🌐 Open-Economy Macroeconomics & Mundell-Fleming

In an open global economy, national economies interact through international trade in goods/services (Current Account) and international capital flows in financial assets (Financial Account).

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1. 💱 Foreign Exchange & Interest Parity Conditions

• Nominal Exchange Rate ($e$): Domestic currency price of foreign currency ($e=\text{USD/EUR}$ or direct/indirect quotations).
• Real Exchange Rate ($ϵ$): Relative price of foreign goods in terms of domestic goods:$$ϵ=\frac{e\cdot P^{\ast }}{P}$$A real depreciation ($ϵ\uparrow$) makes domestic goods relatively cheaper, boosting exports ($X\uparrow$) and reducing imports ($M\downarrow$) if the Marshall-Lerner condition holds.

1.1 Uncovered Interest Parity (UIP)

Under risk-neutral international investors and perfect capital mobility:

$$1+i_t=(1+i_t^{\ast })\frac{\mathbb{E}[e_{t+1}]}{e_t}⟹i_t\approx i_t^{\ast }+\frac{\mathbb{E}[e_{t+1}]-e_t}{e_t}$$

The domestic interest rate equals the foreign interest rate plus expected currency depreciation.

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2. 🌊 The Mundell-Fleming Model

The open-economy extension of the IS-LM framework incorporates the Balance of Payments ($BP$) curve:

$$IS:Y=C(Y-T)+I(i)+G+NX(Y,Y^{\ast },ϵ)$$$$LM:\frac{M}{P}=L(Y,i)$$$$BP:BP=NX(Y,ϵ)+CF(i-i^{\ast })=0$$

Under Perfect Capital Mobility, the $BP$ curve is perfectly horizontal at $i=i^{\ast }$.

  Interest Rate (i)
    ▲
    │                 IS             LM
 i* │──────────────────•──────────────•──── BP = 0 (i = i*)
    │                 / \            /
    │                /   \          /
    └───────────────┴─────┴────────┴──────► Real Output (Y)
                   Y*

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3. ⚖️ Policy Effectiveness Matrix

Exchange Rate Regime	Fiscal Policy ($\mathrm{\Delta }G>0$)	Monetary Policy ($\mathrm{\Delta }M>0$)
Floating Exchange Rates	Completely Ineffective ($\mathrm{\Delta }Y=0$) IS shifts right $⟹i\uparrow ⟹$ capital inflow $⟹$ currency appreciates ($e\downarrow$) $⟹NX\downarrow$ (100% exchange rate crowding out!).	Highly Effective ($\mathrm{\Delta }Y>0$) LM shifts right $⟹i\downarrow ⟹$ capital flight $⟹$ currency depreciates ($e\uparrow$) $⟹NX\uparrow ⟹$ IS shifts right!
Fixed Exchange Rates	Highly Effective ($\mathrm{\Delta }Y>0$) IS shifts right $⟹i\uparrow ⟹$ appreciation pressure $⟹$ central bank buys FX reserves & expands money supply $⟹$ LM shifts right!	Completely Ineffective ($\mathrm{\Delta }Y=0$) Central bank cannot set independent money supply; attempts to cut rates trigger reserve depletion.

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4. 🔺 The Policy Trilemma (The Impossible Trinity)

A sovereign nation cannot simultaneously maintain all three policy objectives:

                            Independent Monetary Policy
                                       ▲
                                      / \
                                     /   \
                         Floating   /     \  Capital Controls
                                   /  TRI- \  (e.g., China)
                                  /  LEMMA  \
                                 /           \
                                /             \
   Free Capital Mobility ◄─────────────────────► Fixed Exchange Rate
                                Currency Peg
                               (e.g., Eurozone)

1. Option A (USA, UK, Japan): Free Capital Mobility + Independent Monetary Policy $⟹$ Floating Exchange Rate.
2. Option B (Eurozone members, Hong Kong): Free Capital Mobility + Fixed Exchange Rate $⟹$ Abandon Monetary Independence.
3. Option C (Bretton Woods, China historically): Fixed Exchange Rate + Independent Monetary Policy $⟹$ Impose Capital Controls.

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5. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Mundell-Fleming Floating Exchange Rate Expansion

Problem: A small open economy operates under floating exchange rates and perfect capital mobility ($i=i^{\ast }=5\mathrm{\%}$).

• $C=100+0.8(Y-T)$, $I=300-1000i$, $G=200$, $T=100$.
• $NX=150-0.1Y-20e$.
• Money Market: $M/P=500$, $L(Y,i)=0.5Y-2000i$.

1. Calculate the equilibrium output level $Y^{\ast }$ and equilibrium nominal exchange rate $e^{\ast }$.
2. Calculate the impact of a fiscal expansion $\mathrm{\Delta }G=100$ on $Y^{\ast }$ and $e^{\ast }$.

Step-by-Step Rigorous Solution:

1. Solve for $Y^{\ast }$ from the Money Market ($i=i^{\ast }=0.05$):

   $$\frac{M}{P}=0.5Y-2000i⟹500=0.5Y-2000(0.05)=0.5Y-100$$$$0.5Y=600⟹Y^{\ast }=\mathrm{\$}1200$$

2. Solve for $e^{\ast }$ from the Goods Market:

   $$Y=C+I+G+NX$$$$1200=100+0.8(1200-100)+[300-1000(0.05)]+200+[150-0.1(1200)-20e]$$$$1200=100+0.8(1100)+[300-50]+200+[150-120-20e]$$$$1200=100+880+250+200+30-20e=1460-20e$$$$20e=1460-1200=260⟹e^{\ast }=13.0$$

3. Impact of Fiscal Expansion ($\mathrm{\Delta }G=100⟹G^{\prime }=300$):

   • In money market: $Y^{\ast }$ remains pinned at $Y^{\ast }=\mathrm{\$}1200$ because $M/P$ and $i^{\ast }$ are unchanged. $\mathrm{\Delta }Y=0$.
   • In goods market:$$1200=1460+100-20e^{\prime }=1560-20e^{\prime }⟹20e^{\prime }=360⟹e^{\prime }=18.0$$

   Economic Finding: Output does not change ($\mathrm{\Delta }Y=0$); currency appreciates sharply ($\mathrm{\Delta }e=+5.0$), crowding out net exports by exactly $\mathrm{\Delta }NX=-20(5)=-\mathrm{\$}100$.