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The complete complementarity of consumption and real balances and the strong superneutrality of money
economics letters ELSEVIER
Economics Letters 48 (1995) 91-97
The complete complementarity of consumption and real balances and the strong superneutrality of money Hiroaki Hayakawa Department of Business Administration and Information Science, Chubu University, 1200 Matsumoto-cho, Kasugai, Aichi-ken 487, Japan
Received 9 November1993; accepted25 August1994 Abstract
In the representative agent model with money in the utility function, if consumption and real balances are perfect complements, steady-state capital intensity and real balance holdings are both invariant to monetary growth, regardless of the nature of time preferences. Keywords: Complete complementarity in utility; Strong superneutrality of money; Variable time preferences;
Cash-in-advance constraint JEL classification: 042
I. Introduction
In the monetary growth literature, a number of alternative sources of nonsuperneutrality of m o n e y have been identified that are attributed to different structural features of intertemporal optimizing models (see, for example, Orphanides and Solow, 1990; Weil, 1991; Wang and Yip, 1992; and Hayakawa, 1994). Central to this literature is Sidrauski's (1967) rational choice model with money in the utility function. In such a model, Epstein and Hynes (1983) pointed out that the superneutrality of money depends critically on the endogenous variability of the rate of time preference. The rationale is that if real balances and consumption are substitutable, real balance holdings decline in response to higher monetary expansion, thereby lowering the steady-state rate of time preference, so that the equality as of steady state between this rate and the net marginal productivity of capital is realized at higher capital intensity (see, also, Hayakawa, 1992, for a graphical exposition). Sidrauski's superneutrality is a special case in which time preferences remain constant. Within constant time preferences, an interesting observation has been made by Asako (1983) that if consumption and real balances are perfect complements in yielding instanta0165-1765/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1765(94)00588-5
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neous utility, both steady-state capital intensity and steady-state real balance holdings remain invariant to monetary growth. Such invariance is stronger than Sidrauski's superneutrality, in which steady-state real balance holdings fall as a result of higher monetary expansion while steady-state capital intensity remains unaffected. We shall refer to such strong invariance as the
strong superneutrality of money. In this paper, we point out that the strong superneutrality under the complete complementarity of consumption and real balances holds even if time preferences are allowed to vary endogenously ~ la Uzawa (1968) and Epstein (1987a,b). This result is stronger than Asako's observation, and it contrasts with Epstein and Hyne's non-superneutrality attributed to the variability of time preferences, which hinges critically on the substitutability assumption. Thus, whether money is a substitute or a perfect complement to consumption is of fundamental importance to the superneutrality-of-money issue. We also point out that the strong superneutrality based on the perfect complementarity is analogous to Stockman's (1981) point that if a cash-in-advance constraint is imposed on consumption alone, steady-state capital intensity and real balance holdings are both unaffected by monetary growth (see, also, Abel, 1985). To the extent that this result extends to the case of variable time preferences, this analogy holds regardless of the nature of time preferences. It is based on the fact that imposing a Clower cash-in-advance constraint on consumption is formally equivalent to specifying an instantaneous utility function after the Leontief form. Such equivalence is a special case of the functional equivalence between entering money into the utility function, on the one hand, and considering liquidity/transaction costs explicitly in budget constraint, on the other, as analyzed by Feenstra (1986). Our point is made in a decentralized, competitive setting in which agents are infinitely-lived and markets clear continuously under perfect foresight.
2. Analysis Let the representative agent's choice space consist of pairs of continuous time paths of consumption and real balances from time 0 to ~, denoted C and M. Time t-images of C and M are denoted c(t) and m(t). On this space are defined recursive preferences of the UzawaEpstein kind represented by the following utility functional: t
f o(c(t).m(t., exp[- I o
.
