- Email: [email protected]
Transactions cost, endogenous labor, and the superneutrality of money
PING WANG The
Pennsylvania University
State Park,
Unioersity
Pennsylvania
CHONG K. YIP Georgia
State Atlanta,
Uniuersity Georgia
Transactions Cost, Enciogenous Labor, and the Superneutrality of Money This paper develops a tractable perfect-foresight-shopping-time model with endogenous labor-leisure choice to investigate the superneutrality of money. Different from the existing literature, we examine the effects of money growth on both employment and capital accumulation. Under the lump-sum money transfer, we show that an increase in money growth rate unambiguously reduces steady-state capital, labor, leisure, consumption, and welfare, contrary to previous work. Since money is introduced into transaction effort technology via time allocation, the positive correlation between steady-state labor and leisure emerges.
1. Introduction This paper develops a neoclassical perfect foresight monetary growth model with an endogenous labor-leisure choice to examine the superneutrality of money. Money interacts with labor effort and hence physical capital accumulation through a shopping-time technology rather than directly in the utility function. The shoppingtime technology proposed in the paper is not only tractable but provides more intuitive results than the conventional ones in the literature. With lump-sum money transfer, an increase in the rate of money growth unambiguously reduces steady-state capital, labor, leisure, and economic welfare, in contrast with previous work. Since the classic papers of Tobin (1965) and Sidrauski (1967), the effect of anticipated inflation on capital accumulation has been one of the central issues in monetary growth theory. Using a portfolio-choice framework, Tobin (1965) shows that higher steady-state inflation would lead to a higher steady-state capital stock. This Tobin effect is questioned by Sidrauski (1967) who, by considering consumers’ rational choice, concludes that the invariance of the steady-state real discount rate to inflation would result in superneutrality of money. However, as pointed out by Fischer (1979), money is not superneutral on transition paths in the Sidrauski model. Further, even in the steady state, money may still have real effects Journal of Macroeconomics, Winter 1991, Vol. Copyright 0 1991 by Louisiana State University 0164-0704/91/$1.50
13, No. Press
1, pp.
183-191
183
Ping Wang and Chong K. Yip in more sophisticated models. For instance, in his pivotal work, Brock (1974) investigates the implications of endogenous labor-leisure choice for the superneutrality of money by putting both real money balances and leisure in the utility function and concludes that the dependence of the marginal utility of leisure on money “destroys the long run neutrality of money.” “[Elven with constant returns the steady-state quantity of real capital will be changed by the rate of growth of money” (Brock 1974, 774). However, his comparative dynamic results depend heavily on the cross partial derivatives of the utility function which generally have little economic intuition. Dornbusch and Frenkel (1973) examine this issue via a shopping effort function with pecuniary transactions costs.’ In their model, money enters the system purely through a positive wealth effect due to a reduction of pecuniary transactions costs (in terms of unit of goods produced). The net effect of the rate of money growth on the steady-state capital stock is thus ambiguous. By considering the transactions service of money, Kimbrough (1986) constructs a shopping-time technology in which. shopping effort per unit of consumption is decreasing and convex in the money-consumption ratio. Nevertheless, due to its analytical difficulty, his model focuses on the analysis of employment and abstracts from capital accumulation. On the other hand, Stockman (1981) generates the nonsupemeutrality of money in the absence of endogenous labor by imposing a cash-in-advance constraint on both consumption and capital goods.2 The contribution of our paper is to develop a tractable shopping-time model of money with endogenous labor-leisure choice through which the effects of money growth on both employment and capital accumulation can be examined. Under lump-sum money transfer, we show that an increase in money growth rate unambiguously reduces steady-state capital,3 in contrast to Dornbusch and Frenkel (1973). The resulting “reversed Tobin effect” is consistent with the empirical finding of Kormendi and Meguire (1985). Contrary to Brock and Tumovsky (1981), we find that the steady-statecapital-labor ratio is independent of monetary factors. Moreover, an
‘For discussions on the properties of shopping transactions costs, the reader is referred to the *For a detailed discussion and survey of the ferred to Wang and Yip (1990). 3Notice that if the money transfer scheme is becomes superneutral in a steady-state analysis,
184
effort functions with pecuniary paper by Gray (1984). above literature, the reader is reproportional even with
(Lucas 1972), money endogenous labor.
