Overshooting model

The Overshooting Model or Exchange rate overshooting, first developed by economist Rudi Dornbusch, aims to explain why exchange rates have a high variance. A key element of the model is that expectations of exchange rate changes are "consistent" — that is, rational — rather than static. The most important insight of the model is that adjustment lags in some parts of the economy can induce compensating volatility in others; specifically, when an exogenous variable changes, the short-run effect on the exchange rate can be greater than the long-run effect, so that in the short run the exchange rate overshoots its new long-run value.

Dornbusch developed this model back when many economists held the view that ideal markets should reach equilibrium and stay there. Volatility in a market, from this perspective, could only be a consequence of imperfect information or adjustment obstacles in that market. Rejecting this view, Dornbusch argued that volatility is in fact a far more fundamental property than that. His model assumes that, while the prices of goods are sticky (slow to adjust), the exchange rate adjusts instantaneously to any change in current financial market conditions. Yet the stickiness of goods prices causes evolution over time in the goods market, and due to market linkages this leads to evolution over time in the foreign exchange market equilibrium. This results in variability in the exchange rate even after a shock to that market has gone away — hence the exchange rate volatility.

For example, when a change in monetary policy occurs (e.g. an increase in the money supply), the market will adjust to a new equilibrium between prices and quantities. Initially, because of the 'stickiness' of prices of goods, the new equilibrium level will first be achieved through shifts in financial market prices. Then, gradually, as prices of goods 'unstick' and shift to the new equilibrium, the foreign exchange market continuously reprices, approaching its new long-term equilibrium level. Only after this process has run its course will a new long-run equilibrium be attained in the domestic money market, the currency exchange market, and the goods market.

In other words, the foreign exchange market will initially overreact to a monetary change, achieving a new short-term equilibrium. Over time, goods prices will eventually respond, allowing the foreign exchange market to dissipate its overreaction, and the economy to land in a new long-run equilibrium pricing structure.

Outline of the model

Assumption 1:	Aggregate demand is determined by the standard open economy IS-LM mechanism

That is to say, the position of the IS curve is determined by the volume of injections into the flow of income and by the competitiveness of Home country output measured by the real exchange rate.

The first assumption is essentially saying that the IS curve (demand for goods) position is in some way dependent on the real effective exchange rate Q.

That is [ IS = C + I + G +Nx(Q)] -> In this case, net exports is dependent on Q (as Q goes up, foreign countries goods are relatively more expensive, and home countries goods are cheaper, therefore there is higher net exports).

Assumption 2:	Financial Markets are able to adjust to shocks instantaneously, and investors are risk neutral.

If financial markets can adjust instantaneously and investors are risk neutral, we can say the uncovered interest rate parity (UIRP) holds at all times. That is, the equation r = r* + Δs^e holds at all times (explanation of this formula is below).

It is clear, then, that an expected depreciation/appreciation offsets any current difference in the exchange rate. If r > r*, the exchange rate (domestic price of one unit of foreign currency) is expected to increase. That is, the domestic currency depreciates relative to the foreign currency.

Assumption 3:	In the short run, goods prices are 'sticky'. That is, aggregate supply is horizontal in the short run, though it is positively sloped in the long run.

In the long run, the exchange rate (s) will equal the long run equilibrium exchange rate,(ŝ).

r: interest rate in home country	r*:interest rate in foreign country	s: exchange rate
Δse: expected change in exchange rate	θ: coefficient reflecting the sensitivity of market participant to the (proportionate) overvaluation/undervaluation of the currency relative to equilibrium.	ŝ: long-run expected exchange rate
m: money supply/demand	p: price index	k: constant term
l: constant term	yd: demand for home country output	h: constant
q: real exchange rate	þ: change in prices with respect to time	π: prices
ŷ: long-run demand for home country output (constant)

Formal Notation

[1] r = r* +Δse (uncovered interest rate parity - approximation)

[2] Δs^e = θ(ŝ – s) (Expectations of market participants)

[3] m - p = ky-lr (Demand/Supply on money)

[4] y^d = h(s-p) = h(q) (demand for the home country output)

[5] þ = π(yd- ŷ)(proportional change in prices with respect to time) dP/dTime

from the above we can derive the following (using algebraic substitution)

[6] p = a - lθ(ŝ - s)

[7] þ = π[h(s-p) - ŷ]

In equilibrium

y^d = ŷ (demand for output equals the long run demand for output)

from this we substitute getting [8] ŷ/h = ŝ - p_hat That is in the long run, the only variable that affects the real exchange rate is growth in capacity output)

Also, Δs^e = 0 (that is, in the long run the expected change of inflection is equal to zero)

when we substitute into [2], we get r = r* sub that into [6] and we get

[9] p_hat = m -kŷ + l r*

taking [8] & [9] together, we get:

[10] ŝ = ŷ(h^-1 - k) + m +lr*

comparing [9] & [10], we can see that the only difference between them is the intercept (that is the slope of both is the same). This tells us that given a rise in money stock pushes up the long run values of both in equally proportional measures, the real exchange rate (q) must remain at he same value as it was before the market shock. Therefore, the properties of the model at the beginning are preserved in long run equilibrium, the original equilibrium we had was stable.

Short run disequilibrium

The standard approach is to rewrite the basic equations [6] & [7] in terms of the deviation from the long run equilibrium). In equilibrium [7] implies 0 = π[h(ŝ-p_hat) - ŷ] If we subtract this from [7] we get

[11] þ = π[h(q-q_hat) The rate of exchange is positive whenever the real exchange rate is above its equilibrium level, also it is moving towards the equilibrium level] - This gives us the direction and movement of the exchange rate

In equilibrium, [9] hold, that is [6] - [9] is the difference from equilibrium.

[12] p - p_hat = -lθ(s-ŝ) This gives us the line upon which the exchange rate must be moving (the line with slope -lθ.

Both [11] & [12] together allow us to demonstrate that the exchange rate will be moving towards the long run equilibrium exchange rate, whilst being in a position that implies that it was initially overshot. From the assumptions above, it is possible to derive the following situation. This demonstrated the overshooting and subsequent readjustment. In the graph on the top left, So is the initial long run equilibrium, S1 is the long run equilibrium after the injection of extra money and S2 is where the exchange rate initially jumps to (thus overshooting). When this overshoot takes place, it begins to move back to the new long run equilibrium S1.

References

Rudiger Dornbusch (1976). "Expectations and Exchange Rate Dynamics". Journal of Political Economy 84 (6): 1161–1176. doi:10.1086/260506. 

• "Dornbusch's Overshooting Model After Twenty-Five Years", 2001 analysis by Kenneth Rogoff, International Monetary Fund.

• Romer, David. Advanced Macroeconomics. Third Edition. p234-236