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Lesson M20.L05: Exchange Rate Overshooting: The Dornbusch Model

Module: Open Economy Macroeconomics Part I Level: intermediate Duration: 30 minutes Learning Objective: Explain how the exchange rate overshoots its long-run equilibrium following a monetary expansion. Data as of: 2024 Provenance: RBA Exchange Rate Research | IMF World Economic Outlook

Explanation

In a simple purchasing power parity (PPP) world, a 10% increase in the money supply should eventually lead to a 10% rise in the price level and a 10% depreciation of the nominal exchange rate. Prices and the exchange rate move proportionally. But real-world exchange rates are far more volatile than price levels. Why?

Dornbusch (1976) answered this with the overshooting model, based on one key asymmetry: goods prices are sticky in the short run; asset prices (including the exchange rate) are fully flexible immediately.

The mechanism:

Consider a permanent increase in the money supply (Ī”M/M = 10%).

Long run (PPP holds): - Price level rises 10%: Ī”P/P = 10% - Real money supply M/P returns to original: M/P unchanged - Exchange rate depreciates 10%: Ī”e/e = 10% (PPP: same real exchange rate) - Domestic interest rate returns to world rate: i = i*

Short run (prices sticky: P fixed): - M rises but P is fixed ā†’ real money supply M/P rises - Money market: M/Pā†‘ ā†’ interest rate i falls below i* to restore money market equilibrium - Uncovered interest parity (UIP): i < i* requires expected future appreciation of domestic currency to compensate investors - Expected appreciation means current exchange rate must overshoot (depreciate more than 10%) so that it can appreciate back to the long-run level

Why the overshoot? The exchange rate must depreciate enough in the short run that investors are willing to hold domestic assets despite their lower yield (i < i*). The only way this works is if they expect future appreciation (i.e., the current rate is "too depreciated" relative to long run). Therefore:

  • Long-run depreciation: 10%
  • Short-run depreciation: more than 10% (overshooting)

Over time, prices gradually rise (sticky adjustment), real money supply falls back, i rises back to i*, and the exchange rate appreciates from its overshooting trough toward the new long-run level.

UIP condition (key equation):

\[i = i^* + \frac{\dot{e}^e}{e}\]

where \(\dot{e}^e/e\) = expected rate of depreciation. If i < i*, then \(\dot{e}^e/e < 0\) (expected appreciation) ā†’ current e must be above its long-run value (i.e., the currency has overshot into depreciation territory).

Australian evidence. The 2014ā€“15 iron ore price crash saw AUD fall from ~USD 0.95 (early 2013) to ~USD 0.70 (early 2016) ā€” a 26% depreciation. PPP estimates suggested the AUD's fair value around USD 0.75ā€“0.80. The additional 5ā€“10% depreciation below fair value is consistent with overshooting. During 2022, the AUD fell to ~USD 0.63 despite strong commodity prices ā€” partly due to USD strength (global dollar tightening) and overshooting of USD.

Notation: - e = nominal exchange rate (units of AUD per USD, or similar; higher = more depreciated domestic currency) - e_LR = long-run equilibrium exchange rate - e_SR = short-run exchange rate (overshooting position) - M = money supply; P = price level; M/P = real money supply - i = domestic interest rate; i* = world interest rate - \(\dot{e}^e/e\) = expected rate of change of exchange rate (UIP)

Worked Example

Question: The Australian money supply increases permanently by 10%. Show algebraically: (a) The long-run effect on the exchange rate (PPP). (b) Why the short-run exchange rate must overshoot. (c) Quantify the overshoot if the interest semi-elasticity of money demand is Ī» = 5 (meaning a 1pp fall in i requires a 5% rise in M/P to restore money market equilibrium).

Step (a) ā€” Long-run PPP effect.

PPP: P = eP* (where P* = foreign price level, assumed constant)

In the long run, neutrality of money: prices rise proportionally to M:

\[\frac{\Delta P}{P} = \frac{\Delta M}{M} = 10\%\]

By PPP: \(\frac{\Delta e}{e} = \frac{\Delta P}{P} - \frac{\Delta P^*}{P^*} = 10\% - 0\% = \mathbf{10\%}\)

The exchange rate depreciates by 10% in the long run.

Step (b) ā€” Short-run overshooting logic.

In the short run, P is fixed. The money supply rises 10%, so real money M/P rises 10%.

Money market equilibrium requires i to fall to absorb the higher real money supply. Let the fall in i be Ī”i.

Using UIP: for investors to hold domestic assets at lower i, they need expected future appreciation:

\[\Delta i = \frac{\dot{e}^e}{e} < 0\]

meaning current e must have depreciated more than its long-run value, so it can appreciate back.

Let the long-run exchange rate = e_LR (depreciated 10%). If current e = e_SR > e_LR (more depreciated), then investors expect appreciation: \(\dot{e}^e/e = (e_{LR} - e_{SR})/e_{SR} < 0\).

