JOURNAL OF ECONOMIC THEORY 4, 103-124( 1972) Expectations and the Neutrality of Money ROBERT E. LUCAS JR* graduate School of industrial Administration, Carnegie-Mellon University Received September 4, 1970 1. INTRODUCTION This paper provides a simple example of an economy in which equ librium prices and quantities exhibit what may be the central feature of the modern business cycle: a systematic relation between the rate of change in nominal prices and the level of real output. The relationship, essentially a variant of the well-known Phillips curve, is derived within a framework from which all forms of "money illusion"are rigorously excluded: all prices are market clearing, all agents behave optimally in light of their objectives and expectations, and expectations are formed optimally (in a sense to be made precise below) Exchange in the economy studied takes place in two physically separated markets. The allocation of traders across markets in each period is in part stochastic, introducing fluctuations in relative prices between the two markets. A second source of disturbance arises from stochastic changes in the quantity of money, which in itself introduces fuctuations in the nominal price level(the average rate of exchange between money and goods). Information on the current state of these real and monetary disturbances is transmitted to agents only through prices in the market where each agent happens to be. In the particular framework presented below, prices convey this information only imperfectly, forcing agents to hedge on whether a particular price movement results from a relative demand shift or a nominal(monetary) one. This hedging behavior results in a nonneutrality of money, or broadly speaking a Phillips curve, similar in nature to that which we observe in reality. At the same time, classical results on the long-run neutrality of money, or independence of real and nominal magnitudes, continue to hold These features of aggregate economic behavior, derived below within a particular, abstract framework, bear more than a surface resemblance to I would like to thank James Scott for his helpful comments C 1972 by Academic Press, Inc
JOURNAL OF ECONOMIC THEORY 4, 103-124 (1972) Expectations and the Neutrality of Money ROBERT E. LUCAS, JR.* Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 Received September 4, 1970 1. INTRODUCTION This paper provides a simple example of an economy in which equilibrium prices and quantities exhibit what may be the central feature of the modern business cycle: a systematic relation between the rate of change in nominal prices and the level of real output. The relationship, essentially a variant of the well-known Phillips curve, is derived within a framework from which all forms of “money illusion” are rigorously excluded: all prices are market clearing, all agents behave optimally in light of their objectives and expectations, and expectations are formed optimally (in a sense to be made precise below). Exchange in the economy studied takes place in two physically separated markets. The allocation of traders across markets in each period is in part stochastic, introducing fluctuations in relative prices between the two markets. A second source of disturbance arises from stochastic changes in the quantity of money, which in itself introduces fluctuations in the nominal price level (the average rate of exchange between money and goods). Information on the current state of these real and monetary disturbances is transmitted to agents only through prices in the market where each agent happens to be. In the particular framework presented below, prices convey this information only imperfectly, forcing agents to hedge on whether a particular price movement results from a relative demand shift or a nominal (monetary) one. This hedging behavior results in a nonneutrality of money, or broadly speaking a Phillips curve, similar in nature to that which we observe in reality. At the same time, classical results on the long-run neutrality of money, or independence of real and nominal magnitudes, continue to hold. These features of aggregate economic behavior, derived below within a particular, abstract framework, bear more than a surface resemblance to * I would like to thank James Scott for his helpful comments. 103 0 1972 by Academic Press, Inc
LUCAS many of the chracteristics attributed to the U. S economy by Friedman and elsewhere]. This paper provides an explicitly elaborated example, to my knowledge the first, of an economy in which some of these propositions can be formulated rigorously and shown to be valid A second, in many respects closer, forerunner of the approach taken here is provided by Phelps. Phelps [8] foresees a new inflation and employment theory in which Phillips curves are obtained within a work which is neoclassical except for "the removal of the postulat all transactions are made under complete information This is pre is attempted her The substantive results developed below are based on a uilibrium which is, I believe, new(although closely related to the principles underlying dynamic programming) and which may be of independent interest. In this paper, equilibrium prices and quantities will e characterized mathematically as functions defined on the space of possible states of the economy, which are in turn chracterized as finite dimensional vectors. This characterization permits a treatment of the relation of information to expectations which is in some ways much more satisfactory than is possible with conventional adaptive expectation hypotheses The physical structure of the model economy to be studied is set out in the following section. Section 3 deals with preference and demand functions; and in scction 4, an exact dcfinition of equilibrium is provided and motivated. The characteristics of this equilibrium are obtained in section 5, with certain existence and uniqueness arguments deferred to the appendix. The paper concludes with the discussion of some of the implications of the theory, in sections 6, 7, and 8 2. THE STRUCTURE OF THE ECONOMY In order to exhibit the phenomena described in the introduction, we shall utilize an abstract model economy, due in many of its essentials to Samuelson [10]. Each period, N identical individuals are born, each of whom lives for two periods(the current one and the next). In cach period then, there is a constant population of 2N: N of age 0 and N of age During the first period of life, each person supplies, at this discretion n, units of labor which yield the same n units of output, Denote the output 1 The usefulness of this model as a framework for considering problems in monetary theory is indicated by the work of Cass and Yaari [1, 2]
