Question

The expected future marginal product of capital is given by

???? ?MPKf=2400-5? .

The depreciation rate is 8% per year; the real interest rate is 3% per year and the price of a unit of capital is

1500.

a. (5 Points) What is the desired capital stock?

b. (6 Points) If the current level of gross investment is It=401, what is the current level of capital Kt?

Answer 1

(a)

Depreciation rate (d) = 8% = 0.08

Real Interest rate (r) = 3% = 0.03

Price of a unit of capital (pk) = 1500

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User cost of capital (UC) = (r +d)pk

=> UC = (0.03 + 0.08)*1500

=> UC = (0.11) * 1500

=>UC = 165

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For desired level of capital, set MPKf = UC

=> 2400 - 5K = 165

=> 2400 - 165 = 5K

=> 2235 = 5K

=> K = (2235 / 5)

=> K* = 447

Desired capital stock (K*) is 447

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(b) Net investment = gross investment - depreciation

Given information:

Gross investment (I_t) = 401

let K_t is current level of capital and K_t+1 is capital in next period

=> Net investment = K_t+1 - K_t

=> Depreciation = dK_t.

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Net investment = gross investment - depreciation

=> K_t+1 - K_t = I_t - dK_t.

=> I_t = K_t+1 - K_t + dK_t .

Note: K_t+1 = K* at optimum point. So, K_t+1 = 447

=> 401 = 447 - K_t + 0.08K_t

=> 401 = 447 - 0.92K_t

=> 0.92K_t = 447 - 401

=> K_t = (46 / 0.92)

=> K_t = 50

Hence, the current level of capital K_t is 50

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