Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Econ 401 Price Theory
Chapter 19: Profit Maximization Problem
Instructor: Hiroki Watanabe
Summer 2009
1 / 49
Intro
SPMP
1
2
3
4
5
6
7
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Introduction
Overview
Short-Run Profit Maximization Problem
Definitions
Short-Run Profit Maximization Problem
Solution to Short-Run Profit Maximization Problem
Example
Interpretation
Comparative Statics
Long-Run Profit Maximization Problem
Solution to Long-Run Profit Maximization Problem
Tangency Condition & Technical Rate of
Substitution
Factor Demand
Returns to Scale and Profit Mazimization Problem
Summary
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Overview
Corresponds to Ch5 utility maximization problem.
()
1 (p, m)
∗
=
ϕ
1
p
ϕ(p, m)
∗
= ϕ2 (p, m)
2
m
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Overview
Q: How many chefs do we need to maximize the
profit?
1
2
You’ll have more revenue as your sales increases.
Hiring too many chefs will reduce the productivity.
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Intro
SPMP
1
2
3
4
5
6
7
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Introduction
Overview
Short-Run Profit Maximization Problem
Definitions
Short-Run Profit Maximization Problem
Solution to Short-Run Profit Maximization Problem
Example
Interpretation
Comparative Statics
Long-Run Profit Maximization Problem
Solution to Long-Run Profit Maximization Problem
Tangency Condition & Technical Rate of
Substitution
Factor Demand
Returns to Scale and Profit Mazimization Problem
Summary
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Definitions
w = (wC , wK ) denotes the factor price (unit price of
inputs).
The total cost associated with the input bundle
(xC , xK ) is
TC(xC , xK ) = wC xC + wK xK .
The total revenue from y is
TR(y) = py
or
TR(xC , xK ) = pf (xC , xK ).
The economic profit generated by the production
plan (xC , xK , y) is
π(xC , xK ) = pf (xC , xK ) − wC xC − wK xK .
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Definitions
The competitive firm takes output price p and all
input prices (w1 , w2 ) as given constants (price taker
assumption).
Output and input levels are typically flows. (To
compute flows, you need to specify a duration of
period on which flows are measured. Stock doesn’t
require that.)
xC = the number of labor units used per hour.
y = the number of cheesecakes produced per hour.
Accordingly, profit is usually a flow.
Other examples: income (f), GDP (f), capital stock
(s), bank balance (s).
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Definitions
Fixed Cost
Fixed cost is a cost that a firm has to pay for the fixed
input.
Kayak’s has to pay the rent (wK ) even when y = 0.
Suppose the size of kitchen if predetermined at x̄K .
FC = wK x̄K .
Fixed cost may or may not be a sunk cost (cost not
recouped, regardless of future actions) depending
on the timing:
1
2
It is sunk after Kayak’s paid the rent.
Not if Kayak’s has not paid the rent.
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Short-Run Profit Maximization Problem
In the short run, the firm solves the short-run profit
maximization problem (SPMP):
Short-Run Profit Maximization Problem (SPMP)
Kayak’s maximizes its short run profit given p, (wC , xK ):
maxxC π(xC , x̄K ) =
pf (xC , x̄K ) −wC xC − wK x̄K
{z
}
| {z } |
total revenue total cost
= pf (xC , x̄K ) − wC xC − FC.
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Short-Run Profit Maximization Problem
Iso-Profit Line
An Iso-profit line at π̄ contains all the production plans
(xC , xK , y) that yield a profit level of π̄.
We do not care if the production plan is actually
feasible.
The iso-profit line simply represents the collection
of plans that yields the same π.
Let’s say xK = 1, wK = 1 and FC = 1 · 1 = 1.
π = py − wC xC − FC
wC
FC + π
⇒ y=
xC +
.
p
p
Higher π means higher y-intercept.
The slope of iso-profit is wpC .
E.g., p = 1, (wC , wK ) = (1, 1) and x̄K = 1.
