Capital Accumulation on the Transition Path in a Monetary Optimizing Model
Stanley Fischer
Econometrica, Vol. 47, No. 6. (Nov., 1979), pp. 1433-1439.
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http://www.jstor.org Wed Oct 3 08:43:20 2007 Econornetrica, Vol. 47, No. 6 (November, 1979)
CAPITAL ACCUMULATION ON THE TRANSITION PATH IN A MONETARY OPTIMIZING MODEL
It is well known that in the Sidrauski monetary intertemporal optimizing model the steady state capital stock is invariant to the rate of inflation. This paper shows that the rate of accumulation of capital is not invariant to the rate of inflation and that, for the constant relative risk aversion family of utility functions (except logarithmic), the rate of capital accumulation is faster the higher the growth rate of money. Money is thus not neutral on transition paths.
THE EFFECTS of anticipated inflation on capital accumulation are central to a number of papers, including those of Tobin [9] and Sidrauski [8].' However, the theoretical validity of the link between inflation and capital accumulation was called into question by Sidrauski's [7] demonstration that money is superneutral (the growth rate of money affects no real variables other than real balances) in his model in which infinitely lived families maximize an intertemporal utility function. In particular, Sidrauski showed that the steady state capital stock for such families is determined by the modified golden rule: the steady state capital stock is such that the real interest rate is equal to the rate of population growth plus the rate of time preference. The rate of inflation or the growth rate of money affects only the steady state level of real balances, but not the per capita steady state levels of the capital stock or consumption. The Sidrauski superneutrality result depends on the infinite horizon assump- tion, since superneutrality does not generally obtain in overlapping generations models in which money and capital are the only assek3 Further, even in the infinite horizon case, relatively simple changes such as including money in the production function, as shorthand for the fact that larger holdings of real balances facilitate transaction^,^ or the inclusion of an endogenous labor supply (Brock [I]), remove the superneutrality result. This note focuses on the transition to the steady state, rather than on the properties of the steady state itself. It shows that even in the original Sidrauski maximizing model, the path the economy takes to the steady state is not in general independent of the growth rate of money. The rate of capital accumulation may be affected by the rate of inflation everywhere but in steady state. The model and the first order conditions are set out in Section 1.Section 2 then shows that there is a unique convergent equilibrium path. Section 3 shows that the conventional
'I am grateful to Andrew Abel, Guillermo Calvo, Rudiger Dornbusch, Robert Solow, and the referees for their comments. Research support from the NSF is acknowledged with thanks. See Dornbusch and Frenkel [3]; recently Lucas [6] and Fischer [5] have explored this effect in rational expectations models. See Drazen [4]. Though this change does not generally imply that a higher capital stock accompanies a higher inflation rate in steady state. 1434 STANLEY FISCHER presumption-that more rapid monetary growth produces more rapid capital accumulation-holds true if utility functions belong to the constant relative risk aversion family.
1. THE MODEL Detailed assumptions are spelled out in Sidrauski [7]. A family, whose membership grows at rate n, maximizes an additive utility function
where ct is per capita consumption, m,is per capita real balances, and 6 >0 is the rate of time discount. The instantaneous utility function u()is concave, with ul,2.42 >0 and ull, u22 <0. The family can hold either money or capital; money is introduced into the system through lump sum taxes or transfers. There is a neoclassical production function where k,is the per capita capital stock. Letting 17 be the expected rate of inflation we can write the first order conditions for an optimal path as:
Equation (5')is the budget constraint that the household faces: x is the per capita rate of real money transfers which are assumed by each household to be independent of its own holdings of money.' In steady state C = 0 = m ;substituting (4)into (3)we obtain the superneutrality result:
On any path on which the actual rate of inflation (@/p)is continuous, and assuming a constant growth rate of money, 8, we have:
We shall be working with perfect foresight equilibrium paths where
If the household regarded x as interest payments on money, and we wrote x = Om,where B is the growth rate of money, then money would be superneutral. Under this assumption we would have (17- 0)in (3)and (4)where 17 appears now. CAPITAL ACCUMULATION 1435
On such paths, the expected and actual inflation rates coincide. In addition, on equilibrium paths, the economy wide constraint that physical output is either consumed or invested binds in the sense that consumption and investment demands, derived from (3)-(59, sum to full employment output. Using the perfect foresight and equilibrium assumptions, we use (7) and (8)to substitute out for 17 in (3) and (4) and m and x( = Om) in (5'). Now (5') becomes
By examining the perfect foresight path, we assure ourselves that our results are not due to any ad hoc expectations assumptions, but rather follow from the logic of the maximizing model itself.
