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A o ?dcnf>fVnP A Series of Occasional Papers in Draft Form Prepared by Members of the Research Department for Review and Comment. 76-1 Money, Prices, and Interest Rates in Stable Monetary Growth Models Thomas A. Gittings Federal Reserve Bank of Chicago ji t i t tu > *♦ ♦ ♦ ♦ ♦ ♦ *< z i i x i z i a ♦♦I Research Paper No. 76-1 MONEY, PRICES, STABLE AND INTEREST MONETARY RATES IN GROWTH MODELS By Thomas A. Department Federal The views and do n ot Reserve material necessarily to s t i m u l a t e permission of of is of discussion, the are or the Research Bank of solely represent Chicago contained of Reserve expressed herein Bank Gittings the Chicago those views of of the the authors Federal Federal Reserve System. a preliminary nature, is and authors. is not to b e The circulated quoted without I. INTRODUCTION* The purpose of this paper is to analyze a series of theoretical macroeconomic models that describe the possible dynamic relationships among money, a price index, and an interest rate. Having been subject to Occam's razor, the models are presented in their simplest form and from a macroscopic point of view. All behavioral equations are assigned specific functional forms so that the models can be solved analytically or simulated numerically. This en ables one to study both the qualitative and quantitative properties of the alternative models. Each of the models is formulated as a system of ordinary differential equations. The first three models are presented as "equilibrium" models where the real quantity of money is inversely related to the rate of inflation in the "long run." Two of these three models have been analyzed extensively in the literature; the third is a subtle extension. After discussing the dynamic relationship between the rate of growth of the quantity of money and the rate of inflation, we extend the third model by including an interest rate equation and a simple unemployment relationship. Finally, a corresponding "disequilibrium" model is derived to show the mathematical equivalence be tween equilibrium and disequilibrium growth models. This equivalency is based on the concept that a system of differential equations is a system of differen tial equations. In the spirit of mathematical growth models, we make a series of heroic assumptions. The models represent a closed economy that produces and consumes a constant amount of a single homogeneous nondurable good. There is no in vestment, depreciation, technological change, or inventories. The population, 2 level of initial are employment, state normalized that any is changes in models. rate, ing second the test path test not the rate initial to of parameter We values begin by hyperinflation the primary cost difference as bonds, in of crease the of money. in the third cash test of be level a the constant. are type of In this constant fiat and money economy without its and a period inflation zero, is causing state on assign the the output is the rate is destabilizing Marshall [9, p. we some shall in his constant change by the force 48] is growth he rate of the the The of in de quantity succinctly states: with such balances summarized where and in prices. dominated cash on compatible reserves, of use study of rate rule models. fairly real that infla feedback model, A initial of forms hyperinflation, the but This rate reasonably of correspond model. following Cagan remains the path. alternative by constant constant simple first alternative exceeds a a the expected the m o d e l formulated durables, of an by alternative at observe steady of in and the grow remains to in of generated shock or is some of m o n e y hyperinflation balances on supply simulate Real consumer of that exception models. work off and amount money two mo d e l s on money of to into period stability period potentially theoretical A ensure or remain price conducted exogenous the our such rate This an With holding equities, During as of time position reviewing return flation. a be nominal money A injected inflation of which [2]. assumptions of reflect disturbance. and assumed the n o minal the zero. is will initial that growth is money rate distribution. rate from could equal and have the the begins displacement to of Money experiments from for one. income is utilization quantity produce assumes economy tion to the beginning time equal basic One capital nominal to costless Several for the and 3 The total fore value cannot increase in repeated, is to the to this lower is fundamental analysis. of of such a by paper increasing which the v a l u e stresses the rapid confidence relationship long-run Given inconvertible seems currency its there quantity; likely to an be of each unit more that this type of than in increase. [8] because