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 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
age-0 person: denote it by F(x, p'l m, P), where p is the current price level Then the decision problem facing an age-0 person is U(e, n)+V( dF(x', p'im (3.7) pn-c)-A≥0. Provided the distribution F is so specified that the objective function continuously differentiable, the Kuhn-Tucker conditions apply to this problem and are both necessary and sufficient. These are U(e,n)-PH≤0, if c>0 Un(, n)+pu 0,(3.1) (x,p'm,p)-≤0, where u is a nonnegative multiplier. We first solve(3. 9)-(3. 11)for c, n, and Pu as functions of A/ p. This is equivalent to finding the optimal consumption and labor supply for a fixed cquisition of money balances. The solution for Pu will have the inter- pretation as the marginal cost (in units of foregone utility from con- sumption and leisure)of holding money. This solution is diagrammed in Fig It is not difficult to show that, as Fig. 1 suggests, for any A/p>0 (3.9)(3. 11)may be solved for unique values of c, n, and pu. As A/p 剑 ( Figure 1
108 LUCAS age-0 person: denote it by F(x’, p’ 1 m, p), where p is the current price level. Then the decision problem facing an age-0 person is: (3.7) subject to: p(n - c) - h > 0. (3.8) Provided the distribution F is so specified that the objective function is continuously differentiable, the Kuhn-Tucker conditions apply to this problem and are both necessary and sufficient. These are: Uck 4 - PP G 0, with equality if c > 0, (3.9) uric, 4 + PP G 0, with equality if n > 0, (3.10) p(n - c) - h 3 0, with equality if p > 0, (3.11) jv($$)$ Wx’, P’ I m, PI - TV 0, where p is a nonnegative multiplier. (3.12) We first solve (3.9)-(3.11) for c, ~1, and pp as functions of h/p. This is equivalent to finding the optimal consumption and labor supply for a fixed acquisition of money balances. The solution for pp will have the interpretation as the marginal cost (in units of foregone utility from consumption and leisure) of holding money. This solution is diagrammed in Fig. 1. It is not difficult to show that, as Fig. 1 suggests, for any h/p > 0 (3.9)-(3.11) may be solved for unique values of c, n, and pp. As h/p FIGURE 1
NEUTRALITY OF MONEY varies, these solution values vary in a continuous and(almost everywhere) continuously differentiable manner. From the noninferiority assumptions (3. 2 ) it follows that as A/p increases, n increases and c decreases. The olution value for pu, which we denote by h(/p) is, positive, increasing, and continuously differentiable. As Aip tends to zero, h(ap) tends to a Substituting the function h into (3. 12), one obtains P%≥p(xAFx,pm,D with equality if A>0. After multiplying through by P, (3. 13)equates the marginal cost of acquiring cash(in units of current utility foregone) to the marginal benefit (in units of expected future utility gained ). Implicitly, (3. 13)is a demand function for money, relating current nominal quantity demanded, A, to the current and expected future price levels 4. EXPECTATIONS AND A DEFINITION OF EQUILIBRIUM Since the two markets in this economy are structurally identical, and since within a trading period there is no communication between them the economys general (current period) equilibrium may be determined by determining equilibrium in each market separately. We shall do so by equating nominal money demand (as determined in section 3)and nominal money supply in the market which receives a fraction 0/2 of the young. Equilibrium in the other market is then determined in the same way,with 8 replaced by 2-0, and aggregate values of output and prices re determined in the usual way by adding over markets. This will be carried out explicitly in section 6. At the beginning of the last section, we observed that money be supplied inelastically in each market. The total money supply, after transfer, is Nmx. Following the convention adopted in section 1, Nmx /2 is supplied in each market. Thus in the market receiving a fraction 8/2 of the young the quantity supplied per demander is(Nmx/2)/(BN/2)= mx/8. Equi- librium requires that X= mx 8, where A is quantity demanded per age-0 person. Since mx/8>0, substitution into(3. 13)gives the equilibrium condition h(mx1 (4.1) Equation (4.1) relates the current period price level to the(unknown)
