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Inflation, human capital accumulation and long-run growth
PAUL PECORINO University of Alabama Tuscaloosa, Alabama
Inflation, Human Capital Accumulation and Long-Run Growth* In a model where money enters the production function, Wang and Yip (1992) find money to be superneutral with respect to the economy's long-run rate of growth. This result is not robust to a generalization of the technology of human capital accumulation. If human capital accumulation requires the input of physical capital, then money is not superneutral with respect to the economy's growth rate. However, for reasonable pararneterizations of the model, inflation is found to have only small effects on growth.
1. Introduction
Wang and Yip (1992) find money to be superneutral with respect to the economy's long-run rate of growth. This result is not robust to a generalization of the technology of h u m a n capital accumulation. 1 If h u m a n capital accumulation requires the input of physical capital, then m o n e y is not superneutral with respect to the economy's growth rate. 2 As in Wang and Yip, this paper utilizes a money in the production function specification. For reasonable values of the model's parameters, the effect of inflation on growth is modest considering the ranges of inflation experienced by the major industrialized countries. The finding that inflation has only small effects on growth is consistent with findings of related papers in the money and growth literature. G o m m e (1993) and Jones and Manuelli (1993) both find small growth rate effects from moderate rates of inflation in the context of cash-in-advance economies. In addition, Wang and Yip (1994) find small effects of m o n e y on growth in a transaction cost economy, as does Ireland (1994) in a model with financial intermediation. The model is set out in Section 2, where it is established that m o n e y is not superueutral with respect to the economy's growth rate. Some sim*I would like to thank John Conlon and two anonymous referees for helpful comments on the paper. All errors are my own. lWang and Yip (1992, 368) acknowledge this possibility. 2Pecorino (1994) shows that allowing for a similar technological change in Lucas's (1990) model a]lows for a significant effect on the growth rate (about 1% per capita per year under the benchmark) from a tax reform under which Lucas found a negligible effect. Thus, the generalization considered here is potentially important with respect to the growth rate effects of inflation.
Journal of Macroeconomics, Summer 1995, Vol. 17, No. 3, pp. 533-542 Copyright © 1995 by Louisiana State University Press 0164-0704 / 95 / $1.50
533
Paul Pecorino
ulations are conducted in Section 3, where the effects of inflation on growth are shown to be small for plausible parameterizations of the model. Section 4 concludes.
2. The Model To facilitate comparison with Wang and Yip's results, and to make sure the assumption on the technology of human capital accumulation is the key factor, the model will, where possible, follow Wang and Yip. As in their model, in the consumption/physical capital sector, real money balances are a productive input. This is reflected in Equation (1).
Y1 = A 1 ((~IK)c~ (u 1H)C~:(M/P1) ~3 = F(~IK,u 1H,M/PI) , where (xa = 1 - (x~ - (x2.
(1)
Output of the joint consumption/physical capital good is denoted Y1, A1 is a technology level parameter, M is nominal money held by the firm and Pj is the price of output produced in sector 1. The stocks of physical and human capital are denoted K and H while (~1 and u I are the percent allocations of each stock of capital to sector 1. Under the Wang and Yip specification, production of human capital requires only time and human capital. Under this specification, the return to capital and hence the growth rate of the economy is determined solely by technology in the human capital sector, and taxation of inputs into that sector, a When physical capital becomes an input in the human capital sector, growth becomes a function of technology and taxation of inputs in the sector that produces physical as well as human capital. Since inflation is a tax on money holdings, it is a tax on an input into production of physical capital and will therefore reduce the growth rate. The production function for human capital is then given by Equation (2) and the economy's resource constraint is given in (3). Y2 = A2(d~2K)Y(u~H) 1-~ = G(~2K,u2H) ;
(2)
~1 + ~2 = 1 ;
(3a)
u) + u2 = 1.
(3b)
In contrast to Wang and Yip, this model will be solved by considering separately the producers' and consumers' problems. Consumers own all assets, including the existing money stock, and rent them out to firms each period. Firms are perfectly competitive on all input and output markets and 3This is discussed in Rebelo (1991, 511) and Mulligan and Sala-i-Martin (1993). 534
Inflation, H u m a n Capital Accumulation and Long-Run Growth
act so as to maximize profits. The first-order conditions for the firms' problems are given by Equation (4a-e). PIFI[~IK, ulH, M/Pi] = rP~ ;
(4a)
PIF2[CplK,u1H, M/P1] = wP.);
(4b)
P1Fz[¢~K, ulH, M/P1] = RP 1 ;
(4c)
P2GI[~,2K, u.2H] = rP 1 ;
(4d)
P2Ge[ipzK, u2H] = w P e .