104 LUCAS many of the chracteristics attributed to the U. S. economy by Friedman [3 and elsewhere]. This paper provides an explicitly elaborated example, to my knowledge the first, of an economy in which some of these propositions can be formulated rigorously and shown to be valid. A second, in many respects closer, forerunner of the approach taken here is provided by Phelps. Phelps [8] foresees a new inflation and employment theory in which Phillips curves are obtained within a framework which is neoclassical except for “the removal of the postulate that all transactions are made under complete information.” This is precisely what is attempted here. The substantive results developed below are based on a concept of equilibrium which is, I believe, new (although closely related to the principles underlying dynamic programming) and which may be of independent interest. In this paper, equilibrium prices and quantities will be characterized mathematically as functions defined on the space of possible states of the economy, which are in turn chracterized as finite dimensional vectors. This characterization permits a treatment of the relation of information to expectations which is in some ways much more satisfactory than is possible with conventional adaptive expectations hypotheses. The physical structure of the model economy to be studied is set out in the following section. Section 3 deals with preference and demand functions; and in section 4, an exact definition of equilibrium is provided and motivated. The characteristics of this equilibrium are obtained in section 5, with certain existence and uniqueness arguments deferred to the appendix. The paper concludes with the discussion of some of the implications of the theory, in sections 6, 7, and 8. 2. THE STRUCTURE OF THE ECONOMY In order to exhibit the phenomena described in the introduction, we shall utilize an abstract model economy, due in many of its essentials to Samuelson [lO].l Each period, N identical individuals are born, each of whom lives for two periods (the current one and the next). In each period, then, there is a constant population of 2N: N of age 0 and N of age 1. During the first period of life, each person supplies, at this discretion n, units of labor which yield the same n units of output. Denote the output 1 The usefulness of this model as a framework for considering problems in monetary theory is indicated by the work of Cass and Yaari [I, 21
NEUTRALITY OF MONEY consumed by a member of the younger generation(its producer) by co, and that consumed by the old by c. Output cannot be stored but can be freely disposed of, so that the aggregate production-consumption pos sibilities for any period are completely described (in per capita terms) by c0+cl≤n,c,c2,n≥0. (21) Since n may vary, it is physically possible for this economy to experience fluctuations in real output In addition to labor-output, there is one other good fiat money, issued by a government which has no other function. This money enters the economy by means of a beginning-of-period transfer to the members of of the older generation, in a quantity proportional to the pretransfer holdings of each. No inheritance is possible, so that unspent cash balances revert, at the death of the holder, to the monetary authority. Within this framework, the only exchange which can occur will involve a surrender of output by the young, in exchange for money held over from the preceeding period, and altered by transfer, by ld. 2 We shal assume that such exchange occurs in two physically separate markets To keep matters as simple as possible, we assume that the older generation is allocated across these two markets so as to equate total monetary demand between them. The young are allocated stochastically, fraction 8/2 going to one and 1-(0/2)to the other. Once the assignment of persons to markets is made, no switching or communication between markets is possible. Within each market, trading by auction occurs, with all trades transcated at a single, market clearing price. 3 The pretransfer money supply, per member of the older generation, known to all agents. Denote this quantity by m. Posttransfer balances a This is not quite right. If members of the younger generation were risk preferrers they could and would exchange claims on future consumption among themselves so as to increase variance. This possibility will be ruled out in the next section his device of viewing traders as randomly allocated over distinct markets serves two purposes. First, it provides a setting in which information is imperfect in a specifie (and hence analyzable) way. Second, random variation in the allocation of traders provides a source of relative price variation. This could as well have been achieved by postulating random taste or technology shifts, with little effect on the structure of the This somewhat artificial assumption, like the abscncc of capital goods and the scrial independence of shocks, is part of an effort to keep the laws governing the transition of the economy from state to state as simple as possible. In general, I have tried to abstract from all sources of persistence of fuctuations, in order to focus on the nature of the initial disturbance
