a model of macro economic activity - fair (ballinger 1974) - copy

8/14/2019 A Model of Macro Economic Activity - Fair (Ballinger 1974) - Copy

1/241

A Model ofMacroeconomicActivityVolume I:The Theoretical Model


2/241


3/241

A Model ofMacroeconomicActivity

IVolume I:The Theoretical Model

Ray C. Fair

Ballinger Publishing Company l Cambridge, Mass.A Subsidiary of J. B. Lippincbrr Company


4/241

Copyright 0 19 74 by Ballinger Publishing Com pany. All rights reserved. Nopart of this publication may be reproduced, stored in a retreival system, ortransmitted in any form or by any means, electronic, mechan ical, photocopy,recording or otherwise, without the prior written consat of the publisher.International Standard Book Num ber: O-8841 0-268-8Library of Congress Catalog Card Num ber: 74-12 199Printed in the United States of Am erica

Library of Congress Cataloging in Publication Da taFair, Ray C

A model of macroeconom ic activity.Bibliography: Y. 1, p.Contents: Y. 1. The theoretical model.1. Macroecon omics-M athematical models. I. Title.

HB171.FZ4 339ISBN O-88410-268-8

74-12199


5/241

Table of Contents

List of Tables ix

Preface

Chapter One Introduction 11.1 The Purpose of the Study 11.2 An Outline of the Mod el 31.3 The Methodology o f the Study 121.4 Suggestions to the Reader 16

Chapter Two Banks 192.12.22.32.42.5

The Basic EquationsThe Formation of ExpectationsBehavioral AssumptionsThe Solution of the Control ProblemSom e Examples of Solving the Control Problemof Bank i

2.6 The Condensed Model for BanksChapter Three Firms

3.1 The Basic Equations

xiii

19222s2929353939


6/241

vi A Model of Macroeconomic Activity Volume I: The Theoretical Model

3.23.33.43.5

The Formation of ExpectationsBehavioral AssumptionsThe Solution of the Control ProblemSom e Examples of Solving the Control Problem

of Finn i

454954

3.6 The Condensed Mode l for Firms5664

Chapter Four Households 7s4.1 The Basic Equations4.2 The Formation of Expectations4.3 Behavioral Assumptions4.4 Ihe Solution of the Control Problem4.5 Som e Examples of Solving the Control Problems

75787981

of the Households4.6 The Condensed Mod el for Households

8288

Chapter Five The Government and the Bond Dealer 975.15.25.3

The Government 97The Bond Dealer 98The Condensed Mod el for the Government and

the Bond Dealer 101Chapter Six The Dynamic Properties of the Model 103

6.16.2

The Com plete Set of Equations for the Modelllx Response of the Mode l to Shocks from a

Position of EquilibriumThe Effects of Policy Changes from a

Disequilibrium PositionThe Long-Run Properties of the Mod elPrice and Wage ResponsesThe Relationship Between Dem and Deposits

and Aggregate Output

103116

6.3

6.46.56.6

147149152153

Chapter Seven A Static-Equilibrium Version of the Model 1577.17.27.37.4

IntroductionThe Static-Equilibrium VersionThe Solution of the Static M odelA Comparison of the Static M odel to the

Textbook Model

157158168177


7/241

Chapter Eight Conclusion

8.1 Summary8.2 Possible Extensions o f t h e M o d e l8 .3 E m p i r i ca l I m p l ica t i on s o f t h e M o d e l8 .4 C o n cl u d i n g R e m a r k s

Appendix The Non-Condensed Version of the Model 19 5

References

Index 22 3About the Author 22 5

18 118 118 61 901 94

21 9


8/241


9/241

2-1 Notation for Banks in Alphabetic Orde r2-2 Para meter Values and Initial Conditions for the

2-32-43-l3-2

3-33-4

4-l4-2

4-3

List of Tables

Control Problem of Bank iResults of Solving the Control Problem of Bank iBank Equations for the Condensed Mod elNotation for Firms in Alphabe tic OrderPara meter Values and Initial Conditions for the

Control Problem of Firm iResults of Solving the Control Problem of Firm iFirm Equations for the Condensed Mod el

Notation for Households in Alphabetic Orde rPara meter Values and Initial Conditions for the

Control Problems of Households 1 and 2Results of Solving the Control Problem of

Household 1

20

30323640

575865

76

83

84i x


10/241

x A Model of Macroeconomic Activity Volume I: The Theoretical Model

4-4 Results o f Solving the Control Problem ofHousehold 2 85

4-5 Results of Solving the Control Problem s ofHouseholds 1 and 2 Based on a Cobb-DouglasUtility Function 87

4-6 Household Equations for the Condensed Mod el 885-l Notation for the Government and the Bond Dealer

in Alphabetic Order 98

5-2 The Government and Bond Dealer Equations for theCondensed Model 1026-1 The Com plete Notation for the Condensed Model

in Alphabetic Order 1046-2 The Com plete Set of Equations for the Condensed

Model 1066-3 Flow-of-Funds Accounts for the Condensed Model:

Stocks of Assets and Liabilities6-4 National Income A ccounts for the Condensed Mod el

114116

6-5 Parameter V alues, Initial Conditions, and GovernmentValues for the Base Run in Table 6-6

6-67-1

7-2

7-3

74

l-5

Results of Solving the Condensed Mode l

119

120Notation for the Static-Equilibrium Model in

Alphabetic OrderThe Equations of the Static-Equilibrium ModelEquations of the Static-Equilibrium Model by Blocks

159160166

Param eter Values, Government Values, and Values ofLF, LH, and SD for the Base Run in Table 7-5 169

Results of Solving the Static-Equilibrium Model forthe Endogenous VBG Case 171


11/241

Tables xi

7-6

7-7

A-l

A-2

A-3

A-4

A-5

A-6

Results of Solving the Static-Equilibrium Mo del forthe Endogenous dj Case

The Equations of the Textbook Mod el

The Com plete Notation for the Non-CondensedMode l in Alphabetic Orde r

The Com plete Set of Equations for the Non-Condensed Model

Flow-of-Funds Accounts for the Non-CondensedModel: Stocks of Assets and LiabilitiesNational Income Accounts for the Non-Condensed

ModelParame ter V alues, Initial Conditions, and Government

Values for the Base Run in Table A-6Results of Solving the Non-Condensed Mod el

175178

195

199

206

207

210212


12/241


13/241

Preface

Ihe work in this volume grew out of both my dissatisfaction with the standardstatic-equilibrium model that is found in most macroeconom ic textboo ks andmy interest in the problem of basing macroeconomic theory on more solidmicroeconomic foundations. I wa s also interested in trying to incorporate into ageneral model o f macroeconomic activity the recent work in economic theorythat has been done on relaxing the assumptions of perfe ct information and theexistence of tatonnement processes that clear markets every period.

It soon becam e apparen t as I began working on this project that themodel that I had in mind wou ld not be capable of being analyzed by standardanalytic methods. I wanted to develop a macroeconomic model that was general,was based on solid microeconomic foundations, and wa s not based on theassumptions of perfect information and the existence of t2tonnement processes .I also wanted the model to account for wealth effects, capital gains effects, andall flow-of-funds constraints. Because of the likely complexity of any model ofthis sort, I decide d at an early stage of the project to use com puter simulationtechniques to help analyze the properties of the model. The methodology that Ifollowed is described in section 1.3.

One o f the main dangers in building a model that is only feasible toanalyze using com puter simulation techniques is that the model becom es toodetailed or complex for anyone other than the model builder to want to spendthe time that it takes to understand the model. I clearly face this danger in thepresent case. However, I have tried to write this volume to make the model asintelligible as possible in as simple a way as possible. First, I have constructed acondensed version of the basic model, with the aim of making the modeleasier to understand. Second, I have constructed a static-equilibrium version

xiii


14/241

xiv Prefaceof the model, with the ho pe that this will put the basic model in a betterperspective. Third, I have organized the discussion so that the different sectorsare each discussed individually before the overall model is put togeth er. Thediscussion of each sector is fairly self-contained, so that the reade r canconcentrate at first on the prope rties of each sector without having tocomprehend the complete model. (I have, howev er, given a brief outline of theoverall m odel in Chapter One.) Finally, I have relied heavily on the use of tablesto present the model and have tried to make the tables fairly self-contained fromthe discussion in the text. One should be able to get a good picture of the overallmodel from a careful reading of the tables. The tables should also be useful forreference purposes.

There are, as discussed in Chapter Eight, many w ays in which thepresent model might be extended. In many cases these extensions were notcarried out here because of the desire not to increase the complexity of themodel anymore than already existed. In future w ork, if the model doe s not turnout to be too unwieldy to com prehend, it wou ld be of interest to carry outmany of the extensions.

