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iugene E Famaimeiifi{*- MFIEr *3 The Theory of Finance Eugene F. Fama Merton H. Miller Graduate Sch00! ofBusiness The University of Chicago DRYDEN PRESS HINSDALE, ILLINOIS PREFACE W . . ' . This book, in various ~versions, has been the main text for courses in finance taught by us overt the last six years at the Graduate School of Business of the University of Chicago. We have generally supplemented ‘it with other materials, ranging all the wayfrom cases and homework problems, on the one hand, to journal articles and research papers on the other, depending on the level of the course and its place in the curriculum. Even in the most elementary courses, however, we have always assigned some journal articles as supplementary reading. We believe an important function of graduate professional education to be that of acquainting future practitioners with leading scientific journals in their field. From our experience, moreover, students at all levels have generally welcomed this chance to confront the literature of finance directly and especially to plunge into the many controversies that have enlivened, if not always enlightened, the subject.“ But for students to be able to grapple with the issues efiectively or to make independent critical evaluations of contending points of view, they must have a more thorough, systematic, and rigorous grounding in the basic theory of s finance than can be obtained from any of the standard, all-purpose texts currently available. Hence, this book. To ~make the essential theoretical framework of the subject stand out sharply, we have primed away virtually all institutional and descriptive material. We also largely pass over suchlpopular standard topics as cash flow forecasting, cash budgeting, ratio analysis, credit management, and similar aspects of managerial finance. .We do not deny that these problems are often of great practical importance, but except perhaps for capital budgetirig, theoretical work in thearea has so far been ad hoc and largely unrelated to the material that we try to 1 cover. In any event, instructors who wish to place more emphasis than we do on these problems of in- ternal financial management and control should have no difficulty in finding suitable supplementary readings. t We have also decided--and here the decision was much more painful-— to omit any review or extensive discussion of empirical tests of the theory. So much high-quality empirical work is now being done and with tech- niques so varied that we despaired of being able to do justice to it without adding excessively to the length of the book and the number of different topics covered. Once again, however, we feel that instructors so inclined t w . t vii _ _ __ A _ _ "'5" ' " 1|kn|ni||lI1*—‘* '7'" "‘ " 7 7 7T 4L 77 ‘*7 7 77 7 viii Preface ' can readily supplement the text with journal articles and, where necessary, with selected chapters from texts in econometrics and statistics. Readers will also note the absence of any detailed examples, of the kind often found in standard texts, purporting to show how to apply the theory, in precise, quantitative terms, to real-world decision prob- lems. This omission is less a move -to save space than a reflection of our belief that the potential contribution of the theory of finance to the de- cision-making process, although substantial, is still essentially indirect. The theory -can often help expose the inconsistencies in existing pro- cedures ; it can help keep the really critical questions from getting lost in the inevitable maze of technical detail; and it can help prevent the too easy, unthinking acceptance of either the old clichcésor new fads. But the theory of ‘finance has not yet been brought, andperhaps never will be, to thecookbook stage. i The cutback that we have made in the amount and kind of material to be covered is also. accompanied by a fairly drastic change, as compared with standard texts, in the way the book is organized. Rather than follow the popular practices of structuring by the balance sheet-—-starting with short-term assets and liabilities and working successively down the right- hand side into longer-term financing—or by the stages in the financial life cycle of a firm--starting with organization and incorporation and working on through to bankruptcy and reorganization--we have tried to present the material in the order of difiiculty and logical priority, although this means that we often are forced to spread the discussion of particular applied problems, such as dividend policy, over several chapters. -Because a complete overview is provided at the start of each main part of the book} we need say little more here about the precise organization of the material beyond indicating that Part, I covers the case of certainty and Part II that of uncertainty. Our treatment of uncertainty is carried out for the most part within the so-called mean-variance or, better, two- parameter risk-return framework. Empirical research in finance has increasingly been conductedwithin this framework and, on the whole, quite successfully. Hence we give only passing attention to the more ele- gant but so far much less operational time-state preference approach. It is assumed throughout that all securities are traded in perfect markets. Originally, we had hoped to have a third majorpart in the book in which, after having dropped the assumption of perfect certainty, wecwent on to drop the assumption of perfect capital markets. After surveying the relevant literature, however, we feel that results in the theory of finance under imperfect capital markets are still too few, too unconnected, and too little tested to justify their incorporation into an introductory text- book. We recognize that there are dangers in arming the student only with comparative static models based on the perfect market assumption, P _ _"-j,- _r "c 7 —7 i * ————’ if 7 7 j 7 W 7 . - Preface ix and we have therefore tried to call attention at various points to some of the pitfalls involved in the unthinking application of these models to real-world problems. But we feel -that in the present state of the art, any extended discussion of financing problems under imperfect markets would run the greater danger of deluding the student, and the practitioner, into believing that the profession had found more solutions for such problems than is infact the case. H e s e s k Not only are analytical results in the area of imperfect markets rela- tively meager, except perhaps in the area of models of cash balance man- agement,‘ but at least insofar as concerns the valuation side of finance, there seem to be no very compelling reasons for extending the analysis in the direction of incorporating imperfections and irrationalities. The most striking single impression that emerges from the mass of empirical work that has been done in the last ten years is how robust the sperfect and efiicient market models are in confrontation with the data, despite what seem to be the outrageous simplifications that have gone intotheir con- struction. - i s H - i Although the discussion to this pointsuggests, and a glance through the book itself will confirm, that we rely much more heavily on the standard apparatus of economic theory than is typically the case in finance, this book is, nevertheless, intended as a text in finance and not as one in eco-. nomics. For. example, in discussions of o decision making, the emphasis throughout is on the microdecision problems of the investor and the corporate manager and not on the macroeconomic problems of social policy that are the main concern of economists. Even at the micro level, we have not hesitated to omit pieces of the standard apparatus that an economist would be likely to regard as essential, such as the distinction between income and substitution effects or between relative and absolute risk aversion, but for which it happens that we have no need. Nor, further- more, has any very substantial previous acquaintance witheconomics. on the part of the reader been assumed. I We have taken as a prerequisite only that amount of introductory economics that has long been the com- pulsory dosage in MBA and even undergraduate business programs--one course in price theory and perhaps one in macroeconomics. Having issued this disclaimer for the benefit of any economist reviewers, we should like to indicate our belief that this book can also be used effectively as at least a "supplementary text in courses in intermediate price theory and capital theory. s s n i ' We have tried to keep the mathematics, requirements as low as possible to avoid excluding those long-since-graduated practitioners of finance who still maintain an active interest in the technical professional litera- ture of their field but who have inevitably become a bit rusty in mathe- matics. Despite any initial impression to the contrary that might be 7_ 7 _ v _ liq’ - 7 Tin i T x Preface i ' gained from casually riffling through the pages, only a working knowledge of high school algebra, geometry, and some elementary statistics is really required to be able to follow the main line of the argument. Where more than this assumed minimum is used, we have so indicatedby starring the particular section concerned or by putting the material in footnotes. Such sections. and footnotes may safely be skipped by readerswho lack the requisite mathematical background, but we should strongly urge even them to try atleast to skim throughthis material. To make doubly sure that there is no lossof continuity, we have taken pains to see that each main point is always covered in at variety of ways: mathematically,’ verbally, graphically, and in a few cases, by numerical examples. e e c e t Finally, we should like to emphasize again that this book was written as a textbook, not a treatise. Virtually all the substantive content, except for a number of new proofs that we have felt called on to supply, can be traced directly back to sources that have long since been published and subjected to criticalscrutiny by the profession. ,We have made no attempi to document these sources in detail or to provide any very elaborate history of doctrine. We have, however, added a brief annotated bibli- ography to the end of each chapter, although itis intended only to indicate some of the high spots and makes no pretense to completeness. J In putting together this book, we have been more than usually fortunate in being able to draw on the thinking and advice of our students and colleagues, some of whom have actually used, this book ininote form as a text in their own courses. Our sincerest thanks to them and to the many others at Chicago and elsewhere whose comments and criticisms have helped shape this book. s k e Some of the proofs and related discussions first appeared in articles in the American Economic Review, the Journal of Business, the Journai of Finance, and the Journal of Political Economy. We wish to thank these journals and their publishers, the American Economic Association, the American Finance Association, and the University of Chicago Press for per- mission to draw on and adapt this materialfor use in the present volume. We also wish to thank Mr. Robert Officer for hishelp in checking the text and in constructing the index. e e Chicago J e i Euonnn F. Fama November 1971 Mnrrrox H. MILLER 7 _,_ —— __,L__ — _ ' --L c ww -n ' '__—r_ "v W" " "—— !