Transactions
Cost
increase in the rate of money growth depresses consumption as well as leisure and hence unambiguously reduces welfare in the steady state, which contrasts with results in the existing literature (for example, see Brock 1974). The remainder of the paper is organized as follows. In the next section, we present the basic model, while Section 3 conducts the steady-state analysis. Finally, there is a fourth and concluding section.
2. The Model Consider a representative-agent neoclassical monetary growth model with endogenous labor. Money enters into the optimization problem through a shopping-time technology with transactions effort, S, depending only on (per capita) real money balances, m, that is, S = S(m). This formulation of shopping effort is not only tractable but provides more intuitive results than the conventional one in the literature. To provide an optimal quantity of real balances, we impose a satiation level of money (m*) which attains the minimum of shopping effort, that is, S(m*) = S,(m*) = 0, where m* < cQ. When real money balances are below the satiation level, S is assumed to be decreasing in m and the technology exhibits diminishing returns to money holdings, that is, S, < 0 and S,, > 0 for m < m*. Finally, to insure the existence of money, we further assume limnrtO S, = --03. Each agent is endowed with one unit of time which must be allocated to leisure (x), labor (I) and transactions: r(t) + Z(t) + S[m(t)] = H ,
0)
where H denotes the (constant) time endowment at period suming well-behaved separable preference and constant-return duction technology,4 the representative agent’s optimization lem can be written as max
x {U[c(t)] + V[x(t)]}e-P’ dt I0
4For the preference, we assume U, > 0, UC, < limUC = lim, V = m, and lim,, U, = lim,, V, = 0 the production technology, we have fk > 0, f; > 0, fkk limbo fk = limb, fi = m, lirnh= fk = lirn&- f; = 0, and
Asproprob-
t.
(2)
0, v, > 0, v,, < 0, for the preference. For < 0, 6, > 0, fU < 0, fi0, I) = f(k, 0) = 0.
185
Ping Wang and Chong K. Yip subject to (1) and the budget c(t) + i(t) + h(t) = f[k(t),
constraint Z(t)] - nk(t) -(7r + n) m(t) + 7(t) ,
(3)
where c and k are (per capita) real consumption and capital stock respectively; n, n and p are the inflation rate, population growth rate and subjective time preference rate respectively; and r denotes the lump-sum real money transfer. The current-value Hamiltonian is then defined by H(k, m, x1, x2, c, 1, z, t) = u(c) + V[I - I - s(m)] + h,[f(k,
1) - nk - (T + n)m
+ 7 - c - z] + A,2) where Ai’s (i = 1, 2) are constate variables and z is a slack variable (z = riL). Applying the Pontryagin’s Maximum Principle to solve for the above optimization problem, we have
-V,+&fi=,O, & = PAI - Adfk - 4 >
(5) (6)
i, = pAz + V,S, + A1(n + n) , (7) -A,+A,=O,
63)
c+&+ti-f(k,Z)+nk+(m+n)m-T=O, :I
(9) emPtAl(t)k(t) = 0,
lim e-“‘A,(t)m(t) t-Pm
= 0;
(11)
together with initial conditions on capital stock and (nominal) money balances as well as nonnegative constraints on c(t), Z(t), k(t), and m(t), for all t L 0. Notice that except for (5), an efficiency condition of labor-leisure choice, and (7), an additional Euler equation of money, all the other conditions are standard in growth theory. Un186
Transactions
Cost
der assumptions of well-behaved preference and technology, the optimal program characterized by (1) and (4)-(11) is the unique, interior competitive perfect foresight equilibrium program.
3. Steady-State Analysis We now construct the steady-state solution from (1) and (4)(11): the particular sohrtion. ((2, 2, 1, ti, hr, A,); IT} such that e = i = 1 = k = ni = Al = A2 = 0. Using the money market equilibrium conditions together with constant money growth presumption, 7 = wm (ex post) and m = m(p - IT - n), where p denotes the money growth rate. Thus, in steady-state, r = t.~ - n, and (9) becomes E=f(k,I)-nl;. Using (6), we have the prototypical (i1 = 0) fi(k
4 =
(159 modified
P +
golden rule equation
(13)
n >
which, under constant-return technology, implies (i/i) is independent of all money matters; this is in contrast to the result in Brock and Turnovsky (1981) but similar to that in Brock (1974) under a money-in-the-utility-function framework. Next, from (4) and (6)-(B), we get -v,s,/u,
= fk
+ Tr .
