Question

2. Assume the following Keynesian model:

C = 400 + .75Yd

I = 200

G = 100

X = 150

M = 50 + .15 Yd

T = 100

a. Find the aggregate expenditure function

b. Find the equilibrium level of GDP.

c. Using a “Keynesian cross” (or 45-degree line) diagram, show graphically the equilibrium in part a).

d. What is the spending multiplier in this model? Tax multiplier?

e. Show that leakages are equal to injections at equilibrium.

f. If government spending increases by $100, find the new equilibrium level of GDP. Show graphically.

Answer 1

a) AD = C + I + G + NX

AD = 400 + 0.75(Y – 100) + 200 + 100 + 150 – 50 – 0.15(Y – 100)

AD = 740 + 0.6Y

b) Y = AD

Y = 740 + 0.6Y

Y* = 740/0.4 = 1850

c) This is shown below

d) Spending multiplier = 1/(MPS + MPM) = 1/(0.25 + 0.15) = 1/0.4 = 2.5. Tax multiplier = -0.6/0.4 = -1.5

e) Leakages = Saving + Import + Tax = (1850 - 100 - 400 - 0.75*(1850 - 100) + 100 - 100) + (50 + 0.15*(1850 - 100)) + 100 = 450

Injections = Investment + Exports + Government spending = 200 + 150 + 100 = 450.

Hence Leakages = Injections

f) When G rises by 100, Y rises by 2.5x100 = 250.

Answer 2

a. The aggregate expenditure function is given by:

AE = C + I + G + (X - M)

Substituting the given values:

AE = (400 + 0.75Yd) + 200 + 100 + (150 - (50 + 0.15Yd)) = 750 + 0.60Yd

b. To find the equilibrium level of GDP, we set aggregate expenditure equal to GDP:

GDP = AE

GDP = 750 + 0.60Yd

c. The equilibrium is represented by the point where the aggregate expenditure function intersects the 45-degree line. At equilibrium, GDP equals aggregate expenditure. We can graphically represent this equilibrium by plotting the aggregate expenditure function (AE = 750 + 0.60Yd) and the 45-degree line (GDP = AE) on a graph.

d. The spending multiplier can be calculated as: Spending Multiplier = 1 / (1 - marginal propensity to consume) In this case, the marginal propensity to consume (MPC) is 0.75, so the spending multiplier is: Spending Multiplier = 1 / (1 - 0.75) = 4

The tax multiplier can be calculated as: Tax Multiplier = - MPC / (1 - MPC) In this case, the MPC is 0.75, so the tax multiplier is: Tax Multiplier = -0.75 / (1 - 0.75) = -3

e. At equilibrium, leakages (saving, S, and taxes, T) are equal to injections (investment, I, and government spending, G): S + T = I + G

In this case, leakages are saving (S) and taxes (T): S = 50 + 0.15Yd T = 100

Injections are investment (I) and government spending (G): I = 200 G = 100

Substituting the values: 50 + 0.15Yd + 100 = 200 + 100

This equation holds true at equilibrium, indicating that leakages are equal to injections.

f. If government spending increases by $100, the new equilibrium level of GDP can be found by adjusting the aggregate expenditure function. The new aggregate expenditure function is:

AE = (400 + 0.75Yd) + 200 + (100 + 100) + (150 - (50 + 0.15Yd)) = 850 + 0.60Yd

To graphically represent the new equilibrium, plot the new aggregate expenditure function (AE = 850 + 0.60Yd) and the 45-degree line (GDP = AE) on the same graph as in part c. The intersection of these two lines will give the new equilibrium level of GDP.