* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Keynes-Wicksell and Neoclassical Models of Money and
Survey
Document related concepts
Business cycle wikipedia , lookup
Economic democracy wikipedia , lookup
Fear of floating wikipedia , lookup
Economic growth wikipedia , lookup
Exchange rate wikipedia , lookup
Long Depression wikipedia , lookup
Inflation targeting wikipedia , lookup
Economic calculation problem wikipedia , lookup
Phillips curve wikipedia , lookup
Rostow's stages of growth wikipedia , lookup
Nominal rigidity wikipedia , lookup
Ragnar Nurkse's balanced growth theory wikipedia , lookup
Monetary policy wikipedia , lookup
Money supply wikipedia , lookup
Transcript
American Economic Association Keynes-Wicksell and Neoclassical Models of Money and Growth Author(s): Stanley Fischer Reviewed work(s): Source: The American Economic Review, Vol. 62, No. 5 (Dec., 1972), pp. 880-890 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/1815206 . Accessed: 18/03/2013 08:00 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review. http://www.jstor.org This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions Neoclassical and Keynes-Wicksell of and Money By Models Growth STANLEY FISCHER* The essential features of Keynes-Wicksell (henceforth KW) monetary growth models, distinguishing them from neoclassical models, are the specification of an independent investment function and the assumption that prices change only in response to excess demand in the goods market.! In neoclassical monetary growth models, by contrast, there is no independent investment function and all markets are continuously in equilibrium. In KW models a steady state of inflation requires persistent excess demand in the goods markets. This suggests that the steady-state properties of such models are unsatisfactory. In neoclassical models, an instantaneous doubling of the quantity of money, however the money is distributed, produces an instantaneous doubling of the price level so long as the expected growth rate of the money supply is the same before and after the "blip" in the money supply. This according to KW theorists suggests that there is something amiss in the short-run dynamics of the price level in such models. In this paper, the price dynamics of both models are discussed, and a modified price determination equation is incorporated into a KW model. The standard comparative dynamic exercises for monetary growth models are undertaken in this modified model; the modification of the price adjustment equation ensures steady state equilibria rather than disequilibria. The properties of the modified KW model are then compared with those of neoclassical models. Essentially, familiar short-run macro-economic conclusions emerge from consideration of short-run behavior in the modified model and neoclassical conclusions emerge from analysis of its long-run behavior. I. Price Dynamics KW models use the Law of Supply and Demand to determine the rate of inflation.2 Specifically, it is assumed in KW models that (1) r = X(D-S), O < X < oc where 7ris the rate of inflation, D and S are aggregate demand for and supply of goods, each in real terms, and X is a constant. It is apparent that there cannot be inflation without excess demand if equation (1) determines the rate of inflation, and thus a steady state with inflation requires persistent excess demand. KW models can accordingly have steady states in which individuals are continuallv frustrated in * Assistant professor, department of economics, University of Chicago. I would like to thank George Borts, Rudiger Dornbusch, and Jerome Stein for their helpful comments on an earlier draft. Thanks for comments and discussion are due, too, to William Brock, Jacob Frenkel, Merton Miller, Michael Mussa, Douglas Purvis, and Richard Zecher. 1 Jerome Stein-who is apparently responsible for the KW designation-has recently provided two very useful expositions of these models (1969, 1970). An earlier article of his (1966), using a KW model which is not so named, provides a full dynamic analysis for the typical KW model. 2 See Kenneth Arrow. It will be assumed that the reader is familar with both types of monetary growth models. A two-asset (money and capital), one-sector model is used as the paradigm of neoclassical models (see James Tobin and Miguel Sidrauski); places where my conclusions would differ if some other neoclassical model were used are footnoted. My paradigmatic KW model is contained in Stein's 1969 article. 