- Email: [email protected]
Examining the long-run effect of money on economic growth
PING WANG The
Pennsylvania University
State Park,
University
Pennsylvania
CHONG K. YIP Georgia
State Atlanta,
Examining the Long-Run Ekt Money on Ewnomic Growth*
University Georgia
of
significant Granger-causal relaChristiano and Ljungqvist (1988) find a statistically tion between money and output when data are measured in log levels, but not when they use data in log differences. To resolve this puzzle, we develop an endogenous growth model with money to study the long-run interactions between real and monetary sectors. Money is regarded as a Hicks-neutral technological factor that improves the efficiency of goods production. We examine the effects of anticipated inflation on the growth rates of real macroeconomic variables and find that money is “superneutral” in the growth rate sense.
1. Introduction The real effects of monetary growth have received increasing attention from macroeconomists since the appearance of the seminal papers of Tobin (1965), Sidrauski (1967) and Brock (1974). The literature generally analyzes the steady-state properties of the system, particularly the impact of an increase in the growth rate of money on the steady-state levels of (per capita) consumption, (per capita) real money balances and capital-labor ratio.’ Given the fact that many countries have experienced sustained growth in per capita consumption and output, it seems desirable to examine the growth rates of economic aggregates rather than their steady-state levels. The recent development of endogenous growth theory holds considerable promise for enhancing our understanding in this as-
*We thank Lam Ljungqvist and Sergio Rebel0 as two anonymous referees for helpful comments. knowledges the financial support provided by the mittee of the Georgia State University. The usual ‘For a detailed discussion on the steady-state the reader is referred to Dornbusch and Frenkel
for valuable discussions as well The second author gratefully acCBA Research Program Comdisclaimer applies. analysis of inflation and growth, (1973).
1992, Vol. Journal of Macroeconomics, Spring Copyright @ 1992 by Louisiana State University 0164-0794/92/$1.50
No. Press
14,
2, pp.
359-369
359
Ping Wang and Chong K. Yip pect.’ However, most of the analyses concentrate on the contribution of real factors on economic growth, and thus the role of money remains open for research. In this paper, we attempt to analyze the effect of an increase in the rate of money growth on the growth rates of economic aggregates. This is done by extending the perfectforesight, balanced-growth model of Lucas (1988) to allow for endogenous accumulation of real balances. An important contribution of this endogenous monetary growth model is to enable us to address the issue of “superneutrality” of money, that is, whether there exist long-run relations between the money growth rate and growth rates of consumption, output and factor inputs. In a recent seminal paper, Christian0 and Ljungqvist (1988) raised an “empirical puzzle” by revisiting the issue of a bivariate Granger-causality test on money and output. Using data in log levels, they found that money strongly Granger-causes output. However, such a causal relation fails to hold up when first differences of the logged data are used. It is not diificult to theoretically derive the non-superneutrality result in the level sense. However, there is, to our knowledge, no existing literature that can lend theoretical support to the superneutrality between the growth rates of money and output. This paper investigates the effects of anticipated inflation on growth rates of consumption and real balances, as well as on the accumulation of human and physical capital. Due to its transactions service, money is regarded as a Hicks-neutral technological factor that improves the efficiency of goods production. Notice that our way to incorporate money essentially captures the money-in-theproduction-function framework developed by Levhari and Patinkin (1968) and Fischer (1974), as well as the transactions costs model constructed by Dornbusch and Frenkel (1973) and Wang and Yip (1991). The view that money is an input into the production process is also consistent with King and Plosser (1984), in which money is introduced into a real business cycle model through a transactive ?‘he generation of endogenous growth without depending upon exogenous changes in population or technology has become one of the central issues in growth theory. This renewal of growth theory offers an explanation for perpetual increases in per capita living standards as an equilibrium outcome of rational choice. The endogenous evolution of human capital is a major force among others driving economic growth. Examples of endogenous growth models include Lucas (1988), Barro (1990) and Rebel0 (1991). Other models in this literature amount to theories of technological progress (Romer 1990) and population change (Becker and Murphy 1988). However, they all abstract from money matters.
380
Examining
the Long-Run
Effect
of Money on Economic
Growth
financial intermediary. Within the money-in-the-production-function framework, it is well documented that money is non-superneutral in the level sense.3 The main novelty of our paper is that we find the growth rate of money creates no long-run effects on the growth rates of consumption, output and human and physical in terms of its imcapital. In other words, money is “superneutral’ pacts on economic growth rates. This, therefore, provides a theoretical interpretation for the empirical puzzle raised by Christian0 and Ljungqvist (1988). The organization of the paper is as follows. In the next section, we develop the basic monetary endogenous growth model. Section 3 studies the balanced-growth equilibrium and examines the role of money in an endogenously growing economy. There is a fourth, concluding section.