(4e)
The subscripts on F and G indicate the partial derivative with respect to the indicated argument. The rate of return to physical and human capital are denoted r and w so that rP 1 and w P 2 represent the return to holding a unit of each type of capital, where P1 and P2 are the relevant prices. The rate of return paid to the owners of the money stock is R which is the nominal interest rate. The consumer seeks to maximize lifetime utility in (5) subject to the constraints which govern the evolution over time of the stocks of human capital~ physical capital and money holdings ill (6a-c). C1-~ e -or dt ;
max
(5)
/~ = It¢;
(6a)
bI = In ;
(6b)
M = wP2H + rP1K + R M - P1C - Pain - P2IH + Z .
(6c)
Consumption is denoted C, p is the rate of time preference, 1/~ is the intertemporal elasticity of substitution, and It< and I H are investment in physical and human capital. Government transfers Z consist of revenue raised from the inflation tax. Along a balanced growth path, all variables will grow at constant (and possibly different) rates. All time allocations will be constant, as will all relative prices. Using the balanced growth path assumption and the firstorder conditions to the consumer's and producer's problems it is possible to establish the following relationships (details on the consumer's problem may be obtained from the author upon request). w=r ;
(7a)
R =r +n ;
(713)
535
Paul Pecorino (7c)
g-C-H-K; 1
g --~- (r - p) ;
(Td)
n = ~t - g.
(7e)
Equation (7a) indicates that the rate of return to holding physical and human capital are equal, while (7b) is the familiar decomposition of the nominal interest rate into its real and inflationary (n denotes the rate of inflation) component. Along the balanced growth path, consumption and the two capital stocks will grow at same constant rate g as shown in (7c). In (7d) we have the familiar relationship between the economy's growth rate g and the real rate of return on capital r. As indicated in (7e), the inflation rate is given by the difference between the money growth rate ~t and the economy's growth rate g. To show that money is not superneutral with respect to the economy's growth rate, it will simply be necessary to show that changes in the money growth rate ~t will affect the return to capital r. Through (7d), this will establish an effect on the growth rate. The solution for r comes from the production side of the economy. In addition to the first-order conditions to the producers' problems, it is necessary to consider the zero profit conditions which must hold in each sector. These are given in Equation (8).
P1 = a~4wP2 + ak rP1 + aiR ;
(Sa)
P2 = a~ wP2 + a~ rPl •
(8b)
The aj are the ratios of the relevant input (j = H,K,M) to output in sector (i = 1,2). The aj may be computed from the first-order conditions in Equation (4) and are given in (9) and (10).
aI=(~IK °qZIIP2]~" ab_Ul H
(9a)
cx2ZI(P2]1-a~ " '
--
;
where, Z1 = r c~lw a2 R a3 and Z2 = A1 a7 ~a7 ~a~ ~• 536
19hi
(9c/
Inflation, Human Capital Accunndation and Long-Run Growth a~ - ( 1 - 7) Z3 (P2] -v
(10a)
wz4
:
:
(10b)
where, Z 3 = r~w 1-~ and Z 4 = A2T~ (1 - 7) l--v. Substitute (9) and (10) into the zero profit conditions to get (11).
Z--2\ ~ 1
= 1;
Z]
- el
(lla)
e2
(11b)
Substitute for P2/P1 in (lla) from (llb), use the definitions o f Z 1 and Za, and Equations (7a) and (7t)) to get Equation (12). (r + n) ~ rV = z2(z4)(~/v),
~2
where ~ = al + cz2 + - - > 0 . 7
(12)
The right-hand side of (12) is a constant. It is straightforward to verify from this equation that dr/dn < 0. We would like however to link changes in r (and therefore growth) to changes in g, the growth rate of the money supply. Differentiating Equations (7d) and (7e) we get the relationships in (13). dg = (1re)dr,
(13a)
dn = dg - dg .
(13b)
Use (13) and totally differentiate (12) to get the solution for dr/dg in (14). dr
- czar
dg - ~ (r + n) + 0~3(1 - 1/c) r '
(14)
Sufficient conditions for this expression to be negative are that (r + n) > 0 and cy > 1. The nominal interest rate, r+n, must be positive in this model because demand for real money balances is infinite at a nominal interest rate of 0. To ensure that utility is bounded, it must be the case that the growth rate is less than the interest rate (g < r).4 A sufficient condition for this is that 4This conditionis stated in Lueas (1988, 10). 537
Paul Pecorino
O > 1. In addition, empirical work indicates that the intertemporal elasticity of substitution is less than 1 (that is, ~ > 1). '~ Thus, money is not superneutral, and some mild restrictions are sufficient to ensure that increased money growth will lower the growth rate. When physical capital is an input into the production of human capital, taxation of any input into production of either physical or human capital, including real money balances, will be growth reducing. Since inflation is a tax on money holdings, increases in inflation will reduce growth in this model. The question addressed in the next section is whether this effect of inflation on growth is likely to be empirically significant.