880 This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions FISCHER: obtaining the goods they demand, even though their demands are based on correct expectations and perceptions of the price level-and they are condemned to be so frustrated forever after. This is an unappealing result and there are two possible lines of attack on the problem: first, demands could be expected to change in response to such frustrations; alternatively, the price determination equation might be inadequate. I pursue the second approach. The question raised by (1) and similar equations is: Whose behavior do such equations describe? The standard Walrasian answer is "the auctioneer"; another frequent answer is "somewhat less than competitive firms." Consider the auctioneer explanation first. In the standard single period exchange model, the auctioneer calls out prices for each good sequentially on the basis of the mechanism: (2) 881 MONEY AND GROWTH Pi,j = Pi-l,j + Xj(pi-1) where i is the iteration number of the current call, j is the number of the good, p is the vector of prices, and xi(pi-1) is an increasing function of excess demand for goodj at the previously called price vector. In intertemporal models an equilibrium price vector is obtained by the above process at the beginning and no further tatonnement is required. If new information is available in each period, as in models including uncertainty, one supposes that there is an "auction" each period. The goal of the auctioneer in each period is to establish market-clearing prices prices at which demands are equal to supplies. Equation (1) is an attempt to use (2) in a temporal context so that the i subscript becomes a t, and to apply (2) to the aggregate price level. But it ignores the motive of the auctioneer. If the auctioneer expects the general price level at time t to be different from that at t- 1, then he might use as his rule of thumb -e (3) Ptj = pt-l, + xj(pi) where fi is the general price level expected to prevail at t, and pt-, is the general price level at t- 1. Aggregating over goods, and in continuous time, an analogue of (3) is: (4) l =r* + X(D-S) where 7r is the actual rate of inflation, and 7r* is the expected rate of inflation. The auctioneer is not present in most markets and it is somewhat unsatisfactory to discuss reasonable behavior for a nonexistent economic agent. Consider alternatively the explanation in terms of the behavior of price-setting firms. As suggested by Arrow, and developed by Robert Barro in a recent and interesting paper, since the existence of disequilibrium is inconsistent with certain assumptions of the perfectly competitive model,3 we may expect pricesetting by firms even in industries for which the competitive model is adequate for comparative static analysis. Barro analyzes optimal price-setting behavior for a monopolistic firm faced with uncertain demand and a fixed cost of adjusting its selling price; the optimal policy is to adjust price only when excess demand or supply reaches certain barriers. He then shows that, by aggregating over firms, the average price may be expected to behave according to (1). Barro confines himself to cases where the aggregate price level is expected to remain constant. Suppose now that all prices but the monopolist's price were expected to increase at the rate 7r*; then costs would be expected to rise at the rate 7r*(since the cost function I In particular, in disequilibrium it cannot be true that each firm can sell as much as it wants at the going price and each consumer can purchase as much as he wants at the going price. This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions 882 THE AMERICAN ECONOMIC REVIEW is homogeneous of degree one in prices), and as of any given price fixed over an interval by the monopolist, the relative price of the monopolist's output would be falling at the rate 7r*. Then, in adjusting prices, the monopolist could be expected to include an adjustment for the trend in prices over the period for which he expects to keep his own price constant. Aggregating over firms, one would expect to reach an equation similar to (4). Thus, on either score, an equation such as (4) is a more adequate representation of price adjustment than is (1). Accordingly, I proceed in Section II to an analysis of a KW model incorporating equation (4). Stein (1970) has in fact suggested that an equation like (4) might be useful in reconciling KW and neoclassical models. Similar equations may be found to describe wage and price adjustment in the literature.4 Before presenting the modified KW model, it is necessary to discuss the price dynamics implicit in the usual neoclassical model. The per capita demand for real balances (md) is a function