Skip to content

M19.L01 — Cost Functions: Short-Run and Long-Run

Module: Module 19 — Costs, Competitive Supply, and Industry Equilibrium Lesson: L01 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | Principles of Microeconomics (Curtis and Irvine)

Learning Objective

Understand the derivation and properties of short-run and long-run cost functions.

Cost Functions: Short-Run and Long-Run

In microeconomics, firms face different cost structures depending on the time horizon. In the short-run, at least one input (typically capital, \(k\)) is fixed, leading to the short-run cost function \(c(w, y, \overline{k})\), where \(w\) is the vector of input prices, \(y\) is output, and \(\overline{k}\) is fixed capital. In the long-run, all inputs are variable, so the cost function is \(c(w, y)\).

The long-run cost function serves as an "envelope" of the short-run cost functions, representing the minimal cost for each output level when all inputs are adjustable. Key properties include: - Non-decreasing in input prices (\(w\)): Higher input prices cannot lower costs. - Homogeneous degree 1 in \(w\): Doubling all input prices doubles total cost. - Concave in \(w\): Cost functions exhibit diminishing marginal costs of input price changes.

Shephard's Lemma states that the derivative of the cost function with respect to an input price gives the conditional demand for that input: \(\frac{\partial c}{\partial w_i} = x_i^*(w, y)\).

Worked Example

[Australian context]

Consider a Sydney-based firm producing coffee with a Cobb-Douglas production function \(y = k^{0.5} l^{0.5}\). In the short-run, capital is fixed at \(\overline{k} = 100\). The wage rate is \(w_l = 20\) AUD/hour, and the rental rate is \(w_k = 10\) AUD/hour.

  1. Short-run cost function: Solve for labor demand \(l\) from the production function: \(l = \frac{y^2}{100}\). Then, \(c(w, y, \overline{k}) = w_l l + w_k \overline{k} = 20 \cdot \frac{y^2}{100} + 10 \cdot 100 = 0.2 y^2 + 1000\).

  2. Long-run cost function: Minimize \(c = w_l l + w_k k\) subject to \(y = k^{0.5} l^{0.5}\). The solution is \(k = y \sqrt{\frac{w_l}{w_k}}\) and \(l = y \sqrt{\frac{w_k}{w_l}}\), leading to \(c(w, y) = 2y \sqrt{w_l w_k} = 2y \sqrt{200} \approx 28.28 y\).

Common Misconception

**"Long-run costs are always lower than short-run costs because firms can adjust all inputs."

Correction: Long-run costs are never higher than short-run costs (due to flexibility), but they can be equal if the fixed input in the short-run is already at its optimal long-run level.

Key Takeaways

  • Short-run cost functions include fixed inputs; long-run functions do not.
  • The long-run cost function is the lower envelope of short-run cost functions.
  • Shephard's Lemma links cost functions to conditional input demands.
  • Cost functions are concave and homogeneous in input prices.

Practice

  1. Derive the short-run cost function for a firm with \(y = 2k^{0.3} l^{0.7}\), \(\overline{k} = 50\), \(w_l = 15\), and \(w_k = 10\).
  2. Explain why the long-run cost function is concave in input prices.
  3. Calculate the conditional labor demand using Shephard's Lemma for \(c(w, y) = 5y w_l^{0.4} w_k^{0.6}\).

Further Resources

Provenance: Intermediate Microeconomics with Excel (Barreto) | Principles of Microeconomics (Curtis and Irvine)