A firm has the production function x = f (L) , where x is output and L is labor input. The firm buys the input in a competitive market.

(a) Assuming the firm sells its output in a competitive market, show that setting output where price equals marginal cost is equivalent to setting labor input where input price equals marginal value product.

(b) Assuming the firm is a monopoly, show that setting output where marginal revenue equals marginal cost is equivalent to setting labor input where input price equals marginal-revenue product.

(c) What restriction do we have to impose on the production function to ensure the second-order conditions in problems (a) and (b) are satisfied?