Profit Maximization Molly W. Dahl Georgetown University Econ 101 - - PowerPoint PPT Presentation

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Profit Maximization

Molly W. Dahl Georgetown University Econ 101 – Spring 2009

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Economic Profit

Suppose the firm is in a short-run

circumstance in which

Its short-run production function is The firm’s profit function is

y f x x = ( ,~ ).

1 2

Π = − − py w x w x

1 1 2 2

~ .

x x

2 2

≡ ~ .

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Short-Run Iso-Profit Lines

A $Π iso-profit line contains all the

production plans that provide a profit level $Π .

A $Π iso-profit line’s equation is

Π ≡ − − py w x w x

1 1 2 2

~ .

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Short-Run Iso-Profit Lines

A $Π iso-profit line contains all the

production plans that yield a profit level of $Π .

The equation of a $Π iso-profit line is Rearranging

Π ≡ − − py w x w x

1 1 2 2

~ . y w p x w x p = + +

1 1 2 2

Π ~ .

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Short-Run Iso-Profit Lines y w p x w x p = + +

1 1 2 2

Π ~

has a slope of

+ w p

1

and a vertical intercept of

Π + w x p

2 2

~ .

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Short-Run Iso-Profit Lines

Π Π ≡ ′

Π Π ≡ ′′

Π Π ≡ ′′′

Increasing profit y x1 Slopes w p = +

1

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Short-Run Profit-Maximization

The firm’s problem is to locate the

production plan that attains the highest possible iso-profit line, given the firm’s constraint on choices of production plans.

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Short-Run Profit-Maximization

x1 Increasing profit Slopes w p = +

1

y

y f x x = ( ,~ )

1 2

Π Π ≡ ′

Π Π ≡ ′′

Π Π ≡ ′′′

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Short-Run Profit-Maximization

x1 y

Π Π ≡ ′

Π Π ≡ ′′

Π Π ≡ ′′′

Slopes w p = +

1

x1

*

y*

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Short-Run Profit-Maximization

x1 y Slopes w p = +

1

Given p, w1 and the short-run profit-maximizing plan is And the maximum possible profit is

x x

2 2

≡ ~ , ( , ~ , ).

* *

x x y

1 2 ′′ Π .

Π Π ≡ ′′

x1

*

y*

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Short-Run Profit-Maximization

x1 y Slopes w p = +

1

At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximal iso-profit line are equal. MP w p at x x y

1 1 1 2

= ( ,~ , )

* *

Π Π ≡ ′′

x1

*

y*

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Short-Run Profit-Maximization

MP w p p MP w

1 1 1 1

= ⇔ × =

p MP ×

1 is the marginal revenue product of

input 1, the rate at which revenue increases with the amount used of input 1. If then profit increases with x1. If then profit decreases with x1.

p MP w × >

1 1

p MP w × <

1 1

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Short-Run Profit-Max: A Cobb-Douglas Example

In class

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Comparative Statics of SR Profit-Max

What happens to the short-run profit-

maximizing production plan as the variable input price w1 changes?

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Comparative Statics of SR Profit-Max y w p x w x p = + +

1 1 2 2

Π ~

The equation of a short-run iso-profit line is so an increase in w1 causes

• - an increase in the slope, and
• - no change to the vertical intercept.

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Comparative Statics of SR Profit-Max

x1

Π Π ≡ ′

Π Π ≡ ′′

Π Π ≡ ′′′

Slopes w p = +

1

y

y f x x = ( ,~ )

1 2

x1

*

y*

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Comparative Statics of SR Profit-Max

x1 Slopes w p = +

1

y

y f x x = ( ,~ )

1 2

x1

*

y* Π Π ≡ ′ Π Π ≡ ′′

Π Π ≡ ′′′

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Comparative Statics of SR Profit-Max

x1 Slopes w p = +

1

y

y f x x = ( ,~ )

1 2

x1

*

y* Π Π ≡ ′ Π Π ≡ ′′

Π Π ≡ ′′′

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Comparative Statics of SR Profit-Max

An increase in w1, the price of the firm’s

variable input, causes

a decrease in the firm’s output level, and a decrease in the level of the firm’s

variable input.

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Comparative Statics of SR Profit-Max

What happens to the short-run profit-

maximizing production plan as the output price p changes?

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Comparative Statics of SR Profit-Max y w p x w x p = + +

1 1 2 2

Π ~

The equation of a short-run iso-profit line is so an increase in p causes

• - a reduction in the slope, and
• - a reduction in the vertical intercept.

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Comparative Statics of SR Profit-Max

x1

Π Π ≡ ′

Π Π ≡ ′′

Π Π ≡ ′′′

Slopes w p = +

1

y

y f x x = ( ,~ )

1 2

x1

*

y*

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Comparative Statics of SR Profit-Max

x1 Slopes w p = +

1

y

y f x x = ( ,~ )

1 2

x1

*

y*

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Comparative Statics of SR Profit-Max

x1 Slopes w p = +

1

y

y f x x = ( ,~ )

1 2

x1

*

y*

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Comparative Statics of SR Profit-Max

An increase in p, the price of the firm’s

• utput, causes

an increase in the firm’s output level, and an increase in the level of the firm’s

variable input.

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Long-Run Profit-Maximization

Now allow the firm to vary both input

levels (both x1 and x2 are variable).

Since no input level is fixed, there are

no fixed costs.

For any given level of x2, the profit-

maximizing condition for x1 must still hold.

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Long-Run Profit-Maximization

The input levels of the long-run profit-maximizing

plan satisfy

That is, marginal revenue equals marginal cost

for all inputs.

Solve the two equations simultaneously for the

factor demands x1(p, w1, w2) and x2(p, w1, w2)

p MP w × − =

2 2

0.

p MP w × − =

1 1

0 and

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Returns-to-Scale and Profit-Max

If a competitive firm’s technology exhibits

decreasing returns-to-scale then the firm has a single long-run profit-maximizing production plan.

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Returns-to Scale and Profit-Max

x y

y f x = ( )

y* x*

Decreasing returns-to-scale

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Returns-to-Scale and Profit-Max

If a competitive firm’s technology exhibits

exhibits increasing returns-to-scale then the firm does not have a profit-maximizing plan.

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Returns-to Scale and Profit-Max

x y

y f x = ( )

y” x’

Increasing returns-to-scale

y’ x” Increasing profit

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Returns-to-Scale and Profit-Max

So an increasing returns-to-scale

technology is inconsistent with firms being perfectly competitive.

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Returns-to-Scale and Profit-Max

What if the competitive firm’s technology

exhibits constant returns-to-scale?

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Returns-to Scale and Profit-Max

x y

y f x = ( )

y” x’

Constant returns-to-scale

y’ x” Increasing profit

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Returns-to Scale and Profit-Max

So if any production plan earns a positive

profit, the firm can double up all inputs to produce twice the original output and earn twice the original profit.

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Returns-to Scale and Profit-Max

Therefore, when a firm’s technology

exhibits constant returns-to-scale, earning a positive economic profit is inconsistent with firms being perfectly competitive.

Hence constant returns-to-scale requires

that competitive firms earn economic profits of zero.

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Returns-to Scale and Profit-Max

x y

y f x = ( )

y” x’

Constant returns-to-scale

y’ x”

Π = 0