can be used to obtain the price index: 1 

1−η

1−η  1−η Pt = (1 − γ ) PH,t + γ PF,t

First, to obtain this index the following problem has to be solved. minimize E = PH,t CH,t + PF,t CF,t subject to the constraint:  n 

n−1

n−1 n−1 1 1 η n n n + γ CF,t Ct = (1 − γ ) CH,t The Lagrangian of this problem is given by: 





n−1

n−1 1 L = PH CH,t +PF,t CF,t +λ Ct − (1 − γ ) CH,t n + γ n CF,t n 1 n



n n−1



The first-order conditions are:

 

n−1 −1 n 1 ∂L η−1 n −1 A n−1 (1 − γ ) n CH,t n = PH,t −λ =0 ∂CH,t n−1 η

  η−1

−1 n 1 ∂L n−1 n A n−1 −1 γ n CF,tη = PF,t − λ =0 ∂CF,t n−1 n where:   1 1

η−1

η−1 η η η η A = (1 − γ ) CH,t + γ CF,t By dividing the first expression by the second we get: PH,t = PF,t

1−γ γ

1 η

CH,t CF,t

− 1 η

In order to obtain each good’s demand equation, we have to solve a two-equation system given by the previous expression and the aggregator

2 The Open Economy Representative Agent Model

33

function of Ct , which can be written as: η−1 η

Ct

  1 1

η−1

η−1 = (1 − γ ) η CH,t η + γ η CF,t η

From the first-order condition, we have:  CH,t =

−η (1 − γ ) PH,t

CF,t



−η

γ PF,t

Combining the two previous equations yields: η−1 η

Ct

  1−η 1−η = (1 − γ ) PH,t + γ PF,t

η−1

CF,tη γ

η−1 η

1−η

PF,t

or:

 η  η  η−1 1−η η−1 1−η 1−η η−1 Ct γ η PF,t = (1 − γ ) PH,t + γ PF,t CF,t Thus: −η

γ PF,t Ct CF,t =   η 1−η 1−η η−1 (1 − γ ) PH,t + γ PF,t which can be written as: ⎤−η

⎡

⎥ ⎢ PF,t ⎥ CF,t = γ ⎢  1 ⎦ ⎣ 1−η 1−η 1−η (1 − γ ) PH,t + γ PF,t

Ct

It is straightforward to obtain: ⎡

⎤−η

⎥ ⎢ PH,t ⎥ CH,t = (1 − γ ) ⎢  1 ⎦ ⎣ 1−η 1−η 1−η (1 − γ ) PH,t + γ PF,t To obtain the price level we use the budget constraint: Et = PF,t CF,t + PH,t CH,t

Ct

34

2 The Open Economy Representative Agent Model

and the two previous expression for CF,t and CH,t . This yields: 1−η

Et = 

1−η

γ PF,t + (1 − γ ) PH,t  η Ct 1−η 1−η − 1−η (1 − γ ) PH,t + γ PF,t

When Ct = 1, Et = Pt . Thus: 1−η

1−η

γ PF,t + (1 − γ ) PH,t Pt =   −η 1−η 1−η 1−η (1 − γ ) PH,t + γ PF,t Since 1 −



−η 1−η



=

1 1−η ,

it follows that:

1 

1−η

1−η  1−η Pt = (1 − γ ) PH,t + γ PF,t

and each good’s demand equation can be written as:

 PH,t −η Ct Pt 

PF,t −η =γ Ct Pt

CH,t = (1 − γ ) CF,t

which solves item (b) of the problem. (II) Show how consumption index: 

1

CH,t =

CH,t (j )

θ−1 θ

θ  θ−1

dj

θ >1

,

0

can be used to obtain: 





1

PH,t =

PH,t (j )1−θ dj

1 1−θ



0

First, we solve the problem:  minimize:

PH,t (j )CH,t (j )dj 0

subject to the constraint:

1

2 The Open Economy Representative Agent Model



35

1

CH,t =

CH,t (j )

θ−1 θ

θ  θ−1

dj

0

The Lagrangean of this problem is: 

1

L= 0

⎧ ⎨



PH,t (j )CH,t (j )dj + λ Ct − ⎩

1

CH,t (j )

θ−1 θ

⎫ θ  θ−1 ⎬ dj

0

⎭

The first-order conditions is ∂L θ = PH,t (j ) − λ ∂CH,t (j ) θ −1



1

CH,t (j )

θ−1 θ

θ  θ−1 −1

dj

0

θ−1 θ −1 CH,t (j ) θ −1 = 0 θ

Thus: 

1

PH,t (j ) = λ

CH,t (j )

θ−1 θ

1  θ−1

dj

CH,t (j )

−1 θ

0

which can be written as: CH,t (j ) =  "1 0

PH,t (j )−θ λθ  θ θ−1 − θ−1 CH,t (j ) θ

which is equivalent to: CH,t (j ) = PH,t (j )−θ λθ CH,t By using the aggregate CH,t , we get:  CH,t =

1

PH,t (j )

−θ θ−1 θ

λ

θθ−1 θ

0

θ−1 θ

CH,t dj

which can be simplified as:  CH,t =

1

 PH,t (j )1−θ λθ−1 dj CH,t

0

Thus:  λ=

1

PH,t (j ) 0

1−θ

1  1−θ

dj

θ  θ−1

36

2 The Open Economy Representative Agent Model

The demand equation for the goods are: CH,t (j ) = PH,t (j )



−θ

1

PH,t (j )

1−θ

1  1−θ

dj

CH,t

0

 CH,t =



1

PH,t (j )

1−θ



1 1−θ

θ−1



1

dj

PH,t (j )

0

1−θ

1  1−θ

dj

0

which can be written as: ⎞−θ

⎛ ⎜ CH,t (j ) = ⎜ ⎝

PH,t (j ) "1

PH,t (j )1−θ dj

0



1 1−θ

⎟ ⎟ ⎠

CH,t

The expenditure is given by: 

1

Et =

PH,t (j )CH,t (j )dj 0

Using the previous expression yields:  Et = 0

1

PH,t (j )  "1 0

PH,t (j )−θ PH,t (j )1−θ dj

 −θ dj Ct 1−θ

When Ct = 1, Et = Pt . Thus:  Pt =



1

1+ PH,t (j )1−θ dj

θ 1−θ

0

or 

1

Pt =





PH,t (j )1−θ dj

1 1−θ

0

Therefore, the demand equations can be written as:

CH,t (j ) =

PH,t (j ) Pt

which answers point (b) of the question.

−θ Ct





CH,t

2 The Open Economy Representative Agent Model

37

Item (III) has the same solution of item (II). (4) The representative agent maximizes the objective function: 

∞

e−

"t 0

ρ(s)ds

u(c)dt

0

subject to constraints a˙ = ra + y − c a(0) given (a) Define:  δ(t) =

t

ρ(s)ds 0

Applying Leibnitz rule, we get: δ˙ =

dδ =ρ dt

(b) Maximize: 

∞

e−ρt e−δ+ρt u(c)dt

0

subject to the constraints: a˙ = ra + y − c δ˙ = ρ a(0) = a0 given The Hamiltonian of this problem is: H = e−δ+ρt u(c) + λ (ra + y − c) + μρ where λ and μ are the costate variables. The first-order conditions are: ∂H = e−δ+ρt u (c) − λ = 0 ∂c ρλ −

∂H = ρλ − λr = λ˙ ∂a

38

2 The Open Economy Representative Agent Model

ρμ −

 ∂H = ρμ − −e−δ+ρt u(c) = μ˙ ∂δ ∂H = ra + y − c = a˙ ∂λ ∂H = ρ = δ˙ ∂μ

(c) The dynamical system for the consumption and wealth variables are obtained as follows. The costate variable is given by: e−δ+ρt u (c) = λ By taking derivatives with respect to time for both sides yields:

λ˙ = e−δ+ρt −δ˙ + ρ u (c) + e−δ+ρt u (c)c˙ Since ρ = δ˙ and e−δ+ρt =

λ u (c)

we obtain:

u (c) λ˙ =  c˙ λ u (c) Because

λ˙ λ

= ρ − r we can write: u (c) c˙ = ρ − r u (c)

or u (c) c˙ = − (r − ρ) u (c) The dynamical system is given by: 



c˙ = − uu(c) (c) (r − ρ) a˙ = ra + y − c

(d) The system is autonomous in spite of the fact that the discount rate (δ) depends on time. (e) The system has a stationary equilibrium. (5) The dynamical system for the representative agent model with variable rate of time preference is:

2 The Open Economy Representative Agent Model

39

˙ =  (ρ(c) − r) c˙ = α (c, ) [ρ (c, ) − r] a˙ = ra + y − c where α (c, ) =

 ucc +

ρ (c, ) = ρ(c)

−U c ρc ρcc

uc − ρc u(c)/ρ(c) 



(a) Show that: α (c, ) < 0,

∂ρ (c, ) 0, then μ < 0. From the Hamiltonian, ∂ 2H = e−s ucc + μρcc ∂c2 2

Since μ < 0, if ρcc > 0, then ∂∂cH2 < 0.  and M are related by (p. 41):  − uc = M = μeS < 0 ρc Since

−uc ρc

< 0 and ρcc > 0 it follows that: α (c, ) =

Let us show that:

 ucc +

−uc ρc ρcc

0. Thus, the steady-state is a saddle point. ∂r ¯ the When c˙ = 0, ρ − r − b ∂b = 0. The solution of this equation gives b, ¯ ¯ steady-state of this variable. Thus, when b > b, c˙ > 0, and b < b, c˙ < 0, as shown in phase diagram of Fig. 2.1. When b˙ = 0, pc c = y − rbQ, and the b˙ = 0 curve is downward sloping. Below this curve b˙ < 0 and above b˙ > 0. The SS curve is the saddle path of the dynamical system. (b) Anticipated permanent change in the real exchange rate depicted in Fig. 2.2. When Q increases, both curves, c˙ = 0 and b˙ = 0, shift, as shown in the phase diagram of Fig. 2.3.Consumption is a jump variable and b is a predetermined variable. At the time the permanent change is announced, consumption decreases but b stays at the same value. The curve c˙ = 0 and b˙ = 0 do not shift at moment t = 0. They will shift at moment T . The ˙ 0 ) = 0. dynamics of the model is given by the curves c(Q ˙ 0 ) = 0 and b(Q At time T the economy will have to reach a new saddle path S1 S1 , and then converge on the new equilibrium Ef . (c) Anticipated transitory change in the real exchange rate depicted in Fig. 2.4.

44

2 The Open Economy Representative Agent Model

Fig. 2.2 An an anticipated permanent increase in the real exchange rate

Fig. 2.3 Dynamic adjustment of the economy to an anticipated permanent increase in the real exchange rate

At time zero a transitory change is announced. The real exchange rate will increase from Q0 to Q1 at time T1 and returns to its previous value at time T2 . Figure 2.5 shows the phase diagram of this experiment. At time zero consumption jumps and the stock b stays the same. The economy travels ˙ 0 ) = 0 system. southwest according to the arrows of the c(Q ˙ 0 ) = 0 and b(Q ˙ ˙ 1 ) = 0, At time T1 the c˙ = 0 and b = 0 curves shift to c(Q ˙ 1 ) = 0 and b(Q respectively, and the dynamics change. The economy goes northwest until time T2 when it reaches the saddle path S0 S0 and converges on the original equilibrium (Fig. 2.5).

2 The Open Economy Representative Agent Model

45

Fig. 2.4 An anticipated transitory increase in the real exchange rate

Fig. 2.5 Dynamic adjustment of the economy to an anticipated transitory increase in the real exchange rate

(7) The functional U is defined by:  U=

∞

u(c)e−

"υ t

ρ(c)ds

dυ

t

(a) Show that U˙ = ρ(c)U − u(c) Leibnitiz rule (Macro Theory, p. 137) is given by: d dr

 β(r) α(r)

 f (x, r) dx = f (β(r), r)

 β(r) ∂f (x, r) dx dβ(r) dα(r) − f (α(r), r) + dr dr ∂r α(r)

Thus "t dU = −u(c)e− t ρ(c)ds + dt

 t

∞

 "υ  u(c) e− t ρ(c)ds ρ(c) dυ

46

2 The Open Economy Representative Agent Model

which can be written as: dU = −u(c) + ρ(c) dt



∞

u(c)e−

"υ t

ρ(c)ds

dυ

t

Therefore: U˙ = −u(c) + ρ(c)U (b) What is the economic interpretation of this differential equation? This differential equation can be interpreted as an arbitrage: U˙ + u(c) U

ρ(c) =

where U˙ is the capital gain (or loss), u(c) is the cash flow and U is the “price”. (8) The representative agent maximizes: 

∞

e

"  t − 0 ρ(c)ds−nt

u(c)dt

0

subject to: k˙ = f (k) − (n + δ) k − c k(0) = k0 (a) Define S =

"t 0

given

ρ(c)ds − nt. Show that: S˙ = ρ(c) − n d dS = dt dt



t

 ρ(c)ds − nt

0

Applying the Leibnitiz rule is straightforward to obtain: dS = ρ(c) − n dt (b) Solve the representative agent’s problem using the new state variable S. 

∞

max 0

e−S u(c)dt

2 The Open Economy Representative Agent Model

subject to: k˙ = f (k) − (n + δ) k − c S˙ = ρ(c) − n k(0) and S(0) given. The Hamiltonian is given by: H = e−S u(c) + λ [f (k) − (n + δ) k − c] + u [ρ(c) − n] The first-order conditions are: ∂H = e−S u (c) − λ + μρ  (c) = 0 ∂c λ˙ = − μ˙ = −

 ∂H = −λ f  (k) − (n + δ) ∂k

∂H = − (−1) e−S u(c) = e−S u(c) ∂S

∂H = f (k) − (n + δ) k − c = k˙ ∂λ ∂H = ρ(c) − n = S˙ ∂μ The first-order conditions can be written as: ⎧ ⎪ λ = e−S u (c) + μρ  (c) ⎪  ⎪ ⎪ ⎪ ⎨ λ˙ = −λ f  (k) − (n + δ) μ˙ = e−S u(c) ⎪ ⎪ ⎪ k˙ = f (k) − (n + δ) k − c ⎪ ⎪ ⎩˙ S = ρ(c) − n Let us define:  = λeS M = μeS We take the time derivative of  to obtain: ˙ S + λeS S˙ ˙ = λe

47

48

2 The Open Economy Representative Agent Model

We substitute out for ˙ and S˙ from the first-order conditions to get: 

˙ =  ρ(c) − f  (k) − δ We take the time derivative of M to obtain: M˙ = μe ˙ S + μeS S˙ We substitute out for μ˙ and S˙ from the first-order conditions to get: M˙ = u(c) + M [ρ(c) − n] The first equation of the first-order conditions can be written as: λeS = u (c) + μeS ρ  (c) or:  = u (c) + Mρ  (c) We take the time derivative of this expression to obtain: ˙  (c) + Mρ  (c)c˙ ˙ = u (c)c˙ + Mρ which can be written as:  ˙  (c) ˙ = u (c) + Mρ  (c) c˙ + Mρ We substitute out for M˙ from the M˙ equation to get:  ˙ = u (c) + Mρ  (c) c˙ + [u(c) + M (ρ(c) − n)] ρ  (c) From:  = u (c) + Mρ  (c) we have: M=

 − u (c) ρ  (c)

we substitute out M to get:   

 − u (c)   ˙  = u (c) + ρ (c) c˙ + u(c)ρ  (c) +  − u (c) (ρ(c) − n)  ρ (c)

2 The Open Economy Representative Agent Model

49

Since: 

˙ =  ρ(c) − f  (k) − δ we substitute out ˙ in the former equation to obtain the differential equation for c:  



 ρ(c) − f  (k) − δ − u(c)ρ  (c) +  − u (c) (ρ(c) − n)   c˙ =  (c)  (c) u (c) + −u ρ  ρ (c) Therefore, we obtain the dynamical system for c, k and : ⎧  [ρ(c,)−(f  (k)−δ )] ⎪ ⎪ −u (c)  ⎨ c˙ =  u (c)+

ρ  (c)

ρ (c)

k˙ = f (k) − (n + δ) k − c ⎪ ⎪ 

⎩˙  =  ρ(c) − f  (k) − δ where:



u(c)ρ  (c) +  − u (c) (ρ(c) − n) ρ (c, ) = ρ(c) − 

(c) This model cannot be reduced to a two of differential equations. Thus, we cannot use a phase diagram with the two variables, c and k. (d) In steady-state ρ(c) = n. When n decreases the steady-state capital stock increases because ρ(c) = f  (k) − δ. (e) In steady-state ρ(c) = f  (k) − δ. If the depreciation rate δ increases the capital stock, at steady-state, decreases. (9) The representative agent maximizes the functional:  ∞ U= e−ρt u (c, z) dt 0

where utility function u(c, z) depends on consumption (c) and on a past consumption index z, according to:  t z(t) = βe−β(t−τ ) c(τ )dτ −∞

(a) Show that: z˙ = β (c − z) Applying Leibnitiz rule we get:   t  t d −β(t−τ ) −β(t−t) βe c(τ )dτ = βe c(t) + β (−β) eβ(t−t) c(t)dt dt −∞ −∞

50

2 The Open Economy Representative Agent Model

which can be written as: z˙ = βc(t) − βz(t) = β (c − z) (b) Establish the first-order condition of the following problem: 

∞

max

e−ρt u (c, z) dt

0

subject to the constraints: a˙ = ra + y − c z˙ = β (c − z) a(0) = a0 and z(0) = z0 given The current value Hamiltonian of this problem is: H = u (c, z) + λ (ra + y − c) + μβ (c − z) The first-order conditions are: ∂H = uc (c, z) − λ + μβ = 0 ∂c λ˙ = ρλ − μ˙ = ρμ −

∂H = ρλ − λr ∂a

∂H = ρμ − [uz (c, z) − μβ] ∂z

∂H = ra + y − c = a˙ ∂λ ∂H = β (c − z) = z˙ ∂μ The first-order conditions can be written as: ⎧ ⎪ uc (c, z) = λ − μβ ⎪ ⎪ ⎪ ⎪ ⎨ λ˙ = λ (ρ − r) μ˙ = (ρ + β) μ − uz (c, z) ⎪ ⎪ ⎪ a˙ = ra + y − c ⎪ ⎪ ⎩ z˙ = β − z) (c (c) From the first equation of the first-order condition, we have:

2 The Open Economy Representative Agent Model

51

λ = ucc (c, z) + μβ Taking the time derivative of this expression yields: λ˙ = ucc c˙ + ucz z˙ + μβ ˙ For stationary consumption we should have: λ˙ = z˙ = μ˙ = 0 Thus, from λ˙ = 0 we conclude that: ρ = r, the rate of the preference should be equal to the real interest rate. (10) The representative agent maximizes: 

∞

e−ρt u(c)dt

0

subject to the constraints: a˙ = ra + y − c a(0) = a0 given Rate r is the foreign interest rate. (a) Assume that r = ρ. Show that: a˙ = y − y p where: 

∞

yp = r

e−rt ydt

0

How would you interpret this result? (b) Assume that r = ρ and u(c) = where:

1

c1− σ 1− σ1

show that a˙ = y − y p + σ (r − ρ) W

52

2 The Open Economy Representative Agent Model



∞

W =

ye−rt dt + a0

0

(c) The Hamiltonian of this problem is: H = u(c) + λ (ra + y − c) The first-order conditions are: ∂H = u (c) − λ = 0 ∂c λ˙ = ρλ −

∂H = ρλ − λr ∂a

If r = ρ, then λ˙ = λ (ρ − r) = 0, and λ is constant. Thus, consumption is also constant. From the budget constraint we obtain: 

∞

a0 =

e−rt (c − y) dt =

0

c − r



∞

e−rt ydt

0

Thus: 

∞

c = ra0 + r

e−rt ydt = ra0 + y p

0

Taking into account the definition of permanent income:  yp = r

∞

e−rt (y)dt

0

and by using the budget constraint it follows that:

a˙ = ra + y − c = ra + y − ra + y p Thus: a˙ = y − y p The intertemporal approach to the balance of payments states that the current account of the balance of payments is equal to the difference between current income and permanent income. If a˙ = y − y p > 0

2 The Open Economy Representative Agent Model

53

there is a surplus, and a deficit when a˙ = y − y p < 0 (d) From the first-order condition and the utility function we obtain: c˙ = σ (r − ρ) c and we can write: c(t) = c0 e(r−ρ)t Inserting this expression into the budget constraint we obtain: 

∞

a0 +

ye−rt dt =

0



∞

e−rt c0 eσ (r−ρ)t dt

0

or:  a0 +

∞

ye−rt dt = c0



0

∞

e−(r−σ (r−ρ))t dt

0

We assume (r − σ (r − ρ)) > 0. Thus: 

∞

e−(r−σ (r−ρ))t dt =

0

1 r − σ (r − ρ)

and the consumption c0 is:   c0 = (r − σ (r − ρ)) a0 +

∞

e−rt ydt



0

Substituting the consumption c0 into the budget constraint results:   a˙ = ra0 + y − (r − σ (r − ρ)) a0 +

∞

e−rt ydt



0

which can be written as:    ∞ e−rt ydtr + σ (r − ρ) a0 + a˙ = y − r 0

0

or: a˙ = y − y p + σ (r − ρ) W

∞

e−rt ydt



54

2 The Open Economy Representative Agent Model

where: 

∞

y =r p

e−rt ydt

0

and: 

∞

W = a0 +

e−rt ydt

0

The current account (a) ˙ depends on two factors. The first is the discrepancy between income and permanent income because the agent wants to smooth consumption. The second factor depends on the degree of impatience of the agent. If the agent is patient (r > ρ) there will be a current account surplus. On the other hand if the agent is impatient (ρ > r) this factor will contribute to a deficit in the current account.

Chapter 3

Overlapping Generations

(1) The budget constraint in this two-periods-of-life model is: c1,t +

c2,t+1 = wt 1 + rt+1

(a) The individual maximizes:

u c1,t +



1 u c2,t+1 1+ρ

subject to the previous budget constraint. The Lagrangean of this problem is:  



1 c2,t+1 L = u c1,t + u c2,t+1 + λ wt − c1,t − 1+ρ 1 + rt+1 The first-order conditions are:

∂u c1,t ∂L = −λ=0 ∂c1,t ∂c1,t

∂L 1 ∂u c2,t+1 = −λ=0 ∂c2,t+1 1 + ρ ∂c2,t+1 (b) Consumption for each period can be obtained by solving the system of two equations: ∂u(c1,t ) ∂c1,t ∂u(c2,t+1 ) ∂c2,t+1

=

1 + rt+1 1+ρ

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_3

55

56

3 Overlapping Generations

Fig. 3.1 Intertemporal allocation of consumption

c1,t +

c2,t+1 = wt 1 + rt+1

Figure 3.1 shows the solution of the problem, with c2,t+1 in the vertical axis, and c1,t+1 in the horizontal axis. When c2,t+1 = 0, c1,t = wt , and when c1,t = 0, c2,t+1 = (1 + rt+1 )wt . Both c1,t and c2,t+1 are functions of wt , rt+1 and ρ. Thus, savings is a function of wt , rt+1 and ρ. That is: st = s (wt , rt+1 ) (c) Assuming that consumption today and consumption tomorrow are normal goods: ∂st >0 ∂wt (d) When the rate of interest shifts it is common knowledge that there are an income effect and a substitution effect. Thus: ∂st 0 ∂rt+1 (e) When the utility function is isoelastic: 1

if σ = 1, u (c) = c− σ

ifσ = 1, u (c) = c−1

3 Overlapping Generations

57

The first-order condition is:

−1 c1,tσ u c1,t 1 + rt+1

= = − σ1 1+ρ u c2,t+1 c2,t+1 Thus:

c2,t+1 =

1 + rt+1 1+ρ

σ c1,t

If we substitute out c2,t+1 in the budget constraint we obtain: c1,t =

wt 1 + (1 + ρ)

−σ

(1 + rt+1 )σ −1

Therefore, the savings function is given by:

st = 1 −

1



1 + (1 + ρ)−σ (1 + rt+1 )σ −1

wt

Taking the derivative of st with respect to rt+1 we obtain: ∂st (σ − 1) (1 + ρ)−σ (1 + rt+1 )σ −2 =  2 wt ∂rt+1 1 + (1 + ρ)−σ (1 + rt+1 )σ −1 Therefore, ∂st > 0, if σ > 1 ∂rt+1 ∂st = 0, if σ = 1 ∂rt+1 ∂st < 0, if σ < 1 ∂rt+1 (2) In the previous question’s OLG economy, the supply side is specified as follows: Production function: Y = F (K, L) Population growth: Lt = (1 + n) Lt−1 Wages: wt = f (kt ) − kt f  (kt ) Interest rate: rt = f  (kt ) − δ where:

58

3 Overlapping Generations

kt =

Kt Lt

Savings equals investment: Kt+1 − Kt + δKt = Lt st (a) Show that: kt+1 =

(1 − δ) kt + st (wt , rt+1 ) 1+n

By dividing both sides by the working population Lt yields: Kt δKt Lt s t Kt+1 − + = Lt Lt Lt Lt Kt+1 Lt+1 − kt + δkt = st Lt+1 Lt kt+1 (1 + n) = (1 − δ) kt + st Thus: kt+1 =

(1 − δ) kt + st (wt , rt+1 ) 1+n

(b) Analyze the model’s equilibrium and dynamics. Since: wt = f (kt ) − kt f  (kt ) and rt+1 = f  (kt+1 ) − δ we can write: kt+1



(1 − δ) kt + st f (kt ) − kt f  (kt ), f  (kt+1 ) − δ = 1+n

This is a nonlinear difference equation in the stock of capital goods per worker. For every value of kt we can obtain kt+1 , but, in general, it cannot be solved in closed form. We take the differential for both sides of this difference equation:   (1 + n) − sr f  (kt+1 ) dkt+1 = (1 − δ) − sw kt f  (kt ) dkt

3 Overlapping Generations

59

Thus: 1 − δ − sw kt f  (k) dkt+1 )) 0 ) = dkt k 1 + n − sr f  (k) The model is stable if: −1
1 1

lim f  (k) = δ − θ = lim

k→∞

k→∞

f (k) k

For the elasticity of substitution greater than one (σ > 1) the Inada condition is not satisfied. (a) The short run rate of growth of labor productivity is given by (4.14, Macro Theory, p. 96). The long run rate of labor productivity is equal to g. (b) The short run rate of growth of labor productivity is given by expression (4.14, Macro Theory, p. 96). The long run rate of growth of labor productivity depends on the value of the elasticity of substitution. (c) When σ > 1, the CES production function yields the same result as an endogenous growth model. There is no difference between the short and long run rates of labor productivity. (2) The Cobb-Douglas production function can be written as

74

4 The Solow Growth Model

Y = kα AL or: Y = k α A0 egt L Taking the logs of both sides: log

Y = log (A0 ) + gt + α log(k) L

In steady-state sk α = (g + n + δ) k or: s = k 1−α g+n+δ Taking the logs of both sides log(s) − log (g + n + δ) = (1 − α) log(k) or: log(k) =

1  log(s) − log (g + n + δ) 1−α

Substitution for log(k) in the equation for the log of the labor productivity yields:

 α Y α = log A0 + gt + log(s) − log (g + n + δ) log L 1−α 1−α (3) Assume a Cobb-Douglas (intensive form) production function: y = Ak α hβ , where α is the share of capital in output and β is the share of human capital in output. The economy, in the model including human capital, is at steady-state: sk f (k, h) = (n + g + δk ) k sh f (k, h) = (n + g + δk ) h Show that the log of labor productivity is given by:

4 The Solow Growth Model

log

75

Y α β = log A0 + gt + log sk + log sh L 1−α−β 1−α−β −

α β log (n + g + δk ) − log (n + g + δh ) 1−α−β 1−α−β

The production function can be written as: Y = Ak α hβ L By hypothesis A = A0 egt Thus:

 Y = log A0 + gt + α log(k) + β log(h) log L At steady-state: sk k α hβ = (n + g + δk ) k sh k α hβ = (n + g + δh ) h which can be written as:  sk + β log(h) = (1 − α) log(k) log n+g+δ k sh log n+g+δ + α log(k) = (1 − β) log(h) h This system of two equations has the solutions: 

sh sk 1−β β log log + 1−α−β n + g + δh 1−α−β n + g + δk 

α sk sh 1−α log(h) = log log + 1−α−β n + g + δk 1−α−β n + g + δh log(k) =

If we substitute out for log(k) and log(h) in the equation of productivity of labor then we get log

Y α = log A0 + gt + log sk L 1−α−β +

β β log sh − log (n + g + δh ) 1−α−β 1−α−β

76

4 The Solow Growth Model

−

α log (n + g + δk ) 1−α−β

(4) The differential equations of the exogenous growth model with human capital are given by: k˙ = sk f (k, h) − (g + n + δk ) k h˙ = sh f (k, h) − (g + n + δh ) h (a) Deduce this system’s Jacobian matrix: The Jacobian matrix is given by:  J =

∂ k˙ ∂k ∂ h˙ ∂k

∂ k˙ ∂h ∂ h˙ ∂h



  sk fh sk fk − (n + g + δk ) = sh fk sh fh − (n + g + δh )

In steady-state: sk f (k, h) = (n + g + δk ) k sh f (k, h) = (n + g + δh ) h Therefore: sk f (k, h) k sh f (k, h) (n + g + δh ) = h (n + g + δk ) =

Thus, we can write: sk fk − (n + g + δk ) = sk fk −

f (k, h) − kf (k) sk f (k, h) = −sk 0. The trace of the Jacobian matrix is: trJss = −

sk (f − kfk ) sh (f − hfh ) − 0 k and we assume that sA > n + δ. (6) The exogenous growth model with government is specified by the following equations: Production Function: Y = F (K, AL) Saving: S = s (Y − T ) Investment: S = I − K + δK

78

4 The Solow Growth Model

Government: G = T Technological Progress: A˙ = gA Population: L˙ = nL (a) Deduce the model’s differential equation for capital accumulation measured in labor efficiency units. The production function can be written as: Y =F AL

 K , 1 = f (k) AL

From the saving-investment equation we get: s (Y − T ) K˙ K = +δ AL AL AL or: s (f (k) − τ ) = k˙ + (g + n) k + δk Thus, the model’s differential equation for capital is: k˙ = s [f (k) − τ ] − (g + n + δ) k (b) The growth rate of capital is: s [f (k) − τ ] k˙ = − (g + n + δ) k k Thus, the government affects the short run rate of growth of output. (c) In the long run the government affects the level of income. (7) The Solow model with money is specified by the following equations: Production function: y = f (k) Assets: a = m + k Saving: S = s (y + τ − mπ ) Investment-Saving: S = k˙ + δk + m ˙ Money demand: m = L(r)k, L < 0 Real interest rate: ρ = f  (k) − δ Monetary policy: m ˙ = m (μ − π − n) M˙ = constant μ= M M where y = YL , k = K L , m = PL We assume that population grows at a constant rate: L˙ = nL. The government makes lump sum (τ ) transfers of money to the private sector. τ=

M˙ M M˙ = = μm PL M PL

4 The Solow Growth Model

79

where μ denotes the growth rate of the stock of money. The asset decision of this model is given by: s (y + τ − mπ ) = k˙ + (n + δ) k + m ˙ + nm Since: m ˙ = m (μ − π − n) the capital accumulation equation can be written as: k˙ = sf (k) − (n + δ) k − (1 − s) (μ − π ) m The inflation rate is the difference between the nominal and real rates of interest according to the Fisher equation: π =r −ρ The inverse of the money demand equation is: r=l

m k

, l < 0

and the real rate is equal the marginal product of capital less the depreciation rate. Thus: m

− f  (k) − δ π =l k The dynamical system of this model is given by: 

k˙ = sf (k) − (n + δ) k − (1 − s) m (μ − π ) m ˙ = m (μ − π − n)

Where π is a function of m, k and δ as specified above. In steady-state k˙ = m ˙ = 0, and μ − π = n. Thus:

sf k¯ = (n + δ) k¯ − (1 − s) mn ¯ Thus, monetary policy affects the stock of capital and real income. Monetary policy is not superneutral in this model. (8) The Cobb-Douglas production function is given by: Y = AK α L1−α

80

4 The Solow Growth Model

(a) Show that: log

1 α K Y = log A + log L 1−α 1−α Y

We can write the production function as: Y α Y 1−α = AK α L1−α Thus:

1−α

α Y K =A L Y Taking logs of both sides and rearranging terms we obtain: log

1 α K Y = log A + log L 1−α 1−α Y

(b) In the short run the productivity of labor increases when the capital-output ratio increases. In the long run the productivity of labor depends on the parameter A.

