Questions: Duality in Production: Profit Function and Hotelling's Lemma

Question 1

A firm's profit function is π(p, w₁, w₂). An economist wants to know how much of input 1 the firm uses at its optimum without solving the full optimization problem. Using duality theory, what should she compute?

• A. The second derivative ∂²π/∂w₁², which gives input demand curvature
• B. -∂π/∂w₁, which by Hotelling's lemma equals the optimal demand for input 1
• C. ∂π/∂p divided by w₁, scaling output supply by the input price
• D. π(p, w)/w₁, the average profit per unit of input cost

Answer: (B)

Hotelling's lemma states that ∂π/∂wᵢ = −xᵢ*(p, w): the partial derivative of the profit function with respect to an input price equals the negative of the profit-maximizing demand for that input. The negative sign is intuitive — higher input prices reduce profit, and the rate of reduction equals how much of that input the firm uses. The power of this result is that you recover the entire factor demand function by a single differentiation of π, rather than re-solving the optimization problem for each price.

Question 2

A firm faces an output price that fluctuates between a high and low value with equal probability. The firm adjusts its production plan in response to prices. Compared to a firm facing the average price with certainty, how do expected profits compare?

• A. Expected profits are lower under price volatility, because uncertainty always hurts firms
• B. Expected profits are equal, because the average price is the same in both cases
• C. Expected profits are higher under price volatility, because the profit function is convex in prices
• D. The comparison depends on the specific functional form of the production function

Answer: (C)

The profit function π(p, w) is convex in prices. By Jensen's inequality, for a convex function f, E[f(x)] ≥ f(E[x]). Therefore, average profits under a fluctuating price exceed profits at the average price: the firm benefits from volatility because it can expand production when prices are high and contract when prices are low, exploiting the upside more than it suffers on the downside. This is a direct consequence of convexity — not a behavioral assumption but a property of optimization.

Question 3

The homogeneity of degree one of the profit function in (p, w) means that if all prices double, the firm's optimal input-output quantities remain unchanged.

• T. True
• F. False

Answer: True

Homogeneity of degree one means π(λp, λw) = λπ(p, w) for all λ > 0. This implies no money illusion: if all nominal prices scale by the same factor, real relative prices are unchanged, so the profit-maximizing plan (output quantity, input quantities) stays the same — only nominal profit doubles. This is analogous to demand being homogeneous of degree zero in prices and income in consumer theory. It is a sanity check on any estimated profit function.

Question 4

Because the profit function is convex in input prices, a firm exposed to volatile input prices is worse off than one facing stable input prices at the same average level.

• T. True
• F. False

Answer: False

This is the opposite of the truth. Convexity in (p, w) means Jensen's inequality applies in all price dimensions, including input prices. A firm facing volatile input prices is better off on average than one facing the average price with certainty, because it can adjust its input mix: when input prices are low, it uses more of that input; when prices are high, it substitutes away. The ability to optimize at each price realization — rather than being locked into a plan based on average prices — is precisely what convexity captures.

Question 5

What does Hotelling's lemma reveal about the relationship between the profit function and the firm's behavioral choices, and why is this practically valuable?

Model answer: Hotelling's lemma states that the firm's optimal output supply and factor demands can be recovered by differentiating the profit function with respect to prices: ∂π/∂p = y*(p, w) and ∂π/∂wᵢ = −xᵢ*(p, w). The profit function encodes all of the firm's optimal behavior — you don't need to re-solve the optimization problem for each new price vector. Practically, this allows empirical economists to estimate a flexible functional form for π from observed price and profit data, then differentiate to obtain supply and demand functions that are automatically consistent with profit-maximizing behavior.

The key insight is that optimization leaves a mathematical fingerprint on the profit function. Because π represents a maximum, its derivatives must equal the optimal quantities being chosen. This is the duality principle: the profit function is not just a number summarizing performance but a complete encoding of the firm's decision rule. Comparative statics (how supply and demands respond to price changes) follow immediately from the second derivatives of π, without ever touching the primal production function.