2. The Model Consider a modified continuous-time, representative-agent, perfect-foresight model of neoclassical monetary growth in which both physical and human capital are endogenously determined. For simplicity, leisure is assumed inelastic.4 The representative agent’s optimization problem can be written as CD maxW=
U(c(t))e+dt I0
subject to c(t) + k(t) + k(t) = F@(t), L(t), m(t)) - d(t)
- (n + 7r(t))m(t) + 7(t) ,
h(t) = 40 - WMt) >
0) (2)
‘This is because the introduction of real balances improves the efficiency of production and allows us to have a permanently higher level of per capita output. ‘This is a common assumption in endogenous growth models; see Lucas (1988), for example. Moreover, it can be shown that endogenizing leisure in the current framework does not alter the growth superneutrality result. Specifically, to allow for variable non-leisure time, one can generalize the instantaneous utility function as U(c, ~.!a) = {[&A)‘-@I’-” - l}/(l - a) and the human capital accumulation equation as h = +(l - t? - X)/I (see Rebel0 1991). Our main result on growth superneutrality still holds, although the equilibrium solutions of some endogenous variables along a balanced growth path may have changed (for example, Equation [17] will be different).
361
Ping Wang and Chong K. Yip where c and m are (per capita) consumption and real money balances, respectively; k and L = he represent (per capita) physical capital and effective labor input, respectively, with h denoting the human capital skill level and 4 indicating the fraction of non-leisure time allocated to production (and hence, 1 - e is the fraction of non-leisure time allocated to education); T is (per capita) lump-sum transfer from the government; 1~ denotes the inflation rate; p, n and 4 are (constant) rates of time preferences, populaton growth and maximal human capital accumulation, respectively. Equation (1) is a modified Sidrauski (1967) budget constraint in which money enters the production function, F. Equation (2) is the Uzawa (RX%)-Rosen (1976)-Lucas (1988) human capital evolution equation. According to (2), if no effort is devoted to education (that is, e = l), then no betterment in human capital can emerge. If one puts all the effort into accumulating human capital (that is, 4 = 0), human capital will grow at its maximal rate, +. In order to obtain feasible steady-state growth, we choose the utility function U to exhibit constant intertemporal elasticity of substitution and the production function F to take the Cobb-Douglas form (see King, Plosser and Rebel0 1988). Specifically, we have U(c) = (P - l)/(l - cx) and F(k, L, m) = A(m)kYLlmY, where cx is the inverse of the intertemporal elasticity of substitution and y is the capital income share.5 Notice that money is introduced via a Hicksneutral production technology, A.6 For tractability, we assume that the marginal technological improvement of money, A,,,(m), is positive and diminishing, that is, A,,, > 0 and A,,,,,, < 0. Moreover, to be consistent with sustained growth in the steady state, we further restrict A to have constant output and marginal output elasticities of money, that is, u = mA,/A and E = -mA,,,,,,/A, are constant (see a discussion in Section 3 below). Let i.~ > 0 be the (constant) rate of money growth. Money market equilibrium requires r(t) = pm(t). Let ti/m = q. By delinition, ti(t)/m(t)
= p - a(t) - n .
(3)
SThese specifications have been often used in endogenous growth models, for instance, see Lucas (1888) and Barre (1990). ‘It seems to us the Hicks-neutral technological progress is a natural choice of modeling money in the production function. It is hard to imagine that money improves the efficiency biased to either one of the production factors as represented by the Harrodand Solow-neutral technological progress. Further, we have assumed that human capital improves production technology in a Harrod-neutral manner.
382
Examining
the Long-Run
Thus, in equilibrium,
Effect
of Money
(1) can be rewritten
on Economic
as
c(t) + k(t) = F(k(t), L(t), m(t)) - nk(t) ,
which
is in fact the goods market
3. Balanced
Growth
Growth
equilibrium
(4)
condition.
Analysis
In this section, we perform a balanced growth analysis to solve for an optimal endogenous monetary growth equilibrium. First denote Al, AZ and A3 as costate variables of the current Hamiltonian associated with (l), (2) and the slack variable identity, z = +x.7 By definition, the rate of growth of each endogenous variable is constant along a balanced growth path.8 Denote 0 as the constant growth that is, e = d/c. Straightforward rate of per capita consumption, application of Pontryagin’s Maximum Principle yields
X,/x, = -ae ,
(5)
Fk = p + n + di ,
(7)
F, = p + n + 7~ + cd,
(8)
A1= A, ,
(9)
and
LA2
= p - : [F, - +(l - 4) , 2
together with (2), (4) and the transversality conditions of k, h and m: lim,,e -@A,(t)q(t) = 0, where 4 = k, h and m for i = 1, 2 and 3, respectively. ‘The
Hamiltonian
of the
control
problem
is then
given
x = U(c) + X,[F(k, L, m) - nk - (n + T)rn + X&l - q/l + h,z ‘We
follow
Lucas’s
(1988,
9) definition
of the
“balanced
as + 7 - c - z]
growth
path.”
363
Ping Wang and Chong K. Yip Equations (5) and (6) are efficiency conditions for goods and time allocation. They imply that goods must be equally valuable in consumption and capital accumulation, and time must be equally valuable in current production and education. Equations (7) and (8) are Keynes-Ramsey rule equations for capital and money that determine real and nominal interest rates (Fk and F,, respectively). We can also learn from these equations that the marginal products of capital and money are constant along a balanced growth path. From (9), the shadow prices of capital and money must be equal because the only function of capital and money is to produce goods. Finally, (10) represents the golden rule equation for human capital. Next, (6) and (10) together give &./A2 = p - t+ . Differentiating
(11)
(6)-(8) with respect to time implies
and l&/F,
= i,/X,
- );,/A, .