inverse an quantity, the p u b l i c ’s l o s s that a its Reform Keynes self-defeating due of increased will proportion In M o n e t a r y be between property consensus, of increase in the the let the us in the inflationary velocity currency. rate of of Friedman inflation and taxation circulation [5] argues real balances theoretical framework now the math e m a t i c a l look at for m o n e t a r y impli cations. II. (1) where cash The first M/P » M tt is the nominal balances, base inflation tt of is quantity the Cambridge of money, instantaneous logarithms, operator Taking spect is defined the n a t u r a l This a modified cash-balance equation, and ot i s P rate a is of the price positive price index, inflation, constant. The M/P exp is is rate real the of as = DP/P = DlnP, the tt is I e x p (-air), is natural where model MODEL = model to the 3DlnP 3InP is is interpreted logarithm the (InP D - of logarithms the of as price equation the right-hand time derivative and InP with re index. (1) yields lnM)/a. unstable logarithm since of the prices derivative is of positive. the rate of inflation 4 When the above nominal its equilibrium inflation. In o t h e r generating al. This is level, words, the is constant the m o del when when there there antithesis this heavenly model stabilizing monetary greater than of m o n e y y money the initial states that is n o t enough money is of and too the much there money Marshall, will one price be level is self-generating one has has self-generating Keynes, self Friedman, Cagan, thesis. In the of inflation, and deflation. et. quantity and the rate where policy of there is to inflation. are perfect increase Let y be foresight nominal the money rate of and no frictions,— balances growth of at the a rate quantity n be the logar i t h m of real cash balances: = DM/M « DlnM, n = In(M/P) - InM - InP. Assume that y is positively related to tt so t h a t y — 0tt. Since Dn = y - tt = (0-1)tt and n = -CX7T, therefore Dn This If = (l-6)n/a» model is 3Dn = 3n (1-0) / a heaven is stable if < and only if 0 stable, (2) M/P alternative = greater than one so that 0. St. Peter must conduct MODEL An is 2/ model— exp(-f3Tr*), used by such an aggressive II Cagan is the following: monetary policy. 5 (3) D tt* - Y ("“ * * ) » ic where it is the expected rate of inflation, p and Y are positive constants. This model is stable if the reaction index (By ) is less than unity. To see why, take the logarithm of equation (2) to get (4) n = - B it*. Differentiating with (5) Dn = y - tt = respect to time yields B D tt*. Use equations (2)-(5) to obtain the following relationships: (6) n* - (7) n * Dn = ( ( B y - l ) v ; + v ) / (By) , ((l-gy)TT-u)/Y, Y(n+3u)/(6Y-l)* When By is less than one, the model is stable and real cash balances are posi tively related to the rate of inflation and negatively related to the rate of change of nominal money balances. In the long run, which corresponds to the particular solution of a differential equation model, the rate of inflation equals the rate of growth of the quantity of money and is inversely related to real cash balances. Notice in equation (6) that the expected rate of inflation is negatively related to the actual rate of inflation and serves as a counter balance. Figure 1 plots the dynamic responses of these two models when the quantity of money is increasing at a constant rate y . Notice that the rate of inflation asymptotically approaches its long-run level from above, so that real cash balances instantaneously fall whenever the nominal quantity of money is increased. This may be a reasonable description of a theoretical economy where new fiat money is introduced by having the monetary authority bid directly for goods and services. When money is introduced via transfers or a bond market, one would expect that real money balances initially increase, reach a maximum, and then 6 Model 2. n = -3ir* Dir* * y (rr— it*) 3 * .5 y = .5 & * Dir = C (7T— 7T ) C = ,5 Dn = a(nd-n) a = 1 7 decline to its new and lower long-run position. In other words, the rate of inflation is initially below its new long-run value. In order for real cash balances to fall eventually, the rate of inflation must at some time rise above its long-run value and then settle onto this new equilibrium position. The minimum mathematical requirement for a model to generate this cyclical process is that it be a second-order differential equation model. The previous two models considered are first-order differential equations linear in the logarithm of real cash balances(n). Let us now analyze some linear second- order models. III. SECOND-ORDER MODELS Assume that the dynamic relationship between money and prices can be re duced to the following linear second-order nonhomogeneous differential equation: (8) D2n + where D 2 pDn + an * -xp + <f)Dp, denotes the second-order time derivative; p, a, x, and <f> are constants. This model is stable if and only if p and a are positive since p is the determinant of the corresponding Jacobian matrix of partial derivatives and o is the negative of the trace of the Jacobian. The thesis that real cash balances are inversely related to the rate of inflation in the long run implies that x is positive. The form of the solution depends upon whether the roots of the characteristic equation are real and distinct, real and equal, or complex. An interesting exercise is to derive the corresponding cash balance equation in logarithmic form. Dn * D2n = p Dp tt - 9 D tr. By definition the following equations exist: 8 Substituting following When rate negatively do + the m o d e l the not equations equation n ■ (ptt (9) to these of to the D-tt - is stable, (p + t an to the and the the rate of third demand some models portional the to expected (10) n = model assumes a weighted rate of real of for and rearranging yields the balances: that change of the of money. MODEL is rate Since balances positively with assume of inflation economists this related form, and is usually it such an cash balances is in equation. III logarithm between The balances implicitly the difference real real that inflation. of growth IV. The (8) (<J>-l)Dp/a. quantity rate explicit analyze equation l o g a r i t h m of ) p ) / cj + inflation related specify structive for into the equations of real actual of rate of 3/ the m o d e l — is inflation are as pro and follows: (7T-g7T*)/a, Dir* » a and Y are assumed to be positive, and 3 is assumed to be greater than one so that in the long run real balances and the rate of anticipated inflation are inversely related. Since the as the quantity the (11) rate of growth of m o ney minus following s y s t e m of of the real rate linear balances of equals inflation, differential the this rate model of can growth of be written simple substi equations: Dn * -an - 3 tt* + p , Dtt* = ayn + (g-l)Y, n‘*« By taking tutions to expressed the time derivative eliminate as a linear the of equation expected second-order rate of (11) inflation differential D 2n + (a+(l-3)Y)Dn + ayn “ ( 1 - 3 ) y p + Dp. and using term, equation: some this model can be 9 The stability condition therefore is that a + (l-g)y > 0. Equation (9) gives the functional form which relates real cash balances to the relevant observable variables. It is interesting to compare this model with one Cagan proposed that in cludes a lag in forming expectations and a lag in adjusting actual levels of real cash balances to their desired levels. variables: He uses two unobservable an expected rate of inflation and the logarithm of ndesiredM real balances or nd - The three basic equations of his model are as follows: nd = -bTT*, D7T* « c(7r-7T*), Dn = a(nd-n), where a, b, and c are positive constants. In a manner similar to our previous analysis, this model can be reduced to a system of two simultaneous differential equations. After substituting to remove the desired real cash balances and the actual rate of inflation terms, the model can be written as follows: (12) Dn = -an - abir*, Dtt = acn + (ab-l)cTT + cp. The corresponding second-order differential equation is D2n + (a+c - abc)Dn + acn = -abcp. This model is locally stable if and only if a + c - abe > 0. By using equation (12) and the definition of Dn> one can obtain the following equation, which relates the logarithm of real balances to the expected rate of inflation: n = (Tr-u)/a - bit*. The easiest way to get a feel for the difference between this model of 1 10 U,tt,T Dir (ir-gir*)/a a « 1. Y (ir-n * ) Y * 10 - 4-1^6 (ir-8ir*)/o a - 1. Y (ir-ir*) Y - -5 6 * 1 .5 e - 1 .5 11 Dir* = c(n-n*) c = .5 Dn = a(nd-n) a = 1. FIGURE 6. n Du * Synthesis Second-order Differential Equation Model (ir-gTr*)/a o-l. Y - -5 6 = 1.5 12 Cagan and of this paper is to compare their response paths following some dis turbance. Figures 2-4 plot the time paths of the actual and expected rates of inflation and real cash balances after the nominal quantity of money be gins to increase at some constant rate. In these simulations the initial expected rate of inflation is assumed to be equal to zero. Figures 5-6 plot the time paths generated when the quantity of money remains constant and the initial expected rate of inflation is equal to some positive value. Since the present model can have real or complex roots, two sets of parameters are simulated and plotted in the first experiment. Caganfs model only can have real roots when the constants are assumed to be positive real numbers. V. INTEREST RATES Up to this point we have concentrated on the dynamic relationship be tween the quantity of money and a price index. The model will not be extended via a combination of traditional arguments to include a macroscopic analysis of an interest rate. A nominal market rate of interest (R) is assumed to be equal to the sum of a natural rate of interest (R*) and a reaction index (r): R = R* + r, where the natural rate of interest is constant and reflects an average rate of time preference. The reaction index is assumed to be a function of the ex pected rate of inflation and real cash balances. It is a proxy variable used to synthesize the arguments of Keynes [7] and Fisher [3].