et + >m (r +m)-y=e A Here et is the standard overshooting result. Once again we see that the initial jump breaks down into a part due to forward- and backward-looking expectations, and a part due to the continuous price-movement restriction. We again present a diagram: 137 FIGURE II An unanticipated increase in the steady-state inflation. P 1 P't C ( e', p')E t B(etA e -4 PA e~~ii) 0% 0 -Pt e't 138 Note that at point B, the initial expectational equilibrium, people are only gradually adjusting to the effect of the higher inflation rate on the long-run equilibrium values of p and e. However, people are fully aware of i', the new higher rate of inflation, and they assume that the current expectational equilibrium also adjusts for that. To illustrate this, differentiate (31), (32): 91 A lel O2kc 1(34) et+km - X(ol-62) (l+ -)e =m ,i62A el 2k>*, pt+k: - x(61-e2) (1+-)e = The first term in each equation represents the adjustment neces- sary to keep pace with current inflation. The second term represents expectational adjustment as people experience a higher inflation rate for a longer period. The generalization to other Dornbusch-type models which embody steady money-supply rules is straight forward. Frankel's model illustrates what Flood has emphasized as the proper concept of overshooting. Here the domestic inflation rate overshoots even the new higher rate of money growth. This overshoot- ing is necessary because the demand for real balances falls in the long run because of the higher rate of inflation. This is a common feature of macro-models with a sticky adjustment equation. 139 It has been shown that overshooting is important in a wide class of models even in the absence of structures on discontinuous price adjustments. Overshooting from sticky, but rational, expectations is always smaller than overshooting due to temporarily fixed prices. In the next section, we consider erratic anticipated shocks, and take a closer look at the form of the Phillips curve. 140 II. Expectational Adjustment and the Neutrality of Perfectly Anticipated Money. It is a feature of the Dornbusch and Frankel models considered in the previous section that anticipated jolts to the money supply and the rate of growth of the money supply will have real effects (i.e., the terms of trade will be affected, at least temporarily). This is true even if we permit discontinuous movements in the price level and the exchange rate at the time of the announcement. The economy can follow a path such that markets will be in equilibrium forever4 after the time of the shock, but then they will be in dis- equilibrium prior to the shock. (See Chapter 2.) These are perfectly sensible features of models which contain expectations-augmented Phillips curves which correct only for "normal" patterns of money- supply change. Agents are perfectly rational in the foreign-exchange market. in the goods market, however, they will only be perfectly rational when the exogenous variables follow stationary patterns through time. Later we will give the appropriate adjustment term (i) for the Phillips curve for which agents will be rational in the goods market, no matter how complicated the (stationary) pattern of the exogenous variables. That nonstationarity of the exogenous variables should permit anticipated real effects is a perfectly reasonable feature of a rational-expectations model where expectations are based on the past as well as the present and the future. Only when price expectations 141 of the future totally dominate the determination of the levels of the levels of current variables will the economy remain in perpetual equilibrium in the presence of nonstationary behavior of the exogenous variables. (To be precise, we mean not homogenously nonstationary.) In Mussa's model, when the domestic price level and the exchange rate can jump, then the current levels of these two variables are based soley on forward-looking expectations. Then even nonstationary behavior of the exogenous variables will not jolt the goods market out of equilibrium. Because his equilibrium model derives from the Sargent-Wallace (1973) work, his pure forward-looking solution shares the criticisms which Blanchard (1979) has made about that strand of the literature. In particular, it is highly sensitive to the sign of the parameter representing the elasticity of money demand with respect to the inflation rate. Also it is sensitive to the assumption of uncertainty versus certainty. For a much more thorough and detailed discussion see Blanchard. In spite of their limitations, the Dornbuszh-Frankel literature stands as an appealing compromise between the old adaptive-expectations models and the newer forward-looking rational-expectations models. This is particularly true for the backward and forward solutions to their models presented here. As pointed out here and in Chapter II their approach of directly assuming adaptive expectations in the foreign exchange market is limited. This