Chapter 5

Economic Growth: Endogenous Saving and Growth

(1) Show that in the RCK model: (a) The saving (investment) rate at stationary equilibrium is given by:

∗ I α (g + n + δ) s = = Y ρ + δ + σ1 g ∗

The dynamical system of the RCK model is given by equation ((5.8), Macro Theory, p. 122): k˙ = f (k) − c − (g + n + δ) k  c˙ g = σ f  (k) − δ − ρ − c σ and the transversality condition (Macro Theory, p. 123) is: ρ−n>g−

g σ

The saving rate in steady-state is: s∗ =

(g + n + δ) k (f (k) − c) = f (k) f (k)

or: s∗ =

g+n+δ f (k) k

From the Cobb-Douglas production function:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_5

81

82

5 Economic Growth: Endogenous Saving and Growth

f (k) = k α the marginal product of capital is: f  (k) = αk α−1 = α

f (k) k

thus: f (k) f  (k) = k α In steady-state, k˙ = c˙ = 0, and: f  (k) = ρ + δ +

g σ

therefore: s∗ =

1 α

α (g + n + δ) g+n+δ

g = ρ + δ + σg ρ+δ+ σ

(b) The stationary equilibrium saving rate is lower than the share of capitaloutput: s∗ < α From the transversality condition, we get: ρ+

g >g+n σ

We add δ to both sides to get: ρ+

g +δ >g+n+δ σ

thus: g+n+δ 0 dρ We conclude that the IS curve may have an upward slope.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_6

99

100

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve

(2) Assume that consumption depends on disposable income (y d ), c = c(y d ), and that disposable income is defined by: yd = y − g where y is the real income and g is government spending. (a) Why could you define disposable income in this way? This definition assumes Ricardian equivalence, e.g., government expenditures have to be paid now or in the future. (b) Would a tax cut, for a given level of g, affect expenditure in this economy? No, because consumption depends on disposable income, and disposable income, as defined in this exercise, is not affected by tax. (3) Assume that consumption depends on disposable income (y d = y − τ ) and the real quantity of money, m = M P : c = c (y − τ, m). (a) Is the IS curve independent from monetary policy? No. Monetary policy affects the real quantity of money. An easy monetary policy increases m and thus consumption. This is known in the literature as the real cash balance effect due to Pigou. (b) Is the full-employment real interest rate independent from monetary policy? The full-employment real interest rate (¯r ), the natural rate of interest, is obtained from the IS curve when output is at its full-employment level, y = y. ¯ Thus: y¯ = c (y¯ − τ, m) + i (¯r ) Therefore, r¯ depends on the real quantity of money m. (4) Consider the model: I S : y = c (y − τ ) + i(i) + g LM :

M = L (y, i) P

MP R : i = i¯ When the Central Bank sets this economy’s interest rate according to this monetary policy rule, is the economy’s price level determined? This model assumes that the expected rate of inflation is equal to zero. Thus, the nominal rate of interest is equal to the real rate of interest. Output is determined by the IS curve: ¯ +g I S : y = c (y − τ ) + i(i)

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

101

Therefore, given this output and the fixed interest rate, the LM curve determines: m = M P :

M = L y, i¯ P and not the price level. Neither the stock of money nor the price level are determined. The economy has no anchor. (5) Assume that the utility function is: 1 u (ct ) = − e−αc , a > 0 α (a) How do you interpret parameter α? The absolute risk premium is defined by: Eu (x + ) = u (x − pr ) where , a stochastic variable has mean zero, E = 0, and variance E 2 = σ 2 . A Taylor expansion of u (x + ) is given by: u (x + ) = u(x) + u (x) +

u (x) 2  2

and a Taylor expansion of u (x − pr ) is: u (x − pr ) = u(x) + u (x) (−pr ) By using the definition of the risk premium we obtain: u(x) +

u (x) 2 σ = u(x) + u (x) (−pr ) 2

and the risk premium is: pr = −

u (x) 2 σ 2u (x)

This premium is known as the absolute risk premium: ar = − For the utility function:

u (x) u (x)

102

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve

1 u(c) = − e−αc α the two derivatives are given by: u (c) = e−αc u (c) = −αe−αc and the absolute risk premium is: −

−αe−αc u (c) = − =α u (c) e−αc

(b) Use the Euler equation to deduce the equation of the IS curve associated with this utility function. The Euler equation is given by: u (ct ) = β (1 + rt ) u (ct+1 ) and u (ct ) = e−αct , u (ct+1 ) = e−αct+1 Substituting this expression into the Euler equation results: e−αct = β (1 + rt ) e−αct+1 By taking the log of both sides we get: −αct = log β (1 + rt ) − αct+1 The economy only has consumer goods. Thus, yt = ct , yt+1 = ct+1 Therefore: yt = yt+1 −

1 log β (1 + rt ) α

which can be written as: yt − y¯t = yt+1 − y¯t+1 + y¯t+1 −

1 (rt − ρ) α

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

where β = as:

1 1+ρ ,

103

and y¯ denotes potential output. We can write this IS curve

xt = xt+1 −

1 (rt − r¯ ) α

where the natural rate of interest is: r¯ = ρ + αy¯t+1 where y¯t+1 is the rate of growth of potential output per-capita. (6) The marginal utility of consumption in period t + 1 may be written as a function of the marginal utility in period t, of the derivative of the marginal utility of period t, and of the difference between consumption tomorrow and consumption today, according to the following Taylor expansion: u (ct+1 ) ∼ = u (ct ) + u (ct ) (ct+1 − ct ) This expansion disregards second-order terms. Show that the Euler equation with continuous variable is: c˙ = −

u (c) (r − ρ) cu (c)

The Euler equation is given by: u (ct ) = β (1 + rt ) u (ct+1 ) Substituting the Taylor expansion into this equation results: u (ct ) =

1 + rt   u (ct ) + u (ct ) (ct+1 − ct ) 1+ρ

which can be written as:   1+ρ u (ct ) = 1+  (ct+1 − ct ) 1 + rt u (ct ) Taking the logs of both sides and using the approximation log (1 + x) ∼ = x, we obtain: ρ − rt = ct

u (ct ) (ct+1 − ct ) u ((ct ) ct

Taking into account that the rate of growth of consumption:

104

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve

(ct+1 − ct ) ∼ d log ct = c˙ = ct dt the Euler equation becomes: c˙ = −

u (c) (r − ρ) cu (c)

(7) Suppose that the output gap depends on a lagged output gap and on the interest rate gap according to: xt = λxt−1 − α (rt − r¯t ) xt = λxt+1 − α (rt − r¯t ) The parameter λ lies between zero and one and the parameter α is positive. (a) Write down the output gap, respectively, as: xt = −α

−∞ +

λi (rt−i − r¯t−i )

i=0

xt = −α

∞ +

λi (rt+i − r¯t+i )

i=0

We use the log operator L, Lzt = zt−1 , to write the first equation as: (1 − λL) xt = −α (rt − r¯t ) which can be written as: xt = −

  α (rt − r¯t ) = −α 1 + λL + λ2 L2 · · · · · · (rt − r¯t ) (1 − λL)

By using the operator L we get:

xt = −α (rt − r¯t ) − αλ rti − r¯t−1 − α 2 λ (rt−2 − r¯t−2 ) or: xt = −α

−∞ + i=0

λi (rt−i − r¯t−i )

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

105

We use the forward operator F , F zt = zt+1 , to write the second equation as: (1 − λF ) xt = −α (rt − r¯t ) which can be written as: xt = −

  α xt = −α 1 + λF + λ2 F 2 · · · · · · (rt − r¯t ) (1 − λF )

By using the operator F we obtain: xt = −α (rt − r¯t ) − αλ (rt+1 − r¯t+1 ) − αλ2 (rt+2 − r¯t+2 ) or: xt = −α

∞ +

λi (rt+i − r¯t+i )

i=0

(b) What happens when the parameter λ is equal to one? Can the output gap be written as a function of the past (future) interest rate gaps? When λ = 1, the first equation is: xt = xt−1 − α (rt − r¯t ) This equation can have either a backward or a forward solution: ,−∞ backward: xt = −α i=0 (rt−i − r¯t−i ) , forward: xt = α ∞ i=0 (rt+1+i − r¯t+1+i ) To check that both are solutions just subtract xt−1 from xt to find that: xt − xt−1 = −α (rt − r¯t ) When λ = 1, the second equation is: xt = xt+1 − α (rt − r¯t ) This equation can , have either a backward or a forward solution: backward: xt = α ,−∞ i=0 (rt−1−i − r¯t−1−i ) forward: xt = −α ∞ i=0 (rt+i − r¯t+i ) To check that both are solutions just subtract xt+1 from xt to find that: xt − xt+1 = −α (rt − r¯t ) (8) The consumer’s utility function is:

106

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve 1

u(c) =

c1− σ − 1 1−

1 σ

Deduce the money demand equation when the utility function of money is specified by: (a) υ(m) = m 1−λ−1 , λ = 1, and υ(m) = log(m), λ = 1 The first-order condition to obtain the demand for money is: 1−λ

υ  (m) =i u (c) From the utility function we get: υ  (m) = m−λ 1

u (c) = c− σ Therefore: i=

υ  (m) m−λ = 1 u (c) c− σ

Taking the logs of both sides we obtain the following demand for money equation: log(m) =

1 1 log(c) − log(i) σλ λ

(b) υ(m) = m (α − β log(m)), β > 0. The marginal utility of money is: 

β υ  (m) = α − β log(m) + m − m or: υ  (m) = α − β log(m) − β Therefore: i= which can be written as:

α − β − β log(m) υ  (m) = 1  u (c) c− σ

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

107

1

c− σ α−β − i log(m) = β β When consumption is constant this is the Cagan money demand equation. (9) When uncertainty is introduced into the transaction cost model, the bank solves the following problem: min {E [iR + c (t, T )]} t

(a) Show that this problem’s first-order condition implies the following turnover ratio:

γ K ∗ t = αET δ 1 where K = E{iT } and γ = 1+β From Macro Theory (p. 174),

t=

α β T , C (t, T ) = t −1 Tδ R β

The problem is:  

δ α β T t −1 T min E i + t t β The first-order condition is:    α δ β−1 1 =0 E iT − 2 + T βt β t Which can be written as:

E

iT t2

 = αt β−1 ET δ

It is straightforward to obtain the optimum turnover ratio: ∗

t =

E (iT ) αET δ



1 β+1

(b) Assume, for simplicity, that T is log-normally distributed. For a normal X, it is known that:

108

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve

1 E exp (τ X) = exp μτ + σ 2 τ 2 2



Show that this expression can be used to calculate the mathematical expectation of T δ : ET δ = E exp (δ log(T )) and obtain:

log(t) = γ log

K α



1 − γ δE log(T ) − γ δ 2 Var log(T ) 2

where Var represents variance. T is log-normally distributed. We may write the expectation of T δ as: δ

ET δ = Eelog T = Eeδ log T = E exp (δ log T ) T is log-normally distributed. Thus, log T is normally distributed, hence: 1

ET δ = eδE log T + 2 γ δ

2 Var log(T )

From the first-order condition:   1 2 E (iT ) = αt β+1 eδE log T + 2 δ Var log T Taking the logs of both sides: log

E (it) 1 = (β + 1) log t + δE log T + δ 2 Var log T α 2

By rearranging terms of this expression we obtain the t ratio equation:



2  K δ 1 δ 1 log − E log T − Var log(T ) log t = 1+β α 1+β 2 1+β (c) In this model, does the volatility of parameter (T ) affect the turnover? From the last equation we obtain: ∂ log t 1 δ2 =− 0 ∂ log i ∂ log i This elasticity is positive because the elasticity of the real quantity of money with respect to the nominal interest rate is negative. (c) Define k = V1 . What is the unit of k? k=

M 1 = V Y

M is measured, say, in dollars and Y in dollars per unit of time. Hence, the unit of k is time. (12) The former German Central Bank (the Bundesbank) carried out monetary programming based on the following identity: MV = P y Assume that potential output of the German economy grows at an annual rate of 2.5%. The Bundesbank annual inflation target was 2.5%. (a) What information did the Bundesbank need to calculate the corresponding growth rate of M? Let a hat denote the rate of growth: xx˙ = x. ˆ From the identity MV = P y we get: Mˆ = Pˆ + yˆ − Vˆ

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

111

The assumptions are Pˆ = 2.5% and yˆ = 2.5%. Hence the Bundesbank would need information to predict the rate of growth of the velocity of money. (b) How would you obtain this information? From exercise (11) the velocity of money is a function of real income and the nominal interest rate: V = V (y, i) To predict V we would need to estimate this function. (c) Assume unstable income velocity. Would you adopt the same method? If V = V (y, i) is unstable we would not have a good forecast for V . Hence, we would not adopt the Bundesbank’s method. (13) Assume that the elasticity of money demand with respect to the interest rate is equal to minus infinity (liquidity trap). (a) Use the identity MV = P y to show why the monetary policy does not affect the economy’s output. Let us assume that the money demand function is specified by: log m = δ +

β , β > 0, γ > 0 i−γ

The elasticity of money demand with respect to the interest rate is given by: ηm,i = i

β ∂ log m = − 2 ∂i i−γ i

i

lim ηm,i = −∞

i→γ

Hence, i = γ is a liquidity trap as shown in Fig. 6.1. Any change in the money stock will be held by the public, without changing the interest rate. Hence, M does not change, if M increases V decreases in the same proportion and vice-versa. (b) Some claim that the liquidity trap is a reasonable assumption when the nominal interest rate approaches zero. Others claim that, under these circumstances, the elasticity must equal zero. How might the issue be resolved? The issue may be resolved by specifying a demand for money function that encompasses the two hypotheses, such as: log m = δ − αi +

β i−γ

112

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve

Fig. 6.1 Money demand: a liquidity trap

The elasticity of money with respect to the nominal interest rate is given by: ηm,i = i

β ∂ log m = −αi + ∂i i−γ

If β = 0, then ηm,i = 0 when i = 0. On the other hand, if α = 0, and γ = 0, ηm,i = −∞ when i = 0. (14) Right, wrong, or maybe. Justify your answer: (a) The balanced budget multiplier (increased government spending=increased taxes) equals zero. The Keynesian IS curve is specified by [Macro Theory, p. 159]:

x = −α (r − r¯ ) + β f − f¯ + g − g¯

(b)

(c)

(d)

(e)

The balanced budget multiplier assumes f = f¯. Thus, the multiplier is the coefficient of g, that is, equal to one. In the short term, the inflation rate depends on monetary policy alone. In the short term demand shocks as well as supply shocks affect the inflation rate. In the long run inflation depends on monetary policy alone. When the monetary policy is expansionary the economy’s real liquidity decreases. When the monetary policy is expansionary the economy’s real liquidity increases in the short run. In the long run, as the nominal rate of inflation increases, real liquidity decreases. Inflation inertia increases the social cost of fighting inflation. If there is inflation, inertia is a backward-looking variable. Hence, the only way to bring inflation down is through recession. The social cost of fighting inflation, under this environment, is unavoidable. The real interest rate is independent from the public deficit in the case of Ricardian equivalence.

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

113

Under Ricardian equivalence the public deficit does not affect any real variable. Hence, the real interest rate does not depend on the public deficit. (f) Increased government spending increases the economy’s real output in both the short and the long run. When there is slack in the economy increased government spending increases real output in the short term. In the long-term there will be crowding out of either consumption or investment. (15) Suppose that an economic model can be represented by the following finite difference equation: yt = αE (yt+1 |It ) + βxt ,

|α| < 1

Show how to obtain this model’s fundamental and bubble solutions, and apply the method to the following cases. (a) Arbitrage between fixed income and variable income (riskless). The difference equation can be written with the forward operator F , F xt = xt+1 , as Et [(1 − αF ) yt − βxt ] = 0 or:  Et yt −

 β xt = 0 1 − αF

The fundamental solution is: f

yt = Et

∞ +

αβ i xt+i

i=0

It is straightforward to verify that: ytb = Cα −t is a solution. This is the bubble solution. The general solution is: f

yt = yt + ytb = Et

∞ +

αβ i xt+i + Cα −t

i=0

The arbitrage equation: E (pt+1 |It ) − pt + dt = rt pt

114

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve

can be written as: pt =

1 dt E (pt+1 |It ) + 1+r 1+r

The general solution is: pt = Et

  ∞ + 1 dt+i + C (1 + r)t 1+r (1 + r)i i=0

The first component is the fundamental solution and the second component is the bubble solution. (b) Determination of the price level according to the Cagan model of money demand: mt − pt = −γ [E (pt+1 |It ) − pt ] Cagan’s demand for money equation can be written as an equation that determines the price level: pt =

γ mt E (pt+1 |It ) + 1+γ 1+γ

The solution of this finite difference equation is: pt = Et

 ∞ + mt+i i=0

(1 + γ )i

+C

γ 1+γ

−t

The first part is the fundamental solution and the second part is the bubble solution. (16) Consider the model: IS: y = −αi + u LM: m = −βi + γ y + υ Where u and v υ are random, non-correlated, variables with averages equal to zero and the respective variances σu2 and συ2 .The Central Bank’s loss function is given by: L = y2 The Central Bank may choose between the interest rate (i) or the quantity of money (m) as a policy instrument. (a) What value of m minimizes the expected value of the loss function? Solving the IS and LM equations we obtain the equation for output:

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

α m+ y= β + αγ

β β + αγ

115

  α u− υ β

If y is output gap m = 0 (deviation from the trend) minimizes the expected value of the loss function. The expected value of the loss function is:  Lm = Ey = Et 2

β β + αγ

 2 α u− υ β

which can be written as: Lm =

β 2 σu2 + α 2 συ2 (β + αγ )2

(b) What value of i minimizes the expected value of the loss function? The IS equation is: y = −αi + u If y is output gap i = 0 (deviation from trend) minimizes the expected value of the loss function, which is given by: L = Ey 2 = σu2 (c) What instrument should the Central Bank choose? The Central Bank should choose the interest rate as its instrument if: σu2
0 The (log of) the price set by the firm in t, when it receives the sign is: 

∞

υt = t

(ps + αxs ) δe−δ(s−t) ds, α > 0

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

119

where ps is the price level and xs is excess demand, both in period s. The (log of) price level (p) is defined by the formula:  pt =

T −∞

υs δe−δ(t−s) ds

(a) Show that the average of the random variable H is given by: 1 δ

EH = The expected value of H is defined by: 

∞

EH =



∞

hf (h)dh =

0

hδe−δh dh

0

We use integration by parts: 

 udυ = uυ −

υdu

to compute the integral: 

∞

hδe

−δh

dh = −he

)∞  ) −

−δh )

0

0

∞

 −e−δh dh

0

Since: )∞ ) −he−δh ) = 0 0

we may write: 

∞

hδe−δh dh =



0

∞ 0

e−δh =

1 δ

Therefore, EH =

1 δ

(b) Make explicit the arguments that justify the expression of υt , and pt . The price υt is given by: 

∞

υt = t

(ps + αxs ) δe−δ(s−t) ds, α > 0

120

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve

The probability that the sign will be received in s − t periods from t is δe−δ(s−t) and ps + αxs is the price set by the firm that receives the sign. The price pt is given by:  pt =

t

−∞

υs δe−δ(t−s) ds,

where υs is the price set at s and e−δ(t−s) is the percentage of firms that set that price. (c) Take the derivative of υt and pt with respect to time, applying Leibnitiz rule, and show that: υ˙ = δ (υ − p − αx) π = p˙ = δ (υ − p) The Leibnitz rule (Macro Theory, p. 337) is given by: 

β(r)

V (r) =

f (x, r) dx α(r)

and: dV (r) dβ(r) dα(r) = f (β(r), r) − f (α(r), r) + dr dr dr



β(r)

α(r)

∂f (x, r) d ∂r

Applying this rule to υt we obtain: dυt = − (p(t) + αx(t)) δ + dt



∞

δ (ps + αxs ) δe−δ(s−t) ds

t

or: dυt = − (p(t) + αx(t)) δ + δυt dt By rearranging terms, we get: dυt = δ (υ(t) − p(t) − αx(t)) dt Applying Leibinitz rule to pt we obtain: dpt = υ(t)δ + dt



t

−∞

υs δe−δ(t−s) (−δ) ds

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

which can be written as: dpt = υ(t)δ − δp(t) dt Hence, π=

dpt = δ (υ(t) − p(t)) dt

(d) Take the derivative of π with respect to time and show that: π˙ = −αδ 2 x The rate of inflation obtained in the previous item is: π = δ (υ − p) We take the time derivative of this expression to get: π˙ = δ (υ˙ − p) ˙ The derivative of υ (item c) is: υ˙ = δ (υ − p − αx) and p˙ = δ (υ − p) We substitute these two expressions in the equation of π˙ to obtain: π˙ = δ (υ˙ − p) ˙ = δ (δ (υ − p − αx) − δ (υ − p)) By simplifying yields: π˙ = −αδ 2 x This is the well known Calvo Phillips curve as derived by him. (e) Suppose that υt is given by: 

∞

υt =

e−ρ(s−t) (ps + αxs ) δe−δ(s−t) ds, α > 0

t

where ρ is the discount rate. Show that:

121

122

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve

π˙ = ρδp + ρπ − αδ 2 x We take the time derivative of υt applying Leibinitz rule to obtain: dυt = −δ (pt + αxt ) + dt



∞

e−ρ(s−t) (ps + αxs ) δe−δ(s−t) (ρ + δ) ds

t

or: υ˙ = −δ (p + αx) + (ρ + δ) υ The inflation rate change and the inflation rate are given by: π˙ = δ (υ˙ − p) ˙ and π = p˙ = δ (υ − p) Hence: υ˙ − p˙ = ρυ − αδx Therefore: π˙ = δρυ − αδ 2 x We use the inflation rate equation to get rid of υ, to obtain: π˙ = ρδp + ρπ − αδ 2 x (f) Is the previous item’s Phillips curve vertical in the long run? When π˙ = 0: x=

ρ ρ p + 2π αδ αδ

Therefore, the Phillips curve is not vertical in the long run. (21) Consider the Phillips curve model: π = π e + δ (y − y) ¯ Assume that the expected inflation rate follows the adaptive expectations mechanism:

6

Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .

123

π˙ e = β π − π e Show that the acceleration of the inflation rate is: π˙ = βδ (y − y) ¯ + δ y˙ when the growth rate of potential output is equal to zero y˙¯ = 0. We take the derivative of both sides of the Phillips curve to obtain: π˙ = π˙ e + δ y˙ By combining the Phillips curve and the adaptive expectation mechanism results:

¯ π˙ e = β π − π e = βδ (y − y) We substitute this into the previous expression to obtain: π˙ = βδ (y − y) ¯ + δ y˙ We may conclude that the acceleration of inflation depends on the output gap and also on the rate of change of the output gap. (22) The new Keynesian Phillips curve with perfect foresight is: πt = βπt+1 + δxt Show that the New Keynesian curve with continuous variable is: π˙ = ρπ − kx δ where ρ = 1−β β and k = − β . The New Keynesian Phillips curve can be written as:

πt = β (πt+1 − πt ) + βπt + δxt which can be rewritten as: (1 − β) πt − δxt = βπt+1 Hence: πt+1 =

1−β δ πt − xt β β

124

6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve 1 The discount factor β = 1+ρ . Thus, 1−β β = ρ. Let us denote New Keynesian Phillips curve with continuous variable is:

π˙ = ρπ − kx

δ β

= k. Hence, the

Chapter 7

Economic Fluctuation and Stabilization

(1) Consider the model: IS: x˙ = σ (i − π − r¯ ) PC: π˙ = δx MPR: i = r¯ + π + φ (π − π¯ ) + θ x IC: Given p(0) and π(0) Analyze the consequences of an unanticipated permanent change in the inflation target. The dynamical system of this model is given by the two linear differential equations: 

x˙ = σ φ (π − π¯ ) + σ θ x π˙ = δx

The first equation was obtained by combining the IS curve and the MPR. The Jacobian of this system is: J =

 ∂ x˙

∂ x˙ ∂x ∂π ∂ π˙ ∂ π˙ ∂x ∂π



  σφ σθ = δ 0

The determinant of this matrix is given by: |J | = −σ θ δ < 0 and it is negative. Thus, the steady-state is a saddle point. Figure 7.1 depicts the phase diagram of the model with the inflation rate on the vertical axis and the output gap on the horizontal axis. The saddle path SS is downward sloping. Figure 7.2 shows an unanticipated permanent change in the inflation target, from π¯ 0 to π¯ 1 < π¯ 0 . Figure 7.3 shows the dynamic adjustment of the economy

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_7

125

126

7 Economic Fluctuation and Stabilization

Fig. 7.1 The phase diagram for the π and x system

Fig. 7.2 An unanticipated permanent decrease in the inflation target

Fig. 7.3 Dynamic adjustment of the economy to an unanticipated permanent decrease in the inflation target

to the inflation target change. To simplify the figure, we just draw the saddle path SS after the inflation target change. The initial conditions of the model state that inflation is a predetermined variable. Thus, the economy jumps from the initial equilibrium (E0 ) to the point Ei on the saddle path. From then onwards the economy converges on the new equilibrium (Ef ). (2) Consider the model: IS: x˙ = −λx − α (r − r¯ ) PC: π˙ = δx

7 Economic Fluctuation and Stabilization

127

MPR: i = r¯ + π + φ (π − π¯ ) + θ x IC: Given p(0) and π(0) (a) Analyze the model’s equilibrium and dynamics on a phase diagram, with the inflation rate (π ) on the vertical axis and the output gap (x) on the horizontal axis. The dynamical system of this model is given by the two linear differential equations: 

x˙ = − (λ + αθ ) x − αφ (π − π¯ ) π˙ = δx

The first equation was obtained by combining the IS curve and the MPR. The Jacobian of this system is: J =

 ∂ x˙

∂ x˙ ∂x ∂π ∂ π˙ ∂ π˙ ∂x ∂π



 =

− (λ + αθ ) −αφ δ 0



The determinant and the trace of this matrix are given by: |J | = αφδ > 0 trJ = − (λ + αθ ) < 0 The determinant is positive and the trace is negative. Thus the model is stable. Figure 7.4 depicts the phase diagram of the model with inflation on the vertical axis and the output gap on the horizontal axis. (b) Use the phase diagram from item (a) to show what happens in this model when the Central Bank raises the inflation rate target. Fig. 7.4 The phase diagram for the π and x system

128

7 Economic Fluctuation and Stabilization

Figure 7.5 depicts the phase diagram when the inflation target increases to π¯ 1 from π¯ 0 < π¯ 1 . In this model the initial conditions stipulate that inflation is a predetermined variable. Thus, at the initial moment when the change occurs, inflation does not change and the output gap becomes positive. The phase diagram of Fig. 7.5 assumes inflation rate overshooting. In the long run it converges on the new inflation target. (c) What happens in this model if the parameter φ is negative? How do you interpret the monetary policy rule in this case? If the parameter φ is negative the determinant of the Jacobian matrix is negative: |J | = αφδ < 0 The equilibrium of the model is a saddle point as depicted in Fig. 7.6. The MPR can be written as: i = r¯ + π¯ + (1 + φ) (π − π¯ ) + θ x Fig. 7.5 The dynamic adjustment of the economy to an unanticipated increase in the inflation target

Fig. 7.6 The phase diagram for the π and x system when φ π¯ 0 from π¯ 0 ? Figure 7.9 depicts the phase diagram of the adjustment of the economy when the inflation target increases. The economy jumps from point E0 to point Ei because the inflation rate is a predetermined variable. From then onwards the economy converges on the new equilibrium (π = π¯ 1 and x = 0).