— I. O LA. I.B. I.B.1 I.B.2. I.B.3. 1.0. LD. "'I.E. II. PP 5"?’ I-l "'II. A.2 II. A.3 II. AA II.A.5. II. A.6 II.B. II. B. 1 II. B.2. II.B.3 II.B.4 II.B.5. "'II. B.6. II. C. I. . I LA CONTENTS Preface O I I CERTAINTY MODELS I Choplor I A Model of tho Accumulation and I Allocation of wmaui by Individuals J e THE ECONOMIC THEORY, OF CHOICE I Opportunities and Preferences t Representation of Preferences: The Utility Function and O Indifference Curves if I ' k O The axioms of choice and the principle of maximum utility’: O Indifference curves and the geometrical representation of I Preferences I I O I I O I The axioms of nonsatiation and convexity if The Opportunity Set Choice Subject toConstraints O v a _ The Solution in Mathematical Form I I THE APPLICATION OF THE THEORY OF CHOICE T0 THE ALLOCATION or FINANCIAL assouacss ovnn TIME e I I The Two-Period Case , J I J J J The objects of choice: standards of living at different points in time e r O The properties of U(C1, c,, . . ., c,, . . .) I it I Opportunities: resources and capital markets g if The opportunityset under perfect capital markets 1 Interest rates and present values c 1 The preferred allocation g g I I k T Extension to Three or More Time Periods O t k i The structure of prices for claims O I I t An equivalent representation in terms of interest rates Multiperiod ratmof interest and the concept of the term ; structure J O J O The equal rate of return principle I I O O The n-period opportunity set ~ O t O O O The optimal allocation in the multiperiod case; mathematical treatment O I I J Conclusion I I O O I O Appendix: An liluclrotlvo Application of tho Wcollh Allocation Modal: Tho life Cycle of Savings _ ' . rrna BASIC rnamnwonx ' O O Smoothing Irregularitiec in the Income Stream I Witty’ i-#?#*i §5§eo~I an ti xii Contents I.B I.C I.D I_.E II.‘ ILA II.B I. I.A. I.B. I.C. I.C.1 I. C.2'. "'I.D. II. I II.A. II.B. II.B.1 "‘II.C. II.D. II.D.1 II.D.2. II.D.3. III. III.A. III.A.1. III. A.2. III.B. III. C. III.D. III.'D.. 1 . III.D.2 III. E. III. E. 1 III. E.2 III. E.3 III. F. III. F. 1. III. F.2. III. F.3. IV. U Some Specifications and Simplifications . Consumption. Saving. and Wealth over the Lire ‘Cycle Consumption and Saving with a Growing Income e i The Eflecta of Changes. in Income and Wealth - _ CONSUMPTION. SAVING. AND THE RATE OF INTEREST Some Additional Assurnptions: Homogeneity and Its Implications Interest Rates and Consumption'Decisiona' over the Life Cycle Choptor 2 Extension oftho Model to Durable Commodities, P Production, ond Corporations s t e I DURABLE COMMODITIES AND INVESTMENT e Representation of the Carry-over Opportunities Provided by Commodity Storage and Production ' r _ O The Opportunity Set When Both Commodity Carry-overs and Capital Markets Are Available P I The Case of Production and Investment a P to The double-tangency solution C Some remaining issues A P V The Solution in the n-Period Case C k t O EXTENSION TO THE CASE OF CORPORATIONS Management Objectives and Stockholder Preferences r The Market Value Criterion P P The case of many owners V " The Market Value Criterion in the General n-Period Case The Market Value Criterion: Some Problems and Limitations ~ A k t C t A The market value rule and profit maximization s I The market value rule and management motivation p The criterion problem under imperfect capital markets r MARKET VALUE, DIVIDENDS, AND C ~ P STOCKHOLDER nnrunns t The Efiect of I Dividend Policy = " P Assumptions, notation, and timing conventions O ~ The equal rate of return principle once again I The Efiectseof Dividend Policyon the Market Value of the Shares a ~ A Graphical Illustration y A v The Independence Proposition Further Considered . The bird-in-the-hand argument r I t A Dividend policy and the internal rate of return Some Equivalent Alternative Valuation Formulas A I The stream of dividends approach e t C t Cash flow and earnings approaches O A p k P Investment opportunities, growth, and valuation Growth Potential and Stockholder Returns- The constant growth model t k P p t A The growth of total earnings and the growth oi dividends and price per share t k 1 . Corporate earnings and investor returns k . t SUMMARY ~ ' _ jg —"'—"';.i> 3 "*7" " 7 I . I I. II. . P V V 1. Contents xiii Appendix: Valuation Formulas: Some Numerical Illustrations FINANCIAL POLICY. RETURNS. AND VAL-UA'.I‘I1ON . AN EXAMPLE USING THE CONSTANT GROWTH MODEL Chapter 3 Criteria for O'pl‘lmG|1|nV§stmQl1l Decisions A S I. I.A. I.B. I.C. I.C.1. I.C.2. I.D. I.E. I.F. I.F.1. I.F.2. II. II.A. II.A.1. II.A.2. II.B. II.C. 1 II.D. II.D.1 II.D.2. 11.0.3 11.11. 11.1". ll UNCERTAINTY MODELS I 1 I THE REPRESENTATION OF OPTIMAL V I INVESTMENT DECISIONS . S I up P 1 . V The Case of a Single Capital Good and Two Time Periods Graphical Representation of the Complete Solution 1 An Alternative Representation Highlighting the Investment Decision 1 1 U 1 1 I Alternative forms for the optimizing conditions ; . . Optimal capital stock and optimal investment Extension to the Case of Many Different Machines V - The Investment Decision and the Transformation Curve I Extension to More than1Two Time Periods 1 1 C The case of perfect markets forcapital goods A 1 V The case of fixed capital 1 P 1 1 INVESTMENT DECISIONS AND CAPITAL BUDGETING Problemsin the Application of the Present Value Criterion Marginal versus average present values ” V I 1 Comparing investments with different lives 1 1 Replacement Policies and the Optimal Economic Life of I Equipment Vi 1 . V . Maximizing Present Value Subject to Constraints. I The Rate of Return Criterion: Usesiand Abuse 1 The discounted cash flow rate of return V 1 V 1 I V Mutually exclusive investments andmultiple rates of 'return~‘ A modified rateof return rule for the mutually exclusive case C Rate of Return, Rankings of Projects, and Financial Constraints Conclusion . U Chapter 4 Financing Decisions, Investment Decisions, and the Costof Copitolj A 1 1 1 . Y11 1 I. II. III. III.A. III.B. III.C. III.C.l. I'II.C.2. III.C.3. III.C.-I. III.C.5. III.D. 7 I I I INTRODUCTION . 1 . A 1 MARKET SETTING V CAPITAL STRUCTURE VAND MARKET VALUES '1?-we-Period Market Equilibrium Model . 1 . Multiperiod Model A I 1 U I Two Partial Equilibrium Treatments I V I Two-period states of the world model The market value of a firm in the two-period risk class model The market value of a firm in a multiperiod risk class model The effects of financing decision on the firm’s bosndholdsers C V and shareholders 7 . I . I 1 U C U Summary" A V A 1 1 A 1 1 Market Imperfections’: The Effects of Tax-Laws I00 I00 104 I ‘I08 109 109 111 115 117 I18 119 121 122 I22 123 126 126 1271 128 130 131 137 131' 139 110 142 143 45 1 1147 1117 143 150 150 153 151 157' 160 16-I 1'67 170 no III.D.2. III.D.3 IV. I e IV.A. IV.B. V. V. A. V. B. VI .- I. II. FF? ow;-> III. III.A. i III.B. -IV. IV. A. IV.B. V. I. II. _ III. Chapter. 6 ""‘|-"-iili»-iii-iFfififififififi ow»my xiv _ Contents The tax deductibility of personal interest payments Some closing comments on marketeimperfeotioins t p THE FIRM'S OBJECTIVE FUNCTION: THE BIARKET VALUE ' RULE . ' ' " ' The Market Value.Rule: Derivation ~ A A The Market Value Ruleilmplementation E O THE COST OF CAPITAL AND THE RETURN ON A FIRM’S SHARES ~ A O - DiscountRates and the Cost of Capital e 2 A E The Expected Return on ComrnoniStock A SUMMARY AND CONCLUSIONS 2 Chapter 5 The Expected Utility Approach totho Problemiof Choice Under Uncertaintye I C A . 0 INTRODUCTION _ A D THE EXPECTED UTILITY MODEL: GENERAL AXIOMATIC TREATMENT _ A The Axiom System “ Derivation of the Expected Utility Rule A O S Some Properties of the Utility Functions. Implied by the Expected Utility Model THE TIMTELESS» EXPECTED UTILITY OF WEALTH MODEL A i C e Obtaining the Utility of Wealth Function from Axiom 4 -A Usual Typeset Utility of Wealth Functions: Risk Aversion, r Risk Preference, andlRiski Neutrality » I A EXPECTED UTILITY AND THE THEORY OF pi ~ O FINANCE P A A The Multiperiod" Expected Utility of Consumption Model A Utilities for Consumption Dollars from Utilities for Consumption Goods A A S CONCLUSION S 2 ~ Appendix: Statigtlcal Rovi'ew.