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1、Chapter NineteenProfit-MaximizationEconomic ProfituA firm uses inputs j = 1,m to make products i = 1,n.uOutput levels are y1,yn.uInput levels are x1,xm.uProduct prices are p1,pn.uInput prices are w1,wm. The Competitive FirmuThe competitive firm takes all output prices p1,pn and all input prices w1,w
2、m as given constants.Economic ProfituThe economic profit generated by the production plan (x1,xm,y1,yn) is p yp yw xwxn nm m1 11 1.Economic ProfituOutput and input levels are typically flows.uE.g. x1 might be the number of labor units used per hour.uAnd y3 might be the number of cars produced per ho
3、ur.uConsequently, profit is typically a flow also; e.g. the number of dollars of profit earned per hour.Economic ProfituHow do we value a firm?uSuppose the firms stream of periodic economic profits is 0 0, , 1 1, , 2 2, , and r is the rate of interest.uThen the present-value of the firms economic pr
4、ofit stream isPVrr 012211()Economic ProfituA competitive firm seeks to maximize its present-value.uHow?Economic ProfituSuppose the firm is in a short-run circumstance in which uIts short-run production function isyf xx (,).12xx22 .Economic ProfituSuppose the firm is in a short-run circumstance in wh
5、ich uIts short-run production function isuThe firms fixed cost isand its profit function isyf xx (,).12 pyw xw x1 12 2.xx22 .FCw x 2 2Short-Run Iso-Profit LinesuA $ iso-profit line contains all the production plans that provide a profit level $ .uA $ iso-profit lines equation is pyw xw x1 12 2.Short
6、-Run Iso-Profit LinesuA $ iso-profit line contains all the production plans that yield a profit level of $ .uThe equation of a $ iso-profit line isuI.e. pyw xw x1 12 2.ywpxw xp 112 2 .Short-Run Iso-Profit Linesywpxw xp 112 2 has a slope of wp1and a vertical intercept of w xp2 2.Short-Run Iso-Profit
7、Lines Increasing profityx1Slopeswp 1Short-Run Profit-MaximizationuThe firms problem is to locate the production plan that attains the highest possible iso-profit line, given the firms constraint on choices of production plans.uQ: What is this constraint?Short-Run Profit-MaximizationuThe firms proble
8、m is to locate the production plan that attains the highest possible iso-profit line, given the firms constraint on choices of production plans.uQ: What is this constraint?uA: The production function.Short-Run Profit-Maximizationx1TechnicallyinefficientplansyThe short-run production function andtech
9、nology set for xx22 .yf xx (,)12Short-Run Profit-Maximizationx1Increasing profitSlopeswp 1yyf xx (,)12 Short-Run Profit-Maximizationx1y Slopeswp 1x1*y*Short-Run Profit-Maximizationx1ySlopeswp 1Given p, w1 and the short-runprofit-maximizing plan is x1*y*xx22 ,(,).*xxy12Short-Run Profit-Maximizationx1
10、ySlopeswp 1Given p, w1 and the short-runprofit-maximizing plan is And the maximumpossible profitis xx22 ,(,).*xxy12 . x1*y*Short-Run Profit-Maximizationx1ySlopeswp 1At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximaliso-profit line areequal. x1*y*
11、Short-Run Profit-Maximizationx1ySlopeswp 1At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximaliso-profit line areequal.MPwpat xxy1112 (,)* x1*y*Short-Run Profit-MaximizationMPwppMPw1111 pMP 1 is the marginal revenue product ofinput 1, the rate at w
12、hich revenue increaseswith the amount used of input 1.If then profit increases with x1.If then profit decreases with x1. pMPw 11pMPw 11Short-Run Profit-Maximization; A Cobb-Douglas ExampleSuppose the short-run productionfunction isyxx 11/321/3.The marginal product of the variableinput 1 isMPyxxx1112
