Given that, 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, we need to set up the national income identity: \[ Y = C + I + G + (X - M) \] ### Step 1: Calculate disposable income (Yd) Disposable income (Yd) is given by: \[ Y_d = Y - T \] where T is tax revenue, given by: \[ T = 0.3Y \] So: \[ Y_d = Y - 0.3Y = 0.7Y \] ### Step 2: Substitute Yd into the consumption function (C) The consumption function is given by: \[ C = 50 + 0.8Y_d \] Substituting for \(Y_d\): \[ C = 50 + 0.8(0.7Y) \] \[ C = 50 + 0.56Y \] ### Step 3: Substitute values into the national income identity Now we can substitute \(C\), \(I\), \(G\), \(X\), and \(M\) into the national income identity. We also need to express M in terms of Y. The imports (M) are given by: \[ M = 20 - 0.25Y \] Now we can rewrite the national income identity: \[ Y = C + I + G + (X - M) \] Substituting the values we have: \[ Y = (50 + 0.56Y) + 100 + 200 + (250 - (20 - 0.25Y)) \] ### Step 4: Simplify the equation We simplify the equation step by step: 1. Combine the constants: \[ Y = (50 + 100 + 200 + 250 - 20) + 0.56Y + 0.25Y \] \[ Y = 580 + 0.81Y \] 2. Rearranging it: \[ Y - 0.81Y = 580 \] \[ 0.19Y = 580 \] ### Step 5: Solve for Y Now solve for Y: \[ Y = \frac{580}{0.19} \] \[ Y = 3052.63 \] ### Conclusion The equilibrium national income \(Y\) is approximately **3052.63**. Update (2024-07-22): To find the equilibrium national income, we can use the Keynesian equilibrium condition, which states that aggregate demand (AD) is equal to aggregate supply (national income, \( Y \)). The aggregate demand (AD) consists of consumption (C), investment (I), government spending (G), and net exports (X - M). 1. **Aggregate Demand (AD) Equation:** \[ AD = C + I + G + (X - M) \] 2. **Expressions for the variables:** - Consumption (\( C \)) is given by \( C = 50 + 0.8Y_d \), where \( Y_d \) (disposable income) is given by \( Y_d = Y - T \). - Taxes (\( T \)) are given by \( T = 0.3Y \). - Therefore, \( Y_d = Y - 0.3Y = 0.7Y \). - Substituting this into the consumption function, we get: \[ C = 50 + 0.8(0.7Y) = 50 + 0.56Y \] 3. **Investment (I)** is given as \( I = 100 \). 4. **Government spending (G)** is given as \( G = 200 \). 5. **Net Exports (NX)**: - Exports (\( X \)) = 250. - Imports (\( M \)) are given by \( M = 20 + 0.25Y \). - Therefore, net exports \( (X - M) \) is: \[ NX = 250 - (20 + 0.25Y) = 230 - 0.25Y \] 6. **Aggregate demand (AD)**: Now we can combine all these components to form the aggregate demand equation: \[ AD = C + I + G + NX \] \[ AD = (50 + 0.56Y) + 100 + 200 + (230 - 0.25Y) \] \[ AD = 50 + 100 + 200 + 230 + (0.56Y - 0.25Y) \] \[ AD = 580 + 0.31Y \] 7. **Equilibrium Condition**: At equilibrium, aggregate demand equals national income (\( Y \)): \[ Y = 580 + 0.31Y \] 8. **Rearranging the equation**: \[ Y - 0.31Y = 580 \] \[ 0.69Y = 580 \] \[ Y = \frac{580}{0.69} \approx 840.58 \] Thus, the equilibrium national income is approximately **\( Y \approx 840.58 \)**. Update (2024-07-22): To find the equilibrium national income, we need to derive the aggregate demand (AD) and set it equal to the national income (Y). The equilibrium condition for national income is given by: \[ Y = C + I + G + (X - M) \] Where: - \( C \) is consumption, - \( I \) is investment, - \( G \) is government spending, - \( X \) is exports, - \( M \) is imports. ### Step 1: Determine the Components of Aggregate Demand 1. Consumption function: \[ C = 50 + 0.8Y_d \] Where \( Y_d \) (disposable income) is given by: \[ Y_d = Y - T \] And taxes are given by: \[ T = 0.3Y \] Therefore: \[ Y_d = Y - 0.3Y = 0.7Y \] Substituting \( Y_d \) into the consumption function: \[ C = 50 + 0.8(0.7Y) = 50 + 0.56Y \] 2. Investment: \[ I = 100 \] 3. Government spending: \[ G = 200 \] 4. Exports: \[ X = 250 \] 5. Imports: \[ M = 20 + 0.25Y \] ### Step 2: Substitute Components into Aggregate Demand Equation Now substituting the components into the equation for \( Y \): \[ Y = C + I + G + (X - M) \] Substituting: \[ Y = (50 + 0.56Y) + 100 + 200 + (250 - (20 + 0.25Y)) \] ### Step 3: Simplify the Equation Now simplify this equation: \[ Y = 50 + 0.56Y + 100 + 200 + 250 - 20 - 0.25Y \] Combine like terms: \[ Y = 580 + (0.56Y - 0.25Y) \] This simplifies to: \[ Y = 580 + 0.31Y \] ### Step 4: Isolate \( Y \) Rearranging gives: \[ Y - 0.31Y = 580 \] \[ 0.69Y = 580 \] \[ Y = \frac{580}{0.69} \approx 840.58 \] ### Conclusion The equilibrium national income \( Y \) is approximately **840.58**.