of the per capita capital stock (k) and the expected rate of inflation (r*): (5) md = L(k, r*, Li > O,L2 < O At any instant of time the capital stock (we omit "per capita" where no confusion is likely to result) and the expected rate of inflation are given, as is the nominal money stock and population. Then, adding to (5) the neoclassical specification that the money market is always in equilibrium (6) is sufficient particular, money will the system M/PN m =md to determine the price level. In a doubling of the stock of double the price level but leave otherwise unaffected.5 See, for example, Edmund Phelps. I In two-sector neoclassical models (e.g., Duncan Foley and Miguel Sidrauski) determination of the price level requires also commodity market clearing, and the 4 Is there any reason to regard this instantaneous neutrality with suspicion? There are circumstances under which it might be regarded as reasonable: for instance, if it was announced that at some point of time every i ndividual's nominal money balances would be doubled, then, given some sophistication by economic agents, it might be realized that this action was analogous to creation of a new unit of account and the price level might simply double. It is, however, a basic assumption of neoclassical models that injections of money are not distributed on the basis of existing holdings of money (since otherwise the transfer payments by which the money supply is expanded would be equivalent to interest payments on money holdings). Given this assumption, increases in the nominal balances of some individuals in the economy can be expected to produce their effects on prices gradually, through real balance effects, rather than instantaneously. Hence the KW objection to this neutrality has force. Using (5) and (6), the rate of inflation in neoclassical models is given by (7) 1 w=j,-X- m [L,Dk + L2Dr*] where ,u is the (assumed constant) rate of expansion of the nominal money supply, n is the rate of population growth, and D denotes the time derivative. In the steady state 7rw- n; thus the rate of inflation will be reduced below its steady-state value by capital accumulation and raised above its steady-state value by increases in the expected rate of inflation. Even leaving aside the expectational factor, D7r*, equation (7) is not analogous to (4). price level cannot be said to be determined by the requirement of portfolio balance. It remains true that in such models, "jumps" in the money stock affect only the aggregate price level. This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions FISCHER: 11. The Modified KW Model In outlining this KW model I shall point to its departures from neoclassical analysis. Both types of model have in common a production function, stock demand functions for assets, a savings function, and an expectations function. I shall specify forms of these functions which could be usecl in either type of model. The per capita output of goods is (8) y = f(k) tained.' There are three assets: money, private bonds, and physical capital. Stock demand functions for real balances, real bonds (the excess demand function, since it is assumed there are no outside bonds), and capital are given by (9), (10), and (11), respectively.7 The assets are assumed to be gross substitutes. The variable y, output, enters to represent the transactions demand for money. Per capita wealth, a=(k+m), 6 For a KW model with variable employment, see Keizo Nagatani. I The demand functions for assets differ from those used in Foley and Sidrauski only in that the price of capital does not enter. It is assumed that production always takes place away from corners of the production possibility frontier so that the relative price of capital and consumption goods remains fixed. I note, quoting David Levhari and Don Patinkin, "that it would be more consistent with general considerations of economic theorv if . . [the demands for assets] . . . were represented as depending upon disposable income . .. This, however, would greatlv complicate the . .. analvsis which follows...." (p. 720). md = L(y, a, f'(k) + r*, p) (10) bI = H(y, a,f'(k) + r*, p) (11) kd = J(y,a,f'(k) + w*,p) enters as the stock budget constraint.8 Bonds and capital are not perfect substitutes so thatf'(k)+7r*, the expected nominal return on capital, may differ from p, the nominal interest rate. The three demand functions are dependent since the sum of the demands for assets is constrained by wealth at each instant. Per capita savings is a function of disposable income and wealth: (12) s f' > 0, f " < 0 where, for convenience, it is assumed