3. Some Simulation Results The extent to which inflation affects the growth rate will depend upon the pararneterization of the model. The key parameters include (z3, the share of real money balances in production, 1/o, the intertemporal elasticity of substitution, and 7, the share of physical capital in the production of human capital. The closer 7is to 0, the closer is the specification of the model to Wang and Yip (1992), where money is superneutral with respect to the growth rate of output. The results of four simulations are reported. The results for this model indicate that, for realistic parameter values, inflation is likely to have only a small negative impact on growth. To evaluate the reasonableness of the choice of(x3, the implied velocity of money will be reported under the benchmark parameterization of the model, where velocity V = P Y / M and P Y = P~Y1 + P2Y2 • Under the benchmark, the technology parameters A 1 and A 2 are chosen so as to give a 1.5% per capita growth rate. The rate of time preference is set to 0.015 and the intertemporal elasticity of substitution is set equal to 1. This value is high relative to empirical estimates. The nominal interest rate under the benchmark is 6%, the inflation rate 3% and the real interest rate 3%. The parameter 7 = 0.2. This is in line with the estimates provided in Bowen (1987). Finally, cz3 is set equal to 0.1, which will imply a low value of velocity relative to those observed empirically. The parameter values under the benchmark are summarized in Table 1. They have been chosen so as to bias the model towards finding a large growth rate effect. As seen in Table 2, however, the effect on the growth rate of increases in inflation are rather moderate. Increasing the inflation rate from 3% to 50% decreases the growth rate from 1.5% to 1.36%. While a reduction of almost 10% in the growth rate is nontrivial, the 50% inflation 5Boskin (1978) estimates the intertemporal elasticity of substitution to be in the range of 0.4. This is considered to be one of the higher estimates of this parameter.
538
Inflation, Human Capital Accumulation and Long-Run Growth T A B L E 1.
Benchmark Values
Rate of time preference Physical capital share parameters
p (~1 7 0~2 1 - y 0~3 1/c n g R V
H u m a n capital share parameters Real money balances share parameter Intertemporal elasticity of substitution Inflation rate Growth rate of output Nominal interest rate Velocity
0.015 0.20 0.20 0.70 0.80 0.10 1 0.03 0.015 0.06 0.8
T A B L E 2.
Benchmark rt 0.03 0.10 0.20 0.30 0.40 0.50
Trial 2 (~ = 2
Trial 3 ct3 = 0.05
Trial 4 ~ = 2, 0t3 = 0.05
g
V
g
V
g
V
g
V
0.0150 0.0145 0.0141 0.0139 0.0137 0.0136
0.8 1.6 2.9 4.1 5.2 6.4
0.0150 0.0147 0.0144 0.0142 0.0141 0.0140
0.9 1.7 2.8 4.0 5.1 6.2
0.0150 0.0147 0.0145 0.0144 0.0143 0.0143
1.6 3.5 6.1 8.6 11.2 13.7
0.0150 0.0148 0.0147 0.0146 0.0146 0.0145
1.8 3.5 5.8 8.2 10.6 12.9
NOTE: Under trials 2 and 4, with n = 0.03, R = 0.075.