(14)
The left-hand side of (14) represents the marginal benefit of money in terms of consumption of utils; the right-hand side indicates the marginal cost of money. Combining (4) and (5) yields
fi = VAJC>
(15)
which equates the marginal benefit of labor, fi, with its marginal cost, V,/U,. We use (14) and (15) to get
the fundamental
relation
between
capital-labor
allocation
and real 187
Ping Wang and Chong K. Yip money balances. Moreover, (16) becomes -f(k,
t.~ = IT + n in the steady state, and so
i) S,@) = p + p .
(17)
The equation system formed by (13), (15), and (17) determines the steady-state values of i, i, and CZ. Then, from (1) and (12), we obtain f and c recursively. To see how monetary growth rate affects steady-state capital, labor and real money balances, we totally differentiate (13), (15), and (17) to obtain
where
Let the production
function j(k, I) satisfy the following envelope -Then det (A) < 0 since sign [det (A)] = sign (dk/dp) = sign (dk/dn) which is negative, as in standard comparative-dyneoclassical growth models.6 Hence straightforward namic analysis provides’
property:f; - kfkl > 0.’
di -= dp
fd’xxSm< o det (A)
‘In the class of CES elasticity of substitution % is straightforward
production is bounded to derive
(19)
’ functions, this envelope below by unity. the following comparative
property statics
holds
if the
results:
di d, =& ~’ c%hv,- s-f,w, +hUc+ U,fXfi- &dl . Since it is well documented in growth theory that di/dp < 0 and dk/dn < 0, the assertion of det(A) < 0 immediately follows given the fact that the numerators in both expressions are negative under the envelope property, fi - & > 0. ‘We have used a property of constant-return technology: fhkf” - f; = 0.
188
Transactions
di -=-dp
< o
.tikVx&
de -=
-
&
(20)
’
det (A)
[uccfik!fk,
Cost
-
fifkk)
-
fkkvxrl
<
0 .
&
The result that dfi/dp < 0 contrasts with the result of Fischer (1979) in which the effects of money growth rate on steady-state real money balances is ambiguous. Intuitively, a higher money growth rate leads to a higher inflation and hence a higher cost of money holding, thus depressing real money balances in the steady state. This then leads to a higher transaction effort, and so steady-state labor effort must decrease. Under the assumption of factor complementarity,’ a lower level of steady-state capital stock results, which gives us the “reversed Tobin effect” (see Stockman 1981 for an analagous result under a cash-in-advance setting). Finally, we examine the effect of the rate of growth of money on welfare in the steady state. Totally differentiating (1) and (12) with respect to i.~ and together with (19)-(21), we have (22)
df -
dp
1 =
-
det (A)
Sm[~ccfrk’h~
-
j&k)]
<
o
.
In contrast to Brock (1974), a higher money growth rate depresses steady-state leisure. This is due to the introduction of money into the transactions effort technology through time allocation rather than the inclusion of both money and leisure in the utility function as in Brock (1974), w h ere labor and leisure have to be negatively correlated (thus a higher money growth rate decreases labor but increases leisure). Accordingly, (22) and (23) together imply that an increase in the rate of money growth reduces welfare in the steadystate unambiguously.
4. Conclusion This paper has developed a tractable shopping-time model with endogenous labor-leisure choice to investigate the superneutrality ‘The
inequality
fk, >
0 has to hold
for any
constant-return
production
functions.
