Econ 400 HW 3

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Econ 400 HW 3

1. Consider an apple farm in the state of Washington. It produces apples q according to production function q = f(L,K, F), where L is labor, K is capital, and F is farmland. Inputs can be purchased at price w, r, and s, with all prices positive. Assume that for each input i = L,K,F, fi(′)  > 0, fi(′′)  < 0.

(a) Write down the firm’s cost minimization problem. 2 points.

(b) Write out the lagrangian and first order conditions for each input choice. 3 points.

(c) The first order conditions you found above have a solution L* (w,r,s,q), K* (w,r,s,q) and F* (w,r,s,q), and associated cost function is the value function of this problem,, C(w,r,s,q).

Now, we will write the profit maximization problem, when costs are given by that cost function. Write down the profit maxi- mization problem, lagrangian, and first order conditions for this problem. 2 points.

(d) We will next reconsider the profit maximization problem, taking the choice variables to be the inputs L,K, andF, rather than out-put q. Write out this version of the profit maximization problem, associated lagrangian, and first order conditions. 5 points.

(e) Finally, using the envelope theorem, argue that the first order conditions from part three are equivalent to the solution to profit maximization problem of part 2, using the minimized cost from part 1. 5 points.

2. We will now consider the case of the above apple farm, where

f(L,K, F) = ALα Kβ Fγ

but farmland F is fixed in the short run atF(¯)

(a) Write down the short run cost minimization problem with re- spect to L and K and the first order conditions with respect to L and K. 2 points.

(b) Solve for L* and K* . 3 points.

(c) How does L vary with w? Compute ∂w/∂L*. Does this make sense? What is effect of an increase in A (techno- logical progress)? 2 points.

(d) Calculate the short run cost function C. and derive the short run marginal cost C′, and short run average cost C/q. 3 points.

(e) Suppose α + β  <  1; sketch the marginal cost function and av-erage cost functions you calculated.  Sketch the firm’s supply curve; is it increasing in p? 2 points.

(f) Suppose α + β  =  1; sketch the marginal cost function and av-erage cost functions you calculated. Calculate the supply curve of the firm exactly; why does this firm never want to produce a positive quantity despite constant returns to scale in labor and capital? 2 points.

(g) Suppose α + β  >  1; sketch the marginal cost function and av-erage cost functions you calculated. Sketch the firm’s supply curve; is it increasing in p? 2 points.