(13)
Thus, along a balanced growth path, only the marginal product of effective labor changes over time with its rate of change being determined by the difference between the shadow prices of human and physical capital. Further, from (7), we have F/k = FJy = (p + n + aO)/y. Then manipulating (4), we obtain the consumptioncapital ratio c/k = (p + n + aO)/y - n - i/k,
(14)
which is constant along balanced growth paths. Thus (per capita) consumption and physical capital have to grow at the same rate, 8. Further denote per capita output as y, that is, y = F(k, L, m). Then the constancy of the marginal product of capital and F/k = Fk/y together imply output and capital have to grow at the same rate, too. Thus we derive the following relation for the growth rates of consumption, capital and output: d/c = i/k 364
= zj/y = 8 ,
(15)
Examining
the Long-Run
Effect of Money on Economic
Growth
which is a standard result in endogenous growth theory (for example, Lucas 1988). Further, as an immediate consequence of (2), 4 has to be constant along a balanced growth path. Since FL = (1 - y)F/L, this together with (5), (ll), (13) and (15) give the following expression for the growth rate of effective labor:
v = i/L Combining
= h/h = 4 - p -(a
- l)e.
(16)
(2) and (16), we have
e = [p + (a - l)c.I]/4.
(17)
The interiority restriction of e E (0, 1) requires that -p < (a - 1)6 < 4 - p which provides upper and lower bounds for 6; the first inequality, p - (1 - ~$3 > 0, also guarantees that the lifetime utility is bounded. Moreover, applying (16), this interiority restriction ensures a positive rate of human capital evolution (that is, v > 0). There are two important implications. First, (17) implies that a wellskilled person (with a higher 4) can work less given the same common balanced growth rate, 9. Second, (16) indicates that whether the growth rate of human capital skill level is positively correlated with the common growth rate of per capita consumption and physical capital depends on the magnitude of the inter-temporal elasticity of substitution (a-‘). Specifically, h/h and 8 will have a positive correlation if ci-’ > 1.’ Next, recall that u = mA,/A and E = -mA,,,,/A,. By the definition of balanced growth, both A/A and &,,/A, must be constant. Since A/A = aq and A,/A, = --ET), u and E must be constant along a balanced growth path. Totally differentiating Fk = yAkY-‘L’-Y and F,,, = A,,,kYLleY with respect to time and manipulating, we obtain
8 = (CT+ ~)7)
(18)
and v = (1 - Y)(P - 4 + 4/u.
(19)
‘The condition seems reasonable: it is consistent with the property of factor complementarity obtained from constant-return production technology (that is, physical capital and human capital are positively correlated).
Ping Wang and Chong K. Yip Combining
(16) with (18) and (19), we get 8 = B-‘(+ - p)(l - y) rl = m+
- PM - rm
F-3 + 4 >
(21)
and u = B-‘(ql - p)[l - y - a/(a + E)] )
(22)
where B = o(1 - y) - ~/(a + E). In Lucas (1988), 4 is estimated to be 5% and thus is higher than standard values of the time preference rates which is around 2% (for example, see Barr-o 1990). Therefore, we expect + - p > 0. In observing the positive rate of economic growth, one should restrict B > 0. Since in general we believe 0~~’ > 1 and B > 0, v is positive as well. In fact, the growth rates of real macroeconomic variables (0, Y, q) depend only on preference and production parameters. Specifically, the higher the rate of time preferences (p) or the degree of diminishing returns to money (E) is, the lower these growth rates will be. Further, these growth rates are positively correlated with the maximal human capital accumulation rate (+), the intertemporal elasticity of substitution (o-l) and the productivity of money (a). In addition, the effects of higher capital income share (y) are to increase capital and real balances accumulation but to reduce human capital evolution. According to (20)-(22), the growth rates of real macroeconomic aggregates are independent of the money growth rate, that is, dv/dp = dtl/dp = dq/dp = 0. Since human capital accumulation is the main engine of growth in this model, the result that dv/dp = 0 implies that money does not affect the growth process. This, therefore, explains why we obtain the superneutrality of money in the “growth rates” sense, although due to the productive efficiency of real money balances, higher money growth would still lower the “level” of per capita quantities. This finding also provides a theoretical framework to resolve the empirical puzzle raised by Christian0 and Ljungqvist (1988). Further, from (3), (8) and q = p - r - n, it can be shown that dr/dF = dF,,,/dp = 1. Thus, in our economy, the Fisherian prediction that the nominal interest rate adjusts completely to anticipated inflation is validated. To search for the optimal growth rate of money, we examine how the lifetime utility is affected by monetary policy. Along the balanced growth path, the growth rate of per capita consumption 366
Examining
the Long-Run
(0) is constant. a constant)
The lifetime
Effect
of Money on Economic
utility
(W) is then given by (aside from
[4w-” w = (1 - c.u)[p- (1 - ($31’ Thus the condition p From (14), we have
(1 - cx)e > 0 ensures utility
40) = W)hl{[p + (1 - rbl + (a - rM s
Growth
(23) is bounded.
