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M18.L04 — Profit Maximisation and the Competitive Firm

Module: Module 18 — Technology, Production, and Profit Maximisation Lesson: L04 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Principles of Microeconomics (Curtis and Irvine) | University of Utah Intermediate Microeconomics

Learning Objective

Derive profit-maximizing conditions for a competitive firm using calculus.

Profit Maximisation

The firm maximizes profit \(\pi = py - w_1x_1 - w_2x_2\), where \(p\) is output price.

Key Conditions:

  1. Input Side: \(p \times MP_i = w_i\) for each input \(i\).
  2. Output Side: \(p = MC\) (marginal cost).
  3. Supply Function: \(y^*(p, w_1, w_2)\) solves the profit-maximization problem.
  4. Hotelling’s Lemma: \(\frac{\partial \pi}{\partial p} = y^*\).
  5. Profit Function Properties:
  6. Non-decreasing in \(p\).
  7. Non-increasing in input prices.
  8. Convex in prices.

Worked Example

[Australian context]

An Australian winery has \(y = x_1^{0.4}x_2^{0.4}\), with \(w_1 = 10\) AUD, \(w_2 = 5\) AUD, and \(p = 20\) AUD/bottle. 1. Profit Maximization: [ p \times MP_1 = 20 (0.4 x_1^{-0.6} x_2^{0.4}) = 10 \implies x_2 = 2x_1 ] [ p \times MP_2 = 20 (0.4 x_1^{0.4} x_2^{-0.6}) = 5 \implies x_1 = 2x_2 ] Contradiction implies corner solution or DRS adjustment. 2. Supply Function (Simplified): If \(y = x_1^{0.5}x_2^{0.5}\) (CRS), supply is infinitely elastic at \(p = MC\).

Common Misconception

"Profit maximization always occurs where \(p = MC\)."

Only true if MC is increasing. For CRS, profits are zero or infinite.

Key Takeaways

  • Profit maximization equates the value of marginal product to input prices.
  • Hotelling’s Lemma links profit sensitivity to output supply.
  • CRS technologies lead to zero profits in competitive equilibrium.

Practice

  1. For \(y = x_1^{0.3}x_2^{0.3}\) and \(w_1 = w_2 = 1\), find \(y^*(p)\).
  2. Verify Hotelling’s Lemma for \(\pi = p^2/4w\).

Further Resources

Provenance: Principles of Microeconomics (Curtis and Irvine) | University of Utah Intermediate Microeconomics