rate is very high relative to developed country experience .6 O f course other countries, particularly in Latin America, have experienced sustained inflation well in excess of 50%. 7 With the intertemporal elasticity of substitution reduced to 0.5 (o = 2), the value chosen by Lucas (1990), the reduction in the growth rate is from 1.5% to 1.40%. The parameter ot3 is also crucial in determining the magnitude of any growth rate effects. Under the benchmark, velocity V = 0.8. This is a very low value for velocity o f M 1 which for the U.S. in recent years has been about ~Recently, the major industrialized countries have all had inflation rates well under 10%. See the economic indicators in the The Economist, February 19, 1994. 7Recently Brazil's annual rate of inflation has approached 2500%. See the emerging market indicators in The Economist, February, 19, 1994. 539
Paul Pecorino 6 and low for velocity of M 2 which has been approximately 1.5. s As can be seen from Table 2, when a.3 is reduced to 0.05, the growth reducing effects of inflation are considerably reduced. Even in these cases the implied value of velocity is very low relative to the velocity of M 1. The sensitivity of the growth rate to inflation depends on the key parameters a 3, and 1/a. Our empirical knowledge of these parameters (or of variables which they imply, such as velocity) indicate that the growth rate is relatively insensitive to the inflation rate. This small effect of inflation on growth is consistent with the findings of Gomme (1993), Jones and Manuelli (1993), Wang and Yip (1994) and Ireland (1994). Jones and Manuelli utilize a cash-in-advance technology in their model. In one version, they do find potentially significant growth rate effects at very high rates of inflation. 9 For example, in one case raising the money growth rate from 0 to 200% decreases the growth rate from 2% to 1.1%, while an increase to 50% lowers the rate to 1.7%. In this version of their model, growth rate effects occur through decreases in labor supply. This in turn reduces the utilization of human capital. Allowing for an endogenous labor supply in the model presented in this paper would provide some increased scope for growth rate effects of inflation through utilization effects on human capital. However, it should remain the case that the effects of inflation on growth will be small over the ranges of inflation currently experienced by the major industrialized nations. 4. Conclusion Extending Wang and Yip's (1992) analysis to allow for physical capital as an input into human capital production leads to a reversal of their result on the superneutrality of money with respect to the economy's growth rate. The growth rate is instead decreasing in the inflation rate. Parameterization of the model however, indicates that the effects of inflation on the growth rate are likely to be small as an empirical matter. This result in an economy where money enters the production function is consistent with other work in the money and growth literature. Received: March 1994 Final version: September 1994
References Baumol William J., and Man S. Blinder. Macroeconomics: Principles and Policy. 6th ed.. Fort Worth: Dryden Press, 1994. SSee Baumol and Blinder (1994, 764). ~l'his result hinges on whether the two consumption goods in the model are complements or substitutes.
540
Inflation, Human Capital Accumulation and Long-Run Growth Boskin, Michael J. "Taxation, Saving and the Rate of Interest." Journal of Political Economy 86 (April 1978, part 2): $3-$27. Bowen, Howard R. The Costs of Higher Education. San Francisco: JosseyBass Publishers, 1987. Gomme, Paul. "Money and Growth Revisited: Measuring the Costs of Inflation in an Endogenous Growth Model." Journal of Monetary Economics 32 (1993): 51-77. Ireland, Peter N. "Money and Growth: An Alternative Approach." American Economic Review 84 (March 1994): 47-65. Jones, Larry E., and Rodolfo Manuelli. "Growth and the Effects of Inflation." NBER working paper 4523, November 1993. Lucas, Robert E. "On the Mechanics of Economic Development." Journal of Monetary Economics 22 (1988): 3-42. --. "Supply Side Economics: An Analytical Review." Oxford Economic Papers 42 (1990): 293-316. Mulligan, Casey B., and Xavier Sala-i-Martin. "Transitional Dynamics in Two Sector Models of Endogenous Growth." Quarterly Journal of Economics 108 (August 1993): 739-73. Pecorino, Paul. "The Growth Rate Effects of Tax Reform." Oxford Economic Papers 46 (July 1994): 492-501. Rebelo, Sergio. "Long Run Policy Analysis and Long Run Growth." Journal of Political Economy 99 (June 1991): 500-21. Wang, Ping, and Chong K. Yip. "Examining the Long-run Effect of Money on Economic Growth." Journal of Macroeconomics 14 (Spring 1992): 359--69. --. "Real Effects of Money and Welfare Costs of Inflation in an Endogenously Growing Economy with Transaction Costs." Manuscript, Georgia State University, Janua~ 1994.
Appendix List of Variables Y1 Y2
Ai q~i ui M K H G~I,~/ ¢2,1 - 7
= output of the consumption/physical capital good. = production of human capital. = technology level parameter in sector i. -- percent of physical capital stock allocated to sector i. = percent of human capital stock allocated to sector i. = nominal money stock. = physical capital. = human capital. physical capital share parameters. = human capital share parameters. =
541
Paul Pecorino O{3
Pi C
IK,IH 1/~ P R r,w g Z l'f
V
549
= share of real money stock in production of good 1. = ratio of input d to output in sector i. = price of good i. = consumption. = investment in physical and h u m a n capital. = intertemporal elasticity of substitution. = rate of time preference. = nominal interest rate. = rate of return to holding physical and h u m a n capital = rate of growth of consumption. = transfers payments from the government. = rate of inflation. = rate of growth of the m o n e y stock. = velocity of money.