13、 321/313 /.The profit-maximizing condition isMRPpMPpxxw1112 321/313 ().*/Short-Run Profit-Maximization; A Cobb-Douglas Examplepxxw312 321/31()*/ Solvingfor x1 gives().*/xwpx12 3121/33 Short-Run Profit-Maximization; A Cobb-Douglas Examplepxxw312 321/31()*/ Solvingfor x1 gives().*/xwpx12 3121/33 That
14、is,()*/xpxw12 321/313 Short-Run Profit-Maximization; A Cobb-Douglas Examplepxxw312 321/31()*/ Solvingfor x1 gives().*/xwpx12 3121/33 That is,()*/xpxw12 321/313 soxpxwpwx121/313 213 221/233*/. Short-Run Profit-Maximization; A Cobb-Douglas Examplexpwx113 221/23*/ is the firmsshort-run demandfor input
15、1 when the level of input 2 is fixed at units. x2Short-Run Profit-Maximization; A Cobb-Douglas Examplexpwx113 221/23*/ is the firmsshort-run demandfor input 1 when the level of input 2 is fixed at units. x2The firms short-run output level is thusyxxpwx*(). 11/321/311/221/23Comparative Statics of Sho
16、rt-Run Profit-MaximizationuWhat happens to the short-run profit-maximizing production plan as the output price p changes?Comparative Statics of Short-Run Profit-Maximizationywpxw xp 112 2 The equation of a short-run iso-profit lineisso an increase in p causes - a reduction in the slope, and - a redu
17、ction in the vertical intercept.Comparative Statics of Short-Run Profit-Maximizationx1 Slopeswp 1yyf xx (,)12x1*y*Comparative Statics of Short-Run Profit-Maximizationx1Slopeswp 1yyf xx (,)12x1*y*Comparative Statics of Short-Run Profit-Maximizationx1Slopeswp 1yyf xx (,)12x1*y*Comparative Statics of S
18、hort-Run Profit-MaximizationuAn increase in p, the price of the firms output, causesan increase in the firms output level (the firms supply curve slopes upward), andan increase in the level of the firms variable input (the firms demand curve for its variable input shifts outward).Comparative Statics
19、 of Short-Run Profit-Maximizationxpwx113 221/23*/ The Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 isyxx 11/321/3ypwx*. 311/221/2and its short-runsupply isComparative Statics of Short-Run Profit-MaximizationThe Cobb-Douglas example: When then the firms short-run
20、demand for its variable input 1 isyxx 11/321/3x1*increases as p increases.and its short-runsupply isxpwx113 221/23*/ ypwx*. 311/221/2Comparative Statics of Short-Run Profit-MaximizationThe Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 isyxx 11/321/3y*increases as
21、 p increases.and its short-runsupply isx1*increases as p increases.xpwx113 221/23*/ ypwx*. 311/221/2Comparative Statics of Short-Run Profit-MaximizationuWhat happens to the short-run profit-maximizing production plan as the variable input price w1 changes?Comparative Statics of Short-Run Profit-Maxi
22、mizationywpxw xp 112 2 The equation of a short-run iso-profit lineisso an increase in w1 causes - an increase in the slope, and - no change to the vertical intercept.Comparative Statics of Short-Run Profit-Maximizationx1 Slopeswp 1yyf xx (,)12x1*y*Comparative Statics of Short-Run Profit-Maximization
23、x1Slopeswp 1yyf xx (,)12x1*y* Comparative Statics of Short-Run Profit-Maximizationx1Slopeswp 1yyf xx (,)12x1*y* Comparative Statics of Short-Run Profit-MaximizationuAn increase in w1, the price of the firms variable input, causesa decrease in the firms output level (the firms supply curve shifts inw
24、ard), anda decrease in the level of the firms variable input (the firms demand curve for its variable input slopes downward).Comparative Statics of Short-Run Profit-Maximizationxpwx113 221/23*/ The Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 isyxx 11/321/3ypwx*
25、. 311/221/2and its short-runsupply isComparative Statics of Short-Run Profit-Maximizationxpwx113 221/23*/ The Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 isyxx 11/321/3x1*decreases as w1 increases.ypwx*. 311/221/2and its short-runsupply isComparative Statics of
