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1、Chapter Nineteen,Profit-Maximization,Structure,Economic profitShort-run profit maximizationComparative staticsLong-run profit maximizationProfit maximization and returns to scaleRevealed profit maximization,Economic Profit,A firm uses inputs j=1,m to make products i=1,n.Output levels are y1,yn.Input
2、 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,Economic Profit,Output and inp
3、ut 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.,Opportunity Costs(机会成本),All inputs must be valued at their mark
4、et value.LaborCapital,Economic Profit,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 is,Profit Maximization,A competitive firm seeks to maximize its present-value.How?,S
5、uppose the firm is in a short-run circumstance in which Its short-run production function isThe firms fixed cost isand its profit function is,Short-Run Profit Maximization,Short-Run Iso-Profit Lines,A$P iso-profit line(等利润线)contains all the production plans that yield a profit level of$P.The equatio
6、n of a$P iso-profit line isI.e.,Short-Run Iso-Profit Lines,has a slope of,and a vertical intercept of,Short-Run Iso-Profit Lines,Increasing profit,y,x1,Short-Run Profit-Maximization,The firms problem is to locate the production plan that attains the highest possible iso-profit line,given the firms c
7、onstraint on choices of production plans.Q:What is this constraint?A:The production function.,Short-Run Profit-Maximization,x1,Technicallyinefficientplans,y,The short-run production function andtechnology set for,Short-Run Profit-Maximization,x1,Increasing profit,y,Short-Run Profit-Maximization,x1,y
8、,Short-Run Profit-Maximization,x1,y,Given p,w1 and the short-runprofit-maximizing plan is,Short-Run Profit-Maximization,x1,y,Given p,w1 and the short-runprofit-maximizing plan is And the maximumpossible profitis,Short-Run Profit-Maximization,x1,y,At the short-run profit-maximizing plan,the slopes of
9、 the short-run production function and the maximaliso-profit line areequal.,Short-Run Profit-Maximization,x1,y,At the short-run profit-maximizing plan,the slopes of the short-run production function and the maximaliso-profit line areequal.,Short-Run Profit-Maximization,is the marginal revenue produc
10、t(边际收益产量)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.,Short-Run Profit-Maximization;A Cobb-Douglas Example,Suppose the short-run productionfunction is,The marginal product of the variableinput 1 is,T
11、he profit-maximizing condition is,Short-Run Profit-Maximization;A Cobb-Douglas Example,Solving,for x1 gives,Short-Run Profit-Maximization;A Cobb-Douglas Example,Solving,for x1 gives,That is,Short-Run Profit-Maximization;A Cobb-Douglas Example,Solving,for x1 gives,That is,so,Short-Run Profit-Maximiza
12、tion;A Cobb-Douglas Example,is the firmsshort-run demand,for input 1 when the level of input 2 is fixed at units.,Short-Run Profit-Maximization;A Cobb-Douglas Example,is the firmsshort-run demand,for input 1 when the level of input 2 is fixed at units.,The firms short-run output level is thus,Compar
13、ative Statics of Short-Run Profit-Maximization,What happens to the short-run profit-maximizing production plan as the output price p changes?,Comparative Statics of Short-Run Profit-Maximization,The equation of a short-run iso-profit lineis,so an increase in p causes-a reduction in the slope,and-a r
14、eduction in the vertical intercept.,Comparative Statics of Short-Run Profit-Maximization,x1,y,Comparative Statics of Short-Run Profit-Maximization,x1,y,Comparative Statics of Short-Run Profit-Maximization,x1,y,Comparative Statics of Short-Run Profit-Maximization,An increase in p,the price of the fir
15、ms 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 of Short-Run Profit-Maximization,The Cobb-Douglas example:When,then t
16、he firms short-rundemand for its variable input 1 is,and its short-runsupply is,Comparative Statics of Short-Run Profit-Maximization,The Cobb-Douglas example:When,then the firms short-rundemand for its variable input 1 is,increases as p increases.,and its short-runsupply is,Comparative Statics of Sh
17、ort-Run Profit-Maximization,The Cobb-Douglas example:When,then the firms short-rundemand for its variable input 1 is,increases as p increases.,and its short-runsupply is,increases as p increases.,Comparative Statics of Short-Run Profit-Maximization,What happens to the short-run profit-maximizing pro