\_ INTRODUCTION ' - EXPECTED VALUES OF WEIGHTED SUMS OF . RANDOM-VARIAB"L-ES A I A THE vanrancne on A IWEIGHTEDUSUM or C O RANDOIPI VARIABLES ' The 'l'w'o-Period Consumption-Investment Model ’ INTRODUCTION 2 ~ A i ‘A O The Mean-Standard Deviation Model: An Overviewi A Familiar Picture T O. 2 THE CONSUMER’S TASTES Portfolio Decisions Based on Edi) and edé) Marginal Expected Utilities and the Eflicient Set Theorem Properties of Indifference Curves in the Two-Parameter Model THE INVESTMENT OPPORTUNITY. SET: A _ TWO-ASSET CASE 174 -175 116 176 11s 181 181 184 187 E I89 189 190 192 194 196 198 199 200 203 203 206 207 209 209 209 2 1 1 215' 215 216 220 222 223 224 226 228 _ 4 7 7 _ Ii__ _' _ _ . __ __ _ _ __ __ _ __ "H" new ~_——— —J-I ~ — 1-I111-III — 4""-"7E _I N N T IV. IV.A. IV.B. "'IV.C. IV.D. 1v.o.1 IV.D.2 IV.D.3 IV.D.4. V. 1 v.A. V.B. $314 $39 I-5 V. C.2 V. C.3 V. C.4 VI. r r o I V Contents THE EFFICIENT SET AND. CONSUMER r EQUILIBRIUM WITH N ASSETS I P A Definitions and Elementary Expressions for the N-Asset Model c 8 Geometric Representation of Consumer Equilibrium ' 2 r Algebraic Representation ofConsumer Equilibrium r 8 Some Odds and Ends of the Mean—Standard Deviation Model _ The investment assets to be includedin portfolio models P I P A The effects of diversification: algebraic treatment R r r Quadratic utility functions P r R I r I Why normality? I 7 I i THE TWO-PARAMETER MODEL WITH ~ SYMMETRIC-STABLE DISTRIBUTIONS OF PORTFOLIO RETURNS r q Properties of Stable Distributions: A Brief Review -P r i Representation of the Consume-r’s Tastes: The Eflicient I S Set Theorem A R . e P The Opportunity Set withStable-Return. Distributions The market model P 8 Diversification. and the dist-riblution of portfolio return I q P I The efficient set P Consumer equilibrium r CONCLUSIONS I IV 233 234 236 243 250 250 253 256 259 261 261 265 267 267 269 271 273 274 Chapter 7 Risk, Return, and Market Equilibrium 276 r 1_ II. III. [II.iA. III. B. IV. IV.‘A.' IV.B. IV. C. INTRODUCTION V t THE MARKET SETTING S A RISK AND EXPECTED RETURN FROM THE _ O.VIEWPOINT or A CONSUMER " The Risks of Assets and Portfolios | I r A I The Relationship between Risk and Expected Return r i RISK AND EXPECTED RETURN FOR THE MARKET Homogeneous Expectations and Portfolio Opportunities I 8 The Role of a Riskless Asset ' I I H Interpretation .2 in A P I IV.C.1. Risk and expected return in the market portfolio ~ i V. I VI. VI.A. VI.B. VI. C. VI.D. ‘VII. VII. A. t VII.B. VII. C. - - hi’ WT’ ii” J i fir“ --_-L r 4 iiiiiiji A--—-‘ff’ AW i r ___-I I i _ . IV.C.2. P Homogeneous expectations and the riskless asset: a closer t Y P r Tlook .. . l _ . . EQUILIBRIUM EXPECTED RETURN AND THE - _. MARKET VALUE OF A FIRM - OPTIMAL PRODUCTION DECISIONS BY FIRMS " The Firm’s Objective Function Optimal Production Decisions: Single-Activity Firms q Optimal Production Decisions: Multiple-Activity Firms Optimal Production Decisions: Some Comments. r H ‘ I I r ALGEBRAIC TREATMENT OF CAPITAL RMARKET EQUILIBRIUM L q Consumer Equilibrium r R i Equilibrium for the Firm Market Equilibrium 276 277 279 279 281 286 287 288 290 290 292 295 299 299 301 we 307 308 309 -311 312 ____ I xvi Contents R VIII. I MARKET EQUILIBRIUM WITH SYMMETRIC- e A I I A STABLE RETURN DISTRIBUTIONS R P I 313 VIILA. The Market Model Return-Generating Process 313 VIJLB. Consumer Equilibrium and the Measurernentof Risk 314% R Risk and Return for the Market A .3172 1x. concwsrons _ 3187 Chapter 8 Multiperlod Models I V 321 A I. ‘INTRODUCTION A A A 321 II. MULTIPERIOD CONSUMPTION-INVESTMENT DECISIONS , 322 The Problem R R A‘ I 322 9-'Ili"'llI'4 2-*2-*2-* ow?- . The Wealth Allocation Model R I 323 . Implications: Bridging the Gap between Two-Period and R ' A Multiperiod Models R Y A . R 326 II.C.1. The utility of money function R R A A A 327 II.C.2. Two-period and multiperiode models: general ‘treatment 327 r P II. 0.3. A multiperiod setting for two-period two-parameter portfolio R R models r . V . r R 329 P II.C.4 Theorems and proofs 332 III. I EFFICIENT CAPITAL MARKETS A 335 lII.A. Expected Return or Fair Game Models R 336 III_.B. A The Submartingale Model A R e e 338 III.C. The Random Walk Model P 339 Index 343 E i I i if W L Z iéiiifi i ‘*7 _ L i é 7: 7_____ V 1‘ . _ . ‘ . i CERTAINTY MODELS v . ‘ . The theoryofflnance is concerned with how individuals and firms C allocate resources through time. ln particular, it seeks to explain how solutions to the problems faced in allocating resources through time are facilitated by the existence of capital markets (which T provide a means for individual economic agents to exchange T resources to be available at different points in time) and of firnis T (which, by their production-investment decisions, provide a. means for individuals to tra-nsformlcurrent resources‘-physically into - ' "' resources to be available in the future). T I r s A central or perhaps the central theme of this book is the role of a capital market that is perfec't,'in a sense to be defined in detail later, in allowing individuals and firms most efficiently to exchange resources to beavailable at different points in time. The procedure in this book is to work from simple models to more complicated ones. Thus, the first part of the book is concerned with ~al world in which there is no uncertainty about future events; the future consequences of actions taken now are assumed -to be perfectly predictable. The second part of the book then moves on to consider models of a world of uncertainty. C C x I The steps to be followed in presenting the theory of finance for a world of perfect certainty are as follows. First, we introduce -a model of the problem of resource allocation through time faced by an individual consumer as a special case of the general theory of choice. The two important elements of this model are ll) some ~ presumptions about the individual's tastes in ordering the obiects about which decisions must be made, and (2) a specification of C the opportunities that are available. _ - Chapter l considers only the opportunities available to the v individual when there is a capital market in which current I resources can -be sold in exchange for resources to be obtained in the future; and vice verso, resources to be obtained in the future can be sold now in exchange for current resources. Chapter 2 t expands the opportunity set toinclude the opportunities for transforming current resources into -future resources provided by the - -investment of current resources in physical production, which e production is -assumed. to be undertaken by firms. Finally, Chapter \ \\ 3 considers some of the problems of detail involved in implementing the decision rules developed in Chapters l and 2, especially the problems that arise in implementing the rules for optimal production-financing-investment decisions by firms. A MODEL OF THE ACCUMULATION AND ALLOCATION OF WEALTH BY INDIVIDUALS In this introductory chapter to Part I, we present one of the fundamental building blocks of the theory of finance, namely. the model of the accumulation and allocation of wealth over time by individuals under conditions of certainty and perfect capital markets. Once the model itself has been set forth in fairly general and abstract terms, we go on in subsequent chapters to con- sider its extensions to a variety of problems in finance. The extension of the model to the case of uncertainty is given in Part II. Because the wealth allocation model itself is merely a special case of the more general economic theory of choice under certainty, we begin by reviewing briefly some of the main concepts and features of this theory. I. THE ECONOMIC THEORY OF CHOICE LA. Opportunities and Preferences The economic theory of choice, like any other body of theory, aims at establishing empirical generalizations 3 4 Certainty Models about the class of phenomena under study. For the theory of choice, this means generalizations about the way in which choices change in response to changes in the circumstances surrounding the choice. The first step in constructing the theory is the simple but important one of classification. We divide the many separate elements bearing on any choice into two classes. One class is called the “opportunity set” or “con- straint set." As the name implies, it is the collection of possible choices available to the decision maker. Its content, depending on the context, may be determined by technological limitations, such as the impossibility of constructing a perpetual-motion machine, or by legal restrictions, such as those prohibiting selling oneself into slavery, or by market restrictions, such as the inability to buy a new automobile for $5 because of the absence of sellers at this price. The other main elements in the decision problem are the decision maker’s “tastes” or preferences. For an individual choosing on his own behalf, these tastes depend on personality, upbringing, education, and so forth. The theory does not propose to say precisely how these factors enter separately into the choice but takes the individual and his preferences as given from outside the problem and proceeds from there. For individuals acting as managers, and hence presumably on behalf of others, the question of precisely whose preferences we are talking about, that is, the agent’s or the principal’s or some combination, must also be faced. We eventually try to do so, but to keep the presentation uncluttered, we defer this issue to Chapter 2. For the remainder of the present chapter, we are concerned only with a single individual acting entirely on his own behalf. ' l.B. Representation of Preferences: The Utility Function and Indifference Curves To develop interesting generalizations about choice behavior, we need a convenient way of representing tastes and opportunities. One possibility is to tabulate the decision maker’s choices under laboratory conditions. We could present the subject with a series of bundles or boxes each containing some of the relevant objects of choice, carefully record the contents of each box, and then note which box he actually chose. By suitably varying the contents of the boxes, we could obtain an extensive picture of his likes and dislikes. Such a table would certainly contain a great. deal of information about the subject’s tastes, but it would be difficult to work with. Any patterns in it would be too hard to see. The question arises, therefore, whether some simpler and more compact way exists for representing or at least approxi- mating the data in the table. Such -a representation is indeed possible, provided that the subject’s preferences satisfy certain conditions. [Ch. 1, Sec. I] Accumulation and Allocation of Wealth 5 0 . l.B. I . The axioms of choice and the principle of . maximum utility c - i i These conditions collectively constitute the axioms of the economic theory of choice. The'word sa:cz'om is not to be taken here in its ~ old-fashioned sense of a' self-evident truth. The axioms are to _ be regarded rather as provisional assumptions, plausible enough, perhaps, as approximations, but whose ultimate justification comes, not from their own truth or plausibility, but from the predictive and descriptive power of the conclusionsito which they lead.