An Economics Toolkit

# Cobb-Douglas Profit Maximization

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

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

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)

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)

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)

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:

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