Exchange Rate Dynamics and the Overshooting Hypothesis
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund
  • | 2 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

THE EVOLUTION of the international monetary system into a regime of flexible exchange rates and the large volatility of these rates during the 1970s have led to a renewed interest in studying the principal determinants of equilibrium exchange rates. The large fluctuations stimulated theories of exchange rate dynamics and have led to the development of various versions of the overshooting hypothesis.1 The explanations of the overshooting phenomenon vary, but they all rely on the short-run fixity of some nominal quantity. Some account for it by the assumption that in the short run commodity prices are slow to adjust relative to asset prices (Dornbusch, 1976); some attribute it to the lack of sufficient speculation in the markets for foreign exchange (McKinnon, 1976); some attribute it to the differential effects of new information on commodity and asset markets (Dornbusch, 1979; Frenkel, 1981 a, 1981 b; Frenkel and Mussa, 1980; Mussa, 1979); and some attribute it to the implications of the process by which asset holders restore portfolio balance in the face of disturbances (Kouri, 1976; Calvo and Rodriguez, 1977; Branson, 1979; Ethier, 1979).2

Abstract

THE EVOLUTION of the international monetary system into a regime of flexible exchange rates and the large volatility of these rates during the 1970s have led to a renewed interest in studying the principal determinants of equilibrium exchange rates. The large fluctuations stimulated theories of exchange rate dynamics and have led to the development of various versions of the overshooting hypothesis.1 The explanations of the overshooting phenomenon vary, but they all rely on the short-run fixity of some nominal quantity. Some account for it by the assumption that in the short run commodity prices are slow to adjust relative to asset prices (Dornbusch, 1976); some attribute it to the lack of sufficient speculation in the markets for foreign exchange (McKinnon, 1976); some attribute it to the differential effects of new information on commodity and asset markets (Dornbusch, 1979; Frenkel, 1981 a, 1981 b; Frenkel and Mussa, 1980; Mussa, 1979); and some attribute it to the implications of the process by which asset holders restore portfolio balance in the face of disturbances (Kouri, 1976; Calvo and Rodriguez, 1977; Branson, 1979; Ethier, 1979).2

THE EVOLUTION of the international monetary system into a regime of flexible exchange rates and the large volatility of these rates during the 1970s have led to a renewed interest in studying the principal determinants of equilibrium exchange rates. The large fluctuations stimulated theories of exchange rate dynamics and have led to the development of various versions of the overshooting hypothesis.1 The explanations of the overshooting phenomenon vary, but they all rely on the short-run fixity of some nominal quantity. Some account for it by the assumption that in the short run commodity prices are slow to adjust relative to asset prices (Dornbusch, 1976); some attribute it to the lack of sufficient speculation in the markets for foreign exchange (McKinnon, 1976); some attribute it to the differential effects of new information on commodity and asset markets (Dornbusch, 1979; Frenkel, 1981 a, 1981 b; Frenkel and Mussa, 1980; Mussa, 1979); and some attribute it to the implications of the process by which asset holders restore portfolio balance in the face of disturbances (Kouri, 1976; Calvo and Rodriguez, 1977; Branson, 1979; Ethier, 1979).2

In this paper the implications of two classes of models for the dynamics of exchange rates are analyzed. It is assumed throughout that expectations are formed rationally and that individuals are fully informed. Therefore, the possibility that the dynamics of exchange rates are due to systematic expectational errors is precluded.

In Section I, the dynamics of exchange rates in a model in which commodity prices are slow to adjust is analyzed. The basic structure of the model is due to Dornbusch (1976). The extension here allows for a finite rate of capital mobility, and it is shown that the speed of adjustment in capital markets plays a critical role in determining whether or not following a monetary disturbance the exchange rate overshoots its equilibrium value. It is shown that, when the degree of capital mobility is low, the exchange rate is likely to undershoot its equilibrium value.

In Section II, the dynamics of exchange rates within the framework of the portfolio-balance model with complete flexibility of prices is analyzed. In this context, earlier results of Calvo and Rodriguez (1977) concerning the effect of monetary growth on the real exchange rate are interpreted. It is shown that the fundamental factors determining the dynamics of the real exchange rate are the specification of the portfolios of assets, the degree of capital mobility, and the relative qualities of the various assets as hedges against inflation. The study of the dynamics of the real exchange rate shows that throughout the adjustment process the nominal exchange rate is changing at a rate that differs from the rate of inflation. It follows that a policy that ties the exchange rate to the rate of inflation would be inconsistent with the equilibrium self-fulfilling expectations adjustment path.

I. Exchange Rate Dynamics and the Speed of Adjustment in Commodity and Asset Markets

In this section, the dynamics of exchange rates and the overshooting hypothesis are analyzed from the perspective that emphasizes the speeds of adjustment in commodity and asset markets. This perspective was developed by Dornbusch (1976), who assumed that, as a first approximation, asset markets clear instantaneously while adjustment in commodity markets is sluggish. Dornbusch showed that, when these assumptions are coupled with the assumption that expectations are rational, a monetary expansion induces an immediate depreciation of the currency in excess of its long-run equilibrium value; that is, a monetary expansion results in an overshooting of the exchange rate. To gain further understanding of the relationship between the dynamics of exchange rates and the speeds of adjustment in goods and asset markets, the Dornbusch model is modified and allowance has been made for a finite speed of adjustment in asset markets. It is shown that under these circumstances the key factors determining whether the exchange rate overshoots or undershoots its equilibrium value are the relationship between the speed of adjustment in asset markets, the interest elasticity of the demand for money, and the effects of relative prices on the balance of trade. The speed of adjustment in commodity markets does not seem to be a fundamental factor.