Inflation, Human Capital Accumulation and Long-Run Growth* In a model where money enters the production function, Wang and Yip (1992) find money to be superneutral with respect to the economy's long-run rate of growth. This result is not robust to a generalization of the technology of human capital accumulation. If human capital accumulation requires the input of physical capital, then money is not superneutral with respect to the economy's growth rate. However, for reasonable pararneterizations of the model, inflation is found to have only small effects on growth.
1. Introduction
Wang and Yip (1992) find money to be superneutral with respect to the economy's long-run rate of growth. This result is not robust to a generalization of the technology of h u m a n capital accumulation. 1 If h u m a n capital accumulation requires the input of physical capital, then m o n e y is not superneutral with respect to the economy's growth rate. 2 As in Wang and Yip, this paper utilizes a money in the production function specification. For reasonable values of the model's parameters, the effect of inflation on growth is modest considering the ranges of inflation experienced by the major industrialized countries. The finding that inflation has only small effects on growth is consistent with findings of related papers in the money and growth literature. G o m m e (1993) and Jones and Manuelli (1993) both find small growth rate effects from moderate rates of inflation in the context of cash-in-advance economies. In addition, Wang and Yip (1994) find small effects of m o n e y on growth in a transaction cost economy, as does Ireland (1994) in a model with financial intermediation. The model is set out in Section 2, where it is established that m o n e y is not superueutral with respect to the economy's growth rate. Some sim*I would like to thank John Conlon and two anonymous referees for helpful comments on the paper. All errors are my own. lWang and Yip (1992, 368) acknowledge this possibility. 2Pecorino (1994) shows that allowing for a similar technological change in Lucas's (1990) model a]lows for a significant effect on the growth rate (about 1% per capita per year under the benchmark) from a tax reform under which Lucas found a negligible effect. Thus, the generalization considered here is potentially important with respect to the growth rate effects of inflation.
Journal of Macroeconomics, Summer 1995, Vol. 17, No. 3, pp. 533-542 Copyright © 1995 by Louisiana State University Press 0164-0704 / 95 / $1.50
533
Paul Pecorino
ulations are conducted in Section 3, where the effects of inflation on growth are shown to be small for plausible parameterizations of the model. Section 4 concludes.
2. The Model To facilitate comparison with Wang and Yip's results, and to make sure the assumption on the technology of human capital accumulation is the key factor, the model will, where possible, follow Wang and Yip. As in their model, in the consumption/physical capital sector, real money balances are a productive input. This is reflected in Equation (1).
Y1 = A 1 ((~IK)c~ (u 1H)C~:(M/P1) ~3 = F(~IK,u 1H,M/PI) , where (xa = 1 - (x~ - (x2.
(1)
Output of the joint consumption/physical capital good is denoted Y1, A1 is a technology level parameter, M is nominal money held by the firm and Pj is the price of output produced in sector 1. The stocks of physical and human capital are denoted K and H while (~1 and u I are the percent allocations of each stock of capital to sector 1. Under the Wang and Yip specification, production of human capital requires only time and human capital. Under this specification, the return to capital and hence the growth rate of the economy is determined solely by technology in the human capital sector, and taxation of inputs into that sector, a When physical capital becomes an input in the human capital sector, growth becomes a function of technology and taxation of inputs in the sector that produces physical as well as human capital. Since inflation is a tax on money holdings, it is a tax on an input into production of physical capital and will therefore reduce the growth rate. The production function for human capital is then given by Equation (2) and the economy's resource constraint is given in (3). Y2 = A2(d~2K)Y(u~H) 1-~ = G(~2K,u2H) ;
(2)
~1 + ~2 = 1 ;
(3a)
u) + u2 = 1.
(3b)
In contrast to Wang and Yip, this model will be solved by considering separately the producers' and consumers' problems. Consumers own all assets, including the existing money stock, and rent them out to firms each period. Firms are perfectly competitive on all input and output markets and 3This is discussed in Rebelo (1991, 511) and Mulligan and Sala-i-Martin (1993). 534
Inflation, H u m a n Capital Accumulation and Long-Run Growth
act so as to maximize profits. The first-order conditions for the firms' problems are given by Equation (4a-e). PIFI[~IK, ulH, M/Pi] = rP~ ;
(4a)
PIF2[CplK,u1H, M/P1] = wP.);
(4b)
P1Fz[¢~K, ulH, M/P1] = RP 1 ;
(4c)
P2GI[~,2K, u.2H] = rP 1 ;
(4d)
P2Ge[ipzK, u2H] = w P e .
