1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009

2 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

3 Short-Run Iso-Profit Lines A $  iso-profit line contains all the production plans that provide a profit level $ . A $  iso-profit line’s equation is

4 Short-Run Iso-Profit Lines A $  iso-profit line contains all the production plans that yield a profit level of $ . The equation of a $  iso-profit line is Rearranging

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

6 Short-Run Iso-Profit Lines Increasing profit y x1x1

7 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.

8 Short-Run Profit-Maximization x1x1 Increasing profit y

9 Short-Run Profit-Maximization x1x1 y

10 Short-Run Profit-Maximization x1x1 y Given p, w 1 and the short-run profit-maximizing plan is And the maximum possible profit is

11 Short-Run Profit-Maximization x1x1 y At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximal iso-profit line are equal.

12 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 x 1. If then profit decreases with x 1.

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

14 Comparative Statics of SR Profit-Max What happens to the short-run profit- maximizing production plan as the variable input price w 1 changes?

15 Comparative Statics of SR Profit-Max The equation of a short-run iso-profit line is so an increase in w 1 causes -- an increase in the slope, and -- no change to the vertical intercept.

16 Comparative Statics of SR Profit-Max x1x1 y

17 Comparative Statics of SR Profit-Max x1x1 y

18 Comparative Statics of SR Profit-Max x1x1 y

19 Comparative Statics of SR Profit-Max An increase in w 1, 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.

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

21 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.

22 Comparative Statics of SR Profit-Max x1x1 y

23 Comparative Statics of SR Profit-Max x1x1 y

24 Comparative Statics of SR Profit-Max x1x1 y

25 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, and  an increase in the level of the firm’s variable input.

26 Long-Run Profit-Maximization Now allow the firm to vary both input levels (both x 1 and x 2 are variable). Since no input level is fixed, there are no fixed costs. For any given level of x 2, the profit- maximizing condition for x 1 must still hold.

27 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 x 1 (p, w 1, w 2 ) and x 2 (p, w 1, w 2 ) and

28 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.

29 Returns-to Scale and Profit-Max x y y* x* Decreasing returns-to-scale

30 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.

31 Returns-to Scale and Profit-Max x y y” x’ Increasing returns-to-scale y’ x” Increasing profit

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

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

34 Returns-to Scale and Profit-Max x y y” x’ Constant returns-to-scale y’ x” Increasing profit

35 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.

36 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.

37 Returns-to Scale and Profit-Max x y y” x’ Constant returns-to-scale y’ x”  = 0