Expectations and Exchange Rate Dynamics

Dornbusch sets out to “develop a theory that is suggestive of the observed large fluctuations in exchange rates while at the same time establishing that such exchange rate movements are consistent with rational expectations formation” (p. 1161, Abstract; p. 1162). The model combines three ingredients: perfect capital mobility (so interest parity holds continuously), a goods market that adjusts only slowly relative to asset markets, and “consistent” (perfect-foresight) expectations – and Dornbusch is explicit that “the dynamic aspects of exchange rate determination in this model arise from the assumption that exchange rates and asset markets adjust fast relative to goods markets” (p. 1162).

Interest parity is imposed as r = r + x, where r is the domestic interest rate, r the given world rate, and x the expected rate of currency depreciation** (Eq. 1, p. 1163), assumed to hold at all times via incipient capital flows. Expected depreciation is assumed proportional to the gap between the long-run exchange rate ē and the current spot rate e: x = θ(ē − e) (Eq. 2, p. 1163) – a rule Dornbusch flags as apparently ad hoc but shows in Part III to be exactly consistent with perfect foresight for one specific value of θ.

The domestic interest rate is determined by money-market equilibrium, with a conventional log-linear money demand equation in real income and the interest rate (Eq. 3, p. 1163); combining this with interest parity and the expectations rule gives a key relationship between the spot exchange rate, the current price level, and the long-run exchange rate (Eq. 4, 6, pp. 1163-1164). Because a stationary money supply implies equal current and expected exchange rates in the long run (so interest rates equalize internationally), the long-run price level satisfies p̄ = m + (λr − θy)*, letting the model separate current dynamics from the long-run, money-neutral equilibrium (Eq. 5, p. 1164).

Demand for domestic output depends on the relative price of domestic goods (e − p), interest rates, and income, and price inflation is assumed proportional to this excess-demand measure (Eq. 7-8, p. 1164), which together imply a long-run equilibrium exchange rate ē that depends, “with the conventional homogeneity properties,” on monetary variables, but also on real variables (Eq. 9, p. 1165). Combining the price-adjustment equation with the money-market relationship yields an exponential convergence path for prices, p(t) = p̄ + (p₀ − p̄) exp(−νt), and for the exchange rate, e(t) = ē + (e₀ − ē) exp(−νt), where ν is a convergence speed determined by the model’s structural parameters (Eqs. 10-13, pp. 1165-1166).

Since the actual rate at which prices and the exchange rate converge to equilibrium is ν, “for the expectations in (2) to correctly predict the actual path of exchange rates it must be true that θ = ν” (p. 1167); Dornbusch solves the resulting quadratic for the unique “consistent expectations coefficient” θ as an explicit function of the interest-elasticity of money demand, the price- and income-elasticities of goods demand, and the price-adjustment speed (Eq. 14-15, p. 1167). He justifies focusing on this perfect-foresight path because “it is the only expectational assumption that is not arbitrary (given the model) and that does not involve persistent prediction errors… the deterministic equivalent of rational expectations” (p. 1167, n. 10).

At every point in time the exchange rate adjusts instantaneously to keep the economy on the QQ schedule (money-market equilibrium plus interest parity), which is positively sloped and flatter than a 45-degree line; goods-market equilibrium, shown by a separate positively sloped ṗ = 0 schedule, is reached only in the long run (p. 1165-1166, Fig. 1). Starting from a point like B, with prices below their long-run level and the exchange rate correspondingly above its long-run level, there is excess demand for domestic goods; prices rise over time while the exchange rate appreciates along the QQ schedule, until the economy reaches the intersection of both schedules at point A, where goods and asset markets both clear and expected depreciation is zero (p. 1166-1167).

Dornbusch derives de/dm = 1 + 1/(λθ) for the impact effect of a monetary expansion on the spot exchange rate under fixed output, a value exceeding 1 – confirming that “in the short run the exchange rate will overshoot” (Eq. 16, p. 1169). The mechanism: at the initial price level, the monetary expansion lowers domestic interest rates, and interest parity then requires an expectation of subsequent currency appreciation large enough to compensate lenders for the lower domestic rate; since expected appreciation is proportional to the gap between the spot and long-run rates, the spot rate must overshoot its long-run depreciation by exactly enough to generate that expectation (pp. 1168-1169). Substituting the rational-expectations solution for θ into this expression gives the overshooting magnitude purely in terms of the economy’s structural parameters (Eq. 17, p. 1170).

A higher interest-elasticity of money demand dampens overshooting, since it means a given real money expansion produces only a small interest-rate decline, which in turn requires only a small expected appreciation (and hence a small excess depreciation) to be offset; more generally, any factor that speeds up the economy’s convergence to long-run equilibrium – high interest-responsiveness of money or goods demand, or high price-elasticity of demand for domestic output – dampens the impact overshooting (pp. 1169-1170). In the limiting case where price adjustment becomes instantaneous, the economy jumps immediately to the new long-run equilibrium with no overshooting at all (p. 1170, n. 12).

As the economy moves from the short-run overshooting point B back toward long-run equilibrium at C, rising domestic prices reduce real money balances, which raises domestic interest rates; by interest parity, rising domestic interest rates must be accompanied by the expectation (and reality) of an appreciating exchange rate to keep expected net yields equalized internationally, so “the model therefore confirms the link between interest rates and exchange rates that is emphasized in popular interpretations of foreign exchange events” – but the mechanism runs through appreciation accompanying rising, not falling, interest rates during this adjustment phase, and Dornbusch stresses this trend behavior of exchange rates “stands in strong contrast” to their cyclical, overshooting-driven short-run behavior (pp. 1162, 1170-1171).

Replacing the fixed-output goods-market equation with a demand-determined output equation and an Okun’s-law/Phillips-curve-style price adjustment equation (Part V, Eqs. 18-19), Dornbusch shows the qualitative exponential-convergence structure survives, but a monetary expansion now also raises short-run output, and the resulting extra money demand can be strong enough to actually raise, rather than lower, domestic interest rates (pp. 1171-1173). The condition determining whether the exchange rate depreciates more or less than proportionately to the money supply – and correspondingly whether interest rates fall or rise – is given explicitly in terms of income- and price-elasticities of money and goods demand (Eq. 20, p. 1173); when it fails, exchange rate overshooting is “no longer a necessary feature of the adjustment process,” and terms-of-trade fluctuations are correspondingly dampened relative to the fixed-output case.

Dornbusch concludes that the exchange rate is “a critical channel for the transmission of monetary policy to aggregate demand for domestic output,” potentially the only such channel in the variable-output extension, since interest rates may actually rise rather than fall during the transition (pp. 1173-1174). He is explicit that the paper’s key mechanism – slow goods-price adjustment relative to fast asset-market adjustment – “lacks… very persuasive theoretical support, but the facts clearly point in this direction,” and that while sticky prices are assumed rather than derived, the resulting behavior of exchange rates is nonetheless “suggestive of recent experience” (p. 1174). He also flags as a direction for future work that the analysis is entirely deterministic (a perfect-foresight path), and that “an extension of this paper would draw in an explicit manner on stochastic elements to provide a rationale for the short-run stickiness of prices,” which would also have implications for how expectations are formed (p. 1175, n. 14).