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Money in a Theory of Finance

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Money in a Theory of Finance Carnegie-Rochester Conference Series on Public Policy 21 119941446 North-Holland MONEY IN A THEORY OF FINANCE Robert E. Lucas, Jr.* The University of Chicago I. INTRODUCTION The title of this essay is taken, of course, from the Gurley/Shaw (1960) monograph to remind the reader at the outset that the objective of constructing a unified theory of money and finance is an old one, one that has challenged theorists at least since J.R. Hicks's (1935) "Suggestion." That the attainment of this objective is still regarded as part of an a- genda for future research suggests that there must be something difficult about the problem that earlier writers either did not see or did not ade- quately face. This paper is an attempt to identify this difficulty and to offer one way of dealing with it. If it is easier today than it was in 1960 to identify exactly in which respects the theory of finance fails as monetary theory, this is largely due to rapid recent progress in the theory of finance. Theoretical re- search in finance is now conducted almost entirely within the contingent- claim general equilibrium framework introduced by Arrow (1964) and Oebreu (1959). This is not an historical statement, for each of the three pillars of modern financial theory--portfolio theory, the Modigliani-Miller Theorem, and the theory of efficient markets--was discovered within differ- ent (and mutually distinct) theoretical frameworks, but all three have since been reformulated in contingent-claim terms, and it was this reformu- lation that revealed their essential unity and set the stage for many further theoretical advances. This paper begins in Section II with a review of a simple version of the Fisherian model of real capital theory in contingent-claim terms and a review of the relationship of this model to various aspects of financial economics. A central feature of this model is that alI trading occurs in a *I am grateful to Nancy Stokey for criticism of an earlier draft and for ccmments by Arthur Kupferman, Allan Meltzer, Bennett McCallum. and Manuel Sanchez. centralized market, with all agents present. In such a setting, the po- sition of each agent is fully described by a single number: his wealth, or the market value of all the claims he owns. The command any one claim has over goods is fully described by its market value, which is to say all claims are equally "liquid." If the point of a theory of money, or of "liquidity preference," is to capture the fact that, in some situations in reality, money has a relative command over other goods in excess of its relative value in centralized se- curities trading, then a successful theoretical model must place agents in such situations, at least some of the time. How, as a matter of modeling strategy, might this best be done? I do not believe we have enough experience with alternative formu- lations to answer this question now, but the monetary model introduced in Section III employs a device used in Lucas (1982), Townsend (1982), and Lucas and Stokey (1983), in which agents alternate between two different kinds of market situations. Each period, they all attend a securities market in which money and other securities are exchanged. Subsequent to securities trading, agents trade in (implicitly) decentralized goods markets in which the purchase of at least some goods is ,assumed subject to the cash-in-advance constraint of the form suggested by Clower (1967). The assumption of this model that agents regularly, if not continu- ously, trade in a centralized securities market admits a theory of securi- ties pricing that is close to the standard barter theory reviewed in Section II. Yet interesting and fully operational modifications are re- quired for a monetary system, so that pricing formulas differ in important ways from the barter versions that have been subjected to much recent em- pirical testing. These are reviewed in Section IV. Section V turns to the question, central to the objectives of monetary theory though traditionally peripheral to the theory of finance, of methods for constructing monetary equilibria under alternative fiscal and monetary regimes. Here a simplified version of the model of Section III is studied to the point where one can begin to see what a full analysis would involve, and various simple examples are fully "solved." Section VI contains con- cluding comments. II. THE THEORY OF FINANCE The theory of finance, as the term is now generally understood, con- 10 sists of various specializations and applications of the Arrow-Debreu con- tingent-claim formulation of a competitive equilibrium for an economy