(4e)
The subscripts on F and G indicate the partial derivative with respect to the indicated argument. The rate of return to physical and human capital are denoted r and w so that rP 1 and w P 2 represent the return to holding a unit of each type of capital, where P1 and P2 are the relevant prices. The rate of return paid to the owners of the money stock is R which is the nominal interest rate. The consumer seeks to maximize lifetime utility in (5) subject to the constraints which govern the evolution over time of the stocks of human capital~ physical capital and money holdings ill (6a-c). C1-~ e -or dt ;
max
(5)
/~ = It¢;
(6a)
bI = In ;
(6b)
M = wP2H + rP1K + R M - P1C - Pain - P2IH + Z .
(6c)
Consumption is denoted C, p is the rate of time preference, 1/~ is the intertemporal elasticity of substitution, and It< and I H are investment in physical and human capital. Government transfers Z consist of revenue raised from the inflation tax. Along a balanced growth path, all variables will grow at constant (and possibly different) rates. All time allocations will be constant, as will all relative prices. Using the balanced growth path assumption and the firstorder conditions to the consumer's and producer's problems it is possible to establish the following relationships (details on the consumer's problem may be obtained from the author upon request). w=r ;
(7a)
R =r +n ;
(713)
535
Paul Pecorino (7c)
g-C-H-K; 1
g --~- (r - p) ;
(Td)
n = ~t - g.
(7e)
Equation (7a) indicates that the rate of return to holding physical and human capital are equal, while (7b) is the familiar decomposition of the nominal interest rate into its real and inflationary (n denotes the rate of inflation) component. Along the balanced growth path, consumption and the two capital stocks will grow at same constant rate g as shown in (7c). In (7d) we have the familiar relationship between the economy's growth rate g and the real rate of return on capital r. As indicated in (7e), the inflation rate is given by the difference between the money growth rate ~t and the economy's growth rate g. To show that money is not superneutral with respect to the economy's growth rate, it will simply be necessary to show that changes in the money growth rate ~t will affect the return to capital r. Through (7d), this will establish an effect on the growth rate. The solution for r comes from the production side of the economy. In addition to the first-order conditions to the producers' problems, it is necessary to consider the zero profit conditions which must hold in each sector. These are given in Equation (8).
P1 = a~4wP2 + ak rP1 + aiR ;
(Sa)
P2 = a~ wP2 + a~ rPl •
(8b)
The aj are the ratios of the relevant input (j = H,K,M) to output in sector (i = 1,2). The aj may be computed from the first-order conditions in Equation (4) and are given in (9) and (10).
aI=(~IK °qZIIP2]~" ab_Ul H
(9a)
cx2ZI(P2]1-a~ " '
--
;
where, Z1 = r c~lw a2 R a3 and Z2 = A1 a7 ~a7 ~a~ ~• 536
19hi
(9c/
Inflation, Human Capital Accunndation and Long-Run Growth a~ - ( 1 - 7) Z3 (P2] -v
(10a)
wz4
:
:
(10b)
where, Z 3 = r~w 1-~ and Z 4 = A2T~ (1 - 7) l--v. Substitute (9) and (10) into the zero profit conditions to get (11).
Z--2\ ~ 1
= 1;
Z]
- el
(lla)
e2
(11b)
Substitute for P2/P1 in (lla) from (llb), use the definitions o f Z 1 and Za, and Equations (7a) and (7t)) to get Equation (12). (r + n) ~ rV = z2(z4)(~/v),
~2
where ~ = al + cz2 + - - > 0 . 7
(12)
The right-hand side of (12) is a constant. It is straightforward to verify from this equation that dr/dn < 0. We would like however to link changes in r (and therefore growth) to changes in g, the growth rate of the money supply. Differentiating Equations (7d) and (7e) we get the relationships in (13). dg = (1re)dr,
(13a)
dn = dg - dg .
(13b)
Use (13) and totally differentiate (12) to get the solution for dr/dg in (14). dr
- czar
dg - ~ (r + n) + 0~3(1 - 1/c) r '
(14)
Sufficient conditions for this expression to be negative are that (r + n) > 0 and cy > 1. The nominal interest rate, r+n, must be positive in this model because demand for real money balances is infinite at a nominal interest rate of 0. To ensure that utility is bounded, it must be the case that the growth rate is less than the interest rate (g < r).4 A sufficient condition for this is that 4This conditionis stated in Lueas (1988, 10). 537
Paul Pecorino
O > 1. In addition, empirical work indicates that the intertemporal elasticity of substitution is less than 1 (that is, ~ > 1). '~ Thus, money is not superneutral, and some mild restrictions are sufficient to ensure that increased money growth will lower the growth rate. When physical capital is an input into the production of human capital, taxation of any input into production of either physical or human capital, including real money balances, will be growth reducing. Since inflation is a tax on money holdings, increases in inflation will reduce growth in this model. The question addressed in the next section is whether this effect of inflation on growth is likely to be empirically significant.
