Question

• Assume a firm' production function is Q = KL Assume price of capital is r= 1, and price of labor is w = 2 • How many units of capital and labor is need to produce Q=200 in cheapest way? O Show your logic using cost minimization condition, and Analyze it graphically Hint: First, you should use the definition of MP and MPK and calculate them

Answer 1

We have Q = KL

This gives MRTS = MPL/MPK = K/L

At the optimum input choice MRTS = w/r which gives K/L = 2/1

Hence we have K = 2L

Now Q = KL becomes Q = 2L*L or L = (Q/2)^0.5

Thus K = 2(Q/2)^0.5 and L = (Q/2)^0.5

Units of capital and labor for Q = 200 are K = 2*(200/2)^0.5 = 20 units and L = (200/2)^0.5 = 10

units.

Hence, the minimum cost to produce 200 units is C = wL + rK or 2*10 + 1*20 = $40.

In the graph, the isoquant showing Q = KL is tangent to the isocost line 40 = 2L + K.

At the tangency point, MRTS = w/r which gives K = 20 and L = 10

Capital 40 MRTS w/r 20 Q 200 10 20 Labor