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 Econometrica is currently published by The Econometric Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/econosoc.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. 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.