(1)
o
where v(c(t), m(t)) and ~(c(t), m(t)) are referred to as the instantaneous utility and discounting functions, respectively (see Uzawa, 1968, and Epstein, 1987a,b). Consider the case in which consumption and real balances are perfect complements. In such a case, v(c(t), m(t)) and ~(c(t), m(t)) can be reexpressed as u(z(t)) and o-(z(t)) where z(t) is defined as z(t)=-min{(1/3~)c(t), m(t)} with 31 being a positive constant defining the complementarity. Hence, with Z denoting a path of z(t), the utility functional (1) can be rewritten as
H. Hayakawa / Economics Letters 48 (1995) 91-97 oo
93
!
0
0
where u(z) and tr(z) are twice continuously differentiable with usual properties: u ( z ) > 0 , u'(z) > O, u"(z) < 0, o-(z) > 0, o"(z) > 0, o-"(z) > 0, and u'(z)tr(z) - u(z)o"(z) > 0; the last condition assures the monotonicity over constant paths. Since consumption and real balances are always chosen to meet the complementarity requirement, c(t)= ym(t), regardless of price conditions, the relevant paths of consumption and real balances for optimization purposes are the ones that maintain this relation. Hence, with this complementarity requirement taken into account, we let the representative agent determine his optimal paths of consumption and real balances by solving the following optimization problem under perfect foresight. oc
max / u(c(t))y(t) dt
(3)
0
subject to dy(t) / dt = - cr(c(t))y(t),
(3a)
da(t)/ dt = [r(/) - n]a(t) + w(t) + to(t) - [r(t) + 7r(t)]m(t) - c(t) ,
(3b)
c( t ) = "ym( t ) ,
(3c)
y(0) = 1 and
a(0) =a0(given ) ,
where a(t) is real asset holdings consisting of capital k(t) and real balances m(t); w(t) is wage earnings; to(t) is monetary transfers from the government in real terms, which equals Ore(t) where 0 is the growth rate of money supply; r(t) is the real interest rate; ~" is the expected (and also actual) inflation rate; n is the rate at which the agent multiplies; y(t) is a discount factor declining endogenously at rate o-(c(t)). The current and future values of wage earnings, the real interest rate, and the expected inflation rate are part of given data. Eq. (3b) is the usual budget constraint; implicit in this specification is an assumption that the agent is endowed with one unit of labor which is supplied inelastically to earn wages w(t). Let the Hamiltonian of this problem be specified as H[c(t); A(/), v(/)] = y(t)[u(c(t)) - A(t)tr(c(t))] + v(t)[{[(r(t) - n]a(t) + w(t) + to(t) - {[r(t) + 7r(t)]/~, + 1}c(t)}],
(4)
where A(t) and v(t) are costate variables for stock variables y(t) and a(t). By Pontryagin's maximum principle, the following conditions hold along the optimal plan: y(t)[u'(c(t)) -A(t)tr'(c(t))] = v(t)[1 + i(t)/'y] ,
(5)
dA(/) /dt = - [u(c(t)) - A(t)tr(c(t))],
(6)
dr(l)/dt = - [r(t) - n] v(t),
(7)
H. Hayakawa / Economics Letters 48 (1995) 91-97
94
where i ( t ) - r(t)+ 7r(t) is the nominal interest rate. And, the transversality conditions are stated as lim A(t)y(t) = 0
(8)
lim v(t)a(t) =lim (y(t)[u'(c(t)) - A(t)tr'(c(t))]/[1 + i(t)/7])a(t) = O.
(9)
t___~ ~t~
t----~ oo
l----~ oc
Differentiating A(t)y(t) with respect to time t and making use of (3a) and (6) gives
(d/ dt)[-A(t)y(t)] = u(c(t))y(t) .
(10)
Hence, integrating this from time s to ~ and utilizing (8), we obtain q
x(s) = u ( , c ) -
fu(c(q))exp[-f S
dq,
(11)
S
where ~C is the right-hand tail path of C from time s onwards (a path whose time z-image equals the time (s + r)-image of C). Thus, costate variable A(t) measures the future lifetime utility from time t onwards in current terms. Thus, with (11) substituted in and with (7) integrated, conditions (5)-(9) are consolidated as
y(t)[u'(c(t)) - U(,C)o-'(c(t))] = v(t)[1 + i(t)/7],
(12)
v(t) = v(O)exp{- f'[r(z) - n] dr} ,
(13)
0
!im v(t)a(t) =!fin { y(t)[u'(c(t)) - U(tC)tr'(c(t))] /[1 + i(t) /7 ]}a(t) = O .