oper- ating through time, subject to stochastic shocks. As background for summa- rizing several of the main results in the theory of finance, and also for considering how this theory might be extended to include monetary elements, it will be useful to state a highly simplified version. We consider an economy subject to exogenously given stochastic shocks, (St), where the realization of the vector st is public knowledge prior to any consumption or production activity in t and where the joint density ft of (Sl,..., St) is known to all agents. Use st = (~1 ,...,st) to denote the full history of shocks up to and including time t. A commodity or good in this setting is idealized as a function ct(st), the value of which denotes the quantity of the good to be exchanged (or consumed or produced) at date t contingent on the occurrence of the history st. I will confine attention here to two consumption goods: a nonstorable, produced good, ct, and leisure, xt. The sequence [c+,,xt} of pairs of functions ct(st), x+,(st), each defined over all possible histories st, provides a catalogue of an individual consumer's consumption for all dates, under all possible circum- stances. 1 Consumers will be taken to maximize expected utility: m z Bt I U(c&), xt(st))ft(st)dst. t=o The shorthand m t t t z 6 I U(ct,xt)f ds (2.1) t=o is taken to mean the same thing and will be used repeatedly below. Firms are assumed to have a technology: Ct + gt + kt+1 = F(k$-x& (2.2) ‘Here and below I am simply setting out a notation useful for discussing technically- elementary aspects of various models. If a mathematically-rigorous exposition were to be provided , it would be necessary to specify the commodity space and functions defined in more detail, and phases like “defined over all possible histories” would need elaboration or replacement. 11 describing the combinations of private consumption goods c+_, government consumption goods gt, and end-of-period capital stocks kt+I that can be produced when beginning-of-period capital stocks are kt, labor is 1-xt, and the shock history is st. At this formal level one could consider many different consumers and firms, but it will economize on subscripts to con- sider one ("representing" many) of each. Questions of government finance of the expenditure stream [gt) will be kept simple, here and throughout the paper, by assuming that government has the ability to levy distortion-free, lump-sum taxes on consumers.* Let et(st) denote contingent tax obligations of consumers at t. To admit the possibility of deficit finance, let 60 denote initial, goods-denominated debt obligations owed consumers by government. To describe the trading possibilities open to these agents, and hence to formulate a definition of equilibrium, it is useful to keep in mind two quite different, but highly complementary scenarios, one of which is standard in general equilibrium theory and the other of which is closer to the traditions of financial and monetary theory. In the first, Arrow- Debreu, scenario, all agents are taken to convene at time 0, knowing SO and the distributions f1,f2 ,... of future shocks, to trade 'in a complete range of sequences (ct,xt] of contingent claims on goods. In this trading the price nt(st) of the contingent consumption claim ct(st) and the price - -nxt(st) of a contingent claim on leisure xt(st) are both dated functions of the shock history st, so that, for example, the value of the claim ct(st) is the product mt(st)ct(st) and the present value of an entire sequence (ct] is3 m z I &)ct(st)dst t=o Here prices are quoted in an abstract unit-of-account, so a normalization like no = 1 is permitted. In this setting, firms choose (ct + gt, l-xt, kt+I), given k. and (nt,nXt), to maximize 2 See Lucas and Stokay (1983) for a normative analysis. in a context similar to this one, of government finance when all taxes distort. 3 t_ Here and below, the normalization /ds - I, all t, is assumed. 12 m 1 ~[~,(st)(ct(st)+gt(st))-nx,(st)(l-xt(st))~dst,(2.3) t=o subject to (2.2) for all t, st. Call the value of this maximized objective function n. Consumers are endowed with one unit of labor-leisure per period, they are liable for taxes, they own the firms, and they hold the outstanding government debt, so their budget constraint is: m z rl$st)(ct(st) + e&H - (2.4) t=o - nxt(st)(I-xt(st))]dst < n + ,,oBo Consumers choose (ct,xt}, given (nt,nXt), [et],", and BO, to maximize (2.1) subject to (2.4). This scenario, in which all equilibrium quantities and prices are set at time 0, conflicts (though very superficially) with the observation that in reality trading goes on all the time, concurrent with consumption and production of goods rather than prior to these activities.
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