NEUTRALITY OF MONEY 109 varies, these solution values vary in a continuous and (almost everywhere) continuously differentiable manner. From the noninferiority assumptions (3.2), it follows that as h/p increases, n increases and c decreases. The solution value for pp, which we denote by h(h/p) is, positive, increasing, and continuously differentiable. As X/p tends to zero, h(h/p) tends to a positive limit, h(O). Substituting the function h into (3.12), one obtains (3.13) with equality if h > 0. After multiplying through by p, (3.13) equates the marginal cost of acquiring cash (in units of current utility foregone) to the marginal benefit (in units of expected future utility gained). Implicitly, (3.13) is a demand function for money, relating current nominal quantity demanded, h, to the current and expected future price levels. 4. EXPECTATIONS AND A DEFINITION OF EQUILIBRIUM Since the two markets in this economy are structurally identical, and since within a trading period there is no communication between them, the economy’s general (current period) equilibrium may be determined by determining equilibrium in each market separately. We shall do so by equating nominal money demand (as determined in section 3) and nominal money supply in the market which receives a fraction S/2 of the young. Equilibrium in the other market is then determined in the same way, with 19 replaced by 2 - 8, and aggregate values of output and prices are determined in the usual way by adding over markets. This will be carried out explicitly in section 6. At the beginning of the last section, we observed that money be supplied inelastically in each market. The total money supply, after transfer, is Nmx. Following the convention adopted in section 1, Nmx/2 is supplied in each market. Thus in the market receiving a fraction 8/Z of the young, the quantity supplied per demander is (Nmx/2)/(8N/2) = mx/8. Equilibrium requires that h = mx/t?, where h is quantity demanded per age-0 person. Since mx/e > 0, substitution into (3.13) gives the equilibrium condition h (z) ; = s v’ (+) 5 dF(x’, p’ I m, p). (4.0 Equation (4.1) relates the current period price level to the (unknown)
LUCAS future price level, P. To"solve " for the market clearing price p(and hence to obtain the current equilibrium values of employment, output, and consumption) p and pmust be linked. This connection is provided in the definition of equilibrium stated below, which is motivated by the following considerations First, it was remarked earlier that in some(not very well defined)sense the state of the economy is fully described by the three variables(m, That is, if at two different points in calendar time the economy arrives at a particular state(m, x, B) it is reasonable to expect it to behave the same way both times, regardless of the route by which the state was attained each time. If this is so, one can express the equilibrium price as a function p(m,x, B)on the space of possible states and similarly for the equilibrium values of employment, output, and consumption Second, if price can be expressed as a function of (m, x, 0), the true probability distribution of next period,'s price, p'=p(m, x, 0) P(mx, x, 8)is known, conditional on m, from the known distributions of x,x', and 8. Further information is also available to traders, however, since the current price, p(m, x, A), yields information on x. Hence, on the basis of information available to him, an age 0 trader should take the expectation in (4. 1) [or (3. 13)] with respect to the joint distribution of (m, x, x',8)conditional on the values of m and p(m, x, 8), or treating m as a parameter, the joint distribution of (x, x, 0) conditional on the value of P(m, x, 0). Denote this latter distribution by G(x, x, 0p(m, x, e)) We are thus led to the following DEFINITION. An equilibrium price is a continuous, nonnegati function p() of (m, x, 0), with mx/p(m, x, 6) bounded and bounded awa from zero, which satisfi tvm,x,6」p(m,x,6 ∫[bmx,时]m,00(x,pm,x,D)(42) Equation (4.2)is, of course simply(4.1)with p replaced by the value of the function p( under the current state, (m, x, 0), and preplaced by The assumption that traders use the correct conditional distribution in forming expectations, together with the assumption that all exchanges take place at the market learing price, implies that markets in this economy are efficien as this term is defined by Roll [9]. It will also be true that price expectations are rational in the sense of Muth