— ^ According to Keynes1 liquidity-preference function, real balances and a composite interest rate are inversely related. Since we have normalized the initial quantity of real balances to be equal to one, a simple liquiditypreference function can be expressed as follows: * r = -Cn, 13 where r* is a Keynesian reaction index. If there is a sustained period of a constant rate of inflation, one would expect this reaction index to equal the rate of inflation. C « In terms of our third model this implies that o/(B-l). Fisher's reaction index in a continuous time model is the expected rate of inflation & “nr . The synthesis reaction index is assumed to be a weighted average of these two alternative indices: r = 6 tt* + (l-6)r*, where 6 is a positive constant. Figure 7 plots the actual and expected rate of inflation, the Keynesian reaction index, and the synthesis index in a simulation where the nominal quantity of money begins to increase at some constant rate. The difference between the actual rate of inflation and the synthesis index is plotted to show the time path of the "real” rate of interest. There is an initial and short-lived decrease in the nominal market rate of interest. The real rate of interest decreases for a longer duration before it eventually increases, overshoots, and then returns to its long-run position. In this simple experiment it is easy to see how unexpected inflation can redistribute income. When the real rate of interest is negative, lenders are subsidizing borrowers and when it is positive, borrowers are subsidizing lenders. If such a monetary policy were conducted, the persons with better foresight would initially borrow and later lend money so as to maximize their effective subsidy. The fiscal analogue of this monetary policy is a tax on those persons with "poorer" foresight and a transfer payment to those persons who have better foresight or "inside" information. VI. AN ALTERNATIVE INTEREST RATE An implicit assumption of the previous method of introducing an interest rate variable is that the fundamental relationship between money and a price 14 FIGURE 7. n Dir* r* r Synthesis Second-order Differential Equation Model (Tr-gir*)/a a = i. Y (iT—TT*) 3 = 1.5 an/ (1-3) Y * 6 = <5tt* + (1-6) r* ? ■ r - tt 15 index is unaltered when borrowing and lending is allowed. The coefficients of such an extended model may change, but the dynamic function between money and prices is assumed to remain a linear second-order differential equation. An alternative approach is to allow this fundamental relationship to change after interest rates are introduced. This change may be of the form of introducing a nonlinear relationship and/or a different order differential equation. An example of the latter possibility is the following model: n = (TT-BTT*-xr)/a, Dir* = y (tt- tt*) , r = tt* Dr = - 4>Dn, ui(r-r), where ip and w are positive constants. r is a reaction index of an actual or a hypothetical interest rate that is instantaneously decreased by an increase in the rate of growth of money. If money is introduced via an open-market operation, this interest rate may be a proxy for the federal funds rate, the reaction index for an interest rate of a longer-term bond. r is This model can be rearranged into the following system of three linear differential equations DlnP = -alnP + Bn* + x* + otlnM, Dir* = -a y ln P 4- y ( B - 1 ) tt* + Dr = -ai/>u)lnP 4- a)(14-B^)ir* 4- a ) ( x ^ l ) r 4- i|>u)(alnM-DlnM). yxr 4* aylnM, Representative simulations of this type of model are presented in the sections on variable output and disequilibrium models.— ^ V I I . AUTOREGRESSIVE FORM When the rate of growth of money is constant, it is possible to derive the corresponding second-order autoregressive equation where the current value of the logarithm of real balances is expressed as a function of real balances 16 in two previous and equally spaced time periods. Suppose that our initial model has real and distinct roots so that the solution equation is the following form: n(t) = c exp(m t) + c exp(m t) 4* (l-3)y/a. 1 c m 1 2 2 and c^ are constants of integration and depend upon the initial conditions. 