Sure! Let’s break down your macroeconomics homework step by step and explain how to approach each part. You’re given a set of equations describing an economy and asked for IS-LM analysis. --- ## **Given Equations** **Goods Market** - \( C = 200 + .65(Y - T) \) *(Consumption as a function of disposable income)* - \( I = 500 - 30r + .1Y \) *(Investment as a function of interest rate and income)* - \( G = 800 \) *(Government spending)* - \( T = 150 + .15Y \) *(Taxes as a function of income)* **Money Market** - \( M_s = 140 \) *(Money supply)* - \( M_d = .64Y - 80r \) *(Money demand as a function of income and interest rate)* **Price Adjustment** - \( P = .5Y + 200 \) *(Price level as a function of income)* --- ## **Tasks Explained** ### 1. **Derive Reduced Form IS Equation in terms of \( Y \) and \( r \)** The **IS curve** comes from equilibrium in the **goods market**: **Aggregate demand = Aggregate output:** \[ Y = C + I + G \] Plug in the equations for \( C \), \( I \), and \( G \): \[ Y = [200 + .65(Y - T)] + [500 - 30r + .1Y] + 800 \] Remember \( T = 150 + .15Y \): \[ C = 200 + .65(Y - (150 + .15Y)) \\ = 200 + .65(Y - 150 - .15Y) \\ = 200 + .65(.85Y - 150) \\ = 200 + .5525Y - 97.5 \\ = 102.5 + .5525Y \] So the IS equation becomes: \[ Y = [102.5 + .5525Y] + [500 - 30r + .1Y] + 800 \] Combine like terms: \[ Y = 102.5 + .5525Y + 500 - 30r + .1Y + 800 \] \[ Y = (102.5 + 500 + 800) + (.5525Y + .1Y) - 30r \] \[ Y = 1402.5 + .6525Y - 30r \] Bring all terms with \( Y \) to one side: \[ Y - .6525Y = 1402.5 - 30r \] \[ .3475Y = 1402.5 - 30r \] \[ Y = \frac{1402.5 - 30r}{.3475} \] This is the reduced form IS equation. --- ### 2. **Derive LM Equation Explicitly** The **LM curve** comes from equilibrium in the **money market**: \[ M_s = M_d \] Plug in the values: \[ 140 = .64Y - 80r \] Solve for \( Y \): \[ .64Y - 80r = 140 \] \[ .64Y = 140 + 80r \] \[ Y = \frac{140 + 80r}{.64} \] That's the explicit LM equation. --- ### 3. **Solve IS-LM System Simultaneously (for \( Y \), \( r \), \( P \))** Set the two equations equal (since both are \( Y \) in terms of \( r \)): \[ \frac{1402.5 - 30r}{.3475} = \frac{140 + 80r}{.64} \] Now solve for \( r \): 1. Cross-multiply: \[ .64(1402.5 - 30r) = .3475(140 + 80r) \] 2. Expand: \[ 897.6 - 19.2r = 485.5 + 27.8r \] 3. Collect \( r \) terms: \[ 897.6 - 485.5 = 27.8r + 19.2r \] \[ 412.1 = 47r \] \[ r = \frac{412.1}{47} \approx 8.77 \] Substitute \( r \) back into either equation to get \( Y \): \[ Y = \frac{1402.5 - 30(8.77)}{.3475} \] \[ Y \approx \frac{1402.5 - 263.1}{.3475} \approx \frac{1139.4}{.3475} \approx 3278.6 \] Now, use the price equation: \[ P = .5Y + 200 \\ P = .5 \times 3278.6 + 200 \approx 1639.3 + 200 = 1839.3 \] --- ### 4. **Compute Fiscal Multiplier from Reduced Form** The **fiscal multiplier** is the change in \( Y \) from a change in \( G \): From IS: \[ Y = \frac{1402.5 - 30r}{.3475} \] If \( G \) increases by 1, the numerator increases by 1, so: \[ \frac{\partial Y}{\partial G} = \frac{1}{.3475} \approx 2.88 \] --- ### 5. **If \( G \) increases by 200, Compute New Equilibrium** Add 200 to \( G \), so numerator in IS becomes: \[ Y = \frac{1402.5 + 200 - 30r}{.3475} = \frac{1602.5 - 30r}{.3475} \] Repeat step 3 with this new numerator. --- ### 6. **Relative Effectiveness: Monetary vs Fiscal Policy** - Fiscal policy shifts IS curve (affects output more if LM is flatter) - Monetary policy shifts LM curve (affects output more if IS is flatter) - Compare the coefficients: If IS is steeper, monetary policy is more effective; if LM is steeper, fiscal policy is more effective. --- ### 7. **Examine Stability Using Partial Derivatives** Check the partial derivatives of \( Y \) with respect to \( r \) in both IS and LM curves: - IS: \( \frac{\partial Y}{\partial r} = -\frac{30}{.3475} \) - LM: \( \frac{\partial Y}{\partial r} = \frac{80}{.64} \) If the system's Jacobian determinant is positive, equilibrium is stable. --- ## **Summary Table** | Step | What To Do | Equation Used | Key Result/Tip | |------|------------------------------|------------------------|-------------------------------| | 1 | IS Curve | Solve for \( Y \) | \( Y = \frac{1402.5-30r}{.3475} \) | | 2 | LM Curve | Solve for \( Y \) | \( Y = \frac{140+80r}{.64} \) | | 3 | Equilibrium \((Y, r, P)\) | Solve IS-LM together | \( Y \approx 3278.6, r \approx 8.77, P \approx 1839.3 \) | | 4 | Fiscal Multiplier | \(\frac{\partial Y}{\partial G}\) | \( \approx 2.88 \) | | 5 | \( G \) increases by 200 | Update IS numerator | Recalculate \( Y, r, P \) | | 6 | Policy Effectiveness | Compare slope values | Steeper curve = less effective | | 7 | Stability | Partial derivatives | Positive Jacobian = stable | --- If you need **step-by-step calculations or numerical answers for steps 5-7**, let me know!