Question

Yes x₁ it is to the power ^a and x₂ is to the power ^b. It is the Cobb Douglas function.

Suppose that production function is given by f(x₁,x₂)=Ax₁^ax₂^b. Suppose the costs of the firm are linear and given by C(x_1xx₂)=(w₁x₁+w₂x₂)/B, where B is a parameter denoting the level of cost-minimizing technology of the firm. Assume the firm can charge the price p for their output.

(a) Set up the profit function and take the first-order conditions.

(b) Suppose that a+b=1, can you solve for the factor demands x_i(w₁,w₂p) for i=1,2? Explain why there may be a problem?

(c) Suppose that a+b<1, solve for your factor demands and the profit as a function of prices.

(d) Show that you can recover your output and factor demands and the profit as a function of prices.

(e) What are the implications of an increase in A and B on factor demands, output and profit. Use the envelope theorem to show this.

Answer 1

a) Profit function = PF = p*(A(x1)^a(x2)^b) - ((w1x1+w2x2)/B)

First order conditions

pAa(x1)^(a-1)(x2)^b = w1/B --- (1)

pAb(x1)^a(x2)^(b-1) = w2/B ----(2)

Multiply both sides of (1) with x1 and both sides of (2) with x2 and ABx1^ax2^b = y

ayp = w1x1

byp = w2x2

x2 = (bw1x1)/aw2 ---- (3)

Plug (3) into (1)

x1 = (pBA)^ (1/1-a-b) a^(1-b/1-a-b)b^(b/1-a-b)w1^(b-1/1-a-b)w2^(-b/1-a-b) ---- (4)

x2 = bw1(pBA)^ (1/1-a-b) a^(1-b/1-a-b)b^(b/1-a-b)w1^(b-1/1-a-b)w2^(-b/1-a-b)/(aw2) ----(5)

b) If a+b =1, then the exponents in equation (4) become 1/0 which is not valid so we cannot solve for x1 and x2.

c) If a+b<1, the values of x1 and x2 are given by equation (4) and (5) respectively.

d) Output = (A(x1)^a(x2)^b)

Profit = p*(A(x1)^a(x2)^b) - ((w1x1+w2x2)/B)

Using the values of x1 and x2 from equations (3) and (4), values of output and profit can be calculated as a function of prices.