1、Chapter Nineteen,Profit-Maximization,Structure,Economic profit Short-run profit maximization Comparative statics Long-run profit maximization Profit maximization and returns to scale Revealed profit maximization,Economic Profit,A firm uses inputs j = 1,m to make products i = 1,n. Output levels are y

2、1,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,Economic Profit

3、,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.,Opportunity Costs (機(jī)會(huì)成本),All inputs must

4、be valued at their market value. Labor Capital,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 max

5、imize its present-value. How?,Suppose the firm is in a short-run circumstance in which Its short-run production function is The firms fixed cost isand its profit function is,Short-Run Profit Maximization,Short-Run Iso-Profit Lines,A $P iso-profit line (等利潤(rùn)線) contains all the production plans that yi

6、eld a profit level of $P . The equation of a $P iso-profit line is I.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 po

7、ssible iso-profit line, given the firms constraint 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,Increasin

8、g profit,y,Short-Run Profit-Maximization,x1,y,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

9、 short-run profit-maximizing plan, the slopes of 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 P

10、rofit-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; A Cobb-Douglas Example,Suppose the short-run productionfunct

11、ion is,The marginal product of the variableinput 1 is,The 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

12、,Solving,for x1 gives,That is,so,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.,Short-Run Profit-Maximization; A Cobb-Douglas Example,is the firmsshort-run demand,for input 1 when the level of input 2 is fix

13、ed at units.,The firms short-run output level is thus,Comparative 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

14、,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,y,Comparative Statics of Short-Run Profit-Maximization,x1,y,Comparative Statics of Short-Run Profit-Maximization,x1,y,Comparative Statics of Sh

15、ort-Run Profit-Maximization,An increase in p, the price of the firms output, causes an increase in the firms output level (the firms supply curve slopes upward), and an increase in the level of the firms variable input (the firms demand curve for its variable input shifts outward).,Comparative Stati

16、cs 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,Comparative Statics of Short-Run Profit-Maximization,The Cobb-Douglas example: When,then the firms short-rundemand for its variable input 1 is,incr

17、eases as p increases.,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,increases as p increases.,Comparative Statics of Short-

18、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,The equation of a short-run iso-profit lineis,so an increase in w1 causes - an increase in the slope, and - no change t

19、o 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 w1, the price of the firms varia

20、ble input, causes a decrease in the firms output level (the firms supply curve shifts inward), and a 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,

21、then the 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,decreases as w1 increases.,and its short-runsupply is,Comparative Stati

22、cs 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.,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

23、x2 are variable. Since no input level is fixed, there are no fixed costs.,The profit-maximization problem is FOCs 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 fun

24、ction for x1 is For a given optimal demand for x1 inverse demand function for x2 is,Inverse Factor 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,

25、Returns-to-Scale and Profit-Maximization,If 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-Maximization,x,y,y*,x*,Decreasingreturns-to-scale,Returns-to-Scale and Profit-Maximizatio

26、n,If 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-Maximization,x,y,y”,x,Increasingreturns-to-scale,y,x”,Increasing profit,Returns-to-Scale and Profit-Maximization,So an increasing return

27、s-to-scale technology is inconsistent with firms being perfectly competitive.,Returns-to-Scale and Profit-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,Return

28、s-to Scale and Profit-Maximization,So 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.,Returns-to Scale and Profit-Maximization,Therefore, when a firms technology exhibits constant returns-to-sc

29、ale, earning a positive economic profit is inconsistent with firms being perfectly competitive. Hence constant 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,

30、Consider a competitive firm with a technology that exhibits decreasing returns-to-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 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