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 firm’s profit function is

4. Short-Run Iso-Profit Lines • A $Piso-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 $Piso-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 Lines y 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-Maximization y Increasing profit x1

10. Short-Run Profit-Maximization y x1

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

12. Short-Run Profit-Maximization Given p, w1 and the short-runprofit-maximizing plan is And the maximumpossible profitis y x1

13. 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

14. 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

15. 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.

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 lineis so an increase in w1 causes -- an increase in the slope, and -- no change to the vertical intercept.

19. Comparative Statics of SR Profit-Max y x1

20. Comparative Statics of SR Profit-Max y x1

21. Comparative Statics of SR Profit-Max y 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 lineis so an increase in p causes -- a reduction in the slope, and -- a reduction in the vertical intercept.

25. Comparative Statics of SR Profit-Max y x1

26. Comparative Statics of SR Profit-Max y x1

27. Comparative Statics of SR Profit-Max y 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-Max y y* Decreasingreturns-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-Max y Increasing profit y” y’ Increasingreturns-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-Max y Increasing profit y” Constantreturns-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-Max y P = 0 y” Constantreturns-to-scale y’ x” x’ x