anticipated shocks and exchange rate dynamics

Anticipated Shocks and Exchange Rate DynamicsAuthor(s): Charles A. WilsonSource: Journal of Political Economy, Vol. 87, No. 3 (Jun., 1979), pp. 639-647Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/1832027 .

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Anticipated Shocks and Exchange Rate Dynamics

Charles A. Wilson University of Wisconsin-Madison

The paper extends Dornbusch's analysis of exchange rate dynamics to include the case where changes in government policy are anticipated before they occur. It is demonstrated that simply the announcement of an expansionary policy will cause the exchange rate to jump, which induces an expansionary impact on the economy even before the policy is implemented.

In "Expectations and Exchange Rate Dynamics" (1976), Dornbusch developed a simple macroeconomic model designed to study how exchange rates respond to unanticipated policy changes or other shocks to the economy. His model focused on the assumption that movements in the exchange rate, after any instantaneous response has occurred, are consistent with expectations. In this paper I extend Dornbusch's analysis to consider the impact of anticipated shocks or policy changes in the economy. Using Dornbusch's example, I ask the question, How will the exchange rate respond if an announcement is made today that the money supply will be increased at some point in the future? This extension has two kinds of benefits. From a policy perspective, it gives us a clearer idea of how a lag in the implementa- tion of a policy will affect the economy both before and after the policy is implemented. From a theoretical perspective, it permits us to disentangle the effect of a change in expectations about future policy variables from the effect of a change in the actual values of those variables.

[Journal oJ Political Economy, 1979, vol. 87, no. 3] ? 1979 by The University of Chicago. 0022-3808/79/8703-0013$00.95

639



640 JOURNAL OF POLITICAL ECONOMY

Beside the assumption that any changes in the exchange rate are correctly anticipated after the money supply has increased, Dorn- busch employed two other critical assumptions. The first is that capital is perfectly mobile. Combined with the perfect foresight assumption, this ties the rate of change in the exchange rate to the domestic and foreign interest rates by a simple arbitrage condition. The second is that the price level adjusts sluggishly to excess demand for domestic goods. This requires the domestic interest rate to respond to any discrete change in the money supply, which in turn induces the effect on the exchange rate.

Dornbusch shows that when the increase in the money supply is not anticipated, equilibrium requires two instantaneous effects at the time the expansion occurs. The first is a decrease in the interest rate to clear the money market. The second is an increase in the exchange rate to make the future path consistent with equilibrium. If real output is assumed fixed, the instantaneous jump in the exchange rate will actually overshoot its long-run value.

When the increase in the interest rate at time T is anticipated at time 0, these two discrete effects occur at different times. Since the potential for arbitrage profits rules out the possibility of any anticipated discrete jump in the exchange rate, any discrete change in the exchange rate must occur at the instant the change in policy is first announced. On the other hand, since the money supply does not actually change at this point, there will be no jump in the interest rate. The net effect is still an expansionary impact, however. The higher exchange rate induces a terms-of-trade shift which increases excess demand for goods. This, in turn, puts upward pressure on prices, which forces the interest rate to gradually rise. When the increase in the money supply actually occurs, the only discrete change is in the interest rate. In order to increase demand for the larger stock of money, the rate of interest must fall. If world output is held fixed, it will fall below the world rate of interest, reversing the rate of change in the exchange rate. Prices continue to rise until the new stationary state is reached.

These implications reinforce the point which has already been emphasized by Brock (1975) in the context of a perfect foresight monetary model. When changes in monetary policy are anticipated, there is an immediate effect on the economy very similar to that which would result if an unanticipated policy were implemented. Thus, if government policy can be anticipated at all, changes in the supply of money may generate leads as well as lags in the level of prices. Similar results are also implied in other recent studies of exchange rate determination in the context of simple monetary models (e.g., Barro 1978; Bilson 1978).



