A small firm produces raw lumber in a competitive market using just capital and labor. It hires L hours of labor at wage w/hour. Capital is expensive, so it rents K units of the equipment (including chainsaws and trucks) at a cost of r/hour. The firm produces felled trees (logs) according to the Cobb-Douglas production function = , where > 0, 0, 0 and + < 1.

(1). Calculate the total cost function, (), and the (inverse) supply function for the firm. Explain if there are fixed costs.

(2). Now, suppose that in order to rent equipment, the firm has to commit to a contract to purchase a fixed amount of capital, 0. That is, it cannot adjust its capital in the short-run, and will have to pay 0 until the contract expires and it can enter into a new contract. Calculate the firm's short-run total cost function and (inverse) supply function. Explain if there are fixed costs. For the rest, assume that = 2, = 0.5, = 0.25.

(3). If wages are w=10, the cost of capital is r=40, and logs sell for p=80, calculate how much capital the firm would want to rent and find the long-run equilibrium.

(4). Suppose now that regulators impose safety regulations that cause the effective wage to increase. If wages increase to w=20, explain what the firm will do in the short-fun and find the short-run equilibrium. In this equilibrium, find the value of the marginal product of capital and of labor. Explain if these values follow the usual profit maximization rule.

(5). Explain how the firm will adjust to the new wages in the long-run and find the long-run equilibrium. Calculate the new short-run marginal cost function. Explain if it is higher or lower than those you found in part (4). Explain intuitively why the firm chose to make it so.

(6). On a single graph, carefully graph the equilibria from parts (3), (4), and (5) in (p,q)-space, including the prices, quantities, and all relevant supply curves. Carefully label the different curves and equilibrium quantities.

(7). Explain intuitively how "sticky" factors of production relate to supply elasticity. Further describe why it is important for policy-makers to consider "stickiness" when implementing policies that affect firms' costs. Explain how policy-makers might alter policies to account for this. Use a concrete example(s) to make your point.