property will never hold in the presence of nonstationary anticipated shocks, and causes the 142 omission of the backward solution. Below we solve for the appropriate form of the steady-state adjustment term in the Phillips curve for the Dornbusch-class model presented earlier. The resulting Phillips curve will give the economy the property that anticipated changes in the exogenous variables will have no real effects, assuming they follow a homoge- nously nonstationary process. For convenience, we repeat equations (1) through (4) from Section I. (1) m-p=-Ar +Oy Money market equilibrium *(2) r=r +x Interest parity condition (3) x=e Rational expectations (4) p=6(e-p)+i(m, y, r , e, p) Expectations- augmented Phillips curve (e-p= goods market disequilibrium (excess demand)) Sp)=(de/dt , dp/dt) First we solve for i, such that in certainty world, where the exogenous variables follow a set pattern, there will be no goods- market disequilibrium. This can be accomplished by equating the solutions for e and p (given in the appendix), and then solving the 143 resulting differential equation in i. An alternative procedure is given below. This method is based on the fact that for goods-market equilibrium to hold, then e and p must follow equivalent second-order differential equations. By equating the forcing functions in these two equations, which will have identical roots, one again obtains a differential equation in z. Under certainty, solve for z s.t. pt=et for all t: 0 * e=P/ 1/ (m+r -t Using D tu denote the differential operator (Dx=Ix) [: (m+xr*-ty) -(6+D)l/x SP - L ej This can be solved to give a second-order differential either p or e. Thus, equation in (5) ) P+6P*/f=Dz-6/ (m+xrt-oy) (6) 144 Naturally, both equations have identical roots: 6 + (62+46/A)2 2 e2= - (6 26/X) For e and p to be equal over time, so that the goods market is always in equilibrium, we require that the right-hand sides of (5) and (6) be equal. Equating the two forcing functions and canceling terms yields: (7) l/ iJD/m+xr*-.y)=Di , or (7') (1/X-D)i=D/X(m+Ar*-Oy) , which is a differential equation in i, the desired goods-market clearing steady-state adjust- ment term. The convergent-expectations assumption yields the solu- tion: (8) i=i/A )(Xm+Ar*-qy)e / A(t-s) ds t (8') =D / (m+Ar*-y)el/A(t-s) ds, where the procedure t of passing the D operator outside the integral sign can be made 145 rigorous by using integration by parts to evaluate (8) and (8'). The form of s hoid not be too surprising. It is the solution of the money-market equation for p when goods-market equilibrium (e=p) is imposed. (5) and (6) have the same backwards and forwards solutions as in Section I. To illustrate (5) and (6), substitute i=A, as in Frankel's model: (5') e+66-6/*= -6/X(m+Ar -@y) (Imposing Frankel's assumption (6') P+Sp.{64/)p= -6/X(m+Ar*-4y)+m The Frankel model needs the assumption that the inflation rate is constant (ii=O), so et=Pt. V t In the Dornbusch-Frankel models, the i term in the Phillips curve provides the necessary price-level adjustment to keep the economy in a steady state, if it is already there. The powerful equilibrium properties of the Mussa model derive from a i term which keep the goods market in equilibrium, if it is already there. Below we derive an assumption analytically equivalent to Mussa's, though somewhat different in interpretation, and apply it to the simple model of this paper. Note that Mussa's assumption accentuates the overshooting phenomenon. 146 In the long run, the domestic and foreign real rates of interest will be equalized. (2") r-r By substiting equation (2') for equations (2) and (3) in the model above, one can obtain a long-run money-market clearing condition: (5") m-p=-x(r*+p)+Oy Given the current actual price level, pt, we can define the real interest rate equalizing rate of inflation, Pt, by: (Note that we are sup- pressing p, and there- for r* is the foreign real rate of interest.) The Phillips curve now becomes: (4") The first term in (4') corrects for current goods-market disequili- brium, and the second term represents the adjustment necessary to keep real interest rates equalized. Actual real interest rates will of course only be equal when the domestic goods market is in equili- brium. (6") vt t m +xr-y) PC=6(et pt)+t 147 The Mussa technique is to take the goods-market equilibrium condition (here, e=p), and then define i by differentiating this equation with respect to time.. The Phillips curve will then be the same as equation (4'). The solution to the Mussa-type model is . given by: t (9") et=cle(/ft+c2e-6t -l/x (m+x r*- y)6 /X)t-s)ds (lol") pt=cle +c2(-Xf)e-6t -'/x w-a0/X)(t-s)ds The backwards-looking integral corresponding to the negative root, -6, is vanishing. Once again the assumption of convergent expectations allows us to set cl=O. If prices can jump discontinu- ously in response to shocks to the forward integral, then the solution curve, where c2=0, insures that no kind of anticipated money