189
Ping Wang and Chong K. Yip of money in a perfect foresight economy. It is shown that in the presence of endogenous labor supply, a higher rate of money growth reduces the steady-state capital stock, labor effort, consumption, leisure, and economic welfare unambiguously. This is due to the fact that an increase in money growth rate decreases steady-state real money balances and, through increasing the transaction effort, decreases steady-state labor effort which consequently reduces steadystate capital under the assumption of factor complementarity. Further, the positive correlation between steady-state labor and leisure is due to the introduction of money into transaction effort technology through time allocation. As a concluding remark, it seems to us that the inclusion of uncertainty, the introduction of a distributional effect under heterogeneous capital, and the consideration of time-varying discount rate, as in the literature of recursive utility functions, are interesting extensions for future work. Received: May 1989 Final version: April 1990
References Brock, William A. “Money and Growth: The Case of Long Run Perfect Foresight.” Znternational Economic Review 15 (October 1974): 750-77. Brock, William A., and Stephen J. Turnovsky. “The Analysis of Macroeconomic Policies in Perfect Foresight Equilibrium.” Znternational Economic Review 22 (February 1981): 179-209. Dornbusch, Rudiger, and Jacob A. Frenkel. “Inflation and Growth: Alternative Approaches.” Journal of Money, Credit, and Banking 5 (May 1973): 141-56. on the Transition Path in Fischer, Stanley. “Capital Accumulation a Monetary Optimizing Model.” Econometrica 47 (November 1979): 1433-39. Gray, Jo Anna. “Dynamic Instability in Rational Expectations Models: An Attempt to Clarify.” International Economic Review 25 (February 1984): 93-122. Kimbrough, Kent P. “Inflation, Employment, and Welfare in the Presence of Transactions Costs.” Journal of Money, Credit, and Banking 18 (May 1986): 127-40. Kormendi, Roger C., and Philip G. Meguire. “Macroeconomic Determinants of Growth: Cross-Country Evidence.” Journal of Monetary Economics 16 (September 1985): 141-63. 190
Transactions
Cost
Lucas, Robert E., Jr. “Expectations and the Neutrality of Money.” Journal of Economic Theory 4 (April 1972): 103-24. Sidrauski, Miguel “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Review Papers and Proceedings 57 (May 1967): 534-44. Stockman, Alan C. “Anticipated Inflation and the Capital Stock in a Cash-In-Advance Economy.” Journal of Monetary Economics 8 (November 1981): 387-93. Tobin, James. “Money and Economic Growth.” Econometrica 33 (October 1965): 671-84. Wang, Ping, and Chong K. Yip. “Alternative Approaches to Money and Growth.” The Pennsylvania State University, University Park, Pennsylvania, 1990. Mimeo.
191
Pennsylvania University
State Park,
Unioersity
Pennsylvania
CHONG K. YIP Georgia
State Atlanta,
Uniuersity Georgia
Transactions Cost, Enciogenous Labor, and the Superneutrality of Money This paper develops a tractable perfect-foresight-shopping-time model with endogenous labor-leisure choice to investigate the superneutrality of money. Different from the existing literature, we examine the effects of money growth on both employment and capital accumulation. Under the lump-sum money transfer, we show that an increase in money growth rate unambiguously reduces steady-state capital, labor, leisure, consumption, and welfare, contrary to previous work. Since money is introduced into transaction effort technology via time allocation, the positive correlation between steady-state labor and leisure emerges.
1. Introduction This paper develops a neoclassical perfect foresight monetary growth model with an endogenous labor-leisure choice to examine the superneutrality of money. Money interacts with labor effort and hence physical capital accumulation through a shopping-time technology rather than directly in the utility function. The shoppingtime technology proposed in the paper is not only tractable but provides more intuitive results than the conventional ones in the literature. With lump-sum money transfer, an increase in the rate of money growth unambiguously reduces steady-state capital, labor, leisure, and economic welfare, in contrast with previous work. Since the classic papers of Tobin (1965) and Sidrauski (1967), the effect of anticipated inflation on capital accumulation has been one of the central issues in monetary growth theory. Using a portfolio-choice framework, Tobin (1965) shows that higher steady-state inflation would lead to a higher steady-state capital stock. This Tobin effect is questioned by Sidrauski (1967) who, by considering consumers’ rational choice, concludes that the invariance of the steady-state real discount rate to inflation would result in superneutrality of money. However, as pointed out by Fischer (1979), money is not superneutral on transition paths in the Sidrauski model. Further, even in the steady state, money may still have real effects Journal of Macroeconomics, Winter 1991, Vol. Copyright 0 1991 by Louisiana State University 0164-0704/91/$1.50
13, No. Press
1, pp.