26、 Short-Run Profit-Maximizationxpwx113 221/23*/ The Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 isyxx 11/321/3x1*decreases as w1 increases.y*decreases as w1 increases.ypwx*. 311/221/2and its short-runsupply isLong-Run Profit-MaximizationuNow allow the firm to va
27、ry both input levels.uSince no input level is fixed, there are no fixed costs.Long-Run Profit-MaximizationuBoth x1 and x2 are variable.uThink of the firm as choosing the production plan that maximizes profits for a given value of x2, and then varying x2 to find the largest possible profit level.Long
28、-Run Profit-Maximizationywpxw xp 112 2 The equation of a long-run iso-profit lineisso an increase in x2 causes - no change to the slope, and - an increase in the vertical intercept.Long-Run Profit-Maximizationx1yyf xx (,)12Long-Run Profit-Maximizationx1yyf xx (,)122yf xx (,)12yf xx (,)123Larger leve
29、ls of input 2 increase theproductivity of input 1.Long-Run Profit-Maximizationx1yyf xx (,)122yf xx (,)12yf xx (,)123Larger levels of input 2 increase theproductivity of input 1.The marginal productof input 2 isdiminishing.Long-Run Profit-Maximizationx1yyf xx (,)122yf xx (,)12yf xx (,)123Larger level
30、s of input 2 increase theproductivity of input 1.The marginal productof input 2 isdiminishing.Long-Run Profit-Maximizationx1yyf xx (,)122yf xx (,)12yf xx (,)123yx*() 2xx12*() xx122*() xx123*() yx*()22 yx*()32 pMPw 110 for each short-runproduction plan.Long-Run Profit-Maximizationx1yyf xx (,)122yf xx
31、 (,)12yf xx (,)123The marginal productof input 2 isdiminishing so .yx*() 2xx12*() xx122*() xx123*() yx*()22 yx*()32 for each short-runproduction plan.pMPw 110Long-Run Profit-Maximizationx1yyf xx (,)122yf xx (,)12yf xx (,)123the marginal profitof input 2 isdiminishing.yx*() 2xx12*() xx122*() xx123*()
32、 yx*()22 yx*()32 for each short-runproduction plan.pMPw 110Long-Run Profit-MaximizationuProfit will increase as x2 increases so long as the marginal profit of input 2uThe profit-maximizing level of input 2 therefore satisfiespMPw 220.pMPw 220.Long-Run Profit-MaximizationuProfit will increase as x2 i
33、ncreases so long as the marginal profit of input 2uThe profit-maximizing level of input 2 therefore satisfiesuAnd is satisfied in any short-run, so .pMPw 110pMPw 220.pMPw 220.Long-Run Profit-MaximizationuThe input levels of the long-run profit-maximizing plan satisfyuThat is, marginal revenue equals
34、 marginal cost for all inputs.pMPw 220.pMPw 110andLong-Run Profit-Maximizationxpwx113 221/23*/ The Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 isyxx 11/321/3ypwx*. 311/221/2and its short-runsupply isShort-run profit is therefore Long-Run Profit-Maximization pyw
35、 xw xppwxwpwxw x*/1 12211/221/2113 221/22233Long-Run Profit-Maximization pyw xw xppwxwpwxw xppwxwpwpww x*/1 12211/221/2113 221/22211/221/21111/22233333Long-Run Profit-Maximization pyw xw xppwxwpwxw xppwxwpwpww xppwxw x*/1 12211/221/2113 221/22211/221/21111/22211/221/22233333233Long-Run Profit-Maximi
36、zation pyw xw xppwxwpwxw xppwxwpwpww xppwxw xpwx*/1 12211/221/2113 221/22211/221/21111/22211/221/222311/22333332334271/222 w x.Long-Run Profit-Maximization 427311/221/222pwxw x.What is the long-run profit-maximizinglevel of input 2? Solve0124272311/221/22 xpwxwto get.*xxpw w22312227 Long-Run Profit-
37、MaximizationWhat is the long-run profit-maximizinginput 1 level? Substitutexpwx113 221/23*/ xpw w2312227* intoto getLong-Run Profit-MaximizationWhat is the long-run profit-maximizinginput 1 level? Substitutexpwx113 221/23*/ xpw w2312227* intoto getxpwpw wpw w113 231221/2312232727*/. Long-Run Profit-
38、MaximizationWhat is the long-run profit-maximizingoutput level? Substitutexpw w2312227* intoto getypwx* 311/221/2Long-Run Profit-MaximizationWhat is the long-run profit-maximizingoutput level? Substitutexpw w2312227* intoto getypwpw wpw w*. 327911/231221/2212ypwx* 311/221/2Long-Run Profit-Maximizati
