Question

Suppose Cool T-Shirts Co produces T-shirts and employs labor (L) and capital (K) in production. Suppose production function for Cool T-Shirts Co is Q=K*L, and Cool T-Shirts Co wants to produce Q=625. Suppose marginal product of labor (MPL) and marginal product of capital (MPK) are as follows: MPL=K and MPK=L. Suppose Cool T-Shirts Co pays workers $10 per hour (w=$10) and interest rate on capital is $250 (r=250). What is the cost-minimizing input combination if Cool T-Shirts Co wants to produce 625 T-Shirts per month. Find total cost. Graph your solutions and label intercepts, axes, isocost lines, isoquant curve and the optimal combination of K and L.

Answer 1

Production function for Cool T-Shirts Co is Q=K*L and target output is Q=625.

This gives marginal product of labor (MPL) = K and marginal product of capital (MPK) = L. Hence MRTS = MPL/MPK = K/L. Given that wage rate = $10 per hour and r = $250. Optimum input mix has MRTS = w/r

K/L = 10/250

L = 25K

Use this in the production function

Q = K*(25K)

625 = 25K^2

K = 25^0.5 = 5 units and L = 25*5 = 125 units.

This is cost-minimizing input combination if Cool T-Shirts Co wants to produce 625 T-Shirts per month.

Total cost is C = wL + rK or C = 10*125 + 250*5 = 2500.