183-191
183
Ping Wang and Chong K. Yip in more sophisticated models. For instance, in his pivotal work, Brock (1974) investigates the implications of endogenous labor-leisure choice for the superneutrality of money by putting both real money balances and leisure in the utility function and concludes that the dependence of the marginal utility of leisure on money “destroys the long run neutrality of money.” “[Elven with constant returns the steady-state quantity of real capital will be changed by the rate of growth of money” (Brock 1974, 774). However, his comparative dynamic results depend heavily on the cross partial derivatives of the utility function which generally have little economic intuition. Dornbusch and Frenkel (1973) examine this issue via a shopping effort function with pecuniary transactions costs.’ In their model, money enters the system purely through a positive wealth effect due to a reduction of pecuniary transactions costs (in terms of unit of goods produced). The net effect of the rate of money growth on the steady-state capital stock is thus ambiguous. By considering the transactions service of money, Kimbrough (1986) constructs a shopping-time technology in which. shopping effort per unit of consumption is decreasing and convex in the money-consumption ratio. Nevertheless, due to its analytical difficulty, his model focuses on the analysis of employment and abstracts from capital accumulation. On the other hand, Stockman (1981) generates the nonsupemeutrality of money in the absence of endogenous labor by imposing a cash-in-advance constraint on both consumption and capital goods.2 The contribution of our paper is to develop a tractable shopping-time model of money with endogenous labor-leisure choice through which the effects of money growth on both employment and capital accumulation can be examined. Under lump-sum money transfer, we show that an increase in money growth rate unambiguously reduces steady-state capital,3 in contrast to Dornbusch and Frenkel (1973). The resulting “reversed Tobin effect” is consistent with the empirical finding of Kormendi and Meguire (1985). Contrary to Brock and Tumovsky (1981), we find that the steady-statecapital-labor ratio is independent of monetary factors. Moreover, an
‘For discussions on the properties of shopping transactions costs, the reader is referred to the *For a detailed discussion and survey of the ferred to Wang and Yip (1990). 3Notice that if the money transfer scheme is becomes superneutral in a steady-state analysis,
184
effort functions with pecuniary paper by Gray (1984). above literature, the reader is reproportional even with
(Lucas 1972), money endogenous labor.
Transactions
Cost
increase in the rate of money growth depresses consumption as well as leisure and hence unambiguously reduces welfare in the steady state, which contrasts with results in the existing literature (for example, see Brock 1974). The remainder of the paper is organized as follows. In the next section, we present the basic model, while Section 3 conducts the steady-state analysis. Finally, there is a fourth and concluding section.
2. The Model Consider a representative-agent neoclassical monetary growth model with endogenous labor. Money enters into the optimization problem through a shopping-time technology with transactions effort, S, depending only on (per capita) real money balances, m, that is, S = S(m). This formulation of shopping effort is not only tractable but provides more intuitive results than the conventional one in the literature. To provide an optimal quantity of real balances, we impose a satiation level of money (m*) which attains the minimum of shopping effort, that is, S(m*) = S,(m*) = 0, where m* < cQ. When real money balances are below the satiation level, S is assumed to be decreasing in m and the technology exhibits diminishing returns to money holdings, that is, S, < 0 and S,, > 0 for m < m*. Finally, to insure the existence of money, we further assume limnrtO S, = --03. Each agent is endowed with one unit of time which must be allocated to leisure (x), labor (I) and transactions: r(t) + Z(t) + S[m(t)] = H ,
0)
where H denotes the (constant) time endowment at period suming well-behaved separable preference and constant-return duction technology,4 the representative agent’s optimization lem can be written as max
x {U[c(t)] + V[x(t)]}e-P’ dt I0
4For the preference, we assume U, > 0, UC, < limUC = lim, V = m, and lim,, U, = lim,, V, = 0 the production technology, we have fk > 0, f; > 0, fkk limbo fk = limb, fi = m, lirnh= fk = lirn&- f; = 0, and
Asproprob-
t.
(2)
0, v, > 0, v,, < 0, for the preference. For < 0, 6, > 0, fU < 0, fi0, I) = f(k, 0) = 0.