The analysis draws on the analytical framework developed by Dornbusch (1976) and, in order to highlight the key point, the model has been simplified by abstracting from many of the inconsequential details. It is assumed that the economy faces a given price of foreign output and a given world rate of interest, that domestically produced goods differ from foreign goods, and that the supply of output is fixed.

the money market

Let the demand for real balances depend on real income, Y, and on the rate of interest, and let the logarithm of the demand be linear in the logarithm of income, y, and in the rate of interest, i. Equilibrium in the money market attains when

mp=φy1bi(1)

where m and p denote, respectively, the logarithms of the nominal quantity of money and the price level. Thus, the equilibrium rate of interest can be written as

i=bφyb(mp)(2)

the goods market

The demand for domestic output, D, is composed of domestic demand and foreign demand. This demand can be expressed as the sum of total domestic absorption (domestic demand for domestic and foreign goods) and the excess of exports over imports (the trade balance surplus). Absorption is assumed to depend on real income, while the trade balance is assumed to depend on the relative price of domestic and foreign goods.3 Total demand for domestic output can therefore be written as

D=A(Y)+T(SP*/P)(3)

where A denotes domestic absorption, T denotes the balance of trade, S denotes the exchange rate (the price of foreign exchange in terms of domestic currency), P* the fixed foreign price level (in terms of foreign currency), and P the price of domestic output. For convenience, units are defined so as to equate the foreign price level to unity; therefore, the trade balance may be viewed as depending on the real exchange rate, s = S/P. Furthermore, since real output is given, absorption is fixed and the demand for domestic output varies with the real exchange rate.

Long-run equilibrium obtains when the demand for domestic output equals the fixed supply, that is, when D = Y. The real exchange rate that is associated with this long-run equilibrium is defined by s. In order to abstract from long-run accumulation of foreign assets, it is assumed that in the long run the trade balance is zero, so that T(s) = 0. This assumption, made for convenience only, implies that absorption equals the given level of output (A(Y) = Y).

Proceeding with the log-linear specification, the trade balance is written as

T=δln(s/s¯)(4)

or equivalently as

T=δ(epk)(5)

where e, p, and k are the logarithms of S, P, and s respectively. Substituting equation (5) into equation (3), and recalling that A(Y) = Y, the demand for domestic output is

D=Y+δ(epk)(6)

The percentage change in the price level, , is assumed to be proportional to excess demand (D – Y):

p˙=π(DY)(7)

where the parameter π measures the speed of adjustment in the goods market.4 Substituting equation (6) into equation (7) yields

p˙=α(epk)(8)

where α = πδ.

At each moment in time, the price level is given, and its evolution is described by equation (8). The coefficient α is the product of two factors: π—the speed of adjustment in the goods market, and δ—the sensitivity of the balance of trade to the real exchange rate. As will be seen below, of these two factors only the latter plays a role in determining whether or not monetary changes result in exchange rate overshooting.

the capital account and the balance of payments

Equilibrium in the world asset market attains when the difference between the rates of interest on domestic and foreign securities, which are identical in all respects except for the currency of denomination, just equals the expected rate of change in the exchange rate. For example, when the domestic currency is expected to depreciate at the percentage rate x, long-run equilibrium requires that

i=i*=x(9)

where i* denotes the rate of interest on assets denominated in foreign currency. It is assumed that expectations concerning the percentage rate of depreciation depend on the relationship between the equilibrium long-run exchange rate, S, and the current rate, S. Expressed logarithmically,

x=θ(e¯e);θ>0(10)

where ē is the logarithm of S and where θ denotes the expectations adjustment coefficient, the determinants of which are analyzed below. Equation (10) states that, when the long-run value, ē, exceeds the current value, e, individuals expect a depreciation of the currency toward ē, that is, the expected depreciation, x, is positive.

The equilibrium that is described in equation (9) is attained through the mechanism of arbitrage that is effected through the international mobility of capital. It is assumed that capital flows are proportional to (ii*x)—the discrepancy between the net rates of return on the various securities. Substituting equation (10) for the expected depreciation of the currency, the international flow of capital can be specified as

C=β{ii*θ(e¯e)}(11)

where C denotes net capital inflow (the surplus in the capital account) and where β denotes the speed of adjustment in asset markets.5 When capital is perfectly mobile, β = ∞, and the mobility of capital ensures that equation (9) holds all the time. At the other extreme, when β = 0, capital is completely immobile, and the mechanism of arbitrage in asset markets is completely inoperative. It is shown below that the magnitude of β is a key factor determining the dynamics of exchange rates.

Equilibrium in the balance of payments in the absence of central bank intervention is attained when the sum of the trade balance and the capital account is zero. Adding equations (5) and (11) and substituting equation (2) for the (equilibrium) domestic rate of interest yields

δ(epk)+β{bφyb(mp)i*θ(e¯e)}=0(12)

as the equilibrium condition for the balance of payments. It should be noted that the equilibrium condition for the balance of payments (which is a flow relationship) is necessary, since the speed of adjustment in asset markets is assumed to be finite. When asset markets clear instantaneously, equation (12) always holds as an identity, and the equilibrium condition is replaced by the interest parity condition as in equation (9) above.6

equilibrium exchange rate, speed of adjustment, and the price level

An analysis of the equilibrium exchange rate and the relationship between the exchange rate, the price level, and the speed of adjustment in the goods and asset markets follows. It is noted, first, that in the long run, given the quantity of money, the exchange rate equals its long-run value7 and i = i*. Substituting i* for i in equation (1)—the condition for money market equilibrium—the long-run price level, p, can be expressed as

p¯=m+1bi*φy(13)

To obtain the relationship between p and ē, it is noted that in the long run excess demand for goods is zero and, therefore, = 0; it follows from equation (8) that

e¯=p¯+k(14)

As may be seen from equations (13) and (14), the system satisfies the homogeneity postulate: a given change in the money supply results in an equiproportional change in the long-run equilibrium price level and the exchange rate. Using equations (13) and (14) in equation (12), the equilibrium in the balance of payments can be written as

δ{(ee¯)(pp¯)}+β{b(pp¯)θ(e¯e)}=0(15)

Equation (15) expresses the various accounts in the balance of payments as functions of the discrepancies between current and long-run values of the price level and the exchange rate. Since at each moment in time the price level is given, equilibrium in the balance of payments is obtained only when the exchange rate is at a level that satisfies equation (15). By collecting terms in equation (15), the equilibrium exchange rate can be written as

e=e¯+ɛ(pp¯)(16)

where ɛ=δβbδ+βθ><0.

