C= 50 + 0.8Yd I = 100 G = 200 X = 250 M = 20 + 0.25Y T = 0.3Y Find the equilibrium national income

To find the equilibrium national income (Y), we can use the basic macroeconomic identity, which states that in equilibrium, aggregate demand (AD) equals aggregate supply (Y). The aggregate demand can be expressed as: \[ AD = C + I + G + (X - M) \] Where: - \( C \) is consumption, - \( I \) is investment, - \( G \) is government spending, - \( X \) is exports, - \( M \) is imports. ### Step 1: Express C and M in terms of Y 1. **Consumption (C)**: \[ C = 50 + 0.8Y_d \] Where \( Y_d \) (disposable income) is defined as: \[ Y_d = Y - T \] And taxes (T) are given by: \[ T = 0.3Y \] So, \[ Y_d = Y - 0.3Y = 0.7Y \] Now substitute \( Y_d \) in the consumption function: \[ C = 50 + 0.8(0.7Y) = 50 + 0.56Y \] 2. **Imports (M)**: \[ M = 20 + 0.25Y \] ### Step 2: Substitute C, I, G, and M into the AD equation Now we can express AD: \[ AD = C + I + G + (X - M) \] Substituting the values: \[ AD = (50 + 0.56Y) + 100 + 200 + (250 - (20 + 0.25Y)) \] Simplifying, \[ AD = 50 + 0.56Y + 100 + 200 + 250 - 20 - 0.25Y \] Combining like terms yields: \[ AD = 580 + (0.56Y - 0.25Y) \] Thus, \[ AD = 580 + 0.31Y \] ### Step 3: Set AD equal to Y for equilibrium In equilibrium: \[ Y = AD \] So, we set \( Y \) equal to the expression for \( AD \): \[ Y = 580 + 0.31Y \] ### Step 4: Solve for Y Rearranging the equation to solve for \( Y \): \[ Y - 0.31Y = 580 \] \[ 0.69Y = 580 \] Now, divide both sides by 0.69: \[ Y = \frac{580}{0.69} \approx 840.58 \] ### Conclusion The equilibrium national income \( Y \) is approximately \( 840.58 \).