An Economics Toolkit
Utility Maximization Problems
Profit Maximization Short-run Profit Maximization
Profit Maximization Short-run Profit Maximization
Profit Maximization Short-run Profit Maximization

Cobb-Douglas Profit Maximization

Production Function (in the form Ax1αx2β)

A: α: β:

Input Prices

Price of input 1: Price of input 2:
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 αx1+βx2)

α: β:

Input Prices

Price of input 1: Price of input 2:
The firm will supply:
To find the cost function, we want to minimize the objective function w1x1+w2x2 subject to the constraint αx1+βx2=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{αx1, βx2})

α: β:

Input Prices

Price of input 1: Price of input 2:
The firm will supply:
To find the cost function, we want to minimize the objective function w1x1+w2x2 subject to the constraint min{αx1, βx2}=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 Ax1αx2β)

A: α: β:
Fixed input:
x1 is fixed at:
x2 is fixed at:

Input Prices

Price of input 1: Price of input 2:
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 αx1+βx2)

α: β:
Fixed input:
x1 is fixed at:
x2 is fixed at:

Input Prices

Price of input 1: Price of input 2:
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{αx1, βx2})

α: β:
Fixed input:
x1 is fixed at:
x2 is fixed at:

Input Prices

Price of input 1: Price of input 2:
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