(14)
The economic meaning of these conditions is clear: condition (12) states that the marginal utility of consumption in the Volterra derivative sense (Volterra, 1959; Wan, 1970; Hayakawa and Ishizawa, 1993a) on the left equals the forgone asset accumulation on the right, 1 + i(t)/7, converted into utility units using the marginal utility of time-t assets measured by v(t), which declines at rate r(t) - n (see Hayakawa and Ishizawa, 1993b, 1994 for details). The reason we need to consider 1 + i(t)/7, rather than just 1, on the right, is that whenever consumption is increased by one unit, real balance holdings increase by 1/7 due to the complementarity, so that the holding cost of real balances also rises by i(t)/7. Therefore, the total forgone asset accumulation inclusive of this holding cost amounts to 1 + i(t)/7. The transversality condition (14) states that the utility value of assets measured by the marginal utility of assets should vanish in the indefinite future. The Keynes-Ramsey rule is obtained by taking a logarithmic time derivative of (12) and substituting (7) into it: [dc(t)/dt]/c(t) = -[1/~(c(t), U(tC))] [r(t) - n - p (t; C)] + [dR (t)/dt]/R(t), where
(15)
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~(c(t), U ( , C ) ) =-[u"(c(t)) - t r " ( c ( t ) ) U ( , C ) ] c ( t ) / [ u ' ( c ( t ) ) - t r ' ( c ( t ) ) U ( , C ) ] , p(t; C ) = [u'(c(t))tr(c(t)) - u ( c ( t ) ) t r ' ( c ( t ) ) ] / [ u ' ( c ( t ) ) - t r ' ( c ( t ) ) U ( , C ) ] , R(t)-
i(t)/3~ + 1 .
~(c(t), U ( , C ) ) is the elasticity of the marginal utility of time l-consumption; p(t; C ) is the rate of time preference at time t (see Epstein, 1987a and Hayakawa, 1991); R ( t ) is the cost of
consumption in terms of total forgone asset accumulation. Since the agent's decisions are made in a competitive and market-clearing environment, the dynamics of the system are represented by the following pair of equations: [dc(t) / dt]/c(t) = - [1/~(c(t), U ( tC))] [ f ' ( k ) - n - p(t; C)] + [dR(t) / dt]/R(t),
(16)
dk(t) / dt = f ( k ( t ) ) - n k ( t ) - c(t) ,
(17)
where R ( t ) = {f'(k) + [0 - n - ( d c ( t ) / d t ) / c ( t ) ] ) / 3 ~ + 1 .
