Question

Given the following model: Y= C + I + G + X – Z C = a + bYd Z = Z0 + zYd Yd = Y – T a) Compute the expression for equilibrium income b) Compute the expression for the tax multiplier c) Suppose there is an autonomous increase in imports (Z0) of 20 units. To counteract this contraction in domestic aggregate demand, assume the government cuts taxes by 20 units. Will equilibrium income rise or fall? By how much? Explain.

Answer 1

(a)

In equilibrium, Y = a + b(Y - T) + I + G + X - Z0 - z(Y - T)

Y = a + bY - bT + I + G + X - Z0 - zY + zT

(1 - b + z)Y = a - bT + I + G + X - Z0 + zT

Y = (a - bT + I + G + X - Z0 + zT) / (1 - b + z)

(b)

Tax multiplier = $\partial$ Y/$\partial$T = - (b - z) / (1 - b + z)

(c)

Autonomous spending multiplier = 1 / (1 - b + z)

Absolute value of Tax multiplier = |$\partial$Y/$\partial$T| = (b - z) / (1 - b + z)

Therefore, Autonomous spending multiplier > Absolute value of Tax multiplier, and hence, an equal increase in import and a decrease in tax will lead to a net decrease in income.

When imports rise by 20, Decrease in Y = 20 / (1 - b + z)..........(1)

When taxes fall by 20, Increase in Y = [- 20 x (b - z) / (1 - b + z)].........(2)

Net Decrease in Y = (1) + (2)

= [20 / (1 - b + z)] - [20 x (b - z) / (1 - b + z)]

= [20 / (1 - b + z)] x (1 - b + z)

= 20 x [(1 - b + z) / (1 - b + z)

= 20