NEUTRALITY OF MONEY 105 consumed by a member of the younger generation (its producer) by co, and that consumed by the old by cl. Output cannot be stored but can be freely disposed of, so that the aggregate production-consumption possibilities for any period are completely described (in per capita terms) by: co + cl < n, co, cl, n > 0. (2.1) Since n may vary, it is physically possible for this economy to experience fluctuations in real output. In addition to labor-output, there is one other good: fiat money, issued by a government which has no other function. This money enters the economy by means of a beginning-of-period transfer to the members of of the older generation, in a quantity proportional to the pretransfer holdings of each. No inheritance is possible, so that unspent cash balances revert, at the death of the holder, to the monetary authority. Within this framework, the only exchange which can occur will involve a surrender of output by the young, in exchange for money held over from the preceeding period, and altered by transfer, by the old.2 We shall assume that such exchange occurs in two physically separate markets. To keep matters as simple as possible, we assume that the older generation is allocated across these two markets so as to equate total monetary demand between them. The young are allocated stochastically, fraction e/2 going to one and 1 - (e/2) to the other. Once the assignment of persons to markets is made, no switching or communication between markets is possible. Within each market, trading by auction occurs, with all trades transcated at a single, market clearing price.3 The pretransfer money supply, per member of the older generation, is known to all agents.4 Denote this quantity by m. Posttransfer balances, 2 This is not quite right. If members of the younger generation were risk preferrers, they could and would exchange claims on future consumption among themselves so as to increase variance. This possibility will be ruled out in the next section. 3 This device of viewing traders as randomly allocated over distinct markets serves two purposes. First, it provides a setting in which information is imperfect in a specific (and hence analyzable) way. Second, random variation in the allocation of traders provides a source of relative price variation. This could as well have been achieved by postulating random taste or technology shifts, with little effect on the structure of the model. 4 This somewhat artificial assumption, like the absence of capital goods and the serial independence of shocks, is part of an effort to keep the laws governing the transition of the economy from state to state as simple as possible. In general, I have tried to abstract from all sources of persistence of fluctuations, in order to focus on the nature of the initial disturbances
106 LUCAS denoted by m, are not generally known (until next period) except to the extent that they are"revealed"to traders by the current period price level Similarly, the allocation variable 8 is unknown, except indirectly via price The development through time of the nominal moncy supply is governed =mx, (2.2) where x is a random variable Let x' denote next periods value of this ransfer variable, and let g be next period s allocation variable It is assumed that x and x are independent, with the common, continuous density function f on(0 ao). Similarly, 8 and 8 are independent, with the common, continuous symmetric density g on(0, 2) To summarize, the state of the economy in any period is entirely described by three variables m, x, and 8. The motion of the economy from state to state is independent of decisions made by individuals in the economy, and is given by (2.2)and the densities f and g of x and 8 3. PRETERENCES AND DEMAND FUNCTIONS We shall assume that the members of the older generation prefer mor consumption to less, other things equal, and attach no utility to the holding of money. As a result, they will supply their cash holdings, as augmented by transfers, inelastically. (Equivalently, they have a unit elastic demand or goods. The young, in contrast, have a nontrivial decision problem to which we now turn The objects of choice for a person of age 0 are his current consumption c, current labor supplied n, and future consumption, denoted by c. All individuals evaluate these goods according to the common utility function (c,n)+F{v()} (The distribution with respect to which the expactation in(3. 1)is taken will be specified later. )The function U is increasing in c, decreasing in n strictly concave, and continuously twice differentiable. In addition, current consumption and leisure are not inferior goods, or Un Unn<o and Ucc+ Uen <o The function V is increasing, strictly concave and continuously wIce
106 LUCAS denoted by m’, are not generally known (until next period) except to the extent that they are “revealed” to traders by the current period price level. Similarly, the allocation variable 0 is unknown, except indirectly via price. The development through time of the nominal money supply is governed by m’ = mx, (2.2) where x is a random variable. Let x’ denote next period’s value of this transfer variable, and let 8’ be next period’s allocation variable. It is assumed that x and X’ are independent, with the common, continuous density functionf on (0, co). Similarly, 8 and 0’ are independent, with the common, continuous symmetric density g on (0, 2). To summarize, the state of the economy in any period is entirely described by three variables m, x, and 8. The motion of the economy from state to state is independent of decisions made by individuals in the economy, and is given by (2.2) and the densities f and g of x and 0. 3. PREFERENCES AND DEMAND FUNCTIONS We shall assume that the members of the older generation prefer more consumption to less, other things equal, and attach no utility to the holding of money. As a result, they will supply their cash holdings, as augmented by transfers, inelastically. (Equivalently, they have a unit elastic demand for goods.) The young, in contrast, have a nontrivial decision problem, to which we now turn. The objects of choice for a person of age 0 are his current consumption c, current labor supplied, n, and future consumption, denoted by c’. All individuals evaluate these goods according to the common utility function: WC, n> + JWV)). (3.1) (The distribution with respect to which the expactation in (3.1) is taken will be specified later.) The function U is increasing in c, decreasing in n, strictly concave, and continuously twice differentiable. In addition, current consumption and leisure are not inferior goods, or: UC, + u,, < 0 and UC, -+ u,, < 0. (3.2) The function V is increasing, strictly concave and continuously twice