‘ . ‘ I t D ~ s i D s In discussing the axioms, we let the letters :2, y, and z represent boxes of objects presented to the subject for choice. These objects, whatever their outward form,are referred to as “ commodities” to which, instead of names, we assign numbers 1, 2, . . ., n. Each box is completely specified by in- dicating the number of units q of each commodity that it contains; that is, the box :1: is represented by an n-tuple of the form (q1F=>l,q2‘=>,.i . .,q,,<="). _ 0 ' e Axiom I (Comparability). For every pair» of boxes it: and y the decision maker can tell ‘us either (A1) that he prefers 2: to y, or (2) that he prefers y to x, or (3) that he is indifferent to having :1: or y. . s The function of the axiom of comparabilityis to rule out cases in which the decision maker refuses or is unable to make a choice, because, for example, he may regard the objects ofchoice as essentially different and hence incomparable. it I Axiorn 2 (Transitivity). Whenever the decision maker prefers :2: to y and y to z, he alsoprefers as to z. Likewise, if the is indifferent to having :c or y and to having y or z, he is also indifferent to having as or z. Basically, the subject behaves consistently in making his choices. D t A If the decision maker’s tastes conform to these axioms, the following important proposition holds: i s The subject's choice behavior may be characterized by saying that he behaves as if he were maximizing the value of a “utility function.” This function assigns a numerical value or “utility index” to each box x, y, . . ., 1 The methodological principle that theories are to be judged by the empirical validity of their consequences rather than by that of their assumptions considered separately has come to be called “positivism.” The classical statement of the positivist position in economics is that of Milton Friedman, “The Methodology of Positive Economics,” in Essays in Positive Chicago: University of Chicago Press, 1956. .4 6 Certainty Models _ I- with the quantities, of each commodity in the box as the arguments of the function. " t 2 2 " s 2 To illustrate, suppose that we somehow knew that a utility function for a particular subjectytook the form t s 2 r 2 T T U 2* F.(q1>q==) = Q1’ + Q2’, T 2 ' Iv where ‘the letter U is the utility index and q; and gs are the quantities respectively of commodities 1 and 2 in any box. Suppose that we also knew that box :2: held 6 units of commodity 1 and 4 units of 2 and that box y held 8 units of commodity 1 and 2 units of 2. Which one does he prefer? Substituting the contents of box :1: into the utility function, we obtain an index of 6’ + 4’ =52; for the second box we -have '8’ +i 2’ === 68. The assignmentof a higher utility index to the second box is equivalent to saying that the subject prefers box y to box j u s o Note that in the statementof the proposition we say that the subject behaves as if he were maximizing the value of a utility function. The economic theory of choice does not assert that the subject performs these calculations on his utility function in making his choices or even that the subject knows that the has at utility‘ function. It says merely that if his preferences are complete and consistent, the utility function provides us, as outside observers, with a way of representing his choices. In what follows we sometimes gloss over this distinction and speak, say, of a decision maker’s increasing or decreasing his utility by_some action. But this should always be considered a stylistic device and must not be taken literally. 2 The reasonableness of the proposition in relation to the axioms is easily seen. The axioms of comparability and -transitivity permit us to rank. all possible boxes in increasing order of preferability. This rank ordering, in turn, can always be expressed by some device that assigns numbers to each box so that higher numbers go with higher ranks. The utility function is just such adevice.’ 2 1 t s p pp It should also be clear that any specific representation of ta subject's utility function, such as in this illustration,is not unique. Ifone such function exists, so domany others, because what matters is whether the number assigned to a box ishigher or lower than that of another box. How muchlhigher or lower is of no consequence so long as the ranking is pre- served. Thus multiplying by pa positive constant or performing any other monotone-increasing transformation of a utility function also yields an ' The more advanced mathematical treatments in the economics literature are also concerned with the continuity of the utility function and the additional requirements necessary to assure it. Readers interested in .a complete and rigorous‘ discussion of these and related issues can find it in the classic treatment of the subject by Gerard Debreu, The Theory of Value. New York: Wiley, 1959. t t 2 H-uu¢¢=-Pa-Iun|n|:un:a\ié.vj»='-I¢..Ini'»-ir-1'-.1-0d~- - J I I‘ - l j i W. L w i WW 7 7 -Q [Ch. 1, Sec.I] p Accumulation and Allocation of Wealth 7 . . equivalent and perfectly consistent representation of the subject’s preferences.‘ t ' A " I s _ - I I Given the utility function to represent the general structure of preferences, the next step in the search for interesting, and testable, generalizations about choice behavior is to in-voke certain additional assumptions with respect to the form and properties of this function. Some of these assump- tions are mainly for mathematical convenience and are noted here, in footnotes, only‘fo1" the: sake of completeness.‘ Others, however, are more substantive and merit further discussion. In introducing and explaining them, it is helpful first to show how choices and utility functions can be represented in geometrical or graphical form. A I I.B.2. Indifference curves and the geometrical I j . representation of preference: ~ I In representing -the utility function U = U(q1,q2,. . .,q,,) graphically, we are, of course, limited to at most three dimensions. Because we must reserve one dimension for U, the value of the utility index, we are thus restricted to only two commodities in making up the boxes between which the subject, is to choose. Although this collapsing down from n distinct commodities to only two may seem a very drastic simplification, the loss of generality is actually quite small. For establishing the kinds of proposi- tions that are our main concern, the two-commodity case is almost always adequate. i r A One way of representing at utility function in three dimensions on a two-dimensional page is with figures of the kind used in solid geometry. Because it is difficult, and expensive, to draw the figures in correct perspec- tive, however, we use a projective method that is considerably simpler but requires, at first, a somewhat greater effort. in interpretation on the part of the reader. To see how it works, imagine a three-dimensional solid repre- sentation of the utility function U = U (q1,q2), with q; and qr,» on two axes in a horizontal plane and U measured on a third axis vertical, and perpen- dicular, to the other two. Imagine now that we pick some particular value ‘i Utility functions with this property are said to be “ordinal” functions, in contrast to “cardinal” utility functions, which are unique up to a _“linear" transformation and which hence can convey some indication of how much higher one bundle is on the sub- jeet’s scale of preferences than another. We have no need for -the stronger, cardinal functions until we reach Part II and introduce uncertainty. A ‘ In particular, we assume that the arguments of the utility function, q,-, can be repre- sented as continuous variables, that is, that all commodities are infinitely divisible, that the utility function itself is a continuous function of its arguments, and that it has continuous derivatives of whatever order we happen to need. I i Another assumption that we have already made implicitly by our distinction between tastes and opportunities is that the two can in fact be separated in the sense that each can be defined independently of the other. 8 Certainty Models a of U, say, U = 7, and pass a knife horizontally through the surface at -this'level. The cut being horizontal, the outline is necessarily always a two-dimensional figure. Now project, that is, drop, this horizontal outline down onto the horizontal q,q2~ plane. If we repeated this process for many different values of U, we should obtain a whole set of such two-dimensional figures; and by labeling each such figure or contour with its value ‘of U, we could mentally reconstruct the entire three-dimensional figure. An example of a utility functionrepresented by such contoursis shown in Figure‘ 1.1a. From the numbering of the contours it can be seen that the utility surface has the form of an irregular hill rising to a summit at the point labeled U = 40. The indentations in the contours on the southwest slope imply a trough or draw at the lower levels running in t-he northeast- southwest direction. The circular area labeled Uw 20 lying between the contours U =-= 10 and U = 15 implies a small isolated knollon the northeast face. And so on for the various other curls and wrinkles that have been drawn into the map. - t s A t In addition -to the utility surface itself, the figure also shows the repre- sentation of some boxes offered to the subjectifor choice and the pattemof his preferences among them. These boxes are indicated by the points labeled w, as, y, and z---the box w containing q1<"’l units of commodity 1 and _q2‘"° of commodity 2, and similarly” for the others. As drawn, the points w, zr, and y lie on the same contour, U = 10. That these boxesallhaveithe same utility index is the same as saying that the subject wascompletely . qz f./*5U . _ U==‘lO 'I . I. - U==15 U"20 t I u-2o \0 t t p uao I71 qstwi -—-— --- ‘” ”'4°: I Q2“) ...........................