Assignment 1: (total 100 points)
1. Compute the partial derivatives ƏU/ Əx (or Ux )and ƏU/ Əy (or Uy) : (6 points in total, 2 points for each question)
2. Firm optimization with one input: (24 points in total)
A firm hires works (L) to produce drills (D) according to a production function D=f(L; A), where A>0 is an exogenous parameter representing firm technology. Assume the market price of a drill is p>0, and the workers are paid a wage of w>0.
2.1) Assuming that f(L; A)= A*f(L), write an equation for the profit function π(w; p, A) involving A*f(L) and the relevant first and second order conditions that determine L*(the optimal choice of L). What do the first and second order conditions imply for the signs of the derivatives of f(L)? (4 points)
2.2) Compute L*(w, p, A) assuming each of the following production functions and check if the second order condition holds: (8 points, 4 points for each question)
a) f(L; A) = A L2/3
b) f(L; A) = A Lb/2 what must be true for b to represent a maximum?
2.3) Assuming a general functional form f(L; A)= A f(L), derive comparative statics showing how the optimal quantities change as functions of the parameters p, w, A. (12 points, 4 points for each question)
a) the change in profit (π)
b) total workers (L)
c) the number of outputs (D)
3. Firm optimization with two inputs: (14 points in total)
For a small firm, the firm maximizes its profit:
π(K, L; p, r, w) = pf(K,L) – rK- wL
The optimal choices of capital and labor are K*(p, r, w) and L*(p,r,w). Please take comparative statics for K* and L* with respect to r and p respectively, and discuss how the optimal choices of capital and labor will be changed when r/p increases.
4. Candy production (24 points in total, 8 points for each question)
A firm makes candies with two inputs sugar (S) and butter (B) following the production function:
The prices of these inputs are PS and PB , and the firm can sell candy for p dollars each.
4.1 Set up the firm’s profit maximization problem. Find the first order conditions, and solve for the optimal ratio of inputs B/S. Find the second order conditions for a maximum.
4.2. Set-up the firm’s cost minimization problem given an output quantity C and solve the Lagrangian. Show that the optimal ratio of inputs B/S is the same as you got in Q1.
4.3. Solve for the optimal level of inputs S* , B* given output quantity C and prices PS ,PB
5. A mining firm has two inputs：axes (X) and bread (B). The miners buy axes at a price r pre axe, and bread at a price m per loaf. With these inputs, the firm produces Q=f(X,B) tons of metal which they sell for a fixed price p per ton.
Suppose f(X, B) = a ln(X) + b ln(B), where a, b>0 （32 points in total）
1) Derive the mine’s cost function C (8 points)
2) Find the derivate of cost with respect to quantity (4 points)
3) Determine if the mine has increasing, decreasing or constant returns to scale. (4 points) If the production function has decreasing returns to scale, then
4) Solve for the optimal X*, B*, Q*. (8 points)
5) Derive the comparative statics of X*, B*, and Q* with respect to r and m; (8 points)