3. Some Simulation Results The extent to which inflation affects the growth rate will depend upon the pararneterization of the model. The key parameters include (z3, the share of real money balances in production, 1/o, the intertemporal elasticity of substitution, and 7, the share of physical capital in the production of human capital. The closer 7is to 0, the closer is the specification of the model to Wang and Yip (1992), where money is superneutral with respect to the growth rate of output. The results of four simulations are reported. The results for this model indicate that, for realistic parameter values, inflation is likely to have only a small negative impact on growth. To evaluate the reasonableness of the choice of(x3, the implied velocity of money will be reported under the benchmark parameterization of the model, where velocity V = P Y / M and P Y = P~Y1 + P2Y2 • Under the benchmark, the technology parameters A 1 and A 2 are chosen so as to give a 1.5% per capita growth rate. The rate of time preference is set to 0.015 and the intertemporal elasticity of substitution is set equal to 1. This value is high relative to empirical estimates. The nominal interest rate under the benchmark is 6%, the inflation rate 3% and the real interest rate 3%. The parameter 7 = 0.2. This is in line with the estimates provided in Bowen (1987). Finally, cz3 is set equal to 0.1, which will imply a low value of velocity relative to those observed empirically. The parameter values under the benchmark are summarized in Table 1. They have been chosen so as to bias the model towards finding a large growth rate effect. As seen in Table 2, however, the effect on the growth rate of increases in inflation are rather moderate. Increasing the inflation rate from 3% to 50% decreases the growth rate from 1.5% to 1.36%. While a reduction of almost 10% in the growth rate is nontrivial, the 50% inflation 5Boskin (1978) estimates the intertemporal elasticity of substitution to be in the range of 0.4. This is considered to be one of the higher estimates of this parameter.
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Inflation, Human Capital Accumulation and Long-Run Growth T A B L E 1.
Benchmark Values
Rate of time preference Physical capital share parameters
p (~1 7 0~2 1 - y 0~3 1/c n g R V
H u m a n capital share parameters Real money balances share parameter Intertemporal elasticity of substitution Inflation rate Growth rate of output Nominal interest rate Velocity
0.015 0.20 0.20 0.70 0.80 0.10 1 0.03 0.015 0.06 0.8
T A B L E 2.
Benchmark rt 0.03 0.10 0.20 0.30 0.40 0.50
Trial 2 (~ = 2
Trial 3 ct3 = 0.05
Trial 4 ~ = 2, 0t3 = 0.05
g
V
g
V
g
V
g
V
0.0150 0.0145 0.0141 0.0139 0.0137 0.0136
0.8 1.6 2.9 4.1 5.2 6.4
0.0150 0.0147 0.0144 0.0142 0.0141 0.0140
0.9 1.7 2.8 4.0 5.1 6.2
0.0150 0.0147 0.0145 0.0144 0.0143 0.0143
1.6 3.5 6.1 8.6 11.2 13.7
0.0150 0.0148 0.0147 0.0146 0.0146 0.0145
1.8 3.5 5.8 8.2 10.6 12.9
NOTE: Under trials 2 and 4, with n = 0.03, R = 0.075.