-_ .......... _.x Y qgll/l “ * * “ * -""""|" "'- _" kZ . qgizl * 1i“* """"""'l' ‘H l _ l | | I 0 v Q v q,(w) q1(x_l_. q,(y) Q1 Q1 (Z) . Figure 1.1a Iso-utility Contours 4- an 1-:-<...-!\uvb¢n£<I..¢zvii‘-ill‘ s I‘ 7ililI|‘~“"'~ -n [Ch. A 1, $60; Il . Accumulation and Allocation of .Wealth 9 indifferent_ when presented with a choice between them-j---hence the term "‘ indifference curves,” which is customarily used in the theory of choice to refer to an iso-utility contour line. Box z, however, lies onan indifference curve with a lower utility index, which is to say that it would be rejected by the subject in any choice involving w, :2, or y. . A A A ~ t.B.3. The axiom: of nonsotiotion and convexity a j ‘ A v U A We turn now to the final two axioms or assumptionsabout choice behavior. ' . 1- ' ' ' - " ' ' . '1! Axiom 8 (Nonsatiation of Wants). The subject would always prefer, or at worst would be indifferent, to have more of any commodityif at the same time he did not have to take less of any other commodity. A This assumption serves to rule out any positively sloped segments of indifference curves, such as the segments mu or kl on curve U -_-,= 5 in Figure 1.1a. For where such segments exist, there are points, suchas j, having the same amount of qi as A-box ubut less Q2 and yet preferred to u, as shown by the higher utility index of its indifference curve. A map of a utility surface that does conform to the axiom—-a hill with no summit or northeast face-1 is shown in Figure 1.1b. v A 1 ‘ Q3. W . \ ._ .\. . J "~ I u-so t A u-20 At y t s u-1o _ UIS _ . 0 Q1 I Figure 1.1bA Convexity and Concavity of Indifierence Curves s v iAs1 I-r '0 — - fir, _ _ + ‘I0 Certainty Models , j A A:z:z'om4 .(Conve:mIty). If a: and 3} are two boxes such that U(:r) = U (Ay) and if z is a combination of boxes :0: and y of the form z-= era: + (1 -- oz) y, 0 '5 oz 5 1, then U (z) Z U (z) =s U (y). In words, if we have two boxes between which the subject is inditferent,and we construct a new box whose contents, commodity by commodity, are a weighted average of those of :1: and y, where the weights are positive and sum tounity, the subject never chooses either 2: or y in preference to the combination box 2.5 t Examples of (local) regions of the utility surface that do and do not meet this convexity assumption are shown in Figure 1.16. If we draw a straight line connecting the points w and 2: lying on the same indifference curve U == 10, this straight line representing all the combinationboxes along the line ax + (1 -t a)'w for 0 3 la. 3 1, we see that a combination box, such as t, would not be preferable to as or w. It lies on a lower indifference curve with index, say, U A=- 5, as we have happened to draw it. In this region, then, the indifference curve does not meet the convexity assump- tion. By contrast, the intermediate points on a straight line between :1: and y all lie on higher indifference curves, and in this region the indifference curve for U = 10 is convex. i j t I A p * To see in more concrete terms what this assumptionof convexity implies about choiceibehavior, imagine that the boxes contain two commodities, such as books and theater tickets. Suppose that we start with a box con- taining 20. books and 3 ticketsand then withdraw 1 ticket. Books and tickets both being desirable commodities for the subject, we should have to add, say, 2 additional books to the box to keep him on the same indifference curve--the additional books, in effect, compensating for the tieket with- drawn. If now we subtract another ticket from theft? remaining, the con- vexity assumption implies that we should have toadd at least another A2 books and possibly more, to keep the utility index of the box unchanged. When the removal of the second item requires more compensation than that of the first, the indifference curve is said to be “strictly convex.” When the compensation required is the same on successive removals, the curve is linear, which is the limiting extreme case of convexity.“ A In most subsequent applications of the theory to problems of finance we work with indifference maps of the kind shown in Figure 1.1c. Such maps have a number of important features and properties that will be invoked -i As anillustration of the contruction of z from :1: and y,'suppose that z has 8 units of commodity 1 and 6 units of commodity 2 and y has 6 units of 1 and 10 units of2; then by taking a === i, z would be ea box with {(8) + i(6) == 6} units of commodity 1 and {-(6) + §( 10) = 9 units of commodity 2. ' For those readers who find it difficult to accept the convexity axiom, even provisionally, it may be well to point out that we use it here mainly to simplify the graphical presenta- tions and to avoid the complication of multiple or “corner” solutions. All the really essential conclusions can be derived from the first three axioms only. , s A- 4 F n-mu-nilblkmn1'-M‘-I\=-In-Al-.lu|$A-n|-@<--=-‘-'-'-"A4". f 1 ,_ 7 —— — it " ’ [Ch. 1, Sec. I] p A t 1 Accumulation and -Allocation of Wealth II U'3O . UP 20 u-1o 0 . qr V I Figure 1.1c A Standard Indifference Map I , repeatedly inideriving propositions about choice behavior. To summarize, the slope of an indifference curve, often called the marginal rate of sub- stitution of qifor Q1, is always negative. The marginal rate of substitution rises inalgebraic or falls in absolute-value as we move from left to right. Curves take on higher utility indexes as we move upward on the __map. And there are never any crossings of indifference curves. To test their understanding, readers for whom this apparatus isunfamiliar may find it useful to imagine the contrary of each of these properties and thenshow that such cases would be inconsistent-with one or more of the basic axioms of comparability, stransitivity, nonsatiation, or convexity.’ . | - ' r T Because these concepts are used repeatedly throughout the book, it is well to note more formally here the distinction between convex “sets” and convex and concave “functions” and “curves.” A set of points is said to be convex if for any two points :r and y in the set, the point z == ax + (1 -- a)y, 0 5 at 5 1, is alsoin the set. Geometric- ally, a set is convex if all points on a straight line between any points in the set are also intheset. A A I s “C. if On the other hand, a function 1 is semi; if for any two points == and ,, in the domain off A I I 2 A f(¢=$'+ (1 - ¢=)y) S <1f(=v) + (1 "-1 ¢=)f(U), 0 SR5 1» and the function is concave if A s '2 1 f(¢=-?¢ + (1 A“ ell!) 2 ¢f(==) + (1 -A ¢)f(t/)5 CA0 S <1 S 1-1 Geometrically, a function or curve is convex if a line between any two points on the _ ____* ._ /77 mm input "' ' 7 _l 2 ACertainty Models y LC. The Opportunity Set i A y ‘ i g A . As noted earlier, the class of choice problems of most concern in economics as opposed to, say, psychology or aesthetics is that in which the decision maker’s choices are limited by external restrictions. A graphical repre-o sentation of such restrictions for a two-commodity case is shown in Figure 1.2. As indicated by the shading, the feasible boxes or combinations of commodities1 and 2 are restricted to those lying within, or on the boundaries of, the irregular figure outlined by OAB. Any box notlying within this opportunity'set is excluded from consideration, _no matter how desirable it might be for the subject to have it. is A g A A A A t i A ‘72 A. x \ 1- _ D “\ W. 1. t A o D E _ ' BAA Q1 A ‘ . Figure 1.2 Opportunity Sets o ' - The precise content or shape of the opportunity set depends, of course, on the problem at hand. There is, however, one general restriction thatis a feature of virtually all those to be considered here, namely, theassumption that the set isconvex; thatis, if :2: and y are any two boxes in the oppor- tunity set, all boxes z of the form z == ax + (1 --Y oz) y, 0, < a < 1, are also in the set. Thus, in terms of Figure 1.2,___the property of convexity of the set 1 _ . function lies everywhere on or above the function; concavity implies that a line between any two points is everywhere on or below the function. s Hence, when added to the nonsatiation axiom, the convexity axiom presented in the text is an assumption that commodity indifference curves are convex. But note care- fully that it is not an assumption that the utility function U is convex. This function is still only ordinal; that is, the only rat:-iction on the numbers assigned to successive indifierence curves are that they are monotone-increasing, so that the curvature of the function across indifierence curves is arbitrary. o s A s nu-.,,__- . :1-ill!-u|\su';.n|‘-It-inlk-¢nIi\~i-ue-Es‘-\- r ‘ . 7____ ___ ____ _ #7 7 r . ' 1-|—| I Ir‘ -1. 0 A[Ch. 1, Sec. I] s A g Accumulation and Allocation of Wealth ‘I3 rules out segments like that between wand :c,*:whereA there are boxes like .t lying on a line joining 3: and w, but not partloffthe allowable choices.