represents a constant-returns-to-scale neoclassical production function. Eq. (17) is obtained from the goods market-clearing condition. Here, C is a perfect foresight path determined by these equations. The expression for R ( t ) is obtained by noting that [ d c ( t ) / d t ] / c(t) = [dm(t)/dt]/m(t) by the complementarity and [dm(t)/dt]/m(t) = 0 - 7r - n. In steady state, therefore, the following conditions hold:
f(k)
f'(k*)
- n = tr(g(k*)) ,
c* = f ( k * ) - n k * =- g ( k * ) ,
(18) (19)
where an asterisk denotes steady state; g ( k ) is a function defined by f ( k ) - n k . Note that the rate of time preference p(t; ,C) reduces in steady state to o-(c*). Condition (18) states that the steady-state capital intensity is determined at the intersection of two schedules, one representing the net marginal productivity of capital and the other representing the (steady-state) rate of time preferences; clearly, it is independent of monetary growth. By the complementarity, steady-state real balance holdings are also invariant to monetary growth. Thus, if consumption and real balances are perfect complements, steady-state capital intensity and real balance holdings are both independent of monetary expansion, determined completely by the properties of the production function f ( k ) and the instantaneous discounting function o-(c), which serves as the rate of time preference in steady state. Obviously, this result holds a fortiori when the rate of time preference is constant as observed by Asako (1983). In this regard, we recall Stockman's point (1981, p. 393) that if a cash-in-advance constraint is imposed on consumption spending alone, steady-state capital intensity and real balance holdings are both unaffected by monetary growth. While this point was demonstrated under constant time preferences, it is straightforward to extend it to a more general case in which time preferences are endogenously variable. Thus, on the superneutrality of money, a useful analogy exists between the perfect complementarity between consumption and real balances in
96
H. Hayakawa / Economics Letters 48 (1995) 91-97
yielding instantaneous utility, on the one hand, and the imposition of a cash-in-advance constraint on c o n s u m p t i o n spending, on the other, irrespective of the constancy or the e n d o g e n o u s variability of time preferences. The analogy reflects the formal equivalence b e t w e e n specifying the instantaneous utility function in the Leontief form along the money-inthe-utility-function approach and imposing a cash-in-advance constraint on c o n s u m p t i o n spending. This formal equivalence is a special case of Feenstra's (1986) functional equivalence b e t w e e n entering m o n e y into the utility function and considering liquidity/transaction costs explicitly in budget constraint.
3. Conclusion
In the representative agent m o d e l with m o n e y in the utility function, w h e t h e r m o n e y is superneutral or not d e p e n d s critically on the substitutability/complementarity of c o n s u m p t i o n and real balances in the instantaneous utility and discounting functions. If c o n s u m p t i o n and real balances are substitutable, m o n e y becomes non-superneutral only if the rate of time preference is allowed to vary endogenously. If this function is constant, Sidrauski's superneutrality obtains. In contrast, if c o n s u m p t i o n and real balances are perfect c o m p l e m e n t s in yielding instantaneous utility, m o n e y turns superneutral irrespective of w h e t h e r time preferences are constant or endogenously variable. To m a k e this superneutrality even stronger, steady-state real balance holdings also remain invariant to m o n e t a r y growth by virtue of the c o m p l e m e n tarity. F u r t h e r m o r e , regardless of the nature of time preferences, a close analogy is observed b e t w e e n imposing the perfect c o m p l e m e n t a r i t y of c o n s u m p t i o n and real balances on the instantaneous utility and discounting functions, on the one hand, and imposing a Clower cash-in-advance constraint on consumption spending as in the S t o c k m a n model, on the other. This analogy reflects Feenstra's functional equivalence between m o n e y in the utility function and liquidity/transaction costs specified in budget constraint.
References
Abel, A.B., 1985, Dynamic behavior of capital accumulation in a cash-in-advance economy, Journal of Monetary Economics 16, 55-71. Asako, K., 1983, The utility function and the superneutrality of money on the transition path, Econometrica 51, 1593-1596. Epstein, L.G., 1987a, A simple dynamic general equilibrium model, Econometrica 41, 68-95. Epstein, L.G., 1987b, The global stability of efficient intertemporai allocations, Econometrica 55, 329-355. Epstein, L.G. and J.A. Hynes, 1983, The rate of time preference and dynamic economic analysis, Journal of Political Economy 91, 611-635. Feenstra, R.C., 1986, Functional equivalence between liquidity costs and the utility of money, Journal of Monetary Economics 17, 271-291. Hayakawa, H., 1991, On the rate of time preference under recursive preferences, Economics Letters 36, 347-354. Hayakawa, H., 1992, The neutrality of money and the optimal monetary growth rule when preferences are recursive: Cash-in-advance vs. money in the utility function, Journal of Macroeconomics 14, 233-266.
