This problem is meant to teach you with a concrete example how to derive the short-run and long-run cost functions and supply curves for a firm in a competitive market, using the two-step approach.

Suppose the firm in question uses two factors, capital (K) and labor (L) for production, and its production function is given by Q=K1/2L1/2. The rent for capital is r, the wage for labor is w, and the market price of its product is p.

1. a)Assume that the firm's capital stock K is fixed at K=1 in the short run. Derive the firm's short-run cost function.
2. b)What is the firm's fixed cost? What are the total variable cost, average variable cost, and marginal cost of producing Q units of output?
3. c)What is the firm's profit s of producing Q units of output? What is the firm's short-run supply curve, i.e., the optimal output as a function of output price, Qs*(p)?

Now let's turn to the firm's long run decision, where both capital and labor inputs are variable. We first

1. d)When K units of capital and L units of labor are used, what is the marginal product of capital and what is the marginal product of labor? What is the cost-minimizing input rule in this context?
2. e)What is the optimal capital and labor inputs to produce Q units of output?
3. f)What is the firm's long-run cost function Cl(Q)?
4. g)In the long run, what is the firm's fixed cost? What are the total variable cost, average variable
5. cost, and marginal cost of producing Q units of output?
6. h)What is the firm's long-run profit l of producing Q units of output? What is the firm's long-run
7. supply curve Ql*(p)?