Question

A firm has the production function F(L, K) = L1/2 + K1/2. The price of labor is $30 and the price of capital is $10. The firm has a production goal of 100 units of output.

a) Carefully write out this firm’s cost minimization problem, using the particulars of this problem.

b) Give two equations—particular to this problem—that the solution satisfies.

c) Solve for the firm’s optimal input bundle.

d) Determine the firm’s cost of producing 100 units of output.

e) Resolve this problem, leaving the production goal Q as a variable in order to get the firm’s cost function C(Q).

Answer 1

as the firm's problem in to minimize 30L + LOK. - sit. 100 = V2 + KY2 Lyk oo At equilibrium: MPL = 1 62 . i.k91 . Q = 100 = _Y2 + KP2..(2) 1. (a) sulestitute (i) in (a): 25 100 = LY2 + 3L%2 =* Y2 ini. * = (25)2 = 625 ... k*ax 625 = 5625 (d) The minimum cost in i C = 306625)+10(5625) - 75000 ces for Q = _Y2+k'? ....(3) sulestituting cis in C3): Q = 172 +3 172 = 4 L72 :.L: .K=982 ::CC) = 30 () +10(982) = (30+20) . - 12082 :.CCQ) = 7.50? 16

as the firm's problem in to minimize 30L + LOK. - sit. 100 = V2 + KY2 Lyk oo At equilibrium: MPL = 1 62 . i.k91 . Q = 100 = _Y2 + KP2..(2) 1. (a) sulestitute (i) in (a): 25 100 = LY2 + 3L%2 =* Y2 ini. * = (25)2 = 625 ... k*ax 625 = 5625 (d) The minimum cost in i C = 306625)+10(5625) - 75000 ces for Q = _Y2+k'? ....(3) sulestituting cis in C3): Q = 172 +3 172 = 4 L72 :.L: .K=982 ::CC) = 30 () +10(982) = (30+20) . - 12082 :.CCQ) = 7.50? 16