Question 1 - A competitive firm uses two inputs, capital k and labour l, in its production process, in order to produce output y. Its production technology is given by: y = l½k¼. Consider here the firm's long-run perspective.
a) Show algebraically that the firm's technology exhibits decreasing returns to scale.
b) If the rate of return on capital is r = 8 and the wage rate is w = 1, represent graphically the firm's cost minimization problem for y = 1. Then, find the firm's cost function, i.e C(y) ∀y ≥ 0.
c) Find an algebraic expression for the firm's average cost function, AC(y), and for its marginal cost function, MC(y).
d) Find an algebraic expression for the firm's long-run supply function, y(p).
Question 2 - A competitive firm faces a two-input production function given by y = 3x1 + 5x2. Input prices are given by w1 = 10 and w2 = 6.
a) Represent graphically the firm's cost minimization problem, and then find algebraic expressions for its conditional factor demand functions, x1(w1, w2, y) and x2(w1, w2, y), as well as its cost function, C(w1, w2, y), for all possible input prices.
b) Given the above input prices, find the firm's long-run supply function, y(p).
c) Suppose that the firm cannot vary input 1 in the short run, which remains set at x1 = 5. Find an algebraic expression for the firm's short-run total cost function, and decompose it between variable and fixed costs. Then, find an expression for the firm's shortrun supply function, yS(p). How does it compare with the firm's long-run supply function?
d) What would happen if input prices were w1 = 6 and w2 = 10 instead? Find the firm's new long-run cost function, and an expression for the firm's long-run supply function, y(p).