Algebra of the income-expenditure model Consider a small economy that is closed to trade, so that its net exports are zero. Suppose that the economy has the following consumption function, where C is consumption, Y is income (real GDP), IP is planned investment, G is government purchases, and T is taxes:. C = $20 billion+0.75×(Y – T) Suppose G=$35 billion, IP=$60 billion, and T=$20 billion. Given the consumption function and the fact that, in a closed economy, planned expenditure can be calculated as Y=C+IP+G, the equilibrium income level is billion. Suppose that government purchases are increased by $100 billion. The new equilibrium level of income will be equal to billion. Based on the effect of the change in government purchases on equilibrium income, you can tell that this economy's multiplier is equal to .

a. The equilibrium income level is $100 billion.

b. The new equilibrium level of income will be equal to $500 billion.

c. This economy's multiplier is equal to 4.

Explanation:

a. Calculation of the equilibrium income level

Since;

Y= C + IP + G (1)

Where;

C = $20 billion + 0.75 × (Y – T)

G = $35 billion

IP=$60 billion

T = $20 billion.

Remove the billion now for simplicity purpose to include later, substitute the values into equation (1) and solve for Y, we have:

Y = $20 + 0.75 * (Y – $120) + $60 + $35

Y = $20 + 0.75Y - (0.75 * $120) + $95

Y - 0.75Y = $20 + $95 - $90

0.25Y = $25

Y = $25 / 0.25

Y = $100

Therefore, the equilibrium income level is $100 billion.

b. Suppose that government purchases are increased by $100 billion. The new equilibrium level of income will be equal to billion.

With this, we now have:

G = $35 billion + $100 billion = $135 billion

Replace this with G in part a and substitute other values as already given in part a into equation (1) and solve for Y, we have:

Y = $20 + 0.75 * (Y – $120) + $60 + $135

Y = $20 + 0.75Y - (0.75 * $120) + $195

Y - 0.75Y = $20 + $195 - $90

0.25Y = $125

Y = $125 / 0.25

Y = $500

Therefore, the new equilibrium level of income will be equal to $500 billion.

c. Based on the effect of the change in government purchases on equilibrium income, you can tell that this economy's multiplier is equal to .

Since the change in government purchases makes equilibrium income level to increase from $100 billion to $500 billion, we can calculate the rate of change in the equilibrium income level as follows:

Rate of change in equilibrium income = (New income – Previous income) / Previous income = ($500 - $100) / $100 = 4

With the rate of change of 4, we can tell that this economy's multiplier is equal to 4.

This can be confirmed using the multiplier formula as follows:

Multiplier = 1 / (1 – MPC) ……………………….. (2)

Where;

MPC = 0.75 from the consumption equation given C = $20 billion + 0.75 × (Y – T).

Substitute for MPC in equation (2), we have:

Multiplier = 1 / (1 – 0.75)

Multiplier = 1 / 0.25

Multiplier = 4

Which is the same as already obtained above.

Therefore, this economy's multiplier is equal to 4.