Fig. 7.8 The phase diagram for the π and x system when φ 0 PC: π˙ = −δx, δ > 0 MPR: i = r¯ BC + π + φ (π − π¯ ) + θ x IC: Given p(0) (a) The dynamical system of this model is given by a system of two equations: 

BC

α x = − 1+αθ r¯ − r¯ − π˙ = δx

αφ 1+αθ

(π − π¯ )

The first equation of this model was obtained by combining the IS curve and the MPR. In the steady-state equilibrium the output gap is zero (x = 0), but the inflation rate π ∗ is given by: π ∗ = π¯ −

 1  BC r¯ − r¯ φ

Thus, if r¯ BC = r¯ , π ∗ = π¯ , as shown in the phase diagram of Fig. 7.10. (b) How would you interpret r¯ = r¯ BC ? There are two ways to interpret this hypothesis. One is that the Central Bank estimate of the natural rate of interest is biased. The other is that the operational procedure uses a real interest rate. (5) Consider the model: IS: x = −α (r − r¯ ) , α > 0 PC: π˙ = −δx, δ > 0 MPR: i = r¯ + π + φ (π − π¯ ) + θ x, φ > 0, θ > 0 IC: Given p(0) and π(0) This exercise is a particular case of exercise (2) when λ = 0. Fig. 7.10 The phase diagram for the π and x system

132

7 Economic Fluctuation and Stabilization

(6) Consider the model: IS: u − u¯ = α (r − r¯ ) , α > 0 PC: π˙ = −δ (u − u) ¯ , δ>0 MPR: i˙ = ϕ (π − π) ¯ − θ u˙ IC: Given p(0) and π(0) The symbols have the following meanings: u = unemployment rate; u¯ = natural unemployment rate, i = di ˙ = du dt , u dt . (a) How would you obtain this model’s IS curve? Okun’s Law (Macro Theory, p. 180, equation 6.66) is given by: y − y¯ = x = −b (u − u) ¯ The IS curve is specified as: x = −a (r − r¯ ) By combining Okun’s Law with the traditional IS curve we obtain: −b (u − u) ¯ = −a (r − r¯ ) which can be written as: u − u¯ = a (r − r¯ ) , α =

a >0 b

(b) The MPR, di ¯ ) − θ u, ˙ does not need any information on nondt = ϕ (π − π observable variables, such as the natural rate of interest and the natural unemployment rate. (c) Analyze the model’s equilibrium and dynamics on a phase diagram with inflation on the vertical axis and the unemployment rate on the horizontal axis. Is this model stable? We start taking derivatives with respect to time of the IS curve:

 di u˙ = α − π˙ dt Next, we use the PC and the MPR to obtain: u˙ = αϕ (π − π¯ ) − αθ u˙ + αδ (u − u) ¯ Therefore, the dynamical system of this model is given by: u˙ =

αϕ αδ (u − u) ¯ + (π − π¯ ) 1 + αδ 1 + αθ π˙ = −δ(u − u) ¯

The Jacobiano of this system is:  ∂ u˙ ∂ u˙   αδ J = ∂∂uπ˙ ∂∂ππ˙ = 1+αθ −δ ∂u ∂π

αϕ 1+αθ

0



7 Economic Fluctuation and Stabilization

133

Fig. 7.11 The phase diagram for the π and x system

The determinant and trace of this matrix are: |J | =

δαϕ >0 1 + αθ

trJ =

αδ >0 1 + αθ

Both the determinant and the trace are positive. Therefore, the system is unstable. Thus, albeit the MPR is interesting for not requiring information on non-observable variables, if the Central Bank does use this MPR the economy will not reach the inflation target. Figure 7.11 depicts the phase diagram of the model. (d) Assume that the model’s initial inflation is an endogenous variable and the parameter δ is negative. Analyze the model’s equilibrium and dynamics under these circumstances. If δ is negative, the determinant of the Jacobian matrix is negative: δαϕ 0 This model has two positive characteristic roots and one negative root. The model is unstable and has just one equilibrium. (e) Would you recommend using this monetary policy rule? If the world were a Keynesian world I would not recommend the monetary rule because the Central Bank would not be able to deliver its inflation target. On the other hand, if the world were New Keynesian the monetary policy rule would achieve the goal of the Central Bank. We would not recommend this monetary policy rule because we do not know what the true model is. (7) Consider the model: IS: x = −α (r − r¯ ) PC: π˙ = δx MPR: i = r¯ + π + φ (π − π¯ ) + θ x (a) In this model, is the long run inflation a monetary phenomenon? In strict sense the long run inflation (π¯ ) is not a monetary phenomenon because we do not need the stock of money to determine inflation, either in the short run or in the long run. (b) Assume that LM curve is specified by:

LM : m − m ¯ = λx − β i − i¯ where m is the real quantity of money, m = M P . Does the inflation rate target equal the growth rate of monetary base in the long run? In the long ¯ Thus m = m run x = 0 and i˙ = i. ¯ =constant. Thus, the rate of growth of money is equal to the rate of growth of the price index. π¯ =

M˙ P˙ = P M

7 Economic Fluctuation and Stabilization

135

(c) Does the monetary policy rule that sets the interest rate imply that inflation is not a monetary phenomenon in the long run? The monetary policy rule that sets the interest rate implies that money is an endogenous variable in the long run. In the long run inflation is a monetary phenomenon in the sense that inflation is equal to the rate of growth of money. (d) Based on this model, might one argue that if the society’s objective is to reduce the interest rate, the interest rate must rise initially? Society does not control the long run real interest rate, the natural rate of interest. Society does control the long run nominal rate of interest. According to the MPR if society desires to have less inflation in the long run it has to increase the interest rate initially. (8) Consider the model: IS: x = −α (r − r¯ ) PC: π˙ = δx MPR: i s = r¯ s + π + θ (π − π¯ ) + φx Credit: i = i s + sp ¯ + β (sp − sp) IC: Given p(0) and π(0) Definitions: i = r − π, r¯ = r¯ s + sp The symbol sp represents the spread of the interest rate on the credit market, and sp is the long run equilibrium spread. Substituting the MPR equation into the credit equation we obtain: i = r¯ + π + θ (π − π¯ ) + φx + β (sp − sp) The interest rate gap provided by this equation can be substituted into the IS curve: x=−

αθ αβ ¯ − (π − π) (sp − sp) 1 + αθ 1 + αφ

(a) Show that this model’s aggregate demand equation is given by: π = π¯ −

1 + αφ β x − (sp − sp) αθ θ

From the previous equation it is straightforward to obtain the aggregate demand equation. (b) Use a phase diagram (π on the vertical axis and x on the horizontal axis) to show what happens to the output gap and the inflation rate when a credit market shock causes sp − sp > 0. Figure 7.12 depicts the phase diagram of the model. The credit shock shifts the curve D0 D0 to D1 D1 . The rate of inflation is a predetermined variable. Thus, at the moment of the shock the economy enters into a recession. The initial output gap becomes negative (x0+ ).

136

7 Economic Fluctuation and Stabilization

Fig. 7.12 Dynamic adjustment of the economy to an unanticipated credit shock

Then, from that point onward the economy converges on full employment (Ef ). (9) Consider the model: IS: x˙ = −α (r − r¯ ) , α > 0 PC: π˙ = −γ (π − π¯ ) + δx, δ > 0 MPR: i = r¯ + π + φ (π − π¯ ) + θ x, φ > 0, θ > 0 IC: Given p(0) and π(0) (a) Analyze the model’s equilibrium and dynamics on a phase diagram with π on the vertical axis and x on the horizontal axis. The dynamical system of this model is given by a system of two linear differential equations: 

x˙ = −αφ (π − π¯ ) − αθ x π˙ = −γ (π − π¯ ) + δx

The first equation of this system was obtained by combining the IS curve and the MPR. The Jacobian of this system is: J =

 ∂ x˙

∂ x˙ ∂x ∂π ∂ π˙ ∂ π˙ ∂x ∂π

 =

  −αθ −αφ δ −γ

The determinant and the trace of this matrix are:

7 Economic Fluctuation and Stabilization

137

Fig. 7.13 The phase diagram for the π and x system

Fig. 7.14 Dynamic adjustment of the economy to an unanticipated decrease in the inflation target

p p0 E0

Ef

.

x=0 x

|J | = αθ γ + αφδ > 0 trJ = − (αθ + γ ) < 0 The system is stable because the determinant is positive and the trace is negative. Figure 7.13 depicts the phase diagram of the model. (b) Use the previous item’s phase diagram to show this model’s dynamics when the Central Bank reduces the inflation target. Figure 7.14 shows the phase diagram describing the economy adjustment to a reduction of the inflation target. The economy enters into a recession and the inflation rate decreases. It is possible that there will be an undershooting of the inflation rate as the economy converges on full employment and the new inflation target. (c) What would happen in this model if the parameter φ were negative? The determinant of the Jacobian matrix is given by:

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7 Economic Fluctuation and Stabilization

|J | = α (θ γ + φδ) If φ < 0 in such way that: θ γ + φδ > 0 The model is stable as shown in Fig. 7.13. If φ < 0 and |J | = α (θ γ + φδ) < 0, the steady-state is a saddle point. (10) Consider the model: IS: x˙ = −α (r − r¯ ) , α > 0 PC: π˙ = δx, δ > 0 MPR: i = i¯ IC: Given p(0) and π(0) (a) Analyze the model’s equilibrium and dynamics on a phase diagram with π on the vertical axis and x on the horizontal axis. The dynamical system of this model is given by a system of two linear differential equations: 



x˙ = −α i¯ − π − r¯ π˙ = δx

The Jacobian of this system is: J =

 ∂ x˙

∂ x˙ ∂x ∂π ∂ π˙ ∂ π˙ ∂x ∂π



  0α = δ 0

The determinant of this matrix is: |J | = −αδ < 0 The steady-state of this model is a saddle point because the determinant is negative. Figure 7.15 depicts the phase diagram of the model. The implicit inflation target is given by the difference between the fixed nominal rate of interest and the natural rate of interest π¯ = i¯ − r¯ . (b) Use the previous item’s phase diagram to show the model’s dynamics when the Central Bank reduces the inflation rate. The Central Bank reduces the nominal rate of interest, implying a cut in the inflation target. The inflation rate is a predetermined variable. The economy jumps from E0 to Ei (Fig. 7.16) yields a recession, the output gap becomes x(0+ ). From that point onwards the economy converges on the new equilibrium (Ef ), with full employment and inflation given by: π¯ 1 = i¯1 − r¯ (Fig. 7.16).

7 Economic Fluctuation and Stabilization

139

Fig. 7.15 The phase diagram for the π and x system

Fig. 7.16 Dynamic adjustment of the economy to an unanticipated decrease in the nominal interest rate

(11) Consider the model: IS: x˙ = γ x + σ (r − r¯ ) PC: π˙ = δx MPR: i = r¯ + π + φ (π − π¯ ) + θ x IC: Given p(0) The dynamical system of this model is given by a system of two differential equations: 

x˙ = (γ + σ θ ) x + σ φ (π − π¯ ) π˙ = δx

The Jacobian of this system is:

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7 Economic Fluctuation and Stabilization

J =

 ∂ x˙

∂ x˙ ∂x ∂π ∂ π˙ ∂ π˙ ∂x ∂π



  γ + σθ σφ = δ 0

The determinant and the trace of this Jacobian are: |J | = −σ φδ trJ = γ + σ θ (a) In equilibrium: π˙ = 0 = x. ˙ This implies π = π¯ and x = 0. If all parameters were positive, the equilibrium would be a saddle point, and π(0) would be a predetermined variable. (b) What are the model’s properties when γ = 0 and δ < 0 (new Keynesian model)? Under these assumptions the determinant and the trace of the Jacobian matrix are: |J | = −σ φδ > 0 trJ = σ θ The model would be unstable and the steady-state would be a unique equilibrium. (c) What are the model’s properties when δ < 0, δ > 0, and σ < 0 (Keynesian model)? Under these assumptions the determinant and the trace of the Jacobian matrix are: |J | = −σ φδ > 0 trJ = σ θ < 0 The model is stable because |J | > 0 and trJ < 0. The inflation rate is a predetermined variable. (12) Consider the model: IS: x˙ = −α (r − r¯ ) , α > 0 PC: π˙ = δx MPR: di ¯ ) + θx dt = φ (π − π (a) What condition must the parameter θ meet for the model to be stable? The dynamical system of this model is given by a system of three linear differential equations:

7 Economic Fluctuation and Stabilization

141

⎧ ⎨ x˙ = −α (i − π − r¯ ) π˙ = δx ⎩ di ¯ ) + θx dt = φ (π − π The Jacobian of this system is given by: ⎡ ∂ x˙ J =

∂x ⎣ ∂ π˙ ∂x ∂i ∂x

∂ x˙ ∂π ∂ π˙ ∂π ∂i ∂π

∂ x˙ ⎤ ∂i ∂ π˙ ⎦ ∂i ∂i ∂i

⎡

⎤ 0 α −α = ⎣δ 0 0 ⎦ θ φ 0

The determinant and trace of this matrix are: |J | = −γ δφ < 0 trJ = 0 As you may recall the determinant is equal to the product of the characteristic roots. The determinant is negative. Thus, the system has either three negative roots or two positive roots and one negative root. The trace, which is equal to the sum of the characteristic roots, is equal to zero. Thus, we can rule out three negative roots. The system has a saddle path because it has two positive roots and one negative root. On the other hand, the model has two predetermined variables, the inflation rate and the nominal interest rate. Therefore, the dynamical system is unstable regardless of the sign of the parameter θ . (b) Assume that α < 0 and δ < 0. Does the model have multiple equilibria? Under these assumptions the determinant of the Jacobian is negative and the trace is equal to zero. Therefore, two roots are positive and one root is negative. Because δ < 0 the inflation rate is a jump variable. The only predetermined variable is the nominal interest rate. Therefore, the model has a stable path and no multiple equilibria. (c) Would you recommend that the Central Bank should use this monetary policy? No, we would not recommend this monetary policy rule because it is not robust to the hypothesis of the model. (13) Consider the New Keynesian model: IS: x˙ = σ (r − r¯ ) PC: π˙ = ρπ − kx MPR: r − r¯ = φ (π − π¯ ) + θ (x − x) ¯ (a) The inflation target π¯ and the output gap x¯ are such that x¯ = ρkπ¯ . The dynamical system of this model has two linear differential equations:

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7 Economic Fluctuation and Stabilization



x˙ = σ φ (π − π¯ ) + σ θ (x − x) ¯ π˙ = ρπ − kx

The first equation was obtained by combining the IS curve and the MPR. The Jacobian of this system is given by: J =

 ∂ x˙

∂ x˙ ∂x ∂π ∂ π˙ ∂ π˙ ∂x ∂π



 =

σθ σφ −k ρ



The determinant and trace of this matrix are: |J | = σ (θρ + kφ) > 0 trJ = σ θ + ρ > 0 The system is unstable and it has two jump variables, the inflation rate and the output gap. Therefore, the model has a unique equilibrium. (b) Is the monetary policy neutral in the long run? In the long run when π˙ = 0, the output gap depends on the inflation rate according to: x¯ =

ρ π¯ k

Therefore, for each inflation target there is an output gap. The trade-off depends on the ratio of the rate of time preference (ρ) and the output gap coefficient of the Phillips curve. (14) *(Discounted Euler Equation): The canonical new Keynesian model implies that forward guidance interest rate policy has implausible effects on current output gap. The “discounted Euler equation” introduces a coefficient less than one in the expected future output gap that discounts the effects of future interest rates. (a) Show how this specification solves the “forward guidance puzzle”. (b) Specify this new Keynesian IS curve using continuous variables and show the difference between forward looking and backward looking behavior. (c) The “discounted Euler equation” approach assumes that there are heterogeneous agents and the Euler equation is given by:   −γ −γ CH,t = β (1 + rt ) Et wCH,t+1 + (1 − w) M −γ where the index H denotes one type of agent, w is probability of the agent to be employed, M is the amount of consumption goods received by the agent

7 Economic Fluctuation and Stabilization

143

when unemployed, and γ is the coefficient of relative risk aversion. Aggregate consumption Ct is given by: Ct = wCH,t + (1 − w) M What is the level of aggregate consumption in stationary equilibrium CH,t = CH,t+1 = CH ? (a) The discounted specification of the IS curve is given by: xt = αEt xt+1 − σ (rt − r¯ ) , 0 < α < 1

(7.1)

The solution of this difference equation is: xt = −σ Et

∞ +

α i (rt+i − r¯ )

i=0

The weight α i goes to zero as i → ∞. When α = 1 the solution of this IS curve is not given by: xt = −σ Et

∞ +

(rt+i − r¯ )

i=0

as shown in exercise C of the Appendix C. (b) With continuous variables the IS curve (7.1) becomes: x˙ = λx + φ (r − r¯ ) , λ > 0, φ > 0

(7.2)

The solution of this differential equation is: 

∞

x=−

φe−λ(τ −t) (r − r¯ ) dτ

t

When λ < 0 and φ < 0 the IS is the old Keynesian curve, which is backward looking:  x=

t

−∞

φeλ(t−τ ) (r − r¯ ) dτ

We conclude that IS reduced form (7.2) can be used to test whether the IS curve is forward or backward looking. In the canonical new Keynesian model λ = 0 and the interest rate affects not the level but the change of output gap. In the new specification the interest rate affects both the level and the change in the output gap, like the old Keynesian IS curve.

144

7 Economic Fluctuation and Stabilization

(c) The discounted Euler equation can be written as: 

   M −γ 1+ρ CH,t+1 −γ = Et w + (1 − w) 1 + rt CH,t CH,t

(7.3)

Next, we use the following approximations:

CH,t+1 CH,t

−γ

M CH,t

=e

−γ

−γ log

=e

CH,t+1 CH,t

−γ log

M CH,t

= e−γ (cH,t+1 −cH,t ) ∼ = 1−γ cH,t+1 − cH,t

= e−γ (m−cH,t ) ∼ = 1 − γ m − cH,t

Substituting these approximations into (7.3) results: 



1+ρ = Et w 1 − γ cH,t+1 − cH,t + (1 − w) 1 − γ m − cH,t 1 + rt By simplifying and rearranging terms we obtain: cH,t = wEt cH,t+1 + (1 − w) m −

1 (rt − ρ) γ

In steady-state, rt = ρ, cH,t = cH,t+1 = cH Therefore: cH = m ⇒ CH = M Aggregate consumption is: Ct = wCH,t + (1 − w) M In steady-state: C=M This conclusion is at odds with consumption theory. The traditional Euler equation is a statement about change in levels, but not about levels. The discounted Euler equation is a statement about level. The former needs the budget constraint to determine the level of consumption. The discounted Euler equation rules out the budget constraint. At best it could be used to determine M.

Chapter 8

Open Economy Macroeconomics

(1) Consider the following model of a small open economy (monetary approach of the balance of payments with a fixed exchange rate): Ms = C + R M d = P L(y, i) i = i∗ y = y¯ P = EP ∗ , E = E¯ = constant (a) What is the effect of a foreign exchange devaluation on the balance of payments? This model assumes that the money market is in equilibrium: Md = Ms Therefore, C + R = P L (y, i) ¯ we obtain: By using the hypothesis that P = EP ∗ , i = i ∗ and y = y, ∗

¯ i −C R = EP ∗ L y, Therefore, ∗

∂R = P ∗ L y, ¯ i ∂E The effect of a foreign exchange devaluation on the balance of payments is to increase the stock of international reserves.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_8

145

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8 Open Economy Macroeconomics

(b) What is the effect of an increase in net domestic credit on the balance of payments? The partial derivative of R with respect to C is: ∂R = −1 < 0 ∂C Therefore, the effect of an increase in net domestic credit on the balance of payments is to decrease the stock of international reserves. (2) Consider the following model of a small open economy (monetary approach of the balance of payments with a flexible exchange rate): Equilibrium in Country A’s money market: M = L (y, i) P Equilibrium in Country B’s money market:

M∗ = L y∗, i∗ ∗ P Exchange rate: E=

P P∗

Comment on the following: (a) The exchange rate depreciates when the country grows more quickly than others. We use the exchange rate definition and the two money market equilibrium equations to write: E=

M L(y,i) M∗ L(y ∗ ,i ∗ )

=

M L (y ∗ , i ∗ ) M ∗ L (y, i)

(8.1)

∗

y When y y > y ∗ the answer to this item depends on the income elasticities in the two countries. If the income elasticities are equal, E decreases (appreciate). (b) The exchange rate appreciates when the money stock grows more quickly M ∗ than other countries. When M M > M ∗ it is straightforward to see from Eq. (8.1) that E increases, e.g., it depreciates.

(3) Consider the following model of a small open economy (monetary approach of the balance of payments with a flexible exchange rate): Equilibrium in Country A’s money market:

8 Open Economy Macroeconomics

147

m − p = αy − βi Equilibrium in Country B’s money market: m∗ − p∗ = αy ∗ − βi ∗ Exchange rate: e = p − p∗ Uncovered interest parity: i = i ∗ + e˙ (a) Deduce the differential equation for exchange rate determination. We use the exchange rate definition and the two money market equilibrium equations to write: e = m − αy + βi − m∗ + αy ∗ − βi ∗ which can be written as:



e = β i − i ∗ + m − m∗ − α y − y ∗ We substitute the uncovered interest parity condition into this equation to obtain:

e = β e˙ + m − m∗ − α y − y ∗ which can be written as: e˙ =

1 e−f β

where: f =

m − m∗ − α (y − y ∗ ) β

(b) Does the solution to this equation include a bubble component? The solution of the exchange rate differential equation is: 

∞

e=

e

− β1 (τ −t)

f dτ

t

and the second component is the bubble solution: 1

eb = Ce β (4) Consider the regression:

t

148

8 Open Economy Macroeconomics

et+1 − et = a0 + a1 (ft − et ) + t or:

et+1 − et = a0 + a1 it − it∗ + t (a) Is the forward market’s exchange rate, or the differential of the interest rate, a good predictor of the future exchange rate on forward market? The UIP is given by: it = it∗ + et+1 − et and the CIP is expressed by: it = it∗ + ft − et By comparing the two equations it follows that: et+1 = ft Therefore, the coefficient a1 of the expression should be equal to one: a1 = 1 Therefore, the forward rate or the interest rate gap should be a good predictor of the future exchange rate. (b) Several empirical studies have obtained negative values for the a1 parameter. How would you interpret this result? One possibility to explain why a1 < 0 is to include a risk premium in the UIP, such as: it = it∗ + et+1 − et + ρt where ρt is a risk premium that changes over time. Thus, if this is the case, the regressions of this exercise have a specification error that bias the estimates. (5) Harberger-Laursen-Meltzler (HLM) Effect. In an open economy, the national product (Y ) equals the sum of absorption (A) and the balance of the current account on the balance of payments (X − Z): Y =A+X−Z The price index of absorption is a geometric average of the price of the domestically product (P ) and the price of the foreign product, converted into

8 Open Economy Macroeconomics

149

the domestic currency at the exchange rate (SP ∗ ):

α Pa = P 1−α EP ∗ = P

EP ∗ P

α = P Sα

where α is the share of the imported good in absorption and S = terms of trade. The product may be written in real terms as:

EP ∗ P

is the

y = d + x − Sz Z where d = PPa a , X = xP and z = SP ∗ . Real absorption (a) depends on the real income as defined by: ya = P y/Pa

(a) Show that the elasticity of expenditure with respect to the terms of trade ηd,s is:

ηd,s = α 1 − ηa,ya where na,ya is the elasticity of absorption with respect to the real income. National output is equal to absorption added to net-exports: Y =A+X−M which can be written as: P y = Pa a + P x − EP ∗ cm or: y=

EP ∗ Pa a +x− cm P P

d=

EP ∗ Pa a and S = P P

We define:

Therefore: y = d + x − Scm Domestic expenditure is the sum of domestically produced consumer goods and imported goods: d = cd + Scm

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8 Open Economy Macroeconomics

The absorption price index is a geometric average of domestic and imported goods: Pa = P

EP ∗ P

α = Sα

Using the price index real domestic expenditure can be written as: d=

Pa a = Sα a P

ya =

Py = S −α y Pa

Real income is defined by:

and we take derivative of d with respect to S to obtain: ∂d ∂a = αS α−1 a + S α ∂S ∂S By taking into account that: ∂a ∂ya ∂a = ∂S ∂ya ∂S we write the previous expression as: d ∂d ∂a ∂ya = α + Sα ∂s S ∂ya ∂S or: ∂d d aS α ∂a ya ∂ya S =α + ∂s S S ∂ya a ∂S ya By rearranging the terms of this expression we obtain: d d ∂a ya ∂ya S ∂d =α + ∂s S S ∂ya a ∂S ya From the definition of ya we get: ∂ya S = −α ∂S ya and we define:

8 Open Economy Macroeconomics

151

ηa,ya =

∂a ya ∂ya a

to write the previous expression as:

S ∂d = α + ηa,ya (−α) = α 1 − ηa,ya d ∂S Therefore:

ηd,s = α 1 − ηa,ya where: ηd,s =

S ∂d d ∂S

(b) Show that saving (s = y − d − τ , where τ represents taxes) varies with the terms of trade:

∂s αd = ηa,ya − 1 ∂S S The partial derivative of savings with respect to the terms of trade is: ∂y ∂d ∂s = − ∂S ∂S ∂S

  S d d S

However, ∂y =0 ∂S and ∂s d = − ηd,S ∂S S From the previous item we get:

d ∂s = − α 1 − ηa,ya ∂S S Therefore:

αd ∂s = ηa,ya − 1 ∂S S

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8 Open Economy Macroeconomics

(c) According to the HLM effect, worsening (improving) terms of trade decrease (increase) the economy’s real income, reducing (increasing) saving. For a given level of investments, the reduction (increase) in saving deteriorates (improves) the current account on the balance of payments. What happens to the current account on the balance of payments if ηa,ya < 1 and the country’s terms of trade improve? From item (b), if ηa,ya < 1,

αd ∂s = ηa,ya − 1 < 0 ∂S S Therefore, in this case, if the country’s terms of trade improve the current account on the balance of payments deteriorates. (6) Consider the model: ⎧ ∗ ˙ W ⎪ ⎪ M = m (i, ∗i + e) ⎪ ⎪ ˙ W ⎪ ⎨ B = b (i, i + e) ˙ W EF = f (i, i ∗ + e) ⎪ ⎪ W = M + B + EF ⎪   ⎪ ⎪ ⎩ F˙ = ϕ EP ∗ + i ∗ F P (a) Discuss the specification of each of the model’s equations and analyze its equilibrium. This exercise deals with the portfolio balance model. It assumes that foreign assets and domestic bonds are not perfect substitutes. Net financial wealth has three components money (M), domestic bonds (B) and foreign assets (F in foreign currency and EF in domestic currency, i is the domestic interest rate, i ∗ is the foreign interest rate and e˙ is the expected d rate of change of the exchange rate e˙ = dt log E . The asset demand equations are represented by m( ), b( ) and f . Its partial derivatives have the following signs: mi < 0, m∗i < 0 bi > 0, bi∗ < 0 fi < 0, fi∗ > 0 where the index denotes a partial derivative, e.g., mi = solution, we specify demand equations as: M ∗ = e−αi e−β (i +e˙) W

∂m ∂i .