Thii volume is one of two . In Volum e II an empirical model w ill bedeveloped that is based on the theoretical model found in this volume. Becausethere is no unique way to specify an empirical version o f the theoretical model,it seemed best to present the theoretical and empirical models in two separatevolumes. The present volume can be read without reference to Volume II.

Ne ither volume has been w ritten specifically as a textboo k. It isp&sible, howev er, that either or both volumes could be used as texts in agraduate level macroeconomics course. B ecause of my unhappiness with thestandard textbook model, I have used for the past two years parts o f the presentvolume in a graduate level macroeconomics course that I have taught atPrinceton.

I would like to thank a number of people for their helpful commentson an earlier draft of this volume. These include Alan S. Blinder, Greg ory C.Cho w, Robert W. Glower, Kenneth D. Garbade, Herschel I. Grossman, EdwinKuh, and Michael Rothschild. I am also grateful to the National ScienceFoundation for financial supp ort.

Ray C. FairMay 1974


15/241

A Model ofMacroeconomicActivityVolumel:The Theoretical Model


16/241


17/241

I hapter OneIntroduction

1.1 THE PURPOSE OF THE STUDY

Mu ch of the work in econom ic theory in the past few years has been concernedwith relaxing two important assumptions of classical economic theory: perfectinformation and the existence of t&nmem ent proce sses to clear ma rkets. Onegroup of studies has followed from the work o f Patinkin [43, Chapter 131 andGlow er [IO] .a Som e of the studies in this group have been concerned with thequestion of wh ether standard, textbook Keynesian theory is different from wh atKeynes 130 1 actually had in mind. Glower [lo] and particularly Leijonhujwd[32] have argued that it is, wh ereas Grossman [25] ha s argued that it is not.Although the question of wha t Keynes meant is primarily of historical interest,the studies of Glow er and others ha ve ma de important advances in ma croeco-nomic theory. By relaxing th e assumption that m arkets are always inequilibrium, these studies hav e provided a mcne solid theoretical basis for theexistence of the Keynesian consumption function and for the existence ofunemployment. The existence of exce ss supply in the labor mark et is ajustification for including income as an explan atory variable in the consum ptionfunction, and the existence of exce ss supply in the comm odity mark et is ajustification for the existence of unemployment.

Another group of studies concerned with relaxing the assumptionof perfect information has followed from the wo rk of Stigler [52] .b The mostprominent studies in thts group are the studies in Phelps et al. [44]. Many of thestudies in this group have been con cerned with the mechanism by which pricesor wag es are determined.c In most cases prices or wa ges are postulated as beingset by firms, as opp osed to, say, by custom ers 01 wo rkers. The price- orwage-setting activities of firms are usually assumed to be guided by profit-maximizing considerations. In particular, Phelps has emp hasized with respect to


18/241

2 A Model of Macroeconomic Activity Volume I: The Theoretical Model

the studies in Phelps et al. [44] tha t . [the theory] sticks dogged ly to theneoclassical postulates of lifetime expe cted utility maximization and net worthmaximization. .[45, p. 31.

Although important progress has been made in relaxing theassumptions of perfect information and t&mnem ent process es, no generaltheoretical mod el has been develope d with these assumptions relaxed. In thedisequilibrium mod el of Barre and Grossman [S] , for exam ple, only output andemploym ent are determined. All other variables, including prices and wag es, aretaken as given. There are no financial and investment sectors in the model. In thefurther study of Grossman [Z6], only investment is determined, and no attemp tis mad e to integrate the investment mode l with the earlier output andemployment model.

In the Solow and Stiglitz model [Sl] , output, employm ent, prices,and wa ges are determined, but there are no financial and investment sectors.Also, as Barre and Grossman point out,* the Solow and Stiglitz m odel is notconstructed on a choice-theoretic basis. Likew ise, the Korliras mode l [31],which is similar to the W ow and Stiglitz m odel but doe2 include financial andinvestment sectors, is not constructed on a choice-theoretic basis. The mode l ofTucker [55] is concerned with short mn fluctuations in output and employ-ment, and prices and wage s are taken as given. In the group of studies concernedwith price-setting behavior,e the price- or wage-setting activities of firms havealso not been considered within the context of a general theoretical mod el. Inthe Maccini m odel [36], for example, which is one of the more general modelsin this group, only prices, output, and inventories are determined. There are noemploym ent, investment, and financial sectors in the mod el.

The studies cited above, with the possible exception of the studyof Korliras 1311 , could be characterize d as partial equilibrium studies if theywere equilibrium studies, but given that the studies are concerned withdisequilibrium phenom ena, they can perha ps best be characterize d as partialdisequilibrium studies. The partial nature of these studies is particularlyrestrictive in a disequilibrium context because of the possible effects thatdisequilibrium in one mark et may have on other m arkets. For exam ple, mod elsin which there is no financial sector rule out any effects th at disequilibrium infinancial mark ets may have on labor and good s markets. The Korltias mod el,while being m ore general in certain respects than the other mode ls, isparticularly restrictive with respec t to the effects of one ma rket on another. Themod el rules out any cross-ma rket effects of disequilibrium and concentrates onlyon within-market disequilibrium effects. Tuckers discussion [56] of Korlllassmod el emphasizes this point.

In addition to the partial nature of the studies c ited above, it is alsothe case tha t the price-setting behavior postulated by the second group ofstudies, in particular that firms set prices and/or wag es to maxim ize profits, hasnot been integrated into the first group of studies. Only in the mod els of Solow


19/241

Introduction 3

and Stiglitz and Korliras are prices and wag es determined, and these models arenot choice-theoretic. The treatment of prices and wag es as exogenous or in an adhoc manner is again particularly restrictive in a disequilibrium context becausedisequilibrium questions are inherently concerned with the problem that pricessom ehow do not get set in such a way as always to clear markets. It is thusparticularly important in a disequilibrium context to determine how prices areset and wh y it is that prices m ay not always clear markets.

The purpose of this study is to develop a theoretical model ofmacroeconomic activity with the following characteristics.

The model should be general enough to incorporate most of the variables ofinterest in a macroeconom ic context.The model should be based on solid microeconomic foundations in thesense that the decisions of the main behavioral units in the model shou ld bederived from the assumption of maximizing behavior.The behavioral units in the model should not be assum ed to have perfectforesight, but instead should be assumed to have to make decisions on thebasis of expectations that may not always turn out to be corre ct.T&xm ement processes that clear markets every period should not bepostulated.

Regarding point 1, the endogenous variables in the present modelinclude sales, production, employm ent, investment, prices, wa ges, interest rates,and loans. The model also accounts for wealth effec ts, capital gains effec ts, allflow-of-funds constraints, and the governmen t budget constraint. The generalnature of the model allows cross-market disequilibrium effec ts to be analyzed,allows one to consider why prices, wages, and interest rates may not always beset in such a way that clears markets every period, and allows the effects ofvarious aggrega te constraints, like the Row-of-funds constraints, to be analyzed.

The rest of this chapter provides an outline of the model anddiscusses various methodolog ical and compu tational issues. The individualbehavioral units are discussed in detail in Cha pters Tw o through Five. Thedynamic properties of the overall model are discussed in Chap ter Six. Astatic-equilibrium version of the dynamic model is presen ted in Cha pter Seven,and this version is com pared to the standard static-equilibrium model found inmost mactoeconomic textbooks. Chapter Eight contains a brief summary of themodel and its properties, a discussion of how the model might be changed orextended, and a discussion of som e of the empirical implications of the model.1.2 AN OUTLINE OF THE MODEL

There are five basic behavioral units in the model: banks, firms, households, a


20/241


bond dealer, and the government. Banks are meant here to include all financialintermediaries, not just comm ercial banks. A t the beginning of each period ea chbank, firm, and household, knowing last periods values, receiving in som e casesinformation from others regarding certain current-period values, and formingexpectations of future values, solves an optimal control problemThe objective function of banks and firms is the present discountedvalue of expe cted future after-tax profits, and the objective function ofhouseho lds is the present d iscounted value of expe cted future utility. The factthat the decisions of the main behav ioral units a re derived by solving optimalcontrol problems places the mode l an a respectable microeconom ic foundation,thus meeting the requirement of point 2 above. Point 3 is also met in the sensethat the decisions are based on expectations of future values, rather than on theactual future values. None of the behavioral units in the mod el has perfectforesight.

The mo del is recursive in the sense that information flows in onedirection from the bond dealer, to banks, to firms, to househo lds. Banks, forexam ple, are not given an opportunity to change their decisions for the currentperiod once firms and househo lds have mad e theirs. A fter all decisions have beenma de at the beginning of the period, transactions take place throughou t the restof the period. The recursive nature of the mode l m eets the requirement of point4 above in the tense that recontracting is not allowed . Banks, for exam ple, onlyfind out wha t th e decisions of firms and househo lds are in the cuuent period bythe transactions that take place during the period. Likew ise, firms only find outwh at the decisions of househo lds are by the transactions that take place.