110 LUCAS future price level, p’. To “solve” for the market clearing price p (and hence to obtain the current equilibrium values of employment, output, and consumption) p and p’ must be linked. This connection is provided in the definition of equilibrium stated below, which is motivated by the following considerations. First, it was remarked earlier that in some (not very well defined) sense the state of the economy is fully described by the three variables (m, x, 0). That is, if at two different points in calendar time the economy arrives at a particular state (m, x, f?) it is reasonable to expect it to behave the same way both times, regardless of the route by which the state was attained each time. If this is so, one can express the equilibrium price as a function p(m, X, 0) on the space of possible states and similarly for the equilibrium values of employment, output, and consumption. Second, if price can be expressed as a function of (m, x, 8), the true probability distribution of next period’s price, p’ = p(m’, x’, 0’) = p(mx, x’, 0’) is known, conditional on m, from the known distributions of X, x’, and 8’. Further information is also available to traders, however, since the current price, p(m, X, Q yields information on x. Hence, on the basis of information available to him, an age-0 trader should take the expectation in (4.1) [or (3.13)] with respect to the joint distribution of (m, X, x’, 0’) conditional on the values of m andp(m, x, 8), or treating m as a parameter, the joint distribution of (x, x’, 0’) conditional on the value of p(m, X, 0): Denote this latter distribution by G(x, x’, Ojp(m, x, 0)). We are thus led to the following DEFINITION. An equilibrium price is a continuous, nonnegative function p(.) of (m, X, Q with mx/Op(m, x, 0) bounded and bounded away from zero, which satisfies: h I e$Z 3 , e) I p(m,lx, e) dG(t, X’, 8’ /Ph X, 8)). (4.2) Equation (4.2) is, of course, simply (4.1) with p replaced by the value of the function p(m) under the current state, (m, x, e), and p’ replaced by ‘The assumption that traders use the correct conditional distribution in forming expectations, together with the assumption that all exchanges take place at the market clearing price, implies that markets in this economy are efficient, as this term is deflned by Roll [9]. It will also be true that price expectations are rational in the sense of Muth t71
NEUTRALITY OF MONEY l11 the value of the same function under next periods state(mx, x', 0). In addition, we have dispensed with unspecified distribution F, taking the expectation instead with respect to the well-defined distribution G In the next section, we show that (4.2) has a unique solution and develop the important characteristics of this solution. The more difficult mathe- matical issues will be relegated to the appendix 5. CHARACTERISTICS OF THE EQUILIBRIUM PRICE FUNCTION We proceed by showing the existence of a solution to(4.2 )of a partic form, then showing that there are no other solutions, and finally by haracterizing the unique solution. as a useful preliminary step we show: LEMMA 1. Ifpo is any solution to(4.2), it is monotonic in x e in the sense that for any fixed m, xo/8o >x,/0, implies p(m, xo, 8o)+p(m, x1, 81) Proof. Suppose to the contrary that xo/Bo>x,/e, and P(m, xo, Bo =p(m, x1, 01)=P(say). Then from(4.2), b(m1)2=jP[m,]1m,6)06x,p (m)几一jP[m,]:,(后x,0 Since h is strictly increasing while V is strictly decreasing, these equalities are contradictory. This completes the proof. In view of this Lemma, the distribution of (x, x, 8) conditional on p(m, x, 8)is the same as the distribution conditional on x e for all solution functions p(), a fact which vastly simplifies the study of (4.2) It is a plausible conjecture that solutions to(4.2)assume the form p(m, x, 0)= mo(x/ 0), where p is a continuous, nonnegative function 8 The restriction, embodied in this definition, that price may be expressed as a function of the state of the economy appears innocuous but in fact is in the models of Cass and Yaari without storage, the state of the economy never changes, so the only sequences satisfying the definition used here are constant sequences (or stationary schemes, in the terminology of [iD 9 To decide whether it is plausible that m should factor out of the equilibrium price function, the reader should ask himself: what are the consequences of a fully announced change in the quantity of money which does not alter the distribution of money over persons? To see why only the ratio of x to e affects price, recall that x/e alone determines the demand for goods facing each individual produce
