Overshooting model

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The Dornbusch Overshooting Model

First established by economist Rudi Dornbusch, the key insight of the model is that lags in some parts of the economy can induce additional volatility in others to compensate. Dornbusch wrote this back when many economists were still claiming that ideal markets should reach equilibrium and stay there. Volatility was thought to be a consequence of imperfect information or market obstacles. But it's not; volatility is more fundamental than that.

The Dornbusch Overshooting Model aims to explain why exchange rates have a high variance. In simple terms, the model begins by observing prices on goods are 'sticky' in the short run, while 'prices' in the financial markets adjust to disturbances quickly. When a change in monetary policy (for example an increase in the money supply) occurs, the market will need to move to a new equilibrium between prices and quantities. Because goods have sticky prices, an initial new equilibrium will probably be achieved through shifts in financial market prices. Then, over time, the goods prices will unstick and shift to a new equilibrium, which will allow the financial market prices to shift back toward (but not all the way back to) the original levels, and a new long-run equilibrium will be attained between money, finance, and the supply of goods. In other words, the financial market will initially overreact to a change in money in order to achieve a new short-term equilibrium, but over time goods prices will also respond, allowing the financial market to dissipate its overreaction and the economy as a whole will settle on a new long-run equilibrium.

[edit] 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 competitivness 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 adject 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 will be expected to go down (that is the price of the currency will appreciate).

Assumption 3:	In the short run, goods prices are 'sticky'. That is, aggregrate supply is horizontal in the sort 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 cournty	s: exchange rate
Δs^e: 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 suppy/demand	p: price index	k: constant term
l: constant term	y^d: 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)(proportinal change in prces with respect to time) ɗP/ɗTime

from the above we can derive the folling (using algerbaic substution)

[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 nly 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 inflaction 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 then 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 the rewrite the basic eqations [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 beingin a position that implies that it was initally overshot. From the assumptions above, it is pobbile 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 S2.