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# AP ECON 2300 F2012 Session 10.doc

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York University

Economics

ECON 2300

Wai Ming Ho

Fall

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AP ECON 2300 F2012 Session 10
Topic: Profit Maximization and Cost Minimization
Reading: Chapters 19 & 20
Ch 19: Profit Maximization:
Economic Profit
• A firm uses inputs j = 1…,m to make products i = 1,…n; Output levels are y ,…,y 1 n
• Input levels are x1,…,x m; Product prices are p ,1,p . n
• Input prices are w 1…,w m.
Economic Profit
• The economic profit generated by the production plan (x ,…,x1,y ,…my 1 is n
Π = p y + + p y − w x − w x .
1 1 n n 1 1 m m
• x1might be the number of labor units used per hour.
• And y 3ight be the number of cars produced per hour.
How do we value a firm?
• Suppose the firm’s stream of periodic economic profits is P , P 0 P1, …2and r is the rate of interest.
• Then the present-value of the firm’s economic profit stream is
Π Π
PV = Π + 1 + 2 +
0 1+r (1+r) 2
• A competitive firm seeks to maximize its present-value.
• How?
Page 1 of 15 • Suppose the firm is in ~ short-run circumstance in which its short-run production function is
y = f (x1, x2).
• The firm’s fixed cost is ~
FC = w x 2 2
• and its profit function is
~
Π = py − w x1−1w x 2 2
Short-Run Iso-Profit Lines
• A $ Π iso-profit line contains all the production plans that provide a profit levΠl $.
~
• A $ Π iso-profit line’s equation Π ≡ py − w x1−1w x 2 2
~ ~
y = w 1x + Π + w 2 2. has a slope of + w 1 and a vertical intercept of Π + w 2 2.
p 1 p p p
Short - Run Iso - Profit Lines
y
x 1
© 2010 W. W. Norton & Company, Inc. 13
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.
• Q: What is this constraint?
• A: The production function.
Short -Run Profit -Maximization
Given p, w and the short -run
y 1
profit -maximizing plan is
And the maximum
possible profit
is
x
1
© 2010 W. W. Norton & Company, Inc. 20
Page 2 of 15 Short-Run Profit -Maximization
At the short -run profit -maximizing plan,
y
the slopes of the short -run production
function and the maximal
iso-profit line are
equal.
x 1
© 2010 W. W. Norton & Company, Inc. 22
w 1
MP =1 ⇔ p × MP = 1 1
p
• p *MP 1 is the marginal revenue product of input 1, the rate at which revenue increases
with the amount used of input 1.
• If p *MP >1w the1 profit increases with x . 1
• If p *MP < w then profit decreases with x .
1 1 1
Short-Run Profit-Maximization; A Cobb-Douglas Example
1/3~ 1/3
• Suppose the short-run production function is y = x 1 x 2 .
MP = ∂ y = 1 x − 2 x1/ .
1 ∂ x1 3 1 2
p * −2 /~1/3 p
MRP =1p × MP = 1 (x1) x2 = w 1 (x 1 −2 /x32/ = w 1
3 3
~1/3
* −2/3 3w 1 * 2/3 px 2
(x1) = ~ 1/3. (x1) =
px2 3w 1
~1/3 3/2 3/2
* px 2 p ~1/2
x 1 = x2 .
3w 1 3w 1
p 3/ 2
x 1 x2/ 2 is the firm’s short-run demand for input 1 when the level of input 2 is fixed
3w 1 at ~ units. x2
x 2
The firm’s short-run output level is thus
1/ 2
y = (x )* 1/3x1/ 3= p x1/ .
1 2 3w 2
1
Comparative Statics of Short-Run Profit-Maximization
Page 3 of 15 • What happens to the short-run profit-maximizing production plan as the output price p changes?
• The equation of a short-run iso-profit line is
w Π + w x ~
y = 1 x1+ 2 2
p p
• so an increase in p causes
-- a reduction in the slope, and
-- a reduction in the vertical intercept.
Comparative Statics of Short -
y Run Profit -Maximization
x 1
© 2010 W. W. Norton & Company, Inc. 34
• 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).
1/3~1/3
The Cobb-Douglas example: When y = x 1 x2 then the firm’s short-run demand for its variable input 1 is
3 / 2 1/ 2
x = p x 1/ 2y = p x1/ .
1 3w 2 3w 2
1 1
x* increases as p increases
1
*
y increases as p increases.
