💱 Foreign Exchange, Balance of Payments & Crises

Global financial stability depends on the interplay between national currency exchange rates, cross-border capital flows, and balance-of-payments equilibrium.

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1. 🏛️ Balance of Payments (BOP) Accounting

The Balance of Payments records all economic transactions between domestic residents and the rest of the world:

$$\text{Current Account (CA)}+\text{Financial Account (FA)}+\text{Capital Account (KA)}+\mathrm{\Delta }\text{FX Reserves}\equiv 0$$

1. Current Account ($CA$): Trade Balance ($NX=X-M$) + Net Primary Income (foreign earnings/remittances) + Net Secondary Income (foreign aid).
2. Financial Account ($FA$): Net acquisition of foreign financial assets minus net incurrence of liabilities (FDI, portfolio equity/debt).
3. Official Reserve Settlement ($\mathrm{\Delta }FX$): Central bank intervention in foreign exchange markets.

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2. 💵 Purchasing Power Parity (PPP)

• Law of One Price (LOOP): In the absence of trade barriers and transport costs, identical tradeable goods must sell for identical prices worldwide when converted to a common currency:$$P_i=e\cdot P_i^{\ast }$$
• Absolute PPP: The general price level of a standardized basket of goods is equal across countries:$$e_{PPP}=\frac{P}{P^{\ast }}$$
• Relative PPP: The percentage change in the exchange rate equals the domestic-foreign inflation differential:$$\frac{\mathrm{\Delta }e}{e}\approx \pi -{\pi }^{\ast }$$

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3. 📉 The Marshall-Lerner Condition & J-Curve

When a nation devalues/depreciates its currency, the nominal price of imports rises immediately, while export volumes take time to expand due to contractual rigidities.

       Trade Balance (NX)
               ▲
               │                     J-CURVE DYNAMICS
               │                                      .--- Long-Run Surplus
          NX_0 │──────────────────────.              /
               │                       \            /
          0 ───┼────────────────────────\──────────/────────► Time (t)
               │    Currency             \________/
               │   Depreciation         Short-Run Deficit
               │

3.1 Marshall-Lerner Condition

A real currency depreciation improves the trade balance ($dNX/de>0$) if and only if the sum of price elasticities of demand for exports (${\eta }_x$) and imports (${\eta }_m$) exceeds unity:

$${\eta }_x+{\eta }_m>1$$

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4. ⚡ Generations of Currency Crisis Models

1. First-Generation Models (Krugman 1979): Inconsistent fundamentals. A government runs persistent fiscal deficits financed by domestic credit creation under a fixed exchange rate peg. Central bank foreign exchange reserves steadily deplete until speculative attack forces sudden abandonment of the peg.
2. Second-Generation Models (Obstfeld 1994): Self-fulfilling expectations. Multiple equilibria occur where government faces a cost-benefit tradeoff (e.g. defending the peg requires high interest rates that worsen domestic unemployment). If speculators attack, the cost of defense becomes unbearable, triggering devaluation.
3. Third-Generation Models (Asian Crisis 1997): Balance-sheet mismatches (currency and maturity mismatches in private banking sectors) causing sudden stops in capital flows.

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5. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Relative PPP and Exchange Rate Forecasting

Problem: The current spot exchange rate is $e_0=1.20\text{ USD/EUR}$. Expected annual inflation in the US is ${\pi }_{US}=5.0\mathrm{\%}$, and expected annual inflation in the Eurozone is ${\pi }_{EU}=2.0\mathrm{\%}$.

1. Using Relative PPP, calculate the expected spot exchange rate $e_1$ after 1 year and 3 years.
2. If nominal 1-year US Treasury bond yields are $i_{US}=6.0\mathrm{\%}$, calculate the equilibrium 1-year European government bond yield $i_{EU}$ according to Uncovered Interest Parity (UIP).

Step-by-Step Rigorous Solution:

1. Calculate Exchange Rate using Relative PPP:

   $$\frac{e_1}{e_0}=\frac{1+{\pi }_{US}}{1+{\pi }_{EU}}=\frac{1.05}{1.02}\approx 1.02941$$$$e_1=1.20\times 1.02941=1.2353\text{ USD/EUR}$$

   After 3 years ($t=3$):

   $$e_3=1.20\times {\left(\frac{1.05}{1.02}\right)}^3=1.20\times (1.0909)\approx 1.309\text{ USD/EUR}$$

   Result: Higher US inflation causes the US Dollar to depreciate relative to the Euro.

2. Calculate European Bond Yield using UIP:

   $$1+i_{US}=(1+i_{EU})\frac{e_1}{e_0}⟹1.06=(1+i_{EU})(1.02941)$$$$1+i_{EU}=\frac{1.06}{1.02941}\approx 1.0297⟹i_{EU}=2.97\mathrm{\%}\approx 3.0\mathrm{\%}$$

   Economic Finding: The nominal interest rate differential ($i_{US}-i_{EU}=6\mathrm{\%}-3\mathrm{\%}=3\mathrm{\%}$) exactly mirrors the inflation differential (${\pi }_{US}-{\pi }_{EU}=3\mathrm{\%}$), confirming Fisher effect equalization in real rates.