Question

8.13) A firms produces a product with labor and capital. its production function is described by Q=L+K. The marginal products associated with this production function are MPL =1 and MPK=1. Let w=1, and r=1 be the prices of labor and capital respectively.

a) fund the equation for the firms long-run total cost curve as a function of quantity Q when the prices of labor and capital are w=1 and r=1.

b) find the solution to the firms short-run cost-minimization problem when capital is fixed at a quantity of 5 units, and w=1 and r=1. derive the equation for the firms short-run total cost curve as a function of quantity Q and graph it together with the long-run total cost curve

c)how do the graphs of the short-run and long run total cost curves change when w=1, and r=2?

d) how do the graphs of the short-run and long-run total cost curves change when w=2 and r=1?

Answer 1

Answer:-

a) With a linear production function, the firm operates at a corner point, so will use all labor or all capital. We want to compare MPL/w with MPk/r. This tells us for another dollar spent on labor how much more product/output would be produced, or for a dollar spent on capital how much more product/output would be produced. Because the marginal products and the prices of the inputs are exactly the, the firm is indifferent among combination of L and K

b) When capital is fixed at 5 units, the firm’s output would be given by Q = 5 + L. If the firm wants to produce Q < 5 units of output, it must produce 5 units and throw away 5 – Q of them. The total cost of producing fewer than 5 units is constant and equal to $5, the cost of the fixed capital. For Q > 5 units, the firm increases its output by increasing its use of labor. In particular, using the production function we can see that to produce Q units of output, the firm uses Q – 5 units of labor (L=Q-5 when we sub in for K=5). Subbing L and K into the Total cost function we get:, STC(Q) = Q – 5 + 5 = Q