There is one good in the economy, which can be used either forconsumption or investment purposes. There are no consumer durables: all good sthat are used for consumption purpose s are consumed in the current period. Alllabor is hom ogenous. Bank loans are one-period loans, government bills areone-period securities, and government bonds are cons& There is no currency inthe system. The decision variables of the government are the various tax rates inthe system, the xserve requirement ratio, the number of good s to purcha se, thenumber of wo rker hours to pay for, the value of bills to issue, and the number ofbonds to have outstanding. The government is subject to the constraint eachperiod that expenditures less revenues must equ al the change in the value of billsplus bonds plus bank reserves (high pow ered money).f The governmentsdecisions are treated as exogenou s in the model.

Banks receive m oney from households in the form of savingsdeposits, on which interest is paid, and from households, firms, and the bonddealer in the form of deman d deposits, on which no interest is paid. Banks lendmoney td househo lds and firms and. buy government bills and bonds. Banks areassume d not to com pete for savings deposits, and the rate paid on all savingsdeposits is assum ed to be the bill rate. Banks hold reserves in the form ofdeposits with the government. Banks do not hire labor and do not buy goods.


21/241

In troduc d on 5

At the beginning of the period, banks receive information from thegovernment on the tax rates and the reserve requirement ratio for the currentperiod and from the bond dealer on the bill and bond rates for the currentperiod. How ever, at this time banks do not know the values of their demand andsavings deposits for the current period, and do not know the demand schedulesfor their loans. Banks must form expectations of these variables for the currentperiod, as well as for the future periods, when making their decisions at thebeginning of the period.

The three main decision variables of each bank are its loan rate, thevalue of bills and bonds to buy, and the maximum amount of money that it tilllend in the period. Once a bank makes its decision on the value of bills andbonds to buy, the bank is assum ed to have to buy this amount in the period. Abank needs to set a maximum on the amount of money that it will lend in theperiod in orde r to prepa re for the possibility that it either overestimates thesupply of funds available to it in the period or underestimates the demand for itsloans at the loan rate that it set. Because of these two possibilities, a bank mayend up with the actual demand for its loans at the loan rate that it set beinggreate r than the amount that it can supply. A bank is assum ed to prepare for thisby setting the maximum amount of money that it will lend in the period lowenough so that the bank is assured , based on its past expectation errors , that itwill end up in the period with at least this much money to lend.

Firms borrow money from banks, hire labor from households, buygoods from other firms for investment purposes, and produce and sell goods toother firm s, households, and the government. At the beginning of the periodeach firm receives information from the government on the profit tax rate forthe current period, and from banks an the loan ra te that it will be charged forthe period and on the maximum amount of money that it will be able to borrowin the period. (Since in general each bank sets a different loan rate, it is notobvious wh ich loan rate any particular firm faces . It also is not obvious how theloan constraints from the banks are translated into the loan constraint facing anyparticular firm. Problems of this sort are discussed in section 1.3.) Firms do notknow at this time the demand schedules for their goods for the current periodand the supply schedules of labor for the current period.

The seven main decision variables of a firm are: (1) its price, (2) itsproduction, (3) its investment, (4) its wa ge rate, (5) its loans from banks, (6) themaximum number of work er hours that it will pay for in the period, and (7) themaximum number of goods that it will sell in the period. Regard ing the lattertwo variables, firms, like banks, must prepa re for the possibility that theirexpectations are incorrect. A firm is assum ed not to want to hire more labor inthe pe riod than it plans at the beginning of the period to hire. S ince a firm mayunderestimate the supply of labor facing it at the wage rate that it set, itprepa res for this possibility by setting a maximum on the amount of labor that itwill hire in the period. This msximum is assum ed to be the amount that the firmplans at the beginning of the pe riod to hire. A firm is also assum ed to set a


22/241

6 A Model of hlacroeconomic Activity Volume I: The Theoretical Model

maximum on the number of goods it will sell in the period, since it cannot sellmore goods in the period than the sum of wh at it produces and has ininventories. The maximum is assumed to be set low enough so that the firm isassure d, based on its past expectation errors , tha t it will end up in the periodwith at least this many goods to sell.Househo lds receive wa ge income from firms and the government,purchase goods from firms, and pay taxes to the government. A household eitherhas a positive amount of savings or is in debt. It it has savings, the savings cantake the form of demand depos its, savings depos its, or stocks. If it is in debt, thedebt takes the form of loans from banks. A household does not both borrowfrom banks and have savings deposits OI stocks at the same time. At thebeginning of the period each household receives eight items of information forthe current period: (1) the tax rates, (2) the rate it will be paid on its savingsdeposits (the bill rate), (3) the loan rate it will be charged, (4) the maximumamount of money it will be allowed to borrow, (5) the price it will be cha rgedfor goods, (6) the wa ge rate it will be paid, (7) the maximum number of hours itwill be allowed to work , and (8) the m aximum number of goods it Will beallowed to purchase. (The question of how this information gets translated toeach particular househ old is discussed in section 1.3 .) The two main decisionvariables of a household are the number of hours to wor k and the number ofgoods to purchase.

The bond dealer represents in the model both the bill and bondmarket and the stock m arket. The bond dealer does not hire labor and does notbuy goods. The decision variables of the bond dealer are the bill rate, the bondrate, and the av erage stock price. The bond dealer is not a profit maximizer;rather , itg tries to set the bill and bond rates for the next period so as to equatethe demand for bills and bonds in that period to the supply of bills and bondsin the period. The bond dealer holds an inventory of bills and bonds, andit absorbs in each period any difference between the supply of bills and bondsfrom the government and the demand for bills and bonds from the banks.Househo lds own the stock of the banks, the firms, and the bonddealer. All after-tax profits of the banks, firms, and bond dealer are paid to thehouseholds in the form of dividends. Banks, firms, and the bond dealer areassumed not to issue any new stocks. The bond dealer sets the average stockprice equal to the present discounted value of expe cted future dividend levels,the discount rates being expected future bill rates. The expectations of thefuture dividend levels and bill rates are form ed by households and arecomm unicated to the bond dealer. All households are assumed to have the sameexpectations regarding these variables.

Because of the w ay the bond dealer sets the stock price, householdsexpect the before -tax, one-period rate of return on stocks, including capital gainsand losses, to be the same for a given period as the expected bill rate for thatperiod. The bill ra te is the ra te paid on savings depos its. No w, capital gains and


23/241

Introduction 7

losses are assumed to be recorded each period and to be taxed as regular income,which means that households also expect the a f te r - taxates of return on stocksand savings deposits to be the same. Households can therefore be assumed to beindifferent between holding their asse ts in the form of stocks or in the form ofsavings depos its. This assumption greatly simplifies the model.Banks are similarly assum ed to be indifferent between holding thenonloan part of their assets in the form of bills or in the form of bonds. Thebond dealer sets the price of a bond, each bond yielding one dollar per periodforeve r, equa l to the present discounted value of a perpetua l stream of one-dollarpayments, the discount rates being the current bill rate and expected future billrates. These expectations of the bill rates are formed by banks and arecommunicated to the bond dealer. All banks are assumed to have the sameexpectations regarding the future bill rates. The bond rate is equal to thereciprocal of the bond price.

Because of the w ay that the price of a bond is set, banks expect thebefore-tax , one-period rate of return on bonds, including capital gains and losses,to be the same for a given period as the expected bill rate for that period. Sincecapital gains and losses are recorded each period and taxed as regular income,banks also expect the a f te r -tax a tes f return on bills and bonds to be the same,which m eans that they can be assumed to be indifferent between the two.

The discussion in the last three p aragraphs can be summ arized to saythat s tocks and savings deposits are assumed to be perfect substitutes and thatbills and bonds are assum ed to be perfect substitutes. These assumptions havethe effect of decreasing the number of decision variables of both households andbanks by one each , and they obviously simplify the model. As will be seen insection 1.3, distributional issues are generally ignored in this study, and theabove assum ptions are in a sense just another exam ple of the ignoring ofdistributional issues. The reason that stocks and bonds w ere included in themodel at all was so that the effects of capital gains and losses on the economycould be analyzed.

The bond dealer is assumed to set the bond price and the stock pricefor the next period at the end of the current period, but before all transactionsfor the current period have been completed. This is assumed to be done so thatcapital gains and losses for the current period can be reco rded during the currentperiod. All stocks in the model are end-of-period stocks. The model is discrete,and no consideration is given to the rate of change of the stock variables duringthe period.