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Short-Run Profit Maximization Problem
Isoprofit
10
9
6
2
8
Cheesecakes (y)
8
7
π=py−wCxC−FC
0
−2
4
6
2
0
−4
−2
6
4
5
2
4
0
−6
−4
−2
3
−8
2
1
0
0
Intro
SPMP
0
−6
−4
−2
0
−1
−8
1
2
3
4
5
6
Chefs (xC)
Comparative Statics
LPMP
7
8
9
10
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Factor Demand
Returns to Scale
Σ
Solution to Short-Run Profit Maximization Problem
Some of the production plans (xC , xK , y) cannot be
chosen (not feasible)
because of the technological constraint:
y ≤ f (xC , xK ).
(1)
Which production plan yields the highest profit level
while satisfying (1)?
p
E.g., y = f (xC , x̄K ) = 8xC .
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Short-Run Profit Maximization Problem
Isoprofit
10
8
6
7
9
0
1
2
3
−1
−2
4
5
Cheesecakes (y)
8
−3
6
7
6
−2
−5
−3
4
0
1
2
3
−4
−1
−2
3
−8
0
1
π = py − wC xC − F C
√
−7
6
−
y = f(xC , x̄K ) = 9 8xC 10
−4
−1
−2
0
0
−7
−6
−5
−3
1
Intro
−4
−1
4
5
5
2
0
1
2
3
−5
−3
1
SPMP
2
3
4
5
6
Chefs (xC)
Comparative Statics
LPMP
7
−
−
−8
8
9
10
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Factor Demand
Returns to Scale
Σ
Solution to Short-Run Profit Maximization Problem
Recall:
1
2
The slope of production function when xK = x̄K
denotes the marginal product of xC .
The slope of iso-profit is wpC .
Kayak’s profit is maximized at (xC , x̄K , y) where the
production function is tangent to the iso-profit
curve.
Tangency Condition
At the optimal production plan (xC , x̄K , y),
wC
p
= MPC (xC , x̄K ).
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Example
Example
Suppose p = 1,p
w = (1, 1), x̄K = 1 and
y = f (xC , x̄K ) = 8xC .
1
What is the fixed cost?
2
Which production plan maximizes the short-run
profit? (MPC (xC , x̄K ) = p2 ).
2xC
Tangency condition: p2
2xC
= 1.
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Interpretation
What does the tangency condition mean?
wC
p
= MPC (xC , x̄K )
⇒ wC = pMPC (xC , x̄K )
∆y
⇒ ∆TC
= p ∆x
∆xC
C
⇒ additional cost of hiring a chef = additional revenue.
What if wC > pMPC (xC , x̄K )?
What if wC < pMPC (xC , x̄K )?
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Interpretation
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Intro
SPMP
1
2
3
4
5
6
7
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Figure:
Introduction
Overview
Short-Run Profit Maximization Problem
Definitions
Short-Run Profit Maximization Problem
Solution to Short-Run Profit Maximization Problem
Example
Interpretation
Comparative Statics
Long-Run Profit Maximization Problem
Solution to Long-Run Profit Maximization Problem
Tangency Condition & Technical Rate of
Substitution
Factor Demand
Returns to Scale and Profit Mazimization Problem
Summary
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Q: How does Kayak’s respond to wage increase or
price reduction?
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
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Intro
SPMP
Comparative Statics
LPMP
Figure:
Factor Demand
Returns to Scale
Σ
↑ wC reduces xC and y .
↓ p reduces xC and y .
Discussion
1
Does the increase in wK affect the optimal production
plan (xC , x̄K , y)?
2
Does the increase in x̄K affect the optimal production
plan (xC , x̄K , y)?
1
2
No effect (profit gets smaller though).
Short-run technology changes. The same amount of
xC produces more y. xC ↓.
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Example
Suppose p = 1,p
w = (2, 1), x̄K = 1 and
y = f (xC , x̄K ) = 8xC . MPC (xC , x̄K ) = p2
2xC
). What is the
optimal production plan (xC , x̄K , y )?
