Consider a perfectly competitive firm that hires quantities x1 and x2 of inputs 1 and 2, at prices p1 and p2, and sells the resulting output in a perfectly competitive market at price p. The production function is given by f(x1, x2) = x1 + x2.

(a) Formulate the profit maximization problem for this firm and find the associated first-order conditions.

(b) Form the cost-minimization problem for this firm, find the contingent demand functions, and find the cost function for the firm.

(c) Use your cost function from (b) to find the profit-maximizing level of output, i.e., to find the value of q that maximizes pq C(q). Show that solving the first-order conditions in (a) leads to the same value of output. The conclusion here is that is does not matter whether we model firms as choosing inputs or output. As long as we keep track of the link between inputs and outputs provided by the production function, these are equivalent ways of looking at the firm, and we can use whichever is the most convenient.