1. Molly W. Dahl Georgetown University Econ 101 – Spring 2008
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 firm’s profit function is

4. 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 line’s equation is

5. 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

6. Short-Run Iso-Profit Lines
has a slope of
and a vertical intercept of

7. Short-Run Iso-Profit Linesy
Increasing profit x1

8. 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.

9. Short-Run Profit-Maximizationy
Increasing profit x1

10. Short-Run Profit-Maximizationy x1

11. Short-Run Profit-Maximization
Given p, w1 and the short-run profit-maximizing plan is y x1

12. Short-Run Profit-Maximization
Given p, w1 and the short-run profit-maximizing plan is And the maximum possible profit is y x1

13. Short-Run Profit-Maximization
At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximal iso-profit line are equal. y x1

14. Short-Run Profit-Maximization
At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximal iso-profit line are equal. y x1

15. Short-Run Profit-Maximization
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.

16. Short-Run Profit-Max: A Cobb-Douglas Example
In class

17. Comparative Statics of SR Profit-Max
What happens to the short-run profit-maximizing production plan as the variable input price w1 changes?

18. Comparative Statics of SR Profit-Max
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.

19. Comparative Statics of SR Profit-Maxy x1

20. Comparative Statics of SR Profit-Maxy x1

21. Comparative Statics of SR Profit-Maxy x1

22. 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 (the firm’s supply curve shifts inward), and
a decrease in the level of the firm’s variable input (the firm’s demand curve for its variable input slopes downward).

23. Comparative Statics of SR Profit-Max
What happens to the short-run profit-maximizing production plan as the output price p changes?

24. Comparative Statics of SR Profit-Max
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.

25. Comparative Statics of SR Profit-Maxy x1

26. Comparative Statics of SR Profit-Maxy x1

27. Comparative Statics of SR Profit-Maxy x1

28. Comparative Statics of SR Profit-Max
An increase in p, the price of the firm’s output, causes
an increase in the firm’s output level (the firm’s supply curve slopes upward), and
an increase in the level of the firm’s variable input (the firm’s demand curve for its variable input shifts outward).

29. 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.

30. 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

31. 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.

32. Returns-to Scale and Profit-Maxy y*
Decreasing returns-to-scale x* x

33. 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.

34. Returns-to Scale and Profit-Maxy
Increasing profit y” y’
Increasing returns-to-scale x’ x” x

35. Returns-to-Scale and Profit-Max
So an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive.

36. Returns-to-Scale and Profit-Max
What if the competitive firm’s technology exhibits constant returns-to-scale?

37. Returns-to Scale and Profit-Maxy
Increasing profit y”
Constant returns-to-scale y’ x’ x” x

38. 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.

39. 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.

40. Returns-to Scale and Profit-Maxy
P = 0 y”
Constant returns-to-scale y’ x’ x” x