the Inada conditions hold and that real balances do not enter the production function. It is also assumed that the labor force, growing at the rate n, is supplied inelastically and that full employment is main- (9) 883 MONEY AND GROWTH = 1> S(ye, a), sl > (, s2 < 0 Expected disposable income, ye consists of factor payments, f(k), plus transfer payments ym, where y is the constant and preannounced rate of expansion of the nominal money supply (it is assumed that the current price level is correctly perceived), minus expected capital losses on money holdings, 7r*m. Thus ye = f(k) + ( - 7r*)m Saving is definitionally equal to desired additions to asset holdings; it is the sum of Id, h'd, and x'1 which are desired additions, per capita, to real balances, bonds and capital, respectively. Consumption demand and savings demand are constrained by dlisposable income: (13) ye = Cd + s It is well known that the stability of dynamic models is heavily dependent on the expectations function. We assume here adaptive expectations: (14) 7r*= f(7r-7r*), 0 < d <0o Thus far we have outlined a fairly stan8 Since there are no outside bonds in the model, the net per capita value of bonds is zero. L1 > O, 1 > L2 > O, L3 H1 < O H, > O, H3 < < O, L4 O,H4 > 0 Ji < O, 1 > J2 > O,J3 > O,J.j This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions < 0 < 0 884 THE AMERICAN ECONOMIC REVIEW dard neoclassical model. A neoclassical analysis would proceed as follows: assume asset market equilibrium and use any two of (9)-(1 1) to determine the price level and the nominal interest rate at each instant of time these are functions of the capital stock and expected rate of inflation. Then assume that consumption demand is always satisfied and obtain the rate of capital accumulation as the residual of output minus consumption. The scene is then set for determining "next instant's" short-run equilibrium; the economy proceeds through these equilibria, and if it is stable, ultimately reaches a steady state in which the capital stock and expected rate of inflation are constant. In fact, the model we have set up is very similar to Levhari and Patinkin's "Money as a Consumer Good" model. The four KW features of the model follow. First, there is the specification of an investment demand function, xd. We assume a stock adjustment demand for investment. (15) xd = nzk + D(kd-k), '> O The flow demand for capital consists of the replacement demand, nk, plus a term which depends on the divergence between the actual capital stock and that demanded at the current levels of wealth and current rates of return and income. The basic justification for (15) lies in the existence of adjustment costs in changing the capital stock: the greater the divergence between actual and desired capital stocks, the greater the costs that can profitablv be incurred in changing the capital stock.9 The investment demand function (15) has the property which is the basis for I See Robert Eisner and Robert Strotz for the derivation of an investment demand function such as (15); see also Marc Nerlove for critical comments on this and subsequent developments. Note that although adjustment costs are invoked in explaining (15), thev are not explicitly incorporated in the model. investment functions in Stein's KW models that an increase in the difference between the expected nominal return on capital, f'(k) +7r*, and the nominal interest rate, p, increases investment demand. 'I'his is the "Wicksell" feature of KW models for (16) f'(k) + * - p = f'(k) - (p -*); the first term on the right-hand side of (16) is the natural rate and the second is the real rate, and differences between these two rates affect investment demand.10 Second, there is the price adjustment equation, in which it remains to specify aggregate demand and supply. (17) 7r = 7r* + X(cd + Xd -f(k)) The demand for goods consists of the demands for consumption and investment; the supply is simply full employment output. Third, it is specified that the bond market be continuously in equilibrium, so that b = bd = 0 (18) This is an assumption of convenience rather than necessity.1" Fourth, there is the question of the allocation of output in periods of excess demand or supply. Here it is assumed that both consumption and investment plans are partially frustrated when there is excess demand; in particular, planned investment is reduced by some positive fraction (1 -y) of excess demand to give the actual rate of investment. x - (1 - y) [cd + xd f(k)j, In general -y could be expected to be an endogenous variable rather than a con10 The "Keynes" part lies in the specification of an independent investment demand function; other Kevnesian features, such as unemployment, can be captured in KW models with variable employment. 