39、onSo given the prices p, w1 and w2, andthe production functionyxx 11/321/3the long-run profit-maximizing productionplan is(,),.*xxypw wpw wpw w123122312221227279 Returns-to-Scale and Profit-MaximizationuIf a competitive firms technology exhibits decreasing returns-to-scale then the firm has a single
40、 long-run profit-maximizing production plan.Returns-to Scale and Profit-Maximizationxyyf x ( )y*x*Decreasingreturns-to-scaleReturns-to-Scale and Profit-MaximizationuIf a competitive firms technology exhibits exhibits increasing returns-to-scale then the firm does not have a profit-maximizing plan.Re
41、turns-to Scale and Profit-Maximizationxyyf x ( )y”xIncreasingreturns-to-scaleyx”Increasing profitReturns-to-Scale and Profit-MaximizationuSo an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive.Returns-to-Scale and Profit-MaximizationuWhat if the competiti
42、ve firms technology exhibits constant returns-to-scale?Returns-to Scale and Profit-Maximizationxyyf x ( )y”xConstantreturns-to-scaleyx”Increasing profitReturns-to Scale and Profit-MaximizationuSo if any production plan earns a positive profit, the firm can double up all inputs to produce twice the o
43、riginal output and earn twice the original profit.Returns-to Scale and Profit-MaximizationuTherefore, when a firms technology exhibits constant returns-to-scale, earning a positive economic profit is inconsistent with firms being perfectly competitive.uHence constant returns-to-scale requires that c
44、ompetitive firms earn economic profits of zero.Returns-to Scale and Profit-Maximizationxyyf x ( )y”xConstantreturns-to-scaleyx” = 0Revealed ProfitabilityuConsider a competitive firm with a technology that exhibits decreasing returns-to-scale.uFor a variety of output and input prices we observe the f
45、irms choices of production plans.uWhat can we learn from our observations?Revealed ProfitabilityuIf a production plan (x,y) is chosen at prices (w,p) we deduce that the plan (x,y) is revealed to be profit-maximizing for the prices (w,p).Revealed ProfitabilityxySlopewp x y(,) x y is chosen at prices
46、(,) w pRevealed Profitabilityxy is chosen at prices so is profit-maximizing at these prices.Slopewp x y(,) x y(,) w p(,) x yRevealed Profitabilityxy is chosen at prices so is profit-maximizing at these prices.Slopewp x y(,) x y(,) w p(,) x y x y(,) xy would give higherprofits, so why is it notchosen
47、?Revealed Profitabilityxy is chosen at prices so is profit-maximizing at these prices.Slopewp x y(,) x y(,) w p(,) x y x y(,) xy would give higherprofits, so why is it notchosen? Because it isnot a feasible plan.Revealed Profitabilityxy is chosen at prices so is profit-maximizing at these prices.Slo
48、pewp x y(,) x y(,) w p(,) x y x y(,) xy would give higherprofits, so why is it notchosen? Because it isnot a feasible plan.So the firms technology set must lie under theiso-profit line.Revealed Profitabilityxy is chosen at prices so is profit-maximizing at these prices.Slopewp x y(,) x y(,) w p(,) x
49、 y x ySo the firms technology set must lie under theiso-profit line.The technologyset is somewherein hereRevealed Profitabilityxy is chosen at prices so maximizes profit at these prices.(,) xy(,) wp y xSlopewp x y(,) xy would provide higherprofit but it is not chosen(,) xyRevealed Profitabilityxy is
50、 chosen at prices so maximizes profit at these prices.(,) xy(,) wp y x x y(,) xy would provide higherprofit but it is not chosenbecause it is not feasible(,) xySlopewp Revealed Profitabilityxy is chosen at prices so maximizes profit at these prices.(,) xy(,) wp y x x y(,) xy would provide higherprof