1 and m 2 are the roots of the characteristic equation m2 4* (a4-(l-B)y)m 4- ay = 0. Therefore m ][ = P 4- q, = P - q3 where P “ -(a+(l-e)y)/2, q = /p2 - ay. The values of n in the time periods t-0 and t-20 are given by the following equations: n(t-0) - c exp(m (t-0)) 4- c exp (m (t-0)) 4- (l-3)u/a, 1 1 2 2 n(t-20) = c exp(m (t-20)) 4- c exp (m (t-20)) 4* (l-3)p/a. 1 1 2 2 The autoregressive equation is n(t) = <f> 4- <p n(t-0) 4- cp n(t-20), 0 1 2 where (13) 1 = <f> exp (-m 0) 4- <j> exp (-2m 0), 1 1 2 1 (14) 1 88 <P exp (-m 0) 4* <j> exp (-2m 0), v J 1 2 2 2 (p = (1-4) ~<P )(l-3)p/a. 0 1 2 Solving equations (13) and (14) and substituting in the functions for the roots yields the following equations for the two autoregressive parameters: (p = 2 cosh(q0)exp (-p0) , 1 <p = -exp(2p0) , 2 17 where cosh first autoregressive <f> = is the hyperbolic cosine. When the roots are real and equal, the term becomes 2 exp(-pG), 1 and w h e n the (J)^ = Notice of 2 that nominal A lates roots complex, cos(q0)exp(-p0). the intercept term exercise current value is to of real a lagged real balances term. creasing at constant rate position. rates a In this included coefficient model has initial and and beginning can be the and to derived = <j> o determine and to the constant roots. increase of at rate When <}> r ( t ) . + example of = -20/(l-(/6-4)0), = exp((2 is initially of growth rate zero, y, the re interest and balances in equation with interest for the the balances and off are long-run Consider real of which its equations equal rate nominal differential initial constant + i current equation that model simple inflation a the coefficient. A): <j> n ( t - e ) the such rate to corresponding assume second-order interest rates we that generate equal the balances (see A p p e n d i x numerical <f> proportional Again the not expected are A case does real n(t) is balances. similar the are first case where equal nominal lag one, the the balances corresponding equation 2 these coefficients yields the following equations 2 <f> / 6 ) 0 ) / ( 1 - ( ./S - - 4)0), <f>0 = — ((1— 2)/2+4>2 )U » when With for a = the the 1.0, 3 = 1.5, appropriate cases where Y = amount there 10-4^6, of are and 6 = cranking, real and .75 . these distinct coefficients roots or can complex be determined roots and 18 where the these relationships change area coefficients of for the when quantity further and initial stochastic of m o n e y is of constant relaxed by our [4] Real unemployment. the inflation. the will tion will change the Phelps to the output actual rate real is was in n o m i n a l are different. introduced to v a r y could What and be an happens the rate to of interesting OUTPUT that the level balances. This accelerationist theory [10]. of The assumed this between real assumption unemployment unemployment the output actual and can as be developed assumed expected rate of is above (below) its long-run level inflation is above (below) the expected rate a is zero, smaller assumed then amount a that to given an be the be output is to to inversely relationship related is remains be inflation by rate of of to real of output decrease possible of VARIABLE difference rate Furthermore increase allowed assumptions Therefore, expected One and related inflation. whenever a introducing Friedman inversely If initial following by elements are study. VIII. One conditions rate of of 6/ nonlinear.— rate equal of inflation rate of defla it. equation— ^ that characterizes this relationship is the following: lnq where Q index is s ln(Q/V0 ) = is the level constant, constants. The q i (k ~ 1- (k + of real is the cash-balance tt- tt* ) " 1) , output, ratio of equation VQ is the Q divided is n o w form: n = ln(M/P) ■ ln(Q/Vc ) + (7r-37T*-xr)/a, or n = i (k ” 1-( k+ tt- tt*)” 1) + (7r-B7r*-xr)/a income by velocity VQ . assumed i and to have when the price < are positive the following 19 This model has which can be bined with the expressed combination rate, of Figures with the is at variable primary to rate output. By purposes for the the of of alternative impact for relatively when at and our the of then begins of the converge run when third-order figures output rate of inflation equation of is inflation numerically, In the rates, output, of the rate 7 and to v a r y Notice in com and using a begin to time. actual of Next, and downwards onto one observes such a prescribed of its 9 the the to the rate real of is rates turning reaches of interest points inflation. The rates a peak inflation its m a x i m u m long-run manner plotted interest output attains that synthesis first reduce expected inflation equation model equation model and is balances 8, figure timing balances nominal differential differential nominal period between expected rate this approximated adjustment. interest Finally, to the second-order short difference a maximum. in comparing