11In Stein (1966), for instance, it is assumed that the money market is in equilibrium. This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions FISCHER: MONEY AND GROWTH stant; while (19) is very much a deus ex machina, theories of allocation under disequilibrium are not well developed and there is no formulation which is obviously theoretically superior at this stage. Note that (19) is equivalent, through (17), to (20) x= -( x" * Before proceeding to an exposition of the short- and long-run properties of the modified model, we use the assumption that the bond market is always in equilibrium (18), to derive the implied relationship between the nominal rate and the capital stock, real balances, and the expected rate of inflation. Given k, m, and 7r*,there is, from (10) and (18), only one nominal interest rate which equilibrates the bond market. Specifically (21) p A (k, m, 7r*) where A1=- -1 H1 [Hif' + H2 +Hf"] < O - H, HA3= - - H3 H4 >0 Ihe only ambiguity in (21) concerns the effects of an increase in the capital stock on the nominal rate: there is, in adldition to the substitution effect (IIff") and wealth effect (11.), an income effect, (11f'); we assume that the substitution and wealth effects dominate and that the reduced real rental on capital resulting from an increase in k leads to a decrease in the nominal rate as of any given r*. T hus, we assume that increases in the capital stock ten(l to reduce the nominal rate; our earlier assumptions imply that increases in real balances tend to reduce the nominal rate 885 while increases in the expected rate of inflation tend to increase the nominal rate of interest. III. The Short and Long Run in the Modified Model We now discuss the behavior of this KW model in the short and long run. Given the assumption that the adjustment coefficient, X, in (17) is finite, the price level is given at any instant-that is, it is inherited from the past. Accordingly, m, real balances per capita, is determined exogenously, for M, nominal balances, is a policy variable. TIhe capital stock and the expected rate of inflation are also inherited from the past. Tlhus, at an instant of time, k, m, and r* are predetermined. Tlhe behavioral relations of the model determine, in the short run, the nominal rate of interest and thence, through the goods market, the rate of inflation. Given the rate of inflation, and k, m, and 7r*, the rate of capital accumulation is determined from (19), and the rate of change of the expected rate of inflation from (14). The stage is then set to determine the capital stock, real balances, and the expected rate of inflation at the next "instant"; the economy proceeds in this way through time, reaching a steady state if. the system is stable. The remainder of this section consists of a more detailed examination of this process.12 Given k, m, and 7r*, the predetermined variables, the nominal interest rate is determined through the requirement of bond market equilibrium, and is given by (21). TIhat nominal rate in turn, together with the predetermined variables, determines the demands for consumption and investment and the consequent rate of inflation. 12The verbal description we give of the dynamic process of this economv corresponds more closely to a difference e(quationsystemnthan to the differential equation system contained in the formal analysis; this is simp)ly a matter of convenience. This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions 886 THE AMERICAN ECONOMIC REVIEW Using (17) and the flow budget constraint, the rate of inflation is (17') 7r= 7r*+X(xd+ a)) (Au-7r*)m-S(ye, Consider now the effects of changes in k, m, and 7r* on the rate of inflation. The effects of changes in k and m occur only insofar as excess demand is affected (recall that xd is a function of the nominal rate, so that effects working through the bond market must also be considered) while a change in r* has an expectational effect on the rate of inflation in addition to excess demand effects. We obtain (22) 7r= G(k, m, *, ) where G1=XQi/ G2=XQ i-) -+n-sif'?S2) (i - \dm + 7r*)1Sl) / G3=1+--X' dJ - S2) -m(1-sSi) >O0 >0 GiA=Xm(l-si)>0 The derivatives of the J function are written as total derivatives to indicate that bond market effects are to be included. Increases in the capital stock have an uncertain effect on excess demand; they reduce the stock excess demand for capital13 but may either increase or decrease savings since the income and wealth