1. Consider a production function of two inputs, labor and capital, given by Q = (√L + √K)². The marginal products associated with this production function are as follows:

MP_L = |L1/2 + K1/2|L1/2
MP_K = |L1/2 + K1/2|k-1/2

Let w = 2 and r = 1.

Find the equation of the firm's long-run total cost curve

2. When a firm uses K units of capital and L units of labor, it can produce Q units of output with the production function Q = K√L. Each unit of capital costs 20, and each unit of labor costs 25. The level of K is fixed at 5 units.

Find the equation of the firm's short-run total cost curve

3. Dave's Fresh Catfish is a northern Mississippi farm that operates in the perfectly competitive catfish farming industry. Dave's short-run total cost curve is STC(Q) = 400 + 2Q + 0.5Q², where Q is the number of catfish harvest per month. The corresponding short-run marginal cost curve is SMC(Q) = 2 + Q.

All of the fixed costs are sunk.

(a) What is the equation for the average variable cost (AVC)?

(b) What is the minimum level of average variable costs'?

(c) What is Dave's short-run supply curve?

4. The wood-pallet market contains many identical firms, each with the short-run total cost function STC(Q) = 400 + 5Q + Q², where Q is the firm's annual output (and all of the firm's $400 fixed cost is sunk). The corresponding marginal cost function is SMC(Q) = 5 + 2Q. The market demand curve for this industry is D(P) = 262.5 - P/2, where P is the market price. Each firm in the industry is currently earning zero economic profit. How many firms are in this industry, and what is the market equilibrium price?