‘ Note that the definitions of convexity includes setswith straight-line outerboundaries, suchas the set bounded by ODE in Figure 1.2,as the limiting special case. In general, A convex opportunity sets with curved boundaries usually arise when the constraints areimposed by~‘o‘ technology” and the curvature represents “diminishing returns” inthe physical possi- bilities of transforrningvvcommodity 1 into commodity 2. The straight-iline cases usually represent market exchange opportunities, thefconstant slope implying that the commodities can be exchanged for each other at given, fixed prices. In some cases,we consider problems with both types of,con- straints operatingsimultaneously. A A t A A A - A i LD. Choice Subject to Constraints ' _ A A V To obtain a representation of a choice in the presence of constraints, there remains now only to bring together the two pieces of the problem, tastes and opportunities, that we havcso farjconsidered separately. A graphical illustration for a two-commodit y case is shown in Figure 1.3. The Aqz _ -. . 3 _ . _ L/"40 f u-so C A U"2O . s U=l0 ' .A ~ u-s O .. - A_ q, Figure 1.3 Representation of a Constrained Choice i g figure shows a subject whose preferences meet the four axioms of choice and can therefore be represented by a utility map with convex indifference ' As with the assumption of convexity of indifference curves (see Afootnote .6 above) , the assumption of convex opportunity sets is much stronger than necessary for most of the importantconclusions and is used throughout mainly to simplify the presentation. 14 Certainty Models , curves increasing intutility indexas one moves up and to the right. The possible boxes available to him lie withinthe figure OBA. Which one of the immense number of possibilities representsthe one that he would actually choose? A A A We can, first of all, eliminateimmediately a large number of boxes as possibilities. Inparticular, no box such as :1: in the interior of the opportunity set could possibly be hiseventual choice, because the boxes flying between a and b on the boundary all have at least as much of one of the two com? modities as :1: and; more of the other. By virtue of the nonsatiation axiom, such dominated interior boxes are ruled out, and only the boxes along the right boundary need be considered. It is for this reason, thatthe right boundary of the opportunity set is often referred to as the “efficiency frontier”or“efiicientset”.° A s A t so A i . Having narrowed the range of possibilitiesto the efiicient set, we need only sweep along it in some systematic manner. If,Afor example, we work up from the lower right-hand corner, steadily decreasing the amount of q, in the box and increasing the amountof qg, we produce a series of choices of the forma versus b, then b versus c, and so on. As drawn, the subject prefers b to 0;, indicated by the fact that b lies on a higher indifference curve, and prefers c to b. Eventually, by repeating thislprocess, we find a point that is preferred to its neighbors on either side. In the figure as drawn, this is the point d, at which an indifference curve is tangent tothe efficiency frontier. All other boxes, whether on the efficiency frontier or within it, lie on lower indifference curves. A This representation of a constrained choice completes our review of the fundamentals of the theory of choice. The importance of this theory for our purposes lies in its simplicity and in the fact that it is both context- and subject-free. No matter what kinds of commodities that we put in the boxes and no matter who the subject or what the details of his personal preferences, his choice can always be represented by a point on the efficiency frontier, provided only thathis tastes meet Axioms 1 to 3. If, in addition, his tastes meet Axiom 4 and if the opportunity set is convex, we can narrow the possible choices along the frontier down to the single one where an indifference curve is tangent to the opportunity set. The four axioms plus the convexity of the opportunity set guarantee both that there must be such a point and that there is no more than one." A ” A ° Because the right boundary of the opportunity set is assumed to be negatively sloping, the assumption that the set is convex implies that this bouAndary must be a concave OUPV8. A _ 1° In what follows, we frequently refer to these tangency points as “optimal solutions” or “equilibrium points.” As in the case of references to utility, however, such terms are normally to be taken in an as if sense and not as implying that the subject is consciously optimizing or equilibrating. A t A 1 wtx.A-'<I'-n-vd\:1.q.;.|.-‘ta,-,,A..n.|,5A,u\-1-»-uw if at1- [Ch. 1, Sec. I] AccumulatAionAandA Allocation of Wealth A15 The task of much of the rest of the book is essentially a filling in of these “empty boxes”. and a showing of how the very general and abstract theory of choice can be specialized to obtain meaningful generalizations about financial choices. But before turning to this task, we present for the record a very brief account of how a choice subject to constraints can be repre- sented in mathematical form. A t _ \ "'I.E. The Solution in Mothemoticol Form“ I T A S A A In partic1.1lar,for an An-commodity-case, the subject’s choice can be expressed as the solution to the problem" . A A A to A A . max U (q1,q~.i.- - -,q..) Q1»?!---"Q8 _ ' subject to a constraint or opportunity set that maybe written in general, implicit formas _ A A A - ' A A I . A , T(q1.qi.--sq») = 0. A A andwhich represents the pointsA (q1,q-3,. . .,q,,) that lie along the efficiency frontier. A s . A A With appropriate assumptions about the continuity of the derivatlives of the U and T functions, we can use the methods of the differential calculus to study properties of the solution to this problem. Specifically, form the lagrangian function A A A A A L “A: U(q1rAq9:_' ' '_iqfl)A """ )\T(q1rq2v- - -sq"): _ and differentiate partially with respect to A and each of theAq,-A. -Setting these derivatives equal to zero yields the following n + 1 equations as first-order or necessary conditions for a maximum?‘ s s t'i~— >\Ti=0 L’; - ATQ = 0 t : ~ r(1-2) U1. - ART;=== 0 T(q1.qA=,- - sq») = 0. " Starred sections here and throughout mark the places in the exposition where some knowledge of the calculus is required. Such sections may safely be skipped by readers lacking the necessary mathematical background, without fear of losing the ‘main thread of the argument. Such readers may nevertheless find it helpful to skim through the sections, because. the discussion surrounding the mathematical rmults may provide additional insights. A A y I A " The expression A - - A max U (qlsqla ° ' .° sq!) A 11.08. ' "til . I is read “choose valum of qr to q,, that maximize the utility index.” A " In general, in the mathematical treatments of various maximisation and 1-6 Certainty Models _ o where Uir and Ti are the partial derivatives of U t and A T with respect t0q¢.A A .A I. A AA To see the relation between this representation and the graphical repre- sentation of the two-commodity case, observe that between any pair of commodities we can eliminate A from the relevant equations of (1.2) to obtain, for example, A A g A A A A A A ._ A T; T; . or U;-T‘. (1.3) The definition of an indifference curve is equivalent to the solution set! satisfying the condition on the differential A A A . A A t dU = U{dq1 + Uidqg + + trying, = 0 ' A A for some specified level of U; Holdingq; toqA,, constant, that is, setting dqa to dgn = 0, this condition implies that A A A A t A m=_fi A t dql U; o A Or, in words, the slope of an indifference curve at any point is equal to (minus) the ratio of the first partial derivatives evaluated at this point. Similarly, for the opportunity set we have t ti A A ~dq1 .TiA Thus the condition (1.3) is equivalent to the statement that at a maximum of A U, the slope of an indifference curve is the same as that of the oppor- tunity set, which is,of course, the familiar tangency condition, in the geometric analysis. t g t A A A ll. THE APP_l.l_CATlO,N OF THE THEORY OF CHOICEATO THE ALLOCATION A OF ‘FINANCIAL RESOURCES OVER TIME s AA A A A ILA. The Two-Period Cose i As noted earlier, the basicapplication of the theory of choice under certainty to the field of finance is the problem of the allocation offinancial resourcessby individuals over time. The rest of the present chapter is tion problems in this book, we do not present sufficient or second-order conditions, be- cause they usually contribute little in the way of economic insight, which, after all, .is the primary concern. Moreover, such “mathematical license” is often characteristic of our mathematical treatments in other ways as well. For example, we ignore,for the most part, the additional complications introduced by nonnegativity conditions A on some or all of the variables. A A A A A A j i if __ __ _m 7 — A [Oh. 1, Sec. II] Accumulation and Allocation of Wealth 17 devoted to the development of this application in its most general and abstract form. A number -of extensions are then taken up in subsequent chapters. ' . e v i 'H~.A.I. The objects of choice: standards of living at h s r different points in time ‘ _ _ i T t In studying the allocation of financial resourcesover time, the objects of choice can -be defined in several ways, dependingon how much detail is to be .shown and precisely how the passage of time is to be represented. Taking the question of time first, we assume throughout that the passage of time is not continuous but occurs in discrete jumps one “unit of time” apart._ Decisions are assumed to be taken andpayments to be madeonly at the start of these discrete time periods, For generating the kind of qualitative generalizations about behaviorthat are our concern, no precise statement need be made about the length of a unit time period in terms of calendar time.