To simplify the

8 Open Economy Macroeconomics

153

EF ∗ = e−γ i eδ (i +e˙) W B M EF =1− − W W W Let us denote letters m and f the following ratios (in logs):

log

M W



= m; log

EF W

 =f

In this model one equation is redundant due to the constraint. Therefore, we use the two equations, one for money and the other for foreign assets, to determine i and i ∗ + e: ˙

−αi − β i ∗ + e˙ = m

−γ i + δ i ∗ + e˙ = f The solution of this linear system is: 

    1 i −δ −β m = i ∗ + e˙ αδ + βγ −γ −α f

Thus: i=− i ∗ + e˙ =

δm + αf αδ + βγ −γ m + αf αδ + βγ ˙

The symbol e is the log of the exchange rate, e = log E. Thus e˙ = E E . From the last equation we obtain the differential equation for the exchange rate: E˙ =

−γ m + αf αδ + βγ

 E

where we use the simplifying hypothesis i ∗ = 0. Therefore, the dynamical system of this model has two equations: E˙ =

−γ m + αf αδ + βγ

 E

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8 Open Economy Macroeconomics

F˙ = ϕ



EP ∗ P

+ i∗F

In equilibrium, E˙ = F˙ = 0, and −γ m + αf = 0

∗

i F = −ϕ

EP ∗ P



The Jacobian of this system is:  J =

∂ F˙ ∂E ∂ E˙ ∂E

∂ F˙ ∂F ∂ E˙ ∂F



 =

i∗ ϕ ∂ E˙ ∂F



P∗ P ∂ E˙ ∂E



For the equilibrium of this model to be a saddle point the following inequality has to be satisfied: i

˙

∗ ∂E

E

1 i∗F P In equilibrium i ∗ F = −ϕ. If F > 0, −ϕ > 0. Therefore, the inequality can be written as an elasticity: ηT B,S =

∂ϕ S >1 ∂S (−ϕ)

8 Open Economy Macroeconomics

155

This inequality is the Marshall-Lerner condition for the trade balance (TB) to increase when the terms of trade increase. Figure 8.1 depicts the phase diagram of the model with the exchange rate on the vertical axis and foreign assets on the horizontal axis. We assume a creditor country. (b) Show what happens to E and F in each of the following circumstances: (1) an increase in M and (2) an increase in B. Figure 8.2 shows what happens when the stock of money increases and Fig. 8.3 what happens to E and F when the stock of bonds increases. In both cases the curve F˙ = 0 does not shift. Figure 8.2 depicts an increase in the money stock that shifts upward the E˙ = 0 curve. There is an overshooting of the exchange rate, which then converges on its long run equilibrium. Figure 8.3 depicts an increase in the stock of bonds that shifts downward the E˙ = 0. There is an undershooting of the exchange rate, which then converges on its long run equilibrium along the saddle path.

Fig. 8.1 The phase diagram for the E and F system

Fig. 8.2 Dynamic adjustment of the economy to an unanticipated increase in the stock of money

156

8 Open Economy Macroeconomics

Fig. 8.3 Dynamic adjustment of the economy to an unanticipated increase in the stock of bonds

(7) Consider the following model of a flexible exchange rate portfolio:

∂E φαδ

170

9 Economic Fluctuation and Stabilization in an Open Economy

x = −α (r − r¯ ) + β (s − s¯ ) + λf where λ > 0 and f stands for the public deficit. Thus, the differential equation for r˙ is now specified as: r˙ = (φγ + θ − φδα) (r − r¯ ) + φδβ (s − s¯ ) + f In the phase diagram of Fig. 9.10 the r˙ = 0 curve shifts up as depicted in Fig. 9.11. The terms of trade (real exchange) jumps from s¯ (0) to s¯ (1), due to the jump of the nominal exchange rate. Fig. 9.10 The phase diagram for the r and s system when φγ + θ < φαδ

Fig. 9.11 Dynamic adjustment to an unanticipated increase of the public deficit

9 Economic Fluctuation and Stabilization in an Open Economy

171

Fig. 9.12 Dynamic adjustment to an unanticipated increase of the foreign interest rate

(d) Use the phase diagram from item (a) to show what happens in this model when the real foreign interest rate increases. When the real foreign interest rate increases the central bank increases the domestic interest rate according to the MPR. The r˙ = 0 curves shifts up. Assuming that the term of trade (real exchange rate) does not change Fig. 9.12 depicts the phase diagram of the model. The economy goes from point E0 to E1 . (4) Consider the model: IS: x = −α (r − r¯ ) + β (s − s¯ ) PC: π˙ = γ s˙ + δx UIP: r = r¯ + s˙ , r¯ = r ∗ MPR: e = e¯ IC: Given p(0) and π(0) We use the same letters of exercise 3 for the parameters of the Phillips Curve (PC). To obtain the dynamical system of this model we start with the terms of trade definition: s = e + p∗ − p We take the time derivatives of both sides of this expression: s˙ = e˙ + p˙ ∗ − p˙ = π ∗ − π We use the MPR so that the nominal exchange rate is constant. Thus, e˙ = 0. It follows from UIP that: s˙ = r − r¯ = π ∗ − π Taking the time derivative of this expression we obtain: r˙ = π˙ ∗ − π˙ = −π˙ because we assume π˙ ∗ = 0. Next, substitution for π˙ from the PC curve implies: r˙ = −γ s˙ − δx

172

9 Economic Fluctuation and Stabilization in an Open Economy

Substituting for s˙ and x for UIP and the IS curve, respectively, yields the following differential equation after rearranging terms. r˙ = (−γ + αδ) (r − r¯ ) − βδ (s − s¯ ) Together with the UIP equation, s˙ = r − r¯ we have a system of two linear differential equations, with the following Jacobian matrix:  ∂ r˙ ∂ r˙    αδ − γ −βδ ∂r ∂s J = ∂ s˙ ∂ s˙ = 1 0 ∂r ∂s The determinant of this Jacobian is positive: |J | = βδ > 0 The trace of this Jacobian is: trJ = αδ − γ and it can be either positive or negative. We assume the trace to be negative: αδ − γ < 0, for the system to be stable. Use a phase diagram with r in the vertical axis and s on the horizontal axis to show this model’s equilibrium and dynamics. Figure 9.13 depicts the phase diagram of the model. The r˙ = 0 curve is downward sloping. The arrows show the dynamics of the model. Fig. 9.13 The phase diagram for the r and s system

9 Economic Fluctuation and Stabilization in an Open Economy

173

(5) Consider the model: IS: x = −α (r − r¯ ) + β (s − s¯ ) PC: π˙ = γ s˙ + δx UIP: r = r¯ + s˙ MPR: i = r¯ + π + θ (π − π¯ ) + φ s˙ IC: Given p(0) and π(0) (a) Analyze the model’s equilibrium and dynamics on a phase diagram with inflation on the vertical axis and the terms of trade on the horizontal axis. From the MPR we obtain the interest rate gap: r − r¯ = θ (π − π¯ ) + φ s˙ Substituting for the interest rate gap into the UIP yields: s˙ = θ (π − π¯ ) + φ s˙ which can be written as: s˙ =

θ (π − π¯ ) 1−φ

Substituting for the output gap from the IS curve into the Phillips Curve results: π˙ = γ s˙ + δ [−α (r − r¯ ) + β (s − s¯ )] which can be written as: π˙ = γ s˙ − αδ (r − r¯ ) + βδ (s − s¯ ) Substituting for s˙ from the previous equation into the π˙ equation after taking into account the UIP equation we get: π˙ =

(γ − αδ) θ (π − π¯ ) + βδ (s − s¯ ) 1−φ

The dynamical system of the model is: π˙ =

(γ − αδ) θ (π − π¯ ) + βδ (s − s¯ ) 1−φ s˙ =

θ (π − π¯ ) 1−φ

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9 Economic Fluctuation and Stabilization in an Open Economy

Fig. 9.14 The phase diagram for the r and s system

Fig. 9.15 Dynamic adjustment to an unanticipated decrease in the inflation target

The Jacobian of this system is: J =

 ∂ π˙

∂ π˙ ∂π ∂s ∂ s˙ ∂ s˙ ∂π ∂s



 =

(γ −αδ)θ 1−φ θ 1−φ

βδ 0



The determinant of this Jacobian is: |J | = −

βδθ 1−φ

This determinant is negative if 1 − φ > 0. We assume that the monetary policy rule is such that φ < 1. Thus, the equilibrium is a saddle path. Figure 9.14 depicts the phase diagram of the model. The inflation rate, a

9 Economic Fluctuation and Stabilization in an Open Economy

175

predetermined variable is on the vertical axis. The terms of trade, a jump variable, is on horizontal axis. SS, the saddle path, is downward sloping. (b) Use the previous item’s phase diagram to show what happens when the Central Bank lowers the inflation target to π¯ 1 (< π¯ 0 ) from π¯ 0 . The dynamic adjustment of the economy for this experiment is depicted in Fig. 9.15. The point E0 is the initial equilibrium of the model before the Central Bank changes its inflation target. When the Central Bank changes its inflation target the curve π˙ = 0 shifts down. The new long run equilibrium of the economy is given by point Ef . Because π is a predetermined variable there is a jump from E0 to E1 when the new policy is announced. Then the economy converges on Ef through the saddle path. (6) Consider the model: IS: x = −α (r − r¯ ) + β (q − q) ¯ PC: π˙ = γ q˙ + δx UIP: r = r¯ + q˙ MPR: i = r¯ + πc + θ (πc − π¯ c ) w ˙ χ = 1−w CPI: πc = π + χ q, FE: i = r + πc IC: Given p(0) and π(0) (a) Analyze the model’s equilibrium and dynamics on a phase diagram with the real interest rate (on the vertical axis) and the real exchange rate (on the horizontal axis). In this Keynesian model the real exchange rate is defined by q = e + p ∗ − pc and the consumer price is a weight average of the domestic price and the imported price:

pc = (1 − w) p + w e + p∗ From the MPR we can write: r = r¯ + θ (πc − π¯ c ) Taking the derivatives with respect to time of both sides of this expression results in: r˙ = θ π˙ c From the consumer price index (CPI) definition we get π˙ c = π˙ + χ q¨ = π˙ + χ r˙

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9 Economic Fluctuation and Stabilization in an Open Economy

since the UIP implies r˙ = q¨ If we substitute out for π˙ c in the r˙ equation we get r˙ = θ π˙ + θ χ r˙ which can be written as r˙ =

θ π˙ 1 − θχ

If we substitute out π˙ from the Phillips equation r˙ =

θ [γ q˙ + δx] 1 − θχ

If we substitute out q˙ and x from the UIP and the IS curve, respectively, we get the differential equation for the real interest rate: r˙ =

θ (γ − αδ) βθ δ ¯ (r − r¯ ) + (q − q) 1 − θχ 1 − θχ

The second differential equation of the model is the UIP equation: q˙ = r − r¯ The Jacobian of this system of differential equations is given by:  J =

∂ r˙ ∂r ∂ q˙ ∂r

∂ r˙ ∂q ∂ q˙ ∂q



 =

θ(γ −αδ) βθδ 1−θχ 1−θχ

1



0

The determinant of this Jacobian is negative, |J | = −

βθ δ 1 − θχ

if 1 − θ χ > 0. We assume that the MPR is such that this assumption holds. Figure 9.16 depicts the phase diagram of this model. SS is the saddlepath. It should be noticed that both variables, r and q, are jump variables.

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177

Fig. 9.16 The phase diagram for the r and s system

(b) Use the previous item’s phase diagram to show what happens in this economy when the real interest rate increases to r1∗ > r0∗ from r0∗ . Figure 9.17 depicts the adjustment of the economy to an increase in the international rate of interest. The real exchange rate depends on the international real rate of interest. When the international rate of interest increases the real exchange rate decreases. The economy jumps from E0 to E1 because the nominal exchange rate adjusts instantaneously. Fig. 9.17 Dynamic adjustment to an unanticipated increase in the foreign interest rate

(c) Analyse the model’s equilibrium and dynamics with the inflation rate on the vertical axis and real output on the horizontal axis.

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9 Economic Fluctuation and Stabilization in an Open Economy

Substitution of the UIP equation into the PC curve results π˙ = γ (r − r¯ ) + δx We substitute the interest rate gap given by MPR into this equation to obtain: π˙ = γ θ (πc − π¯ c ) + δx From the definition of the consumer price index it is straightforward to show that π˙ c =

1 π˙ 1 − χθ

Combining the two last equations yields the differential equation for the consumer inflation rate: π˙ c =

γθ δ x (πc − π¯ c ) + 1 − χθ 1 − χθ

Taking the derivatives with respect to time of the IS curve we get x˙ = −α r˙ + β q˙ = −α r˙ + β (r − r¯ ) The expression after the second sign of this equality was obtained using the UIP equation. We substitute the interest rate gap from the MPR into the x˙ equation to obtain x˙ = −α r˙ + βθ (πc − π¯ c ) Taking the time derivatives of the MPR yields r˙ = θ π˙ c Thus, we substitute this expression into the x˙ equation to get: x˙ = −αθ π˙ c + βθ (πc − π¯ c ) By substituting the π˙ c equation into this expression and rearranging terms we obtain the equation for x. ˙ Therefore, the dynamical system is given by:

 αθ δ αθ γ x˙ = − x+θ β − (πc − π¯ c ) 1 − χθ 1 − χθ

9 Economic Fluctuation and Stabilization in an Open Economy

π˙ c =

179

γθ δ x (πc − π¯ c ) + 1 − χθ 1 − χθ

The Jacobian matrix of this system is given by:  J =

∂ π˙ c ∂ π˙ c ∂πc ∂x ∂ x˙ ∂ x˙ ∂πc ∂x



 =

 θ

γθ 1−χ θ  αθγ β − 1−χ θ

δ 1−χ θ αθδ − 1−χ θ

The determinant of this Jacobian is: |J | =

Fig. 9.18 The phase diagram for the πc and x system

Fig. 9.19 Dynamic adjustment to an unanticipated decrease in the inflation target

−θ δβ 0



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9 Economic Fluctuation and Stabilization in an Open Economy

We assume that 1 − χ θ > 0. Figure 9.18 depicts the phase diagram of the model with the inflation rate on the vertical axis and the output gap on the horizontal axis. SS is the saddle path, which is downward sloping. (d) Use the previous item’s phase diagram to show what happens when the Central Bank lowers the inflation target to π¯ 1 (< π¯ 0 ) from π¯ 0 . Phase diagram 9.19 shows what happens to the inflation rate and to the output gap. The inflation rate is a predetermined variable while the output gap is a jump variable. Figure 9.19 does not show the complete phase diagram, just the saddle path SS after the inflation target change. The inflation rate is a predetermined variable. Therefore, at the initial moment, the economy jumps to point E0 yielding a recession, the output gap becomes negative, and the inflation rate converges on the new equilibrium at π¯ 1 . (7) Consider the model: IS: x = −α (r − r¯ ) + β (q − q) ¯ PC: π˙ = γ q˙ + δx UIP: r = r¯ + q˙ MPR: e = e¯ CPI: πc = π + χ q˙ FE: i = r + πc IC: Given p(0) and π(0) (a) Analyze the model’s equilibrium and dynamics on a phase diagram with the real interest rate (on the vertical axis) and the real exchange rate (on the horizontal axis). Substituting the IS curve and the UIP equation into the Phillips Curve we obtain: π˙ = (γ − αδ) (r − r¯ ) + βδ (q − q) ¯ From the definition of the real exchange rate, q = e + p∗ − pc , we get: q˙ = π ∗ − πc Substituting the CPI definition in this equation results: q˙ = π ∗ − π − χ q˙ or (1 + χ ) q˙ = π ∗ − π Taking the time derivatives of both sides of this expression yields: (1 + χ ) q¨ = −π˙

9 Economic Fluctuation and Stabilization in an Open Economy

181

assuming that π˙ ∗ = 0. From the UIP it follows that: r˙ = q¨ By combining the two last equations we obtain: r˙ = −

1 π˙ 1+χ

Plugging the π˙ equation, derived at the beginning of this item, into this expression we obtain the differential equation for the real interest rate: r˙ =

αδ − γ βδ ¯ (r − r¯ ) − (q − q) 1+χ 1+χ

together with the UIP differential equation for q q˙ = r − r¯ , we obtain the dynamical system of the model. The Jacobian matrix is:  J =

∂ r˙ ∂r ∂ q˙ ∂r

∂ r˙ ∂q ∂ q˙ ∂q



 =

αδ−γ 1+χ

1

βδ − 1+χ 0



The determinant and the trace of this matrix are given by: |J | =

βδ >0 1+χ

trJ = αδ − γ If αδ − γ < 0 the system is stable. We assume that this inequality holds. Figure 9.20 is the phase diagram of the model: (b) Use the previous item’s phase diagram to show what happens in this economy when the real interest rate increases to r1∗ > r0∗ from r0∗ . Figure 9.21 depicts the phase diagram of the model when the real interest rate increases. The arrowed path shows the dynamic adjustment of the economy towards the new equilibrium (point Ef ). The real exchange rate starts to fall until it reaches the new q˙ = 0 curve. From that point it rises and then, in a cyclical way, converges on the new equilibrium where the real interest rate is equal to the new world rate of interest. (c) Analyze the model’s equilibrium and dynamics with the inflation rate on the vertical axis and real output gap on the horizontal axis. From the definition of the real exchange rate, q = e + p∗ − pc , we obtain

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9 Economic Fluctuation and Stabilization in an Open Economy

Fig. 9.20 The phase diagram for the r an q system

Fig. 9.21 Dynamic adjustment to an unanticipated increase in the foreign interest rate



Ef

1* q˙̇ 1*  0 0*

E0

q˙̇ 0*  0 ˙̇  0 q

q˙ = e˙ + π ∗ − πc = π ∗ − πc The last equality is due to the fact the exchange rate is fixed: e˙ = 0. Substituting the consumer price into this expression results: q˙ = π ∗ − π − χ q˙ Thus, q˙ =

π∗ − π 1+χ

Substituting this into the Phillips curve we obtain the differential equation for the rate of inflation: π˙ = −

γ π − π ∗ + δx 1+χ

9 Economic Fluctuation and Stabilization in an Open Economy

183

The differential equation for the output gap can be obtained as follows. First, we take the derivatives with respect to time of the IS curve: x˙ = −α r˙ + β q˙ From the UIP equation we get: r˙ = q¨ = −

π˙ 1+χ

because π ∗ − π˙ 1+χ

q¨ =

and we assume π˙ ∗ = 0. Thus the differential equation for the output gap is: αδ 1 x˙ = x− 1+χ 1+χ



αγ β+ π − π∗ 1+χ

The system of linear differential equation has the following Jacobian: J =

 ∂ π˙

∂ π˙ ∂π ∂x ∂ x˙ ∂ x˙ ∂π ∂x





γ − 1+χ  = 1 β+ − 1+χ

αγ 1+χ



δ



αδ 1+χ

The determinant and the trace of this matrix are: |J | =

βδ >0 1+χ

trJ =

αδ − γ 1+χ

We assume that αδ − γ < 0 for for the system to be stable. The phase diagram of this dynamical system is depicted in Fig. 9.22. The slope of the x˙ = 0 line is less than the slope of π˙ = 0 because αδ(1+χ ) β(1+χ )+αγ δ(1+χ ) δ

=

αγ q¯ inflation increases and if q < q¯ inflation decreases. Thus, the fact that q˙ = 0 does not imply that q = q, ¯ it says that q is constant. (b) Show what happens in this economy when the natural interest rate increases. If the natural rate of interest (which is equal to the foreign rate of interest) increases, the natural real exchange rate will decrease. If q were equal to q¯ before the change in foreign interest rate, the MPR does not yield adjustment in the real exchange rate. (9) Assume that imports and labor are used as inputs in the production of a domestic product. The economy’s real output is then equal to the sum of consumption and exports: yt = ω1 ct + ω2 ext The variables are in log form and the weight ωt is the respective variable proportion in the steady-state. The demand for export equation is the demand equation for an input that depends on the world output and the relative price: ext = yt∗ + ηst + k The parameter η is elasticity of substitution between the input and labor, k is a constant and s is the terms of trade as defined by: st = et + pt∗ − pt . The Euler equation is given by: ct = −σ (rt − ρ) + ct+1 (a) Show that the IS curve for the open economy model is the following: xt = xt+1 − ω1 σ (rt − r¯t ) − ω2 η (st+1 − s¯t+1 ) By substituting the Euler equation into the output equation we obtain: yt = −ω1 σ (r + ρ) + ω1 ct+1 + ω2 ext The output equation for next period is: yt+1 = ω1 ct+1 + ω2 ext+1

9 Economic Fluctuation and Stabilization in an Open Economy

187

Subtracting this equation from the previous one results: yt = yt+1 − ω1 (rt − ρ) − ω2 ext+1 + ω2 ext The export equation for period t and t + 1 are: ext = yt∗ + ηst + κ ∗ + ηst+1 + κ ext+1 = yt+1

Thus, ∗ − yt∗ + η (st+1 − st ) ext+1 − ext = yt+1

Taking into account this expression the output equation is given by: ∗ − ω2 ηst+1 yt = yt+1 − ω (rt − ρ) − ω2 yt+1

where ∗ ∗ = yt+1 − yt∗ yt+1

and st+1 = st+1 − st This output equation can be written as: ∗ yt − y¯t = yt+1 − y¯t+1 + (y¯t+1 − y¯t ) − ω1 σ (rt − ρ) − ω2 yt+1

− ω2 η (st+1 − ¯st+1 ) − ω2 η¯st+1 where y¯ is potential output, and we added and subtracted the term ω2 η¯st+1 , where the bar over the variable is its full employment value. By using the notation x = y − y¯ , we write the IS curve as: ∗ xt = xt+1 − ω1 σ (rt − ρ) + y¯t+1 − ω2 yt+1 − ω2 η¯st+1

−ω2 η (st+1 − ¯st+1 ) This equation can be rearranged as: 

ω2 yt+1 ω2 η y¯t+1 + + ¯st+1 xt = xt+1 − ω1 σ rt − ρ − ω1 σ ω1 σ ω1 σ − ω2 η (st+1 − ¯st+1 )



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9 Economic Fluctuation and Stabilization in an Open Economy

(b) Show that the natural rate of interest is given by the expression: r¯t = ρ +

∗ ω2 yt+1 ω2 η y¯t+1 − − ¯st+1 ω1 σ ω1 σ ω1 σ

This expression is inside the terms in brackets in the equation above. Thus, the IS curve of this model is: xt = xt+1 − ω1 σ (rt − r¯t ) − ω2 η (st+1 − ¯st+1 ) (c) Show that this expression of the natural rate can be simplified, using a little algebra, to obtain the same formula as in a closed economy: r¯t = ρ +

1 y¯t+1 σ

If the natural rate of interest is given by this expression and by the previous one, it follows that: ∗ ω2 yt+1 ω2 η¯st+1 1 y¯t+1 y¯t+1 = − − σ ω1 σ ω1 σ ω, σ

It follows that: ∗ ω1 y¯t+1 = y¯t+1 − ω2 yt+1 − ω2 η¯st+1

which can be written as: ∗ + ω2 η¯st+1 (1 − ω1 ) y¯t+1 = ω2 yt+1

We can write the output equation for full employment in periods t and t +1: y¯t = ω1 c¯t + ω2 ex¯t y¯t+1 = ω1 c¯t+1 + ω2 ex¯t+1 Subtracting the former from the latter we obtain: y¯t+1 = ω, c¯t+1 + ω2 ex¯t+1 The export equations for periods t and t + 1 are: ex ¯ t = yt∗ + η¯st + κ ∗ ex ¯ t+1 = yt+1 + η¯st+1 + κ

9 Economic Fluctuation and Stabilization in an Open Economy

189

thus, ∗ + η¯st+1 ex¯t+1 = yt+1

In full employment: y¯t+1 = c¯t+1 By taking into account these two expressions we can write the rate of growth of potential output as: ∗

+ η¯st+1 y¯t+1 = ω1 y¯t+1 + ω2 yt+1 which can be written as: ∗ + ω2 η¯st+1 (1 − ω1 ) y¯t+1 = ω2 yt+1

Therefore, the natural rate of interest in this open economy model is the same rate of the closed economy. (d) What is your verdict regarding this IS curve model of a small open economy? The verdict is that this open economy IS curve has no theoretical coherence because according the UIP the natural rate of interest is equal to the international rate of interest plus the change in the real exchange rate: r¯ = rt∗ + q¯t+1 (10) *(Tradable and nontradable goods). In a small open economy there are two goods, a tradable and a nontradable good. The tradable good price is given and fixed in international markets. The endowments of the two goods are given: yN = y¯N , yT = y¯T , where yN and yT denote output of nontradable and tradable goods, respectively, and a bar over a variable stands for the given endowment. The demand for nontradable goods relative to tradable goods is specified: cN = cT

PN PT

−η

= S −η

where PN and PT are, respectively, the prices of nontradable and tradable goods, the letter c denotes consumption with indexing stating for the type of good. We assume both goods are normal. S is the relative price of nontradable goods in terms of tradable goods, and η is the elasticity of substitution between the two goods.

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9 Economic Fluctuation and Stabilization in an Open Economy

The price index is approximated by:

1−γ γ PN P = EPT∗ where PT = EPT∗ , E is the nominal exchange rate and PT∗ is the international price of tradable goods. For simplicity we normalize PT∗ = 1. Thus, PT = E. We assume perfect international capital markets. This small open economy takes as given the international interest rate. (a) Assume that prices are flexible. What is the natural exchange rate? (b) Assume that prices of nontradable goods are sticky and change according to a Phillips Curve. Suppose that the Central Bank uses a Taylor rule to set the interest rate. Analyze the equilibrium and stability of this model under two hypotheses: (i) the inflation rate is predetermined, and (ii) the inflation rate is free to change. (a) The accumulation of a foreign asset, the current account of the balance of payments, is: B˙ = iB + PT yT + PN yN − PT cT − PN cN where B is the stock of foreign assets, the current account of the balance of payments, is: iB B˙ = + yT + SyN − cT − ScN PT PT We define the stock of foreign assets in terms of tradable goods by: b=

B PT

We take the time derivatives of both sides of this equation to obtain: B˙ − bπT b˙ = PT where πT is the inflation rate of tradable goods: πT =

P˙T PT

By substituting the equation for b˙ into the current account yields: b˙ = πT b + yT + SyN − cT − ScN

9 Economic Fluctuation and Stabilization in an Open Economy

191

where the real interest rate in terms of tradable goods is: rT = i − πT The real interest rate in terms of prices of both goods is: r =i−π The price index can be written as: 1−γ

P = PT

γ

PN = PT

PN PT

γ = PT S γ

Therefore π = πT + γ s˙ where s = log(S). It is straightforward to show that: r = rT − γ s˙ The equilibrium of the nontradable goods market is given by: cN = y¯N The domestic output of tradable goods is equal to the endowment: yT = y¯T The last two equations imply that the current account is given by: b˙ = rT b + y¯T − cT Integrating forward this international flow constraint yields the international stock budget constraint: 

∞

b(0) =

e−rT t (cT − y¯T ) dt

0

In equilibrium the consumption of tradables is constant. Thus, from this stock constraint results: c¯T = rT b(0) + y¯T

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9 Economic Fluctuation and Stabilization in an Open Economy

The relative price of nontradable goods in equilibrium is obtained from the demand equation for nontradable: y¯N = S¯ η rT b(0) + y¯T which can be written in logs as: s¯ =

1 log (rT b(0) + y¯T ) − log (y¯N ) η

The real exchange rate is defined by: Q=

EP ∗ P

We assume, for simplicity, that the foreign price level has the same weights of the domestic price level, thus: ∗1−γ

∗γ

EPT PN EPT∗ EP ∗ = Q= = 1−γ P PT PT

S∗ S

γ

=

S∗ S

γ

By taking logs results:

q = γ s∗ − s Again, for simplicity, and no loss of generality, s ∗ = log S ∗ = 0. Thus: q = −γ s By substituting s¯ into this expression we obtain the natural exchange rate: q¯ =

γ  log (y¯N ) − log (rT b(0) + y¯T ) η

The small open economy takes the international interest rate as given. Thus, uncovered real interest rate parity yields: r = r ∗ + q˙ In equilibrium q˙ = s˙ = 0, and r¯T = r¯ = r ∗

9 Economic Fluctuation and Stabilization in an Open Economy

193

Therefore, we conclude that the natural exchange rate is given by: q¯ =

γ  log (y¯N ) − log r ∗ b(0) + y¯T η

The natural exchange rate depends on y¯N , r ∗ , b(0) and y¯T with the following partial derivatives: ∂ q¯ ∂ q¯ ∂ q¯ ∂ q¯ < 0, > 0, < 0 and ∗  0 if b(0)  0 ∂ y¯N ∂b(0) ∂ y¯T ∂r (b) Nontradable goods prices are sticky and change according to the Phillips curve: π˙ = κ (cN − y¯N ) The parameter κ can be either positive or negative. It is positive in the Keynesian model, which is backward looking. It is negative in the New Keynesian model, which is forward looking. From the demand curve for nontradables we obtain: cN − cT = −ηs In full employment this equation becomes: y¯N − cT = −η¯s Subtracting one expression from the other results: cN − y¯N = −η (s − s¯ ) and taking into account the relationship between the relative price s and the real exchange rate we obtain: cN − y¯N =

η ¯ (q − q) γ

Therefore, the Phillips curve depends on the exchange rate gap: π˙ N =

κη ¯ (q − q) γ

The Central Bank sets interest rates following a simplified Taylor rule: i = r ∗ + π + φ (πN − π¯ N )

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9 Economic Fluctuation and Stabilization in an Open Economy

By combining this rule with the interest rate arbitrage condition results: q˙ = r − r ∗ = φ (πN − π¯ N ) The model of this item can be abridged into a system of two differential equations: 

π˙ N = κη ¯ γ (q − q) q˙ = φ (πN − π¯ N )

The Jacobian matrix of this system is given by:  ∂ π˙ J =

N

∂πN ∂ q˙ ∂πN

∂ π˙ N ∂q ∂ q˙ ∂q





0 κη γ = φ 0



The determinant and the trace of J are: |J | = −

φκη γ

trJ = 0 If κ > 0, |J | < 0. Thus, the equilibrium of the Keynesian model is a saddle point, as depicted in the phase diagram of Fig. 9.25. If κ < 0, |J | > 0, trJ = 0. Thus the equilibrium is unstable, as depicted in phase of diagram of Fig. 9.26. It should be mentioned that the trace is equal to zero and the determinant of the Jacobian matrix is positive. Thus the two roots are complex conjugates and the real part is equal to zero, such as ±bi, where i 2 = −1. Fig. 9.25 The phase diagram for the πN and q system when k > 0

9 Economic Fluctuation and Stabilization in an Open Economy

195

Fig. 9.26 The phase diagram for the πN and q system when k < 0

(c) (Natural exchange rate: Keynesian and Australian models). Compare the natural exchange rates obtained from the Keynesian model and the Australian model and interpret the differences between the two rates. The current account (ca) is defined as the sum of the trade balance (tb) and the income balance (ib): ca = tb + ib The trade balance is the net exports of goods and services. The income balance is the net factor payment received from (or paid to) abroad. Net assets can be either positive or negative, depending on the country being a creditor or a debtor country. Thus, we can write: tb = ca − r ∗ b where r ∗ is the foreign interest rate and b is the total stock of foreign assets (b > 0) or foreign debt (b < 0). The Keynesian model has two goods, a domestic one and an imported good. When the Marshall-Lerner condition [Macro Theory, p. 249] holds the trade balance depends on the real exchange rate according to: tb = tb (q) ,

∂tb >0 ∂q

The economy is in equilibrium when savings (s) is equal to the sum of investment (i), public deficit (f ) and the current account [Macro Theory, p. 254]:

s (y¯ − τ¯ ) = i r ∗ + f¯ + tb (q) ¯ + r ∗b

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9 Economic Fluctuation and Stabilization in an Open Economy

Fig. 9.27 The natural real exchange rate

Savings depends on the real disposable income and investment depends on the real interest rate. A bar over a variable denotes full employment. The domestic real interest rate is, by arbitrage, equal to the real foreign interest rate. It follows from this expression that the trade balance in the long run equilibrium, can be written as:

tb (q) ¯ = s (y¯ − τ¯ ) − i r ∗ − f¯ − r ∗ b = tb Figure 9.27 depicts this equation, with the real exchange rate on the vertical axis and the trade balance on the horizontal axis. Given the full employment trade balance we obtain the natural exchange rate. The full employment trade balance depends on the foreign interest rate (r ∗ ), on the full employment fiscal deficit (f¯), on the net foreign assets (debt) (b), and the full employment real disposable income. The Australian model has two goods, a tradable good and a nontradable good. The trade balance is the difference between the domestic production of tradable goods (yT ) and the domestic consumption of tradable goods (cT ). Thus, the current account is given by: ca = tb + r ∗ b = yT − cT + r ∗ b The domestic production of tradable goods depends on the relative price of nontradable goods in terms of tradable goods since its marginal cost of production is upward sloping. The consumption of tradable goods is a function of the relative price of nontradable goods. Therefore, the trade balance depends on the relative price of nontradable goods according to: tb = yT (S) − cT (S) = tb(S)

9 Economic Fluctuation and Stabilization in an Open Economy

197

and ∂tb 0 ∂S ∂S In the first part of this exercise we have shown that the real exchange rate and the relative price of nontradable goods in terms of tradable goods are related by:

Q=

S∗ S

γ

It follows that Q and S are negatively correlated and the trade balance and the real exchange rate are positively correlated: tb = tb (q) ,

∂tb >0 ∂q

where q = logQ. The natural real exchange rate is the rate consistent with the long run equilibrium of the current account: ca = tb (q) ¯ + r ∗b Thus, tb (q) ¯ = ca − r ∗ b = tb Figure 9.27 shows this equation curve. Both models, the Keynesian and the Australian, yield a positive correlation between the trade balance and the real exchange rate. The framework is very much alike as Fig. 9.27 depicts albeit the different specification of each model.