ANTICIPATED SHOCKS 641

The Model

I will confine my attention to the simplest version of the model. The demand for domestic goods, D, is given by In D = u + 8(e - p) + yy - or, where e, p, and y are the logs of the exchange rate, the domestic price level, and the real domestic income, respectively, and r is the domestic interest rate. The foreign price level and domestic real income are assumed to be constant. The rate of change in the price level is then assumed to be proportional to the log of the ratio of demand to supply of domestic goods:

p = r ln(D/y) = 4r[u + 8(e -p) + (y- l)y - r]. (1)

Since output prices adjust slowly, equilibrium in the domestic money market is obtained by instantaneous adjustments in the domestic interest rate. Letting m be the log of real balances, Dornbusch assumes that

-Xr + 4y =m-p. (2)

In analyzing the asset market, I will deviate somewhat from Dornbusch's approach. He assumed that the expected rate of change in the exchange rate is a linear function of the current and long-run values of the exchange rate and then verified later in the paper that the actual path will satisfy this relation. As we shall see, however, when an announcement of a money supply increase precedes the actual increase, the change of the exchange rate will be a linear function of the current and long-term rate only after the increase in money supply actually takes place. Therefore, I will assume from the outset that the rate of change in e is correctly anticipated and derive the equilibrium path of e, p, and r simultaneously. The world rate of interest is assumed fixed at r*. Since capital is perfectly mobile, the rate of change in the exchange rate must satisfy a simple arbitrage condition.

r=* + e. (3)

Fixing m, we may use equations (1)-(3) to solve for the stationary state value of p and e:

p(m)= m - oy+Xr* (4)

and

e(m) = m- [u + (&I + y-1)y- +r)r*].l (5)

By appropriate normalization of units, we may assume that p (0) = e(O) = 0. This assumption is implicit in fig. 1.




P

450 Q

A

p(T) 7_1_

p~~mO)I I I Io

eCmo) e(Md) e(O) e(T) e' e

FIG. 1

Notice that the stationary state values of both the price level and the exchange rate are homogeneous of degree one in the money supply. Substituting equations (2), (4), and (5) into (1) and (3) then yields the general expression for the change in p and e as a function of only the current and stationary state values of p and e:

p = -V 8 + -) p fi-p(m)] + N[e - e(m)] (6)

and

e = ( [p-p(m)] (7)

The general solution to equations (6) and (7) has the form2

p(t) - p(m) = c 1X~Ae'Lit + c2 X2e 92t (8)

and

e(t) - e(m) = cle"it + c2e'J2t, (9)

where c1 and c2 are arbitrary constants, tl = -i7((o + X6)/2X +

{Lir(o + X8)/2X]2 + IT8//X}112 > 0, and P'2 = -7T(o + X8)/2X - {Lir(S +

X8)/2X]2 + 7T/X} 1/2 < 0.

2 See, e.g., Hurewicz 1958, p. 56.




The Effect of an Anticipated Monetary Expansion

In the remainder of the paper, I will be concerned with the following question. Suppose the economy is initially in a stationary state associated with money supply mi. The government announces at time 0 that the stock of money will increase to m at time T. What is the instantaneous change in the economy at time 0, and what is the path of p, e, and r thereafter?

Consider first the path of the economy after time T. Following Dornbusch, I will assume that p(t) and e(t) converge to p(m 1) and e(m,).3 Therefore, from equation (8), ,at > 0 implies c1 = 0. This yields two important implications. Differentiating equation (9) and substituting, we obtain

e = p.2[e(t) - e(m 1)]. (1 0)

This is precisely Dornbusch's result. After the stock of money has increased (whether anticipated or not), the rate of change in the exchange rate which is consistent with expectations must be a linear function of the difference between the current rate and the stationary state value.

Second, if we divide equation (8) by (9) (still setting c1 = 0), we obtain

p (t)- p(m l) = XA2[e (t) - e(m 1)] , for all t ' T. ( 11)

Evaluating this expression at t = T yields the relation between e(T) and p(T) which must be satisfied in a perfect foresight equilibrium. Fol- lowing Dornbusch, this relationship is represented in figure 1 by the curve labeled QQ. Note that it is linear and negatively sloped. Starting at any point on this curve at time T, p and e will converge exponen- tially top (m 1) and e-(m 1).

