You are given the data on the following variables in an economy: Government spending = 300, Planned investment = 200, Net exports = 50, Autonomous taxes = 250, Income-tax rate = 0.1, Marginal propensity to consume = 0.5

Calculating Equilibrium Income using the Keynesian Model

The Keynesian model expresses equilibrium income (Y) as the sum of aggregate expenditure (AE):

Y = C + I + G + NX

Where:

• Y = National Income
• C = Consumption
• I = Investment
• G = Government Spending
• NX = Net Exports

However, consumption is not autonomous; it depends on income through the marginal propensity to consume (MPC). Therefore, we can express consumption as:

C = a + bY

Where:

• a = Autonomous Consumption
• b = Marginal Propensity to Consume (MPC)

1. Determining Autonomous Consumption (a)

We are given government spending (G = 300), planned investment (I = 200), net exports (NX = 50), autonomous taxes (T = 250), income tax rate (t = 0.1), and MPC (b = 0.5). To find autonomous consumption (a), we need to consider the disposable income and the consumption function.

Disposable Income (YD) = Y - T

Consumption (C) = a + bYD = a + b(Y - T)

We can rewrite the equilibrium income equation as:

Y = a + b(Y - T) + I + G + NX

Rearranging to solve for Y:

Y = a + bY - bT + I + G + NX

Y - bY = a - bT + I + G + NX

Y(1 - b) = a - bT + I + G + NX

Y = (a - bT + I + G + NX) / (1 - b)

Since we don't have Y directly, we need to find 'a' first. However, we can proceed by assuming an initial value for Y and iterating, or by recognizing that the tax component affects disposable income and thus consumption. Let's proceed by substituting the given values and solving for 'a' assuming we are looking for the equilibrium income.

2. Calculating the Multiplier

The Keynesian multiplier (k) is calculated as:

k = 1 / (1 - b)

k = 1 / (1 - 0.5) = 1 / 0.5 = 2

3. Calculating Equilibrium Income (Y)

First, we need to calculate the autonomous expenditure (A):

A = I + G + NX - bT

A = 200 + 300 + 50 - (0.5 * 250)

A = 550 - 125 = 425

Now, we can calculate the equilibrium income (Y):

Y = k * A

Y = 2 * 425 = 850

4. Considering Taxes

The inclusion of taxes requires a slight adjustment. The actual disposable income is Y(1-t). Therefore, the consumption function becomes C = a + bY(1-t). Substituting this into the equilibrium income equation:

Y = a + bY(1-t) + I + G + NX

Y = a + 0.5Y(1-0.1) + 200 + 300 + 50

Y = a + 0.45Y + 550

0.55Y = a + 550

To find 'a', we need to assume a value for Y. However, we can also express the equilibrium income in terms of autonomous expenditure, adjusted for taxes. The tax multiplier is -b/(1-b). The change in autonomous expenditure is I + G + NX - tT. Since we are given autonomous taxes, we need to adjust for the tax rate. The effective autonomous expenditure is:

A = I + G + NX - tT

A = 200 + 300 + 50 - (0.1 * 250)

A = 550 - 25 = 525

Now, the equilibrium income is:

Y = k * A = 2 * 525 = 1050

Therefore, the equilibrium income is 1050.