# Cobb-Douglas Profit Maximization

## Production Function (in the form Ax_{1}^{α}x_{2}^{β})

## Input Prices

The firm will supply:

First, we set up the cost minimization problem to find the conditional input demands.

Lagrangian:

First order conditions:

Solving the system:

Thus, the firm's cost function is as follows:

Then, for profit maximization, we have the following objective function:

Taking the first order condition with respect to y:

# Perfect Substitutes Profit Maximization

## Production Function (in the form αx_{1}+βx_{2})

## Input Prices

The firm will supply:

To find the cost function, we want to minimize the objective function w

_{1}x_{1}+w_{2}x_{2}subject to the constraint αx_{1}+βx_{2}=y.Calculus will not help here; instead, it is intuitive to realize that the firm will only use the input with the lower marginal cost, given by

Thus, the firm's cost function is as follows:

Then, for profit maximization, we have the following objective function:

Taking the first order condition with respect to y:

Since the perfect substitutes (linear) production function exhibits constant returns to scale, this implies that the firm only produces when price equals marginal cost, or

# Perfect Complements Profit Maximization

## Production Function (in the form min{αx_{1}, βx_{2}})

## Input Prices

The firm will supply:

To find the cost function, we want to minimize the objective function w

_{1}x_{1}+w_{2}x_{2}subject to the constraint min{αx_{1}, βx_{2}}=y.Calculus will not help here; instead, it is intuitive to realize that the firm's cost is minimized at the kink of each isoquant, where

Thus, the firm's cost function is as follows:

Then, for profit maximization, we have the following objective function:

Taking the first order condition with respect to y:

Since the perfect complements (Leontief) production function exhibits constant returns to scale, this implies that the firm only produces when price equals marginal cost, or

# Short-run Profit Maximization (Cobb-Douglas)

## Production Function (in the form Ax_{1}^{α}x_{2}^{β})

Fixed input:
x

_{1}is fixed at:
x

_{2}is fixed at:## Input Prices

The firm will supply:

Thus, the firm's short-run cost function is

Then, for profit maximization, we have the following objective function:

Taking the first order condition with respect to y:

# Short-run Profit Maximization (Perfect Substitutes)

## Production Function (in the form αx_{1}+βx_{2})

Fixed input:
x

_{1}is fixed at:
x

_{2}is fixed at:## Input Prices

The firm will supply:

Then, for profit maximization, we have the following objective function:

Taking the first order condition with respect to y:

# Short-run Profit Maximization (Perfect Complements)

## Production Function (in the form min{αx_{1}, βx_{2}})

Fixed input:
x

_{1}is fixed at:
x

_{2}is fixed at:## Input Prices

The firm will supply:

Then, for profit maximization, we have the following objective function:

Taking the first order condition with respect to y:

Contact information: sxshi@uchicago.edu