effects on savings work in opposite directions. If the system is near the golden rule, then and the term (n-s1f'-s2) n-sif'>O will be positive. Thus the sign of G1 is ambiguous. Increases in real balances are inflation13 This may be shown by computing the derivative dJ/dk- 1 and using the stock budget constraint. ary; they increase both consumption and investment demand. Increases in the expected rate of inflation have a direct effect on actual inflation throughout the expecalso increase investtations effect-they ment demand but reduce consumption demand by reducing the value of expected transfer payments. Thus, whether the actual rate of inflation increases by more or less than the expected rate depends on whether increases in the expected rate produce an excess supply or excess demand for goods; in other words, on whether the reduction in consumption demand is greater than or less than the increase in investment demand. It later turns out that this is an important factor in determining the stability of the system, and it may be seen that the smaller is 4V' the more slowly is the capital stock adjusted the more likely is (G3- 1) to be negative. Finally, an increase in the rate of growth of the money stock increases transfer payments and is inflationary. The "short-run" position of the economy is determined by (21) and (22). Its behavior through time is determined by the capital accumulation equation (20), the rate of change of real balances equation which can be derived by differentiating m with respect to time, and the expectations equation (14). For convenience we rewrite and renumber these equations here: (23) Dk = 44J(y, a,f'(k) + (24) Dm = - A( ))-k] [G(k, m, 7r*, ) -7 - r*, n - m, G(k, 7m, (25) D7r* = O[G(k, m, 7r*,,) -r ,,)]m -7r*] Consider now the steady state for this economy. In the steady state, Dwr*= 0, and so the actual rate of inflation is equal to the expected rate, and, from (24), each is equal to (,u-n). From (23), the demand for the This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions FISCHER: MONEY AND GROWTH capital stock is equal to the existing capital stock, and there is no excess demand for capital; from the stock budget constraint it follows that there is no excess demand for real balances either. From (17), the excess demand for goods is also zero and since investment demand is satisfied, so is consumption demand. As in the neoclassical model, there are no unsatisfied demands in the steady state of the modified KW model. The reformulation of the price adjustment equation is thus sufficient to remove the unsatisfactory feature of previously published KW models the persistence of excess demand in the steady state. IV. Changes in the Stock of Money and in the Rate of Growth of Money Stock Suppose the economy is in the steady state and there is an increase in the money stock, but no change in the rate of growth of the money supply. T hen since , is the only exogenous variable in the system in the long run, it is apparent that if the system is stable, it will return to the same steady state. However, this economy, unlike our earlier neoclassical system, will be forced out of equilibrium by the increase in the money stock, and will take time to return to its steady state. T he steady-state neutrality is of course neoclassical but the dynamics is not. Consider now the impact effects of an increase in the money stock. The nominal interest rate is reduced, and the rate of inflation is increased because excess demand is increased. The increase in the rate of inflation increases the expected rate of inflation and begins to reduce real balances. The effects of the increase in the money stock on capital accumulation are ambiguous: the demand for both investment goods and consumption goods is increased, and investment is more likely to increase the relatively greater are real balance effects on investment demand and 887 the more fully are investment plans, rather than consumption plans, realized. This short-run story is very Keynesian insofar as the effects of the change in the money stock manifest themselves in the bond market and result in an increase in investment demand through the lowering of the nominal rate. If we had been dealing with a model with unemployed resources, the story would have been even more Keynesian for the increase in both consumption and investment demand could have called forth more output, rather than resulting in inflation. The path followed by the economy thereafter depends on its stability properties, which are analyzed in the Appendix. It is shown in the Appendix that if the steady state