Profit Maximization and Cost Minimization

This week, Microeconomics focused on Profit Maximization and Cost Minimization. Profit maximization implies cost minimization; however, cost minimization doesn't necessarily mean profit maximization. Profit maximization is a combination of both revenue maximization and cost minimization because profits(denoted by pi) are equal to the difference between revenues and costs.

π = P*Y-w₁x₁-w₂x₂

Y=output
P=price of output
w₁=cost of input 1

w₂= cost of input 2

x₁=input 1

x₂=input 2

In the short run economy, one of the two hypothetical inputs has to be fixed. In order to solve the profit maximization problem in the short run, therefore, requires marginal benefit to equal marginal cost, with respect to the variable input. 

P*MP₁=w₁

(x₂=fixed)

A profit maximizing choice can only be found with decreasing returns to scale(DRTS); with increasing returns to scale(IRTS) or constant returns to scale(CRTS), no profit maximizing choice can be calculated. 

 Increasing returns to scale: Double the inputs results in more than double the output.
(With Cobb-Douglas production functions, the sum of the exponents is greater than 1)

Constant returns to scale: Doubling inputs results in double the output.
(With Cobb-Douglas production functions, the sum of the exponents is 1)

Decreasing returns to scale: A doubling of inputs results in less than double the output.
(With Cobb-Douglas production functions, the sum of the exponents is less than 1)

Increasing Returns to Scale

Constant Returns to Scale

Decreasing Returns to Scale

Long run profit maximization problems are solved by setting the Technical Rate of Substitution, the TRS, equal to the ratio of the input costs. The TRS is the slope of an isoquant, which is the function that includes all the combination of the inputs that can produce a given level of output. The TRS is equal to the marginal product of input 1 divided by the marginal product of input 2.

The amounts of each input which result in maximum profits are referred to as factor demands; they depend upon the price of output and both input costs.

X₁^*(w₁,w₂,P)

X₂^*(w₁,w₂,P)

Cost minimization problems are solved by finding the intersection of the isocost and isoquant lines. An isocost line includes all combinations of inputs which cost the same amount(a given amount). The amounts of each input which minimize costs are known as conditional factor demands and depend upon a given fixed level of output and both input costs.

X₁^*(w₁,w₂,Y)

X₂^*(w₁,w₂,Y)

Cost minimization problems(unlike those of profit maximization) can always be solved, regardless of the returns to scale. Short run cost minimization problems are easier to solve because one of the inputs has to be fixed. Cost functions are used to solve this type of problem, therefore it is helpful to be able to distinguish returns to scale based on average cost graphs. The average cost curve in the long run looks something like this.

a=IRTS  b=CRTS  c=DRTS