Profit Maximization

1
Profit Maximization

• Molly W. Dahl
• Georgetown University
• Econ 101 Spring 2008

2
Economic Profit

• Suppose the firm is in a short-run circumstance
  in which
• Its short-run production function is

3
Economic Profit

• Suppose the firm is in a short-run circumstance
  in which
• Its short-run production function is
• The firms profit function is

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Short-Run Iso-Profit Lines

• A P iso-profit line contains all the production
  plans that provide a profit level P .
• A P iso-profit lines equation is

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Short-Run Iso-Profit Lines

• A P iso-profit line contains all the production
  plans that yield a profit level of P .
• The equation of a P iso-profit line is
• Rearranging

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Short-Run Iso-Profit Lines
has a slope of
and a vertical intercept of
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Short-Run Iso-Profit Lines
y
Increasing profit
x1
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Short-Run Profit-Maximization

• The firms problem is to locate the production
  plan that attains the highest possible iso-profit
  line, given the firms constraint on choices of
  production plans.

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Short-Run Profit-Maximization
y
Increasing profit
x1
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Short-Run Profit-Maximization
y
x1
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Short-Run Profit-Maximization
Given p, w1 and the
short-runprofit-maximizing plan is
y
x1
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Short-Run Profit-Maximization
Given p, w1 and the
short-runprofit-maximizing plan is And the
maximumpossible profitis
y
x1
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Short-Run Profit-Maximization
At the short-run profit-maximizing plan, the
slopes of the short-run production function and
the maximaliso-profit line areequal.
y
x1
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Short-Run Profit-Maximization
At the short-run profit-maximizing plan, the
slopes of the short-run production function and
the maximaliso-profit line areequal.
y
x1
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Short-Run Profit-Maximization
is the marginal revenue product
ofinput 1, the rate at which revenue
increaseswith the amount used of input 1. If
then profit increases with
x1. If then profit decreases
with x1.
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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
The equation of a short-run iso-profit lineis
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
y
x1
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Comparative Statics of SR Profit-Max
y
x1
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Comparative Statics of SR Profit-Max
y
x1
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Comparative Statics of SR Profit-Max

• An increase in w1, the price of the firms
  variable input, causes
• a decrease in the firms output level (the firms
  supply curve shifts inward), and
• a decrease in the level of the firms variable
  input (the firms demand curve for its variable
  input slopes downward).

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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
The equation of a short-run iso-profit lineis
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
y
x1
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Comparative Statics of SR Profit-Max
y
x1
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Comparative Statics of SR Profit-Max
y
x1
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Comparative Statics of SR Profit-Max

• An increase in p, the price of the firms output,
  causes
• an increase in the firms output level (the
  firms supply curve slopes upward), and
• an increase in the level of the firms variable
  input (the firms demand curve for its variable
  input shifts outward).

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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)

and
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Returns-to-Scale and Profit-Max

• If a competitive firms 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
y
y
Decreasingreturns-to-scale
x
x
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Returns-to-Scale and Profit-Max

• If a competitive firms 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
y
Increasing profit
y
y
Increasingreturns-to-scale
x
x
x
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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 firms technology
  exhibits constant returns-to-scale?

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Returns-to Scale and Profit-Max
y
Increasing profit
y
Constantreturns-to-scale
y
x
x
x
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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 firms 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
y
P 0
y
Constantreturns-to-scale
y
x
x
x