Part III

Monetary and Fiscal Policy Models

Chapter 10

Government Budget Constraint

(1) The fiscal policy rule is given by: f = g − τ + ib = a + αb > 0 From the budget constraint it follows that b˙ + (n + π ) b + μm = a + αb which can be written as: b˙ = a − μm + (α − n − π ) b If α − n − π > 0 and μm − a > 0 we can write:  ∞ b= e−(α−n−π )t (μm − a) dt 0

(2) Consider the model: Bonds-and-money-financed public deficit: g − τ + rb = b˙ + μm Tax dependent on public debt: τ = τ (b), τ  (b) > 0 Money demand: m = f (i), f  < 0

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_10

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From this demand we can express the services of money as a function of the real quantity of money: s(m) = im = f −1 (i)m = s(m) and: s  (m) =

ds(m) 0 dm

Monetary policy rule: m ˙ = m (μ − π ) , μ = constant Constant real interest rate: r = constant The monetary policy rule can be written as: m ˙ = mμ − mπ − rm + rm which is equivalent to: m ˙ = (r + μ) m − (r + π ) m = (r + μ) m − im = (r + μ) m − s(m) The model can be reduced to two differential equations: b˙ = g − τ (b) + rb − μm m ˙ = (r + μ) m − s(m) The Jacobian of this dynamical system is:  J =

∂ b˙ ∂b ∂m ˙ ∂b

∂ b˙ ∂m ∂m ˙ ∂m



 =

 −μ −τ  (b) + r 0 (r + μ) − s  (m)

The determinant and the trace of this matrix are:



|J | = r − τ  (b) r + μ − s  (m) trJ = r − τ  (b) + (r + μ) − s  (m) There are four possibilities for stability of this model as shown in Table 10.1:

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203

Table 10.1 Stability of the dynamical system

r + μ − s  (m) r − τ  (b) +

+ |J | > 0 trJ > 0 |J | < 0

−

− |J | < 0 |J | > 0 trJ < 0

The equilibrium can be unstable (fourth cell) or a saddle point (second and  third cells), according to the sign of r + μ − s  (m) and r − τ  (b) . (3) Consider the model: Money demand: m = α −βπ e , α > 0, β > 0, m ≤ m. ¯ Money-financed public deficit: f =

M˙ , f constant, m ˙ = f − mπ P

π˙ e = θ π − π e

(a) We take the time derivative of the money demand equation and we take into account the expected inflation rate to obtain:

m ˙ = −β π˙ e = −βθ π − π e and we use the money demand function to get: m ˙ = −βθ π +

βθ (α − m) β

which can be written as: m ˙ = −βθ π + θ (α − m) = f − mπ where we use the hypothesis of money-financed public deficit. From this expression we obtain the inflation rate as: π=

f − θ (α − m) , m − βθ

m = βθ

Substituting the inflation rate into the public deficit financed by money yields:

f − θ (α − m) m ˙ = f − mπ = f − m m − βθ



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which can be written as:

θ m2 − αm + βf m ˙ =− m − βθ This is the dynamic equation of the model. In steady-state m ˙ = 0. Thus: m2 − αm + βf = 0 The roots of this equation are: α α 4βf m= ± 1− 2 2 2 α We assume: 1−

4βf ≥ 0 → α 2 − 4βf ≥ 0 α2

Figure 10.1 shows the phase diagram of the model. The high inflation (H ) equilibrium is stable and the low-inflation equilibrium is unstable. (b) When θ → ∞, π e = π . The money demand equation is: m = α − βπ and: π=

Fig. 10.1 The phase diagram of the model: high inflation equilibria is stable and the low inflation equilibria is unstable

α−m β

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205

Fig. 10.2 The phase diagram of the model: high inflation equilibrium is stable and the low inflation equilibria is unstable

Substituting this inflation rate into the public deficit financed by money equation yields: m ˙ = f − mπ = f −

m (α − m) β

which can be written as: 2

m − αm + βf m ˙ = β This is the dynamic equation of this model under the hypothesis π e = π . When m ˙ = 0, the two roots are the same as in the previous item. Figure 10.2 depicts the phase diagram of the model. The high inflation (H ) equilibrium is stable and the low inflation equilibrium is unstable. (c) The inflation tax revenue of the item (a) model is:

τ = πm =

 f − θ (α − m) m m − βθ

which can be written as: τ=

θ m2 + (f − αθ ) m m − βθ

The real quantity of money that maximizes the inflation tax revenue is obtained by solving the equation  θ m2 − 2βθ m + β (αθ − f ) dτ = =0 dm (m − βθ )2

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The inflation tax revenue of the item (b) model is given by: τ = π m = π (α − βπ ) which can be written as: τ = απ − βπ 2 The inflation rate that maximizes the inflation tax revenue is obtained by solving the equation: dτ = α − 2βπ = 0 dπ (4) Assume an economy described by: π˙ = F (π, m, α) and two economic policy regimes: MP Ra : m ˙ = m (μ − π ) MP Rb : m ˙ = f − mπ (a) The specification of the equation for π˙ is given in (Macro Theory, page 228), by combining the PC, IS and LM curves. It should be noticed that: ∂F = Fπ > 0 ∂π and: ∂F = Fm > 0 ∂m (b) In the first regime we have the following model: 

π˙ = F (π, m, α) m ˙ = m (μ − π )

The Jacobian of this dynamical system is: J =

 ∂ π˙

∂ π˙ ∂π ∂m ˙ ∂m ˙ ∂m ∂π ∂m





Fπ Fm = −m μ − π



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207

In the steady-state μ − π = 0, and the determinant and trace of the matrix are: |J | = mFm > 0 trJ = Fπ > 0 Thus, this model is unstable. The second monetary regime is given by the two differential equations system: 

π˙ = F (π, m, α) m ˙ = f − mπ

The Jacobian of this dynamical system is: J =

 ∂ π˙

∂ π˙ ∂π ∂m ˙ ∂m ˙ ∂m ∂π ∂m



 =

Fπ Fm −m −π



In the steady-state μ − π = 0, and the determinant and trace of the matrix are: |J | = −Fπ π + mFm  0 trJ = Fπ − π  0 Thus, this system can be either stable or unstable depending on parameters and inflation rate. (c) When the fiscal deficit f decreases, the inflation rate decreases and this can affect the stability of the model. (d) When the monetary expansion rate (μ) decreases the steady-state inflation rate decreases. (5) Consider the model: Aggregate demand: y = k + α (m − p) + βπ PC: π = π e + δ (y − y) ¯ Expectations: π e = μ M˙ = μ = constant or: m ˙ = m (μ − π ) MPR: M (a) In this economy the Phillips curve is: π = μ + δ (y − y) ¯ Thus, when π = μ, y = y, ¯ and the output differs from full employment output.

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(b) By taking the time derivatives of the aggregate demand and the Phillips curve, we obtain: y˙ = α (μ − π ) + β π˙ = α (μ − π ) + βδ y˙ π˙ = δ y˙ The first equation can be written as: y˙ =

α (μ − π ) 1 − βδ

Substituting this value into the equation of π˙ we obtain: π˙ =

αδ (μ − π ) 1 − βδ

From the Phillips curve μ − π = −δ (y − y). ¯ Thus the differential equation of y˙ is: y˙ = −

αδ ¯ (y − y) 1 − βδ

The Jacobian of the dynamical system is: J =

 ∂ y˙

∂ y˙ ∂y ∂π ∂ π˙ ∂ π˙ ∂y ∂π





αδ − 1−αδ 0 = αδ 0 − 1−βδ



The determinant and trace of this matrix are:

2 αδ |J | = − , 1 − βδ trJ =

2αδ 1 − βδ

Thus the stability of the system depends on the values of the parameters. (c) The monetary policy rule can be written as: m ˙ = f − mπ From the aggregate demand curve we have:

¯ 

 M M − log + β (π − π¯ ) = c + α log m + βπ y − y¯ = α log P P

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209

Fig. 10.3 The phase diagram of the model: a) 1 − δβ < 0 b) 1 − δβ > 0

  From the Phillips curve and taking into account that get: π=

M˙ M

=μ=

f + α (y − y) ¯ m

By combining the three previous equations, we obtain: m ˙ =−

δ  f + αm log m 1 − δβ

Figure 10.3 shows the phase diagram of model. (6) Consider the following budget constraint: f + rb = b˙ +

M˙ P

and the following rule: M˙ = αf P (a) By substituting this rule into the budget constraint we obtain: f + rb = b˙ + αf or: b˙ = rb + (1 − α) f

f m,

we

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Solving this differential equation, we get: 

∞

b=

e−rt (α − 1) f dt

0

Thus, if α > 1 the public debt can be sustainable. (b) From the fiscal rule we obtain: m ˙ = αf − mπ + rm − rm or: m ˙ = rm + αf − (π + r) m where r is the real rate of interest, i = r + π is the nominal rate of interest and s(m) = im is the value of the services of money. We write the differential equation as: m ˙ = rm + αf − s(m) Solving this differential equation, we obtain: 

∞

m=

e−rt (s(m) − αf ) dt

0

If a financial innovation occurs in such way that s(m) − αf becomes negative a hyperinflation will occur. (7) The government budget constraint is: Gt + it−1 Bt−1 = Tt + Bt − Bt−1 Define: dt =

dt−1 , t = 1, 2, · · · 1 + rt−1 d0 = 1

(a) Show that: +

dt Tt =

+

dt Gt + B0 − lim dT BT T →∞

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211

From the budget constraint we can write: B0 =

T1 B1 G1 + − 1 + i0 1 + i0 1 + i0

Taking into account that: d0 1 = 1 + i0 1 + i0

d1 =

the previous expression becomes: B0 = d1 T1 + d1 B1 − d1 G1 The stock B1 is given by: B1 =

T2 B2 G2 + − 1 + i1 1 + i1 1 + i1

Substituting this expression in the former equation yields: B0 = d1 T1 − d1 G1 +

d1 T2 d1 G2 d1 B2 − + 1 + i1 1 + i1 1 + i1

since: d2 =

d1 1 + i1

we can write: B0 = d1 T1 + d2 T2 − d1 G1 − d2 G2 + d2 B2 It follows from recursive substitution that: B0 =

T+ −1

dt Tt −

t=1

T+ −1

dt Gt + dt BT

t=1

Thus, B0 =

∞ + t=1

dt Tt −

∞ + t=1

dt Gt + lim dT BT > 0 T →∞

If limT →∞ dT BT > 0 the public debt is not sustainable and I would not buy government bonds.

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10 Government Budget Constraint

(8) The government finances public deficit by printing money according to: Gt − Tt = Mt − Mt−1 We divide both sides by the price index Pt , Gt − Tt Mt Mt−1 Mt−1 Mt−1 = − − + Pt Pt Pt Pt−1 Pt−1 and we added and subtracted

Mt−1 Pt−1 .

We use the notation:

Gt − Tt Mt Mt−1 = ft ; = mt ; = mt−1 Pt Pt Pt−1 and we write: ft = mt − mt−1 +

Mt−1 Mt−1 − Pt−1 Pt

The rate of inflation is defined by: Pt = 1 + πt Pt−1 therefore: ft = mt + mt−1 −

Mt−1 Pt−1 Pt−1 Pt

which can be written as: ft = mt + mt−1 −

mt−1 1 + πt

or: ft = mt +

πt mt−1 1 + πt

The inflation tax with discrete variables is measured by: inflation tax =

πt mt−1 1 + πt

˙ is given by: (9) The balance of the current account of the balance of payments (B)

 B B˙ TB =− +i Y Y Y

10 Government Budget Constraint

We define b =

B Y

213

and we take the time derivatives of both sides:

  ˙ dY ˙b = B − B 1 Y dt Y2

The gross domestic product is equal to the price level P ∗ times the real product y. Thus:

B˙ b˙ = − b π ∗ + g Y where: π∗ =

P˙ ∗ y˙ and g = ∗ P y

When we combine this expression with the current account balance we obtain:

b˙ = −tb + i − π ∗ − g b where: tb =

TB Y

The real rate of interest r ∗ is defined by: r∗ = i − π∗ Thus the differential equation can be written as:

b˙ = r ∗ − g b − tb The solution of this differential equation is: b = lim e−(r

∗ −g

T →∞

)T b(T ) +



∞

e−(r

∗ −g

)t tbdt

0

The foreign debt is sustainable when: lim e−(r

T →∞

∗ −g

)T b(T ) = 0

This is equivalent to saying that the foreign debt should grow at a rate less than the difference between the international rate of interest and the rate of growth of the economy. b˙ < r∗ − g b

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(10) According to the Ricardian equivalence theorem the “public debt is not regarded as part of the private sector’s wealth because the value of public debt corresponds to a liability equal to the present value of future taxes needed to repay the debt” (Macro Theory, p. 330). Thus, if the government defaults on its public debt, there will not be future taxes to be paid. Therefore, consumer spending will not be affected and the default has no consequence for the economy. (11) The government budget constraint is given by: G T B B˙ M˙ − +i = + Y Y Y Y Y The government program has the following objectives: (a) To keep the tax burden stable: T = stable Y (b) To increase the ratio of government spending to output: G = increases Y (c) To reduce the debt service: i

B = reduces Y

(d) To keep the inflation rate stable.

 M˙ M˙ M = = stable Y M Y Thus the deficit will be financed by issuing treasury debt: B˙ = increases Y At the same time the government wants to reduce the debt service. Since B Y will increase, the nominal rate of interest has to decrease. On the other hand, the government wants to keep inflation stable. Therefore, this government program is not consistent. (12) The government budget constraint is given by: G−T B B˙ M˙ +i = + Y Y Y Y

10 Government Budget Constraint

215

(a) The government has a nominal deficit of 3% as an objective. The value of the primary surplus should be: G−T B + i = 0.03 Y Y thus: B T −G = i − 0.03 Y Y (b) The government sets its primary surplus according to the following rule: T −G = αrb Y Public debt is sustainable when: b˙ 0 public debt is sustainable. (c) Suppose that the primary surplus is given by: T −G = αb Y From equation (10.13), [Macro Theory, p. 312], we have: T −G = −αb b˙ − (r − n) b = − Y thus: b˙ =r −n−α b If α > 0, public debt is sustainable.

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(13) Consider the model: Bonds-and-money-financed public deficit: b˙ = f + rb − μm Monetary policy: m ˙ = m (μ − π ) Money demand: m < 0

m = m(i), Fisher equation:

i =r +π Assumption: public deficit is constant. From the money demand equation, we obtain the services of money as a function of the real quantity of money: im = s(m) From the Fisher Equation we write the monetary policy as: m ˙ = μm − (i − r) m = (μ + r) m − im which can be written as: m ˙ = (r + μ) m − s(m) We have the following dynamical system: 

b˙ = f + rb − μm m ˙ = (r + μ) m − s(m)

(a) The Jacobian of this dynamical system is given by:  J =

∂ b˙ ∂b ∂m ˙ ∂b

∂ b˙ ∂m ∂m ˙ ∂m



 r −μ = 0 r + μ − s  (m) 

The determinant and trace of this Jacobian are:  |J | = r μ + r − s  (m)

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217

Fig. 10.4 The phase diagram of the b and m system

trJ = r + μ + r − s  (m) If s  (m) < 0 both the determinant and the trace of the Jacobian are positive. Under this hypothesis the model is unstable. When b˙ = 0, we can write: b=−

μ f + m r r

This curve is depicted in Fig. 10.4. When m ˙ = 0, ¯ = s(m) ¯ (r + μ) m and the real quantity of money is constant and equal to m, ¯ as shown in Fig. 10.4, which shows the phase diagram of the dynamical system, with b on the vertical axis and m on the horizontal axis. (b) Figures 10.5 and 10.6 depict this question’s experiment. Figure 10.5 shows that the Central Bank reduces the monetary expansion, at time zero, from μ0 to μ1 until time T . Figure 10.6 shows that there is a public debt ceiling equal to b(T ). In equilibrium, b˙ = m ˙ = 0. Thus, f + r b¯ = μm ¯ ¯ = s (m) ¯ (μ + r) m By combining these two equations, we obtain: f + r b¯ = s (m) ¯ − rm ¯

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10 Government Budget Constraint

Fig. 10.5 Unanticipated decrease in the rate of growth of the stock of money

μT ?

μ

μ0 μ1

T Fig. 10.6 Public debt ceiling at time T : b(T )

b bT

T

or: 1 b¯ = [−f + s (m) ¯ − r m] ¯ r We take the derivative of b¯ with respect to m: ¯ 1  d b¯ = s (m) ¯ −r dm ¯ r The sign of this derivative can be positive, zero or negative according to: ¯ 0 s  (m) We assume that d b¯ 0, c > 0 where τ stands for taxes and c for cost. The government chooses the path of taxes to minimize the present value of the costs: 

∞ 0

e−rt c(τ )dτ

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227

subject to the flow budget constraint: b˙ = rb + g − τ, b(0) given where b is the stock of public debt, r the real rate of interest and g government expenditures. The real rate of interest is constant and the path of government expenditures is exogenous. (a) What is the optimal path of taxes? (b) Define permanent government expenditures and show how transitory government expenditures are financed. (a) The government minimizes: 

∞

e−rt c(τ )dτ

0

subject to: b˙ = rb + g − τ b(0) given The current-value Hamiltonian is: H = c(τ ) + λ (rb + g − τ ) The first-order conditions are: ∂H = c (τ ) − λ = 0 ∂c λ˙ = rλ −

∂H = rλ − λr ∂b

∂H = rb + g − τ = b˙ ∂λ From the second equation λ is constant: λ˙ = rλ − λr = 0 Thus, the marginal cost of distortions should be constant: c (τ ) = λ = constant

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10 Government Budget Constraint

We conclude that the best tax policy for the government is to smooth taxes, namely the tax rate should be constant in this perfect foresight environment. (b) The stock government budget constraint is obtained by integrating the flow budget constraint, which yields: 

∞

b(0) +

e−rt τ dt =



0

∞

ert gdt

0

Permanent government spending (g) ¯ is defined as the annuity value that has the same present value of the path of government expenditures, 

∞

e

−rt

 gdt ¯ =

0

∞

e−rt gdt

0

It follows that: 

∞

g¯ = r

e−rt gdt

0

By combining this equation with the stock government budget constraint, taking into account that tax rate is constant and rearranging terms, results: τ = g¯ + rb(0) Therefore, the optimum tax rate should be set equal to the sum of the permanent expenditure with the interest payment on public debt. By substituting the optimal tax value in the flow government budget constraint we obtain: b˙ = r (b − b(0)) + g − g¯ This equation states that transitory government expenditures should be financed by issuing government securities.

Chapter 11

Monetary Theory and Policy

(1) Consider the model: IS-Curve: x = −α (r − r¯ ) , α > 0 Phillips Curve: π˙ = δx δ > 0 MPR: i˙ = λ (i ∗ − i) , λ > 0, i ∗ = r¯ + π + φ (π − π¯ ) From the Fisher equation i = r + π , taking the derivatives with respect to time yields: r˙ = i˙ − π˙ We substitute in this expression the MPR and PC to obtain: r˙ = λ [¯r + π + φ (π − π¯ ) − (r + π )] − δx Using the IS and collecting terms yields r˙ = (αδ − λ) (r − r¯ ) + λφ (π − π¯ ) The second equation of the dynamical system combines the Philips Curve and the IS equation: π˙ = −αδ (r − r¯ ) The Jacobian of this dynamical system is: J =

 ∂ r˙

∂ r˙ ∂r ∂π ∂ π˙ ∂ π˙ ∂r ∂π

 =

  (αδ − λ) λφ −αδ 0

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_11

229

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11 Monetary Theory and Policy

Fig. 11.1 The phase diagram for the π and r system

Fig. 11.2 Dynamic adjustment to an decrease in the inflation target

π π0

π–1

π˙̇  0 E0

˙̇  0

Ef

The determinant and trace of this matrix are: |J | = αδλφ > 0 trJ = (αδ − λ) < 0, if λ > αδ Since the determinant is positive and the trace is negative the dynamical system is stable. (a) Figure 11.1 depicts the phase diagram of the dynamical system, with π on the vertical axis and r on the horizontal axis. (b) Figure 11.2 shows what happens when the inflation target is lowered from π¯ 0 to π¯ 1 . The economy converges on the new equilibrium along the path E0 EF . First, inflation starts declining and the real rate of interest increases. The path depicted Fig. 11.2 assumes an oscillating path.

11 Monetary Theory and Policy

231

(2) Consider the model: IS-Curve: x = −α (i − π − r¯ ) , α > 0 Phillips Curve: π˙ = δx MPR: i = i¯ (a) Substituting the MPR into IS equation, and this into the Phillips Curve we obtain:

π˙ = −αδ i¯ − r¯ + αδπ Figure 11.3 shows the phase diagram of this equation. This system is unstable. (b) When the parameter δ is negative, the equation of π˙ is downward sloping as shown in Fig. 11.4. The system is stable, but in this forward looking Phillips Curve inflation at the initial moment, π0 is not given. Thus, there are infinite solutions. (c) This monetary policy rule should not be recommended because inflation would not converge on the implicit inflation target.

Fig. 11.3 The phase diagram of the model: δ > 0

Fig. 11.4 The phase diagram of the model: δ < 0

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11 Monetary Theory and Policy

(3) The Central Bank’s loss function is: L=

α 2 ¯ π − y + y, 2

α>0

This economy’s Phillip curve is given by: π = π e + β (y − y) ¯ ,

β>0

(a) The Central Bank minimizes: L=

α 2 π − (y − y) ¯ 2

subject to: π = π e + β (y − y) ¯ When we substitute out y − y¯ in the loss function we obtain L=

α 2 1 π − π − πe 2 β

The first-order condition of this problem is: 1 ∂L = απ − = 0 ∂π β Thus: π∗ =

1 αβ

(b) Would a monetary policy rule that could be enforced be better for this economy? Yes, if a monetary policy rule has a target inflation π = 0. (c) Would a conservative central banker produce better results than an ad-hoc one at the Central Bank? A conservative Central Bank would have a very larger parameter α (α → ∞). Thus: lim π =

α→∞

1 =0 αβ

(d) Would a conservative central banker’s loss function be the one specified in this exercise?

11 Monetary Theory and Policy

233

No, the loss function specified in this exercise is appropriate for a populist government that wants to use monetary policy to increase output beyond potential output. A conservative central bank would have a loss function such as: L=

α 2 ¯ 2 π + (y − y) 2

(4) Consider the model: IS: yt = y¯t − α (rt − r¯ ) + t PC: πt = πt−1 + β (yt − y) ¯ + υt The random variables t and υt are uncorrelated zero average, constant variance. The Central Bank’s loss function is: ¯ 2 L = γ (πt − π¯ )2 + (yt − y) When we substitute out for yt − y¯t in the PC equation from the IS equation we get: πt = πt−1 − αβ (rt − r¯t ) + βt + υt Using this equation and the IS equation the loss function can be written as: L = γ [πt−1 − αβ (rt − r¯t ) + βt + υt − π¯ ]2 + [−α (rt − r¯t ) + t ]2 or:

  L = γ (πt−1 − π¯ − αβ (rt − r¯t ))2 + (βt + υt )2 + 2 (πt−1 − π¯ − αβ (rt − r¯ )) (βt + υt )   + α 2 (rt − r¯ )2 + t2 − 2α (rt − r¯ ) t

The expected value of this loss-function is: . / Et L = Et γ (πt−1 − π¯ − αβ (rt − r¯t ))2 + α 2 (rt − r¯ )2 + constants This result is obtained taking into account that: Et (βt + υt )2 = β 2 σ2 + στ2 Et (πt−1 − π¯ − αβ (rt − r¯t )) (βt + υt ) = 0 Et t2 = σ2 Et (rt − r¯t ) t = 0

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11 Monetary Theory and Policy

The first-order condition to minimize the expected value of the loss function is: ∂Et Lt = 2γ [(πt−1 − π¯ ) − αβ (rt − r¯t )] (−αβ) − 2α 2 (rt − r¯t ) = 0 ∂rt Collecting terms and simplifying, we obtain: rt − r¯t =

βγ

(πt−1 − π¯ ) α β 2γ + 1

(5) Consider the following model of the bank reserves market: R d = R0 − αi R S = BR + NBR   BR = β i − i d Rd = RS Figure 11.5 shows the model with the interest rate on the vertical axis and total reserves on the horizontal axis. ¯ Figure 11.6 (i) Consider the following operational procedure: setting i = i. shows how this operational procedure works, when the demand curve shifts from D0 D0 to D1 D1 . ¯ This (ii) Consider the following operational procedure: setting BR = BR. ¯ of the first item. procedure is equivalent to set i = i, ¯ (iii) Consider the following operational procedure: setting NBR = NBR. Figure 11.7 shows how this operational procedure works when the demand curve shifts from D0 D0 to D1 D1 .

Fig. 11.5 The market bank reserves

11 Monetary Theory and Policy

235

Fig. 11.6 The market for bank reserves operational procedure: fixing the rate of interest

Fig. 11.7 The market for bank reserves operational procedure: sitting non-borrowed reserves (NBR)

(6) Consider the following model of the bank reserves market: e Rtd = α − βrt + δit+1

RtS = R Rtd = RtS Graphically show what happens today when the market expects the Central Bank to raise the interest rate tomorrow. When the market expects the Central Bank to raise the interest rate tomorrow the demand curve shifts from D0 D0 to D1 D1 , as depicted in Fig. 11.8. The interest rate rises from r0 to r1 .

