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Profit Maximization. Molly W. Dahl Georgetown University Econ 101 – Spring 2008. Economic Profit. Suppose the firm is in a short-run circumstance in which Its short-run production function is. Economic Profit. Suppose the firm is in a short-run circumstance in which

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Profit maximization

Profit Maximization

Molly W. Dahl

Georgetown University

Econ 101 – Spring 2008


Economic profit
Economic Profit

  • Suppose the firm is in a short-run circumstance in which

  • Its short-run production function is


Economic profit1
Economic Profit

  • Suppose the firm is in a short-run circumstance in which

  • Its short-run production function is

  • The firm’s profit function is


Short run iso profit lines
Short-Run Iso-Profit Lines

  • A $Piso-profit line contains all the production plans that provide a profit level $P .

  • A $P iso-profit line’s equation is


Short run iso profit lines1
Short-Run Iso-Profit Lines

  • A $Piso-profit line contains all the production plans that yield a profit level of $P .

  • The equation of a $P iso-profit line is

  • Rearranging


Short run iso profit lines2
Short-Run Iso-Profit Lines

has a slope of

and a vertical intercept of


Short run iso profit lines3
Short-Run Iso-Profit Lines

y

Increasing profit

x1


Short run profit maximization
Short-Run Profit-Maximization

  • The firm’s problem is to locate the production plan that attains the highest possible iso-profit line, given the firm’s constraint on choices of production plans.


Short run profit maximization1
Short-Run Profit-Maximization

y

Increasing profit

x1



Short run profit maximization3
Short-Run Profit-Maximization

Given p, w1 and the short-runprofit-maximizing plan is

y

x1


Short run profit maximization4
Short-Run Profit-Maximization

Given p, w1 and the short-runprofit-maximizing plan is And the maximumpossible profitis

y

x1


Short run profit maximization5
Short-Run Profit-Maximization

At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximaliso-profit line areequal.

y

x1


Short run profit maximization6
Short-Run Profit-Maximization

At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximaliso-profit line areequal.

y

x1


Short run profit maximization7
Short-Run Profit-Maximization

is the marginal revenue product ofinput 1, the rate at which revenue increaseswith the amount used of input 1.

If then profit increases with x1.

If then profit decreases with x1.



Comparative statics of sr profit max
Comparative Statics of SR Profit-Max

  • What happens to the short-run profit-maximizing production plan as the variable input price w1 changes?


Comparative statics of sr profit max1
Comparative Statics of SR Profit-Max

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 sr profit max5
Comparative Statics of SR Profit-Max

  • An increase in w1, the price of the firm’s variable input, causes

    • a decrease in the firm’s output level (the firm’s supply curve shifts inward), and

    • a decrease in the level of the firm’s variable input (the firm’s demand curve for its variable input slopes downward).


Comparative statics of sr profit max6
Comparative Statics of SR Profit-Max

  • What happens to the short-run profit-maximizing production plan as the output price p changes?


Comparative statics of sr profit max7
Comparative Statics of SR Profit-Max

The equation of a short-run iso-profit lineis

so an increase in p causes -- a reduction in the slope, and

-- a reduction in the vertical intercept.





Comparative statics of sr profit max11
Comparative Statics of SR Profit-Max

  • An increase in p, the price of the firm’s output, causes

    • an increase in the firm’s output level (the firm’s supply curve slopes upward), and

    • an increase in the level of the firm’s variable input (the firm’s demand curve for its variable input shifts outward).


Long run profit maximization
Long-Run Profit-Maximization

  • Now allow the firm to vary both input levels (both x1 and x2 are variable).

  • Since no input level is fixed, there are no fixed costs.

  • For any given level of x2, the profit-maximizing condition for x1 must still hold.


Long run profit maximization1
Long-Run Profit-Maximization

  • The input levels of the long-run profit-maximizing plan satisfy

  • That is, marginal revenue equals marginal cost for all inputs.

  • Solve the two equations simultaneously for the factor demands x1(p, w1, w2) and x2(p, w1, w2)

and


Returns to scale and profit max
Returns-to-Scale and Profit-Max

  • If a competitive firm’s technology exhibits decreasing returns-to-scale then the firm has a single long-run profit-maximizing production plan.


Returns to scale and profit max1
Returns-to Scale and Profit-Max

y

y*

Decreasingreturns-to-scale

x*

x


Returns to scale and profit max2
Returns-to-Scale and Profit-Max

  • If a competitive firm’s technology exhibits exhibits increasing returns-to-scale then the firm does not have a profit-maximizing plan.


Returns to scale and profit max3
Returns-to Scale and Profit-Max

y

Increasing profit

y”

y’

Increasingreturns-to-scale

x”

x’

x


Returns to scale and profit max4
Returns-to-Scale and Profit-Max

  • So an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive.


Returns to scale and profit max5
Returns-to-Scale and Profit-Max

  • What if the competitive firm’s technology exhibits constant returns-to-scale?


Returns to scale and profit max6
Returns-to Scale and Profit-Max

y

Increasing profit

y”

Constantreturns-to-scale

y’

x”

x’

x


Returns to scale and profit max7
Returns-to Scale and Profit-Max

  • 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 max8
Returns-to Scale and Profit-Max

  • Therefore, when a firm’s technology exhibits constant returns-to-scale, 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 max9
Returns-to Scale and Profit-Max

y

P = 0

y”

Constantreturns-to-scale

y’

x”

x’

x


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