What is Hotelling’s Lemma: How to Compute Profit Function and Competitive Supply (microeconomics tutoring)

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Hotelling’s Lemma: Profit Function and Competitive Supply (microeconomics tutoring)
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Hotelling’s Lemma is a central result in microeconomics and producer theory. It states that the derivative of the profit function with respect to price yields the firm’s supply function. This connects duality, optimization, and observable firm behavior.

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Hotelling’s Lemma: Supply q*(p) = ∂π(p, w) / ∂p, where π(p, w) is the profit function.

The lemma shows that competitive supply can be recovered directly from the profit function, without solving the firm’s optimization problem again. It is the producer‑side analogue of Shephard’s Lemma in cost minimization.

Why Hotelling’s Lemma Matters

Hotelling’s Lemma is powerful because it:

  • links profit maximization to observable supply behavior
  • provides a dual representation of the firm’s problem
  • simplifies comparative statics and welfare analysis
  • appears frequently in graduate micro and prelim exams

How to Derive Hotelling’s Lemma Step by Step

  1. Start with the profit maximization problem:
    π(p, w) = maxₓ [p·f(x) − w·x].
  2. Let x*(p, w) be the optimal input choice.
    This solves the firm’s first‑order conditions.
  3. Apply the envelope theorem:
    ∂π/∂p = f(x*(p, w)) = q*(p). The derivative with respect to price picks up only the direct effect.
  4. Interpretation:
    The slope of the profit function with respect to price equals the firm’s supply.
  5. Duality insight:
    Just as Shephard’s Lemma recovers Hicksian demand from the cost function, Hotelling’s Lemma recovers supply from the profit function.

Numerical Example

Suppose a competitive firm has production function q = √x and input price w.

Profit function: π(p, w) = maxₓ [p√x − wx].

First‑order condition: (p / (2√x)) − w = 0 → x* = (p² / (4w²)).

Output: q* = √x* = p / (2w).

Now compute ∂π/∂p:

π(p, w) = p·(p / (2w)) − w·(p² / (4w²)) = p²/(2w) − p²/(4w) = p²/(4w).

∂π/∂p = (2p)/(4w) = p/(2w) = q*.

Hotelling’s Lemma holds exactly.

Common Mistakes

  • Confusing supply with marginal cost
  • Forgetting that the lemma requires profit maximization
  • Applying the lemma to non‑competitive firms
  • Ignoring the envelope theorem’s role

Why This Matters in Producer Theory

Hotelling’s Lemma is essential for:

  • deriving supply functions quickly
  • understanding duality in production
  • computing welfare effects of price changes
  • building general equilibrium models

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