NEUTRALITY OF MONEY differentiable. The function v(c)c is increasing, with an elasticity bounded away from unity, or v"(c)c'+v(c)>0, (33) c'(c) ≤-a<0 (34) ondition(3. 3)essentially insures that a rise in the price of future goods will, ceteris paribus, induce an increase in current consumption or that the substitution effect of such a price change will dominate its income effect The strict concavity requirement imposed on V implies that the left term of (3.4)be negative, so that(3. 4)is a slight strengthening of concavity Finally we require that the marginal utility of future consumption be high enough to justify at least the first unit of labor expended, and ultimately tend to zero im V'(c)=0 (3.6) Future consumption, c, cannot be purchased directly by an age 0 individual. Instead, a known quantity of nominal balances A is acquired in exchange for goods. If next period's price level (dollars per unit of ouptut) is p and if next period's transfer is x', these balances will then purchase x'mlp' units of future consumption. 6 Although it is purely formal at this point, it is convenient to have some notation for the distribution function of (x', p), conditioned on the information currently available to the 6 The restrictions(3. 2)and (3.3)are similar to those utilized in an econometric study of the labor market conducted by Rapping and myself, [5]. Their function here is the same as it was in [5]: to assure that the Phillips curve slopes the"right way. " 6 There is a question as to whether cash balances in this scheme are "transactions balances"or a"store of value i think it is clear that the model under discussion is not rich enough to permit an interesting discussion of the distinctions between these, or other, motives for holding money. On the other hand all motives for holding money require that it be held for a positive time interval before being spent there is no reason o use money (as opposed to barter) if it is to be received for goods and then instan- ' yields utility. "Certainly the answer in this context is yes, in the sense that ir ey aneously exchanged for other goods. There is also the question of whether m imposes on an individual the constraint that he cannot hold cash, his utility under ar ptimal policy is lower than it will be if this constraint is removed. It should be equally lear, however, that this argument does not imply that real or nominal balances should be included as an argument in the individual preference functions. The distinction the familiar one between the utility function and the alue of this function under a
NEUTRALITY OF MONEY 107 differentiable. The function V’(c’)c’ is increasing, with an elasticity bounded away from unity, or: vyc’) c’ + V’(c’) > 0, (3.3) c’ V”(c’) ~ < -a < 0. V(d) (3.4) Condition (3.3) essentially insures that a rise in the price of future goods will, ceteris paribus, induce an increase in current consumption or that the substitution effect of such a price change will dominate its income effect.5 The strict concavity requirement imposed on V implies that the left term of (3.4) be negative, so that (3.4) is a slight strengthening of concavity. Finally, we require that the marginal utility of future consumption be high enough to justify at least the first unit of labor expended, and ultimately tend to zero: lim V(c’) = +co, C’--0 (3.5) lim v’(c’) = 0. c’+m (3.6) Future consumption, c’, cannot be purchased directly by an age 0 individual. Instead, a known quantity of nominal balances X is acquired in exchange for goods. If next period’s price level (dollars per unit of ouptut) is p’ and if next period’s transfer is x’, these balances will then purchase x’h/p’ units of future consumption.6 Although it is purely formal at this point, it is convenient to have some notation for the distribution function of (x’, p’), conditioned on the information currently available to the 6 The restrictions (3.2) and (3.3) are similar to those utilized in an econometric study of the labor market conducted by Rapping and myself, [5]. Their function here is the same as it was in [5]: to assure that the Phillips curve slopes the “right way.” e There is a question as to whether cash balances in this scheme are “transactions balances” or a “store of value.” I think it is clear that the model under discussion is not rich enough to permit an interesting discussion of the distinctions between these, or other, motives for holding money. On the other hand, all motives for holding money require that it be held for a positive time interval before being spent: there is no reason to use money (as opposed to barter) if it is to be received for goods and then instuntaneously exchanged for other goods. There is also the question of whether money “yields utility.” Certainly the answer in this context is yes, in the sense that if one imposes on an individual the constraint that he cannot hold cash, his utility under an optimal policy is lower than it will be if this constraint is removed. It should be equally clear, however, that this argument does ll~t imply that real or nominal balances should be included as an argument in the individual preference functions. The distinction is the familiar one between the utility function and the value of this function under a particular set of choices