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Hayakawa, H., 1994, Time cost of transactions and the superneutrality of money: A reformulation of the Stockman-Abel cash-in-advance model under recursive preferences, Economic Studies Quarterly 45, 14-31. Hayakawa, H. and S. Ishizawa, 1993a, On the role of generating functions when preferences are recursive, Economics Letters 41, 41-45. Hayakawa, H. and S. Ishizawa, 1993b, The fundamental principle of intertemporal optimization, Economics Letters 41, 273-280. Hayakawa, H. and S. Ishizawa, 1994, The fundamentals of intertemporal optimization in the continuous time modeling of consumer behavior, unpublished manuscript. Orphanides, A. and R.M. Solow, 1990, Money, inflation and growth, in: B.M. Friedman and F.H. Hahn, eds., Handbook of monetary economics 1, North-Holland, New York, 223-261. Sidrauski, M., 1967, Rational choice and patterns of growth in a monetary economy, American Economic Review 57, 534-544. Stockman, A.C., 1981, Anticipated inflation and the capital stock in a cash-in-advance economy, Journal of Monetary Economics 8, 387-393. Uzawa, H., 1968, Time preferences, the consumption function, and optimum asset holdings, in: J.N. Wolfe, ed., Value, capital, and growth: Papers in honour of Sir John Hicks (University of Edinburgh Press, Edinburgh), 485-504. Volterra, V., 1959, Theory of functionals and of integral and integro-differential equation (Dover, New York). Wang, P. and C.K. Yip, 1992, Alternative approaches to money and growth, Journal of Money, Credit, and Banking 24, 553-562. Wan, H.Y., Jr, 1970, Optimal saving programs under intertemporally dependent preferences, International Economic Review 11, 521-547. Weil, P., 1991, Is money net wealth?, International Economic Review 32, 37-53.
Economics Letters 48 (1995) 91-97
The complete complementarity of consumption and real balances and the strong superneutrality of money Hiroaki Hayakawa Department of Business Administration and Information Science, Chubu University, 1200 Matsumoto-cho, Kasugai, Aichi-ken 487, Japan
Received 9 November1993; accepted25 August1994 Abstract
In the representative agent model with money in the utility function, if consumption and real balances are perfect complements, steady-state capital intensity and real balance holdings are both invariant to monetary growth, regardless of the nature of time preferences. Keywords: Complete complementarity in utility; Strong superneutrality of money; Variable time preferences;
Cash-in-advance constraint JEL classification: 042
I. Introduction
In the monetary growth literature, a number of alternative sources of nonsuperneutrality of m o n e y have been identified that are attributed to different structural features of intertemporal optimizing models (see, for example, Orphanides and Solow, 1990; Weil, 1991; Wang and Yip, 1992; and Hayakawa, 1994). Central to this literature is Sidrauski's (1967) rational choice model with money in the utility function. In such a model, Epstein and Hynes (1983) pointed out that the superneutrality of money depends critically on the endogenous variability of the rate of time preference. The rationale is that if real balances and consumption are substitutable, real balance holdings decline in response to higher monetary expansion, thereby lowering the steady-state rate of time preference, so that the equality as of steady state between this rate and the net marginal productivity of capital is realized at higher capital intensity (see, also, Hayakawa, 1992, for a graphical exposition). Sidrauski's superneutrality is a special case in which time preferences remain constant. Within constant time preferences, an interesting observation has been made by Asako (1983) that if consumption and real balances are perfect complements in yielding instanta0165-1765/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1765(94)00588-5
H. Hayakawa / Economics Letters 48 (1995) 91-97
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neous utility, both steady-state capital intensity and steady-state real balance holdings remain invariant to monetary growth. Such invariance is stronger than Sidrauski's superneutrality, in which steady-state real balance holdings fall as a result of higher monetary expansion while steady-state capital intensity remains unaffected. We shall refer to such strong invariance as the
strong superneutrality of money. In this paper, we point out that the strong superneutrality under the complete complementarity of consumption and real balances holds even if time preferences are allowed to vary endogenously ~ la Uzawa (1968) and Epstein (1987a,b). This result is stronger than Asako's observation, and it contrasts with Epstein and Hyne's non-superneutrality attributed to the variability of time preferences, which hinges critically on the substitutability assumption. Thus, whether money is a substitute or a perfect complement to consumption is of fundamental importance to the superneutrality-of-money issue. We also point out that the strong superneutrality based on the perfect complementarity is analogous to Stockman's (1981) point that if a cash-in-advance constraint is imposed on consumption alone, steady-state capital intensity and real balance holdings are both unaffected by monetary growth (see, also, Abel, 1985). To the extent that this result extends to the case of variable time preferences, this analogy holds regardless of the nature of time preferences. It is based on the fact that imposing a Clower cash-in-advance constraint on consumption is formally equivalent to specifying an instantaneous utility function after the Leontief form. Such equivalence is a special case of the functional equivalence between entering money into the utility function, on the one hand, and considering liquidity/transaction costs explicitly in budget constraint, on the other, as analyzed by Feenstra (1986). Our point is made in a decentralized, competitive setting in which agents are infinitely-lived and markets clear continuously under perfect foresight.
