HOMEWORK

Q1. A differentiable supply function Q(p) maximizes a competitive firm's profit function given by

π(q; p) = pq - c(q)

for all p > 0, where p is the unit price of a single output, q ∈ R₊ is the level of this output, and c(q) a twice-differentiable cost function. State and derive a necessary condition for profit maximization in this instance.

Q2. Ken hires a single input x at a competitive price w. Four observations of Ken's behavior are made: (w¹, x¹) = (2, 9), (w², x²) = (5, 3), (w³, x³) = (3, 6), and (w⁴, x⁴) = (8, 1). The question arises whether these data could have been drawn from the profit-maximizing activities of Ken who always chooses a unique profit-maximizing input for each wage.

(a) Define a notion of profit rationality for observations {(wⁱ, xⁱ)}_i=1⁴ that is relevant to answering that question.

(b) Is the given data profit-rational in the sense you defined in (a) above? Explain clearly.

Q3. Tjalling purchases a single competitive input and is observed to choose the following input levels at different input prices: (w₁, x₁) = (10, 450), (w², x²) = (50, 330) and (w³, x³) = (70, 270). The question arises whether these data could have been drawn from the profit-maximizing activities of Tjalling who always chooses a unique profit-maximizing input for each wage level.

(a) Define a notion of profit rationality for observations {(wⁱ, xⁱ)}_i=1³ that is relevant to answering that question.

(b) Find a profit function (of the usual kind) that rationalizes this data in the sense you defined in part (a). Be sure to demonstrate that this profit function rationalizes this data!

Q4. Consider a producer who purchases a single competitive input and is observed to choose different input levels at different input prices.

Supposed the observed data set is S = {(w¹, x¹), (w², x²), (w³, x³)}, where w¹ ≠ w² ≠ w³ ≠ w¹, (i.e., the input prices are distinct).

Assume that for any two observations, the Weak Axiom of Revealed Profit (WARII) holds, i.e., (wⁱ - w^j)(xⁱ - x^j) < 0 for i, j = 1, 2, 3, i ≠ j. Prove that this implies the Strong Axiom of Revealed Profit (SARII), i.e.,

w¹(x¹ - x²) + w²(x² - x³) + w³(x³ - x¹) < 0.

Hint: Without loss of generality, suppose w¹ > w² > w³. Show that if x¹ ≠ x² ≠ x³ ≠ x¹, then x¹ < x² < x³. Now suppose that the SARII condition above does not hold. Add and subtract w²(x² - x¹) and show a contradiction.