講義利潤最大化課件

1、講義利潤最大化課件講義利潤最大化課件Economic ProfitA 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. Economic ProfitA firm uses inpThe Competitive FirmThe competitive firm takes all output prices p1,pn and all input pric

2、es w1,wm as given constants.The Competitive FirmThe competEconomic ProfitThe economic profit generated by the production plan (x1,xm,y1,yn) isEconomic ProfitThe economic prEconomic ProfitOutput and input levels are typically flows.E.g. x1 might be the number of labor units used per hour.And y3 might

3、 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 ProfitOutput and inpuOpportunity Costs (機(jī)會成本)All inputs must be valued at their market value.LaborCapitalOpportunity Costs (機(jī)會成本)All inEconomic Profit

4、How do we value a firm?Suppose the firms stream of periodic economic profits is P0, P1, P2, and r is the rate of interest.Then the present-value of the firms economic profit stream isEconomic ProfitHow do we valueProfit MaximizationA competitive firm seeks to maximize its present-value.How?Profit Ma

5、ximizationA competitiShort-Run Profit MaximizationSuppose the firm is in a short-run circumstance in which Its short-run production function isThe firms fixed cost isand its profit function isShort-Run Profit MaximizationSShort-Run Iso-Profit LinesA $P iso-profit line (等利潤線) contains all the product

6、ion plans that yield a profit level of $P .The equation of a $P iso-profit line isI.e.Short-Run Iso-Profit LinesA $PShort-Run Iso-Profit Lineshas a slope ofand a vertical intercept ofShort-Run Iso-Profit Lineshas Short-Run Iso-Profit LinesIncreasing profityx1Short-Run Iso-Profit LinesIncrShort-Run P

7、rofit-MaximizationThe 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.Q: What is this constraint?A: The production function.Short-Run Profit-MaximizationTShort-Run Profit-Maximizationx1Technic

8、allyinefficientplansyThe short-run production function andtechnology set for Short-Run Profit-MaximizationxShort-Run Profit-Maximizationx1Increasing profityShort-Run Profit-MaximizationxShort-Run Profit-Maximizationx1yShort-Run Profit-MaximizationxShort-Run Profit-Maximizationx1yGiven p, w1 and the

9、short-runprofit-maximizing plan is Short-Run Profit-MaximizationxShort-Run Profit-Maximizationx1yGiven p, w1 and the short-runprofit-maximizing plan is And the maximumpossible profitis Short-Run Profit-MaximizationxShort-Run Profit-Maximizationx1yAt the short-run profit-maximizing plan, the slopes o

10、f the short-run production function and the maximaliso-profit line areequal.Short-Run Profit-MaximizationxShort-Run Profit-Maximizationx1yAt the short-run profit-maximizing plan, the slopes of the short-run production function and the maximaliso-profit line areequal.Short-Run Profit-MaximizationxSho

11、rt-Run Profit-Maximization is the marginal revenue product(邊際收益產(chǎn)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. Short-Run Profit-Maximization Short-Run Profit-Maximization; A Cobb-Douglas ExampleSup

12、pose the short-run productionfunction isThe marginal product of the variableinput 1 isThe profit-maximizing condition isShort-Run Profit-Maximization;Short-Run Profit-Maximization; A Cobb-Douglas ExampleSolvingfor x1 givesShort-Run Profit-Maximization;Short-Run Profit-Maximization; A Cobb-Douglas Ex

13、ampleSolvingfor x1 givesThat is,Short-Run Profit-Maximization;Short-Run Profit-Maximization; A Cobb-Douglas ExampleSolvingfor x1 givesThat is,soShort-Run Profit-Maximization;Short-Run Profit-Maximization; A Cobb-Douglas Exampleis the firmsshort-run demandfor input 1 when the level of input 2 is fixe

14、d at units. Short-Run Profit-Maximization;Short-Run Profit-Maximization; A Cobb-Douglas Exampleis the firmsshort-run demandfor input 1 when the level of input 2 is fixed at units. The firms short-run output level is thusShort-Run Profit-Maximization;Comparative Statics of Short-Run Profit-Maximizati

15、onWhat happens to the short-run profit-maximizing production plan as the output price p changes?Comparative Statics of Short-RComparative Statics of Short-Run Profit-MaximizationThe equation of a short-run iso-profit lineisso an increase in p causes - a reduction in the slope, and - a reduction in t

16、he vertical intercept.Comparative Statics of Short-RComparative Statics of Short-Run Profit-Maximizationx1yComparative Statics of Short-RComparative Statics of Short-Run Profit-Maximizationx1yComparative Statics of Short-RComparative Statics of Short-Run Profit-Maximizationx1yComparative Statics of

17、Short-RComparative Statics of Short-Run Profit-MaximizationAn 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 shift