increasing be simulation allowing period can the When the 8 / comparison. first a of for form. for R u n g e - K u t t a ’s a l g o r i t h m s . — a rate lengthen for solution 9 plot equation implicit equations constant effect equation the in N e w t o n ’s a n d 8 and a differential only differential interest increase a nonlinear is level value. IX. DISEQUILIBRIUM APPROACH A parallel way of looking at the dynamics of a monetary growth model is the disequilibrium apprach, where the rate of inflation is a function of the excess demand for goods and the expected rate of inflation. This excess demand is the difference between "planned*1 investment and "planned" savings. Actual investment usually is assumed to be a weighted average of planned investment and savings. These hypothetical variables are assumed to be functions of real balances, an interest rate and/or the ex pected rate of inflation when the real output and quantities of capital 20 * TT,TT ,D U.lnq Time FIGURE 8. Synthesis Model Third-order with Variable Differential Equation Output lnq + (ir-8ir*-xr) fa a lnq i (ic-1-(K+ir-ir*)-1) 3 = 1.5 D tt* Y (it- tt*) Y = S - ip - w = * IT - l^rDn l =s .015 u>(r-r) K = .15 n r Dr 1.0 X - 0. V •1 = 21 Time Synthesis Third-order Differential Equation Model with Variable Output n * lnq + (n-Sir*-xr)/“ a = 1.0 lnq = l(le e = 1.5 Dtt* ss y (tt-■if*\) * r = TT ■ ipDi) y = 6 = Dr = aj(r- r) f = r - ir r* » an/(l-e) r" = 6n* + (l-(S)r* i = .015 K = .15 X 0. u .1 to 22 and labor are constant. References to these growth models can be found in the works by J. Stein, [11], Burmeister and Dobell [1] , Urrutia [12], and L. Johnson [6]. To see how this method is equivalent to the previous approach, one need only derive the corresponding disequilibrium model. Suppose the cash- balances equation model is the following: (15) n = In (M/P) = (ir-8ir*-xr)/o. The market rate of interest is assumed to equal a natural rate plus a reaction index: (16) R = R* + r. These equations can be combined into the following differential equation for the logarithm of the price index: it = DlnP ■ alnM - alnP + $tt* + - xR* or 7T = an + 3 tt* + x**. A representative disequilibrium form for this differential equation is 7T = X (I*(* )“S*(* )) + A where A is a positive parameter. I*(«) and S*(») are the unobservable functions for planned investment and savings, respectively. The actual level of in vestment is assumed to equal zero and is a weighted average of the planned levels of investment and savings: I = al*(0 + (l-a)S*(.) = 0, where a is a nonnegative constant that is less than one. Simple substitutions yield the following equations: (17) S* = -a(an + (8-1)it* + XR - xR*)A, (18) I* = (l-a)(an + (8-l)ir* + XR - xR*)A. The usual assumption that the partial derivative of planned savings with respect to the interest rate is positive and the partial derivative of 23 planned investment with respect to the interest rate is negative implies that X is less than zero. This assumption means that the partial deriva tive of real balances with respect to the interest rate is positive. In x affects the dynamic properties order to get an idea of how the sign of of such a model, several simulations were run using the third-order model with a constant level of output. One simulation has a negative value for X; the other simulation uses a positive value. The coefficient of the price expectations variable is adjusted so that the simulations have the same long-run, or particular, solutions. The dynamic paths of the rate of inflation and the reaction index for the rate of interest are plotted in Figure 10. While these simulations show the sign of the interest rate coefficient is not necessarily of particular importance in determining the dynamic response of a model, most disequilibrium models specify that planned savings is positively related to the interest rate, while real balances and planned investment are negatively related to the interest rate. In order for all these conditions to be satisfied, it is necessary to impose an explicit relationship between real balances and the rate of interest. This means that the partial derivative of planned savings and investment with respect to the interest rate cannot treat real balances and the in terest rate as independent variables. For example, the partial deriva tive of planned savings in equation becomes 41^ = -a(oiH + x)/A. The explicit relationship between real balances and the interest rate can be used to express planned savings and investment as functions of only 24 Dir = r = y (tt—TT*) tt* y m u M .5 - ljiDn Dr = w(r-r) Run i. 6 = 1.0 X = .5 Run 2. 