Question

a firm produces a product with labor and capital as inputs. The production function is described by Q=LK. the marginal products associated with this production function are MPL=K and MPK=L. let w=1 and r=1 be the prices of labor and capital, respectively

a) find the equation for the firms long-run total cost curve curve as a function of quantity Q

b) solve the firms short-run cost-minimization problem when capital is fixed at a quantity of 5 units (ie.,K=5). 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 long-run and short-run total cost curves change when w=1 and r=4?

d) how do the graphs of the long run and short run total cost curves change when w=4, and r=1?

Answer 1

Answer:-

a) Cost-minimizing quantities of inputs are equal to L= √Q√(r/w) and K= √Q/ √(r/w). Hence, in the long-run the total cost of producing Qunits of output is equal to TC(Q) = 2√(Qrw). For w= 1 and r= 1 we have TC(Q) = 2√Q.

b) When capital is fixed at a quantity of 5 units (i.e., K= 5) we have Q= K*L= 5 L. Hence, in the short-run the total cost of producing Qunits of output is equal to STC(Q) = 5 + Q/5.

d)Answer:-

When w= 4 and r= 1 we have TC(Q) = 4√Qand STC(Q) = 5+ 4Q/5