In a nontatonnement model the orde r in wh ich information flowsand transactions take place is obviously quite important. In a t2onnementmodel the orde r is not important because recontracting is allowed and notransactions take place until the equilibrium prices and quantities have beendetermined. One must also be concerned in a nont&nnement model with wh atdetermines the actual quantities traded when the quantities dem anded do not


24/241


25/241

Househo lds make their decisions knowing the hours constraints fromfirms and the government, thus the constrained maximization processes ofhouseholds will always result in the constrained supply of labor being less thanOT equal to the sum of the governments demand and the maximum set by thefirms. Th e constrained supply of labor will thus always be the actual quantity oflabor sold in the period . If the hours constraints are not binding on thehouseholds, so that the unconstrained and constrained supplies of labor are thesam e, then the supply of labor will be less than the sum of the governmen tsdemand and the maximum set by the firms. In this case the government isassumed to get all the labor that it demanded, so that the firms are the ones whoend up with less labor than they expected. (Remem ber that the maximum set bya firm is its expected supply.) In this case the timx may be forced to produceless output than they had planned, depending on how much exc ess labor theyhad planned for. (The concept of excess labor is discussed at the end of thissection.)

Because households make their decisions knowing the goodsconstraints from firms, the constrained maximization processes of householdswill always result in the constrained demand for goods being less than or equalto the maximum set by the firms. The demand for goods includes the demandby households, the demand by the government, and the demand by firms (in theform of investment). Firms and the government are assumed always to get thenumber of goods that they w ant, so that households are the ones who aresubject to a goods constraint.

Since firms are assumed to set the maximum low enough so thatthey are assured of having this many goods to sell in the period , it will always bethe case that the constrained demand for goods is less than or equal to theavailable supply. Any difference between the number of goods produced andsold by the firms results in a change in inventories. If it happens that the actualdemand for a firms goods exceeds the demand the firm expected? the firm isassum ed not to receive this information quickly enough for it to be able toincrease its production and employm ent plans for the period.

This com pletes the discussion of som e of the main transactions inthe model. It is obvious that the particular orde r of information flows andtransactions postulated in the model is some what arbitrary and that other ord erscould be postulated. l%e particular order chosen here was designed to try tocapture possible credit rationing effec ts from the financial secto r to the realsector and possible employm ent constraints from the business sector to thehousehold sector. This order seemed to be the most natural one, although infuture wo rk it wou ld be of interest to see how sensitive the conclusions of thisstudy are to the postulation of different orde rs.

The assumptions that firms do not retain any earnings and do not.issue any bonds and new stocks a re not as restrictive in the present context as


26/241

10 A Model of Macroeconomic Activity Volume /: The Theoretical Model

one might think. What the model is trying to capture are aggregate financialrestrictions facing the firm se ctor, and if in practice at least som e firms areconstrained at times from being able to borrow as much money as they wouldlike at the current interest rates (i.e., either constrained in their borrowing fromfinancial intermediaries, in their issung of bonds, or in their issuing of newstocks), the specification of the model may not be too unrealistic. In theaggregate , only so much m oney is available in any given period to borrow, and ifinterest rates do not get set in such a way as to clear the financial markets everyperiod, then in periods of too-low interest rates some potential borrowers mustgo unsatisfied.

The model does account for all aggregate flow-of-funds constraints,and so the most important financial restrictions in a macroeconom ic contexthave been taken into account. It should also be emphasize d that banks in themodel are meant to include commerc ial banks, savings and loan associations,mutual savings banks, life insurance companies, and other financial interme-diaries, which makes it less unrealistic to aswn e that all borrowing takes placefrom the banks. Also, many corp orate bond issues, are in practice privatelyplaced-m ostly to life insurance compan ies-and this again lessens the restrictive-ness of the assumption that all borrowing in the model takes p lace from thebanks.

Before concluding this section, it will be useful to describe themodel of firm behavior in som ewhat more detail. It is usually the case that theprice, production, investment, and employm ent decisions of a firm are analyzedseparately rather than within the context of a complete behavioral model. A fewstudies have analyzed two of the decisions at a time. Holt, M odigliani, Muth, andSbnon[29], for exam ple, have considered the joint determination of productionand employm ent decisions within the cqntext of a quadra tic cost minimizingmodel. Lucas [34] has recently postulated a general stock adjustment model inwh ich the stock of one input may influence the demand for another input, andNad iri and Rosen [41] have used this basic model in an empirical study ofemploym ent and investment decisions. Coen and Hickman[l l] have wo rkedwith a model that takes into account the interrelationship of employm ent andinvestment decisions. Mills [38] , Hay [27], and Maccini [36] have consideredthe joint determination of price and production decisions. In the model of firmbehavior in this study, all four of the decisions are determined simultaneously~

The underlying technology of a firm is assum ed to be of aputty-clay type, w here at any one time there a re a number of different typesof machines that can be purch ased. The machines differ in price, in the numberof work ers that must be used with e ach machine per unit of time, and in theamount of output that can be produced per machine per unit of time. Theworker-machine ratio is assumed to be fixed for each type of machine.

One important premise of this study regarding the production,employm ent, and investment decisions of a firm is that there are costs involved


27/241

Introduction 11

in changing the size of the wo rk force and in changing the sire of the capitalstock. Beca use of these costs, a firm is likely to choose to operate some of thetime below capacity and off its production function. This means that some ofthe time the number of work er hours paid for may be greater than the numberof hours that the workers are effectively working. Similarly, som e of the timethe number of machine hours available for use may be greater than th e numberof machine hours actually used.

The evidence presented in Fair 114, Cha pter 31 rather stronglyindicates that firms do spend som e of the time off of their production functions,and the mode l of employmen t decisions develope d in [14] was based on thedistinction between hours paid for and hours work ed. The difference betweenhours paid fo r by a firm and hours wo rked will be referred to as excess labor.kSimilarly, the difference between the number of machines on hand and thenumber of machines required to produc e the output will be referred to asexcess capital. Tw o important constraints facing a firm are that the number ofworker hours paid for must be greater than or equal to the number of workerhours work ed and that the number of machine hours used must be less than orequal to the number available for use.

Another important premise of this study concerns the firms pricedecision. A firm is assume d to have a certain amount of monopoly pow er in theshort run in the sense that raising its price above prices charged by other firmswill not result in an imm ediate loss of all its custome rs and lowering its pricebelow prices cha rged by other firms will not result in an imm ediate gain ofeveryone elses custom ers. There is assume d, how ever, to be a tendency in thesystem for a high price firm to lose customers over time and for a low price firmto gain custom ers. This assumption-that a firms ma rket share is a function ofits price relative to the prices of other firms-is comm on to the studies ofMortensen [39], Phelps [46], Phelps and Winter [47], and Maccini [36]. Themode l dev eloped here, howe ver, differs from or expands on the mod els in thesestudies by postulating that a firm also expe cts that the future prices of otherfirms are in part a function of its own past prices. As will be seen in Cha ptersTw o and Three, this postulate has an important influence on the final propertiesof the model.

The tendency for firms to lose 01 gain custom ers depending onwhe ther their prices a re high or low can be justified by assuming that custom erssearch. If during each period some customers search, and if each customer whosearche s buys from the lowe st price firm that he or she finds, then there will be atendency for high price firms to lose custom ers and vice versa. AIthough thistendency can be justified by assuming that cu stomers do search, in the presentcase the search activities of custom ers are not explained within the mod el. In thespecification of the behavior of househo lds, for exam ple, the possible gains andcosts of search arepot considered, and search is not considered a decisionvariable of househo lds. If search w ere treated as a decision variable, it would be


28/241

72 A Model of Macroeconomic Activity Volume /: The Theoretical Model

necessary to specify a much m ox complicated model than has been done. Suchan undertaking is beyond the scope of the present study.

A firms market share of labor supplied to it is treated in a mannersimilar to its market share of goods sold: a firms m arket share of labor isassumed to be a function of its wag e rate relative to the wag e rates of otherfirms. Also, a firm is assumed to expect that the future wa ge rates of other firmsare in part a function of its own past w age rat:s.

Finally, a banks market share of loans is treated in a manner similarto a firms m arket share of goods: a banks market share of loans is assumed tobe a function of its loan rate relative to the loan rates of other banks. Likew ise, abank is assumed to expect that the future loan rates of other banks are in part afunction of its own past loan rates.1.3 THE METHODOLOGY OF THE STUDY

The me thodology of this study is unusual enough to require so me discussion.The most important aspect of the methodology is the use of computersimulation to analyze the behavior of the banks, firm s, and households and toanalyze the properties of the overall m odel. The behavior of each bank, firm,and household was analyzed in the following way.1.2.3.4.

5.

The basic equations were specified and the optimal control problem wasformulated for the behavioral unit.Assumptions regarding the formation of expect& m were made.Using the information from I and 2, algorithms we re written to solve theoptimal control problem of the behavioral unit.Particular values for the param eters and initial conditions we re chosen , anda base run wa s obtained by using the algorithms to solve the optimalcontrol problem for these particular values. The param eter values and initialconditions we re chosen so that the optimal paths of the decision variablesfor the base run would be roughly flat.Various changes in the initial conditions from those used for the base nmwere made, and for each change the control problem was resolved to obtainthe optimal paths of the decision variables corresponding to the change .These new paths were then comp ared to the base run paths to see how thebehavioral unit modified its decisions as a result of the change. A flat baserun w as chosen in 4 to ma ke it easier to com pare the behavioral unitsmod ified decisions to its original decisions.

