Questions: Producer Duality: Cost and Profit Functions

Question 1

A firm's cost function is C(w₁, w₂, y) = 2y · w₁^(1/2) · w₂^(1/2). According to Shephard's lemma, what is the firm's conditional demand for input 1?

• A. ∂C/∂y = 2w₁^(1/2)·w₂^(1/2) — the rate at which cost rises with output
• B. ∂C/∂w₁ = y·w₁^(−1/2)·w₂^(1/2) — the derivative of cost with respect to the price of input 1
• C. C/w₁ = 2y·w₂^(1/2)·w₁^(−1/2) — total cost divided by the input price
• D. The production function must be specified; the cost function alone cannot identify factor demands

Answer: (B)

Shephard's lemma states: x*ᵢ(w, y) = ∂C(w, y)/∂wᵢ. Differentiating C = 2y·w₁^(1/2)·w₂^(1/2) with respect to w₁ gives ∂C/∂w₁ = 2y·(1/2)·w₁^(−1/2)·w₂^(1/2) = y·w₁^(−1/2)·w₂^(1/2). This is the conditional demand for input 1 — obtained by a single differentiation, without re-solving the constrained optimization problem. Option D is precisely the misconception Shephard's lemma refutes: by duality, the cost function contains exactly the technological information needed to recover factor demands.

Question 2

An applied economist has estimated a firm's cost function from market data on input prices and expenditures, but has never directly observed the firm's production technology. She claims she can derive the firm's input demand functions for any set of input prices from this data alone. Is her claim valid?

• A. No — the production function is the fundamental object; the cost function is derived from it and cannot contain more information
• B. Yes — by Shephard's lemma, differentiating the cost function with respect to each input price directly yields the conditional factor demand functions
• C. Partially — she can find input quantities but not substitution elasticities, which require the production function
• D. No — she also needs data on the firm's output prices to identify factor demand functions

Answer: (B)

This is the practical payoff of producer duality. The cost function contains exactly the same technological information as the production function — duality establishes they are two equivalent representations of the same technology. Shephard's lemma gives a direct route from cost function to factor demands: x*ᵢ(w, y) = ∂C/∂wᵢ. Cost functions are typically easier to estimate from observable market data than production functions, which require direct observation of physical input-output relationships that firms rarely report. The claim is valid and represents duality theory's most powerful empirical application.

Question 3

A cost function that is homogeneous of degree one in input prices means that if all input prices double, total minimum cost exactly doubles.

• T. True
• F. False

Answer: True

Homogeneity of degree one in input prices is not an assumption — it is a theorem, a consequence of cost minimization. If all input prices scale by factor t, the cheapest way to produce a given output is unchanged (the same input combination minimizes cost), but every unit of input costs t times as much. So total cost scales by exactly t: C(tw, y) = t·C(w, y). This property provides a useful consistency check: if an estimated cost function predicts that costs more than double when all prices double, the estimate is inconsistent with cost-minimizing behavior and should be rejected.

Question 4

Duality in producer theory means the cost function is a simplified summary of the production function, so working directly with the production function usually provides more complete technological information.

• T. True
• F. False

Answer: False

Duality establishes that the cost function and the production function contain *exactly the same technological information* — they are dual representations of the same technology, not one a simplification of the other. Every regularity of the production function (curvature, returns to scale, factor substitutability) has a precise mathematical counterpart in the cost function, and vice versa. In empirical work, cost functions are often *preferred* because input prices and expenditure data are observable, while production function estimation requires controlled conditions or strong identifying assumptions that are rarely available in field data.

Question 5

State Shephard's lemma precisely and explain why it is practically powerful for economists studying firm behavior.

Model answer: Shephard's lemma states that the conditional factor demand for input i equals the partial derivative of the cost function with respect to the price of input i: x*ᵢ(w, y) = ∂C(w, y)/∂wᵢ. It is practically powerful because it allows researchers to recover the firm's entire input demand system by differentiating a single estimated cost function — without re-solving the optimization problem and without ever directly observing the production function. Since cost functions can be estimated from market data on prices and expenditures, economists can infer how firms substitute between inputs as prices change, and by extension the shape of the underlying production technology, purely from observed cost behavior.

Industrial organization economists and regulators routinely estimate flexible cost functions (translog, CES) from accounting or market data and differentiate to get factor demand elasticities and Allen-Uzawa substitution elasticities. This is more feasible than estimating production functions directly, which require detailed data on input quantities and physical output that firms rarely report. Shephard's lemma is what makes the cost-function approach complete: you lose nothing by switching from the primal (production) to the dual (cost) representation.