As promised, here is a concise (and also thorough) explanation concerning how to solve problem 8 in the Sample Midterm Exam:
Question #8: If output is produced according to Q = 12K.5L.5, the price of K is $2, the price of labor is $2, and the price of Q is $100, the marginal profit at the optimal combination of inputs that cost $100 is ___________.
Solution: In chapter 4, we learned that the optimal combination of inputs for a given cost occurs at the point of tangency between the isocost line and the isoquant. In the graph shown below, the optimal combination of labor and capital inputs are given by the point (K1, L1).
The key to finding this point is to set the ratios of marginal products to prices for labor and capital equal to each other (see equation (4.6), p. 110 in the textbook); i.e., set
MPL/PL = MPK/PK —> MPL/2 = MPK/2 —> MPL= MPK.
Applying the power rule, the product of labor is MPL = .5(12)K.5L-.5,and the marginal product of capital is MPK = .5(12)K-.5L.5. Thus,
MPL= MPK—> .5(12)K.5L-.5 = .5(12)K-.5L.5 —> K.5L-.5 = K-.5L.5 —> K = L.
Since the amount budgeted for labor and capital is $100 and each unit of capital and labor costs $2, this means that we will optimally employ 25 units of capital and 25 units of labor. This will enable us to produce Q = 12K.5L.5 = 12(25.5)(25.5) = 300 units of output.
The question calls for determining marginal profit at the optimal combination of inputs that cost $100. We know that marginal revenue from selling one more unit of output is $100; this is given in the problem. Since marginal profit is the difference between marginal revenue and marginal cost, we must determine marginal cost in order to answer this question. On page 135 of the textbook, it is shown that the marginal cost of an input can be determined by dividing the cost per unit of input by the input’s marginal product. It follows then that the marginal cost of output at the optimal input combination is given by the price of the variable input (in this case, labor) divided by its marginal product. Thus MC = PL/MPL= $2/6K.5L-.5. However, since the optimal combination of inputs involves one unit of capital per unit of labor, we substitute K in place of L in the marginal cost equation and find that MC = $2/6K.5L-.5 = $1/3K.5K-.5 = $0.33. Thus marginal profit is MR – MC = $100 – $0.33 = $99.67.