NEUTRALITY OF MONEY 111 the value of the same function under next period’s state (mx, x’, 0). In addition, we have dispensed with unspecified distribution F, taking the expectation instead with respect to the well-defined distribution G.8 In the next section, we show that (4.2) has a unique solution and develop the important characteristics of this solution. The more difficult mathematical issues will be relegated to the appendix. 5. CHARACTERISTICS OF THE EQUILIBRIUM PRICE FUNCTION We proceed by showing the existence of a solution to (4.2) of a particular form, then showing that there are no other solutions, and finally by characterizing the unique solution. As a useful preliminary step, we show: LEMMA 1. If p(s) is any solution to (4.2), it is monotonic in x/8 in the sense thatfor anyjixedm, x,/B,, > x,/O, impliesp(m, x0 , 19,) # p(m, xl , 9,). Proof. Suppose to the contrary that x,/B, > x1/9, and p(m, x0, 8,) = p(m, xi , 0,) = pO (say). Then from (4.2), and Since h is strictly increasing while v’ is strictly decreasing, these equalities are contradictory. This completes the proof. In view of this Lemma, the distribution of (x, x’, 0’) conditional on p(m, x, 0) is the same as the distribution conditional on x/O for all solution functions p(.), a fact which vastly simplifies the study of (4.2). It is a plausible conjecture that solutions to (4.2) assume the form p(m, x, 0) = my(x/@>, where v is a continuous, nonnegative function.g * The restriction, embodied in this definition, that price may be expressed as a function of the state of the economy appears innocuous but in fact is very strong. For example, in the models of Cass and Yaari without storage, the state of the economy never changes, so the only sequences satisfying the definition used here are constant sequences (or stationary schemes, in the terminology of [l]). 8 To decide whether it is plausible that m should factor out of the equilibrium price function, the reader should ask himself: what are the consequences of a fully announced change in the quantity of money which does not alter the distribution of money over persons? To see why only the ratio of x to fI affects price, recall that x/6 alone determines the demand for goods facing each individual producer
112 LUCAS If this is true, the function p satisfies(multiplying(4. 2) through by mx 8 ind substituting) q(x6)]6q(x/6) ∫p[)](x5)( Let us make the change of variable z= xB, and z'=x 8,and H(z, 0) be the joint density function of z and 0 and let H(z, e)be the density of A conditional on z. Then(5. 1)is equivalent to p(z)]p(z) j[05]0c.B,0)mhm(2) Equations(4.2)and (5. 2)are studied in the appendix. The result of nterest is THEOREM 1. Equation(5.2) has exactly one continuous solution p(z) on(0, oo) with z/ o(z) bounded. The function (z) is strictly positive and continuously differentiable. Further, mop(x/@)is the unique equilibrium price Proof. See the appendix, We turn next to the characteristics of the solution function It is convenient to begin this study by first examining two polar cases, one in which 8=1 with probahility one, and a second in which x= I with probability one The first of these two cases may be interpreted as applying to an economy in which all trading place in a single market, and no nonmonetary dis turbances are present. Then z is simply equal to x and, in view of Lemma I the current value of x is fully revealed to traders by the equilibrium price It should not be surprising that the following classical neutrality of money theorem holds THEOREM 2. Suppose 6=1 with probability one. Let y* be the un h(y)=v(y) Then P(m, x, 8=mx/y* is the unique solution to(4.2)
112 LUCAS If this is true, the function qz satisfies (multiplying (4.2) through by mx/% and substituting): h [ %I&%) 1 e&e) ~- dG (E, x’, 8’ I$]. (5.1) Let us make the change of variable z = xl%, and z’ = x’/%‘, and let H(z, %) be the joint density function of z and % and let ii(z, %) be the density of % conditional on z. Then (5.1) is equivalent to: = s v’ iI ii(z, %) H(z’, %‘) d% dz’ d%‘. (5.2) Equations (4.2) and (5.2) are studied in the appendix. The result of interest is: THEOREM 1. Equation (5.2) has exactly one continuous solution y(z) on (0, 00) with z/v(z) bounded. The function y(z) is strictly positive and continuously dtzerentiable. Further, my(x/%) is the unique equilibrium price function. Proof See the appendix, We turn next to the characteristics of the solution function v. It is convenient to begin this study by first examining two polar cases, one in which % = 1 with probability one, and a second in which x = 1 with probability one. The first of these two cases may be interpreted as applying to an economy in which all trading place in a single market, and no nonmonetary disturbances are present. Then z is simply equal to x and, in view of Lemma 1, the current value of x is fully revealed to traders by the equilibrium price. It should not be surprising that the following classical neutrality of money theorem holds. THEOREM 2. Suppose % = 1 with probability one. Let y* be the unique solution to h(y) = V’(Y). (5.3) Then p(m, x, %) = mx/y* is the unique solution to (4.2)