shock will have real. effects (that is, assuming that the exogenous variables do not increase by an order of magnitude greater than exponential). If the domestic price level cannot jump discontinuously in response to shocks, then c2 will be non-zero, and expectations will have a backwards-looking component. As shown in Section I, Dornbusch models always have a backwards-looking component. Overshooting is greater in the Mussa model for the classic overshooting exercise than in our earlier model.. That is because exchange-rate appreciation itself hinders the necessary price adjustments after the shock through 148 the augmented Phillips curve. Assume that m, r , y have been at a constant level. Let an unanticipated pennanent increase in the stock of money take place. Overshooting in the Mussa model is:5 e -e=Am(l+X )>AM(l+ 1t Adt 16(Ad. (62+46/XY/ where the right-hand expression gives overshooting in the Dornbusch model. Overshooting in both cases is small when X is large. 149 III. Uncertainty This section briefly considers some of the issues which arise when uncertainty is introduced explicitly. The generalization of the earlier models to cases where the exogenous variables follow sta- tionary stochastic processes is straight-forward. The entire analysis can be shifted to expectations space. As an example, a special case using the Mussa model is considered below. As in Chapter II, the level of the money supply will occasionally be subjected to random shocks. (This does not lead to a mixed difference/differential equation, but simply to a differential equation with a discontinuous forcing function.) The linear restriction place on the exchange rate and price level at the time of each shock is that the domestic price level cannot jump. Assume that in the Mussa model, which has the solution given in (9') and (10') of Section II, future values of the exogenous variable are unknown. As a particular example, assume r , y are known and constant. m is subjected to occasional shocks, with mean zero. Suppose the most recent shock occurred at time t. The solution to the Mussa model given at the end of Section II is now: t+k (9") tEet+k=c2e k) )(t+k-s)ds 150 t+k (10") t Ept+k=- X6c2e-6(t+k)-1/x t E(+Axr*-y)e(I/)t+k-s)ds where tExt+k represents the expected value of xt+k at time t. m is the uncertain money supply. c2 depends on what type of initial con- dition we impose after the shock. (The domestic price level could be fixed, the terms of trade could be fixed, goods-market equilibrium could be reached, etc.) Now impose the restriction that the domestic price level cannot "jump" in response to the occasional increases or decreases in the permanent level of the money supply. Define: - - * /X)(h-s) Pheh=-/X E(m+Xr-y)Ods Assume that at time t-k, the economy was in equilibrium. Finally, define: j j (11) APh jl/x E( n+Xr*y)/)j-s)ds+l/X E(E+Xr y)eO/X)(j-s)ds 1im(h-cj) 1 im(j+h) where the first integral is the right-hand side limit and the second integral is a left-hand side limit. Thus Afh reflects the shock occurring at time h. The shock may represent an increase in the money supply at time h (unanticipated), or a shock which brings news 151 of (a) future money-supply shock (s). This is the same exercise as performed in Chapter II. Here, the general solution has a neat closed- form solution. (12) tEpt=_&-kk 1tk+ie -pt i=0 The exchange rate is always buffeted by each shock to a greater degree that the price level, when all shocks are to the money supply (Aet=APt) k (13) tEet -Aet-k+i e +et i=0 Equation (11) is the proper formalization of the idea of "new information available at time h," under the assumptions made here. Equations (12) and (13) show how disequilibrium can be generated when uncertainty is explicitly introduced. Here it is true that money- supply information buffets the exchange rate, and may cause over- shooting. Uncertainty poses technical problems when we try to use the assumption of convergent expectations to form the appropriate bounds of integration in the expectations integrals corresponding to each of the roots of the model. As an example, consider a stationary process which is sufficiently damped that we cannot throw out the forward 152 solution associated with a negative root simply through the convergent- expectations assumption. Take the case where the expected value of the log of the money supply is zero. Set xr - y=O. While we know that the actual money supply will almost never be zero in the future, the expected-value integral is zero. In fact, as long as the money supply has expected value zero beginning at some point in the future (say, the money supply follows a damped stochastic process), then the forward integral may be finite. Yet in the certainty world it is never finite. A topic for further research is to show how uncertainty provides a motivation for using past levels of the exogenous variables in de- termining the present levels of endogenous variables, even in the absence of strictures on discontinuous price and exchange-rate movements. Finally, the introduction of uncertainty can lead to severe problems when the assumption of continual money-market clearing is relaxed. See Burmeister-Flood-Turnovsky (1978). 