Comparative Statics with Production

This module considers how agents reason about changes in the prices at which they can sell their goods, as well as changes in the costs of producing those goods. In particular, we can test how this affects their optimal choice of inputs to their production process (e.g., how many people to hire or robots to lease). We test comparative statics on the prices of inputs to the production function, changes to the underlying production technology, and substitution between goods for classic production functions such as Cobb-Douglas and Leontief. Analogous to the relationship between Deriving Demand and Comparative Statics of Demand, these tests involve comparative statics of the argmax from the profit maximization of Deriving Factor Demand—i.e., using an Envelope theorem and perturbing factor prices.

Elements

1. Response of Optimal Labor to the Wage

   Finding how a price-taking firm's profit-maximising labor changes with the wage, for a Cobb-Douglas or a Leontief technology.

2. Price Elasticity of Supply

   Computing the price elasticity of supply at a given price for a linear or quadratic supply function.

3. Total Factor Productivity

   Recovering a firm's total factor productivity A from its production function and either its output or the input prices it faces.

4. Shephard's Lemma

   Finding a firm's cost-minimising labor or capital as the derivative of its cost function with respect to that input's price.