Question

1. Assume that a firm has future marginal productivity of capital given by MPK-A(100-K). The price of capital (machine) is $20,000, the real interest rate is 10%, and capital depreciates at a 10% rate. Assume further that each unit of output sells for $50. A) Calculate the user cost of the capital (in real term) that the firm faces. B) Assume A-1, then calculate the desired capital stock. What is the firm's gross investment if the firm currently has 10 machines? (Assume the machines depreciate at the same rate as the usual rate above.)

Answer 1

Here, we have: warnival Prodvetivity of capital : MPK = A (100-K). price of capital $20,000 (+ PK). Real interest rate : r = 10% = 0.1 hice of Output : P = $50. Luciahbu Raleis = 10% = 0.1 A). The user cost of capital (in real terus) is given by! ng I putting the respective values, we get : c 20000 (0.1 +0.) 50 = 80 Lo Real cost of capital is 80 units of output. 16) Given the cost of capital, the firm's profit is maximused I at a point where, Marginal Productivity of capital = Real cost of capital -) MPK - PR (r + 8) We how 4:1. :. Putting the respective values in the above equation, we 100 - K = 80 K 1600-80-20. >> K220 The desired capital stock is: K= 20. I

Now, it is given that K = 10 Gross In westurent of a firm is the sum of Net investment and the replacement of tem depriciates capital Further, Net investment is the difference between MPK and the real cost of capital Here, we have MPK = A (100 = 100-10=90 (since, k = 10). e = 80 units of output. . SK = Replacement of depriciated capital => SK = 0.1x10 of ? Depriciated * Gross Juvestment - Net Investment + Replacement Capital => Boross Investment = [upk-c] + SK - 190-80)+1; - 10+1= Thus, Gross Investment, of capital when A=1 and K=10, . is ll units of capital C) Now, A = 2; K=10 At A = 2, and K=10, MPK = A (100-k) = 2 (100 -10) u Cand SK would remain unchanged.h

:, Gross Investureet = [upk-c7+ 8K = (180–80] + 1 = 100+1=101 Thus, Giross investment when A = 2 and K = 10, is 101 units of capital