QUIZ - COMBINATION OF MULTIPLECHOICE AND FILL IN THE BLANK 21 QUESTIONS
Chapter Eighteen 9/27/2015 1 Chapter Nineteen Profit-Maximization Economic Profit A firm uses inputs j = 1…,m to make products i = 1,…n. Output levels are y1,…,yn. Input levels are x1,…,xm. Product prices are p1,…,pn. Input prices are w1,…,wm. The Competitive Firm The competitive firm takes all output prices p1,…,pn and all input prices w1,…,wm as given constants. Economic Profit The economic profit generated by the production plan (x1,…,xm,y1,…,yn) is p y p y w x w xn n m m1 1 1 1 . Economic Profit Output and input levels are typically flows. E.g. x1 might be the number of labor units used per hour. And y3 might be the number of cars produced per hour. Consequently, profit is typically a flow also; e.g. the number of dollars of profit earned per hour. Economic Profit How do we value a firm? Suppose the firm’s stream of periodic economic profits is 0, 1, 2, … and r is the rate of interest. Then the present-value of the firm’s economic profit stream is PV r r 0 1 2 21 1( ) 9/27/2015 2 Economic Profit A competitive firm seeks to maximize its present-value. How? Economic Profit Suppose the firm is in a short-run circumstance in which Its short-run production function is y f x x ( , ~ ).1 2 x x2 2 ~ . Economic Profit Suppose the firm is in a short-run circumstance in which Its short-run production function is The firm’s fixed cost is and its profit function is y f x x ( , ~ ).1 2 py w x w x1 1 2 2 ~ . x x2 2 ~ . FC w x 2 2 ~ 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 py w x w x1 1 2 2 ~ . 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 I.e. py w x w x1 1 2 2 ~ . y w p x w x p 1 1 2 2 ~ . Short-Run Iso-Profit Lines y w p x w x p 1 1 2 2 ~ has a slope of w p 1 and a vertical intercept of w x p 2 2 ~ . 9/27/2015 3 Short-Run Iso-Profit Lines y x1 Slopes w p 1 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. Q: What is this constraint? 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. Q: What is this constraint? A: The production function. Short-Run Profit-Maximization x1 Technically inefficient plans y The short-run production function and technology set for x x2 2 ~ . y f x x ( , ~ )1 2 Short-Run Profit-Maximization x1 Slopes w p 1 y y f x x ( , ~ )1 2 Short-Run Profit-Maximization x1 y Slopes w p 1 x1 * y* 9/27/2015 4 Short-Run Profit-Maximization x1 y Slopes w p 1 Given p, w1 and the short-run profit-maximizing plan is x1 * y* x x2 2 ~ , ( , ~ , ). * *x x y1 2 Short-Run Profit-Maximization x1 y Slopes w p 1 Given p, w1 and the short-run profit-maximizing plan is And the maximum possible profit is x x2 2 ~ , ( , ~ , ). * *x x y1 2 . x1 * y* Short-Run Profit-Maximization x1 y Slopes w p 1 At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximal iso-profit line are equal. x1 * y* Short-Run Profit-Maximization x1 y Slopes w p 1 At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximal iso-profit line are equal. MP w p at x x y 1 1 1 2 ( , ~ , ) * * x1 * y* Short-Run Profit-Maximization MP w p p MP w1 1 1 1 p MP 1 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. p MP w 1 1 p MP w 1 1 Short-Run Profit-Maximization; A Cobb-Douglas Example Suppose the short-run production function is y x x 1 1/3 2 1/3~ . The marginal product of the variable input 1 is MP y x x x1 1 1 2 3 2 1/31 3 / ~ . The profit-maximizing condition is MRP p MP p x x w1 1 1 2 3 2 1/3 1 3 ( ) ~ .* / 9/27/2015 5 Short-Run Profit-Maximization; A Cobb-Douglas Example p x x w 3 1 2 3 2 1/3 1( ) ~* / Solving for x1 gives ( ) ~ . * /x w px 1 2 3 1 2 1/3 3 Short-Run Profit-Maximization; A Cobb-Douglas Example p x x w 3 1 2 3 2 1/3 1( ) ~* / Solving for x1 gives ( ) ~ . * /x w px 1 2 3 1 2 1/3 3 That is, ( ) ~ * /x px w 1 2 3 2 1/3 13 Short-Run Profit-Maximization; A Cobb-Douglas Example p x x w 3 1 2 3 2 1/3 1( ) ~* / Solving for x1 gives ( ) ~ . * /x w px 1 2 3 1 2 1/3 3 That is, ( ) ~ * /x px w 1 2 3 2 1/3 13 so x px w p w x1 2 1/3 1 3 2 1 3 2 2 1/2 3 3 * / /~ ~ . Short-Run Profit-Maximization; A Cobb-Douglas Example x p w x1 1 3 2 2 1/2 3 * / ~ is the firm’s short-run demand for input 1 when the level of input 2 is fixed at units. ~x2 Short-Run Profit-Maximization; A Cobb-Douglas Example x p w x1 1 3 2 2 1/2 3 * / ~ is the firm’s short-run demand for input 1 when the level of input 2 is fixed at units. ~x2 The firm’s short-run output level is thus y x x p w x* *( ) ~ ~ . 1 1/3 2 1/3 1 1/2 2 1/2 3 Comparative Statics of Short-Run Profit-Maximization What happens to the short-run profit- maximizing production plan as the output price p changes? 9/27/2015 6 Comparative Statics of Short-Run Profit-Maximization y w p x w x p 1 1 2 2 ~ 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. Comparative Statics of Short-Run Profit-Maximization x1 Slopes w p 1 y y f x x ( , ~ )1 2 x1 * y* Comparative Statics of Short-Run Profit-Maximization x1 Slopes w p 1 y y f x x ( , ~ )1 2 x1 * y* Comparative Statics of Short-Run Profit-Maximization x1 Slopes w p 1 y y f x x ( , ~ )1 2 x1 * y* Comparative Statics of Short-Run Profit-Maximization 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). Comparative Statics of Short-Run Profit-Maximization x p w x1 1 3 2 2 1/2 3 * / ~ The Cobb-Douglas example: When then the firm’s short-run demand for its variable input 1 is y x x 1 1/3 2 1/3~ y p w x* ~ . 3 1 1/2 2 1/2 and its short-run supply is 9/27/2015 7 Comparative Statics of Short-Run Profit-Maximization The Cobb-Douglas example: When then the firm’s short-run demand for its variable input 1 is y x x 1 1/3 2 1/3~ x1 * increases as p increases. and its short-run supply is x p w x1 1 3 2 2 1/2 3 * / ~ y p w x* ~ . 3 1 1/2 2 1/2 Comparative Statics of Short-Run Profit-Maximization The Cobb-Douglas example: When then the firm’s short-run demand for its variable input 1 is y x x 1 1/3 2 1/3~ y* increases as p increases. and its short-run supply is x1 * increases as p increases. x p w x1 1 3 2 2 1/2 3 * / ~ y p w x* ~ . 3 1 1/2 2 1/2 Comparative Statics of Short-Run Profit-Maximization What happens to the short-run profit- maximizing production plan as the variable input price w1 changes? Comparative Statics of Short-Run Profit-Maximization y w p x w x p 1 1 2 2 ~ 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. Comparative Statics of