Question

3. Suppose that the expected future marginal product of capital is MPK^f = 20 – 0.02K, where K is the future capital stock. The depreciation rate of capital, d, is 20% per period. The current capital stock is 900 units of capital. The price of a unit of capital is 1 unit of output. Firms pay taxes equal to 50% of their output. The consumption function in the economy is C= 100 + 0.5Y-200r, where C is consumption, Y is output, and r is the real interest rate. Government purchases equal 200, and full-employment output is 1000.

                a. suppose that the real interest rate is 10% per period. What are the values of the tax-adjusted user cost of capital, the desired future capital stock, and the   desired level of investment?

                b. Now consider the real interest rate determined by goods market equilibrium. This part of the problem will guide you to this interest rate.

                                i. Write the tax-adjusted user cost of capital as a function of the real interest rate r. also write the desired future capital stock and desired investment as functions of r.

                                ii. Use the investment function derived in Part (i) along with the consumption function and government purchases, to calculate the real interest rate that clears the goods market. What are the goods market-clearing values of consumption, saving, and investment? What are the tax-adjusted user cost of capital and the desired capital stock in this equilibrium?

Answer 1

Total value of the capital = 21*900*0.1 = $1,890

Depreciation = 378

r = 0.10

We have user cost of capital = uc/(1 - τ)

ð (r + d)p K /(1 - τ)

ð [(.1 + .2) x 1] / (1 - .5) = 0.6

ð MPK f = uc/(1 - τ), so 20 - .02K = .6

ð K * = 970 (the desired future capital stock)

ð K * - K = I – dK

ð I = K * - K + dK

ð 970 - 900 + (.2 x 900)

ð 250.

Answer - uc/(1 - τ) = (r + d)p K / (1 - τ)

[(r + .2) x 1] / (1 - .5) = .4 + 2r.

MPK f = uc/(1 - τ)

ð 20 - .02K = .4 + 2r

ð K * = 980 - 100r

ð I = K * - K + dK

ð 980 - 100r - 900 + (.2 x 900) = 260 - 100r.

Answer - Y = C + I + G

1000 = [100 + (.5 x 1000) - 200r] + (260 - 100r) + 200 1000
1060 - 300r
300r = 60
r = 0.2
C = 560
I = 240 = S
uc/(1 - τ) = .4 + (2 x .2) = 0.8
K * = 960