The results in 5 are analogous to partial-derivative results in analytic wo rk in thetense that one obtains the change in one variable corresponding to a change insome other variable. In Chapters Two , Three, and FOU L, tables of results o fcarrying out the procedure in 5 are presented for banks, firms, and households,and from these tables one can get an understanding of how each unit behaves.


29/241

After the b ehavior of each unit wa s analyzed separately, the entiremode l wa s put to gether and solved. One solution of the overall mod el for onetime period correspond s to the solution of an optimal control problem for eachbehavioral unit and to the computation of the transactions that take place afterall the decisions have been ma de. After the transactions have all been com puted,time switches to the beginning of the next period, and the behavioral units solvetheir con trol problems again, the new solutions being based on the newinformation that has resulted from the previous periods transactions. After thenew solutions have been obtained, the new transactions based on these solutionsare com puted, and then time switches to the next period. This proces s can berepea ted for as many periods as one is interested in.

One important point to keep in mind about th e solution of theoveraU mode l is that although the solution of the optimal control problem foreach behavioral unit correspond s to optimal t imepaths of the decision variablesbeing co mpu ted, only the v alues for the current period are used in computingthe transactions that take place. Each period new time paths are computed foreach decision variable, and so the optimal values of the decision variables fo rperiods other than the current period are of importance only insofar as theyaffect the optimal values for the current period.

The optimal control problem of each behavioral unit is stochastic,nonlinear, and subject to equality and inequality constraints. In order tosimplify the problem som ewh at, each behavioral unit wa s assume d to convert itsstochastic control problem into a deterministic control problem by setting all ofthe values of the stochastic variables equal to their expec ted values beforesolving. This is a comm on proced ure in the control literature (see, for exam ple,Athans [3]). The solution values that result from such a procedu re must, o fcourse, be interpreted as being only approximations to the true solution valuesof the complete stochastic control problem. Only in the linear case would thedecision values for the current period that result from this proced ure be thesam e as the decision values that result from solving the com plete stochasticcontrol problem.

There is also another source of inaccuracy in this study regarding thesolutions of the control problems. Cos t considerations prevented the writing ofhighly accurate algorithms to solve the deterministic control problems, and thereis no guarantee that th e optim a found by the algorithms are in fact the trueoptima of the deterministic control problem s. Particular attention was concen-trated, how ever, on searching over values of the decision variables for the firstfew periods of the horizon, so that som e confidence could be placed on theassumption that the values chosen for the current period are close to the truesolution values of the deterministic control problem for the current period. T healgorithms that h ave been used to solve each particular control problem arediscussed in the following chapte rs. The length o f the decision horizon for eachbehavioral unit wa s always assume d to be 30 periods in the programm ing of themodel.


30/241


Because of the assumption that the behavioral units replacestochas tic variables with their expec ted values, the model is presented in the textusing expected values directly rather than density functions. A superscr ipt eon a variable is always used to denote the expe cted value of the variable.

Another important aspect of the methodology of this study is thetreatment of the aggregation problem. There are at least two basic ways in whichone might put a model of the sort developed in this study together. One waywould be to specify a number of different banks, firms, and househo lds; haveeach one solve its control problem; and then have them trade with each other insome way. T o do this, one would have to specify mechanisms for deciding whotrades with whom and would have to keep track of each individual trade in themodel. Questions of search behavior invariably arise in this context, as dodistributional questions. This way of putting the model together is considerablybeyond the scope of the present study.

The other basic way of putting the model together is to ignoresearc h and distributional questions. Even within this context, how ever, the re areat least two ways in which sea rch and distributional questions can be ignored.One way would be to postulate only one bank and one firm and treat the two asmonopolists. The othe r way is to postulate more than one bank and one firm,but treat all banks as identical and all firms as identical. I%is second way is theapproach taken in this study. The advantage of postulating more than one bankand one firm is that models can be specified in wh ich the behavior of anindividual bank or firm is influenced by its expectations of the behavior of otherbanks OT firms. M odels o f this type, in which m arket share considerations canplay an important role, seem more reasonable in a macroeconom ic context thando models of pure monopoly behavior.

An apparen t disadvantage of postulating more than one bank andfirm and yet treating all banks and firms as identical is that w henever, say, a firmexpe cts other firms to behave differently than it plans to behave, the firm isalways wrong. If all firms are identical, they obviously always behave in the sam eway, even though they almost always expect that they will not all behave in thesam e way . Firms never learn, in other wo rds, that they are all identical.Fortunately, this disadvantage is more apparent than real. If one is ignoringsearch and distributional questions anyway, there is no real difference (as far asignoring these questions is concerned) wh ether one postulates only one firm ormany identical firms. B oth postulates are of the same order of approximation,namely the com plete ignoring of searc h and distributional questions. and if onefeels that a richer model can be specified by postulating more than one firm, onemight as well do so. One will gain the added richness without losing any moreregarding search and distributional issues than is already lost in the monopolymodel.

The fact that distributional issues are ignored in the model mak esthe treatment of stock prices and shares of stock much easier than it otherwise


31/241

would be. The economy can be treated as if there wtxe only one share of stockin existence, of which individual credito r house holds own certain fractions. Theprice of this share of stock is set by the bond dealer. The bond dealer usesexpectations of future aggrega te dividend levels in setting the price, whe re theaggrega te dividend level in any period is the sum of all of the dividends from thefirms, the banks, and the bond dealer. Th e households are, of course, the oneswho form the expectations of the future aggrega te dividend levels, which thenget communicated to the bond dealer.

Two versions of the overall model have actually been used in thisstudy, one called the non-condensed version and one called the condensedversion. The non-condensed version postulates two identical banks, two identicalfirms, and two households. The two households are not identical; one is acredito r and one is a debto r. This version is solved in exactly the mannerdescribed above. Since the non-condensed version is large, costly to solve, andsom ewhat difficult to com prehend in its entirety, an alternative and smallerversion was also specified. This condensed version was specified as follows.1. The behavior of the banks, Arms, and households was examined by looking

at the tables o f results obtained by the procedure described in 1 through 5above (p. 12).

2. Using the results in these tables and a general knowledge of the optimalcontrol problems of the behavioral units, the behavior of the banks, firms,and households was approxim ated either by equations in closed form or bysimple algorithms. In the process of making these approximations, the bankswex aggrega ted and the firms were aggregated, so that one ended up withequations or algorithm s pertaining only to a bank secto r and a firmsector.

3. The transactions equations for the non-condensed model were thenmodified appropriately to correspond to the more simplified nature of thecondensed model.

The advantage of the condensed version is that one can see more directly whatinfluences the decisions of the behavioral units. In the non-condensed versionthe influences are buried in the optimal control problem s of the behavioral units,and many times one cannot see directly what affects wha t. No optimal controlproblems have to be solved in computing the solution of the condensed versioneach period since the optimal control problems have in effect been approx-imated by equations in closed form or by simple algorithms.

For the analysis of the properties of the overall model in Chap terSix, the condensed version has been used. The analysis of the non-condensedversion is relegated to the Appendix. Since the properties of the two versions arevirtually the same-one merely being an approximation of the other-it seemedbest to concen trate on the simpler version in the text. The Appendix contains


32/241


the results of a few runs and enough discussion to show how the non-condensedversion is solved.

There is also a static-equilibrium version o f the mod el, and thisversion should not be confused with either th e condensed or non-condensedversions, which are both dynamic. The static-equilibrium version is discussed inCha pter Seven. T he GaussSeidel algorithm is used to solve the static-equilibriumversion in Cha pter Seven, and again this algorithm should not be confused witheither the algorithms used to solve the optimal control problems OI thealgorithms used in the condensed version of the dynamic mod el.

The advantage of u& g com puter simulation techniques overstandard analytic meth ods to analyze mod els is that one can deal with muchlarger and mope complete models. More than merely one or two decisionvariables of a behavioral unit can be considered at the sane time, multiperioddecision problems can be considered, and in general one can get by with makingless restrictive assumptions. It should be stressed, how ever, that the simulationwo rk in this study is not meant to be a test of the validity of the mod el, butonly an aid to understanding its properties. The param eter values and initialconditions have all been m ade up and have not been estimated from any data.

It should be obvious by now that the mod el develope d in this studyis based on numerous assumptions that can in no way be verified or refuteddirectly. As with most econom ic mo dels, the model is highly abstract. Thephilosophy that underlies the construction of the present mo del goes somethingas follows. The author lboks on a theoretical mod el of the sort develope d in thisstudy as not so much true OI false as useful or not useful. The mode l is useful ifit aids in the specification of empirical relationships that one would not alreadyhave thought of from a simpler mode l and that are in turn confirmed by thedata. It is not useful if it either does not aid in the specification of empiricalrelationships that one would not have thought of from a simpler m od& or aids inthe specification of empirical relationships that are in turn refuted by the data.