is near the golden rule capital stock, then a necessary condition for stability is that increases in the expected rate of inflation reduce excess demand this, as discussed above, is helped by the slow adjustment of investment demand to changes in the desired capital stock, and damaged by a great sensitivity of the demand for capital to the expected rate of inflation. It is also shown that slow adjustment of expectations as in the neoclassical model and rapid adjustment of prices to eliminate excess demands are conducive to stability. However, the conditions for a the rapid adjustment of expectations large 0 to produce instability are less stringent than they are in neoclassical models. Finally, we consider the comparative steady-state properties of the modified KW model. An increase in the growth rate of the nominal money supply ultimately increases the expected rate of inflation by the same amount as the increase in the monetary growth rate. T he higher expected rate of inflation increases the demand for capital and reduces the demand for real balances; one of the factors determining the new stea(ly state is thus the asset demand This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions 888 THE AMERICAN ECONOMIC REVIEW functions and the fact that there will be no excess demands in the long run; the other factor determining the new steady state is savings behavior. Working with our full system of differential equations, we obtain dk* (26) - = d,u - Om'VX dJ Z3 *- - dr* (n(1-S1) -S2) and (27) dm* -- d,u fmc'X = __ Z3 dJ * (n-Slf'-S2) dr* where Z3 is the determinant of the matrix in the Appendix which has to be negative for stability. This negativity is assured if (n-slf'-s2) > 0 Thus we can say that if the system is in a stable steady state, increases in the rate of growth of money unambiguously inincrease the equilibrium capital intensity; and if that steady state is near the golden rule capital stock, increases in the rate of growth of money reduce equilibrium real balances. In any event, if increases in the capital stock reduce savings, so s1f'-s2 <0, then increases in ,u increase k* and reduce m*. These results are familiar and early comparative steady state neoclassical propositions. We obtain them, of course, because this KW system has the same steady-state properties as our neoclassical model of Section JI, which was set up to be very similar to earlier neoclassical monetary growth models."4 Although we chose to represent our steady state by using (23)-(25) we could equally well have been neoclassical and described the steady state in terms of asset market equilibrium and the requirement that savings be just sufficient to maintain real per capita assets constant. 14 In particular, our use of output rather than disposab)le income in the asset demand functions, and the omission of imputed interest on real balances enable us to avoid several pitfalls. It is, incidentally, interesting to use (20) to examine the impact effect on investment of an increase in the growth rate of the money supply. The demand for investment goods Xd is unaffected by increases in ,. Thus the impact effect of a change in , depends only on its effect on the rate of inflation. The rate of inflation increases with ,, so that the actual rate of investment falls when , is increased. The increase in , increases consumption demand but not investment demand and so some investment is displaced. Thus, initially the capital stock falls when the rate of growth of money is increased, though ultimately the capital stock increases. This is similar to the behavior of the capital stock following an increase in , in Sidrauski. V. Conclusions The purpose of this paper has been to modify a KW model in a way which removes the feature of steady-state excess demand in such models and to compare the resulting model with a neoclassical model based on the same demand functions for assets and savings. The paper has made it clear that the element producing the unsatisfactory features of KW models is the price adjustment equation, and arguments have been presented for using an alternative adjustment equation in which prices may change because they are expected to change, as well as because there is excess demand. The potential of KW models for a useful theory of short-run dynamics, emphasized by others, has been demonstrated in the context of the modified model. It has also been shown that there is no inherent reason for the longrun properties of KW and neoclassical models to differ, so long as the KW investment demand function is consistent with the neoclassical stock demand function for