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M19.L02 — The Competitive Firm's Supply Decision

Module: Module 19 — Costs, Competitive Supply, and Industry Equilibrium Lesson: L02 of 05 Duration: ~30 minutes Level: Intermediate Provenance: MIT OCW 14.01 Principles of Microeconomics | University of Utah Intermediate Microeconomics

Learning Objective

Derive a competitive firm's supply function from profit maximization.

The Competitive Firm's Supply Decision

In a perfectly competitive market, firms are price takers and maximize profit by choosing output \(y\) such that price \(p\) equals marginal cost \(MC\): \(p = MC(y, w)\). The supply function \(y^*(p, w)\) is derived from this condition.

Key properties of the supply function: - Non-decreasing in \(p\): Higher prices incentivize greater output. - Homogeneous degree 0 in \(p, w\): Doubling all prices leaves output unchanged.

The profit function \(\pi(p, w)\) represents maximal profits and is convex in \(p\). Hotelling's Lemma states that \(\frac{\partial \pi}{\partial p} = y^*(p, w)\), linking profit to supply.

Comparative statics: How does \(y^*\) change with \(p\) and \(w\)? - \(\frac{\partial y^*}{\partial p} \geq 0\) (upward-sloping supply). - \(\frac{\partial y^*}{\partial w} \leq 0\) (higher input prices reduce supply).

Worked Example

[Australian context]

A Melbourne bakery produces bread with cost function \(c(y) = 2y^2 + 10\). The market price is \(p = 24\) AUD/loaf.

  1. Supply decision: Set \(p = MC\), where \(MC = \frac{dc}{dy} = 4y\). Thus, \(24 = 4y\) ⇒ \(y^* = 6\).
  2. Profit: \(\pi = py - c(y) = 24 \cdot 6 - (2 \cdot 6^2 + 10) = 144 - 82 = 62\) AUD.
  3. Hotelling's Lemma: Verify \(\frac{\partial \pi}{\partial p} = y^* = 6\).

Common Misconception

**"Firms always profit in competitive markets."

Correction: Firms earn zero economic profit in long-run equilibrium due to free entry/exit. Short-run profits can be positive, negative, or zero.

Key Takeaways

  • Competitive firms supply where \(p = MC\).
  • Supply functions are non-decreasing in price and homogeneous degree 0.
  • Hotelling's Lemma links profit and supply functions.
  • Higher input prices reduce supply.

Practice

  1. Find the supply function for \(c(y) = y^3 + 5y + 10\).
  2. Show that \(y^*(p, w)\) is homogeneous degree 0.
  3. Calculate \(\frac{\partial y^*}{\partial w}\) for \(c(y) = w y^2\).

Further Resources

Provenance: MIT OCW 14.01 Principles of Microeconomics | University of Utah Intermediate Microeconomics