"Rational expectations may differ from static expectations under a fixed exchange rate if there is some probability of a change in the exchange rate. In addition, there are cases that fall between floating and fixed exchange rates. One that has attracted considerable attention is the target band, such as those used in the European Monetary System. See Krugman (1991), for example.

Finally, with a fixed exchange rate, the exchange rate itself is a policy instrument. For example, a devaluation an increase in the fixed exchange rate, f stimulates net exports and thus increases aggregate demand.

Rational Exchange-Rate Expectations and Overshooting

The Mundell-Fleming model assumes that exchange-rate expectations are static. But with a floating exchange rate, it turns out that when plausible assumptions about the dynamics of prices and output are added to the model, there are predictable changes in exchange rates. Thus static expectations are not rational: an investor with static expectations is making systematic errors in his or her exchange-rate forecasts. Such an investor can therefore earn a higher average rate of return by using information that helps to forecast exchange-rate movements. Thus it is natural to ask what happens if investors form their expectations concerning movements in the exchange rate using all of the available information-that is, if they have rational expectations. Since static expectations are rational when the exchange rate is fixed and likely to remain so, we focus on a floating exchange rate.*

When expectations are not static, perfect capital mobility no longer necessarily implies that domestic and foreign interest rates are equal. Consider an investor at some time t deciding where to hold his or her wealth. If the investor puts a dollar into a domestic asset that earns a continuously compounded rate of return of i, at time f -b he or she will have e dollars. Suppose the investor instead invests in foreign assets. At f, the investors dollar can be used to purchase foreign assets that are worth l/£(t) units of foreign currency; after At these assets are worth e*4£(t) units of foreign currency; and this foreign currency can be used to buy f(f + At)e*/e{t) dollars.

Under perfect capital mobility, these two ways of investing the dollar must have the same expected payoffi s(t), i, and z * are known, but f (t + M) may be uncertain. Thus we have

Equation (5.17) holds for aU values of At. The derivatives of both sides with respect to Af are therefore equal:

When this expression is evaluated at Af = 0, it simplifies to

f(f) "

(5.19)

Equation (5.19) states that under perfect capital mobility, interest-rate differences must be offset by expectations of exchange-rate movements. The domestic interest rate can exceed the foreign interest rate, for example, only if the domestic currency is expected to depreciate at a rate equal to the interest-rate differential. Equation (5.19) is known as uncovered interest-rate parity}

The possibility of expected exchange-rate movements associated with interest-rate differences gives rise to the possibihty of exchange-rate overshooting (Dornbusch, 1976). "Overshooting" refers to a situation where the initial reaction of a variable to a shock is greater than its long-run response. To see how the exchange rate can overshoot, consider an increase m the money supply starting from a situation where i = i* and where the exchange rate is not expected to change. As stressed later in the chapter, Keynesian models generally imply that monetary disturbances have no real effects in the long run. Thus the long-run effect of the shock is just to cause both the price level and the exchange rate to rise proportionally with the increase in money.

Now consider the short-run effect of the shock. If the monetary expansion reduces the interest rate, then (5.19) implies that must be negative: if i is less than i , investors will hold domestic assets only if they expect the domestic currency to appreciate. But this means that the domestic currency is worth less now than it will be in the long run; that is, it must have depreciated by so much at the time of the shock that it has overshot its expected long-run value.

This leaves the question of whether the monetary expansion reduces the domestic interest rate. A particularly simple case occurs in a variant of the model where producers cannot change output in the very short rtm, so that the IS equation, (5.12), need not be satisfied at every moment. With both prices and output fixed, the only variable that can adjust to ensure that the LM equation, (5.1), is satisfied is the interest rate. Thus i must fall in response to an increase in M, and so there must be exchange-rate overshooting.

The intuition for this result is straightforward. If at the time of the shock the exchange rate merely depreciates to its new long-run equilibrium level, the interest-rate differential causes all investors to want to purchase foreign

The parity is "uncovered" because although positive expected profits can be made by purchasing one countrys assets and selling the others when (5.19) fails, these profits are not riskless. The alternative is covered interest-rate parity, which refers to the relationship m (5.18) with the expected future exchange rate replaced by the price in futures markets of commitments to buy or sell foreign currency at a later date. Failure of covered interest-rate parity would imply a riskless profit opportunity.

Imperfect Capital Mobility

The assumptions that there are no barriers to capital movements between countries and that investors are risk-neutral are surely too strong. Transaction costs and the desire to diversify, for example, cause investors not to put all of their wealth into a single countrys assets in response to a small difference in expected returns. It is therefore natural to consider the effects of imperfect capital mobility. We focus on the case of a floating exchange rate, and for simplicity we revert to the assumption of static exchange-rate expectations.

A simple way to model imperfect capital mobility is to assume that capital flows depend on the difference between domestic and foreign interest rates. Specifically, define the capital flow, CF, as foreigners purchases of domestic assets minus domestic residents purchases of foreign assets. Our assumption is

CF = CF(i - i *), Cf(«) > 0. (5.20)

The capital flow, CF, and net exports, NX, must sum to zero. If net exports are negative, for example, this means that the countrys sales of goods and services to foreigners are not sufficient to pay for its imports. The country must therefore be paying for the excess by selling assets to foreigners- that is, CF must be equal and opposite to NX. Thus:

CF(i + NX(Y,i - 77G, T,£P*IP) = 0. (5.21)

The aggregate demand side of the model now consists of the IS equation, (5.12), the LM equation, (5.1), and the balance-of-payments equation, (5.21). If net exports are the only component of planned expenditure that is

"See Frankel (1979) and Engel and Frankel (1984) for empirical investigations of overshooting.

With perfect capital mobility, CF is minus infinity if i is less than i *, plus infinity if i is greater than j *, and can take on any value-since investors are indifferent about which countrys assets to hold-if i equals j*. Thus (5.21) can hold in this case only if j = i*.

currency to obtain the higher-yielding foreign assets. This cannot be an equi-hbrium. Instead, the price of domestic currency is bid down until it is sufficiently below its expected long-run level that the expected appreciation just balances the lower interest rate on domestic assets.

When the IS equation is assumed to hold continuously, an increase in M no longer necessarily reduces z. Thus in this case there can be either undershooting or overshooting. Which occurs turns out to be a comphcated function of the parameters of the model (see Dornbusch, 1976, and Problem 5.10).!