Equation (16), which is central to the analysis, relates the equilibrium exchange rate to its long-run value and to the discrepancy between current and long-run prices. This is a reduced form relationship that holds at each moment in time. It is pertinent to note that, depending on the sign of the parameter ε, the relationship between the price level and the exchange rate may be positive or negative. As may be seen, the sign of ε depends on the degree of capital market integration, which has been characterized here in terms of the speed of adjustment β. When the speed of adjustment is low, δ > βb and ε > 0. In that case, given ē and p, the exchange rate and the price level must move in the same direction. When the speed of adjustment is high, δ < βb and the opposite holds. In the extreme case for which β = ∞, the price level and the exchange rate are inversely related, since in that case ε = – (b/θ) < 0.8 The determinants of the relationship between the price level and the exchange rate may be interpreted in terms of equation (12) or equation (15). For given long-run values of prices and exchange rates, a rise in the price level worsens the balance of trade and improves the capital account. The improvement in the capital account results from the rise in the rate of interest necessary to restore money market equilibrium in the face of a higher price level. The extent of the required rise in the rate of interest depends on b—the interest (semi) elasticity of the demand for money. When the speed of asset market adjustment is high, a given rise in the rate of interest results in a large improvement in the capital account, which is likely to more than offset the deterioration in the balance of trade. To restore equilibrium in the balance of payments, the domestic currency will have to appreciate (i.e., e will have to fall). The reduction in e serves to restore equilibrium by worsening the trade balance and by creating expectations of currency depreciation and, thereby, worsening the capital account. In this case, the exchange rate and the price level move in opposite directions. If, on the other hand, the speed of adjustment in asset markets is low (or more precisely, if β is low relative to δ/b), the deterioration of the trade balance following the rise in the price level would outweigh the improvement in the capital account, and balance of payments equilibrium would require a depreciation of the currency, that is, a rise in the exchange rate. In that case, the dynamics of adjustment are characterized by a situation in which prices and exchange rates move in the same direction.

the effect of monetary expansion: overshooting and undershooting

In the previous section the relationship between the exchange rate, the price level, and the speed of asset market adjustment was characterized. This relationship can be illustrated with the aid of Figures 1 and 2, which will then be used to analyze the effects of monetary changes. In these figures, the = 0 schedule shows a combination of exchange rates and price levels for which there is no excess demand for domestic output. The schedule plots equation (8), and its slope is unity.9 The intercept of the schedule corresponds to the (logarithm of the) equilibrium long-run real exchange rate, k. Also, along this schedule trade must be balanced, so that T = 0. All points to the right of the = 0 locus correspond to an excess demand for goods and to a trade balance surplus. Consider next the schedule along which the capital account is balanced. From equation (15), the capital account can be written as

Figure 1.
Figure 1.

High Capital Mobility

Citation: IMF Staff Papers 1982, 001; 10.5089/9781451956634.024.A001

Figure 2.
Figure 2.

Low Capital Mobility

Citation: IMF Staff Papers 1982, 001; 10.5089/9781451956634.024.A001

C=β[b(pp¯)θ(e¯e)]

Thus, the slope of the schedule along which the capital account is balanced (C = 0) is –θ/b < 0, as is shown in Figures 1 and 2. Points below the C = 0 schedule correspond to a deficit in the capital account.

The equilibrium relationship between the price level and the exchange rate (which must hold at each moment in time) is summarized by equation (16), plotted as the QQ schedule in Figures 1 and 2. The slope of the schedule is 1/ε, which may be positive or negative depending on the sign of ε, which in turn depends on whether β is smaller or larger than δ/b. When the degree of capital mobility is low, β < δ/b, and the slope of the schedule is positive and larger than unity since ε < 1. When asset markets adjust relatively fast, the QQ schedule is negatively sloped. Since along the equilibrium path the balance of payments is balanced, the QQ schedule must pass in a region that is characterized by a surplus in the balance of trade and a deficit in the capital account (T > 0, C < 0) or by a deficit in the balance of trade and a surplus in the capital account (T < 0, C > 0). Thus, when the QQ schedule is negatively sloped as in Figure 1, it must be steeper than the C = 0 locus.10

Consider point B in Figures 1 and 2. At this point, there is an excess demand for goods, and > 0. Also at point B there is a trade balance surplus (T > 0) and a capital account deficit (C < 0). The path of adjustment is described by the arrows along the equilibrium schedule QQ, and long-run equilibrium is reached at point D, where prices and exchange rates reach their equilibrium values p1 and ē1. At this point, expected depreciation of the currency is zero, domestic and foreign rates of interest are equalized so that the capital account is balanced and the goods market clear so that p = 0 and the balance of trade is balanced. As is evident, in this model the trade balance and the capital account play a symmetric role in determining the equilibrium exchange rate. It may not be argued, therefore, that the exchange rate is determined exclusively in asset markets and not in the commodity markets. In the extreme case, however, when β = ∞, as in the Dornbusch (1976) model, this dichotomy does exist. The exchange rate is determined at that level that ensures that the rates of return on domestic and foreign securities are equalized at each instant so as to satisfy equation (9). The size of the capital account, in turn, is determined by the balance of trade so as to ensure equilibrium in the balance of payments.

To analyze the effects of a monetary expansion, consider an initial long-run equilibrium at point A with p0 and ē0 as the corresponding price level and exchange rate. Through point A passes a QQ schedule (not drawn) that corresponds to the initial quantity of money. A rise in the money supply raises the long-run equilibrium values of prices and the exchange rate equiproportionally to p1 and ē1 and thus moves the long-run equilibrium combination from point A to point D. The initial QQ schedule moves to the right to the position that is drawn in Figures 1 and 2. Upon the change in the money supply, the price level is given at its initial value p0. Equilibrium in the balance of payments requires that the exchange rate jump immediately to e1 and the short-run equilibrium is attained at point B. As may be seen, when the speed of adjustment in asset markets is relatively high, the exchange rate overshoots its equilibrium long-run value (as in Figure 1); when the speed of adjustment is relatively low, the exchange rate undershoots its long-run equilibrium value (as in Figure 2).