2. UNIQUENESS OF THE OPTIMAL PATH Substituting (7) and (8)into (3) and (4), and linearizing around the steady state, we obtain:
where we further assume6
and
Asterisks in (9) denote steady state values of variables, and all derivatives are evaluated as those steady state values, c*, m*, and k*. The first step is to establish that the matrix in (9) has a unique negative root and thus that the steady state is a saddle-point. The product of the roots of the system is given by the determinant
This establishes that there are either three negative roots (two of which may be complex with real parts negative) or one. To establish that there is only one negative root, we turn to a necessary condition for stability of a 3 x 3 system,
These assumptions ensure that both consumption and real balances are normal goods. 1436 STANLEY FISCHER namely that the trace of the matrix (2)be negative.
If Z is to be negative, then
has to be negative. However
is of the same sign as uIIu~~-uf2,which, by virtue of the concavity of u( ), is positive. Accordingly the differential equation system (9)is unstable, implying it has at least one positive root. But since we know it has either zero or two positive roots, it has two. There is therefore a unique negative root, and a unique perfect foresight path satisfying the necessary conditions (3)-(5)that converges to the steady state.7 Brock [I]shows in a similar context that the convergent path is the unique equilibrium perfect foresight path, and we henceforth, consider only that path.8
3. CHANGES IN THE GROWTH RATE OF MONEY To establish that the dynamic path of the economy is affected by the growth rate of money, we have to show that the unique negative root of the dynamic system is affected by the growth rate of money. Rather than work with a general utility function, which would require us to examine explicitly third derivatives of the utility function, we specialize to the constant relative risk aversion family:
For R = 1,we have the logarithmic utility function, u(c, m)= a In c +p In m. The characteristic equation for the system becomes:
'An alternative proof of the saddle-point property is given in Calvo [Z],under the assumption ul2>0. However, I have not established that the unique convergent perfect foresight path is the only equilibrium path under the assumptions of this paper. CAPITAL ACCUMULATION 1437
We have taken care to write (13) as a function of c* and k* (through f"), which remain invariant across steady states as 19 changes, and of 19 itself. We will thus be able to examine the effects of changes in 8 on the negative root. Note that (8+6)>0 by assumption, so that we have less than the optimum quantity of money.9 We may write the characteristic equation equivalently as
We are interested in
where it is understood the derivatives are to be evaluated at the negative root. The sign of d+/aA can be obtained by noting from (14)that $(O, 8) is negative, so that $0must be decreasing in A at the unique negative root.1° Accordingly dA a (16) sign -= sign -.* de ae
We shall now show that a+/ae <0, except for R = 1. Differentiating cj/ with respect to 8 we obtain:
From (14) and (17):
Noting the term -A (A2 -AS)in (18),we proceed to:
For 8 +IS = 0, there may not be an interior solution. 10 Alternatively, draw a graph of $0as a function of A ; note that $(-m, 8)>0, and there is a unique negative root. 1438 STANLEY FISCHER
Since we are concerned with A <0, the left hand side of (19) has the same sign as a~+b/ae,which is therefore negative, except when R = 1 (the logarithmic utility function), in which case aJ,/ae = 0. We have thus shown that in general the greater the growth rate of money, the greater in absolute value is the relevant root of the system. As will be shown below, it follows that, as of any given level of the capital stock, the rate of investment is more rapid the higher is the growth rate of money. The solution of the linearized system for the behavior of the capital stock is: (20) k,=k*-(k*-ko)eA', whence (21) kt= -A (k*- k,). Accordingly, at any k, For given k, where It follows from (23) that on the optimal path with the capital stock increasing, real balances too increase towards their steady state value. The growth rate of real balances, mlm falls over time on the optimal path, implying, from (7),that the rate of inflation increases over time towards its steady state value (8 -n)." 4. CONCLUDING COMMENT The basic result of this note is that the rate of capital accumulation is not invariant to the rate of monetary growth, even in a model in which the steady state capital stock is independent of the growth rate of money. The relationship between the growth rate of money and the rate of capital accumulation is a feature of the transition to the steady state. It would be useful to explain this result informally. One explanation that is tempting is not quite right. That is, that the higher the growth rate of money, the higher the inflation rate, the lower the value of real balances, and thus, given the 11 One would guess that amlae <0, but I have not been able to show that (or the converse) from (23). CAPITAL ACCUMULATION 1439 capital stock, of wealth, and