2. Analysis Let the representative agent's choice space consist of pairs of continuous time paths of consumption and real balances from time 0 to ~, denoted C and M. Time t-images of C and M are denoted c(t) and m(t). On this space are defined recursive preferences of the UzawaEpstein kind represented by the following utility functional: t
f o(c(t).m(t., exp[- I o
.
(1)
o
where v(c(t), m(t)) and ~(c(t), m(t)) are referred to as the instantaneous utility and discounting functions, respectively (see Uzawa, 1968, and Epstein, 1987a,b). Consider the case in which consumption and real balances are perfect complements. In such a case, v(c(t), m(t)) and ~(c(t), m(t)) can be reexpressed as u(z(t)) and o-(z(t)) where z(t) is defined as z(t)=-min{(1/3~)c(t), m(t)} with 31 being a positive constant defining the complementarity. Hence, with Z denoting a path of z(t), the utility functional (1) can be rewritten as
H. Hayakawa / Economics Letters 48 (1995) 91-97 oo
93
!
0
0
where u(z) and tr(z) are twice continuously differentiable with usual properties: u ( z ) > 0 , u'(z) > O, u"(z) < 0, o-(z) > 0, o"(z) > 0, o-"(z) > 0, and u'(z)tr(z) - u(z)o"(z) > 0; the last condition assures the monotonicity over constant paths. Since consumption and real balances are always chosen to meet the complementarity requirement, c(t)= ym(t), regardless of price conditions, the relevant paths of consumption and real balances for optimization purposes are the ones that maintain this relation. Hence, with this complementarity requirement taken into account, we let the representative agent determine his optimal paths of consumption and real balances by solving the following optimization problem under perfect foresight. oc
max / u(c(t))y(t) dt
(3)
0
subject to dy(t) / dt = - cr(c(t))y(t),
(3a)
da(t)/ dt = [r(/) - n]a(t) + w(t) + to(t) - [r(t) + 7r(t)]m(t) - c(t) ,
(3b)
c( t ) = "ym( t ) ,
(3c)
y(0) = 1 and
a(0) =a0(given ) ,
where a(t) is real asset holdings consisting of capital k(t) and real balances m(t); w(t) is wage earnings; to(t) is monetary transfers from the government in real terms, which equals Ore(t) where 0 is the growth rate of money supply; r(t) is the real interest rate; ~" is the expected (and also actual) inflation rate; n is the rate at which the agent multiplies; y(t) is a discount factor declining endogenously at rate o-(c(t)). The current and future values of wage earnings, the real interest rate, and the expected inflation rate are part of given data. Eq. (3b) is the usual budget constraint; implicit in this specification is an assumption that the agent is endowed with one unit of labor which is supplied inelastically to earn wages w(t). Let the Hamiltonian of this problem be specified as H[c(t); A(/), v(/)] = y(t)[u(c(t)) - A(t)tr(c(t))] + v(t)[{[(r(t) - n]a(t) + w(t) + to(t) - {[r(t) + 7r(t)]/~, + 1}c(t)}],
(4)
where A(t) and v(t) are costate variables for stock variables y(t) and a(t). By Pontryagin's maximum principle, the following conditions hold along the optimal plan: y(t)[u'(c(t)) -A(t)tr'(c(t))] = v(t)[1 + i(t)/'y] ,
(5)
dA(/) /dt = - [u(c(t)) - A(t)tr(c(t))],
(6)
dr(l)/dt = - [r(t) - n] v(t),
(7)
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where i ( t ) - r(t)+ 7r(t) is the nominal interest rate. And, the transversality conditions are stated as lim A(t)y(t) = 0
(8)
lim v(t)a(t) =lim (y(t)[u'(c(t)) - A(t)tr'(c(t))]/[1 + i(t)/7])a(t) = O.