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11 Monetary Theory and Policy

Fig. 11.8 The market for bank reserves and expected interest rates

(7) Assume that the long-term security is a perpetuity paying $ 1 per period. This security’s price P is the inverse of the long-term interest rate: P =

1 iL

(a) Show why the short-term interest rate must satisfy the equation: iS =

1 + P˙ P

This equation is an arbitrage equation. The short-term security return is is . The long-term security return is: 1 + P˙ P Thus, the two returns should be equal. (b) Show that: i˙L = i L − iS iL The yield on the long-term security is: iL =

1 P

Thus: i˙L P˙ = − 2 iL

11 Monetary Theory and Policy

237

Which can be written as: i˙L P˙ =− P iL From the arbitrage condition (a) 1 P˙ P˙ + = iL + = iS P P P or: P˙ = iS − iL P Thus: i˙L = i L − iS iL (8) Consider the following model: IS: x = −α (r − r¯ ) , α > 0 PC: π˙ = δx, δ > 0 TSIR: i˙s = β (r − rs ) , β > 0, rs = is − π MPR: is = r¯ + π + φ (π − π¯ ) + θ x IC: given p(0) and π(0) (a) How do you interpret the TSIR (term structure of interest rates)? The TSIR equation states that when the short-term interest rate is expected to rise the long-term is greater than the short-rate. Thus, i˙s > 0 ⇒ r > rs and: i˙s < 0 ⇒ r < rs (b) Analyze the model’s equilibrium and dynamics on a phase diagram with π on the vertical axis and x on the horizontal axis. We start by taking the time derivatives of both sides of the MPR: i˙s = π˙ + φ π˙ + θ x˙ = (1 + φ) π˙ + xθ ˙ when we substitute out i˙s from the TSIR we get β (r − rs ) = (1 + φ) π˙ + θ x˙

238

11 Monetary Theory and Policy

which can be written as: β (r − (is − π )) = (1 + φ) π˙ + θ x˙ We substitute out is from the monetary policy rule (MPR) to obtain: β [r − r¯ − φ (π − π¯ ) − θ x] = (1 + φ) π˙ + θ x˙ Next, we substitute out r − r¯ from the IS curve and π˙ from the Phillips Curve (PC) to get: x˙ = −

β θ

1 (1 + φ) δ +θ + α β

 x−

βφ ¯ (π − π) θ

The dynamical system of this model has two equations, this equation for x˙ and the Phillips Curve: π˙ = δx The Jacobian of this system is given by: J =

 ∂ π˙

∂ π˙ ∂π ∂x ∂ x˙ ∂ x˙ ∂π ∂x



 =

0 β − βφ θ −θ



δ 1 α

+θ +

(1+φ)δ β





The determinant and trace of this matrix are: |J | = trJ = −

β θ

βφδ >0 θ

1 (1 + φ) δ +θ + α β

 0 Otherwise there will be no unique equilibrium because π(0) is not given. (b) When δ < 0 the PC curve is backward looking and π(0) is given. Thus, if: 1 − αλ (1 + φ) δ < 0 Fig. 11.14 The phase diagram of the model when 1 − αλ(1 + φ)δ > 0

π˙̇

π–

π

242

11 Monetary Theory and Policy

Fig. 11.15 The phase diagram of the model when 1 − αλ(1 + φ)δ < 0

π˙̇

A

π–0

π–

π

the phase diagram is depicted in Fig. 11.15. When the initial inflation is π(0), the economy will not converge on π¯ . In this case, the model has no equilibrium, unless π0 = π¯ . (10) *(Consumption Asset Pricing Model): The Euler equation under uncertainty states that the agent is indifferent between consuming one unit of the good today or saving to consume the expected proceeds of the investment tomorrow: u (ct ) = βEt (1 + rt ) u (ct+1 ) 1 and Et is the expected value of the variable conditional where β = 1+ρ on information available at time t. Assume that the agent’s utility function is isoelastic:

u(c) =

c1−γ 1−γ

where γ is the relative risk aversion coefficient. (a) The risk-free rate of interest is given by the Euler equation:   f u (ct ) = β 1 + rt Et u (ct+1 ) What is the expression for the risk-free rate of interest? (b) The rate of returns on stocks is uncertain and given by the Euler equation:

u (ct ) = βEt 1 + rts u (ct+1 ) What is the expression for the rate of return on stocks? (c) What is the equity premium puzzle and the risk-free rate puzzle?

11 Monetary Theory and Policy

243

(d) In a stochastic environment the risk-free rate is the natural rate of interest. How can the risk-free rate explain a negative natural rate of interest? Note: To answer this question assume that the distribution of the variables can be described by a multivariate normal distribution and its moment generating function is given by: 

1 , t



M(t) = Eet X = eμ t+ 2 t

where t  = = ,[t1 , 0t2 , . 1. . , . . . , tn ] is a vector, μ = EX and VarX a matrix: = σij , i, j = 1, . . . , n, σij = σj i , σii = σi2 .

, , , is

(a) The Euler equation for the risk-free rate can be written as:   c −γ t+1 f 1 + ρ = 1 + rt Et ct To compute the expected value in this expression we write:

Et

ct+1 ct

−γ

= Et e

log

c

t+1 ct

−γ

= Et e

c  −γ log t+1 c t

  has a normal distribution. It follows from We assume that gt = log ct+1 ct the moment generating function that: 1

Et e−γ gt = e−γ Egt + 2 γ

2σ 2 g

where σg2 is the variance of gt . Substituting this result into the Euler equation we obtain:   1 2 2 f 1 + ρ = 1 + rt e−γ Egt + 2 γ σg We take lags of both sides of this equation, use the approximation log (1 + χ ) ∼ = χ , and rearrange terms to get the expression for the risk-free rate of interest: 1 f rt = ρ + γ Et gt − γ 2 σg2 2 (b) When the rate of interest is uncertain at time t, the Euler equation is: 



ct+1 −γ s 1 + ρ = Et 1 + rt ct The expected value of this equation can be written as:

244

11 Monetary Theory and Policy c  



ct+1 −γ log(1+rts )−γ log t+1 s ct Et 1 + rt = Et e ct

   , has a bivariate normal We assume that log (1 + rt ) , gt , gt = log ct+1 ct distribution. Thus, we use the multivariate normal moment generating function to compute this expected value. The bivariate normal distribution moment generating function is: M (t1 , t2 ) = Eet1 X1 +t2 X2 = et1 μ1 +t2 μ2 +

t1 σ12 +2t1 t2 σ12 +t22 σ22 2

Therefore Et elog(1+rt )−γ gt = eE log(1+rt )−γ Egt + s

s

σr2 −2γ σr,g +γ 2 σg2 2

Substituting this expression into the Euler equation, taking logs and rearranging yields: γ 2 σg2

σ2 + γ cov (rt , gt ) − r E log 1 + rts = ρ + γ Et (gt ) − 2 2

where σg2 is the variance of gt , σr2 is the variance of rts , cov rts , gt = σr,g is the covariance between rts and gt . The first three terms on the right-hand side of this expressions are the risk-free rate of interest. Therefore:

σ2 f E log 1 + rts = rt + γ cov rts , gt − r 2 By Jensen’s inequality:

σ2 E log 1 + rts + r ∼ = Erts 2 Thus, the rate of return on stocks is:

f Erts = rt + γ cov rts , gt Note: Jensen’s inequality. If f is concave (convex) Ef (X) ≤ f (EX) [Ef (X) ≥ f (EX)]. The function f (X) = log (1 + X) is 1 concave, f  (X) = 1+X , f  (X) = − 1 2 < 0. A first-order expansion, (1+X) around the point X = 0, gives: 1 f (X) = f (0) + f  (0)X + f  (0)X2 2

11 Monetary Theory and Policy

245

which is equal to: X2 log (1 + X) ∼ =X− 2 Thus, E log (1 + X) = EX −

Var(X) EX2 EX2 = EX − + 2 2 2

and we can use the following approximation: E log (1 + X) = EX −

VarX 2

(c) The risk premium between the risky asset (stock) and the risk-free asset (bonds) is given by:

f Erts − rt = γ cov rts , gt which depends on the risk aversion coefficient (γ ) and the covariance between the rate of return on the risky asset and the rate of growth of consumption. There are two stylized facts about the financial market prices: (i) the rate of return on risky assets (stocks) is in the range of 6–8% per year, and (ii) the risk-free rate of return on government treasury bills is 1% per year. The premium risk is around 6% per year. This premium risk should be explained by two parameters, the risk aversion coefficient (γ ) and the covariance between rts and gt . If this covariance were 0.03 the risk aversion coefficient would have to be equal to 20. This is considered too high, to describe economic behavior. The equity premium puzzle is a misnomer for the rejection of this model to explain stylized facts of the financial markets. The risk-free rate puzzle is the difficulty of the consumption asset pricing model, with isoelastic utility function, to match the data, reproducing a low rate, such as 1% per year. This match would occur if one is willing to accept unreasonable parameters of the model. The risk-free rate is the natural rate in a stochastic environment. For this rate to be negative the following condition has to be satisfied: 1 2 2 γ σg > ρ + γ Eg 2 Likewise, the equity premium and the risk rate puzzles, this condition would be satisfied for unreasonable values of the parameters.

246

11 Monetary Theory and Policy

(11) *(FED Operational Procedure After the 2007/2008 Financial Crisis). Before the 2007/2008 financial crisis the Federal Open Market Committee (FOMC) of the American Central Bank (FED) set a target for the Fed funds, implemented by the operational desk (The Desk) at the Federal Reserve Bank of New York. After the crisis, it has changed the operational procedure due a superabundant level of reserves balance in the banking system as a result of the huge amount of securities bought by the FED. Since October 2008 the FED pays interest rate on reserves balances as follows: (i) interest rate on required reserves (IORR), and (ii) interest rates on excess reserves (IOER). These interest rates have been kept at the same level. The Fed has a primary credit rate (PCR), which is the upper bound of the FED funds rate. Thus, the interest rate corridor of the reserves market is given by: i s = P CR and i l = I OER (a) Show the operational procedure in the market for federal funds prior to the 2007/2008 financial crisis. (b) Show the operational procedure in the market for federal funds after the 2007/2008 financial crisis. (a) Figure 11.16 shows the demand for FED funds with three segments: (i) the upper bound, (ii) the lower bound, which was zero until 2008, and (iii) the segment in between. The supply of funds is vertical (SS) because the Central Bank is monopolistic in this market. The DESK would implement the interest rate target (Target) by buying and selling securities.

Fig. 11.16 The demand for FED reserves when reserves were not paid

(b) Figure 11.17 shows the demand for FED funds with three segments. Now, the difference is that the lower bound interest rate can be different from

11 Monetary Theory and Policy

247

Fig. 11.17 The demand for FED reserves when reserves are paid

zero (i l = I OER). With abundant excess reserves the main tool of the American Central bank is the lower bound interest rate. It can shift the lower bound to raise (i1e ) or to lower (i2e ) the FED funds as depicted in Fig. 11.17.

Appendix A

Differential Equations

(1) With help from operator Dx = x, ˙ the differential equation system can be written as: Dx = a11 x + a12 y + b1 Dy = a21 x + a22 y + b2 Collecting terms on the two variables, the system becomes: (D − a11 ) x − a12 y = b1 −a21 x + (D − a22 ) y = b2 or using matrix notation:      D − a11 −a12 x b = 1 −a21 D − a22 y b2 Thus: −1      b1 x D − a11 −a12 = −a21 D − a22 b2 y The inverse of the matrix in the left-hand side of this equation is:   1 D − a22 a12 a21 D − a11 (D − a11 ) (D − a22 ) − a21 a12 From this inverse it is straightforward to obtain the answer to item (a). The solution of item (a) can be written as two equations: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2

249

250

A Differential Equations

[(D − a11 ) (D − a22 ) − a21 a12 ] x = (D − a22 ) b1 + a12 b2 [(D − a11 ) (D − a22 ) − a21 a12 ] y = a21 b1 + (D − a11 ) b2 Since (D − a11 ) (D − a22 ) − a21 a12 = D 2 − (a11 + a22 ) D + a11 a22 − a21 a21 and: tr(A) = a11 + a22 |A| = a11 a22 − a21 a12 where: A=

  a11 a12 a21 a22

It follows that: D 2 − (a11 + a22 ) D + a11 a22 − a21 a12 = D 2 − tr(A)D + |A| Using this expression, we obtain the system: x¨ − tr(A)x˙ + |A|x = a12 b2 − a22 b1 y¨ − tr(A)y˙ + |A|y = a21 b1 − a11 b2 taking into account that: Db1 = 0 Db2 = 0 since b1 and b2 are constants. (2) A good’s market model is exemplified by the equations: Demand: q d = α − βp Supply: q s = γ + δp Adjustment: p˙ =

  dp = φ qd − qs , φ > 0 dt

A Differential Equations

251

The differential equation of the model is: p˙ = φ (α − βp − γ − δp) or p˙ = φ (α − γ ) − φ (β + δ) p (a) When p˙ = 0, we obtain the equilibrium price φ (α − γ ) − φ (β + δ) p¯ = 0 Thus p¯ =

α−γ (β + δ)

α>γ

Fig. A.1 The phase diagram of the model

(b) The solution of the differential equation is given by: p = p¯ + Ce−φ(β+δ)t Given the initial price p(0) = p0 , we obtain the solution ¯ e−φ(β+δ)t p(t) = p¯ + (p0 − p) (c) The phase diagram of the model is shown in Fig. A.1.

252

A Differential Equations

(3) The Harrod-Domar economic growth model has three equations: ⎧ ⎨ S = sY I = υ Y˙ ⎩ I =S Thus: sY = υ Y˙ The rate of growth of the economy is given by: s Y˙ = Y υ (a) The path of this economy’s output is: s

Yt = Y0 e υ t (b) The phase diagram of Harrod-Domar is given by Fig. A.2. (4) The asset price model of this question is given by: r=

υ + p˙ p

which yields the differential equation: p˙ = rp − υ (a) When p˙ = 0, the equilibrium price is: p¯ =

Fig. A.2 The phase diagram of the Harrod-Domar model

υ r

A Differential Equations

253

(b) The solution of the differential equation is given by: p = p¯ + Cert This solution has two components: (i) the fundamental solution p¯ and (ii) the bubble component Cert . When C = 0 the market price and the equilibrium price differ. (c) This model with discrete variables is given by: r=

e − pt υt + pt+1

pt e pt+1

= pt+1

which yields: pt =

vt pt+1 + 1+r 1+r

(5) An economy’s model is specified by the equations: ⎧ ⎪ y˙ = α (d − y) ⎪ ⎪ ⎪ ⎪ ⎨d = c + i + g c = βy ⎪ ⎪ ⎪ i = i¯ ⎪ ⎪ ⎩ g˙ = −γ − y) (y ¯ (a) In equilibrium y˙ = g˙ = 0. Thus, y = y¯ and: d = β y¯ + i¯ + g¯ = y¯ It follows that y¯ =

i¯ + g¯ , 1−β

β 0 trJ = −α (1 − β) < 0 Thus, the system is stable and the fiscal policy rule will lead its real output to full employment. (6) An economy’s model is specified by the equations: ⎧ ⎪ y˙ = α (d − y) ⎪ ⎪ ⎪ ⎪ ⎨d = c + i c = βy

⎪ ⎪ ⎪ r˙ = γ md − m ⎪ ⎪ ⎩ d m = δy − λr ¯ The dynamical system is (a) Assume that investment is constant: i = i. given by:  y˙ = −α (1 − β) y + α i¯ r˙ = γ δy − γ λr − γ m In equilibrium y˙ = r˙ = 0. Thus: y¯ =

i¯ 1−β

and r¯ =

m δ y¯ − λ λ

The Jacobian of the dynamical system is: J =

 ∂ y˙

∂ y˙ ∂y ∂r ∂ r˙ ∂ r˙ ∂y ∂r

 =

  −α (1 − β) 0 γδ −γ λ

The determinant and the trace of this matrix are: |J | = α (1 − β) γ λ > 0 trJ = − [α (1 − β) + γ λ] < 0 We assume β < 1. Since the determinant is positive and the trace is negative the system is stable.

A Differential Equations

255

(7) Consider the model: ⎧ ⎪ P˙ = α (d − y) , α > 0 ⎪ ⎪ ⎨ r˙ = β L (y, r) − M , β > 0 P > 0, ∂d d = d (y, r) , ∂d ⎪ ⎪ ∂y ∂r < 0 ⎪ ⎩ y = y¯ (a) The dynamical system of this model is given by: 

P˙ = α [d (y, r) − y] ¯  r˙ = β L (y, ¯ r) − M P

In equilibrium P˙ = r˙ = 0. Thus, the interest rate and the price level are given by: d (y, ¯ r¯ ) = y¯ M L (y, ¯ r¯ )

P¯ =

The Jacobian of the dynamical system is:  J =

∂ P˙ ∂P ∂ r˙ ∂P

∂ P˙ ∂r ∂ r˙ ∂r



 =

0 β PM2

α ∂d ∂r β ∂L ∂r



The determinant and the trace of this matrix are: |J | = −α

∂d M β >0 ∂r P 2

trJ = β

∂L 0 ⎪ ⎪ ⎪ ⎨ r˙ = α [i + g − t − s] , α > 0 ∂i 0 ⎪ ⎪ ⎩ y = y¯

256

A Differential Equations

The dynamical system of the model is: 

 P˙ = β M ¯ r) P − L (y, r˙ = α [i(r) + g − t − s(y)] ¯

In equilibrium P˙ = r˙ = 0, and the steady-state for P and r, P¯ and r¯ are given by: M L (y, ¯ r¯ )

P¯ =

i(¯r ) + g − t = s(y) ¯ The Jacobian of the dynamical system is:  J =

∂ P˙ ∂P ∂ r˙ ∂P

∂ P˙ ∂r ∂ r˙ ∂r





−β PM2 −β ∂L ∂r = ∂i 0 α ∂r



The determinant and the trace of this matrix are: |J | = −β trJ = −β

M ∂i α >0 P 2 ∂r

∂i M +α 0, ⎪ ⎪ M ⎪ ⎨ P = L (y, r) ∂d d = d (y, r) , ∂d ∂y > 0, ∂r < 0 ⎪ ⎪ ⎪ M = M¯ ⎪ ⎪ ⎩ P = P¯ This model can be expressed as two equations: 

y˙ = α [d (y, r) − y] M P = L (y, r)

In equilibrium y˙ = 0: d (y, r) = y

A Differential Equations

257

The LM curve of this model jointly with IS curve can be solved for y and r. The LM curve has an implicit function for the rate of interest: r=f

 M y, , P

∂r >0 ∂y

From the differential equation for y we obtain:

 ∂ y˙ ∂d ∂d ∂r = −α 1 − +α

∂d ∂y .

Thus the model is stable.

(10) Consider the model: ⎧ ⎪ y˙ = φ (d − y) ⎪ ⎪ ⎨ π = π e + δ (y − y) ¯ e = θ (π − π e ) π ˙ ⎪ ⎪ ⎪ ⎩ d = d (y, π e ) , ∂d > 0, ∂d > 0 ∂y ∂π e Combining the Phillips curve and the adaptive expectation mechanism we obtain: π˙ e = θ δ (y − y) ¯ The differential equation of output is given by:

 y˙ = φ d y, π e − y In equilibrium π˙ e = y˙ = 0. Thus, y = y¯ and: d (y, ¯ π¯ ) = y¯ which gives the inflation rate of equilibrium (π). ¯ The Jacobian of the dynamical system is:  J =

∂ π˙ e ∂ π˙ e ∂π e ∂y ∂ y˙ ∂ y˙ ∂π e ∂y



 =

0 φ πde

 θδ  φ ∂d ∂y − 1



The determinant of this matrix is: |J | = −θ δφ

∂d 0, ∂d > 0 ∂y ∂π e Combining the first, third and fourth equation we obtain the differential equation for the expected rate of inflation.

 π˙ e = θ φ d y, π e − y The differential equation for real output is given by: y˙ =  (y¯ − y) In equilibrium π˙ e = y˙ = 0. It follows that y = y¯ and π = π e , d(y, ¯ π¯ ) = y. ¯ From this equation we obtain the steady-state rate of inflation (π¯ ). The Jacobian of the dynamical system is:  J =

∂ π˙ e ∂ π˙ e ∂π e ∂y ∂ y˙ ∂ y˙ ∂π e ∂y



 =

∂d θ φ ∂π e −θ φ 0 −



The determinant of this matrix is: |J | = −θ φ

∂d f0 . This increase was not anticipated. This increase shifts upward the y˙ = 0 curve as shown in Fig. A.4. The long run interest rate is a jump variable, and at the announcement of the new fiscal policy it jumps to the new saddle path. From that point onward the economy converges on the new equilibrium (point Ef ). (b) What is the effect of an increase in the quantity of money on real output? The quantity of money increases from m0 to m1 > m0 . This increase shifts downward the curve R˙ = 0 as depicted in Fig. A.5. The variable R jumps at the time of the announcement of the new monetary policy to R(0+ ) in such way that the economy will be in the new saddle path. Then, the

260

A Differential Equations

Fig. A.4 Dynamic adjustment to an unanticipated increase in government spending

Fig. A.5 Dynamic adjustment to an unaticipated increase in the money stock

economy converges on the new equilibrium where output is larger than in the previous equilibrium. (13) Consider the model: ⎧ dM ⎨ I − S = −α (r − r ∗ ) = dt , α > 0 p˙ = β (I − S) , β > 0 ⎩ r˙ = γ (p − p) ¯ + δ p, ˙ δ > 0, γ > 0 This model can be reduced to two differential equations: 

p˙ = −βα (r − r ∗ ) r˙ = γ (p − p) ¯ − αβδ (r − r ∗ )

The Jacobian of the dynamical system is: J =

 ∂ p˙

∂ p˙ ∂p ∂r ∂ r˙ ∂ r˙ ∂p ∂r





0 −βα = γ −αβδ



A Differential Equations

261

The determinant and the trace of this matrix are: |J | = βαγ > 0 trJ = −βαδ < 0 The determinant is positive and the trace is negative. Thus, the model is stable. In equilibrium p˙ = r˙ = 0. Therefore, r = r ∗ , p = p¯ and dM dt = 0. (14) The model has three equations: an aggregate demand, a Phillips curve and perfect foresight according to: M

⎧ ⎨ y = k + α log P + βπ e + γf ¯ π = π e + δ (y − y) ⎩ e π =π From the Phillips curve and the hypothesis of perfect foresight, it follows that output is equal to potential output: y = y¯ Thus, the aggregate demand curve can be written as: y¯ = k + α (m − p) + β p˙ + γf where m = log(M) and p = log(P ). This is a first-order differential equation in the price level: p˙ =

α p−υ β

where: υ=

αm + γf + k − y¯ β

The solution of this first-order differential equation is given by:  p(t) =

∞

e

  − βα (τ −t)

υdτ

t

assuming that there is no bubble. (a) Analyze the effects of a change in the government fiscal policy in the following situations:

262

A Differential Equations

(i) (ii) (iii) (iv)

Permanent, unanticipated; Permanent, anticipated; Transitory, unanticipated; Transitory, anticipated.

(i) Permanent, unanticipated The fiscal policy change is given by: 

υ0 , τ ≤ t υ1 , τ ≥ t

υ= The price level is obtained by: 

∞

p(t) =

e

− βα (τ −t)

υ1 dτ = υ1 e

α βt



∞

e

t

− βα τ

dτ

t

Thus: p(t) = υ1 e

α βt

e

− βα t

=

α β

β υ1 α

where: υ1 =

αm + γf1 + k − y¯ β

(ii) Permanent, anticipated The fiscal policy change is given by:  υ=

υ0 , t ≤ τ ≤ T υ1 , τ ≥ T

The price level is obtained by:  p(t) =

T

e

− βα (τ −t)



∞

υ0 dτ +

t

e

− βα (τ −t)

υ1 dτ

T

The first integral on the right-hand side is: 

T

e t

− βα (τ −t)

α βt



υ0 dτ = e υ0



T

e

− βα τ

α βt

dt = e υ0

t

 α β  − βα T t −αt = e β υ0 −e +e β α

−e

− βα τ )T )

α β

)

t



A Differential Equations

263

Therefore: 

T

e

− βα (τ −t)

υ0 dτ =

t

 β  − α −t) υ0 1 − e β (T α

The second integral on the right-hand side is: 

T

e

− βα (τ −t)



α βt



∞

υ1 dτ = e υ1

e

t

− βα τ

α βt

dτ = e υ1

−e

α β

T α

t

=

β − α −t) υ1 e β (T α

υ1 dτ =

β − α −t) υ1 e β (T α

= e β υ1 e

− βα T

− βα τ )∞ )

)

T

Thus, 

∞

e

− βα (τ −t)

T

Collecting the two integrals just obtained we get the price level:  β β  − α −t) − α −t) υ0 1 − e β (T + υ1 e β (T α α

p(t) =

(iii) Transitory, unanticipated  υ=

υ1 , t ≤ τ ≤ T υ0 , τ ≥ T

The price level is obtained by the following integral:  p(t) =

T

e

− βα (τ −t)



∞

υ1 dτ +

t

e

− βα (τ −t)

υ0 dτ

T

From the previous item it is straightforward to obtain: 

T

e

− βα (τ −t)

υ1 dτ =

t

 β  − α −t) υ1 1 − e β (T α

and: 

∞

e T

− βα (τ −t)

υ0 dτ =

 β  − α −t) υ0 1 − e β (T α

The price level is obtained adding these two expressions:



264

A Differential Equations

p(t) =

 β α β  − α (T −t) −t) + υ0 e β (T υ1 1 − e p α α

(iv) Transitory, anticipated ⎧ ⎨ υ0 , t ≤ τ ≤ T1 υ = υ1 , T1 ≤ τ ≤ T2 ⎩ υ0 , τ ≥ T2 The price level is obtained by:  p(t) =

T1

e

− βα (τ −t)

 υ0 dτ +

t

T2

e

− βα (τ −t)

 υ1 dτ +

T1

∞

e

− βα (τ −t)

υ0 dτ

T2

From the previous item, it is easy to get: 

T1

e

− βα (τ −t)

υ0 dτ =

t

 β  −α −t) υ0 1 − e β (T1 α

and: 

∞

e

− βα (τ −t)

υ0 dτ =

T2

β −α −T υ0 e β (T2 ) α

The second integral on the right-hand side of the price level equation is given by: 

T2

e

− βα (τ −t)

υ1 dτ = υ1 e

− βα t

T1





T2

e

− βα τ

dτ = υ1 e

α βt

−e

T1

=

− βα τ )T ) 2

α β

)



T1

 β  − βα (T1 −t) −α −t) υ1 e − e β (T2 α

By collecting the three integrals we obtain the price level:



α  β β β − α (T −t) − (T −t) − α (T −t) − α (T −t) + υ1 e β 1 −e β 2 + υ0 e β 2 p(t)= υ0 1−e β 1 α α α

(b) Analyze the effect of a change in the growth rate of money stock  d log M in the following situations: μ = dt (i) (ii) (iii) (iv)

Permanent, unanticipated; Permanent, anticipated; Transitory, unanticipated; Transitory, anticipated.

