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 <o, with equality if (3.10) pn-c)-A≥0, with equality ifμ>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, with equality if h > 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