(24)
It is apparent from (20) and (24) that both 8 and c(0) are independent of the money growth rate, IL. Therefore, in the steady state, our model does not yield an optimal growth rate of money. Finally, it is worth noting that although we have found superneutrality (in the growth rate sense) in the steady state, it is not clear whether this result can emerge in transition. An increase in the growth rate of money may alter the ratio of physical to human capital and thus the growth neutrality result may break down in transition. It is, however, extremely difficult to derive the transitional dynamics in the present model since endogenous variables do not grow at a common rate. lo
4. Concluding
Remarks
This paper provides a first attempt to incorporate money into an endogenous growth model via a Hicks-neutral production technology. We show that higher money growth rate increases the inflation rate proportionately. This then implies the absence of a real balance effect on the rates of physical and human capital accumulation and consumption and output growth. Therefore, money is “superneutral” along the balanced growth path although it is nonsuperneutral in the “level” sense. This helps us in understanding the empirical puzzle of Christian0 and Ljungqvist (1988) that there exists a statistically significant Granger-causal relation between money
“‘The standard method of analyzing transitional dynamics in sustained-growth models is to transform all endogenous quantity variables in the unit of the factor that drives common growth. Subsequently, one can linearize the dynamic system around the steady state to study local dynamics (King, Plosser and Rebel0 1968). This method cannot be applied here because we do not have a common growth component.
367
Ping Wang and Chong K. Yip and output when data are measured in log-level, but not when first differences of the logged data are used. Interesting extensions of the present model include the consideration of a productive government sector and the construction of a more general form for the evolution of human capital.” The former may allow us to investigate the interactions between fiscal and monetary policies and their long-run impacts on the real economy. The latter may help enhance the long-run responses of consumption growth to money growth. However, an adequate treatment of the above suggestions will require further simplification of the current framework in order to obtain a closed-form solution and unambiguous comparative dynamic results. Finally, we would like to caution the readers that the growth neutrality result may not emerge under different setups. For instance, as shown in Rebel0 (1988) using a cash-in-advance (CIA) model, if the CIA constraint is imposed on the purchase of a broadly defined investment good, an increase in the growth rate of money will retard the growth process. Nevertheless, whether this non-neutrality effect on economic growth is quantitatively important remains uninvestigated. Received: October 1990 Final version: July 1991
References Barro, Robert J. “Government Spending in a Simple Model of Endogenous Growth.” Journal of Political Economy 98 (October 1999): S103-25. Becker, Gary S., and Kevin M. Murphy. “Economic Growth, Human Capital and Population Growth.” Working Paper, University of Chicago, 1988. Brock, William A. “Money and Growth: The Case of Long Run Perfect Foresight.” Znternational Economic Review 15 (October 1974): 750-77. Christiano, Lawrence J., and Lars Ljungqvist. “Money Does Granger-Cause Output in the Bivariate Money-Output Relation.” Journal of Monetary Economics 22 (September 1988): 217-35. Dombusch, Rudiger, and Jacob A. Frenkel. “Inflation and Growth:
“The reader is referred to government into the endogenous incorporation of physical capital
368
Barro (1990) for the introduction of a productive growth framework and to Rebel0 (1991) for the into the human capital evolution equation.
Examining
the Long-Run
Effect
of Money
on Economic
Growth
Alternative Approaches.” Journal of Money, Credit, and Banking 5 (May 1973): 141-56. Economic Fischer, Stanley. “Money and the Production Function.” Inquiry 12 (November 1974): 517-33. King, Robert G., and Charles I. Plosser. “Money, Credit, and Prices in a Real Business Cycle.” American Economic Review 74 (June 1984): 363-80. King, Robert G., Charles I. Plosser, and Sergio T. Rebelo. “Production, Growth and Business Cycles: I. The Basic Neoclassical Growth Model.” Journal of Monetary Economics 21 (March/May 1988): 195-232. Levhari, David, and Don Patinkin. “The Role of Money in a Simple Growth Model.” American Economic Review 58 (September 1968): 713-53. Lucas, Robert E., Jr. “On the Mechanics of Economic Development.” Journal of Monetary Economics 23 (July 1988): 3-42. Rebelo, Sergio T. “Essays on Growth and Business Cycles.” Ph.D. diss., University of Rochester, 1988. -. “Long-Run Policy Analysis and Long-Run Growth. ” Journal of Political Economy 99 (June 1991): 500-21. Romer, Paul M. “Endogenous Technological Change.” Journal of PoZiticaZ Economy 98 (October 1990): S71-102. Rosen, Sherwin. “A Theory of Life Earnings.” Journal of Political Economy 84 (August 1976): S45-67. Sidrauski, Miguel. “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Review: Papers and Proceedings 57 (May 1967): 534-44. Tobin, James. “Money and Economic Growth.” Econometrica 33 (October 1965): 671-84. Uzawa, Hirommi. “Optimal Technical Change in an Aggregate Model of Economic Growth.” International Economic Review 6 (January 1965): 18-31. Wang, Ping, and Chong K. Yip. “Transactions Cost, Endogenous Labor, and the Superneutrality of Money.” Journal of Macroeconomics 13 (Winter 1991): 183-91.
