义利润最大化课件

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1、义利润最大化课件义利润最大化课件Chapter NineteenProfit-Maximization义利润最大化课件义利润最大化课件StructureuEconomic profituShort-run profit maximizationComparative staticsuLong-run profit maximizationuProfit maximization and returns to scaleuRevealed profit maximization义利润最大化课件义利润最大化课件Economic ProfituA firm uses inputs j=1,m to

2、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.义利润最大化课件义利润最大化课件The Competitive FirmuThe competitive firm takes all output prices p1,pn and all input prices w1,wm as given constants.义利润最大化课件义利润最大化课件Economic ProfituThe economic pr

3、ofit generated by the production plan(x1,xm,y1,yn)is p yp yw xw xn nm m1 11 1.义利润最大化课件义利润最大化课件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 hour.uConsequently,profit is typically a

4、 flow also;e.g.the number of dollars of profit earned per hour.义利润最大化课件义利润最大化课件Opportunity Costs(机会成本)uAll inputs must be valued at their market value.uLaboruCapital义利润最大化课件义利润最大化课件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 t

5、he rate of interest.uThen the present-value of the firms economic profit stream isPVrr 012211()义利润最大化课件义利润最大化课件Profit MaximizationuA competitive firm seeks to maximize its present-value.uHow?义利润最大化课件义利润最大化课件uSuppose the firm is in a short-run circumstance in which uIts short-run production function

6、isuThe firms fixed cost isand its profit function isyf xx (,).12 pyw xw x1 12 2.xx22.FCw x 2 2Short-Run Profit Maximization义利润最大化课件义利润最大化课件Short-Run Iso-Profit LinesuA$iso-profit line(等利润线等利润线)contains all the production plans that yield a profit level of$.uThe equation of a$iso-profit line isuI.e.p

7、yw xw x1 12 2.ywpxw xp 112 2.义利润最大化课件义利润最大化课件Short-Run Iso-Profit Linesywpxw xp 112 2 has a slope of wp1and a vertical intercept of w xp2 2.义利润最大化课件义利润最大化课件Short-Run Iso-Profit Lines Increasing profityx1Slopeswp 1义利润最大化课件义利润最大化课件Short-Run Profit-MaximizationuThe firms problem is to locate the produc

8、tion 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.义利润最大化课件义利润最大化课件Short-Run Profit-Maximizationx1TechnicallyinefficientplansyThe short-run production function andtechnology set f

9、or xx22.yf xx (,)12义利润最大化课件义利润最大化课件Short-Run Profit-Maximizationx1Increasing profitSlopeswp 1yyf xx (,)12 义利润最大化课件义利润最大化课件Short-Run Profit-Maximizationx1y Slopeswp 1x1*y*义利润最大化课件义利润最大化课件Short-Run Profit-Maximizationx1ySlopeswp 1Given p,w1 and the short-runprofit-maximizing plan is x1*y*xx22,(,).*xxy

10、12义利润最大化课件义利润最大化课件Short-Run Profit-Maximizationx1ySlopeswp 1Given p,w1 and the short-runprofit-maximizing plan is And the maximumpossible profitis xx22,(,).*xxy12 .x1*y*义利润最大化课件义利润最大化课件Short-Run Profit-Maximizationx1ySlopeswp 1At the short-run profit-maximizing plan,the slopes of the short-run produ

11、ction function and the maximaliso-profit line areequal.x1*y*义利润最大化课件义利润最大化课件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*义利润最大化课件义利润最大化课件Short-Run Profi

12、t-MaximizationMPwppMPw1111 pMP 1 is the marginal revenue product(边际边际收益产量收益产量)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.pMPw 11pMPw 11义利润最大化课件义利润最大化课件Short-Run Profit-Maximization;A Cobb-Douglas Ex

13、ampleSuppose the short-run productionfunction isyxx 11/321/3.The marginal product of the variableinput 1 isMPyxxx1112 321/313 /.The profit-maximizing condition isMRPpMPpxxw1112 321/313 ().*/义利润最大化课件义利润最大化课件Short-Run Profit-Maximization;A Cobb-Douglas Examplepxxw312 321/31()*/Solvingfor x1 gives().*/