A Differential Equations

265

The stock (log) of money grows at a constant rate μ according to: m = μτ When τ is equal to zero, m = 0(M = 1 and log(M) = 0). (i) Permanent, unanticipated The change in the growth rate of money stock is specified as follows:  μ0 , τ ≤ t μ= μ1 , τ ≥ t The fundamentals of the price is given by: αm + γf + k − y¯ β

υ=

we choose units in such a way that: γf + k − y¯ = 0 Thus, υ=

α α m = μτ β β

The price level is given by the following integral:  p(t) =

∞

e

− βα (τ −t) α

β

t

μ1 τ dτ

which can be written as: α α t p(t) = μ1 e β β



∞

e

− βα τ

τ dτ

t

" "

We use the method of integration by parts udυ = uυ − υdu to solve the integral in the price level formula: 

∞

e t

− βα τ

τ dτ = −

e

− βα τ )∞ ) α β

)

t



∞

− t

−ατ

e β β2 α β α   dτ = e− β t t + 2 e− β t α α − βα

By substituting this integral into the equation for p(t) we obtain: p(t) = μ1 t + μ1

β α

266

A Differential Equations

(ii) Permanent, anticipated The monetary policy change is given by:  μ=

μ0 , τ ≤ T μ1 , τ ≥ T

The price level is obtained by: 

T

p(t) =

e

− βα (τ −t) α

β

t



∞

μ0 τ dτ +

e

− βα (τ −t) α

β

T

μ1 τ dτ

which can be written as: α

t

p(t) = e β μ0



α β

T

e

− βα τ

α

t

τ dτ + e β μ1

t

α β



∞

e

− βα τ

τ dτ

T

To compute the second integral on the right-hand side we use the result from the previous item to write: 

∞

e

− βα τ

T

β −αT τ dτ = e β α

 β T + α

The first integral on the right-hand side is: 

T

e t

− βα τ

β − α τ ))T β 2 − α τ ))T τ dτ = − e β ) − 2 e β ) t t α α

It is straightforward to verify that: 

T

e t

− βα τ

  β β β − βα t β − βα t t+ T + − e τ dτ = e α α α α

Substituting the two integrals into the price level formula, we obtain, after some algebra, the following expression:

  β β − α −t) T + + (μ1 − μ0 ) e β (T p(t) = μ0 t + α α (iii) Transitory, unanticipated ⎧ ⎨ μ0 , τ ≤ t μ = μ1 > μ0 , t ≤ τ ≤ T ⎩ μ0 , τ ≥ T

A Differential Equations

267

The price level is given by: 

T

p(t) =

e

− βα (τ −t) α

β

t



∞

μ1 τ dτ +

e

− βα (τ −t) α

β

T

μ0 τ dτ

From the previous item the first integral on the right-hand side of this expression is: 

T

e

  β β −αt μ1 t + − μ1 dτ = e β T + β α α

− βα (τ −t) α

t

By the same token, the second integral of the right-hand side of the price level expression was derived in the previous item. Thus: 

∞

e

 β μ0 τ dτ = μ0 T + β α

− βα (τ −t) α

T

Adding the two last integrals we obtain the price level:

    β β − βα (T −t) + μ0 − μ1 e T + p(t) = μ1 t + α α (iv) Transitory, anticipated The monetary policy change is specified as: ⎧ ⎨ μ0 , t ≤ τ ≤ T1 μ = μ1 , T1 ≤ τ ≤ T2 ⎩ μ0 , τ ≥ T2 The price level is given by:  p(t) =

T1

e t

 +

∞

e T2

− βα (τ −t) α

β − βα (τ −t) α

β

 μ0 τ dτ +

T2

e T1

− βα (τ −t) α

β

μ1 τ dτ

μ0 τ dτ

From the previous item, the first and the third integral on the right-hand side are:

   T1 β β − βα (τ −t) α − βα (T1 −t) T1 + μ0 τ dτ = μ0 t + − μ0 e e β α α t

268

A Differential Equations

and: 

∞

e

− βα (τ −t) α

β

T2

μ0 τ dτ = μ0 e

− βα (T2 −τ )

 β T2 + α

It takes just a little bit of algebra to obtain the second integral on righthand side of the price level expression. This integral is: 

T2

e T1

− βα (T −t) α

β

μ1 τ dτ = μ1 e

− βα (T1 −t)

  β β − α 2−t) T1 + T2 + − μ1 e β (T α α

By collecting the three integrals we write the price level as:

    β β −α −t) + μ1 − μ0 e β (T1 T1 + p(t) = μ0 t + α α 

  β −α −T − μ0 − μ1 e β (T2 1 ) T2 + α (15) The housing market model has three equations: ⎧ ⎨ A = (δ + r − π + τ ) P − P˙ dA 0, otherwise the integral would not exist. It follows that  y c(0) = [r (1 − σ ) + σρ] a(0) + r when σ = 1,  y c(0) = ρ a(0) + r (6) (Tobin’s q) The firm solves the following problem:  max 0

∞

  αI 2 dt e−ρt pQ (K, L) − wL − I − K

subject to: K˙ = I − δK K(0) = K0 ,

given

The Hamiltonian of this problem is: H = Q (K, L) − wL − I −

αI 2 + q(I − δK) 2K

We assume, to simplify, that p ≡ 1 and we use The first-order conditions are:

α 2

⎧ ∂H ⎪ ∂L = QL − w = 0 ⎪ ⎪ ∂H αI ⎨ 0 ∂I = −1 − K + q =  ∂H ⎪ ⎪ q˙ = ρq − ∂K = ρq − QK + ⎪ ⎩ ∂H ˙ ∂q = I − δK = K where QK =

∂Q ∂K

and QL =

∂Q ∂L .

instead of α.

α 2

I 2 K

− qδ



These first-order conditions can be written as: QL ≡ w

290

B Optimal Control Theory

q =1+

αI K

α q˙ = (ρ + δ) q − QK − 2

I K

2

K˙ = I − δK By substituting q into the first and fourth equation we obtain the dynamical system with two differential equations: ⎧ ⎨ q˙ = (ρ + δ) q − Q − K   ⎩ K˙ = q−1 − δ K

α 2



q−1 α

2

α

In equilibrium q˙ = K˙ = 0, and: q¯ = 1 + αδ ¯ K = (ρ + δ) q − α Q 2

q −1 α

2

The Jacobian of this dynamical system is:  ∂ q˙ J =

∂ q˙ ∂q ∂K ∂ K˙ ∂ K˙ ∂q ∂K





K ρ + δ − dQ dK = q−1 K α α −δ



In steady-state the determinant of the Jacobian is: |J | =

K dQK α dK

K This determinant is negative if dQ dK < 0. We show below that for a constant return to scale production function this determinant is equal to zero. We proceed assuming that such is not the case.

(a) The phase diagram of the dynamical system with q on the vertical axis and k on the horizontal axis is depicted in Fig. B.12. The equilibrium is a saddle point and the saddle path is downward sloping. K To obtain the derivative dQ dK we take into account that QL = w. The differential of this expression is: QLL dL + QLK dK = dw and the differential of QK is:

B Optimal Control Theory

291

Fig. B.12 The phase diagram of the Tobin’s q model with q on the vertical axis and the stock of capital K on the horizontal axis

dQK = QKK dK + QKL dL Combining these two expressions, we obtain: dQK = QKK dK +

QKL (dw − QLK dK) QLL

which can be written as: dQKK

QKK QLL − Q2KL dK QKL = + dw QLL QLL

The production function Q(K, L) is concave and obeys the following property: QKK < 0, QLL < 0, QKL > 0, QKK QLL − Q2KL > 0 When the production function Q(K, L) has constant returns to scale its Hessian H is singular: H =

  QKK QKL QLK QLL

|H | = 0. Therefore, the model in this case cannot be analyzed on a phase diagram with q on the vertical axis and K on the horizontal axis: (b) The Marginal Value of Capital.

292

B Optimal Control Theory

Let: 

t1

max

f (t, x, u) dt = V (t0 , x0 )

t0

subject to: x˙ = g (t, x, u) The maximum value can be written as:  t1 V (t0 , x0 ) = ˙ dt [f (t, x, u) dt + λ (g (t, x, u) − x)] t0

We integrate by parts: 

t1 t0

λxdt ˙ = λx|tt10 −



t1

x λ˙ dt = λ (t1 ) x (t1 ) − λ (t0 ) x (t0 ) −

t0



t1

x λ˙ dt

t0

Substituting this expression into the V (t0 , x0 ) equation, we obtain:  V (t0 , x0 ) =

t1



H + x λ˙ dt + λ (t0 ) x (t0 )

t0

where H is the Hamiltonian and we use the transversality condition λ(t1 ) = 0. Taking the derivative with respect to x0 yields: ∂V (t0 , x0 ) = ∂x0 where Hx =

∂H ∂x , Hu



Hx t0

=

∂V (t0 , x0 ) = ∂x0

t1

∂H ∂u .



t1 t0

λ˙ d x˙ dx du + Hu + dx0 dx0 dx0

 dt + λ (t0 )

This expression can be written as: 

dx du dt + λ (t0 ) Hx + λ˙ + Hu dx0 dx0

From the first-order condition: Hx + λ˙ = 0 Hu = 0

B Optimal Control Theory

293

we obtain: ∂V (t0 , x0 ) = λ (t0 ) ∂x0 Therefore, Tobin’s q can be interpreted as the marginal value of capital: ∂V = q0 ∂K0 The Average Value of Capital The differential equation for q is given by: α q˙ = (ρ + δ) q − QK − 2

I K

2

We multiply both sides of this equation by the stock of capital K, α qK ˙ = (ρ + δ) qK − QK K − 2

I2 K



and we assume a constant return to scale production function. Thus: Q = QK K + QL L Therefore: QK K = Q − QL L = Q − wL taking into account the fact that the marginal product of labor is equal to the real wage. Substituting this expression into the qK ˙ equation yields: α qK ˙ − ρqK + q K˙ = δqK − Q + wL − 2

I2 K



+ q K˙

where we added q K˙ to both sides. Since K˙ = I − δK we can write this equation as: qK ˙ − ρqK + q K˙ = δqK − Q + wL − In equilibrium:

 αI q = 1+ K

α 2

I2 K

 + qI − qδK

294

B Optimal Control Theory

and: 

αI 2 qI = I + K By substituting this expression into the former equation we obtain: α qK ˙ − ρqK + q K˙ = −Q + wL + 2

I2 K

 +I

Next, we use the fact that: d −ρt ˙ e qK = −ρe−ρt qK + e−ρt qK ˙ + e−ρt Kq dt to write: −



 α I2 d −ρt e qK = e−ρt Q − wL − I − dt 2 K

Thus: 

∞

−

de

−ρt



∞

qK =

0

e

−ρt

0



∞

−



 α I2 Q − wL − I − dt 2 K

de−ρt qK = q(0)K(0) − lim e−ρt qK t→∞

0

From the transversality condition: lim e−ρt qK = 0

t→∞

Thus, average value of capital is equal to Tobin’s q: q(0) =

V (0) K(0)

(c) When the parameter α equals zero, q is equal to one, and there is no investment because capital adjusts instantaneously. (7) The agent maximizes:  max 0

∞

e−ρt μmdt

B Optimal Control Theory

295

subject to constraint: m ˙ = μm − τ (m) The current value Hamiltonian is given by: H = μm + λ (μm − τ (m)) which is linear in the rate of growth of money, the control variable: H = (1 + λ) μm − λτ (m) A singular control requires that: 1+λ=0 The costate equation is given by: λ˙ = ρλ −

 ∂H = ρλ − (1 + λ) μm − λτ  (m) ∂m

or: λ˙ = ρλ + λτ  (m) Taking into account that 1 + λ = 0, λ = −1. Thus, λ˙ = 0, and ρ + τ  (m) = 0. (a) From this equation we obtain m, ¯ as: τ  (m) ¯ = −ρ Since m = m, ¯ μ = π and: μ¯ =

τ (m) ¯ m ¯

The initial price level is given by: P (0) =

M(0) m ¯

Figure B.13 depicts the inflation, tax curve and the equilibrium value of m, m. ¯ This policy is inconsistent because the Central Bank can increase seigniorage by moving to point A of Fig. B.13.

296

B Optimal Control Theory

Fig. B.13 The inflation tax curve and the inconsistent monetary policy

(8) Using the constraint: m ˙ = μm − τ (m) we can write: 

∞



e−ρt μmdt =

0

∞

e−ρt (m ˙ + τ (m)) dt

0

By integration by parts we obtain: 

∞

e−ρt mdt ˙ = e−ρt m|∞ 0 −

0



∞

m (−ρ) e−ρt dt

0

Thus: 

∞

e−ρt mdt ˙ = lim e−ρt m − m(0) + t→∞

0



∞

ρme−ρt dt

0

The limit in this expression is equal to zero. Therefore: 

∞

e−ρt mdt ˙ =

0

∞

ρ (m − m(0)) e−ρt dt

0

It follows that:   ∞ e−ρt μmdt = 0



∞ 0

e−ρt τ (m)dt +



∞ 0

ρ (m − m(0)) e−ρt dt

B Optimal Control Theory

297

(9) The Central Bank maximizes: 

∞ 0

e−ρt μmdt + m(0) − m(0− )

where m(0) − m(0− ) is the change of the real stock of money at the initial moment. From the previous exercise we have: 

∞

e

−ρt



∞

μmdt =

0

e

−ρt



∞

τ (m)dt +

0

ρ (m − m(0)) e−ρt dt

0

By substituting this expression in the Central Bank’s objective function, we obtain:  ∞  ∞ −ρt e τ (m)dt + e−ρt ρ (m − m(0)) dt + m(0) − m(0− ) 0

0

Due to the fact that:  m(0) − m(0− ) =

∞ 0

e−ρt ρ (m(0) − m(0− )) dt

we get: 

∞

e 0

−ρt

 μmdt+m(0) − m(0− )=

∞

e

−ρt



∞

τ (m)dt+

0

e−ρt ρ (m−m(0)) dt

0

Thus, the Central Bank does not take into account the instantaneous change in the real stock of money at the initial moment in its new monetary policy. Thus, there is no dynamic inconsistency. (10) (a) Taking the derivatives with respect to time of the money demand equation we obtain:

m ˙ = −α π˙ e = −αβ π − π e m where we use the adaptive expectation mechanism. From the definition of the real quantity of money: m = M P , we get: M˙ P˙ m ˙ = − =μ−π m M P Thus, π =μ−

m ˙ m

298

B Optimal Control Theory

and πe = −

1 log m α

Taking this expression into the first equation yields:



 m ˙ m ˙ 1 = −αβ μ − + αβ − log m m m α It is straightforward to obtain: m ˙ =−

β αβ μm − m log m 1 − αβ 1 − αβ

(b) The optimal control problem is to maximize:  ∞ e−ρt μmdt max 0

subject to the constraint: m ˙ =−

αβ β μm − m log m 1 − αβ 1 − αβ

The current value Hamiltonian is: 

β αβ μm + m log m H = μm − λ 1 − αβ 1 − αβ which can be written as: 

λβ λαβ μm − m log m H = 1− 1 − αβ 1 − αβ A singular control exists when: 1−λ

αβ =0 1 − αβ

or: 1 − αβ =λ αβ The first-order condition for the costate variables is: λ˙ = ρλ −

∂H λβ = ρλ + (1 + log m) ∂m 1 − αβ

B Optimal Control Theory

299

Since λ is constant, λ˙ = 0. Thus: 

β λ ρ+ (1 + log m) = 0 1 − αβ and: (1 + log m) = −

ρ (1 − αβ) β

When: m ˙ = 0, −

βm log m αβ μm − =0 1 − αβ 1 − αβ

It follows that: μ=−

1 log m α

or:   ρ (1 − αβ) 1 −1 − μ=− α β Therefore:



μ¯ =

1 α

+ρ

m ¯ = e−α μ¯



1 αβ

 −1

If m0 = m ¯ there is no optimal control. (c) If the Central Bank can inject or remove money at the initial moment such that m(0) = m(0− ) there is an optimal monetary policy, which is inconsistent. (11) (a) From the government budget constraint m ˙ =

M˙ − mπ = −x − mπ P

and from the money demand equation π =−

1 log m α

It follows that m ˙ =

m log m −x α

300

B Optimal Control Theory

The representative agent maximizes  ∞ e−ρt [u(c) + υ(m)] dt 0

subject to c = f (x) m ˙ =

m log m −x α

Thus, the control problem is:  ∞ max e−ρt [u (f (x)) + υ(m)] dt 0

subject to: m ˙ =

m log m −x α

The initial condition for m is not given since the price level at moment zero, P (0), is not a predetermined variable. We assume that the initial stock of money is given: M(0) given The current-value Hamiltonian is:

m log m −x H = u (f (x)) + υ(m) + λ α



The first-order conditions are: ∂u ∂c ∂H = −λ=0 ∂x ∂c ∂x   ∂H λ  ˙λ = ρλ − = ρλ − υ (m) + (log m + 1) ∂m α m log m ∂H = −x =m ˙ ∂λ λ and the transversality condition is: lim λme−ρt = 0

t→∞

B Optimal Control Theory

301

From the first-order condition: ∂u ∂c =λ ∂c ∂x We obtain x as a function of λ: x = x(λ) The dynamical system for the two variables, x and λ, is given by: m log m − x (λ) α   1 + log m λ˙ = −υ  (m) − λ −ρ α m ˙ =

The Jacobian of this system is:  J =

∂m ˙ ∂m ∂ λ˙ ∂m

∂m ˙ ∂λ ∂ λ˙ ∂λ



 =

1+log m −x  (λ) α λ 1+log m  −ρ −υ (m) − αm α



In steady-state, λ˙ = 0, m ˙ = 0, and: m log m = x (λ) α υ  (m) λ = − 1+log m −ρ α The determinant of the Jacobian evaluated at the steady-state point is negative:

|J | =

1 + log m α



−υ  (m) λ



  λ − x  (λ) υ  (m) + mα

The dynamical system equilibrium is a saddle point. (b) The phase diagram of the dynamical system is depicted in Fig. B.14 with costate variable λ on the vertical axis and the real quantity of money on the horizontal axis. SS is the saddle path. (c) The initial real quantity of money is not given. Thus, the costate variable must be zero at the initial moment, λ(0) = 0, and the economy converges on the equilibrium point E. Time inconsistency arises in this model because if government wants to solve the maximization in the future, the

302

B Optimal Control Theory

Fig. B.14 Time inconsistent monetary policy: the phase diagram of the model

initial point of the real quantity of money will be m(0) and not the level prevailing at that time. (12) (a) The Central Bank minimizes:  0

∞

e−ρt



 ϕ 2 1 2 x + π dt 2 2

subject to: π˙ = α (π − μ) + βx x˙ = γ (π − μ) + δx π(0) = μ0 x(0) = 0 For a given rate of growth of the monetary base we assume that the dynamical system for π and x is stable. The matrix of this system  A=

αβ γ δ



is such that its determinant is positive and its trace is negative: |A| = αδ − γβ > 0 trA = α + β < 0

B Optimal Control Theory

303

The current-value Hamiltonian of the control problem is given by: H =

ϕ 2 1 2 x + π + λ [α (π − μ) + βx] + θ [γ (π − μ) + δx] 2 2

where λ and θ are costate variables. This Hamiltonian can be written as: H =

ϕ 2 1 2 x + π + λ [απ + βx] + θ [γ π + δx] − (αλ + γ θ ) μ 2 2

The Hamiltonian is linear on μ. Thus, to have a singular optimal control the coefficient of μ should be zero: αλ + γ θ = 0 The first-order conditions are: ∂H = ρλ − π ∂π ∂H θ˙ = ρθ − = ρθ − ϕx − βλ − δθ ∂x ∂H = π˙ = α (π − μ) + βx ∂λ ∂H = x˙ = γ (π − μ) + δx ∂θ λ˙ = ρλ −

We have to solve the following system: ⎧ ⎪ ⎪ αλ + γ θ = 0 ⎪ ⎪ ⎪ ⎨ λ˙ = ρλ − π θ˙ = ρθ − ϕx − βλ − δθ ⎪ ⎪ ⎪ π˙ = α (π − μ) + βx ⎪ ⎪ ⎩ x˙ = γ (π − μ) + δx Eliminating λ and θ We start by taking the time derivative of the first equation: α λ˙ + γ θ˙ = 0 Then, substitute λ˙ and θ˙ , the second and the third equations to obtain: (αδ − γβ) λ = απ + γ ϕx Next, we take the derivative with respect to time of this equation: (αδ − γβ) λ˙ = α π˙ + γ ϕ x˙

304

B Optimal Control Theory

When we substitute the expression of λ˙ in this equation we obtain: [α (ρ − δ) + γβ] π + ργ ϕx = α π˙ + γ ϕ x˙ Thus, we have a three-equation system: ⎧ ⎨ [α (ρ − δ) + γβ] π + ργ ϕx = α π˙ + γ ϕ x˙ π˙ = α (π − μ) + βx ⎩ x˙ = γ (π − μ) + δx Eliminating μ We multiply the π˙ equation by γ and the x˙ equation by α, and subtract one equation from the other to obtain: γ π˙ − α x˙ = (γβ − αδ) x Therefore, we have a two-equation system: 

α π˙ + γ ϕ x˙ = [α (ρ − δ) + γβ] π + ργ ϕx γ π˙ − α x˙ = (γβ − αδ) x

This system is solved to obtain:     π˙ π =J x˙ x where the matrix J is:   2 1 α (ρ − δ) + αγβ αργ ϕ + γ ϕ (γβ − αδ) J = 2 α + γ 2 ϕ γ α (ρ − δ) + γ 2 β γργ ϕ − α (γβ − αδ) The determinant of this matrix is negative: |J | =

− (αδ − γβ − αρ) (αδ − γβ) α2 + γ 2ϕ

if αδ − γβ − αρ > 0. (b) The two equations of the dynamical system are: π˙ =

−α (αδ − γβ − αρ) π γ ϕ (αδ − γβ − γρ) x − 2 2 α +γ ϕ α2 + γ 2ϕ

x˙ =

γ 2 ρϕ + α (αδ − γβ) x −γ (αδ − γβ − αρ) π + 2 2 α +γ ϕ α2 + γ 2ϕ

B Optimal Control Theory

305

Fig. B.15 The phase diagram of the π and x system

The phase diagram of this system is depicted on Fig. B.15, with π on the vertical axis and x on the horizontal axis. SS is the saddle path. (13) (a) The consumer maximizes:  ∞ e−ρt u(c)dt T

subject to the constraints: x˙ = −c x(0) = S x(T ) = 0 The current value-Hamiltonian is given by: H = u(c) − λc The first-order conditions are: λ˙ = ρλ −

∂H = ρλ − 0 = ρλ ∂x

∂H = u (c) − λ = 0 ∂c ∂H = x˙ = −c ∂λ Taking derivative with respect to time of the second equation yields: u (c)c˙ = λ˙

306

B Optimal Control Theory

which can be written as: u (c)c˙ = ρλ or: c˙ =

ρu (c) u (c)

The rate of growth of consumption is: c˙ u (c) =  ρ c cu (c) if the utility function is given by: 1

u(c) =

c1− σ 1−

1 σ

the rate of growth of consumption is: c˙ = −σρ c and: c(t) = c(0)e−σρt We know that: 

T

cdt = S

0

Thus: 

T

c(0)e−σρt dt = S

0

It is straightforward to obtain: c(0) =

σρS 1 − e−σρT

B Optimal Control Theory

307

(b) What happens to the path when ρ = 0? From the equation: c˙ =

ρu (c) u (c)

when ρ = 0, c˙ = 0, e.g., consumption is constant: ct = c¯ Thus: 

T

cdt ¯ =S

0

and: c¯ =

S T

(14) The firm maximizes: 

T

e−ρt (pq − c (q, x)) dt

0

subject to the constraints: x˙ = −q x(0) = S 0≤q≤S

(a) When c(q, x) = 0, the current value-Hamiltonian is: H = (p(x) − λ) q which is linear in q. The switching function σ (x, λ) is: σ (x, λ) = p − λ Thus, the optimal control is given by:

308

B Optimal Control Theory

⎧ ⎨ qmin , if p − λ < 0 q = q¯ ∈ [qmin , qmax ] , if p − λ = 0 ⎩ qmax , if p − λ > 0 The other necessary conditions are: λ˙ = ρλ −

∂H = ρλ − 0 = ρλ ∂x

and the transversality condition: λ(T ) = 0 If there is a singular control p − λ = 0. In this case, λ is constant and λ˙ = 0. It follows from the first-order condition, that λ = 0. Thus, there is no singular control. If p > λ, the solution is q = qmax = S, and the interest rate is not relevant for the firms decision. (b) When c (q, x) = c(x)q the current-value Hamiltonian is: H = pq − c (x) q − λq which is linear in the control variable: H = (p − c (x) − λ) q The switching function σ (x, λ) gives the following solution σ (x, λ) = p − c(x) − λ Thus, the optimal control is given by: ⎧ ⎨ qmin , if p − c(x) − λ < 0 q = q¯ ∈ [qmin , qmax ] , if p − c(x) − λ = 0 ⎩ qmax , if p − c(x) − λ > 0 The other necessary conditions are: λ˙ = ρλ −



∂H = ρλ − −c (x)q = ρλ + c (x)q ∂x

and the transversality condition: λ(T ) = 0

B Optimal Control Theory

309

Fig. B.16 The singular central solution

If there is a singular control p − c(x) − λ = 0. We take the derivatives with respect to time of this expression to obtain: −c (x)x˙ − λ˙ = 0 which can be written as: λ˙ = c (x)q by taking into account the transition equation: x˙ = −q. ˙ yields: Equating the two expression for λ, λ˙ = ρλ + c (x)q = ρλ + λ˙ Therefore, λ=0 and there is a stock of non-renewable resource x ∗ such that:

p = c x∗ Since c (x) < 0 and c (x) > 0, Fig. B.16 shows that p > c(x) if x > x ∗ , and p < c(x) if x < x ∗ . (15) A firm solves the following problem:  max 0

∞

e−ρt [(p − c(x))] qdt

310

B Optimal Control Theory

subject to the following constraints: x˙ = f (x) − q x(0) = 0 0≤q≤S (a) The current value Hamiltonian is: H = (p − c(x)) q + λ (f (x) − q) which is linear in the control variable: H = (p − c(x) − λ)q + λf (x) The switching function σ (x, λ) gives the following solutions: ⎧ ⎨ qmax , if σ (x, λ) > 0 q = q¯ ∈ [qmin , qmax ] , if σ (x, λ) = 0 ⎩ qmin , if σ (x, λ) < 0 There is a singular control when the switching control is equal to zero for some time interval. p − c (x) − λ = 0 The other necessary conditions are: λ˙ = ρλ −

 ∂H = ρ − −c (x)q + λf  (x) ∂x ∂H = f (x) − q = x˙ ∂λ

and the transversality condition: lim xe−ρt = 0

t→∞

Taking derivatives with respect to time of the switching function we obtain:

λ˙ = −c (x)x˙ = −c (x) f  (x) − q

B Optimal Control Theory

311

By equating two expressions for λ˙ yields:

λ˙ = ρ − f  (x) (p − c(x)) + c (x)q = −c (x) f (x) + c (x)q which can be simplified as: ρ = f  (x) −

c (x) f (x) p − c (x)

The solution of this equation is x ∗ , which is constant. Thus x˙ = 0 and the optimal control is: q ∗ = f (x ∗ ) The initial condition x(0) may be different from x ∗ . If x(0) > x ∗ , σ (x, λ) > 0 and q = qmax = q. ¯ If x(0) < x ∗ , σ (x, λ) < 0 and q = qmin = 0. Figure B.17 shows the solutions for the three cases: x(0) = x ∗ , x(0) > x ∗ , and x(0) < x ∗ . (b) When  x f (x) = αx 1 − S the optimal solution is obtained by solving the equation:    c (x)  x 2x − 1− ρ =α 1− S p − c (x) S

Fig. B.17 The three solutions of the central problem

Appendix C

Difference Equations

(1) Consider the model: I S : xt = Et xt+1 − σ (it − Et πt+1 − r¯t ) P C : πt = βEt πt+1 + kxt MP R : it = r¯t The model is specified by the two equations: 

xt = Et xt+1 + σ Et πt+1 πt = βEt πt+1 + kxt

which can be written in matrix notation:       1 σ Et xt+1 1 0 xt = 0 β Et πt+1 −k 1 πt or     Ex xt = A t t+1 πt Et πt+1 where: −1     1σ 1 0 1 σ A= = 0β −k 1 k σk + β 

The trace and the determinant of the matrix A are given by: trA = 1 + σ k + β |A| = σ k + β − σ k = β

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2

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314

C Difference Equations

Since 1 − trA + |A| = −σ k < 0, 1 + trA + |A| = 1 + σ k + β + β > 0 and |A| < 1, the roots will not be less than one in absolute value. (a) The model is unstable. (b) The solution is not unique. (2) Consider the model: P C : πt = βEt πt+1 + kxt I S : xt = Et xt+1 − σ (it − Et πt+1 − r¯t ) MP R : it = r¯t + πt + φπt + θ xt The model is specified by the two equations: 

xt = Et xt+1 − σ (πt + φπt + θ xt − Et πt+1 ) πt = βEt πt+1 + kxt

or: 

(1 + σ θ ) xt + σ (1 + φ) πt = Et xt+1 + σ Et πt+1 −kxt + πt = βEt πt+1

This system can be written in matrix notation:       1 + σ θ σ (1 + φ) xt 1 σ Et xt+1 = −k 1 πt 0 β Et πt+1 The solution of this system is given by:     xt Et xt+1 =A πt Et πt+1 where the matrix A A=

   −1   1 1 σ − βσ (1 + φ) 1σ 1 + σ θ σ (1 + φ) = 0β −k 1  k kσ + (1 + σ θ ) β

and  = 1 + σ θ + σ k (1 + φ). The trace and the determinant of the matrix A are: trA =

1 + σ k + β (1 + σ θ ) Δ |A| =

β 

C Difference Equations

315

(a) Since 1 + trA + |A| > 0, and |A| < 1, 1 − trA + |A| = σ [θ(1−β)+kφ] > 0,  the model has a unique equilibrium. (b) The condition θ (1 − β) + kφ > 0 does not require that the parameter φ must always be positive. It can be negative, but must satisfy the condition: kφ > −θ (1 − β). (3) Consider the new Keynesian model: P C : πt = βEt πt+1 + kxt I S : xt = Et xt+1 − σ (it − Et πt+1 − r¯t ) MP R : it = r¯t + πt + φEt πt+1 + θ xt After substituting the MP R into the I S cure the model can be reduced to a system of two equations:  (1 + σ θ ) xt + σ πt = Et xt+1 + σ (1 − φ)Et πt+1 −kxt + πt = βEt πt+1 which can be written in matrix form as follows:       1 + σ θ σ xt 1 σ (1 − φ) Et xt+1 = 0 β Et πt+1 −k 1 πt The solution of this system is given by:     xt Et xt+1 =A πt Et πt+1 where A 

1 + σθ σ A= −k 1

−1 

   1 1 1 σ (1 − φ) σ (1 − φ) − σβ = 0 β Δ k kσ (1 − φ) + (1 + σ θ )β

and  = 1 + σ θ + σ k The trace and the determinant of the matrix are: trA = |A| =

1 + kσ (1 − φ) + (1 + σ θ ) β Δ

kσ (1 − φ) + (1 + σ θ ) β − k [σ (1 − φ) − β] β = Δ Δ2

> 0, (a) Since 1 + trA + |A| > 0, and |A| < 1, 1 − trA + |A| = σ [θ(1−β)+kφ]  the model has a unique equilibrium. (b) Therefore, θ (1 − β) + kφ > 0 even if φ is negative such that kφ > −θ (1 − β). (c) Assume that monetary policy rule is given by: it = r¯t + πt + φEt πt+1 + θ Et xt+1

316

C Difference Equations

We use this MP R to eliminate the nominal interest rate in the I S equation to obtain the first equation of the system: 

xt + σ πt = (1 − σ θ ) Et xt+1 + σ (1 − φ) Et πt+1 −kxt + πt = βEt πt+1

which can be written in matrix form as follows:       1 − σ θ σ (1 − φ) Et xt+1 1 σ xt = 0 β Et πt+1 −k 1 πt The solution of this system is given by:     xt Ex = A t t+1 πt Et πt+1 where the matrix A is given by:  A=

1 σ −k 1

    −1  1 1 − σ θ σ (1 − φ) 1 −σ 1 − σ θ σ (1 − φ) = 0 β 0 β 1 + σk k 1

Thus 1 A= 1 + σk



1 − σ θ σ (1 − φ) − σβ k (1 − σ θ ) kσ (1 − φ) + β



The trace and the determinant of A are: trA =

1 − σ θ + kσ (1 − φ) + β 1 + σk |A| =

(1 − σ θ )β 1 + σk

> We assume 1+trA+|A| > 0, and |A| < 1, 1−trA+|A| = σ [θ(1−β)+kφ]  0. Thus, φ can be negative and still we can get: 1 − trA + |A| > 0. (4) (a) The equation xt = Et xt+1 − σ (it − Et πt+1 − r¯t ) has two solutions, one forward looking: xt = −σ Et

∞ +

it+j − Et πt+1+j − r¯t+j i=0

C Difference Equations

317

and another backward looking: xt = −σ

−∞ +

it−j − Et πt+1−j − r¯t−j i=0

(b) The forward solution states that when the elasticity of substitution is equal to one, consumption in t will be affected by present and future interest rates. This is an incorrect statement because consumption in period t does not depend on the interest rate. (5) Consider the new Keynesian Phillips Curve: πt = Et πt+1 + kxt It is important to notice that in this specification β = 1 and the future is not discounted. (a) There are two solutions to this difference equation. One is forward looking: πt = Et

∞ +

(kxt+i )

i=0

and one backward looking: πt =

−∞ +

kxt−i

i=0

(b) There are no criteria to choose the solution. (6) Consider the new Keynesian model: I S : xt = Et xt+1 − σ (it − Et πt+1 − r¯ ) P C : πt − π¯ = βEt (πt+1 − π¯ ) + kxt MP R : it = r¯t + πt + φEt (πt+1 − π¯ ) + θ xt (a) This Phillips curve is vertical in the long run because, when: πt = πt+1 = π¯ the output gap is equal to zero. (b) Eliminating the nominal interest rate of the I S curve with the MP R we obtain the first equation of the following system of difference equations: 

(1 + σ θ ) xt + σ πt = Et xt+1 + σ (1 − φ)Et πt+1 + σ φ π¯ −kxt + πt = βEt πt+1 − β π¯

318

C Difference Equations

This system can be expressed in matrix form as:         1 + σ θ σ xt 1 σ (1 − φ) Et xt+1 σφ = + π¯ 0 β Et πt+1 −k 1 πt −β which can be solved to get:     xt Et xt+1 =A +b πt Et πt+1 where   −1  1 + σθ σ 1 σ (1 − φ) A= 0 β −k 1   1 1 σ (1 − φ) − σβ = 1 + σ θ + kσ k kσ (1 − φ) + (1 + σ θ ) β  b=

1 + σθ σ −k 1

−1 

   1 σφ σ (φ + β) π¯ π¯ = −β 1 + σ θ + kσ θ + kσ kσ φ − β (1 + σ θ )

The trace and the determinant of A are given by: trA =

1 + kσ (1 − φ) + (1 + σ θ ) β 1 + σ θ + kσ |A| =

β 1 + σ θ + kσ

The model has a unique solution when |A| < 1, 1 + trA + |A| > 0, and 1 − trA + |A| = σ [θ(1−β)+kφ] > 0. Thus, the following restriction has to 1+σ θ+kσ satisfied: θ (1 − β) + kφ > 0. This condition does not imply that kφ > 0. Indeed kφ can be negative as long as kφ > −θ (1 − β).