(9)
t___~ ~t~
t----~ oo
l----~ oc
Differentiating A(t)y(t) with respect to time t and making use of (3a) and (6) gives
(d/ dt)[-A(t)y(t)] = u(c(t))y(t) .
(10)
Hence, integrating this from time s to ~ and utilizing (8), we obtain q
x(s) = u ( , c ) -
fu(c(q))exp[-f S
dq,
(11)
S
where ~C is the right-hand tail path of C from time s onwards (a path whose time z-image equals the time (s + r)-image of C). Thus, costate variable A(t) measures the future lifetime utility from time t onwards in current terms. Thus, with (11) substituted in and with (7) integrated, conditions (5)-(9) are consolidated as
y(t)[u'(c(t)) - U(,C)o-'(c(t))] = v(t)[1 + i(t)/7],
(12)
v(t) = v(O)exp{- f'[r(z) - n] dr} ,
(13)
0
!im v(t)a(t) =!fin { y(t)[u'(c(t)) - U(tC)tr'(c(t))] /[1 + i(t) /7 ]}a(t) = O .
(14)
The economic meaning of these conditions is clear: condition (12) states that the marginal utility of consumption in the Volterra derivative sense (Volterra, 1959; Wan, 1970; Hayakawa and Ishizawa, 1993a) on the left equals the forgone asset accumulation on the right, 1 + i(t)/7, converted into utility units using the marginal utility of time-t assets measured by v(t), which declines at rate r(t) - n (see Hayakawa and Ishizawa, 1993b, 1994 for details). The reason we need to consider 1 + i(t)/7, rather than just 1, on the right, is that whenever consumption is increased by one unit, real balance holdings increase by 1/7 due to the complementarity, so that the holding cost of real balances also rises by i(t)/7. Therefore, the total forgone asset accumulation inclusive of this holding cost amounts to 1 + i(t)/7. The transversality condition (14) states that the utility value of assets measured by the marginal utility of assets should vanish in the indefinite future. The Keynes-Ramsey rule is obtained by taking a logarithmic time derivative of (12) and substituting (7) into it: [dc(t)/dt]/c(t) = -[1/~(c(t), U(tC))] [r(t) - n - p (t; C)] + [dR (t)/dt]/R(t), where
(15)
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~(c(t), U ( , C ) ) =-[u"(c(t)) - t r " ( c ( t ) ) U ( , C ) ] c ( t ) / [ u ' ( c ( t ) ) - t r ' ( c ( t ) ) U ( , C ) ] , p(t; C ) = [u'(c(t))tr(c(t)) - u ( c ( t ) ) t r ' ( c ( t ) ) ] / [ u ' ( c ( t ) ) - t r ' ( c ( t ) ) U ( , C ) ] , R(t)-
i(t)/3~ + 1 .
