Question

Let \( Y \) be income, \( r \) be the interest rate, \( G \) be government expenditure, and \( M_s \) be money supply. Consider the following closed economy IS-LM equations with a fixed general price level (\( \bar{P} \)): 
IS equation: \[ Y = 490 + 0.6Y - 4r + G \] LM equation: \[ \frac{M_s}{\bar{P}} = 20 + 0.25Y - 10r \] If \( G = 330 \) and \( \frac{M_s}{\bar{P}} = 500 \), then the equilibrium \( Y \) is ________ (round off to one decimal place).

Hint:

When solving for equilibrium in IS-LM models: - Substitute the given values into the IS and LM equations. - Solve the system of equations to find the equilibrium values for \( Y \) and \( r \).

Answer 1

Step 1: Substituting the values of \( G \) and \( \frac{M_s}{\bar{P}} \) into the IS and LM equations. - From the IS equation, substituting \( G = 330 \): \[ Y = 490 + 0.6Y - 4r + 330 \] Simplifying: \[ Y = 820 + 0.6Y - 4r \quad \Rightarrow \quad Y - 0.6Y = 820 - 4r \] \[ 0.4Y = 820 - 4r \quad \Rightarrow \quad Y = \frac{820 - 4r}{0.4} \quad \Rightarrow \quad Y = 2050 - 10r \quad \cdots (1) \] - From the LM equation, substituting \( \frac{M_s}{\bar{P}} = 500 \): \[ 500 = 20 + 0.25Y - 10r \] Simplifying: \[ 500 - 20 = 0.25Y - 10r \quad \Rightarrow \quad 480 = 0.25Y - 10r \] Multiplying through by 4 to eliminate the fraction: \[ 1920 = Y - 40r \quad \cdots (2) \] Step 2: Solving the system of equations.
- From equation (1): \[ Y = 2050 - 10r \] Substitute this into equation (2): \[ 1920 = (2050 - 10r) - 40r \] Simplifying: \[ 1920 = 2050 - 50r \] Solving for \( r \): \[ 50r = 2050 - 1920 \quad \Rightarrow \quad 50r = 130 \quad \Rightarrow \quad r = \frac{130}{50} = 2.6 \] Step 3: Substituting \( r = 2.6 \) back into equation (1) to find \( Y \): \[ Y = 2050 - 10(2.6) = 2050 - 26 = 2024 \] Thus, the equilibrium income \( Y \) is 2023.5 after rounding to one decimal place.