Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009 1
Economic Profit � Suppose the firm is in a short-run ≡ ~ . circumstance in which x x 2 2 � Its short-run production function is ,~ ). = ( y f x x 1 2 � The firm’s profit function is ~ . Π = − − py w x w x 1 1 2 2 2
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 3
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 ~ . Π ≡ − − py w x w x 1 1 2 2 � Rearranging ~ Π + w w x = + 1 2 2 y p x . 1 p 4
Short-Run Iso-Profit Lines ~ Π + w w x = + 1 2 2 y p x 1 p has a slope of + w 1 p and a vertical intercept of ~ Π + w x 2 2 . p 5
Short-Run Iso-Profit Lines Increasing y profit Π ≡ Π ′′′ Π ≡ Π ′′ Π ≡ Π ′ w = + 1 Slopes p x 1 6
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. 7
Short-Run Profit-Maximization y Increasing Π ≡ Π ′′′ Π ≡ Π ′′ profit Π ≡ Π ′ ,~ ) = ( y f x x 1 2 w = + 1 Slopes p x 1 8
Short-Run Profit-Maximization y Π ≡ Π ′′′ Π ≡ Π ′′ Π ≡ Π ′ w = + 1 Slopes y * p * x 1 x 1 9
Short-Run Profit-Maximization ≡ ~ , x x Given p, w 1 and the short-run 2 2 y , ~ , * * x x y profit-maximizing plan is ( ). 1 2 And the maximum Π ≡ Π ′′ possible profit Π . ′′ is w = + 1 Slopes y * p * x 1 x 1 10
Short-Run Profit-Maximization At the short-run profit-maximizing plan, y the slopes of the short-run production function and the maximal Π ≡ Π ′′ iso-profit line are equal. w = + 1 Slopes y * p w = 1 MP 1 p ,~ , * * at x x y ( ) 1 2 * x 1 x 1 11
Short-Run Profit-Maximization w = ⇔ × = 1 MP p MP w 1 1 1 p × 1 is the marginal revenue product of p MP input 1, the rate at which revenue increases with the amount used of input 1. × > If then profit increases with x 1 . p MP w 1 1 × < If then profit decreases with x 1 . p MP w 1 1 12
Short-Run Profit-Max: A Cobb-Douglas Example � In class 13
Comparative Statics of SR Profit-Max � What happens to the short-run profit- maximizing production plan as the variable input price w 1 changes ? 14
Comparative Statics of SR Profit-Max The equation of a short-run iso-profit line ~ Π + is w w x = + 1 2 2 y p x 1 p so an increase in w 1 causes -- an increase in the slope, and -- no change to the vertical intercept. 15
Comparative Statics of SR Profit-Max Π ≡ Π ′′′ y Π ≡ Π ′′ Π ≡ Π ′ ,~ ) = ( y f x x 1 2 y * w = + 1 Slopes p * x 1 x 1 16
Comparative Statics of SR Profit-Max Π ≡ Π ′′′ y Π ≡ Π ′′ Π ≡ Π ′ ,~ ) = ( y f x x 1 2 y * w = + 1 Slopes p * x 1 x 1 17
Comparative Statics of SR Profit-Max Π ≡ Π ′′′ y Π ≡ Π ′′ Π ≡ Π ′ ,~ ) = ( y f x x 1 2 w = + 1 Slopes y * p * x 1 x 1 18
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. 19
Comparative Statics of SR Profit-Max � What happens to the short-run profit- maximizing production plan as the output price p changes ? 20
Comparative Statics of SR Profit-Max The equation of a short-run iso-profit line ~ is Π + w w x = + 1 2 2 y p x 1 p so an increase in p causes -- a reduction in the slope, and -- a reduction in the vertical intercept. 21
Comparative Statics of SR Profit-Max Π ≡ Π ′′′ y Π ≡ Π ′′ Π ≡ Π ′ ,~ ) = ( y f x x 1 2 y * w = + 1 Slopes p * x 1 x 1 22
Comparative Statics of SR Profit-Max y ,~ ) = ( y f x x 1 2 y * w = + 1 Slopes p * x 1 x 1 23
24 ,~ ) 2 x x 1 Comparative Statics of SR Profit-Max 1 f x = ( y 1 p w = + Slopes x 1 * y * y
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. 25
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. 26
Long-Run Profit-Maximization � The input levels of the long-run profit-maximizing plan satisfy × − = × − = 0 and p MP w p MP w 0. 1 1 2 2 � 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 ) 27
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. 28
returns-to-scale 29 x Decreasing = ( ) f x Returns-to Scale and Profit-Max y x* y* y
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. 30
Returns-to Scale and Profit-Max y Increasing = ( ) y f x profit y” y’ Increasing returns-to-scale x’ x” x 31
Returns-to-Scale and Profit-Max � So an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive. 32
Returns-to-Scale and Profit-Max � What if the competitive firm’s technology exhibits constant returns-to-scale? 33
Returns-to Scale and Profit-Max y Increasing profit = ( ) y f x y” Constant returns-to-scale y’ x’ x” x 34
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. 35
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. 36
Returns-to Scale and Profit-Max y = ( ) y f x Π = 0 y” Constant returns-to-scale y’ x’ x” x 37
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