As discussed in Cha pter Eight, th e present mod el does imply thatmacroe conom etric mode ls ought to be specified quite differently from the waythey now are. The model does appear, therefore, to meet the requirement that itlead to new empirical specifications, and so it does appear to be possible,according to the above philosophy, to decide whether the model is mope usefulthan other theoretical mo dels. (Volum e II will carry out such an analysis.)1.4 SUGGESTIONS TO THE READERBeca use of the models size and the reliance on com puter simulation to analyzeits properties, the overall mode l is not particularly easy to com prehend. Thereade r should have a good understanding of the behavior of the individual unitsin the model from the d isc ussion in Cha pters Tw o through Five beforeproceeding to the discussion of the com plete mode l in Cha pters Six through


33/241

I n t r o d u c t i o n 17

Eight and in the Appendix. Of particular importance in Cha pters Tw o, Three,and Four are the tables of simulation results (Tables 2-3, 3-3, 4-3, and 4.4),wh ere one can see how the behavioral units respond to various c hanges in theinitial conditions. The tables presenting the equations of the condensed mode lfor each behavioral unit (Tables2-4, 3-4, and 4-6) should also help one tounderstand the behavior of each unit.

The tw o m ost im portant tables in the book are Table 6-2 andTable A-2, wh ere the complete sets of equations for the condensed andnon-condensed mo dels are presented, respectively. Since the condensed mod el isa close approximation to the non-condensed mod el and is easier to com prehend,it is advisable for mo st purpose s to study Table 6-2 rather than Table A-2. Afterhaving studied Table 6-2 carefully, the simulation results for the com plete mode lin Table 6-6 and the related discussion should be understandable. In general, thediscussion in the tex t relies heavily on the use of tables, and in most cases it isnecessary to study the tables carefully in order to follow the discussion in thetext. In order to mak e C hapte rs Tw o through Five a little m ore self-contained,som e of the discussion of the behav ioral units in section 1.2 in this chapte r isrepea ted in the following chapte rs.NOTES

~Examplerof these studies are the studies of Leijjonhujvud [ 321, (331, Tucker1531 , 1541 , [55], Barre and Grossman [S], and Grossman 1241 , 1251 , (261. See atw thestudies of Sotow and Stietitz 151 1 and Korttras 1311 .cSee, for example, Atcbian [ 11, Diamond [t 21, Fisher [18] , [ 191, Gepts[201, Gordon and Hynes [22], Lucas and Rapping 1351 , Maccini 1361 , Mortemen [%I,1401 , Phetps [46], Phelps nd Winter [47], and Rothschild [49] See also an early paper byCtower [9], in which an attempt is made to provide a general theory of price determinationthat is applicable to att types of market stiuctues.dBa,o and Grossman [S], pp. 83-84, fn. 6.%e footnote c.%ee, for example, Christ [S] for a discussion of the government budgetUJStraiM. ST he bond dealer tilt be referred to as an it, rather than as a he or B she.hUnles otbewise stated, the phrase demand for OI supply or in the textis meant to refer to the quantity demanded or supplied, not to a demand or a supplyschedule. Since in general a firm plans to end up w ith a positive level of inventories atthe end of the period, the firms expected demand for its eoods is usually less than the

maximum nu%$w of goods that it is witting to sell.JA firms employment decision in the present context corresponds to itswage-rate decision and its decision on the maximum amo unt of labor to hire.k Excess labor was detined in a sti&ly different way in [ 141 as thedifference between standard hours and hours worked. Under this definition excess labor canbe negative if hours worked exceed standard hours. For purposes of the present study it ismore convenient to refer to the difference between hours paid for and hours worked as-excess labor.


34/241


35/241

Chapter Two

Banks

2.1 THE BASIC EQUATIONS

In Table 2-l th e important symbols used in this chapter are listed in alphabe ticorder. The first half of the table presents the notation for the non-condensedmodel, and the second half presents the notation for the condensedmodel. Thenotation for the condensed model pertains only to the discussion in Section 2.6.

Each bank, say bank i, receives money from households in the formof savings deposits (SD& ), on which interest is paid, and from firms,households, and &e bond dealer in the form of demand deposits @D&t), onwhich no interest is paid. Each bank lends money to tirms and households (L& )and buys gOVeINIXW bills (VBIL LBi~) and bonds (B ON DB it). Bank loans XCone-period loans, bills are one-period securities, and bonds are consols. Eachbank holds reserves in the form of deposits with the government (B& ). Eachbank sets its own loan rate (R&). he three main decision variables of each bankare its loan rate, the value of bills and bonds to purchase (V~& ), and the max-imum amount of money that it will lend in the period (LLW fA& ). Banks areassumed not to compete for wings deposits, and the rate paid on all savings de-posits is assum ed to be the bill rate (rr)I

79


36/241

20 A Model of Macroeconomic AC t iv ity Volume I: The Theoret ical Model

Table 2-I. Notation for Banks in Alphabetic Order

Subscript i denotes variable far bank i. Subscript j denotes variable for bank j. Subscript fdenotes variable for period f. An e superscript in the text denotes an expected value of the= number of bonds held, each bond yielding one dollar per period= actual reswves= required mserves= prclfit tax rate= penalty tax mte on the composition of banks portfolios= demand deposits= dividends paid= largest error the bank expects to mak e in overestimating its demand depositsfor any period= latgest error the bank expects to mak e in overestimating its savings depositsfor any period= amo unt that the bank knows it will have available to lend to households andfirms and to buv bi l ls and bonds even if it overestimates its demand andsavings depositiby the max imum am ounts= reserve xquirement ratio= no-tax proportion of banks portfolios held in bills and bonds= total value of loans of the bank sactor= value of loans= maximum value of loans that the bank will make= toial unconsua ined demand for loans= bill rate= bond mte= loan rate (of bank i)= loan fat e (of bankj)= average loan rate in the economy= savings deposits= taxes paid= length of decision horizon= value of bills and bonds that the bank chooses to purchase[ VBILLBit + 80NDBit /R ?]= value of bills held= before-ax profits

Subscript I denotes variable for period f O nly notation that differs from the notation forthe non-condensed model is presented here.EM AX DD = largest enor the bank sector expects to mak e in overestimating its demanddeposits for any periadEMAX SD = largest erra the bank sector expects to mak e in overestimating its savingsFUNDS+

deposits for any period= amount the bank sector knows it will have available to lend to households

and firms and to buy bills and bonds even if it overestimates its demandLBluAX*RLtSD,TAXB,VBB,VBILLB,w

and savings deposits by the maxim um amou nts= maximum value of loans that the bank sector will make= loan rate of the bank sector= savings deposits of the bank sector= taxes paid by the bank sector= value of bills and bonds that the bank S~CLOI hooses to purchase( VBILLB, + BONDB


37/241

Banks 21

The basic equations for bank i for period t are the following:

VBB,, = VBILLB,, + BOND BJR,, [value o f bills and bonds held] (2.1)II& =RBi,LBi, + I? VBILLB ,, + BON DBit - r,SDBi,+ (BONDBir/Rr+I - BOND BJR,), (before-tax profits] (2.2)

TAXB , = d, Wit fdZ [VSS,, - gz(VBBi, + LBit)], [taxes paid] (2.3)DIV& = llBi, - TAXBi,, [dividends paid] (2.4)BR,, =DD Bit + SDB,, - LBi, - VB B,, - (BOND13JRr+l - BOND BBR*!, [actual reserves m ust be greater than OI equal to required

IeRKVeS] (2.7)

Equation (2.1) merely defines the value of bills and bonds held.Since bonds are consols and since each bond is assum ed to yield one dollar ea chperiod, the value of bonds held is merely the number held divided by the bondrate, BON DBi,/R,. Equation (2.2) defines before -tax profits. The first threeterms on the right-hand side of the equation are the interest revenue received onloans, bills, and bonds, respective1y.a The fourth term is the interest paid onsavings deposits. llw last term is the capital gain or loss made on bonds held inperiod t. Taxes are defined in Equation (2.3), w here dl is the profit tax rate.With respect to the second term on the right-hand-side of the equation, thegovernment is assum ed through its taxing policy to try to induce banks to hold acertain proportion, g2, of their asse ts in bills and bonds. In practice, commerc ialbanks and other financial intermediaries are under ce rtain pressures to holdparticular kinds of securities, and here these pressures are assumed to take theform of government taxing po licy. If, in the model, banks w ere not induced insom e way to hold bills and bonds, they w ould never want to hold bills andbonds as long as their loan rates we re higher than the bill rate. The introductionof government taxing policy is a simple way of explaining why banks hold nvxethan one kind of asset.