153 SUMMARY e first section of this paper proposed a backward- and forward- looking expectational equilibrium as a solution to a well-known model of prices and exchange rates. The solution to the Dornbusch model, or a similar one, generally requires an initial condition in addition to the assumption of convergent expectations. One natural condition is to assume that the domestic price level cannot respond instanta- neously to a shock. Another is that prices and exchange rates immediately jump to their new long-run equilibrium. While any linear restriction on prices and the exchange rate will leave the economy on a stable path, most writers consider other assumptions less plausible, or even totally arbitrary. However, the assumption that prices are able to respond instantaneously allows a third natural solution, the expectational equilibrium. There agents choose a general solution to the dynamic model governing the economy, which would make Sense even in the face of anticipated, erratic shocks. The expectational equilibrium rationally weighs the past and the future in the determination of current variables. Overshooting may occur to some extent, even if prices can jump. As long as agents do not forget what happened in the past, or assume that the goods market will automatically clear in the face of an unanticipated shock (even though it cannot always remain in equilibrium in the face of an anticipated shock), they may be assumed to choose the expectational (temporary) equilibrium. In the long-run, as 154 as expectations adjust, the economy will reach the normal long-run equilibrium. The standard exercise, which assumes sticky prices, leads to overshooting which can be decomposed into overshooting due to the tenporarily fixed price-level assumption, and overshooting due to the stickiness of the perceived current expectational equilibrium. This interpretation is extended to a wide class of Dornbusch models. The second section examines the role of the expectations-aug- mented Phillips curves in Dornbusch-Frankel models. It derives from the appropriate expectations augmentation term corresponding to any homogenously nonstationary distribution for the exogenous variable. In the face of other types of nonstationary behavior of the exogenous variables, the real effects of perfec.tiy anticipated monetary shocks cannot be assumed away, except in the limiting case of a pure for- ward-looking solution. The final section gives some results under uncertainty. A general solution to a forward-looking model with continuous price adjustment, which experiences money-supply shocks at discrete intervals, is derived. 155 FOOTNOTES The solution to the unanticipated money-supply shock in Dornbusch's "Expectations and Exchange Rate Dynamics" is usually given as: F1) (pt-') =X02c2ee2t (et-e') =c2ee2t where F' and V' represent the new forward-looking long-run equilibria. We have already imposed the condition c=0, so the root 6 1 has no unstable effect. Dornbusch solves these equations using the initial condition p t=T, the old price-level equilibrium. Inspection of (F) seems to indicate that if prices and exchange rates are freely flexible, the only natural assumption is c2=0. From the perspective of this exercise, the text argues that there is an alternative initial condition and corresponding c2 value which is not ad hoc, and also leads to some overshooting. The reason this choice of c2 is rational is suppressed in the pure forward-looking 7', F'. However, the logic of the text obviously would apply to the equations (F). When we include anticipated shocks, a solution of the form (F1) does not exist. Then we must use the more general answers given in the text. 156 2 Even in "equilibrium" in this system, the current account is going to be in surplus or deficit, and wealth is changing hands. The effects of this transfer mut be considered in the long run. See Henderson. 3 See footnote (1), where the more common representation of the solution is given. There, the assumption that under "flexible" prices and exchange rates the economy would move to the current per- ceived expectational equilibrium amounts to assigning a specific non-zero value to c2. The text explains why this particular value has a natural interpretation as the rational-expectations weighting of the forward and backward solution at time t. 4 The money markets and foreign exchange markets are always in equilibrium in these models by assumption. It is the goods market which experiences disequilibrium. By "equilibrium forever," we mean the goods market was actually in equilibrium in the past, or at least it had fully adjusted to all previous shocks by time t. In the future, expected disequilibrium is zero, although it is known that