The impact effect of the monetary change can be analyzed in terms of equation (16). By the homogeneity postulate, dm = = dp and thus, given the price level, the short-run elasticity of the exchange rate with respect to the money supply is

dedm=1ɛ(17)

and substituting equation (16) for ε, the elasticity can be written as

dedm=1δβbδ+βθ><1(18)

When the speed of adjustment is relatively high, so that β > δ/b, ε < 0 and the short-run elasticity exceeds unity as in Figure 1. This is the overshooting phenomenon that corresponds to the Dornbusch case. On the other hand, when the speed of adjustment is relatively low, so that β < δ/b, ε is positive but smaller than unity, the short-run elasticity is smaller than unity, and the exchange rate undershoots its new long-run equilibrium value. In the border case for which β = δ/b, the QQ schedule is vertical and the exchange rate reaches immediately its long-run value, as e1 coincides with ē1. It may also be noted from equation (18) that the extent of overshooting or undershooting depends on the magnitude of θ—the speed of adjustment of expectations, the determinants of which are analyzed below. As is clear from equation (18), other things being equal, the short-run elasticity gets closer to unity as the value of θ increases, thereby reducing the extent to which the current exchange rate differs from its long-run value. It is also evident that as long as θ is not negative its magnitude is irrelevant for determining whether the short-run elasticity is larger or smaller than unity, and, therefore, the analysis is consistent with a variety of assumptions concerning the formation of expectations, ranging from the assumption of static expectations (for which θ = 0) to the assumption of perfect foresight, which is analyzed below.

As is evident, the key factor determining whether the exchange rate overshoots or undershoots its long-run equilibrium value is the relationship between β and δ/b. These parameters characterize the speed of adjustment in asset markets, the sensitivity of aggregate demand (the balance of trade) to relative prices, and the interest (semi) elasticity of the demand for money. As long as the speeds of adjustment are finite, the speed of commodity price adjustment, π, does not determine whether or not there is overshooting.

Turning to the effect of the monetary expansion on the international accounts, it is noted that, at point B, independent of whether the exchange rate overshoots or undershoots, the rise in e improves the balance of trade and the rise in m deteriorates the capital account by lowering the rate of interest. The deterioration in the capital account due to the interest rate effect is mitigated when the exchange rate overshoots its long-run equilibrium value, since, in that case, expectations are for an appreciation of the currency (a decline in e). In the case of undershooting, the expectations for a further depreciation reinforce the interest rate effect in deteriorating the capital account. The transition toward the long run (the path between points B and D) is characterized by a rising price level and by a decline in the real exchange rate. The decline in (e – p) results in a deterioration of the balance of trade and, therefore, equilibrium in the balance of payments implies that during the transition the capital account improves.

The above analysis implies that the qualitative characteristics of the dynamics of the price level, the trade balance, the capital account, and the real exchange rate are independent of whether there is overshooting or undershooting of the exchange rate. These given qualitative paths may be associated with a path along which the nominal exchange rate is rising, as well as with a path along which the nominal exchange rate is falling. The ambiguous relationship between the path of the nominal exchange rate and the other paths should not be taken to imply that the exchange rate does not exert a definite effect on the trade balance, the capital account, and the path of prices. Rather, it implies that one should not expect to observe, independent of the speed of adjustment, a unique qualitative relationship between the equilibrium paths of the exchange rate and those of prices and the international accounts. This lack of a unique general relationship between the exchange rate and the various balance of payments accounts may be responsible for the view that in recent years exchange rates have shown erratic and unpredictable movements.11

In this section the effects of a once-and-for-all unanticipated change in the money supply have been analyzed. The analysis can be easily extended to examine the effects of other parametric changes such as changes in output, the foreign price level, and the foreign rate of interest. Similarly, the analysis can be extended along the lines suggested by Wilson (1979) and Gray and Turnovsky (1979) to examine the effects of an anticipated future change in the supply of money or in another parameter. For example, it can be shown that an anticipated future rise in the money supply induces an immediate adjustment of the exchange rate, which jumps, for example, to point E in Figures 1 and 2. The extent of the instantaneous jump in the exchange rate is smaller than the change that would have taken place had the money supply been expected to rise at the present (in which case the exchange rate would have adjusted to point B). Following the initial jump in e, both the exchange rate and prices proceed to rise gradually, and their path converges to the new QQ schedule (corresponding to the new quantity of money) at the point in time at which the rise in the money supply actually occurs. Thereafter, the convergence proceeds along the (new) QQ schedule toward the new long-run equilibrium.12

perfect foresight and the coefficient of expectations

So far it has been assumed (in equation (10)) that the expected percentage change in the exchange rate is proportional to the discrepancy (ē—e), with θ > 0 being the proportionality factor. Dornbusch (1976) showed that in a model of perfect foresight the coefficient θ cannot be chosen arbitrarily but, rather, that it must be consistent with the structure of the entire model. An analysis of the determinants of θ follows.

Using equation (14) in equation (8), the rate of inflation may be expressed as

p˙=α{(ee¯)(pp¯)}(19)

and using equation (16), the rate of inflation can be written as

p˙=α(11ɛ)(ee¯)(20)

The relationship between the equilibrium exchange rate and the price level that is described by equation (16) must always be satisfied. Therefore, given the long-run values of ē and p, changes in the exchange rate and in the price level must be related such that

e˙=ɛp˙(21)

Substituting equation (20) for yields

e˙=α(1ɛ)(e¯e)(22)

Equation (22) describes the actual change in the exchange rate, while equation (10) describes the expected change. It is clear that, under perfect foresight, consistency requires that the two be equal and, therefore,

θ=α(1ɛ)(23)

Substituting for ε from equation (16) results in

θ=α{1δβbδ+βθ}(24)

from which it follows that the coefficient of expectations can be obtained by solving the quadratic equation

θ2+(πδαββ)θαβ=0(25)

From equation (25), the solution for 0 (obtained by taking the positive root) is

θ=12[(δαββ){(δαββ)2+4αb}1/2]>0(26)

which expresses the coefficient of expectations as a function of the various parameters of the model.