Step (c) ā€” Quantify the overshoot.

Money market equilibrium in the short run (with P fixed, M/P rises 10%):

By the interest semi-elasticity: a 10% rise in M/P requires i to fall by:

\[\Delta i = -\frac{\Delta(M/P)/ļ¼ˆM/P)}{\lambda} = -\frac{10\%}{5} = -2\text{pp}\]

So i falls by 2 percentage points below i*.

UIP then requires: expected appreciation = Ī”i = 2pp per period.

If the speed of adjustment means the exchange rate fully adjusts back to e_LR over 5 periods, the annual expected appreciation rate = 2pp. The initial overshoot must be large enough that 2pp of appreciation per period brings e back to e_LR.

If e_LR represents 10% depreciation from the original rate (call it eā‚€ = 1.0, so e_LR = 1.10), then:

\[e_{SR} = e_{LR} \times \left(1 + \lambda \cdot |\Delta i|\right) = 1.10 \times (1 + 5 \times 0.02) = 1.10 \times 1.10 = 1.21\]

Short-run overshoot: 21% depreciation vs. long-run 10% depreciation.

\[\text{Overshoot} = e_{SR} - e_{LR} = 1.21 - 1.10 = 0.11 \quad (11\text{pp above long-run})\]

The exchange rate overshoots its long-run value by 11 percentage points ā€” more than doubling the long-run impact in the short run.

Common Misconception

Misconception: "If a country's money supply rises 10%, the exchange rate depreciates by exactly 10% ā€” no more, no less."

Correction: This holds only in the long run under PPP, where prices fully adjust. In the Dornbusch model, the short-run exchange rate overshoots because goods prices are sticky. The exchange rate must do all the short-run adjustment that prices cannot ā€” it absorbs the full shock immediately. Only gradually, as prices rise to their new long-run level, does the exchange rate appreciate back toward the PPP-consistent long-run value. Empirically, this explains why exchange rates are far more volatile than price levels ā€” a well-documented stylised fact that PPP alone cannot explain.

Practice Prompts

  1. Conceptual: In the Dornbusch model, why must the exchange rate appreciate over time after the initial overshoot? What condition guarantees this path?

ā†’ Answer: After overshooting, the exchange rate is "too depreciated" relative to its long-run PPP value. The domestic interest rate is below the world rate (i < i*). Uncovered interest parity requires that investors receive compensation for holding low-yielding domestic assets ā€” this compensation comes in the form of expected appreciation of the domestic currency. The UIP condition i āˆ’ i* = expected depreciation rate implies expected depreciation < 0 (i.e., expected appreciation). As goods prices gradually rise (sticky adjustment), real money supply falls, i rises back toward i*, and the expected appreciation decreases. The exchange rate path is a smooth appreciation from e_SR back to e_LR ā€” consistent with UIP throughout the adjustment.

  1. Numerical: The RBA increases M by 8% permanently. Long-run depreciation = 8% (PPP). Interest semi-elasticity Ī» = 4. Calculate: (a) the fall in i (pp); (b) the short-run exchange rate overshoot percentage; (c) total short-run depreciation.

ā†’ Answer: - (a) Ī”i = āˆ’(Ī”M/M)/Ī» = āˆ’8%/4 = āˆ’2pp (i falls by 2 percentage points) - (b) Overshoot = Ī» Ɨ |Ī”i| Ɨ e_LR factor = 4 Ɨ 2% = 8% above e_LR - (c) Total short-run depreciation = long-run depreciation + overshoot = 8% + 8% = 16%

The exchange rate depreciates 16% in the short run, then appreciates 8% back to the long-run 8% depreciation as prices adjust.

  1. Application: During 2014ā€“15, the AUD fell from ~0.95 to ~0.70 USD (a 26% depreciation). Long-run PPP analysts estimated fair value around USD 0.78. How much of the total depreciation might be attributed to overshooting, and what drove it?

ā†’ Answer: If the PPP/long-run value fell to ~0.78 (a 18% depreciation from 0.95), then the actual rate of 0.70 represents an additional 8 percentage points of overshoot below fair value (0.78 ā†’ 0.70 = ~10% further depreciation). This overshoot is consistent with Dornbusch: the iron ore price crash reduced Australian terms of trade ā†’ investors expected lower AUD long-run value ā†’ capital outflows ā†’ i differential (RBA held rates above the Fed, but expectations changed) ā†’ overshoot. Additional factors: commodity-currency re-rating, global risk-off following Chinese growth fears, and carry-trade unwinding all amplified the depreciation beyond PPP predictions. The AUD then partially recovered toward 0.76 by 2016ā€“17, consistent with the Dornbusch appreciation path back toward long-run equilibrium.

Further Resources