The longrun e¤ects in the model are derived in Section 3.1. Thereafter, in Section 3.2, the exchange rate overshooting phenomenon is investigated. Specifically, we will examine whether the exchange rate “overshoot the overshooting equilibrium”, i.e., if the magnitude of exchange rate overshooting in the present model is larger than in the Dornbusch (1976) model?

The adjustment path to longrun equilibrium, when market expectations are characterized by perfect foresight, is derived in Section 3.3, and the perfect foresight planning horizon and the perfect foresight overshooting effect[6] are scrutinized in Section 3.4. Since the longperiod moving average in (7)( 8) is a function of all past exchange rates, the model is not easy to analyze formally. But by assuming that the economy has, for a long time, been in longrun equilibrium before a monetary disturbance occurs, the moving average in (7)( 8) is (approximately) equal to the longrun equilibrium exchange rate:

(10)

The assumption in (10) simplify the analysis considerably, but will be relaxed in Section 3.5, where a small simulation study is accomplished in order to illustrate the behavior of the model.

Longrun equilibrium

Since it is assumed in this section that the exchange rate is in longrun equilibrium,

(11)

Substituting (10) (assuming equality in the equation) and (11) into the expectations formation in (6), the expectations formed by technical analysis become

(12)

i.e., it is expected that the exchange rate is in longrun equilibrium. Moreover, substituting (11) into the expectations formation in (9), the expectations formed by fundamental analysis become

(13)

i.e., it is expected that the exchange rate is in longrun equilibrium. Then, substitute the expectations formations in (12)( 13) into market expectations in (4):

(14)

i.e., the market expects that the exchange rate is in longrun equilibrium. Recall that the results in (12)( 14) are based on the assumption that the economy has, for a long time, been in longrun equilibrium before a monetary disturbance occurs.

(The logarithm of) the relative price level[7] in longrun equilibrium, p, can be solved for by using the equations that describe the money and the international asset markets in equilibrium, i.e., (1)( 2):

(15)

since, according to (11) and (14),

(16)

Thus, the quantity theory of money holds in the longrun since, according to (15),

(17)

Moreover, since (16) as well as

(18)

hold in longrun equilibrium, the price adjustment mechanism in (3) reduces to

(19)

Thus, purchasing power parity holds in the longrun since, according to (19),

(20)

Finally, the quantity theory of money and purchasing power parity, i.e., (17) and (20), implies that

(21)

which is the longrun e¤ect on the exchange rate of a change in money supply. It should be stressed that the quantity theory of money and purchasing power parity results are not dependent on the simplifying assumption that the economy has, for a long time, been in longrun equilibrium before a monetary disturbance occurs. But even if the quantity theory of money and purchasing power parity hold in the longrun, there are shortrun deviations from these onetoone relationships. This is demonstrated in the next section on exchange rate overshooting.

Exchange rate overshooting

Using (10) (assuming equality in the equation) in the expectations formation in (6), and substituting the resulting equation as well as the expectations formation in (9) into market expectations in (4), we have that

(22)

Then, combine the equations that describe the money and the international asset markets in equilibrium, i.e., (1)( 2), and substitute market expectations in (22) into the resulting equation:

(23)

Di¤erentiating (23) with respect to m[t], s [t] and s gives

(24) or, if (21) is substituted into (24),

(25)

The current price level is held constant when deriving (25) since it is assumed to be sticky. Thus, (25) is the shortrun e¤ect on the exchange rate, near longrun equilibrium, of a change in money supply. A sticky price level also means that the quantity theory of money, i.e., (17), does not hold in the shortrun since the price level is not a¤ected by a monetary disturbance. Moreover, purchasing power parity, i.e., (20), does not either hold in the shortrun since the exchange rate is a¤ected by a monetary disturbance while the price level is not. In order to have exchange rate overshooting, it must be true that

(26)

which means that the planning horizon must satisfy

(27)

where the weight function in (5) is utilized in the derivation. Thus, in the shortrun, before goods prices have time to react, the exchange rate will rise (fall) more than money supply, and, consequently, more than is necessary to bring the exchange rate to longrun equilibrium. It will turn out in the next two sections that (27) is also the stability condition for the model when it is assumed that market expectations are characterized by perfect foresight.

By letting τ --> ∞ in (25), an equation describing exchange rate overshooting that corresponds to Dornbusch (1976) is obtained:

(28)

(28) corresponds to Dornbusch (1976) since, by letting τ --> ∞ in (4)( 5), market expectations coincide with the expectations formed by fundamental analysis. In this case, the magnitude of exchange rate overshooting depends on the nominal interest rate response of real money demand (α), and the expected adjustment speed of the exchange rate according to fundamental analysis (δ).

Moreover, the magnitude of exchange rate overshooting depends inversely on the planning horizon:

(29)

i.e., for shorter planning horizons, more weight is placed on technical analysis, and since technical analysis is a destabilizing force[8] in the foreign exchange market, the extent of exchange rate overshooting depends inversely on the planning horizon. This also means that the magnitude of overshooting is even larger in this model than in the Dornbusch (1976) model:

(30)

i.e., the exchange rate “overshoots the overshooting equilibrium”. Finally, the magnitude of exchange rate overshooting depends on the structural parameters α, β, γ and δ in the following way[9] :

(31)

if exchange rate overshooting is assumed, i.e., if it is assumed that (27) holds,

Thus, the extent of exchange rate overshooting is larger, the less sensitive real money demand is to changes in the nominal interest rate (α), the faster the expected adjustment speed of the exchange rate is according to technical analysis (γ), and the slower the expected adjustment speed of the exchange rate is according to fundamental analysis (δ). The magnitude of exchange rate overshooting is not a¤ected by changes in the degree of stickiness of goods prices (β).

---

6 That is, the planning horizon and the overshooting e¤ect when market expectations are characterized by perfect foresight.

7 Henceforth, it will not be emphasized that a macroeconomic variable is expressing the di¤erence between, for example, the domestic and foreign price levels.

8 Technical analysis is a destabilizing force since chartists expect that the exchange rate will diverge from longrun equilibrium. To see this, substitute (10) into (6).

9 The planning horizon in currency trade (¿ ) is given below in (31)( 34), but is endogenously determined when market expectations are characterized by perfect foresight. See the corresponding equations (69)( 72) below.

By Mikael Bask and Carina Selander

Next: Adjustment path to longrun equilibrium

Summary: Index