In the case analyzed by Dornbusch, T = 0. His result then follows immediately from equation (11). Since p(O) = p(mo), e must jump immediately to the corresponding point on the QQ curve, e'. Since the price level has not changed, equilibrium in the money market requires an instantaneous drop in r. The potential for profitable arbitrage then requires that the expected rate of change in e satisfies equation (3). In the face of excess demand (resulting from the increase in e and the drop in r), p will gradually increase, inducing an increase in r which slows the rate of decrease in e. These interactions

3Note that this is an assumption. There is also a one-parameter family of solutions to eqq. (8) and (9) which would satisfy the conditions of an equilibrium for which e and p diverge. These hyperinflation (deflation) equilibria can only be ruled out if we interpret eqq. (1)-(3) as local approximations around the stationary state values of e and p and introduce assumptions which imply that e and p cannot diverge indefinitely.




result in the movement of p and e up the QQ curve toward the stationary state values of p(m 1) and e(m 1).

Now consider the case where T > 0. Given the foregoing analysis, we need consider only the path of the economy from time 0 to T. Again, start with general expressions for p and e given by equations (8) and (9) and consider the implications of the end-point conditions which must be satisfied.

Since we are assuming that the economy is initially in the stationary state associated with m = mi, we have p(O) =p(m0). Equation (8) then implies ciu + c2,u2 = 0. Dividing equation (9) by (8), we obtain

V(t) X-A11L2 [ E1f I-Lt - ie2t e (p(t) -e(mo))

Evaluating V at t = T defines an equilibrium relation between p(T) - p(mO) and e(T) - e(mo) which is independent of m 1 and depends only on T. It can be shown that V(oo) < 1 and V'(t) > 0.

Second, at time T, [p(T), e(T)] must satisfy equation (11). Let Am =

mlI - mo, and recall from equations (4) and (5) thatp(m1) = p(mno) + Am and e(m,) = e(mo) + Am. Therefore, equation (11) can be transformed to yield a relation between p(T) -p(mo) and e(T) - e(m0) which is a function of e(T) and Am:

W[e(T), Am] XA2 + -(T) -A(m i) e (T) p(mO)

Since the potential for arbitrage profits prohibits a jump at T, equilibrium then requires that

V(T) - W[e(T), Am] = 0. (12)

This determines the value of e(T). Since W[e(T), Am] _ 1 for all e(T) ? e(ml) and V(T) < 1, we may

conclude that e(T) > e(m 1). Note the similarity to Dornbusch's result. Regardless of what the response is at time 0, at the moment the money supply actually increases, the exchange rate must exceed the long-run value e(ml). There must be overshooting since [p(T), e(T)] must lie on the QQ line. It also follows that p(T) will lie below the long-run value, p(m1). In addition, equation (12) implies two useful comparative statics results. Since V'(T) > 0 and OW/9e(T) < 0, it follows that

de(T)/dT < 0. (13)

Differentiating the expression for W[e(T), Am] yields

d[ee(T) - e(T)e() > 0. (14)




It remains to determine the relation between T, an, and e(O). Using Clul = -c2/12, we have from equation (9)

e(t) - e(mo) = C1[I2e Alt - AleII2t], for t T. (15) /J2

Evaluating equation (15) at t = 0 and t = T yields

e(0) - J(m0) = [(p2 - p1)I(,u2eM'T - e"2T)][e(T) - e(mo)]. (16)

It then follows from inspection of equations (16) and (13) that

de(O)< 0 (17) dT

and from equation (14) that

de(0) - e(0) - e(mo) > 0. (18) dAm AM

The reader may also verify that e(O) - J(mo) -O 0 as T -- o or as Am 0. Finally, V(O) = 0 implies from equation (12) that when T = 0,

e(0) - e(mo) = - 1 ) Am. (19) X92

The general solution is also illustrated in figure 1. Again the QQ line represents the restriction imposed by equation (11) on the values of p and e for t > T. Starting at any point on the QQ curve, T, p, and e will converge to [p(m,), eJ(ml)]. T heAA curve represents the path of e andp before the increase in m occurs. It corresponds to the relation defined by V(t). For each value of e(O), there corresponds a differentAA curve. From the definition of V (t), the slope of the ray from [P (MO), e(mO)] to any point on theAA curve increases as we move to the right. However, V(t) < 1 implies that for any choice of e(0) > e(m,), theAA will lie below the 450 line. The equilibrium value of eo found by adjusting the AA curve until the intersection with the QQ curve occurs at time T. From equation (16), the corresponding value of e(O) is unique.