“ _ T - s T T * As for the objects of choice, we could continue to work in the standard economic framework. of individual commodities, giving each commodity a time index to indicate the period which it was being presentedto the subject. The utility function would then be of the form V(qn,qn,. . ., q,,1,q1-;,q2-2, . . .,q,.g,. . .,q1,,q-_»,,, . . . ,q,.,, . . . . But? although formally unobjec- tionable, and really no great complication mathematically, such an ap- proach makes it difficultt to focus sharply on the efiects of time per se and on the purely financial aspects of the allocation decision. To highlight these aspects of the problem, therefore, we make the following simplifications in the utility function. t ' i s First, because the function in its most general form includes in principle commodities that represent income-earning activities, for example, hours of labor, we assume that these can be separated out and that a utility function can be defined over the set of consumer goods ;sthat is, we assume that preferences between boxes of consumer goods depend only on the contents of the boxes and not on the amount or nature of the income-earning ac- tivities that might accompany them. This creates some problems with respect to what might be called “leisure goods,” but they are minor for our purposes and are neglected. We also neglect the details of the decision with respect to occupational choice and hours of work. The subject's occupation and earnings are taken as given, determined somehowoutside the model. “ Despite assertions to the contrary occasionally encountered. in the literature, there is little basis other than taste or convenience in choosing between a discrete-time formula- tion of the kind to be followed here and a continuous-time formulation of the kind often found in economists’ expositions of capital theory. Substantive results that can be developed under the one convention can always be translated into the other under conditions of certainty (see, in this connection, footnote 27, below.) T T s ~ 18 Certainty Models i .¢z e. ' U‘ ‘I-O U"5 - 0 _ . C1 ' s Figure 1.4 Indifference Curves for Patterns of Total Consumption Second, we replace the individual consumer goods in any period tby a single composite commodity ct, called the subject’s “total consumption” or “standard of living” in period t. This composite commodity is defined as ___ as 1: 0 0 1: . . 0: -- Puqu + ;P2:q2c + ' ‘ ' + Pmqvm where the p.-, are the prices of the commodities“ in period it, assumed to be known and fixed in our perfect certainty framework, and the eqii are the amounts of the commodities that would be purchased by the consumer, given the prices and a pattern of total resources over all periods that would permit him to spend c, in period t. Operationally, in terms of the earlier imaginary choice experiments, we now picture ourselves T confronting the subject with different patterns of standards of living, such as, say, $10,000 for period 1 and $8000 thereafter versus $6500 for period 1, $6000 for period 2, and $9000 thereafter. The subject decides how he would allocate these amounts among commodities in each period and then announces his 1‘ Although we speak of prices and commodities, it should be remembered that the q,- represent not stocks of commodities‘ but services rendered by the stocks. In the case of durable goods, for example, the pa are to be interpreted not as the purchase prices but as the one-period rentals for the equipment. I 0 Note also that the pr, are to be regarded as being measured in terms of some standard commodity or numéraire whose price per unit is arbitrarily set equal to 1 in period 1 and every period thereafter. We shall often refer to this standard commodity as “money” and hence speak of dollars of income or consumption. It is important to keep in mind, however, that this is again only a stylistic device, and the concept of money in our sense should not be equated with money in its more familiar sense, in which it servos as a medium of exchange and a store of value as well as the unit of account. p rt: _ W i [Ch. 1, Sec. II] _ A p Accumulation and Allocation of Wealth I9 preference ordering for time sequences of standards of living. This prefer- ence orderingover time sequences of standards of living is summarized in a utility ftmction as U (c1,c2,. . ..,c,,. ). s t s t a * It is shownin the next» (starred) section that this function, stated in terms of total consumptions optimally allocated, satisfies the axioms assumed for the underlyingsutility function defined gin terms of the com- modities q,-,. This implies that the indifierencei map for such. af utility function for a two-period case must have the general properties shown in Figure 1.4. In particular, the nonsatiation axiom implies that any given indifference curve for total consumption must be negatively sloped and that utility must increase as we move upward and to the right ontolhigher indifference curves. The axiom of transitivity implies that indifierence curves cannot cross; the axioms of nonsatiation and convexity together imply convex indifference curves. A e . , A s *l[A.2. The properties of Ulc1,c¢,. .-,c;,. . .) A A _ A a Forsimplicity, the analysis is carried out for a two-period, two-com- modity case. But the method is perfectly general and can readily be ex- tended to the multiperiod, multicommodity case. p - s If U is a utility function for dollars of consumption and V a utility function for consumption.commodities, the relationship betweenethe two functions can be expressed as i I ' ~ i . s .U(c1,c.;) -..= max V(q11,qs1,q12,q-22) _' (1.40) - - 911.921.412.922 ' M . - subject to e A p 61 =_ P111111 + 1121921 H 8-Rd 6:: = Prsqis + Pesqe-.2-i (1.45) We first show that if the convexityaxiom applies to V, it also applies to U. Let .(qi'i,q£'i,qfi,q§'§) be the optimal quantities of commodities consumed when the dollar levels of is consumption are (-01,62), and let (qt,qt.qs,qs) be optimal for (¢,,a,). Assume in addition that . i ¢ s s V(q?i,qa,q1‘¢‘,q:'£)=- V(qi'i,<i="i.,q1"é,<i="£) so that is s U(c1,c2) = U.(c':1,¢¢). Thus (c1,c¢) and (é1,c'==.) are onthe same indifference curve. A A For0 5 or $1,let j (ea) = (ac1+ (1 —- antes + <1 - ea.) ' (§11:é21;§12;Q22) ” + """ I-!)q.f.1; - - -1 ' - '1? “Q3: -"' S0 that - 51 = Pufiu + Pztfizr, A 32 = P12§'12. + P22§22, A and (§u,<j,1,§m,§2¢) is a feasible consumption pattern for i(i:1,5z) . l l L E|. I 20 Certainty Models From the axiom of convexity, the utility function for consumption commodities satisfies s s s i AV(_&11:é21'sé12y&22) + "" Q) Or, equivalently, A A s A t s - ~ . 0 V(§11,§21,§12,§22) Z ¢1U(G1,62)A+ (1 ""' 11) U(¢1,¢-.2)--f Althoughfeasible,‘ the allocation of ,(2‘:;,E:¢) implied b}’,(iAl11,§e1,§m,§'22)A is not necessarily optimal. The decision maker, if given ti; and is directly, might choose some other allocation among commodities in preference to thatofei 0 it (cqfi + <1-- amt, . . .. -as +<1 -— ewe. s Because the more constrained choice can never be preferable to the less constrained one, we thus must have A 0 0 i ' A 1 U(fAA51,¢AA3‘2) Z V(§11:§e1:§12,§:2c)-1 A . Hence U(C1,Cg) 2 C\!(J'(C1,Cg) -' O!) U(é1,0g), 0 S O! s 1, ' which implies that the convexity axiom also applies to U(c1,c§) . p To show that the nonsatiation axiom also applies to U(c;,c2) , note that if the dollars available for consumption in either, period are increased, consumption of at least one commodity can be increased without reducing consumption of any other commodity in either period. Thus, the non- satiation of wants assumed in deriving the commodity utility function V must carry over directly to the utility function U. As in the case of com- modity indifference curves, the nonsatiation and convexity axioms together imply convexity of the indifference curves. s a It is also clear that the axiom of comparability carries lover directly from V to U. Establishing that the preference ordering implied by U (c1,c-3) satisfies the transitivity axiom is left as an exercise for the reader. I Finally, it is well to note that the function U (01,02)., like the function V(q11,q21,qn,q=,-1), provides only a rank orderingofconsumption boxes; that is, it tells us onlyswhether one box is preferred to another, so that U and V are ordinal utility functions, as distinct from cardinal functions that would also tell us unambiguously by how much one box is preferred to another. s A t A t a A i v A |l.A.3. Opportunities: resources and capitol markets i . . . The resources that an individual can draw on for his consumption in any period are of several kinds. Most households, for example, carry over stocks of durable consumer goods from previous periods, so that they may either consume the services of these stocks directly or may rent or sell the goods and consume the proceeds. For simplicity, however, we defer all [Ch. 1, Sec. II] d Accumulation and Allocation of Wealth 21 consideration of durable goods until Chapter 2, after we have first sketched the main features of the wealth allocatAionmodel.AFor concreteness, at this stage, the reader may perhaps find it helpful to think of the household as renting its housing, automobile, television set, and any other durables from specialized rental firms. s s - s A e A A 0‘ Also generally available to individuals to support their current con- sumption are, of course, any wages, salaries, or other similar payments that they receive as compensation for current labor services provided. We denote such payments during any period t as y, and refer to them, somewhat loosely, as the individual’s“income.”j In addition to his current income, the individual typically can look forward to further income in future years. Future earnings obviously cannot be directly consumed today, but they may still be able to support current consumption, provided that the individual can arrange to transfer them to someone else in exchange for resources to be made available to him immediately. A . e I_n what follows, we assume that such exchanges can in fact be made and that they take place in a “capital market.” The term market, of course, is to be taken, not inthe narroAw sense of a physical place wherebuyers and sellersgather, although some real-world capital markets have this property as well, but rather in its broader economic sense of the whole collection of legal, moral, and physical arrangements that make it possibleA to effect exchanges of current and future incomes. p ' , , The precise form that the household’s opportunity set takes in the presence of ta capital market depends on the additional specifying char- acteristics that we choose to attribute to the market. An___extreme, but particularly fruitful, special set of attributes are those which constitute a “perfect capital market.” Insuch a market; we assume the following: s .1. All traders have equal and cost-less access -to information about the ruling prices and all other relevant properties of the securities traded. 