6 = 2.0 X = -.5 25 the actual and expected rates of inflation. Substituting equations (15) and (16) into equations (17) and (18) gives the following equations for planned savings and investment: S* « -a(TT-TT*)/X, I* - (l-a)(ir-ir*)/A, This makes the disequilibrium form for the differential equation of the price index a tautology. Let us now consider the more general disequilibrium model where we be gin by specifying the following functions for planned savings and investment: S* = f(M/P, R, tt*), I* = g(M/P, R, TT*), where the level of output is assumed to remain constant. Next, one makes the initial assumption that real balances and the interest rate are inde pendent variables and that the partial derivatives have the following signs: as* -< 0, a(M/p) It 3R r * 31' > 0, 3(M/P) > 0. 9I* < 0. 3R By the implicit-function rule these assumptions imply that IMP! 3R U# In order to avoid this conclusion, we impose the restriction that real balances and the interest rate are dependent and inversely related. A representative restriction is the following explicit function for the rate of interest: R = h(M/P), where the derivative is assumed to be negative, or dR d(M/P) < °* The differential equation for the rate of inflation can now be expressed as TT = A(g(M/P, h(M/P), TT*) - f ( M / P , h ( M / P ) , it*)) + tt* 26 or ■n A = F(M/P, tt*) semi-logarithmic approximation of this general function is n = ln(M/P) ■ (tt— 3 tt*)/a, which is the cash-balance equation (10) of the third model. The semi- logarithmic approximation of the interest rate equation is the Keynesian version of the synthesis second-order differential equation model. X. CONCLUSION This paper has analyzed a series of dynamic macroeconomic models designed to reflect an inverse relationship between the rate of inflation and real balances in the long run. differential equations. These models were presented as systems of ordinary The particular solutions of differential equation models correspond to the economic concept of the long run. Each of the alternative models has been simulated using parameter values that ensure stability. This enables one to study the subtle differences in the quantita tive responses of dynamic models that have the same qualitative properties. The synthesis models proposed are shown to be compatible with a broad spectrum of economic models. Some possible areas for future studies would be to relax some of the initial and very restrictive assumptions, to develop a microeconomic framework for the synthesis models with an interest rate in cluded, and to analyze the stochastic behavior of the models using different monetary policies. 27 FOOTNOTES *1 am grateful for comments and suggestions offered by my colleagues at the Federal Reserve Bank of Chicago and by participants at a talk given to the Special Studies Section of the Board of Governors. Bob Laurent, Vince Snowberger, and my brother Mike provided challenging discussions that influenced my research. The motivation for this paper stems from some of the difficult and fundamental questions raised by my former teacher Nicholas Schrock. All biases and errors are, of course, mine. i^An alternative model of such a perfectly perfect world is n = -ap. For a discussion of this type of model, see Stein [11]. — ^A variation of this model is n - -B p *, Du* = y(u-u*) where B and Y are assumed to be positive. P is the expected rate of change of nominal balances and is used as a proxy for the expected rate of inflation. In a manner similar to that used in analyzing Cagan’s model, real balances can be expressed as a function of the actual rate of inflation and the rate of change of nominal balances. n = (tt-( i+ b y )p )/y . This model is stable when Y is positive. —3/ An extension of this model is the following cash-balances equation: n - (u - g T r* - 3 * u ) / a , where B* is a positive constant. This model implies that an increase in the rate of growth of nominal balances instantaneously causes an increase in the rate of inflation. — ^It is often difficult, if not impossible, to represent the logical patterns in the arguments of Keynes and Fisher by elementary functions. The equations used should be viewed as stylized constructs of broader theories. — ^When testing feedback rules where the rate of growth of money is a function of the rates of inflation and/or the interest rates, a fourth dif ferential equation must be added to this type of model. The values of the parameters in the feedback rule should be selected so as to maintain the stability of the model. Another way of extending this model is to specify a production function, where the level of real output is related to the capital stock, and an in vestment function that is the differential equation for capital. Care should 28 be exercised when formulating the implicit assumption about "neutral" model is considered is independent of the rate of the investment function since it will include the neutrality or nonneutrality of money. A to be one in which the long-run capital stock growth of nominal balances. 