As may be verified, the coefficient of expectations decreases with 1/b, the interest (semi) elasticity of the demand for money, while it increases with π and β, the speeds of adjustment in the goods and asset markets, respectively. These propositions are independent of the degree of capital mobility. In contrast, however, it is noteworthy that the dependence of the coefficient of expectations on the parameter δ (the sensitivity of aggregate demand to relative prices) is ambiguous and depends on whether πβ—the product of the speeds of adjustment in the goods and asset markets—is smaller or larger than a critical value that is equal to 1/(1 + b/θ). For a given value of π, a high value of β, such that πβ exceeds the critical value, yields a positive relationship between the coefficient of expectations and the parameter δ. On the other hand, when β is small, such that πβ is smaller than the critical value, a high value of δ reduces the size of the coefficient of expectations. Since the relationship between the speed of adjustment of expectations and δ is ambiguous, it follows that the effect of a higher value of δ on the extent of the overshooting or undershooting is also ambiguous. Finally, it might be noted that in the extreme case for which asset markets clear instantaneously, so that β = ∞, equation (24) becomes

θ=α(1+b/θ)(27)

which corresponds to equation (14) in Dornbusch (1976, p. 1167) and, as noted above, in that case δ and θ are positively related.

The dynamics of exchange rates within a rational expectations model in which commodity prices adjust slowly has been analyzed in this section. The analysis was based on a modified version of a model due to Dornbusch (1976). The model was modified to allow for a finite speed of adjustment in asset markets, and it was shown that the short-run effects of a monetary expansion depend on the degree of capital mobility. When capital is highly mobile the exchange rate must overshoot its long-run value, 13 but when capital is relatively immobile the exchange rate undershoots its long-run value. The key reason for the nonneutrality of money in the short run in this model arises from the assumption that prices are not fully flexible. The dynamics of exchange rates in a portfolio-balance model in which all prices are assumed to be fully flexible also in the short run is analyzed in the next section.

II. Exchange Rate Dynamics and Portfolio Balance

The development of a variety of portfolio-balance models that emphasize the effects of asset substitution on exchange rate determination are the subject of numerous recent articles (e.g., Kouri (1976), Branson (1979), Dornbusch (1979), Girton and Roper (1981), Frenkel and Clements (1981), and the references therein). The purpose here is to highlight the implications of alternative assumptions concerning the degree of substitution among assets on the dynamics of exchange rates. It will be shown that the effects of a monetary expansion on the dynamics of exchange rates and in particular on whether exchange rates overshoot or undershoot their equilibrium path depend critically on the specification of asset choice. To contrast the analysis with the one in the previous section, it will be assumed that goods markets clear instantaneously and that all prices are perfectly flexible.

the currency substitution model

The Calvo-Rodriguez (1977) model of currency substitution analyzes a fully employed small, open economy in which residents are assumed to hold portfolios of domestic and foreign currencies. The key building blocks of the model are the specifications of the markets for assets and goods.

The Asset Market

Asset holders are assumed to hold portfolios of domestic money, M, and foreign money, F. The value of assets in terms of foreign currency is denoted by a:

a=M+F(28)

where M' denotes the value of domestic currency holdings in terms of foreign exchange, that is, M'M/S.

The desired ratio of domestic to foreign money holdings is assumed to depend on the expected percentage change in the exchange rate, which measures the expected difference between the rates of return on the two assets. The assumption of rational expectations (which in this model amounts to perfect foresight) permits us to identify the expected change in the exchange rate with the actual change. Using a circumflex (^) to denote the percentage change in a variable, portfolio equilibrium can be written as

MF=L(S^);L<0(29)

which indicates that the desired ratio of domestic money to foreign money declines when the domestic currency is expected to depreciate. It will prove useful to express the portfolio-balance relationship in terms of its inverse:

S^=l(M/F);l<0(30)

The Goods Market

The economy is assumed to produce two classes of goods, tradable and nontradable. For a given state of technology and factor endowment, the rates of production of the two composite commodities depend on their relative price. Denoting the domestic currency price of traded goods by PT and of nontraded goods by PN, the relative price that is relevant for production decisions is PT/PN. The domestic price of traded goods is linked to the foreign price of that good, PT*, through international arbitrage, so that PT = SPT*. The small country is assumed to face a given foreign price for traded goods, which, for convenience, is normalized to unity. Thus, the relative price which governs the allocation of productive resources can be written as sS/PN—the “real exchange rate”—and the output of the two goods can be specified as

QT=QT(s);QT/s>0QN=QN(s);QN/s<0(31)

where QT and QN denote, respectively, the output of traded and nontraded goods.

The demand for the two goods is assumed to depend on their relative price and on the value of assets according to

CT=CT(s,a);CT/s<0,CT/a>0CN=CN(s,a);CN/s>0,CN/a>0(32)

where CT and CN denote, respectively, the demand for traded and nontraded goods.

At each point in time the stock of domestic holdings of foreign assets, F, is given, and the assumption that the small country’s currency is not held by foreigners ensures that F cannot be adjusted instantaneously. Asset holders can, however, alter the stock of foreign assets gradually by running a surplus or a deficit in the balance of trade. Thus,

QT(s)CT(s,a)=F˙(33)

where dF/dt denotes the rate of change in F. Equilibrium in the market for nontraded goods requires that the rate of domestic production equal the domestic demand, so that

QN(s)CN(s,a)=0(34)

Equation (34) implies that there is a specific relationship between the real exchange rate and the value of assets that is consistent with equilibrium in the market for nontraded goods. A rise in the value of assets must be accompanied by a decline in the real exchange rate, since the former creates an excess demand for nontraded goods while the latter induces an excess supply. This relationship is summarized by

s=s(a);s<0(35)

Since equation (35) must hold at each moment in time because of the assumed flexibility of prices, it can be substituted into equation (33) to yield a relationship between the rate of change in F and the value of assets:

F˙=f(a);f(QTsCTs)sCTa<0(36)

The value of assets is uniquely related to the equilibrium real exchange rate and thereby to the rates of production and consumption of goods and to the rate of accumulation of foreign currency. Therefore, knowledge of the time path of assets is necessary for determining the time path of these variables.