What happens to the short-run profit-maximizing production plan as the variable input price w changes? 1
The equation of a short-run iso-profit line is
~
w 1 Π + w x2 2
y = p x1+ p
so an increase in w ca1ses
-- an increase in the slope, and
-- no change to the vertical intercept.
Page 4 of 15 Comparative Statics of Short -
Run Profit -Maximization
y
x
1
© 2010 W. W. Norton & Company, Inc. 43
• An increase in w , 1he 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).
• Example:
3 / 2 1/ 2
1/3~ 1/3 * p ~1/ 2 * p ~1/ 2
PF: y = x 1 x 2 D: x1= 3w x2 S: y = 3w x2 .
1 1
x1* decreases as w in1reases, y* decreases as w increase1
Long-Run Profit-Maximization
• Now allow the firm to vary both input levels.
• Since no input level is fixed, there are no fixed costs.
• Both x 1nd x ar2 variable.
• Think of the firm as choosing the production plan that maximizes profits for a given value of x , and the2 varying
x2to find the largest possible profit level.
• The equation of a long-run iso-profit line is
w Π + w x
y = 1x1+ 2 2
p p
• so an increase in x c2uses
-- no change to the slope, and
-- an increase in the vertical intercept.
Page 5 of 15 Long -Run Profit -Maximization
y
The marginal product
of input 2 is
diminishing.
Larger levels of input 2 increase the x 1
© 2010 W. W. Norton & Company, Inc. 53
Long -Run Profit -Maximization
y for each short -run
production plan.
the marginal profit
of input 2 is
diminishing.
x
1
© 2010 W. W. Norton & Company, Inc. 57
• Profit will increase as x2increases so long as the marginal profit of input 2
p × MP 2 w > 2.
• The profit-maximizing level of input 2 therefore satisfies
p × MP 2 w =20.
p × MP − w = 0
1 1
• That is, marginal revenue equals marginal cost for all inputs.
The Cobb-Douglas example: When
y = x 1/3x1/3
1 2
• then the firm’s short-run demand for its variable input 1 is
3 / 2 1/ 2
x = p x 1/ 2 y = p x 1/.2
1 3w 1 2 3w 1 2
Page 6 of 15 Π = py − w x 1 1 x 2~ 2
1/ 2 3/ 2
p 1/ 2 p 1/ 2
= p x2 − w x2 − w 2~2
3w 1 3w 1
1/ 2 1/ 2
p ~ 1/ 2 p p ~
= p x2 − w 1 − w 2 2
3w 1 3w 1 3w 1
1/ 2
2 p p ~ 1/ 2 ~
= x 2 − w 2 2
3 3w 1
3 1/ 2
4 p ~1/ 2 ~
= x2 − w 2 .2
27w 1
4 p3 1/ 2
Π = x 1/ − w x .
27 w 1 2 2 2
What is the long-run profit-maximizing level of input 2? Solve
3 1/ 2
∂ Π 1 4 p ~ −1/ 2
0 = ~ = x2 − w 2
∂ x 2 2 27 w 1
3
~ * p
x2= x =2 2.
27w w1 2
3 / 2
* p ~ 1/ 2
x1= x 2
3w 1
3/ 2 3 1/ 2 3
x = p p = p .
1 3w 27w w 2 27w w2
1 1 2 1 2
1/ 3 1/ 2 2
y = p p = p .
3w 27 w w 2 9w w
1 1 2 1 2
y = x 1/3x1/3
1 2
p 3 p 3 p 2
(x , x , y ) = , , .
1 2 27w w 2 27w w 2 9w w
1 2 1 2 1 2
Returns-to-Scale and Profit-Maximization
• If a competitive firm’s technology exhibits decreasing returns-to-scale then the firm has a single long-run profit-
maximizing production plan.
• If a competitive firm’s technology exhibits exhibits increasing returns-to-scale then the firm does not have a profit-
maximizing plan.
• So an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive.
What if the competitive firm’s technology exhibits constant returns-to-scale?
• 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.
Page 7 of 15 • 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.
Revealed Profitability: Not included: Page 356 Chapter 19.11.
Ch 20: Cost Minimization
• A firm is a cost-minimizer if it produces any given output level y ≥ 0 at smallest possible total cost.
• c(y) denotes the firm’s smallest possible total cost for producing y units of output.
• c(y) is the firm’s total cost function.
• When the firm faces given input prices w = (w ,w ,…1w 2 the tntal cost function will be written as
c(w 1…,w ,yn.
The Cost-Minimization Problem
• Consider a firm using two inputs to m

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