369
Pennsylvania University
State Park,
University
Pennsylvania
CHONG K. YIP Georgia
State Atlanta,
Examining the Long-Run Ekt Money on Ewnomic Growth*
University Georgia
of
significant Granger-causal relaChristiano and Ljungqvist (1988) find a statistically tion between money and output when data are measured in log levels, but not when they use data in log differences. To resolve this puzzle, we develop an endogenous growth model with money to study the long-run interactions between real and monetary sectors. Money is regarded as a Hicks-neutral technological factor that improves the efficiency of goods production. We examine the effects of anticipated inflation on the growth rates of real macroeconomic variables and find that money is “superneutral” in the growth rate sense.
1. Introduction The real effects of monetary growth have received increasing attention from macroeconomists since the appearance of the seminal papers of Tobin (1965), Sidrauski (1967) and Brock (1974). The literature generally analyzes the steady-state properties of the system, particularly the impact of an increase in the growth rate of money on the steady-state levels of (per capita) consumption, (per capita) real money balances and capital-labor ratio.’ Given the fact that many countries have experienced sustained growth in per capita consumption and output, it seems desirable to examine the growth rates of economic aggregates rather than their steady-state levels. The recent development of endogenous growth theory holds considerable promise for enhancing our understanding in this as-
*We thank Lam Ljungqvist and Sergio Rebel0 as two anonymous referees for helpful comments. knowledges the financial support provided by the mittee of the Georgia State University. The usual ‘For a detailed discussion on the steady-state the reader is referred to Dornbusch and Frenkel
for valuable discussions as well The second author gratefully acCBA Research Program Comdisclaimer applies. analysis of inflation and growth, (1973).
1992, Vol. Journal of Macroeconomics, Spring Copyright @ 1992 by Louisiana State University 0164-0794/92/$1.50
No. Press
14,
2, pp.
359-369
359
Ping Wang and Chong K. Yip pect.’ However, most of the analyses concentrate on the contribution of real factors on economic growth, and thus the role of money remains open for research. In this paper, we attempt to analyze the effect of an increase in the rate of money growth on the growth rates of economic aggregates. This is done by extending the perfectforesight, balanced-growth model of Lucas (1988) to allow for endogenous accumulation of real balances. An important contribution of this endogenous monetary growth model is to enable us to address the issue of “superneutrality” of money, that is, whether there exist long-run relations between the money growth rate and growth rates of consumption, output and factor inputs. In a recent seminal paper, Christian0 and Ljungqvist (1988) raised an “empirical puzzle” by revisiting the issue of a bivariate Granger-causality test on money and output. Using data in log levels, they found that money strongly Granger-causes output. However, such a causal relation fails to hold up when first differences of the logged data are used. It is not diificult to theoretically derive the non-superneutrality result in the level sense. However, there is, to our knowledge, no existing literature that can lend theoretical support to the superneutrality between the growth rates of money and output. This paper investigates the effects of anticipated inflation on growth rates of consumption and real balances, as well as on the accumulation of human and physical capital. Due to its transactions service, money is regarded as a Hicks-neutral technological factor that improves the efficiency of goods production. Notice that our way to incorporate money essentially captures the money-in-theproduction-function framework developed by Levhari and Patinkin (1968) and Fischer (1974), as well as the transactions costs model constructed by Dornbusch and Frenkel (1973) and Wang and Yip (1991). The view that money is an input into the production process is also consistent with King and Plosser (1984), in which money is introduced into a real business cycle model through a transactive ?‘he generation of endogenous growth without depending upon exogenous changes in population or technology has become one of the central issues in growth theory. This renewal of growth theory offers an explanation for perpetual increases in per capita living standards as an equilibrium outcome of rational choice. The endogenous evolution of human capital is a major force among others driving economic growth. Examples of endogenous growth models include Lucas (1988), Barro (1990) and Rebel0 (1991). Other models in this literature amount to theories of technological progress (Romer 1990) and population change (Becker and Murphy 1988). However, they all abstract from money matters.
380
Examining
the Long-Run
Effect
of Money on Economic
Growth
financial intermediary. Within the money-in-the-production-function framework, it is well documented that money is non-superneutral in the level sense.3 The main novelty of our paper is that we find the growth rate of money creates no long-run effects on the growth rates of consumption, output and human and physical in terms of its imcapital. In other words, money is “superneutral’ pacts on economic growth rates. This, therefore, provides a theoretical interpretation for the empirical puzzle raised by Christian0 and Ljungqvist (1988). The organization of the paper is as follows. In the next section, we develop the basic monetary endogenous growth model. Section 3 studies the balanced-growth equilibrium and examines the role of money in an endogenously growing economy. There is a fourth, concluding section.