Dynamics

Changes in asset holdings arise from changes in the domestic and the foreign asset components of the portfolio:

a˙=M˙+F˙(37)

Recalling that M'M/S, it follows that ' = M'(μ - Ŝ), where μ denotes the percentage change in the nominal money supply, that is, μ ≡ /M. Since M' = a – F, the change in M' can be expressed as ' = (a – F) (μ – Ŝ). Thus, equation (37) can be written as

a˙=(aF){μl(aFF)}+f(a)(38)

where, using equations (30) and (36), l(aFF) was substituted for Ŝ and f(a) was substituted for .

Equations (36) and (38) characterize the dynamics of the system. As is clear, in the steady state, when = ȧ = 0, f(a) = 0 (equation (36)) and μ = Ŝ (equation (38)). The steady-state values of a and F are denoted by ā and F and, using equation (35), the implied steady-state real exchange rate is denoted by s. It is also noteworthy that the system satisfies the homogeneity postulate: a once-and-for-all rise in the nominal quantity of money results in an instantaneous equiproportional rise in the money price of non-traded goods and in the nominal exchange rate (and thereby in the money price of traded goods). These changes leave all real variables (including the real exchange rate) unchanged. In contrast with the discussion in Section I, where due to the assumed slow price adjustment a once-and-for-all change in the nominal quantity of money exerted real effects (by inducing in the short run a rise in the real quantity of money), the flexibility of prices assumed in the present section rules out such effects. In the present model, however, changes in the percentage rate of growth of the money supply are not neutral in the long run and do result in a gradual transition period.

Figure 3 describes the dynamics of the system. In the panel on the right, the = 0 and the ȧ = 0 schedules describe combinations of a and F that satisfy equations (36) and (38), respectively. The slope of the ȧ = 0 schedule is drawn on the assumption that (around the steady state) ȧ/∂a < 0. As is evident, the system exhibits a saddle-path stability, and the motion of the variables is described by the arrows that are implied by the signs of the partial derivatives of equations (36) and (38) around the steady state. None of the qualitative conclusions is altered in the case for which ȧ/∂a > 0. In that case, the ȧ = 0 schedule is positively sloped and is steeper than the saddle path. In general, along the perfect-foresight path (which is the unique path that converges to the steady state and satisfies the laws of motion and the initial conditions) a higher value of F is associated with a higher value of a. The panel on the left in Figure 3 presents the combinations of the value of assets and the real exchange rate that satisfy equation (35). These are the combinations that are consistent with equilibrium in the nontraded goods market for which a rise in the value of assets must be associated with a decline in s.

Figure 3.
Figure 3.

Exchange Rate Dynamics in the Currency Substitution Model

Citation: IMF Staff Papers 1982, 001; 10.5089/9781451956634.024.A001

Consider a rise in the rate of monetary expansion. From equation (38) it is seen that the higher value of μ shifts the ȧ = 0 schedule, and hence shifts the saddle path to the right, and results in new, higher steady-state holdings of foreign assets. In the new steady state, the real value of assets remains unchanged so as to ensure that = 0 (equation (36)). Since the equilibrium real exchange rate is uniquely related to the value of assets, also the steady-state real exchange rate remains unchanged and, thus, production and consumption of both goods remain unchanged. Finally, since in the new steady state the new rate of monetary expansion equals the rate of depreciation, Ŝ, individuals will lower the desired ratio of domestic to foreign currency holdings, which, given the unchanged value of assets, implies that F rises and M' falls.

The dynamics of adjustment are shown in Figure 3, where it is assumed that the initial steady-state position was at point A, with F0, ā0, and s0 as the initial equilibrium values of foreign currency holdings, total assets, and the real exchange rate, respectively. The schedules that are drawn correspond to the new higher rate of monetary expansion. Upon the rise in μ, the instantaneous equilibrium jumps to point B along the new saddle path. Since F cannot change instantaneously, point B is the only position of short-run equilibrium that is consistent with the perfect-foresight path that converges to the new steady state. At point B, the value of assets falls from ā0 to a1. This decline in the value of assets is necessary, since the rise in the expected relative cost of holding domestic money induces a reduction in the desired ratio of domestic to foreign monies, which, given the initial value of foreign currency holdings, F0, can be brought about only by a decline in M' and thus in a. Since at the initial point in time the nominal stock of domestic money, M, is given, the decline in M'M/S is brought about through a rise in the nominal exchange rate, S (i.e., through a depreciation of the domestic currency). Finally, as indicated in the panel on the left in Figure 3, equilibrium in the nontraded goods market implies that when the value of assets falls to a1 the real exchange rate rises to s1. Since the real exchange rate is the ratio of the nominal exchange rate, S, to the price of non-traded goods, PN, it follows that S^>P^N. If the aggregate price level is a weighted average of S and PN, it follows that the exchange rate changes by more than the overall price level. This is the overshooting phenomenon in the Calvo-Rodriguez (1977) model.

The transition toward the new steady state is characterized by the path between B and C, along which s declines (so that S^<P^N), and the value of assets rises as does the value of foreign currency holdings. The model implies that during the transition period a higher depreciation of the currency is associated with a trade balance surplus and an accumulation of foreign assets. This characteristic is typical of the currency substitution model, reflecting the desired change in the composition of assets.14

The key feature of the Calvo-Rodriguez model is the specification of the portfolio of assets, where it is assumed that the two alternative assets are domestic and foreign currencies. Since the foreign currency price of traded goods is assumed to be given, the accumulation of foreign currency is equivalent to an accumulation of claims on stocks of traded goods. With this perspective, the currency substitution model can be specified in terms of choice and substitution between domestic money and traded goods. It is intuitively clear, therefore, that following the rise in the rate of monetary expansion asset holders wish to hedge against the expected inflation by shifting the composition of portfolios toward the inflation hedge (traded goods), a shift which results, in the short run, in a rise in s—the relative price of traded goods.

This interpretation of the currency substitution model suggests that, when the inflation hedges are stocks of nontraded goods instead of traded goods, the rise in the rate of monetary expansion might result in an undershooting of the exchange rate as the relative price of nontraded goods rises. A formal analysis of this possibility follows.

nontraded goods as inflation hedge

The structure of the model in which nontraded goods are used as the inflation hedge is very similar to the one described in the currency substitution model. In what follows the minimal modifications are introduced.