6 / — A linear relationship between real output and the difference between the actual and expected rates of inflation may be simulated by just changing the parameters of the second-order model. If lnq e ln(Q/VQ) = i (tt- tt*) and n = ln(M/P) = lnq + (7r-07r*)/a, then n “ ((l+ia)7r - (B+ia)7r*)/a, or n = (ir-8*ir*)/a* where a* = a/ (l+i a) , 0* = (0+ia)/(l+ia). — ^This equation was derived from a translated rectangular hyperbola. Assume that the nonlinear relationship between lnq and the difference be tween the actual and expected rates of inflation is the following: (k+ tt-tt ) (lnq - Q) = -i, where - k and Q are the asymptotes. The model is normalized so that lnq equals zero when v equals tf*. Therefore, Q= \/<> and lnq = x Ck -1 - (k + tt- tt*)"1) 8 / — The implicit differential equation is evaluated using Newton’s method for solving a nonlinear equation. The system of simultaneous differential equations is numerically integrated using Gill’s modification of a fourthorder Runge-Kutta’s algorithm. Variable step integration was used with an absolute error criterion of exp(-8). The subroutines used were XCNLSB and XCRKGM in the CDC Library of Mathematical Subprograms (publication number 86614900). 29 BIBLIOGRAPHY [1] Burmeister, E. and A. R. Dobell, Mathematical Theories of Economic Growth, New York: Macmillan Company, 1970. [2] Cagan, P., "The Monetary Dynamics of Hyperinflation," in Studies in the Quantity Theory of Money, edited by M. Friedman, Chicago: University of Chicago Press, 1956. [3] Fisher, I., The Theory of Interest, New York: 1954. [4] Friedman, M., "The Role of Monetary Policy," American Economic Review, 58 (March 1958), 1-17. [5] Friedman, M . , "A Theoretical Framework for Monetary Analysis," Journal of Political Economy, 78 (March/April 1970), 193-238. [6 ] Johnson, L., "Portfolio Adjustment and Monetary Growth," Review of Economic Studies, (forthcoming). [7] Keynes, J. M., The General Theory of Employment, Interest, and Money, New York: Harcourt, Brace & World, Inc., 1965. [8] Keynes, J. M., Monetary Reform, New York: 1924. [9] Marshall, A., Money Credit and Commerce, London: Kelley & Millman, Inc., Harcourt, Brace and Company, Macmillan & Co., 1923. [10] Phelps, E. S., "Phillips Curves, Expectations of Inflation and Optimal Unemployment over Time," Economica, 34 (August 1967), 254-81. [11] Stein, J. L., Money and Capacity Growth, New York: Press, 1971. Columbia University [12] Urrutia, J. E., "Optimality in a Growing Monetary Economy," Ph.D. dis sertation, University of Colorado, 1972. 30 APPENDIX A Structural equations of the model: (1 ) n = (TT-eTT*)/ct, (2) (3) (4) Dtt* ■ y ( tt- tt* ) , r* = an/(l-S), r *» 6tt* + (1-6) r , where n = it = y = R = ln(M/P), DlnP, DlnM, R* + r. a, 6, y, 6, R*, and u are assumed to be constant. Initial conditions: n(0) = tt*(0) = r*(0) = r(0) = 0. By the general rules of differentiation: Dn = y - it, D 2n * Du - D u = - D tt. Model as a system of differential equations: & Dn « -an - $7r + p , Dtt* ■ ayn + (B-I)ytt*. Two corresponding second-order differential equations: D2n + aDn + bn = (l-B)yu, D2tt* + aDTT* 4- bTr* = ayy, where a - a + (l-B)y, b = ay. Stability condition when a , B , and y are positive: a > 0. CASE I: ROOTS REAL AND EQUAL When roots are real and equal, a2 » 4b, p = -a/2 , 31 where p is the common root. Solutions of the nonhomogeneous second-order differential equations: (5) n(t) « A exp(pt) + A t exp(pt) + (l-B)y/ot, 1 (6) 2 7r*(t) * B^exp(pt) + t exp(pt) + y. Constants of integration given the initial conditions: Ai = (B-l)y/a, A2 « y - pAj, Bi * -y, B2 ■ py. From equations (1), (5), and (6): 7T = an + Btt* (7) tt(t) - C} exp (pt) + C2t exp(pt) + y, where C - -y, 1 C - (a+p)y. 2 From equations (1), (3), and (4) (8) r =s eiT + (l-e)ir*, where e = (1-6)/ (1-3). Solution for interest rate’s reaction index from equations (6), (7), and (8) r(t) ■ D exp(pt) + D 0t exp(pt) 4- y, l * where Di = -y, D2 = (ae+p)y. Lagged value of n: (9) n(t-0) = E 1exp(pt) + E21 exp(pt) + (l-B)y/a, where E 1 “ (A r 6A2)exp(-p0), E2 = A2exp(-p6). n(t) as a function of r(t) and l(t-0): n(t) = <f>0 + <}>J n (t— 0) + 4>2r(t) , 32 where (10) Aj - + ♦jjD j , (11) A2 - ♦ jE j + 4>2D2, +0 » ((1-3)(1-*1)/o -*2)p . Solve equations (10) and (11) using Cramer's rule: Let A = E 2D2 - D xE2 = ((X-e(l-pX))(ae+p) + (l-pX))y2exp(-pe), where X = (e-D/a, 4>1 * (A1D2-d 1a 2)/^ - ((0-l)e+l)u2/A, *2 ■ ff.A j-A jE jjM “ -0(l-pX)2y2exp(-p0)/A. When <*=1, 3=1.5, 6=.75 and the roots are real and equal Y “ 10-4>^6, e = -.5, p ■ 2-/6, X - .5, A - 75((^6-4)0-l)w2exp(-p0) , d>2 “ =1.50p2exp(-p0)/A, - 20/((^-4)0-l), ♦j - .75 P2/A » -exp(p0)/((v^6-4)0-l) " -<t>2exp(2-v^60)/26, ♦0 - -((l-^^/a+^jw.