future surprises may come. 5 In a world with no secular changes, the Mussa model and Dornbusch model are virtually equivalent. Overshooting for the Mussa-type model is only greater, given the parameters 6 and A. 157 APPENDIX I The basic model of the text is given by: A1) -;= -(D+6)p+6e 1/X(m+Xr* -y)=p/X-De where D.x=x This type of equation may be solved by the method of variation of parameters. (See Srepley L. Ross, Introduction to Ordinary Differen- tial Equations (Waltham: Ginn and Co., 1966).) Burmeister, Flood and Turnovsky, "Perfect Forsight and the Stability of Monetary Models," carefully derives the appropriate bounds of integration for such systems. We freely employ their solution technique. The solution to the homogenous equations is given by: Olt o2t2) etC 1el +c2 e pt=Aelcle3t+A02c2e02t , where 01>0, o2<0 are the roots of the characteristic equation: 02+60-6/A=O We wil l set c1=0 by the assumption of convergent expectations. A particular solution to (A,) may be found by noting that: 158 A3 ) elt pt=xev1 (t)eltX+02v2(t)e62t is a particular solution if and only if: A4) Aelv1 (t)e lt+x22(t)e dt=i To verify the necessity of the conditions imposed in (A4) dif- ferentiate e , p in (A3), and substitute into the equations in (A.).ti ny The equations in (A 4) may be solved to yield: -elt e v3 ( - o2) 0 -e2t 12(t) .( 1-2) v-(t)= (i+ei(m+xr -qy)) - (z+0l(m+Xr -jy)) . Thus: t (i+e2 (m+Xr*-Oy))eel (t-s)ds a t (z+el(m+xr *-0y))e 62(t-s~d b epv(t)eelt+vte t2(te1~ S(t)e' It v2(p e2t-+ g g 159 Because el>0, and 02<0, the appropriate bounds of integration are a=+oc, b=-aC, under the assumption of convergent expectations. The general solution to (A 1 ) is given by: Olt 02tA5) et=c 1e +c2 2e tv1I(t)+v2(t) pt=Aolc I eelt+A2c2 e 2t+Aov (t)+xe2v2 (t) Under convergent expectations, c1=0. 160 APPENDIX II We confirm that if m, y and r have been constant through all time and are expected to remain constant, then -p=p=e=e=m+xr*y. Setting m+Ar*- y=z,, We have: t t A6) # - (ztO2eOl(t-s)dsl e2- ( zt Ole 2 (ts)ds +0 - -02+1) = Z The first integral in = looks forward in time. The second looks backwara t t 1 r oIel(t-.s)s -- ( - 0le2(t-s) +C7t2 t =Zxo 2) 2(e +q2) Cancelling and noting 61-62=6/x and 01+2=-6, we see p=e=z. If we integrate backwards only to zero instead of minus Infinity, then A8) et=z-72 (Az)e-e2t 161 where here Az=z A9) pt=z-xQ1 (Az)e- 2 t This problem can easily be negotiated by setting the arbitrary constant, c2, appropriately, though that again imposas an implicit economic assumption. When the rate of money growth has been constant through all times and is expected to remain constant into the future, then the expectational equilibrium is equivalent to the pure forward solution Frankel gives. We first confirm that p=m=e. Integrate v1 and v2 by parts to obtain: / t A,0 v(t- 1( 022 - (mdxr-Oy)+ n12)e" (t-s) ds A 11 ) v2 (t)Y-X( 0l-02 ) Q + -(mxr-y)t+ 5 -mI+4) e02(ts)ds Differentiating Pt=xelv 1(t)+xolv2(t), and setting m= = (Frankel's implicit assumption that only m changes and that it changes at a constant rate over time) A 12) Pt (e-2) 1[el - e2]=i Similarly =1 (t)+%(t) 162 A13) t 1-o22]_m 't(So-o2) where the proof is identical to the homogeneity proof in the model with no steady-state monetary growth. Similarly, we expect 6 0 a )=e=mi+x(r*+*)-y. This can be seen by setting F=e=m, and solving for p in the money-market equation. We confirm this below: Integrate (28') of the text by parts, noting rnc*-==O, and group terms: t A 14) et(m t ( -+)( r(+ )e l(t-s)ds t + -m(1I )e2(t)ds] -C0 = m+ 4) -02) -[ 01- +A-+ 201 =(Mt my ]-m t ) 01 -2 [ T 1 (1 2 02 -5757 ] =(M Xrcjy+r - 2t[.....i 7 ] = t +X(r +m)-4Y) 6A2 Similarly, 02 01 A15) j -1-5-i-+1+57]15) P =mt y)+* (01-02) t] =-O as one would expect. 163 BIBLIOGRAPHY Blanchard, Olivier-Jean, "Backward and Forward Solutions for Economies with Rational Expectations," American Economic Review (1979). Blinder, Alan A. and Stanley Fischer, "Inventories, Rational Expectations and the Business Cycle," Working Paper #220 (MIT Department of Economics: June 1978). Burmeister, E., fR. Flood, and S. Turnovsky, "Perfect Foresight and the Stability of Monetary Models," Discussion Paper (University of Virginia: 1977). Burmeister, E., R. Flood, and S. Turnovsky, "Rational Expectations and Stability in a Stochastic Monetary Model of Inflation," Working Paper (University of Virginia: May, 1978). Dornbusch, Rudiger, "The Theory of Flexible Exchange Rate Regimes and Macroeconomic Policy," The Scandinavian Journal of Economics, Vol. 78, 1976, pp. 255-75. , "Expectations and Exchange Rate Dynamics," Journal of Political Economy, Vol. 84, No. 6, December 1976, pp. 1161-76. Fischer, Stanley, "Anticipations and the Non-Neutrality of Money, II," Working Paper #207 (MIT Department of Economics: 1977). Flood, Robert P., "Exchange Rate Overshooting," University of Virginia: 1977). , "Exchange Rate Expectations in Dual Exchange Markets," Journal of International Economics, #8, 1978, pp. 65-77.