capital. This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions \ -/dJ ~b (Al)~ i -d G1 K1' l- -Gim G2 --- 1-y dJ 1-y dJ 1-z y )- 889 MONEY AND GROWTH FISCHER: - G2m (31 4td*-- - G3m gcG1 i3G2 APPENDIX Stability Conditions The matrix involved in determining the local stability of the system (23)-(25) is shown in (Al) above. Let Z1 be the trace of Z, Z2 the sum of its second-order principal minors, and Z3 its determinant. Necessary and sufficient conditions for local stability are z1 < 0 Z3 < 0 (A2) Z1Z2 Z3 - < 0 A necessary condition implied by (A2) is that Z2 be positive. Now, (A3) Z, = -Om(G3-1) Gs[ --1) O(G3- 1) _ virtue of the gross substitute assumption, the whole expression and if (n-s1lf-s2)>0, will be negative. Now, at low levels of the capital stock, f' is very large and the above expression may be negative unless y is close to unity; for higher levels of the capital stock, and certainly when it is near the golden rule, we are assured that 3(Dk) 8k is negative. We shall assume that the steady state about which we are examining the dynamics is such that nt-s1f'- s2>O and hence &(Dk)ak< O. Given this, it is necessary that (G3-1) be negative, or that the direct effects of an increase in the expected rate of inflation in the requires that goods market be negative-this the adjustment coefficient in the investment equation, V', be sufficiently small and/or that dJ/d7r* be small. Second (A5) 1- -yG <0 -G2 From the derivatives given in (22), we know that G2 is positive; it follows that the product of (G3 - 1) /dJ \ - -1 / dk --AX4'm < <O dJ _- _dm (n-sif'-s2) and [ -(- (dJ 1 (1 - Y)(n (n( 1- SI) -S2)] Given the assumption n-s1f'negative. The value of Z2 iS Z29=mt3G2+I3KP' s2>0, (G3-1) this is /dJ\ (kk-1 (A6) a(Dk) = - - 1 yq/ /dJ \dk 9k - -1) 1 -y must be positive. These are, respectively, the terms a(Dr*)/a7r* and &(Dk)1ak. Consider first &(Dk)/ k which is (A4) dJ G1_dm Z3= -flC/m -Gi- Slf - -->O 2) The first term in parentheses is negative by The sign of the bracketed term is ambiguous: after substitution the term becomes This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions THE AMERICAN ECONOMIC REVIEW 890 -f3AV> L - i)( si)m (- (!J) +-- (n - dir*J slf' - S2) While the first term within the brackets is negative, the second is positive and potentially destabilizing. Note that I have already discussed the size of (IJ/(1dr*,for if this term is large, there may be trouble with price level stability (see the discussion after equation (22) and above (A5)). Note also that the more slowly do expectations smaller is d-the more likely is this stability conadapt-the dition (A6) to be met. Finally, Z1Z2-Z3 ( ZFm)G .m3G2+V ((G3-1 (-(1k -/ )]Z3 [ l+Zl <0O -G, Evidently, the larger is Z1, in absolute value, the more likely is the system to be stable prothe greater (in absolute value) vided Z2>0; are O(Dk)/Ok, OG am and (dOGC1r*- 1), the more likely is the system to be stable. The last two of these derivatives are increasing functions of X, and thus the faster does the price level adjust in response to excess demand, the more likely is stability. REFERENCES K. J. Arrow, "Toward a Theory of Price Adjustment," in M. Abramowitz, ed., The Al- location of Economic Resources, Stanford 1959, 44-51. R. J. Barro, "A Theory of Monopolistic Price Adjustment," Rev. Econ. Stud., Jan. 1972, 39, 17-26. R. Eisner and R. H. Strotz, "Determinants of Business Investment," in D. B. Suits et al., eds., Impact of Monetary Policy, Englewood Cliffs 1963, 59-337. D. K. Foley and M. Sidrauski, "Portfolio Choice, Investment and Growth," Amer. Econ. Rev., Mar. 1970, 60, 44-63. D. Levhari and D. Patinkin, "The Role of Money in a Simple Growth Model," Amer. Econ. Rev., Sept. 1968, 58, 713-53. K. Nagatani, "A Monetary Growth Model with Variable Employment," J. Money, Credit, Banking, May 1969, 1, 188-206. M. Nerlove, "On Lags in Economic Behavior," Econometrica, forthcoming. E. S. Phelps, "Money Wage Dynamics and Labor Market Equilibrium," in E. S. Phelps, ed., Microeconomic Foundations of Employment and Inflation Theory, New York 1970, 124-66. M. Sidrauski, "Inflation and Economic Growth," J. Polit. Econ., Dec. 1967, 75, 796-810. J. L. Stein, "Money and Capacity Growth," J. Polit. Econ., Oct. 1966, 74, 451-65. , "Neoclassical and Keynes-Wicksell Monetary Growth Models," J. Money, Credit, Banking, May 1969, 1, 153-71. "Monetary Growth Theory in Perspective," Amer. Econ. Rev., Mar. 1970, 60,85-106. J. Tobin, "Money and Economic Growth," Econometrica, Oct. 1965, 33, 671-84. This content downloaded on Mon, 18 Mar 2013 08:01:00 AM All use subject to JSTOR Terms and Conditions