14、xwpx12 3121/33 义利润最大化课件义利润最大化课件Short-Run Profit-Maximization;A Cobb-Douglas Examplepxxw312 321/31()*/Solvingfor x1 gives().*/xwpx12 3121/33 That is,()*/xpxw12 321/313 义利润最大化课件义利润最大化课件Short-Run Profit-Maximization;A Cobb-Douglas Examplepxxw312 321/31()*/Solvingfor x1 gives().*/xwpx12 3121/33 That is,

15、()*/xpxw12 321/313 soxpxwpwx121/313 213 221/233*/.义利润最大化课件义利润最大化课件Short-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.x2义利润最大化课件义利润最大化课件Short-Run Profit-Maximization;A Cobb-Douglas Examplexpwx113 221/

16、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*().11/321/311/221/23义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-MaximizationuWhat happens to the short-run profit-maximizing production plan as the out

17、put price p changes?义利润最大化课件义利润最大化课件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 reduction in the vertical intercept.义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-

18、Maximizationx1 Slopeswp 1yyf xx (,)12x1*y*义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-Maximizationx1Slopeswp 1yyf xx (,)12x1*y*义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-Maximizationx1Slopeswp 1yyf xx (,)12x1*y*义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-Maximiza

19、tionuAn 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).义利润最大化课件义利润最大化课件Comparative Statics of Short-Run

20、 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 is义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-MaximizationThe Cobb-Douglas example:When then the firms short-rund

21、emand for its variable input 1 isyxx 11/321/3x1*increases as p increases.and its short-runsupply isxpwx113 221/23*/ypwx*.311/221/2义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-MaximizationThe Cobb-Douglas example:When then the firms short-rundemand for its variable input 1 isyxx 11/321/3y*

22、increases as p increases.and its short-runsupply isx1*increases as p increases.xpwx113 221/23*/ypwx*.311/221/2义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-MaximizationuWhat happens to the short-run profit-maximizing production plan as the variable input price w1 changes?义利润最大化课件义利润最大化课件Co

23、mparative Statics of Short-Run Profit-Maximizationywpxw 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.义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-Maximizationx1 Slopeswp 1yyf xx (,)12x1

24、*y*义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-Maximizationx1Slopeswp 1yyf xx (,)12x1*y*义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-Maximizationx1Slopeswp 1yyf xx (,)12x1*y*义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-MaximizationuAn increase in w1,the price of the

25、 firms variable input,causesa decrease in the firms output level(the firms supply curve shifts inward),anda decrease in the level of the firms variable input(the firms demand curve for its variable input slopes downward).义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-Maximizationxpwx113 221

26、/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 is义利润最大化课件义利润最大化课件Comparative Statics of Short-Run Profit-Maximizationxpwx113 221/23*/The Cobb-Douglas example:When then the firms short-rundemand for its v

27、ariable input 1 isyxx 11/321/3x1*decreases as w1 increases.ypwx*.311/221/2and its short-runsupply is义利润最大化课件义利润最大化课件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 11/321/3x1*decreases as

28、w1 increases.y*decreases as w1 increases.ypwx*.311/221/2and its short-runsupply is义利润最大化课件义利润最大化课件Long-Run Profit-MaximizationuNow allow the firm to vary both input levels,i.e.,both x1 and x2 are variable.uSince no input level is fixed,there are no fixed costs.义利润最大化课件义利润最大化课件uThe profit-maximizatio

29、n problem isuFOCs are:Long-Run Profit-Maximization12121 122,max(,).x xpf x xw xw x1211(*,*)0.f xxpwx1222(*,*)0.f xxpwx义利润最大化课件义利润最大化课件Factor Demand FunctionsuDemand for inputs 1 and 2 can be solved as,11122212(,)(,)xx w wpxx w wp义利润最大化课件义利润最大化课件Inverse Factor Demand FunctionsuFor a given optimal dem

30、and for x2,inverse demand function for x1 isuFor a given optimal demand for x1 inverse demand function for x2 is1112(,*)wpMP x x2212(*,)wpMP xx义利润最大化课件义利润最大化课件Inverse Factor Demand Curvesx1w1112(,*)pMP x x义利润最大化课件义利润最大化课件Example:C-D Production FunctionThe production function is1/31/312yxxFirst-order