2. The Model Consider a modified continuous-time, representative-agent, perfect-foresight model of neoclassical monetary growth in which both physical and human capital are endogenously determined. For simplicity, leisure is assumed inelastic.4 The representative agent’s optimization problem can be written as CD maxW=
U(c(t))e+dt I0
subject to c(t) + k(t) + k(t) = F@(t), L(t), m(t)) - d(t)
- (n + 7r(t))m(t) + 7(t) ,
h(t) = 40 - WMt) >
0) (2)
‘This is because the introduction of real balances improves the efficiency of production and allows us to have a permanently higher level of per capita output. ‘This is a common assumption in endogenous growth models; see Lucas (1988), for example. Moreover, it can be shown that endogenizing leisure in the current framework does not alter the growth superneutrality result. Specifically, to allow for variable non-leisure time, one can generalize the instantaneous utility function as U(c, ~.!a) = {[&A)‘-@I’-” - l}/(l - a) and the human capital accumulation equation as h = +(l - t? - X)/I (see Rebel0 1991). Our main result on growth superneutrality still holds, although the equilibrium solutions of some endogenous variables along a balanced growth path may have changed (for example, Equation [17] will be different).
361
Ping Wang and Chong K. Yip where c and m are (per capita) consumption and real money balances, respectively; k and L = he represent (per capita) physical capital and effective labor input, respectively, with h denoting the human capital skill level and 4 indicating the fraction of non-leisure time allocated to production (and hence, 1 - e is the fraction of non-leisure time allocated to education); T is (per capita) lump-sum transfer from the government; 1~ denotes the inflation rate; p, n and 4 are (constant) rates of time preferences, populaton growth and maximal human capital accumulation, respectively. Equation (1) is a modified Sidrauski (1967) budget constraint in which money enters the production function, F. Equation (2) is the Uzawa (RX%)-Rosen (1976)-Lucas (1988) human capital evolution equation. According to (2), if no effort is devoted to education (that is, e = l), then no betterment in human capital can emerge. If one puts all the effort into accumulating human capital (that is, 4 = 0), human capital will grow at its maximal rate, +. In order to obtain feasible steady-state growth, we choose the utility function U to exhibit constant intertemporal elasticity of substitution and the production function F to take the Cobb-Douglas form (see King, Plosser and Rebel0 1988). Specifically, we have U(c) = (P - l)/(l - cx) and F(k, L, m) = A(m)kYLlmY, where cx is the inverse of the intertemporal elasticity of substitution and y is the capital income share.5 Notice that money is introduced via a Hicksneutral production technology, A.6 For tractability, we assume that the marginal technological improvement of money, A,,,(m), is positive and diminishing, that is, A,,, > 0 and A,,,,,, < 0. Moreover, to be consistent with sustained growth in the steady state, we further restrict A to have constant output and marginal output elasticities of money, that is, u = mA,/A and E = -mA,,,,,,/A, are constant (see a discussion in Section 3 below). Let i.~ > 0 be the (constant) rate of money growth. Money market equilibrium requires r(t) = pm(t). Let ti/m = q. By delinition, ti(t)/m(t)
= p - a(t) - n .
(3)
SThese specifications have been often used in endogenous growth models, for instance, see Lucas (1888) and Barre (1990). ‘It seems to us the Hicks-neutral technological progress is a natural choice of modeling money in the production function. It is hard to imagine that money improves the efficiency biased to either one of the production factors as represented by the Harrodand Solow-neutral technological progress. Further, we have assumed that human capital improves production technology in a Harrod-neutral manner.
382
Examining
the Long-Run
Thus, in equilibrium,
Effect
of Money
(1) can be rewritten
on Economic
as
c(t) + k(t) = F(k(t), L(t), m(t)) - nk(t) ,
which
is in fact the goods market
3. Balanced
Growth
Growth
equilibrium
(4)
condition.
Analysis
In this section, we perform a balanced growth analysis to solve for an optimal endogenous monetary growth equilibrium. First denote Al, AZ and A3 as costate variables of the current Hamiltonian associated with (l), (2) and the slack variable identity, z = +x.7 By definition, the rate of growth of each endogenous variable is constant along a balanced growth path.8 Denote 0 as the constant growth that is, e = d/c. Straightforward rate of per capita consumption, application of Pontryagin’s Maximum Principle yields
X,/x, = -ae ,
(5)
Fk = p + n + di ,
(7)
F, = p + n + 7~ + cd,
(8)
A1= A, ,
(9)
and
LA2
= p - : [F, - +(l - 4) , 2
together with (2), (4) and the transversality conditions of k, h and m: lim,,e -@A,(t)q(t) = 0, where 4 = k, h and m for i = 1, 2 and 3, respectively. ‘The
Hamiltonian
of the
control
problem
is then
given
x = U(c) + X,[F(k, L, m) - nk - (n + T)rn + X&l - q/l + h,z ‘We
follow
Lucas’s
(1988,
9) definition
of the
“balanced
as + 7 - c - z]
growth
path.”