18、duction plan as the variable input price w1 changes?,Comparative Statics of Short-Run Profit-Maximization,The equation of a short-run iso-profit lineis,so an increase in w1 causes-an increase in the slope,and-no change to the vertical intercept.,Comparative Statics of Short-Run Profit-Maximization,x
19、1,y,Comparative Statics of Short-Run Profit-Maximization,x1,y,Comparative Statics of Short-Run Profit-Maximization,x1,y,Comparative Statics of Short-Run Profit-Maximization,An increase in w1,the price of the firms variable input,causesa decrease in the firms output level(the firms supply curve shift
20、s 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-Run Profit-Maximization,The Cobb-Douglas example:When,then the firms short-rundemand for its variable input 1 is,and its short-runsupply is,Co
21、mparative Statics of Short-Run Profit-Maximization,The Cobb-Douglas example:When,then the firms short-rundemand for its variable input 1 is,decreases as w1 increases.,and its short-runsupply is,Comparative Statics of Short-Run Profit-Maximization,The Cobb-Douglas example:When,then the firms short-ru
22、ndemand for its variable input 1 is,decreases as w1 increases.,decreases as w1 increases.,and its short-runsupply is,Long-Run Profit-Maximization,Now 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.,The profit-maximizat
23、ion problem isFOCs are:,Long-Run Profit-Maximization,Factor Demand Functions,Demand for inputs 1 and 2 can be solved as,Inverse Factor Demand Functions,For a given optimal demand for x2,inverse demand function for x1 isFor a given optimal demand for x1 inverse demand function for x2 is,Inverse Facto
24、r Demand Curves,x1,w1,Example:C-D Production Function,The production function is,First-order conditions are:,Solving for x1 and x2:,Plug-in production function to get:,Example:C-D Production Function,Returns-to-Scale and Profit-Maximization,If a competitive firms technology exhibits decreasing retur
25、ns-to-scale then the firm has a single long-run profit-maximizing production plan.,Returns-to Scale and Profit-Maximization,x,y,y*,x*,Decreasingreturns-to-scale,Returns-to-Scale and Profit-Maximization,If a competitive firms technology exhibits exhibits increasing returns-to-scale then the firm does
26、 not have a profit-maximizing plan.,Returns-to Scale and Profit-Maximization,x,y,y”,x,Increasingreturns-to-scale,y,x”,Increasing profit,Returns-to-Scale and Profit-Maximization,So an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive.,Returns-to-Scale and P
27、rofit-Maximization,What if the competitive firms technology exhibits constant returns-to-scale?,Returns-to Scale and Profit-Maximization,x,y,y”,x,Constantreturns-to-scale,y,x”,Increasing profit,Returns-to Scale and Profit-Maximization,So if any production plan earns a positive profit,the firm can do
28、uble up all inputs to produce twice the original output and earn twice the original profit.,Returns-to Scale and Profit-Maximization,Therefore,when a firms technology exhibits constant returns-to-scale,earning a positive economic profit is inconsistent with firms being perfectly competitive.Hence co
29、nstant returns-to-scale requires that competitive firms earn economic profits of zero.,Returns-to Scale and Profit-Maximization,x,y,y”,x,Constantreturns-to-scale,y,x”,P=0,Revealed Profitability,Consider a competitive firm with a technology that exhibits decreasing returns-to-scale.For a variety of o
30、utput and input prices we observe the firms choices of production plans.What can we learn from our observations?,Revealed Profitability,If 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 Profitability,x,y,is
31、chosen at prices so is profit-maximizing at these prices.,Revealed Profitability,x,y,is chosen at prices so is profit-maximizing at these prices.,would give lowerprofits,so,Revealed Profitability,x,y,is chosen at prices so,(x”,y”)is chosen at prices(w”,p”)so(x”,y”)is profit-maximizing at these price
32、s.,Weak Axiom of Profit Maximization,x,y,is chosen at prices so,is chosen at prices so,Revealed Profitability,and,so,and,Adding gives,Revealed Profitability,so,That is,is a necessary implication of profit-maximization.,Revealed Profitability,is a necessary implication of profit-maximization.Suppose
33、the input price does not change.Then Dw=0 and profit-maximizationimplies;i.e.,a competitivefirms output supply curve cannot slopedownward.,Revealed Profitability,is a necessary implication of profit-maximization.Suppose the output price does not change.Then Dp=0 and profit-maximizationimplies;i.e.,a competitivefirms input demand curve cannot slopeupward.,