The Asset Market

To sharpen the contrast with the currency substitution model, it is assumed that the portfolios of assets are composed of domestic money and stocks of nontraded goods, N. The value of assets in terms of foreign exchange is

a=M+N/s(39)

where N/s = PNN/S. As before, the desired ratio of money to inventories of goods is assumed to depend on the difference between their rates of return, that is, on the expected percentage change in the price of nontraded goods. Assuming that expectations are always realized, the desired portfolio composition depends on P^N, and analogously to equation (30) the portfolio-balance relationship can be written as

P^N=g(MN/s);g<0(40)

The Goods Market

The specification of production and consumption is assumed to be the same as in the previous section and is summarized by equations (31) and (32). In the present case, however, an allowance is made for an accumulation of inventories of nontraded goods that must equal the excess of production over consumption of these goods.

QN(s)CN(s,a)=N˙(41)

where N denotes the rate of accumulation of nontraded goods.

Since in this case no allowance is made for capital mobility, equilibrium in the market for traded goods requires that

QT(s)CT(s,a)=0(42)

equation (42) implies that there is a specific relationship between the real exchange rate and the value of assets that guarantees trade balance equilibrium. A rise in the value of assets raises the demand for traded goods and, therefore, to eliminate the excess demand, it must be accompanied by a rise in the real exchange rate, s. This relationship, which must be satisfied at all times, can be written as

s=v(a);v<0(43)

For later use, it is noted that

s^=γa^(44)

where γ denotes the elasticity of the real exchange rate with respect to the value of assets. Substituting equation (43) into equation (41) yields a relationship between the rate of accumulation of nontraded goods and the value of assets:

N˙=h(a);h(QNsCNs)vCNa<0(45)

Dynamics

The rate of change in the value of assets can be written, using equation (39), as

a˙=MM^+(N/s)(N/^s)(46)

Noting that PN=M(M/S)(S/PN), it follows that P^N=μM^s^, and using equation (40) yields

M^=μg(v(a)aN1)γa^(47)

where (ν(a)a/N) – 1 was substituted for M'/(N/s) by using equations (39) and (43) and where γâ was substituted for ŝ by using equation (44).

Using equations (45) and (47) in equation (46) yields

a˙=11+γ{(aNs){μg(v(a)aN1)}+h(a)v(a)}(48)

Equations (45) and (48) characterize the dynamics of the system. In the steady state, = ȧ = 0, the value of assets is ā0, the holdings of nontraded goods are N0, and the rate of monetary expansion, μ, equals g(.)—the rate of change of the price of nontraded goods, P^N. Equilibrium in the balance of trade implies that ā0 is associated with a specific steady-state real exchange rate, s0. Since in the steady state the percentage change in the nominal price of nontraded goods equals the rate of monetary expansion, and since the real exchange rate is given at s0, the percentage change in the nominal exchange rate, Ŝ, must also equal the rate of monetary expansion.

The dynamics of the system are described in Figure 4. In the panel on the right, the = 0 and the ȧ = 0 schedules show combinations of a and N that satisfy equations (45) and (48), respectively. The slope of the ȧ = 0 schedule is drawn on the assumption that around the steady state ȧ/∂a < 0. As in the currency substitution model, the system exhibits a saddle-path stability and the motion of the variables is described by the arrows, which reflect the signs of the partial derivatives of equations (45) and (48) around the steady state. When ȧ/∂a > 0, the ȧ = 0 schedule is positively sloped and is steeper than the saddle path. As before, none of the qualitative conclusions are altered in that case. The key difference between the analysis in Figure 4 and that in Figure 3 lies in the panel on the left, which shows the equilibrium relationship between the real exchange rate and the value of assets. In contrast with the currency substitution model, here s and a must be positively related so as to ensure trade balance equilibrium (equation (42)).

Figure 4.
Figure 4.

Exchange Rate Dynamics When Portfolios Contain Money and Nontraded Goods.

Citation: IMF Staff Papers 1982, 001; 10.5089/9781451956634.024.A001

A rise in the rate of monetary expansion results (from equation (48)) in a rightward shift of the ȧ = 0 schedule and hence of the saddle path. The new steady state is associated with larger holdings of nontraded goods, whose attractiveness has risen with the accelerated inflation. Suppose that the initial equilibrium is described in Figure 4 by point A and that the schedules drawn correspond to the new higher rate of monetary expansion. Since at the initial point in time the stocks of nontraded goods are given at N0, the instantaneous equilibrium is reached at point B with a lower value of assets, a1. The reduced value of assets lowers the demand for traded goods and, therefore, to restore trade balance equilibrium, the real exchange rate must fall to s1 as indicated in the panel on the left.

The higher expected rate of inflation lowers the desired ratio of money to inventories of nontraded goods, MN/s=M/PNN. At the initial moment the quantities of M and N are given, and the desired ratio is attained, therefore, through a rise in PN. Since, however, the value of assets declined and since N/s rose, it follows that the nominal exchange rate, S, must have risen (i.e., the domestic currency must have depreciated) so as to yield a decline in M'M/S. The rise in S must be sufficiently large so as to more than offset the effect of the rise in N/s on the value of assets. While both the nominal exchange rate, S, and the price of nontraded goods, PN, jump upon the rise in the rate of monetary expansion, the fact that the real exchange rate falls to s1 implies that P^N > Ŝ. Thus, in this model the exchange rate undershoots the price level.

The transition toward the steady state is described by the path from B to C, along which the value of assets and the holdings of nontraded goods increase and the real exchange rate rises toward its initial level. Since along the path s rises, the nominal exchange rate must rise faster than the price of nontraded goods.

The fact that the rise in the rate of monetary expansion results in an instantaneous rise in the price level that exceeds the rise in the depreciation of the currency is consistent with the general principles that were outlined at the end of the discussion of the currency substitution model. In this class of the portfolio-balance model, the key factor determining whether following a monetary disturbance the exchange rate overshoots or undershoots the domestic price level is the specification of the inflation hedge. When the domestic rate of inflation accelerates, asset holders substitute away from domestic money into alternative assets. To the extent that the alternative assets are traded goods, their relative price will rise and the nominal exchange rate will overshoot the domestic price level. To the extent that the alternative assets are domestic nontraded goods, their relative price will rise and the exchange rate will undershoot the domestic price.