~(c(t), U ( , C ) ) is the elasticity of the marginal utility of time l-consumption; p(t; C ) is the rate of time preference at time t (see Epstein, 1987a and Hayakawa, 1991); R ( t ) is the cost of
consumption in terms of total forgone asset accumulation. Since the agent's decisions are made in a competitive and market-clearing environment, the dynamics of the system are represented by the following pair of equations: [dc(t) / dt]/c(t) = - [1/~(c(t), U ( tC))] [ f ' ( k ) - n - p(t; C)] + [dR(t) / dt]/R(t),
(16)
dk(t) / dt = f ( k ( t ) ) - n k ( t ) - c(t) ,
(17)
where R ( t ) = {f'(k) + [0 - n - ( d c ( t ) / d t ) / c ( t ) ] ) / 3 ~ + 1 .
represents a constant-returns-to-scale neoclassical production function. Eq. (17) is obtained from the goods market-clearing condition. Here, C is a perfect foresight path determined by these equations. The expression for R ( t ) is obtained by noting that [ d c ( t ) / d t ] / c(t) = [dm(t)/dt]/m(t) by the complementarity and [dm(t)/dt]/m(t) = 0 - 7r - n. In steady state, therefore, the following conditions hold:
f(k)
f'(k*)
- n = tr(g(k*)) ,
c* = f ( k * ) - n k * =- g ( k * ) ,
(18) (19)
where an asterisk denotes steady state; g ( k ) is a function defined by f ( k ) - n k . Note that the rate of time preference p(t; ,C) reduces in steady state to o-(c*). Condition (18) states that the steady-state capital intensity is determined at the intersection of two schedules, one representing the net marginal productivity of capital and the other representing the (steady-state) rate of time preferences; clearly, it is independent of monetary growth. By the complementarity, steady-state real balance holdings are also invariant to monetary growth. Thus, if consumption and real balances are perfect complements, steady-state capital intensity and real balance holdings are both independent of monetary expansion, determined completely by the properties of the production function f ( k ) and the instantaneous discounting function o-(c), which serves as the rate of time preference in steady state. Obviously, this result holds a fortiori when the rate of time preference is constant as observed by Asako (1983). In this regard, we recall Stockman's point (1981, p. 393) that if a cash-in-advance constraint is imposed on consumption spending alone, steady-state capital intensity and real balance holdings are both unaffected by monetary growth. While this point was demonstrated under constant time preferences, it is straightforward to extend it to a more general case in which time preferences are endogenously variable. Thus, on the superneutrality of money, a useful analogy exists between the perfect complementarity between consumption and real balances in
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yielding instantaneous utility, on the one hand, and the imposition of a cash-in-advance constraint on c o n s u m p t i o n spending, on the other, irrespective of the constancy or the e n d o g e n o u s variability of time preferences. The analogy reflects the formal equivalence b e t w e e n specifying the instantaneous utility function in the Leontief form along the money-inthe-utility-function approach and imposing a cash-in-advance constraint on c o n s u m p t i o n spending. This formal equivalence is a special case of Feenstra's (1986) functional equivalence b e t w e e n entering m o n e y into the utility function and considering liquidity/transaction costs explicitly in budget constraint.
3. Conclusion
In the representative agent m o d e l with m o n e y in the utility function, w h e t h e r m o n e y is superneutral or not d e p e n d s critically on the substitutability/complementarity of c o n s u m p t i o n and real balances in the instantaneous utility and discounting functions. If c o n s u m p t i o n and real balances are substitutable, m o n e y becomes non-superneutral only if the rate of time preference is allowed to vary endogenously. If this function is constant, Sidrauski's superneutrality obtains. In contrast, if c o n s u m p t i o n and real balances are perfect c o m p l e m e n t s in yielding instantaneous utility, m o n e y turns superneutral irrespective of w h e t h e r time preferences are constant or endogenously variable. To m a k e this superneutrality even stronger, steady-state real balance holdings also remain invariant to m o n e t a r y growth by virtue of the c o m p l e m e n tarity. F u r t h e r m o r e , regardless of the nature of time preferences, a close analogy is observed b e t w e e n imposing the perfect c o m p l e m e n t a r i t y of c o n s u m p t i o n and real balances on the instantaneous utility and discounting functions, on the one hand, and imposing a Clower cash-in-advance constraint on consumption spending as in the S t o c k m a n model, on the other. This analogy reflects Feenstra's functional equivalence between m o n e y in the utility function and liquidity/transaction costs specified in budget constraint.
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