31、 conditions are:2/31/3112131/32/311223pxxwpxxw义利润最大化课件义利润最大化课件uSolving for x1 and x2:xpw w2312227*3*1212.27pxw wypwpw wpw w*.327911/231221/2212Plug-in production function to get:Example:C-D Production Function义利润最大化课件义利润最大化课件Returns-to-Scale and Profit-MaximizationuIf a competitive firms technology

32、exhibits decreasing returns-to-scale then the firm has a single long-run profit-maximizing production plan.义利润最大化课件义利润最大化课件Returns-to Scale and Profit-Maximizationxyyf x ()y*x*Decreasingreturns-to-scale义利润最大化课件义利润最大化课件Returns-to-Scale and Profit-MaximizationuIf a competitive firms technology exhibit

33、s exhibits increasing returns-to-scale then the firm does not have a profit-maximizing plan.义利润最大化课件义利润最大化课件Returns-to Scale and Profit-Maximizationxyyf x ()y”xIncreasingreturns-to-scaleyx”Increasing profit义利润最大化课件义利润最大化课件Returns-to-Scale and Profit-MaximizationuSo an increasing returns-to-scale tec

34、hnology is inconsistent with firms being perfectly competitive.义利润最大化课件义利润最大化课件Returns-to-Scale and Profit-MaximizationuWhat if the competitive firms technology exhibits constant returns-to-scale?义利润最大化课件义利润最大化课件Returns-to Scale and Profit-Maximizationxyyf x ()y”xConstantreturns-to-scaleyx”Increasin

35、g profit义利润最大化课件义利润最大化课件Returns-to Scale and Profit-MaximizationuSo 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.义利润最大化课件义利润最大化课件Returns-to Scale and Profit-MaximizationuTherefore,when a firms

36、 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 competitive firms earn economic profits of zero.义利润最大化课件义利润最大化课件Returns-to Scale and Profit-Maximizationxyyf x ()y”x

37、Constantreturns-to-scaleyx”=0义利润最大化课件义利润最大化课件Revealed ProfitabilityuConsider a competitive firm with a technology that exhibits decreasing returns-to-scale.uFor a variety of output and input prices we observe the firms choices of production plans.uWhat can we learn from our observations?义利润最大化课件义利润最

38、大化课件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).义利润最大化课件义利润最大化课件Revealed Profitabilityxy is chosen at prices so is profit-maximizing at these prices.Slopewp x y(,)x y(,)w p(,)x y义利润最大化课件义利

39、润最大化课件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 lowerprofits,so p yw xp yw x.义利润最大化课件义利润最大化课件Revealed Profitabilityxy y x x y(,)wp is chosen at prices so(,)xy(,)wp p yw xp yw x.(x”,y”)is chosen at prices(w”,

40、p”)so(x”,y”)is profit-maximizing at these prices.义利润最大化课件义利润最大化课件Weak Axiom of Profit Maximizationxy y x x y(,)w p(,)wp is chosen at prices so(,)x y(,)w p p yw xp yw x.is chosen at prices so(,)xy(,)wp p yw xp yw x.义利润最大化课件义利润最大化课件Revealed Profitability p yw xp yw x p yw xp yw xandso p yw xp yw x p y

41、w xp yw x.andAdding gives()()()().ppywwxppywwx义利润最大化课件义利润最大化课件Revealed Profitability()()()()ppywwxppywwxso()()()()ppyywwxxThat is,p yw x is a necessary implication of profit-maximization.义利润最大化课件义利润最大化课件Revealed Profitability p ywx is a necessary implication of profit-maximization.Suppose the input

42、price does not change.Then w=0 and profit-maximizationimplies ;i.e.,a competitivefirms output supply curve cannot slopedownward.p y 0义利润最大化课件义利润最大化课件Revealed Profitability p ywx is a necessary implication of profit-maximization.Suppose the output price does not change.Then p=0 and profit-maximizationimplies ;i.e.,a competitivefirms input demand curve cannot slopeupward.0 wx