363
Ping Wang and Chong K. Yip Equations (5) and (6) are efficiency conditions for goods and time allocation. They imply that goods must be equally valuable in consumption and capital accumulation, and time must be equally valuable in current production and education. Equations (7) and (8) are Keynes-Ramsey rule equations for capital and money that determine real and nominal interest rates (Fk and F,, respectively). We can also learn from these equations that the marginal products of capital and money are constant along a balanced growth path. From (9), the shadow prices of capital and money must be equal because the only function of capital and money is to produce goods. Finally, (10) represents the golden rule equation for human capital. Next, (6) and (10) together give &./A2 = p - t+ . Differentiating
(11)
(6)-(8) with respect to time implies
and l&/F,
= i,/X,
- );,/A, .
(13)
Thus, along a balanced growth path, only the marginal product of effective labor changes over time with its rate of change being determined by the difference between the shadow prices of human and physical capital. Further, from (7), we have F/k = FJy = (p + n + aO)/y. Then manipulating (4), we obtain the consumptioncapital ratio c/k = (p + n + aO)/y - n - i/k,
(14)
which is constant along balanced growth paths. Thus (per capita) consumption and physical capital have to grow at the same rate, 8. Further denote per capita output as y, that is, y = F(k, L, m). Then the constancy of the marginal product of capital and F/k = Fk/y together imply output and capital have to grow at the same rate, too. Thus we derive the following relation for the growth rates of consumption, capital and output: d/c = i/k 364
= zj/y = 8 ,
(15)
Examining
the Long-Run
Effect of Money on Economic
Growth
which is a standard result in endogenous growth theory (for example, Lucas 1988). Further, as an immediate consequence of (2), 4 has to be constant along a balanced growth path. Since FL = (1 - y)F/L, this together with (5), (ll), (13) and (15) give the following expression for the growth rate of effective labor:
v = i/L Combining
= h/h = 4 - p -(a
- l)e.
(16)
(2) and (16), we have
e = [p + (a - l)c.I]/4.
(17)
The interiority restriction of e E (0, 1) requires that -p < (a - 1)6 < 4 - p which provides upper and lower bounds for 6; the first inequality, p - (1 - ~$3 > 0, also guarantees that the lifetime utility is bounded. Moreover, applying (16), this interiority restriction ensures a positive rate of human capital evolution (that is, v > 0). There are two important implications. First, (17) implies that a wellskilled person (with a higher 4) can work less given the same common balanced growth rate, 9. Second, (16) indicates that whether the growth rate of human capital skill level is positively correlated with the common growth rate of per capita consumption and physical capital depends on the magnitude of the inter-temporal elasticity of substitution (a-‘). Specifically, h/h and 8 will have a positive correlation if ci-’ > 1.’ Next, recall that u = mA,/A and E = -mA,,,,/A,. By the definition of balanced growth, both A/A and &,,/A, must be constant. Since A/A = aq and A,/A, = --ET), u and E must be constant along a balanced growth path. Totally differentiating Fk = yAkY-‘L’-Y and F,,, = A,,,kYLleY with respect to time and manipulating, we obtain
8 = (CT+ ~)7)
(18)
and v = (1 - Y)(P - 4 + 4/u.
(19)
‘The condition seems reasonable: it is consistent with the property of factor complementarity obtained from constant-return production technology (that is, physical capital and human capital are positively correlated).
Ping Wang and Chong K. Yip Combining
(16) with (18) and (19), we get 8 = B-‘(+ - p)(l - y) rl = m+
- PM - rm
F-3 + 4 >
(21)
and u = B-‘(ql - p)[l - y - a/(a + E)] )
(22)
where B = o(1 - y) - ~/(a + E). In Lucas (1988), 4 is estimated to be 5% and thus is higher than standard values of the time preference rates which is around 2% (for example, see Barr-o 1990). Therefore, we expect + - p > 0. In observing the positive rate of economic growth, one should restrict B > 0. Since in general we believe 0~~’ > 1 and B > 0, v is positive as well. In fact, the growth rates of real macroeconomic variables (0, Y, q) depend only on preference and production parameters. Specifically, the higher the rate of time preferences (p) or the degree of diminishing returns to money (E) is, the lower these growth rates will be. Further, these growth rates are positively correlated with the maximal human capital accumulation rate (+), the intertemporal elasticity of substitution (o-l) and the productivity of money (a). In addition, the effects of higher capital income share (y) are to increase capital and real balances accumulation but to reduce human capital evolution. According to (20)-(22), the growth rates of real macroeconomic aggregates are independent of the money growth rate, that is, dv/dp = dtl/dp = dq/dp = 0. Since human capital accumulation is the main engine of growth in this model, the result that dv/dp = 0 implies that money does not affect the growth process. This, therefore, explains why we obtain the superneutrality of money in the “growth rates” sense, although due to the productive efficiency of real money balances, higher money growth would still lower the “level” of per capita quantities. This finding also provides a theoretical framework to resolve the empirical puzzle raised by Christian0 and Ljungqvist (1988). Further, from (3), (8) and q = p - r - n, it can be shown that dr/dF = dF,,,/dp = 1. Thus, in our economy, the Fisherian prediction that the nominal interest rate adjusts completely to anticipated inflation is validated. To search for the optimal growth rate of money, we examine how the lifetime utility is affected by monetary policy. Along the balanced growth path, the growth rate of per capita consumption 366
Examining
the Long-Run
(0) is constant. a constant)
The lifetime
Effect
of Money on Economic
utility
(W) is then given by (aside from
[4w-” w = (1 - c.u)[p- (1 - ($31’ Thus the condition p From (14), we have
(1 - cx)e > 0 ensures utility
40) = W)hl{[p + (1 - rbl + (a - rM s
Growth
(23) is bounded.