In this section the two polar cases in which the substitutes for domestic money were either foreign currency (or equivalently traded goods) or stocks of domestic nontraded goods were analyzed. A more general model would allow for portfolios that consist of domestic and foreign monies as well as of domestic goods. The general principles, however, are likely to be the same. Whether exchange rates overshoot or undershoot the domestic price level will depend on the relative degree of substitution among domestic money, traded goods, and nontraded goods. The relative degrees of substitution will reflect the degree of capital mobility and the qualities of the various assets as hedges against inflation.

III. Concluding Remarks

This paper has analyzed the determinants of the evolution of exchange rates within the context of alternative models of exchange rate dynamics. It has examined the overshooting hypothesis in models that emphasize differential speeds of adjustment in asset and goods markets as well as in models that emphasize portfolio-balance considerations. It has been seen that exchange rate overshooting is not an intrinsic characteristic of the foreign exchange market and that it depends on a set of specific assumptions. It has been shown that the overshooting is not a characteristic of the assumption of perfect foresight, nor does it depend in general on the assumption that goods and asset markets clear at different speeds. As long as the speeds of adjustment in the various markets are less than infinite (which is obviously the case), the key factor determining the short-run effects of a monetary expansion is the degree of capital mobility. When capital is highly mobile, the exchange rate overshoots its long-run value, and when capital is relatively immobile, the exchange rate undershoots its long-run value.

Within the context of the portfolio-balance model, it has been shown that the effects of a monetary expansion on the dynamics of exchange rates and in particular on whether exchange rates overshoot or undershoot their equilibrium path depend critically on the specification of asset choice, on the degree of substitution among assets, and on the quality of the various assets as a hedge against inflation.

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*

Mr. Frenkel is a Professor of Economics at the University of Chicago, an editor of the Journal of Political Economy, and a Research Associate of the National Bureau of Economic Research. He is a graduate of the Hebrew University of Jerusalem and of the University of Chicago.

Mr. Rodriguez is a Professor of Economics at the Center for Macroeconomic Studies, Buenos Aires, and was a member of the faculty of Columbia University. He is a graduate of the University of Chicago.

This paper is a revised version of an earlier paper written in the fall of 1977 and entitled “The Anatomy of the Overshooting Hypothesis in the Market for Foreign Exchange.” The authors are indebted to Joshua Aizenman, Kent Kim-brough, and Michael Mussa, and to participants of the joint international economics workshop of Tel Aviv University and the Hebrew University of Jerusalem for useful comments. Mr. Frenkel acknowledges research support from the National Science Foundation, grant SOC 78-14480. This research is part of the National Bureau of Economic Research’s Program in International Studies.

Work on this paper was done while the authors were consultants with the Fund. The views expressed are not necessarily those of the sponsoring organizations.

1

For a classification of the various versions of the overshooting hypothesis, see Levich (1981).

2

In addition to these studies, a partial list of recent studies includes Bilson (1979), Boyer and Hodrick (1982), Flood (1979), Henderson (1980), Kimbrough (1980), Liviatan (1981), Mussa (1982), and Shafer (1980). Further references are included in the following sections of this paper.

3

This specification abstracts from the effects of the rate of interest on absorption, the effects of income on the trade balance, and the distinction between gross national product and gross domestic product that results from interest income on net ownership of foreign securities. These abstractions are made for simplicity, since they do not affect the nature of the argument. Allowance is made for some of these factors by Dornbusch (1976).

4

The assumption that prices change at a finite speed is taken as a stylized fact and there is no attempt to rationalize it here. For an interesting theory of sticky prices, see Mussa (1981). The specification of the rate of inflation as a function of excess demand is analogous to the specification in Dornbusch (1976), equation (8).

5

The theoretical deficiencies of the capital flow equation (equation (11)) are well known. Since the purpose here is to highlight the role of alternative assumptions concerning the speed of adjustment in asset markets, it has been chosen to specify the simplest formulation and to abstract from many other issues.

6

For a discussion of the role of flow equilibrium when asset markets adjust slowly, see Kouri (1976) and Niehans (1977). For a discussion of the conceptual issues involved in the formulation of flow equilibrium in asset markets, see Mussa (1976).

7

In the present framework, the long-run value of the exchange rate is fixed. In general, the equilibrium value may be specified in terms of a movement along an equilibrium path; see Mussa (1982).

8

The case for which β = ∞ is the Dornbusch case. equation (16) here coincides with equation (6) in Dornbusch (1976, p. 1164).

9

The = 0 schedule would be flatter than a 45° line if the demand for goods were to depend negatively on the rate of interest as in Dornbusch (1976). In that case, a rise in the price level lowers the demand via the higher relative price and via the rate of interest that must be higher in order to restore money market equilibrium.

10

The two schedules tend to coincide in the limit, when the magnitude of β tends to infinity.

11

The relationship between the exchange rate and the trade balance is analyzed in detail in many recent contributions. See, for example, Kouri (1976), Dornbusch and Fischer (1980), Mussa (1980), and Rodriguez (1980).

12

It follows that when capital is highly mobile, so that the QQ schedule is negatively sloped, the path of prices will be monotonic while the path of the exchange rate will exhibit a turning point, that is, e will initially rise and then decline; when capital is less mobile, so that the QQ schedule is positively sloped, both prices and the exchange rate will approach their new, higher values monotonically.

13

This conclusion is based on the assumption that output is fixed. Dornbusch (1976) shows that, when the monetary expansion induces a short-run rise in output, the exchange rate may undershoot its equilibrium level even if capital is highly mobile. This will occur if the rise in output raises the demand for money sufficiently so as to result in a rise in the rate of interest.

14

This characteristic may not be necessary in a model with intertemporal utility maximization; see Liviatan (1981).

IMF Staff papers: Volume 29 No. 1
Author: International Monetary Fund. Research Dept.