rate is very high relative to developed country experience .6 O f course other countries, particularly in Latin America, have experienced sustained inflation well in excess of 50%. 7 With the intertemporal elasticity of substitution reduced to 0.5 (o = 2), the value chosen by Lucas (1990), the reduction in the growth rate is from 1.5% to 1.40%. The parameter ot3 is also crucial in determining the magnitude of any growth rate effects. Under the benchmark, velocity V = 0.8. This is a very low value for velocity o f M 1 which for the U.S. in recent years has been about ~Recently, the major industrialized countries have all had inflation rates well under 10%. See the economic indicators in the The Economist, February 19, 1994. 7Recently Brazil's annual rate of inflation has approached 2500%. See the emerging market indicators in The Economist, February, 19, 1994. 539
Paul Pecorino 6 and low for velocity of M 2 which has been approximately 1.5. s As can be seen from Table 2, when a.3 is reduced to 0.05, the growth reducing effects of inflation are considerably reduced. Even in these cases the implied value of velocity is very low relative to the velocity of M 1. The sensitivity of the growth rate to inflation depends on the key parameters a 3, and 1/a. Our empirical knowledge of these parameters (or of variables which they imply, such as velocity) indicate that the growth rate is relatively insensitive to the inflation rate. This small effect of inflation on growth is consistent with the findings of Gomme (1993), Jones and Manuelli (1993), Wang and Yip (1994) and Ireland (1994). Jones and Manuelli utilize a cash-in-advance technology in their model. In one version, they do find potentially significant growth rate effects at very high rates of inflation. 9 For example, in one case raising the money growth rate from 0 to 200% decreases the growth rate from 2% to 1.1%, while an increase to 50% lowers the rate to 1.7%. In this version of their model, growth rate effects occur through decreases in labor supply. This in turn reduces the utilization of human capital. Allowing for an endogenous labor supply in the model presented in this paper would provide some increased scope for growth rate effects of inflation through utilization effects on human capital. However, it should remain the case that the effects of inflation on growth will be small over the ranges of inflation currently experienced by the major industrialized nations. 4. Conclusion Extending Wang and Yip's (1992) analysis to allow for physical capital as an input into human capital production leads to a reversal of their result on the superneutrality of money with respect to the economy's growth rate. The growth rate is instead decreasing in the inflation rate. Parameterization of the model however, indicates that the effects of inflation on the growth rate are likely to be small as an empirical matter. This result in an economy where money enters the production function is consistent with other work in the money and growth literature. Received: March 1994 Final version: September 1994
References Baumol William J., and Man S. Blinder. Macroeconomics: Principles and Policy. 6th ed.. Fort Worth: Dryden Press, 1994. SSee Baumol and Blinder (1994, 764). ~l'his result hinges on whether the two consumption goods in the model are complements or substitutes.
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Inflation, Human Capital Accumulation and Long-Run Growth Boskin, Michael J. "Taxation, Saving and the Rate of Interest." Journal of Political Economy 86 (April 1978, part 2): $3-$27. Bowen, Howard R. The Costs of Higher Education. San Francisco: JosseyBass Publishers, 1987. Gomme, Paul. "Money and Growth Revisited: Measuring the Costs of Inflation in an Endogenous Growth Model." Journal of Monetary Economics 32 (1993): 51-77. Ireland, Peter N. "Money and Growth: An Alternative Approach." American Economic Review 84 (March 1994): 47-65. Jones, Larry E., and Rodolfo Manuelli. "Growth and the Effects of Inflation." NBER working paper 4523, November 1993. Lucas, Robert E. "On the Mechanics of Economic Development." Journal of Monetary Economics 22 (1988): 3-42. --. "Supply Side Economics: An Analytical Review." Oxford Economic Papers 42 (1990): 293-316. Mulligan, Casey B., and Xavier Sala-i-Martin. "Transitional Dynamics in Two Sector Models of Endogenous Growth." Quarterly Journal of Economics 108 (August 1993): 739-73. Pecorino, Paul. "The Growth Rate Effects of Tax Reform." Oxford Economic Papers 46 (July 1994): 492-501. Rebelo, Sergio. "Long Run Policy Analysis and Long Run Growth." Journal of Political Economy 99 (June 1991): 500-21. Wang, Ping, and Chong K. Yip. "Examining the Long-run Effect of Money on Economic Growth." Journal of Macroeconomics 14 (Spring 1992): 359--69. --. "Real Effects of Money and Welfare Costs of Inflation in an Endogenously Growing Economy with Transaction Costs." Manuscript, Georgia State University, Janua~ 1994.
Appendix List of Variables Y1 Y2
Ai q~i ui M K H G~I,~/ ¢2,1 - 7
= output of the consumption/physical capital good. = production of human capital. = technology level parameter in sector i. -- percent of physical capital stock allocated to sector i. = percent of human capital stock allocated to sector i. = nominal money stock. = physical capital. = human capital. physical capital share parameters. = human capital share parameters. =
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Paul Pecorino O{3
Pi C
IK,IH 1/~ P R r,w g Z l'f
V
549
= share of real money stock in production of good 1. = ratio of input d to output in sector i. = price of good i. = consumption. = investment in physical and h u m a n capital. = intertemporal elasticity of substitution. = rate of time preference. = nominal interest rate. = rate of return to holding physical and h u m a n capital = rate of growth of consumption. = transfers payments from the government. = rate of inflation. = rate of growth of the m o n e y stock. = velocity of money.