185
Ping Wang and Chong K. Yip subject to (1) and the budget c(t) + i(t) + h(t) = f[k(t),
constraint Z(t)] - nk(t) -(7r + n) m(t) + 7(t) ,
(3)
where c and k are (per capita) real consumption and capital stock respectively; n, n and p are the inflation rate, population growth rate and subjective time preference rate respectively; and r denotes the lump-sum real money transfer. The current-value Hamiltonian is then defined by H(k, m, x1, x2, c, 1, z, t) = u(c) + V[I - I - s(m)] + h,[f(k,
1) - nk - (T + n)m
+ 7 - c - z] + A,2) where Ai’s (i = 1, 2) are constate variables and z is a slack variable (z = riL). Applying the Pontryagin’s Maximum Principle to solve for the above optimization problem, we have
-V,+&fi=,O, & = PAI - Adfk - 4 >
(5) (6)
i, = pAz + V,S, + A1(n + n) , (7) -A,+A,=O,
63)
c+&+ti-f(k,Z)+nk+(m+n)m-T=O, :I
(9) emPtAl(t)k(t) = 0,
lim e-“‘A,(t)m(t) t-Pm
= 0;
(11)
together with initial conditions on capital stock and (nominal) money balances as well as nonnegative constraints on c(t), Z(t), k(t), and m(t), for all t L 0. Notice that except for (5), an efficiency condition of labor-leisure choice, and (7), an additional Euler equation of money, all the other conditions are standard in growth theory. Un186
Transactions
Cost
der assumptions of well-behaved preference and technology, the optimal program characterized by (1) and (4)-(11) is the unique, interior competitive perfect foresight equilibrium program.
3. Steady-State Analysis We now construct the steady-state solution from (1) and (4)(11): the particular sohrtion. ((2, 2, 1, ti, hr, A,); IT} such that e = i = 1 = k = ni = Al = A2 = 0. Using the money market equilibrium conditions together with constant money growth presumption, 7 = wm (ex post) and m = m(p - IT - n), where p denotes the money growth rate. Thus, in steady-state, r = t.~ - n, and (9) becomes E=f(k,I)-nl;. Using (6), we have the prototypical (i1 = 0) fi(k
4 =
(159 modified
P +
golden rule equation
(13)
n >
which, under constant-return technology, implies (i/i) is independent of all money matters; this is in contrast to the result in Brock and Turnovsky (1981) but similar to that in Brock (1974) under a money-in-the-utility-function framework. Next, from (4) and (6)-(B), we get -v,s,/u,
= fk
+ Tr .
(14)
The left-hand side of (14) represents the marginal benefit of money in terms of consumption of utils; the right-hand side indicates the marginal cost of money. Combining (4) and (5) yields
fi = VAJC>
(15)
which equates the marginal benefit of labor, fi, with its marginal cost, V,/U,. We use (14) and (15) to get
the fundamental
relation
between
capital-labor
allocation
and real 187
Ping Wang and Chong K. Yip money balances. Moreover, (16) becomes -f(k,
t.~ = IT + n in the steady state, and so
i) S,@) = p + p .
(17)
The equation system formed by (13), (15), and (17) determines the steady-state values of i, i, and CZ. Then, from (1) and (12), we obtain f and c recursively. To see how monetary growth rate affects steady-state capital, labor and real money balances, we totally differentiate (13), (15), and (17) to obtain
where
Let the production
function j(k, I) satisfy the following envelope -Then det (A) < 0 since sign [det (A)] = sign (dk/dp) = sign (dk/dn) which is negative, as in standard comparative-dyneoclassical growth models.6 Hence straightforward namic analysis provides’
property:f; - kfkl > 0.’
di -= dp
fd’xxSm< o det (A)
‘In the class of CES elasticity of substitution % is straightforward
production is bounded to derive
(19)
’ functions, this envelope below by unity. the following comparative
property statics
holds
if the
results:
di d, =& ~’ c%hv,- s-f,w, +hUc+ U,fXfi- &dl . Since it is well documented in growth theory that di/dp < 0 and dk/dn < 0, the assertion of det(A) < 0 immediately follows given the fact that the numerators in both expressions are negative under the envelope property, fi - & > 0. ‘We have used a property of constant-return technology: fhkf” - f; = 0.
188
Transactions
di -=-dp
< o
.tikVx&
de -=
-
&
(20)
’
det (A)
[uccfik!fk,
Cost
-
fifkk)
-
fkkvxrl
<
0 .
