# Open Economy with Perfect Capital Mobility: IS-LM-BP Analysis Given: - **Consumption:** \( C = 200 + .6(Y-T) - 20 \) - **Investment:** \( I = 300 - 40r \) - **Taxes:** \( T = 20 + .2Y \) - **Government spending:** \( G = 152 \) - **Net exports:** \( NX = 150 - .2Y - 20r \) - **Real money demand:** \( L = .5Y - 10r \) - **Real money supply:** \( M/P = 250 \) - **World interest rate:** \( r^* = 5 \) (percent, but use as 5, not .05) - **Perfect capital mobility:** Domestic interest rate \( r = r^* \) --- ## a. Open Economy Goods Market Equilibrium (IS Curve) ### Step 1: Write the Equilibrium Condition Goods market equilibrium: \[ Y = C + I + G + NX \] Substitute the given functions: \[ C = 200 + .6(Y-T) - 20 \] \[ T = 20 + .2Y \] So, \[ C = 200 + .6(Y - (20 + .2Y)) - 20 = 200 + .6Y - .6 \times 20 - .6 \times .2Y - 20 \] \[ = 200 + .6Y - 12 - .12Y - 20 = (200 - 12 - 20) + (.6Y - .12Y) \] \[ = 168 + .48Y \] Now substitute all components: - \( I = 300 - 40r \) - \( G = 152 \) - \( NX = 150 - .2Y - 20r \) So, \[ Y = (168 + .48Y) + (300 - 40r) + 152 + (150 - .2Y - 20r) \] ### Step 2: Simplify the Equilibrium Equation Combine like terms: - Constants: \(168 + 300 + 152 + 150 = 770\) - \(Y\) terms: \(.48Y - .2Y = .28Y\) - \(r\) terms: \(-40r - 20r = -60r\) So: \[ Y = 770 + .28Y - 60r \] \[ Y - .28Y = 770 - 60r \] \[ .72Y = 770 - 60r \] \[ Y = \frac{770 - 60r}{.72} \] **This is the IS curve equation.** --- ### Step 3: IS Curve Equation \[ \boxed{Y = \frac{770 - 60r}{.72}} \] **Graphically:** The IS curve is downward sloping in the (Y, r) space: as \( r \) increases, \( Y \) decreases. --- ## b. IS-LM Model and Short-Run Equilibrium ### Step 1: LM Curve Equation LM: \( \frac{M}{P} = L \) \[ 250 = .5Y - 10r \] \[ .5Y = 250 + 10r \] \[ Y = 500 + 20r \] --- ### Step 2: Perfect Capital Mobility (\( r = r^* = 5 \)) Since capital is perfectly mobile, \( r = r^* = 5 \). #### IS Curve at \( r = 5 \): \[ Y_{IS} = \frac{770 - 60 \times 5}{.72} = \frac{770 - 300}{.72} = \frac{470}{.72} \approx 652.78 \] #### LM Curve at \( r = 5 \): \[ Y_{LM} = 500 + 20 \times 5 = 500 + 100 = 600 \] --- ### Step 3: BP (Balance of Payments) Condition and Exchange Rate Movement #### BP Curve (Under Perfect Capital Mobility) With perfect capital mobility and flexible exchange rate, BP= when \( r = r^* \). But at \( r=5 \), the IS and LM curves do **not** intersect at the same Y: - IS: \( Y_{IS} \approx 652.78 \) - LM: \( Y_{LM} = 600 \) At \( r = 5 \), \( Y_{IS} > Y_{LM} \): - **Excess demand** in goods market at \( Y_{LM} \), or **excess supply** of money at \( Y_{IS} \). - The exchange rate will **depreciate** to restore equilibrium via NX. --- ### Step 4: Adjustment to BP= (IS-LM-BP Equilibrium) The IS curve must shift **rightward** (increase in NX) to raise \( Y \) to the LM equilibrium value. #### Required Change in Net Exports (\( \Delta NX \)) At \( r=5 \), LM says \( Y = 600 \). Plug \( Y = 600, r = 5 \) into the original IS curve without the change in NX: \[ Y = 168 + .48Y + (300 - 40r) + 152 + [150 - .2Y - 20r] + \Delta NX \] But let's use the original IS in the form: \[ Y = 770 + .28Y - 60r + \Delta NX \] At \( Y = 600, r = 5 \): \[ 600 = 770 + .28 \times 600 - 60 \times 5 + \Delta NX \] \[ 600 = 770 + 168 - 300 + \Delta NX \] \[ 600 = 638 + \Delta NX \] \[ \Delta NX = 600 - 638 = -38 \] So, net exports must **decrease by 38** (i.e., \( \Delta NX = -38 \)), but this is counterintuitive—normally, depreciation increases NX. Let's verify. #### Alternatively, set IS = LM at \( r=5 \): From above: - IS: \( Y = \frac{770 - 60 \times 5 + \Delta NX}{.72} \) - LM: \( Y = 600 \) Set equal: \[ 600 = \frac{770 - 300 + \Delta NX}{.72} \] \[ 600 \times .72 = 470 + \Delta NX \] \[ 432 = 470 + \Delta NX \] \[ \Delta NX = 432 - 470 = -38 \] So, yes: **NX must decrease by 38** at equilibrium \( Y = 600 \). --- ### Step 5: Final Answers #### a. **Goods market equilibrium (IS curve):** \[ \boxed{Y = \frac{770 - 60r}{.72}} \] #### b. **IS-LM Equilibrium:** - **IS at \( r=5 \):** \( Y \approx 652.78 \) - **LM at \( r=5 \):** \( Y = 600 \) - **Exchange rate:** Will adjust (depreciate) to shift IS to the right to meet LM at \( Y = 600 \). - **Required change in NX:** \( \Delta NX = -38 \) - (In this setup, the IS curve must shift left, i.e., NX must fall by 38, which may indicate that at \( Y_{LM} \), the original NX is higher than consistent with BP=, so the exchange rate appreciates until NX falls by 38. If this seems counterintuitive, check sign conventions in your model context.) #### **Short-run equilibrium income at IS-LM-BP:** \[ \boxed{Y = 600} \] --- ## Summary Table | Curve | Equation | Value at \( r=5 \) | |-------|----------|--------------------| | IS | \( Y = \frac{770 - 60r}{.72} \) | \( Y \approx 652.78 \) | | LM | \( Y = 500 + 20r \) | \( Y = 600 \) | | IS-LM-BP Equilibrium | \( Y = 600 \), \( r=5 \), \( \Delta NX = -38 \) | | --- ## Notes - **Exchange rate movement:** The model predicts the exchange rate will adjust to ensure BP=. In this setup, at the LM equilibrium income, NX must be lower, so the currency appreciates until net exports fall by 38. - **Perfect capital mobility:** Ensures \( r = r^* \). If IS-LM do not intersect at BP=, the exchange rate adjusts to shift NX and IS horizontally. --- **If you need a graphical representation or have further questions about the intuition behind the direction of exchange rate movement, let me know.**