2. Buyers and sellers, or issuers, of securities take the prices of securities as given; that is, they do, and can justifiably, act asif their activities in the market had no detectable effect on the ruling prices. A - 3. There are no brokerage fees,transfer taxes, or other transaction costs < incurred when securities are bought, sold, or issued." A A t Needless to say, no such marketexists in the real world, nor could it. Rather, what we have here is an idealization of the same kind and function as that of a perfect gas or a perfect vacuum in the physical sciences. Such " Although perhaps not strictly an attribute of the niarket proper, we also assume that there are no income taxes on the earning from securities; or, if thereare, that there are at least no income tax differentials between income in the form of capital gains and dividends or interest. i A 22 Certainty Models A p c c idealizations permit us to focus more sharply on a limited number of aspectsof the problem and usually greatly facilitate both the derivation and statement of the sought-for empirical generalizations. In the nature of the case, however, the generalizations so obtained can never be anything more than approximations to the real phenomena that they are supposedto represent. The question is whether, considered as approximations, they are close enough; and this, of Acourse, is a question that can only be answered empirically and in the light of the specific uses towhich the approximations are Plit." A s AA I A- .|l.A.4. The opportunity set under perfect capitol i markets . - An immediate implication of a perfect market, _and one of the main reasons forusing this concept, is that at any one time only one price may rule in the market. For if two different prices ruled simultaneously, no one whose preferences obey the nonsatiation axiom would be willing to sell at the lower of the two prices or buy at the higher. Only whena common, single price had been restored could transactions take place." p A i s In the capital markets the commodities currently being bought and sold are sums of money to_be delivered at various future points in time. We regard the delivery contracts for each such future point as a separate (perfect) market and represent the single, current or spot price atthe beginning of the 1-th period for delivery of $1 at the beginning of the tth period as apt; that is, i,.p, is the amount of money.that must be paid at period -r for $1 to be obtained, for certain, at t. The advantages of the double subscript notation become clear in later extensions of the analysis." 1 Given these market prices or rates of exchange between future and current resources, what can be said about the form of the opportunity set confronting the decision maker? Let us consider first a two-period case, and ask how much the decision maker can consume in the second of the 1' To make use again of the analogy from physics, laws of motion derived under at per- fect-vacuum assumption may be closAe enough appproximations for many engineering purposes when dealing with heavy objects, but not for some light objects when the neglect of air resistance could lead to a breakdown of the mechanism. i 1' An equivalent, alternative proof often useful in showing the implications of the perfect capital market assumption in more complicated cases involves the notion of “arbitrage” in the sense of a sure profit at no risk or expense; that is, if for ignorance or some other reason there did exist some willing sellers at the lower price and some willing buyers at the higher, knowledgeable arbitragers would buy and rmell until one or the other group had been driven from the market. ~ ' 1' The prices . mi are obviously somehow related to interest rates. But to stress the similarity p between capital markets and other kinds of markets, it is convenient to present the analysis initially in terms of the ,p¢. The formal relationships between thme prices and interest rates are shown later. s A A . A F \._/ ‘R [Ch. 1, Sec. II] - Accumulation and Allocation of Wealth 23 two periods. Because the second period is the -“last” period, no resources can be obtained in period 2 by drawing on future periods. The decision maker’s resources would be limited to his income forperiod 2, ya, plus-any financial" assets carried roverfromperiod 1, as, or minus any net liabilities incurred in period 1 that must now be repaid, in which ascasrepa;-, would be a negative number; that is, the period 2 consumption would be s ' r . 62.=s!}2+fle- 2 v . e ' , ('1-5)“ Of the two components ya is taken for our purposes as a fixed and unalterable amount determined somehow from outside the context of the problem, but as, within limits at least, is under the decision maker’s control. The less he consumes in period 1, the more funds he can bring to the capital market to purchase funds deliverable at the start of period 2. In particular, his net worth at period 2 will be s s v - 1»: "= [(111 + 012). "5 6211"!‘ - (1-6) 2 1P2 , The term in brackets represents the difference between his resources in period 1 and his consumption in this period andis the amount used to purchase resources to be delivered at period 2. The total numberof dollars to be received at period 2 is then just the amount invested divided by the price r;p2.’° Substituting Equation (1.6) for as in the period 2 constraint (1.5), we obtain g r 2 g p 2 s 2 r1 -1 s t . -62 = 312 +' (Z/1 + 01) — s" 61"" r (1-7)t 1P2 1P2 2 as the relationdescribing the allowable, efficient combinations of c1 and Cg that can beobtained by an individual whose income plus initial wealth consists of yl + o1 in the first period and whose income is ysin the second. In graphical terms (seeFigure 1.5). Equation (1.7) is a straight line in the C1Cg plane with a slope of --1/1p2and running through the “endowment point” :2: whose coordinates are ((3/1 + a1), ya). A movement along this efficiency frontier away from this pointto point iv would represent a purchase of funds for future delivery, that dis, lending, and from :0 to z a sale of future funds, that is, borrowing. The maximum attainable value of cs, that is, ‘° Alternatively, if the amount invested at period lgis [(y1 p+ a1) - crjand the price of a dollar to be delivered at period 2 is lpg, the number of period 2 dollars purchased at period 1 must be the value of as such that s . t y ' [(1/1 + 61) "' (=11 "'=g1P=°=» from which we easily obtain Equation (rs). Thus 1/mg is the number of dollars at period 2 that can be obtained by investing a dollar at period 1. . f ../ 14:17‘riziggt:41 I J 24 Certainty t Models e *2. 1 Y3 ‘*-ll/‘1 '|'l1, Q. lg "“"""""‘*. . tI-1 :_ I _...l........2. I== n -~-- --- l 02.. niuninulnlnuluvnn u 's| -- ---t ope- 192 J . . _ C: l yi-+31 _ e .. ‘C1 2 . 2 ' Y1"'*'1*Y2'1P2i'i"'12 If . ‘Figure 1.5 Opportunity Set in a Perfect Gapistal Market the intercept on the C2 axis, is 2 g i t i . 2 2 2 2 , 312+ (U1+a1)'“"', i a. 1 1272 1 . v - 1 which occurs when all the resources available at period 1 are used to pur- chase dollars for delivery at period 2. The maximum attainable value of c1, that is, the interceptonraatahe c1 axis, 2. t 2 e _ s a .3/1 + a1 + ;l/rips, r t 2 2 2 occurs when all income to be received at period.2 is sold at the beginning of period 1. This maximum attainable c1 can also be interpreted as the consumer’s wealth 1.81 at period 1; it is the market valueyof all his current and future resources. 2 t r 2 r V s ||.A..5. Interest rule: and present values V . . , 2 1 2 We have chosen to express the opportunities for carrying over resources in terms of current market prices for dollars to be delivered in the future partly to stress the fundamentalsimilarity betweencapital markets and any of the other markets considered ineconomicsyand partly to lay the groundwork for future extensions in which this approach isreally the only feasible one. For the present class of problems, however, there is another way o_f expressing the rateof exchange between current and future sums. Suppose, for example, that we have the current sum of P dollars andask what will be the number of dollars A that we shall have at the beginning of the next period if we purchase contracts for the future delivery of dollars _: - ~— ————4-r_ [Ch. 1, Sec. II] - Accumulation and Allocation of Wealth 25 at. the current market price of 1P3. The answer is A ' l if-A#Pi. u& 1P2 But we can always express the terminal amount A as a sum of the initial value P, plus, or minus, the difference between A and P, say, AP. Hence. we can rewrite Equation (1.8) as , 2 t 2 2 2 i_g.flwe_ g 1p2.- P E -— l+ P . (1.9) The term AP/P_is the rate of growth of the capital sum invested during the period, to be denoted by 11', and referred to as the one-period, spot rate of interest or, for short, just “rate of interest.” The term 1 + AP/Pg E 1 + In is often referred to'in the economics literature as the i“force of interest” and in the actuarial literature as the one-period.“accumulation factor” at the rate pa.“ ' 2 Note that in an equation such as (1.8) we can also perform the inverse =1 Here and throughout, unless otherwise noted, interest rates, such as m, are assumed to be those on securities denominated in the numéraire
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