Question

Let f:R+n​→R+​be a a firm's differentiable production function satisfying the usual assumptions. Suppose that x∈R+n​ is a vector of inputs and that w∈R+n​ are the factor prices. Let p∈R+​denote the output price. The profit of the firm is given by: π(p,w)=maxx​[pf(x)−wx] Derive and explain Shephard's lemma: ∂wi​∂π​=−xi​(p,w).

Answer 1

Shephard's lemma shows the relationship between input demand and input prices within a firm's profit-maximizing behavior, indicating that the change in profit with respect to an input price is equal to the negative quantity of input used.

Step-by-step explanation:

The student's question pertains to Shephard's lemma, a result in microeconomics that relates the change in cost with respect to changes in input prices to the demand for that input. Specifically, the question asks to derive and explain why the partial derivative of the profit function with respect to the price of input i (∂π/∂w_i) is equal to the negative quantity of input i used (-x_i(p,w)).

Addressing the student's question, we begin with the profit maximization problem π(p,w) = max_x [pf(x) - wx]. It implies that the firm chooses the inputs x to maximize the difference between total revenue (pf(x)) and total cost (wx). From the first-order condition of this optimization problem, we know that the marginal product of each input, weighted by the output price p, must equal its price (w_i).

Applying the envelope theorem to this optimization problem, we derive that the partial derivative of the profit function with respect to w_i, holding the output price p constant, equals the negative of the input quantity x_i. Shephard's lemma thus states that the shadow price of an input (its value in the profit function) is given by the amount of the input used in production. This reflects the firm's input demand and is crucial for understanding how changes in input prices affect a firm's choice of inputs and its production decisions.