In Equation (2.3), bank i is assum ed to be taxed at rate dz on thesquare of the difference between the value of bills and bonds held and &!z imesthe value of loans issued plus bills and bonds held. Since capital gains and lossesare, included in the definition of profits, Equation(2 .3) also reflects theassumption that capital gains and losses are taxed a$ regular income. Bank i is


38/241


assum ed not to retain any earnings, so that the level of dividends, as defined inEquation (2.4), is merely the difference between before-tax profits and taxes.

Bank reserves are defined in Equation (2.5). Slice bank i pays out inthe form of taxes and dividends any capital gains made in the period (andconversely for capital losses), and yet does not receive any actual cash flow fromthe capital gains, capital gains take away from (and conversely capital losses addto) bank reserves, as specified in (2.5). Req uired reserves are defined inEquation (2.6). For simplicity, no rewve requirements ax placed on savingsdeposits. Actual reserves m ust be greater than or equal to required reserves, asindicated in (2.7).2.2 THE FORMATION OF EXPECTATIONSLet T+I be the length of the decision horizon. In order for the bank to solve itscontrol problem at the beginning of period t, it must form expectations of anumber of variables for periods r through t+T Bank i is assumed to form thefollowing expectations.b

bankW)

rate ofqtq-1 ,z [expected loanj for period t], [expected loan rate of bank j for period r+k

(k=l,Z,...,T)] (2.9)q+k =(R& r+k * qr+,):, [expected average loan rate for period t+k

(k=O,I,. .,7)] (2.10)

a3 > 0, [expected aggregate unconstraineddemand for loans for period f] (2.11)

LUIY;;, =Luq+k_, , [expected aggregate unconstraineddemand for loans for period r+k(k = I,&. .,7J] (2.12)

L;+k = Luiv !+k, [expected aggrega te constrained demand for loans for periodwk (k=O,l,. .,TJ] (2.13)

, a4 < 0, [expected market share of loans for period t](2.14)


39/241

Banks 23

[expected market share of loans for periodt+k(k=I,Z,. ..7)] (2.15)

DD%+kDDB,f_I, [expected level of demand deposits for period t+k(k=O,l,. .,Tf] (2.16)sD%+k SDB ,t_l, [expected level of savings deposits for period f+k

(k=O,I,. ..T)] (2.17)I& =rr, [expected bill rate for period t+k (k=I,Z,. .)] (2.18)

I 1 I 1R& = %) + (1+&k) (l+


40/241

24 A Model of Macroeconomic Act iv it v Volume /: The Theoretical Model

t-l (LUV _~)is what would have been the demand for loans on the part of firmsand households had they not been subject to any constraints. Eac h bank isassumed to be awa re of this demand. The aggregate constrained demand forloans @_I) is the actual value of loans made in period f-l. Equation (2.13 )states that bank i expe cts that firms and households will not be constrained intheir borrowing behavior in periods t and beyond. The expected awega teconstrained demand for loans is assum ed in Equation (2.13 ) to be equal to theexpected aggregate unconstrained demand for loans for each period. As will beseen below, bank i does not itself expe ct to turn any custom ers away, and soEquation (2.13 ) merely states that bank i also does not expect any custom ers inthe aggregate to be turned away.

Equation (2.14 ) determ ines bank is expectation of its market sharefor period t and reflects the assumption that a bank exp ects that its market shareis a function of its rate relative to the rates of other banks. The equation statesthat bank is expected market share for period t is equal to last periods marketshare times a function of the ratio of bank is rate for period t to the expectedrate of bank j for period t. Equation (2.15 ) is a similar equation for periods WIthrough t+7:

It should be noted that the market share for period r-l on theright-hand side of Equation (2.14 ) is the ratio of the actual value of loans ofbank i in period t-1 to the actual value of aggregate loans in period t-I(L&_l/Lt_l) and is not the ratio of the actual value of bank is loans to theaggregate unconstrained demand for loans (LB,+I/LUN~_~). Since bank i isassumed to know both Lt._1 and LUNt-1, the latter specification is a possibility.

The justification for the use of LB,,_~/L,_I is as follows. qis bankIs expectation of the aggregate unconstrained (and constrained) demand forloans for period t. Of the potential custom ers represented by this amount, som ewill come to bank i during the period. How many com e depends on how large apart bank i is of the market in period t-1 and on the relative loan rates . Now , agood measm e of how large a part bank i is of the market in period t-l is itsactual market share in period t-l. This measure is a better measure thanLBit-I/LUNt_l , since the latter does not represent in any direct sense bank isparticipation in the market. If LIJNt-1 is greate r than Lt-1, only a part of theunsatisfied custom ers represen ted by this amount are likely to have been turnedaway by bank i. The rest of the custom ers wou ld not have sampled bank i in theperiod. The refore, it seem s more in the spirit of the search literature to use theactual market share on the right-hand side of Equation (2.14) .

As should be evident from the discussion in the next section,Equations (2.8)-(2 .15) are quite important in determining the rate settingbehavior of bank i. Two similar sets of equations are also postulated in Cha pterThree regarding the price setting and wage setting behavior of a firm. The twomost important assumptions underlying Equations (2.8) -(2X) are that banki expects that its rate setting behavior has an effect on bank js rate setting


41/241

B a n k s 25

behavior and that bank i expe cts that its market share is a function of its raterelative to bank 1s rate. The equations can be easily modified if there is morethan just one other bank in existence. Equations (2.8) and (2.9) would hold foreach bank. Equation (2.10) would be the geometric average over all banks.Equations (2.1 I)-(2.13) would remain the same, and Equations (2.14) and(2.15 ) wou ld be changed either to include all the ratios of bank rs rate to theother banks rates, or to include the ratio of bank is rate to the average of theother banks rates.

In Equations (2.16) -(2.18 ) bank i is assumed to expect that thevalues of demand deposits, savings deposits, and the bill rate for all futureperiods will be the same as the last observed values of these variables. Equation(2.19) determines the expected bond rate. The right-hand side of the equation isthe present discounted value of a perpetual stream of one-dollar payments, thediscount x& es being the expected future bill rates. The right-hand side of theequation can thus be considered to be the expected price of a bond for periodw k , and so the reciprocal of this expression can be considered to be theexpected bond rate for period w k . This assumption, of course, ignores the factthat the expected value of a ratio is not equal to the ratio of the expected values,.but this type of problem is ignored all the way through this study by theconverting of stochastic control problems into deterministic control problems inthe manner discussed in Section 1.3.

The assumptions in (2.16) and (2.17), that bank i expects no changein its demand and savings deposits from the last observed values, are importantand typical of many expectational assumptions made in the model. Whenever anexpectational assumption had to be made that was either not concerned withma rket share situations OI for wh ich no obvious assum ption was available, thesimple assumption of no change from the last observed value was made. The aimwa s not to complicate the model any more than seem ed necessary to captureimportant expectational issues.

As long as lagged values have some effect on expectations of currentand future values, assumptions like (2.16 ) and (2.17 ) should not be toounrealistic. It should also be noted that because of the assumption in (2.18 ), thatbank i expects no change in the future bilI rates from the last observed rate, theexpected bond rates in (2.19 ) are simply equal to the current bill rate. It wa smentioned in Section 1.2 that the only reason bonds we re included in the modelat all was to account for the effec ts of capital gains and losses, and so nothing isreally lost in the m odel by having the bill rate and bond rate always be equal.2.3 BEH AVIORA L ASSUMPTION SIhe objective of a bank is to maxim ize the present discounted value of expe ctedfuture after-tax profits. The discount rate is assum ed to be the bill rate. Theobjective function of bank i at the begInning of period t is:


42/241

26 A M o d e l o f M a c ro e co n o m i c A c t i v i t y Vo l u m e /: T h e T h e o re t i ca l M o d e l

OBJBi , = l IB$ - TA X $t lSB$+, - TA X $t+I(JhtJ + (lh *)(I+r;+l) +...%+, - TAX%+T

+ Jht) (J+


43/241

Banks 27

With respect to a banks decision regarding bills and bonds, equation(2.19) means that a bank expects that the before-tax, one-period rate of returnon bonds, including capital gains and losses, for a given period will be the sam eas the expec ted bill rate for that period. Since capital gains and losses are taxedat the same rate as other iixome, the expected after-tax rates of return on billsand bonds are also the same. Because of this, banks are assumed to be indifferentbetween holding bills and bonds, and so instead of determining two variables,VBILLBi, and BOND Bit, a bank can be considered, given Rt, as determiningOd y VBBi,.