therefore the lower the rate of consumption. This cannot be quite right because it also implies that the nonneutrality would hold for the logarithmic utility function, even though we know from (19)that it does not. Presumably, the effect on capital accumulation results from the influence of holdings of real balances on the marginal utility of consumption, as reflected in equation (3), though even here there is no simple connection, since u12changes sign as the coefficient R moves through unity. In brief, a convincing intuitive explanation of the basic result is not yet available. The empirical significance of the effect investigated here in a simple two asset model obviously cannot be evaluated on a purely a priori basis. Massachusetts Institute of Technology Manuscript received April, 1978. REFERENCES [I] BROCK, WILLIAMA,: "Money and Growth: The Case of Long Run Perfect Foresight," International Economic Review, 15 (1974), 750-777. [2] CALVO,GUILLERMO: "On Models of Money and Perfect Foresight," International Economic Review, 20 (1979), 83-104. [3] DORNBUSCH, RUDIGER, AND JACOB A. FRENKEL;"Inflation and Growth: Alternative Approaches," Journal of Money, Credit and Banking, 5 (1973), 141-156. [4] DRAZEN, ALLAN: ''Essays on the Theory of Inflation," unpublished Ph.D. dissertation, Massachusetts Institute of Technology, 1976. [5] FISCHER, STANLEY:LLAnti~ipation~ and the Non-Neutrality of Money," Journal of Political Economy, 87 (1979), 225-252. [6] Luc~s,ROBERT E., JR.: "An Equilibrium Model of the Business Cycle," Journal of Political Economy, 83 (1975), 1113-1144. [7] SIDRA~SKI,MIGUEL: "Rational Choice and Patterns of Growth in a Monetary Economy," American Economic Review, Papers and Proceedings, 57 (1967), 534-44. [a] -: "Inflation and Economic Growth," Journal of Political Economy, 75 (1967), 796-810. [9] TOBIN,JAMES: "Money and Economic Growth," Econometrica, 33 (1965), 671-684. http://www.jstor.org LINKED CITATIONS - Page 1 of 3 - You have printed the following article: Capital Accumulation on the Transition Path in a Monetary Optimizing Model Stanley Fischer Econometrica, Vol. 47, No. 6. (Nov., 1979), pp. 1433-1439. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28197911%2947%3A6%3C1433%3ACAOTTP%3E2.0.CO%3B2-N This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR. [Footnotes] 2 An Equilibrium Model of the Business Cycle Robert E. Lucas, Jr. The Journal of Political Economy, Vol. 83, No. 6. (Dec., 1975), pp. 1113-1144. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197512%2983%3A6%3C1113%3AAEMOTB%3E2.0.CO%3B2-5 2 Anticipations and the Nonneutrality of Money Stanley Fischer The Journal of Political Economy, Vol. 87, No. 2. (Apr., 1979), pp. 225-252. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197904%2987%3A2%3C225%3AAATNOM%3E2.0.CO%3B2-I 7 On Models of Money and Perfect Foresight Guillermo A. Calvo International Economic Review, Vol. 20, No. 1. (Feb., 1979), pp. 83-103. Stable URL: http://links.jstor.org/sici?sici=0020-6598%28197902%2920%3A1%3C83%3AOMOMAP%3E2.0.CO%3B2-X References NOTE: The reference numbering from the original has been maintained in this citation list. http://www.jstor.org LINKED CITATIONS - Page 2 of 3 - 1 Money and Growth: The Case of Long Run Perfect Foresight William A. Brock International Economic Review, Vol. 15, No. 3. (Oct., 1974), pp. 750-777. Stable URL: http://links.jstor.org/sici?sici=0020-6598%28197410%2915%3A3%3C750%3AMAGTCO%3E2.0.CO%3B2-J 2 On Models of Money and Perfect Foresight Guillermo A. Calvo International Economic Review, Vol. 20, No. 1. (Feb., 1979), pp. 83-103. Stable URL: http://links.jstor.org/sici?sici=0020-6598%28197902%2920%3A1%3C83%3AOMOMAP%3E2.0.CO%3B2-X 5 Anticipations and the Nonneutrality of Money Stanley Fischer The Journal of Political Economy, Vol. 87, No. 2. (Apr., 1979), pp. 225-252. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197904%2987%3A2%3C225%3AAATNOM%3E2.0.CO%3B2-I 6 An Equilibrium Model of the Business Cycle Robert E. Lucas, Jr. The Journal of Political Economy, Vol. 83, No. 6. (Dec., 1975), pp. 1113-1144. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197512%2983%3A6%3C1113%3AAEMOTB%3E2.0.CO%3B2-5 7 Rational Choice and Patterns of Growth in a Monetary Economy Miguel Sidrauski The American Economic Review, Vol. 57, No. 2, Papers and Proceedings of the Seventy-ninth Annual Meeting of the American Economic Association. (May, 1967), pp. 534-544. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28196705%2957%3A2%3C534%3ARCAPOG%3E2.0.CO%3B2-9 8 Inflation and Economic Growth Miguel Sidrauski The Journal of Political Economy, Vol. 75, No. 6. (Dec., 1967), pp. 796-810. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28196712%2975%3A6%3C796%3AIAEG%3E2.0.CO%3B2-B NOTE: The reference numbering from the original has been maintained in this citation list. http://www.jstor.org LINKED CITATIONS - Page 3 of 3 - 9 Money and Economic Growth James Tobin Econometrica, Vol. 33, No. 4. (Oct., 1965), pp. 671-684. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28196510%2933%3A4%3C671%3AMAEG%3E2.0.CO%3B2-E NOTE: The reference numbering from the original has been maintained in this citation list.