18、s outward).Comparative Statics of Short-RComparative Statics of Short-Run Profit-MaximizationThe Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 isand its short-runsupply isComparative Statics of Short-RComparative Statics of Short-Run Profit-MaximizationThe Cobb-D

19、ouglas example: When then the firms short-rundemand for its variable input 1 isincreases as p increases.and its short-runsupply isComparative Statics of Short-RComparative Statics of Short-Run Profit-MaximizationThe Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 i

20、sincreases as p increases.and its short-runsupply isincreases as p increases.Comparative Statics of Short-RComparative Statics of Short-Run Profit-MaximizationWhat happens to the short-run profit-maximizing production plan as the variable input price w1 changes?Comparative Statics of Short-RComparat

21、ive Statics of Short-Run Profit-MaximizationThe 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-RComparative Statics of Short-Run Profit-Maximizationx1yComparative Statics of Short

22、-RComparative Statics of Short-Run Profit-Maximizationx1yComparative Statics of Short-RComparative Statics of Short-Run Profit-Maximizationx1yComparative Statics of Short-RComparative Statics of Short-Run Profit-MaximizationAn increase in w1, the price of the firms variable input, causesa decrease i

23、n 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).Comparative Statics of Short-RComparative Statics of Short-Run Profit-MaximizationThe Cobb-Douglas example: When the

24、n the firms short-rundemand for its variable input 1 isand its short-runsupply isComparative Statics of Short-RComparative Statics of Short-Run Profit-MaximizationThe Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 isdecreases as w1 increases.and its short-runsuppl

25、y isComparative Statics of Short-RComparative Statics of Short-Run Profit-MaximizationThe Cobb-Douglas example: When then the firms short-rundemand for its variable input 1 isdecreases as w1 increases.decreases as w1 increases.and its short-runsupply isComparative Statics of Short-RLong-Run Profit-M

26、aximizationNow allow the firm to vary both input levels, i.e., both x1 and x2 are variable.Since no input level is fixed, there are no fixed costs.Long-Run Profit-MaximizationNoThe profit-maximization problem isFOCs are:Long-Run Profit-MaximizationThe profit-maximization probleFactor Demand Function

27、sDemand for inputs 1 and 2 can be solved as,Factor Demand FunctionsDemand Inverse Factor Demand FunctionsFor a given optimal demand for x2, inverse demand function for x1 isFor a given optimal demand for x1 inverse demand function for x2 isInverse Factor Demand FunctionInverse Factor Demand Curvesx1

28、w1Inverse Factor Demand Curvesx1Example: C-D Production FunctionThe production function isFirst-order conditions are:Example: C-D Production FunctiExample: C-D Production FunctionSolving for x1 and x2 :Plug-in production function to get:Example: C-D Production FunctiReturns-to-Scale and Profit-Maxim

29、izationIf a competitive firms technology exhibits decreasing returns-to-scale then the firm has a single long-run profit-maximizing production plan.Returns-to-Scale and Profit-MaReturns-to Scale and Profit-Maximizationxyy*x*Decreasingreturns-to-scaleReturns-to Scale and Profit-MaReturns-to-Scale and

30、 Profit-MaximizationIf a competitive firms technology exhibits exhibits increasing returns-to-scale then the firm does not have a profit-maximizing plan.Returns-to-Scale and Profit-MaReturns-to Scale and Profit-Maximizationxyy”xIncreasingreturns-to-scaleyx”Increasing profitReturns-to Scale and Profi

31、t-MaReturns-to-Scale and Profit-MaximizationSo an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive.Returns-to-Scale and Profit-MaReturns-to-Scale and Profit-MaximizationWhat if the competitive firms technology exhibits constant returns-to-scale?Returns-to

32、-Scale and Profit-MaReturns-to Scale and Profit-Maximizationxyy”xConstantreturns-to-scaleyx”Increasing profitReturns-to Scale and Profit-MaReturns-to Scale and Profit-MaximizationSo if any production plan earns a positive profit, the firm can double up all inputs to produce twice the original output

33、 and earn twice the original profit.Returns-to Scale and Profit-MaReturns-to Scale and Profit-MaximizationTherefore, when a firms technology exhibits constant returns-to-scale, earning a positive economic profit is inconsistent with firms being perfectly competitive.Hence constant returns-to-scale r

34、equires that competitive firms earn economic profits of zero.Returns-to Scale and Profit-MaReturns-to Scale and Profit-Maximizationxyy”xConstantreturns-to-scaleyx”P = 0Returns-to Scale and Profit-MaRevealed ProfitabilityConsider a competitive firm with a technology that exhibits decreasing returns-t

35、o-scale.For a variety of output and input prices we observe the firms choices of production plans.What can we learn from our observations?Revealed ProfitabilityConsiderRevealed ProfitabilityIf 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 ProfitabilityIf a proRevealed Profitabilityxy is chose