When the government announces at time 0 the increase in m at time T, the instantaneous response is ajump in the exchange rate from e(mo) to e(O). Since the supply of money has not yet changed and prices cannot adjust instantaneously, there is no immediate impact on the interest rate, r. Therefore, after the instantaneous jump in the values of e, the rate of change in e is initially zero. The depreciation in domestic currency, however, does generate an increase in the demand for domestic goods. As a consequence, the price level begins to rise. Equilibrium in the money market then requires an increase in r, which in turn requires that the expected change in the value of the exchange rate becomes positive. The net effect is a movement along




the AA curve, further stimulating the increase in the price level, increasing the interest rate even more, and speeding up the increase in e. This process continues until time T when the money supply actually increases. At this point, the potential for arbitrage profits rules out any discrete movement in the exchange rate, but equilibrium in the money market will require a drop in r below the world interest rate r*. This will induce the economy to move along the QQ line toward [p(mD), e(m1)]. Excess demand will be stimulated even further, causing the price level to continue to rise. However, zero arbitrage profits will imply a falling exchange rate e. The argument is then identical with that presented by Dornbusch.

Note that the implications stated in equations (17) and (18) and in the comments which follow are precisely what one should expect. The larger the increase in the money supply, the larger the instantaneous impact on the exchange rate. On the other hand, the further into the future it is expected that the increase will occur, the less is the current impact. As the expected date of increase in m goes to infinity, the instantaneous change in e goes to zero.

Conclusion

There are two lessons to be learned from this exercise. The first is substantive but rather specific to the model. It is that simply the announcement of an expansionary policy may have an expansionary impact (or in this case withy held constant, an inflationary impact) on the economy. Even if we ignore the effect of such announcements stemming from changes in investment or resulting from anticipated changes in prices, an immediate increase in demand may result from a shift in the terms of trade following the adjustment in the exchange rate. It is important to recognize, however, that this conclusion, as well as some of the other properties of the adjustment process, may be sensitive to the particular specification of the model. The announcement of an expansionary policy can generally be expected to have an impact, but the direction or extent of the impact may be different for different specifications.

For example, we might have supposed that rather than remaining constant real income always adjusts to the level of demand. As Dornbusch demonstrates, this may eliminate the overshooting phe- nomenon at the time of the increase in the stock of money. Otherwise, the analysis is unaffected. An anticipated increase in the money supply will still induce a discretejump in the exchange rate, followed by a continuous rise in both the price level and the exchange rate. Another possibility, however, is to have included the exchange rate as an argument in the demand for money. In this case, the overshooting is




reinforced, but the path of the economy before the anticipated increase in the money supply occurs may be quite different. There will still be a jump in the exchange rate. This will stimulate demand through the terms-of-trade effect. However, in order to clear the money market, the interest rate must rise, depressing demand. If the interest rate effect on demand dominates the terms-of-trade effect, the impact of an announcedfuture increase in the money supply will be deflationary. The price level will fall until the money supply is actually increased, at which time the interest rate falls below the foreign rate, reinforcing the terms-of-trade effect on the demand for domestic goods.

The second point of this exercise is technical. Simple models of this type incorporating rational expectations can generally be used to analyze the impact of the effect of a policy change not only at the time the policy change actually occurs but also at the time the policy change becomes anticipated. Furthermore, as long as the model can be re- duced to the form of equations (6) and (7), the general method of solution remains essentially unchanged. As we have seen, however, if the change in expectations does not occur simultaneously with the actual change in the parameters, it will generally not be possible to express "rational" expectations as a function of only current values of the parameters of the economy.

References

Barro, Robert J. "A Stochastic Equilibrium Model of an Open Economy under Flexible Exchange Rates." QJ.E. 92 (February 1978): 149-64.

Bilson, John F. 0. "Rational Expectations and the Exchange Rate." In The Economics of Exchange Rates, edited by Jacob A. Frenkel and Harry G. Johnson. Reading, Mass.: Addison-Wesley, 1978.

Brock, William A. "A Simple Perfect Foresight Monetary Model."J. Monetary Econ. 1 (1975): 133-50.

Dornbusch, Rudiger. "Expectations and Exchange Rate Dynamics."J.P.E. 84, no. 6 (December 1976): 1161-76.

Hurewicz, Witold. Lectures on Ordinaiy Differential Equations. Cambridge, Mass.: M.I.T. Press, 1958.



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