Question

Consider a firm that used only two inputs, capital (K) and labor (L), to produce output....

Consider a firm that used only two inputs, capital (K) and labor (L), to produce output. The production function is given by: Q = 60L^(2/3)K^(1/3) .

a.Find the returns to scale of this production function.

b. Derive the Marginal Rate of Technical Substitutions (MRTS) between capital and labor. Does the law of diminishing MRTS hold? Why? Derive the equation for a sample isoquant (Q=120) and draw the isoquant. Be sure to label as many points as you can.

c. Compute and interpret the elasticity of substitution.

d. Set up the optimization problem of the firm and, for arbitrary values of w, r, and Q, solve for the (long run) input demand curves.

e. Derive the own price elasticity of demand for labor and capital.

f. For arbitrary values of w, r, and Q, solve for the long run cost function. Graph this cost function when w=$16 and r=$64.

g. Derive the long run average cost function when w=$16 and r=$64. Does this cost function exhibit economies of scale, diseconomies of scale, or constant returns to scale? Why?

h. Suppose that the firm's capital is fixed in the short run at K=9. For arbitrary values of w, r, and Q, find the firm's short run demand for labor and the corresponding cost function.

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