&
The result that dfi/dp < 0 contrasts with the result of Fischer (1979) in which the effects of money growth rate on steady-state real money balances is ambiguous. Intuitively, a higher money growth rate leads to a higher inflation and hence a higher cost of money holding, thus depressing real money balances in the steady state. This then leads to a higher transaction effort, and so steady-state labor effort must decrease. Under the assumption of factor complementarity,’ a lower level of steady-state capital stock results, which gives us the “reversed Tobin effect” (see Stockman 1981 for an analagous result under a cash-in-advance setting). Finally, we examine the effect of the rate of growth of money on welfare in the steady state. Totally differentiating (1) and (12) with respect to i.~ and together with (19)-(21), we have (22)
df -
dp
1 =
-
det (A)
Sm[~ccfrk’h~
-
j&k)]
<
o
.
In contrast to Brock (1974), a higher money growth rate depresses steady-state leisure. This is due to the introduction of money into the transactions effort technology through time allocation rather than the inclusion of both money and leisure in the utility function as in Brock (1974), w h ere labor and leisure have to be negatively correlated (thus a higher money growth rate decreases labor but increases leisure). Accordingly, (22) and (23) together imply that an increase in the rate of money growth reduces welfare in the steadystate unambiguously.
4. Conclusion This paper has developed a tractable shopping-time model with endogenous labor-leisure choice to investigate the superneutrality ‘The
inequality
fk, >
0 has to hold
for any
constant-return
production
functions.
189
Ping Wang and Chong K. Yip of money in a perfect foresight economy. It is shown that in the presence of endogenous labor supply, a higher rate of money growth reduces the steady-state capital stock, labor effort, consumption, leisure, and economic welfare unambiguously. This is due to the fact that an increase in money growth rate decreases steady-state real money balances and, through increasing the transaction effort, decreases steady-state labor effort which consequently reduces steadystate capital under the assumption of factor complementarity. Further, the positive correlation between steady-state labor and leisure is due to the introduction of money into transaction effort technology through time allocation. As a concluding remark, it seems to us that the inclusion of uncertainty, the introduction of a distributional effect under heterogeneous capital, and the consideration of time-varying discount rate, as in the literature of recursive utility functions, are interesting extensions for future work. Received: May 1989 Final version: April 1990
References Brock, William A. “Money and Growth: The Case of Long Run Perfect Foresight.” Znternational Economic Review 15 (October 1974): 750-77. Brock, William A., and Stephen J. Turnovsky. “The Analysis of Macroeconomic Policies in Perfect Foresight Equilibrium.” Znternational Economic Review 22 (February 1981): 179-209. Dornbusch, Rudiger, and Jacob A. Frenkel. “Inflation and Growth: Alternative Approaches.” Journal of Money, Credit, and Banking 5 (May 1973): 141-56. on the Transition Path in Fischer, Stanley. “Capital Accumulation a Monetary Optimizing Model.” Econometrica 47 (November 1979): 1433-39. Gray, Jo Anna. “Dynamic Instability in Rational Expectations Models: An Attempt to Clarify.” International Economic Review 25 (February 1984): 93-122. Kimbrough, Kent P. “Inflation, Employment, and Welfare in the Presence of Transactions Costs.” Journal of Money, Credit, and Banking 18 (May 1986): 127-40. Kormendi, Roger C., and Philip G. Meguire. “Macroeconomic Determinants of Growth: Cross-Country Evidence.” Journal of Monetary Economics 16 (September 1985): 141-63. 190
Transactions
Cost
Lucas, Robert E., Jr. “Expectations and the Neutrality of Money.” Journal of Economic Theory 4 (April 1972): 103-24. Sidrauski, Miguel “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Review Papers and Proceedings 57 (May 1967): 534-44. Stockman, Alan C. “Anticipated Inflation and the Capital Stock in a Cash-In-Advance Economy.” Journal of Monetary Economics 8 (November 1981): 387-93. Tobin, James. “Money and Economic Growth.” Econometrica 33 (October 1965): 671-84. Wang, Ping, and Chong K. Yip. “Alternative Approaches to Money and Growth.” The Pennsylvania State University, University Park, Pennsylvania, 1990. Mimeo.
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