(24)
It is apparent from (20) and (24) that both 8 and c(0) are independent of the money growth rate, IL. Therefore, in the steady state, our model does not yield an optimal growth rate of money. Finally, it is worth noting that although we have found superneutrality (in the growth rate sense) in the steady state, it is not clear whether this result can emerge in transition. An increase in the growth rate of money may alter the ratio of physical to human capital and thus the growth neutrality result may break down in transition. It is, however, extremely difficult to derive the transitional dynamics in the present model since endogenous variables do not grow at a common rate. lo
4. Concluding
Remarks
This paper provides a first attempt to incorporate money into an endogenous growth model via a Hicks-neutral production technology. We show that higher money growth rate increases the inflation rate proportionately. This then implies the absence of a real balance effect on the rates of physical and human capital accumulation and consumption and output growth. Therefore, money is “superneutral” along the balanced growth path although it is nonsuperneutral in the “level” sense. This helps us in understanding the empirical puzzle of Christian0 and Ljungqvist (1988) that there exists a statistically significant Granger-causal relation between money
“‘The standard method of analyzing transitional dynamics in sustained-growth models is to transform all endogenous quantity variables in the unit of the factor that drives common growth. Subsequently, one can linearize the dynamic system around the steady state to study local dynamics (King, Plosser and Rebel0 1968). This method cannot be applied here because we do not have a common growth component.
367
Ping Wang and Chong K. Yip and output when data are measured in log-level, but not when first differences of the logged data are used. Interesting extensions of the present model include the consideration of a productive government sector and the construction of a more general form for the evolution of human capital.” The former may allow us to investigate the interactions between fiscal and monetary policies and their long-run impacts on the real economy. The latter may help enhance the long-run responses of consumption growth to money growth. However, an adequate treatment of the above suggestions will require further simplification of the current framework in order to obtain a closed-form solution and unambiguous comparative dynamic results. Finally, we would like to caution the readers that the growth neutrality result may not emerge under different setups. For instance, as shown in Rebel0 (1988) using a cash-in-advance (CIA) model, if the CIA constraint is imposed on the purchase of a broadly defined investment good, an increase in the growth rate of money will retard the growth process. Nevertheless, whether this non-neutrality effect on economic growth is quantitatively important remains uninvestigated. Received: October 1990 Final version: July 1991
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“The reader is referred to government into the endogenous incorporation of physical capital
368
Barro (1990) for the introduction of a productive growth framework and to Rebel0 (1991) for the into the human capital evolution equation.
Examining
the Long-Run
Effect
of Money
on Economic
Growth
Alternative Approaches.” Journal of Money, Credit, and Banking 5 (May 1973): 141-56. Economic Fischer, Stanley. “Money and the Production Function.” Inquiry 12 (November 1974): 517-33. King, Robert G., and Charles I. Plosser. “Money, Credit, and Prices in a Real Business Cycle.” American Economic Review 74 (June 1984): 363-80. King, Robert G., Charles I. Plosser, and Sergio T. Rebelo. “Production, Growth and Business Cycles: I. The Basic Neoclassical Growth Model.” Journal of Monetary Economics 21 (March/May 1988): 195-232. Levhari, David, and Don Patinkin. “The Role of Money in a Simple Growth Model.” American Economic Review 58 (September 1968): 713-53. Lucas, Robert E., Jr. “On the Mechanics of Economic Development.” Journal of Monetary Economics 23 (July 1988): 3-42. Rebelo, Sergio T. “Essays on Growth and Business Cycles.” Ph.D. diss., University of Rochester, 1988. -. “Long-Run Policy Analysis and Long-Run Growth. ” Journal of Political Economy 99 (June 1991): 500-21. Romer, Paul M. “Endogenous Technological Change.” Journal of PoZiticaZ Economy 98 (October 1990): S71-102. Rosen, Sherwin. “A Theory of Life Earnings.” Journal of Political Economy 84 (August 1976): S45-67. Sidrauski, Miguel. “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Review: Papers and Proceedings 57 (May 1967): 534-44. Tobin, James. “Money and Economic Growth.” Econometrica 33 (October 1965): 671-84. Uzawa, Hirommi. “Optimal Technical Change in an Aggregate Model of Economic Growth.” International Economic Review 6 (January 1965): 18-31. Wang, Ping, and Chong K. Yip. “Transactions Cost, Endogenous Labor, and the Superneutrality of Money.” Journal of Macroeconomics 13 (Winter 1991): 183-91.
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