Bibliography

1. Auernheimer, L.: The honest government’s guide to the revenue from the creation of money. J. Polit. Econ. 82, 598–606 (1974) 2. Barbosa, F.: Exploring the Mechanics of Chronic Inflation and Hyperinflation. Springer, Cham, Switzerland (2017) 3. Bailey, M.J.: The welfare costs of inflationary finance. J. Polit. Econ. 64, 93–110 (1956) 4. Barro, R.J.: On the determination of public debt. J. Polit. Econ. 87, 940–971 (1979) 5. Barro, R.J., Gordon, D.B.: Rules, discretion and reputation in a model of monetary policy. J. Monet. Econ. 12, 101–120 (1983b) 6. Blanchard, O.J.: Output, the stock market and interest rates. Am. Econ. Rev. 71, 132–143 (1981) 7. Branson, W.H., Henderson, D.W.: The specification and influence of asset markets. In: Jones, R.W., Kenen, P.B. (eds.) Handbook of International Economics, Vol. II, Chapter 15. North Holland, Amsterdam (1985) 8. Calvo, G.A.: On the time inconsistency of optimal policy in a monetary economy. Econometrica 46, 1411–1428 (1978) 9. Calvo, G.A.: Staggered prices in a utility maximization framework. J. Monet. Econ. 12, 983–998 (1983) 10. Calvo, G.A.: Real exchange rate dynamics with nominal parities, structural change and overshooting. J. Int. Econ. 22, 141–155 (1987) 11. Cass, D.: Optimum growth in an aggregative model of capital accumulation. Rev. Econ. Stud. 32, 233–240 (1965) 12. Corden, W.M.: Booming sector and Dutch disease economics: survey and consolidation. Oxf. Econ. Pap. 36, 359–380 (1984) 13. Diamond, P.: National debt in a neoclassical growth model. Am. Econ. Rev. 55, 1126–50 (1965) 14. Dornbusch, R.: Expectations and exchange rate dynamics. J. Polit. Econ. 84, 1161–76 (1976) 15. Drazen, A.: Tight money and inflation further results. J. Monet. Econ. 15, 113–120 (1984) 16. Frenkel, J.A., Johnson, H.G. (eds.): The Monetary Approach to the Balance of Payments. Allen and Unwin, London (1976) 17. Harberger, A.C.: Currency depreciation, income and the balance of trade. J. Polit. Econ. 58, 47–60 (1950) 18. Hayashi, F.: Tobin’s marginal q and average q: A neoclassical interpretation. Econometrica 50, 213–224 (1982) 19. Heller, D., Lengwiler, Y.: Payment obligations, reserve requirements and the demand for Central Bank balances. J. Monet. Econ. 50, 419–432 (2003)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2

319

320

Bibliography

20. Hotelling, H.: The economics of exhaustible resources. J. Polit. Econ. 39, 137–175 (1931) 21. Ihrig, J.E., Meade, E.E., Weinbach, G.C.: Rewriting monetary policy 101: What’s the Fed’s preferred post crisis approach to raising interest rates? J. Econ. Perspect. 29, 177–198 (2015) 22. Kouri, P.J.K.: The exchange rate and the balance of payments in the short run and in the long run: a monetary approach. Scand. J. Econ. 78, 280–304 (1976) 23. Liviatan, N.: Tight money and inflation. J. Monet. Econ. 13, 5–15 (1984) 24. Lucas, R.E., Jr.: On the mechanics of economic development. J. Monet. Econ. 22, 3–42 (1988) 25. Lucas, R.E., Jr.: Inflation and welfare. Econometrica 68, 247–264 (2000) 26. Mankiw, N.G.: The growth of nations. Brook. Pap. Econ. Act. 1, 275–326 (1995) 27. Mankiw, N.G.: The savers-spenders theory of fiscal policy. Am. Econ. Rev. 90, 120–125 (2000) 28. Mankiw, N.G., Romer, D., Weil, D.N.: A contribution to the empirics of economic growth. Q. J. Econ. 107, 407–37 (1992) 29. Mehra, R., Prescott, E.C.: The equity premium in retrospect. In: Constantinides, G.M., Harris, M., Stultz, R. (eds.) Handbook of the Economics of Finance. North Holland, Amsterdam (2003) 30. McCallum, B.T., Nelson, E.: An optimizing IS-LM specification for monetary policy and business cycle analysis. J. Money Credit Bank. 31, 296–316 (1999) 31. McCallum, B.T., Nelson, E.: Monetary policy for an open economy: An alternative framework with optimizing agents and sticky prices. Oxf. Rev. Econ. Policy 16, 74–91 (2000) 32. McKay, A., Nakamura, E., Steinsson, J.: The discounted Euler equation: A note. NBER, Working Paper #22129 (2016) 33. Metzler, L.A.: Wealth, saving and the rate of interest. J. Polit. Econ. 59, 93–116 (1951) 34. Pearce, I.: The problem of the balance of payments. Int. Econ. Rev. 2, 1–28 (1961) 35. Poole, W.: Optimal choice of monetary policy instrument in a simple stochastic macro model. Q. J. Econ. 84, 197–216 (1970) 36. Romer, P.M.: Increasing returns and long run growth. J. Polit. Econ. 94, 1002–1037 (1986) 37. Salter, W.: Internal and external balance the role of price and expenditure effects. Economic Record 35, 226–238 (1959) 38. Samuelson, P.A.: Interaction between the multiplier and the principle of acceleration. Rev. Econ. Stat. 21, 75–78 (1939) 39. Sargent, T.J., Wallace, N.: The stability of models of money and growth with perfect foresight. Econometrica 41, 1043–1048 (1973) 40. Sargent, T., Wallace, N.: Some unpleasant monetarist arithmetic. Fed. Reserve Bank Minneapolis Q. Rev. 5, 1–17 (1981) 41. Solow, R.M.: Growth Theory an Exposition, 2nd edn. Oxford University Press, New York (2000) 42. Strotoz, R.H.: Myopia and inconsistency in dynamic utility maximization. Rev. Econ. Stud. 23, 165–180 (1956) 43. Tobin, J.: Money and economic growth. Econometrica 33, 671–684 (1965) 44. Tobin, J.: Keynesian models of recession and depression. Am. Econ. Rev. 65, 195–202 (1975) 45. Tobin, J., Brainard, W.C.: Asset markets and the cost of capital. In: Belassa, B., Nelson, R. (eds.) Economic Progress, Private Values and Public Policy: Essays in Honour of William Fellner. North-Holland, Amsterdam (1977) 46. Uzawa, H.: Time preference, the consumption function, and optimum asset holdings. In: Wolfe, J.N. (ed.) Value, Capital and Growth: Papers in Honor of Sir John Hicks. Aldine Publishing Company, Chicago (1968) 47. Weil, P.: The equity premium puzzle and the risk-free rate puzzle. J. Monet. Econ. 24, 401–421 (1989)

Macro Theory: Errata

Chapter 1 (1) Page 31, 4th line from bottom should read: “b(t) = b(T )e−ρ(T −t) +



T

f e−ρ(τ −t) dτ ”

t

(2) Page 32, 2nd line of text should read: “where f is now the real deficit. Show that: b(T ) − b(t) = f T and that” (3) Page 32, 4th line of text should read: “(d) what conclusion can you draw from items (b) and (c).”

Chapter 2 (1) Page 42, 2nd line from bottom should read: “Thus, a variable rate of time preference cannot solve the representative agent model in a small open economy.” (2) Page 58, 2nd line of the text should read: “qt = st + pt∗ − pt . Use this definition to show that:” (3) Page 61, 4th line" from the bottom should read: ∞ “where y p = r 0 e−rt ydt. How would you interpret this result?” Page 61, 3rd line from the bottom should read: “(b) Assume that r = ρ and u(c) =

1

c1− σ 1− σ1

. Show that:”

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2

321

322

Macro Theory: Errata

Chapter 3 (1) Page 83, 9th line from the bottom should read: r = f  (k) − δ (2) Page 85, 2nd 1line from the

bottom should read: “u c1,t + 1+ρ u c2,t+1 , u > 0, u < 0” (3) Page 86, 8th line from the bottom should read: “wt = f (kt ) − kt f  (kt )” Page 86, 2nd line from the bottom should read: “kt+1 =

(1−δ)kt +St (wt ,rt+1 ) ” 1+n

Chapter 4 (1) Page 100, 7th line from the bottom should have a space between line 7 and  (n+g+δ)f  (k ∗ )k ∗ − + g + δ) − line 8: “k˙ = (n (k k ∗ ) placing (n + g + δ) in f (k ∗ ) evidence yields” (2) Page 115, 2nd line from bottom should read: “It measures the workforce’s human capital. Thus, the production function is given by:” (3) Page 115, Table 4.1 the 5th and the 6th line of this table should read: “ K/Y Lˆ

2.5 1.5%

(4) Page 117,

7th line of the text should read: “log YL = log A0 + gt + 1−ααkk−αh log Sk + 1−ααkh−αh log Sh − log (n + g + δk ) − 1−ααkh−αh log (n + g + δh )” (5) Page 117, 11th and 12th line of the text should read: k˙ = sk f (k, h) − (n + g + δk ) k h˙ = sh f (k, h) − (n + g + δh ) h (6) Page 118, 8th line of the text should read: M˙ = constant” “Monetary policy: m ˙ = m (μ − π ) , μ = M

αk 1−αk −αh

Macro Theory: Errata

323

Chapter 5 (1) Page 132, 16th line of the text should read: “a public good.” (2) Page 133, 13th line of the text should read: “Y = K α (φ (K/L) L)1−α = φ 1−α K (3) Page 138, 6th line of the text should read: “For the consumption-capital ratio not to be negative, the following inequality must be satisfied:” (4) Page 140, equation (5.50) should read: ⎡ BI = φ −1 γ 1−τ ⎣



M

p(z)1− dz

1  1−

⎤τ ⎦

(5.50)

0

(5) Page 150, 7th line of the text should read: “(c) What would be the answer to the previous item be if the parameter φ were equal a zero?”

Chapter 6 (1) Page 169, table Central Bank Balance Sheet, item (b) liabilities, should read: “Reserves (R)” (2) Page 175, Fig. 6.6. The reserve market: Fig. 6.6 The reserve market

(3) Page 177, 1st line from the bottom should read: dy P dy P “ε = dP y ; |ε| = − dP y ” (4) Page 179, equations (6.63) and (6.64) should read: “

324

Macro Theory: Errata

π=

W˙ P˙ = P W

(6.63)

W˙ = π e − a (u − u) ¯ W

(6.64)

(5) Page 190, 13th line of the text should read: u (c) c˙ = −  (r − ρ) c cu (c) (6) Page 193, 1st line of the text should read: E (pt+1 /It ) − pt + dt = rt pt (7) Page 195, 2nd line of exercise (22) should read: πt = βπt+1 + δxt (8) Page 195, the last line of exercise (22) should read: −δ “Where ρ = 1−β β and k = β ”

Chapter 7 (1) Page 197, equation (7.2) should read: “PC:π˙ = δx” (2) Page 198, 1st line of the text should read: “IC: given p(0) and π(0)” (3) Page 201, equation (7.7) should read: “IC:i = r¯ BC + π + φ (π − π¯ ) + θ x” (4) Page 214, Fig. 7.19 a) and b) should read the figure below.

Fig. 7.19 (a) Keynesian

Fig. 7.19 (b) New Keynesian

Macro Theory: Errata

325

(5) Page 223, equation (7.31) should read: M “MPR:μ = d log = constant” dt (6) Page 229, 12th line of the text should read: “π˙ = −k + β log m + γ π ” (7) Page 230, Fig. 733 (8) Page 231, 10th line from the bottom: “real if  > 0 and complex if  < 0. If the roots are complex and the trace is equal to” (9) Page 235, 5th line of the text should read: “IC: Given p(0)” (10) The symbols have the following meanings: u = unemployment rate;

Chapter 8 (1) Page 241, 4th line from the bottom should read:

P =

PT1−w PN1−w

= PT

PN PT

ω (8.12)

(2) Page 251, 9th line of the text should read: “Thus, the deviation of the log of terms of” (3) Page 254, 13th line of the text should read: “At the long run equilibrium of an open economy with perfect capital” (4) Page 255, text of Fig. 8.4 should read: “The effects of an increase in the public deficit on the current account: the twin deficits” (5) Page 263, 3rd line from the bottom should read: “(b) Show that saving (s = y − d − τ where, τ represents taxes) varies with the terms of” (6) Page 264, 20th line of the text should read: “(b) show what happens to E and F in each of the following circumstances: (i) an”

Chapter 9 (1) Page 270, 5th line from the bottom of the text should read: “Figure 9.4 describes an experiment in which the foreign inflation rate rises to π1∗ from π0∗ .” (2) Page 275, 15th line of the text should read: “the components of the characteristic vector determined by the solutions of the linear system:”

326

(3) Page 278, 2nd line from the bottom should read: “given by [equation (3.22), p.77]:” (4) Page 280, 2nd line of the text should read: ¯ “(r − n) a¯ + Ey¯x = 0” (5) Page 280, 11th line of the text should read: ¯ αs = ε” “αa = ρ¯ − n, αr = α, (6) Page 301, exercise 4, this equation should be read: “UIP: r = r¯ + s˙ , r¯ = r ∗ ” (7) Page 302, exercise 6, third line of the text should read: “=inflation rate,π˙ = dπ ¯ c =inflation rate target; q˙ = dt ; π

Macro Theory: Errata

dq dt ; inominal”.

Chapter 10 (1) Page 308, equation (10.2) should read: “M˙ = B˙ BC ” (2) Page 328, 2nd line from the bottom should read: “For a Ponzi game not to occur, the following limit must be satisfied:” (3) Page 333, exercise 2, the monetary policy rule should read: “Monetary policy rule: m ˙ = m (μ − π ) , μ constant” (4) Page 334, exercise 7, 2nd of the bottom should read: dt−1 “dt = 1+r , t = 1, 2, · · · ” t−1 (5) Page 335, 1st line of exercise 8 should read: “8. The government finances public deficit by printing money according to:” (6) Page 336, 2nd line of exercise 14 should read: “lim m(T )e−r(T −t) = ert ” (7) Page 337," exercise 18, equation of the item (a) should read: ∞ “a(t) = t [fs + s(m)] e−(r−n)(υ−t) dυ” (8) Page 338, 7th line of the text should read: “economy’s nominal output Y = P y, P is the price level and y is real output.” (9) Page 338, 11th line of the text should read: ˙ “where π = PP , s˙ = ss˙ , n = yy˙ ”

Chapter 11 (1) Page 341, 9th line from the bottom should read: “both on the legal framework that establishes the legal condition for the” (2) Page 364, 4th line of the test should read: “interest rates may be easily deduced from the above expression, when iL > iS , i˙L > 0.”

Macro Theory: Errata

327

(3) Page 367, exercise 1, 3rd equation should read: “MPR i = λ (i ∗ − i) , λ > 0, i ∗ = r¯ + π + φ (π − π¯ )” (4) Page 369, exercise 8, 3rd equation should read: “TSIR: i˙S = β (r − rS ) , β > 0, rS = iS − π ”

Appendix A (1) Page 374, 9th line of the text should be read: “root is positive and the other is negative r1 > 0 > r2 ”. (2) Page 374, 2nd line from the bottom should read: “Let r1 = α + βi and r2 = α − βi be the two complex roots where i 2 = −1”. (3) Page 389, exercise 8, first equation should read: “P˙ = β

 M − L (y, r) , β > 0 P

(4) Page 391, exercise 391, 4th line should read: “perfect foresight: π e = π ” (5) Page 392, 4th line should read: “H˙ = S(P ) − δH,

Appendix B (1) Page 405, Fig. B.8 caption: “An unanticipated transitory increase in parameter α” (2) Page 409, 2nd line should read: “L =

1 ϕ  ¯ 2 + (y − y) ¯ 2 (π − π) 2 2

(3) Page 411, 1st line should read:  “ max

∞

e−ρt μmdt 

0

(4) Page 411, exercise 8, 2nd equation should read: 

∞

“ 0

e−ρt μmdt =



∞ 0

e−ρt τ (m)dt +



∞ 0

e−ρt (m − m(0)) dt 

328

Macro Theory: Errata

(5) Page 412, 11th and 12th lines should read: “(b) Show . . . . (c) Suppose . . . .” (6) Page 413, 6th line of the text should read: 

∞

“

e−ρt [u (f (x)) + υ(m)] dt 

0

(7) Page 415, 5th line from the bottom should read: “0 ≤ q ≤ S” (8) Page 415, 1st line from the bottom should read:  x  “f (x) = αx 1 − S

Appendix C (1) Page 418, 9th line of the text should read: “solution of Eq. (C.1) is backward looking. When |α| = 1, Eq. (C.1) has two solutions,” (2) Page 419, 8th line from the bottom should read: “|λ2 | > 1” (3) Page 422, equation (C.22) of the text should read: “IS:xt = Et xt+1 − σ (rt − r¯t )” (4) Page 424, equation (C.32) of the text should read:   1 (1 + αθ ) β + σ k k “A = 1 + σ θ + σ (1 + φ) k −σ (1 + φ) β + σ 1 (5) Page 427, equation (C.41) of the text should read: “yt = af Et yt+1 + ab Et yt−1 + bxt , yt−1 given, y0 free (C.41)” (6) Page 435, exercise 2, 3rd equation of the text should read: “MPR:it = r¯t + πt + φπt + θ xt ” (7) Page 435, exercise 3, the 1st and 3rd equation of the text should read: “PC:πt = βEt πt+1 + kxt MPR:it = r¯t + πt + φEt πt+1 + θ xt ” (8) Page 435, exercise 6, the 3rd equation of the text should read: “MPR:it = r¯t + πt + φEt (πt+1 − π¯ ) + θ xt ”

Index

A Absolute risk premium utility function, 101 Adaptive expectations, 122, 123, 257, 297 Aggregate demand curve, 208, 261 Aggregate supply curve, 108 AK Model Share of Labor in Output, 87, 89 Asset demands, 152 Asset pricing model/present value model, vii, 242, 245 Average value of capital, 293, 294

Consumption habit formation, vii Covered interest parity (CIP), 148 Credit market, 135 Crowding Out, 225 Current value Hamiltonian, 5, 50, 91, 94, 227, 295, 298, 300, 303, 305, 307, 308, 310

B Backward and forward IS curve, 143 Backward and forward Philips curve (PC), 317 Backward IS curve, 142, 143 Balanced budget multiplier, 112 Bank reserves market, 234, 235 Bubbles, vii, 8, 10, 113, 114, 147, 159, 219, 253, 261

E Endogenous growth, vi, vii, 73, 81–96 Endogenous rate of time preference, 23, 38, 142, 321 Equity premium puzzle, 242, 245 Euler equation, v, vi, vii, 102–104, 142, 144, 186, 242–244, 278, 280, 281, 288 Exchange rate determination, 147 Exchange rate overshooting, 155, 166, 167 Exogenous growth model, 76, 77

C Cagan money demand equation, 107 Calculus of variations, 277, 280, 281 Calvo Philips curve, vii, 121 Canonical new Keynesian model, 142, 143 Cash in advance constraint (CIA), vii, 11, 12 Central bank operational procedure, 131 Central bank’s loss function, 114, 232, 233 Cobb-Douglas production function, 71–74, 79, 81 Conservative central bank, 232, 233 Consumption asset pricing model, vii, 242, 245

D Dynamic inefficiency, 67, 68

F FED operational procedure, 246 Finite-life OLG model, 66 Fiscal policy rule, 21, 129, 201, 254, 259, 261, 262 Fiscal rule, 129, 201, 210, 254, 259, 261, 262 Fisher equation, 79, 216, 229 Fixed exchange rate, 145 Fixed exchange rate MFD model, 145 Flexible exchange rate MFD model, 146

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2

329

330 Foreign interest rate premium, 41, 51, 152, 171, 186 Forward exchange rate market, 148 Forward guidance puzzle, 142 Forward IS curve, 143 Fully funded social security system, vii, 69 Fundamental and bubble solutions of finite difference equations, 113

G Golden-rule, 59, 61, 67

H Hamiltonian, 3–5, 11, 12, 14, 22, 37, 39, 47, 50, 52, 83, 86, 88, 91, 94, 227, 277, 278, 282, 284, 289, 292, 295, 298, 300, 303, 305, 307, 308, 310 Harberger-Laursen-Meltzer (HLM) effect, vii, 148, 152 Harrod-Domar model, 252 Housing market model, 268, 269 Human capital with externality, vii, 91, 93 Hyperdeflation bubbles, 10 Hyperinflation, vi, 8, 10, 210 Hyperinflation bubble, 8, 10

I Increasing Returns with Endogenous Growth, 93–96 Inflation target, 110, 125, 126, 128–130, 133, 137, 138, 141, 142, 174, 177, 178, 180, 230, 231, 238–240 Inflation target change, 125, 126, 175, 180 Inflation target monetary policy rule, 133, 231 Inflation tax, 205, 206, 212, 223, 295, 296 Intertemporal approach to the balance of payments, vii, 52 IS curve, v, vi, 99, 100, 103, 112, 125, 127, 129, 131, 132, 136, 142, 143, 169, 172, 173, 176, 178, 180, 183, 185–189, 229, 231, 238, 257, 317 IS curve in an open economy, 186, 189 IS curve with positive slope, 99

J Jensen’s inequality, 244

Index K Keynesian model, v, vi, vii, 99–124, 133, 141–143, 175, 193, 194, 315, 317 Keynesian Phillips curve, 123, 124, 317

L Leibinitz rule, 120, 122, 220 Linear optimal control, 277–311 Liquidity trap, 111 LM curves, 99–124, 134, 206, 257 Long run money neutrality, 142 Lucas model with leisure, 85

M Marginal value of capital, 291, 293 Marshall-Lerner condition, 155 Monetary approach to the balance of payments, vii, 145, 146 Monetary policy rule (MPR), vi, 15, 100, 125, 127–129, 131–136, 138–142, 158, 168, 171, 173–176, 178, 180, 184–186, 202, 207, 208, 229, 231, 232, 237, 238, 240, 241, 313–317, 322, 325, 327, 328 Monetary regime, 207 Money and the natural rate of interest, 100, 103, 131, 132, 135, 138, 186, 188, 189, 242, 284 Money financed public deficit, 201, 203, 216 Money neutrality, 13, 142 Money superneutrality, vii Multiple equilibria, 141 Mundell-Fleming-Dornbusch (MFD) model, 161

N Natural exchange rate, 190, 192, 193 Natural rate of interest in a small open economy, vii, 157, 190, 192 Negative natural rate of interest puzzle, 242 Net domestic credit, 146 New Keynesian model, v–vii, 141–143, 315, 317 New Keynesian Phillips curve, 123, 124, 317

O Okun’s Law, 132 Old Keynesian IS curve, 143 OLG models, 59, 61, 83

Index Open economy representative agent model, v, vi, 29–54 Optimal control, vi, 277–311 Optimal monetary policy, 299 Overlapping generations diamond model (OLG), 55–70

P Pay-as-you-go (PAYG) social security system, 69 Perfect foresight, 99, 123, 228, 261, 327 Permanent anticipated change, 262, 264, 266, 269 Permanent government expenditures, 227, 228 Permanent income, 52, 54 Permanent unanticipated change, 262, 264, 265 Perpetuity, 236 Perverse monetarist arithmetic, 219 Phillips curve (PC), vi, vii, 99–126, 129, 131, 132, 134–136, 138–142, 168, 171, 173, 175, 178, 180, 182, 184, 185, 190, 193, 207–209, 229, 231, 233, 237, 238, 240, 241, 257, 261, 313–315, 317, 324, 328 Phillips curve in an open economy, vi, 207 Portfolio balance model, 152 Present value Hamiltonian, 5, 50, 91, 94, 227, 295, 298, 300, 303, 305, 307, 308, 310 Price index, 29–32, 134, 148, 150, 163, 175, 178, 185, 190, 191, 212, 219, 223 Price Level Indeterminacy, 158, 162, 163 Primary deficit, 19 Private and social labor productivity, 91

R Ramsey/Cass/Koopmans model, vii, 93 Rational expectations, 108, 109 RCK Model, 81 Real business cycle (RBC), 19, 23–25 Real cash balance effect, 100, 109, 158 Real deficit, 18, 19, 321 Real interest rate and the real exchange rate, 175, 180 Representative agent model, v, vi, 3–27, 29–54 Ricardian equivalence, vi, 100, 112, 113, 214 Risk-free rate puzzle, 242, 245

331 S Short Run Money Neutrality, 13 Singular optimal control, 303 Social security systems, vii, 69, 70 Solow model with CES production function, vii, 71 Solow model with Cobb-Douglas production function, 71–74, 79, 81 Solow model with endogenous growth, vii, 73 Solow model with money, 78 Sustainability of foreign debt, 213

T Tax smoothing, vii, 226 Taylor rule, 99–124, 129, 158, 159, 190, 193 Taylor rule in an open economy, 158 Terms of trade, 149, 151, 152, 155, 168–171, 173, 175, 186 Term structure of interest rates (TSIR), vi, 237, 241, 327 The “Discounted Euler Equation”, vi, vii, 142, 144 Time inconsistency, vii, 301, 302 Time Inconsistent Monetary Policy, 223, 301 Tobin’s q, vii, 289, 291, 293, 294 Tobin’s q with installation costs, vii Tradable and nontradable goods, vii, 189–193 Transaction cost model, 107 Transitory anticipate change, 262, 264, 267, 270, 271 Transitory unanticipated change, 262–266, 269, 271, 272, 274, 275, 285

U UIP risk premium, 148 Uncovered interest parity (UIP), 147, 148, 157, 159, 168, 169, 171–173, 175, 176, 178, 180, 181, 183–185, 189, 326

V Variable discount rate, 38, 121 Velocity of money, 109–111