The main constraint facing a bank is the reserve requirementconstraint (2.7). A bank expects to receive in funds in period f+k, DDBf,+, fSDB;t+k, of which g$DB,e,+, is needed to meet the reserve requirement.Therefore, (1 -gI)DDBft+k + SDB;t+k is the expe cted amount available forperiod t+ k to lend to households and firms and to buy bills and bonds. A bank isassumed, howev er, to have to prepare for the possibility that it overestimates itsdemand and savings deposits. A bank is assumed from past experience to have agood idea of the largest error it is likely to make in overestimating its demandand savings deposits. Call the error for demand depositsEM AX DD i and the errorfor savings deposits EMAXSDi. For simplicity, these expected n~axim~m ~TIO ISare assumed not to change ova time. The quantity (1-g*) (DDB$+k -EMAX DDi) + (SDB$+,- EM4XSDi) is the amount that bank i knows it willhave available in period ttk to lend to households and firms and to buy bills andbonds even if it overestim ates its demand and savings depos its by the maximumamoun ts. Denote this quantity as FIJNDS;+k:

FVND$+, = (I-gI)(DDB$+k - EMAX DD,) + (SDBft+k - EMAXSD ,). (2.21)

Now, given a path of bank ~3 loan rate, it was seen from 1 abovethat bank i can compute the path of its expected loans (LB ,& k=O,l, ,7).In order to make sure of meeting the reserve requirement constraint, bank i isassumed to behave by choosing the path of its loan rate and the path of thevalue of bills and bonds to buy (VBBjt+k, k= 0.1, ,7) so as to satisfy theconstxaint that

LBft+k + VBBi,+k = FUND &., k=O,l,. .,T. (2.22)

By satisfying equation (2.22) , bank i is assured that it will have enough funds tomeet the expected loan demand each period, given its path of the value of billsand bonds to buy. Once a bank decides a t the beginning of period t the value ofbills and bonds to purchase in the period, it is assum ed that the bank m ustpurchase this value.


44/241


This is still not the end of the sto ry, how ever, for bank i must alsoprepare for the possibility that it underestimates the demand for its loans at theloan rate path that it has chosen. Bank i is assumed to prepare for this possibilityby announcing to households and firms the maximum amount of money that itwill lend in each period, in addition to announcing the loan rate. The maxim umamount each period (LBM A&+~, k = OJ, .7) is assumed to.be equal to theexpected loan demand for that period:

LBM AXit+k = Lqt+k. k=O,I,. .,T. (2.23)

Bank i is now assured of meeting its reserve requirement. It will always have atleast amount FlJNDSi;+k at its disposal, and it will never use more than thisamount to lend to households and firms and to buy bills and bonds. Theprocedure just described means, of course, that a bank ex pects to hold someamount of excess reserw most of the time. Only in the extreme case whe re itoverestimates its demand and savings deposits by the full amounts EM aXD Diand EM AX X+, and also lends to households and firms the maximum amount ofmoney that it set, will it en d up with zero excess reserves.

Although in practice comm ercial banks and some other kinds offinancial intermediaries can usually meet unexp ected situations by borrowingfrom a monetary authority, the procedure just described by which banksaccoun t for unexp ected situations in the model is not necessarily unrealistic.Com mercial banks and other financial intermediaries are under basic constraintsof the kind considered above, and it is not unreasonable to assume that theseconstraints play an important role in their decision making proc esses. Also, if abank can hold negative excess reserves in the short run by borrowing from amonetary authority, all this really means in the present contex t is that the bankwould maximize (2.20) subject to the constraint that LB &+ k + VH3 if+k in(2.22) be equal to FUND Sf,+, plus wme positive number. The positive numbermight be, for examp le, the maximum that the bank could expec t to borrowfrom the monetary authority ln an emergency situation.

It is likewise not necessarily unrealistic to assum e that banks mustbuy in the period the value of bills and bonds that they chose to buy at thebeginning of the period. Althoug h in practice one bank can sell bills and bondsto another bank to get more funds to lend to households and firms, in theaggrega te this cannot be done. In the aggrega te the government determines thenumber of bills and bonds to have outstanding, and the private secto r mustbehave w ithin this constraint. In the model the bond dealer absorbs each periodthe difference between the supply of bills and bonds from the government andthe demand from the banks, so the assumption that banks cannot change theirdecisions on the value of bills and bonds to buy during the period merelysimplifies the specification of the way that transaction takes place during the


45/241

Banks 29

period. Any discrepancy between the supply from the government and thedemand from the banks in the current period affec ts the bill and bonds rates setby the bond dealer for the next period.2.4 THE SOLUTION OF THECONTROL PROBLEM

It wa s seen in the last section that given the paths of the loan ra te and the valueof bills and bonds to buy, the axresponding value of the objective function canbe computed. In order to solve the control problem of bank i, an algorithm waswritten to search over various loan rate paths. The base path, from which otherpaths w ere ttied, was taken to be the path in which the proportion of bills andbonds held each period wa s equal to the no-penalty-tax proportiong*. The loanrate path corresponding to this situation is com puted as follows.

First, VBBit+k is set equal to g2FUNDSt+, , and LB,efek is set equalto FCIND S;+~ - VRBit+k (k=O,l, . ,T). Now, for period t, given the values forperiod t-2, Equations (2.Q (2.10), (2.1 l), (2.13), and (2.14) form a system offive equations in six unknowns: RB $, m:, LUJ$, LF, LB$, and R&t, Given avalue for LB$, the system reduces to a system of five equations in five un-knowns, wh ich can be solved recursively to obtain a value for RBi,. For periodt+l, given the values for period t, Equations (29), (Z..lO), (2.12), (2.13), and(2.15 ) likewise fo rm a system of five equations in six unknowns. Given a valueforL%+l Ia value fo r RBjr+] can be obtained. This process can then be repeatedfor periods ~2, ,t+T to obtain the base loan rate path.

Given the base loan rate path, it is straightforward to searc h over al-ternative paths . Given a value ofR & and given values for period t-2,equations(2X), (2.10), (2.1 l), (2.13), and (2.14) can be solved for RBiq, fiB;, LU@, L;,and LB& Once I&t has been determined in this way , the value of VBBj,is merely the difference between FUNDS$ and LE$ Values for periods t+l andbeyond can be obtained in the sam e way by solving Equations (2.9), (2.10 ),(2.12), (2.13), and (2.15). The algorithm was program med to search in one di-rection until the value of the objective function decreased and then to try otherdirections. Particular importance wa s attached to searching over values ofRBt,,since this is the value actually used in the solution of the overall m odel.2.5 SOME EXAMPLES OF SOLVING THECONTROL PROBLEM OF BANK i

PARAMETER VALUES AND INITIAL CONDITIONS

Ike param eter values and initial conditions that we re used for the first exam pleare presented in Tabie 2-2. The most important parameters are d2, the penaltytax rate on portfolio composition, ai, the measure of the extent to which bank i


46/241


47/241

Banks 31

ratio g2. The aggre gate unconstrained demand for loans in period i-1 is the fameas the constrained deman d. The length of the decision horizon is 30 periods.THE RESULTS

The results of solving the control problem of bank i for the param eter values andinitial conditions in Table 2.2 are presented in the first row of Table 2-3. Only asmall subset of the results is presented in Table 2-3, as it is not feasible topresent all 30 values for each variable. Values for the first two periods arepresented for bank is loan rate, its expectation of bank js loan rate, itsexpectation of the aggre gate demand for loans, its expectation of the deman dfor its own loans, its expectation of its mark et share, the value of bills and bondsto purcha se, and its expectation of the ratio of the value of bills and bonds heldto the value of loans plus bills and bonds held. The values in the first row ofTable 2-3 for each variable are equal to the corresponding initial value inTable 2-2, which reflects the way the parameter values and initial conditionswere chosen.

One important reaction of a bank is how the bank responds to achange in its demand or savings deposits. For the results in row 2 in Table 2-3,FUNDS:, was increased by 5.0 percent. An increase in FUNDSff can come aboutby an increase in period t-l of either dem and deposits or savings deposits or bya decrea se in the resew s requirement ratio. B ecause of the expectationalassumptions regarding deman d and savings deposits, a 5.0 percent increase inFCJNDSie, meam that bank i expe cts all future values of this variable to be 5.0percent higher as well.

From the results in row 2 it can be seen that this change caused banki to lower its loan rate for periods f and beyond in an attemp t to increase thedeman d for its loans. Since bank i expe cted that bankjs rate would not respondto this change in bank is rate until period WI, bank i expe cted to increase itsshare o f the market from 0.5000 to OS24 1 in period t. The aggregate demandfor loans was expe cted to increase slightly in period t from 810.2 to 811.3because of the low er averag e loan rate caused by bank i lowering its rate. Bank ia l so chose to raise its ratio of bills and bonds to loans plus bills and bonds from0.2956 to 0.2960 . This slight substitution into bills and bonds from the no-taxamount wa s caused in effect by the lower loan rate relative to the bill rate.

The values of all of the variables for period t+l were essentially thefame as the values for period t excep t for the value of t

a model of macro economic activity - fair (ballinger 1974) - copy

Documents