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SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Isoprofit
10
8
6
7
0
1
2
3
9
−1
−2
4
5
8
Cheesecakes (y)
Intro
−3
6
7
6
3
−4
−1
−2
4
5
−5
−3
5
0
1
2
3
4
−4
−1
−2
3
2
0
1
2
−5
−3
−8
0
1
−4
−1
−2
1
0
0
−7
−6
−5
−3
1
2
3
π = py − wC xC − F C
√
−7
−6 f(xC , x̄K ) = 8xC 0
y=
1
9
4
5
6
Chefs (xC)
7
−
−
−8
8
9
10
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Isoprofit
−5
−1
0
0
−1
5
5
4
0
0 0.5
π = py − wC xC − F C
√
y = f(xC , x̄K ) = 8xC
−2
−5
−1
0
0
−1
5
2
0
Cheesecakes (y)
−5
−1
0
0
5
10
2
10
Chefs (xC)
Intro
SPMP
1
2
3
4
5
6
7
Comparative Statics
LPMP
25 / 49
Factor Demand
Returns to Scale
Σ
Introduction
Overview
Short-Run Profit Maximization Problem
Definitions
Short-Run Profit Maximization Problem
Solution to Short-Run Profit Maximization Problem
Example
Interpretation
Comparative Statics
Long-Run Profit Maximization Problem
Solution to Long-Run Profit Maximization Problem
Tangency Condition & Technical Rate of
Substitution
Factor Demand
Returns to Scale and Profit Mazimization Problem
Summary
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Long-Run Profit Maximization Problem
Long Run
A long-run is the circumstance in which a firm is
unrestricted in its choice of input levels.
Decision-making process in which you can change
the size of the store as well as amount of cheese.
x̄K now becomes xK .
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Long-Run Profit Maximization Problem
Long-Run Profit Maximization Problem (LPMP)
Given w and p, in the long run, Kayak’s solves
max π(xC , xK ) = pf (xC , xK ) − wC xC − wK xK .
xC ,xK
The same condition applies to xK :
wK
p
= MPK (xC , xK ).
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Long-Run Profit Maximization Problem
Example
Kayak’s production function is given by
p
p
f (xC , xK ) = xC + xK .
Price of a cheesecake is p = 2 and w = (1, 1).
MPC (xC , xK ) =
MPK (xC , xK ) =
1
p .
xC
1
p .
2 xK
2
What is the optimal long-run production plan (xC , xK , y)?
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Long-Run Profit Maximization Problem
Isoprofit
5
6
4
4
Cheesecakes (y)
4
8
6
2
4
2
3
0
4
2
0
2
−2
2
0
1
−2 py − wC xC − wK xK−4
π=
√
y = xC + 1
0
0
0
−2
1
−4
2
3
Chefs (xC)
4
5
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Long-Run Profit Maximization Problem
Isoprofit
5
4
8
6
4
4
2
Cheesecakes (y)
6
4
2
3
4
2
0
2
−2
2
0
1
π =−2py − wC xC − wK x−K4
√
y = 1 + xK
0
−2
0
0
Intro
SPMP
0
1
−4
2
3
Size of Kitchen (xK)
Comparative Statics
LPMP
4
5
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Factor Demand
Returns to Scale
Σ
Tangency Condition & Technical Rate of Substitution
Tangency conditions for long-run profit
maximization problem:
wC
p
wK
p
⇒
= MPC (xC , xK )
= MPK (xC , xK ).
wC
⇒
MPC (xC , xK )
=
wK
MPK (xC , xK )
−wC
= TRS(xC , xK ).
wK
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Tangency Condition & Technical Rate of Substitution
If Kayak’s fires one chef, they can expand the
wC
kitchen area by w
.
K
If Kayak’s fire one chef, they need to expand the
kitchen area by TRS(xC , xK ).
The factor market’s idea of chef’s worth coincides
with Kayak’s idea of chef’s worth.
More details in Ch20: cost minimization problem.
33 / 49
Intro
SPMP
1
2
3
4
5
6
7
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Introduction
Overview
Short-Run Profit Maximization Problem
Definitions
Short-Run Profit Maximization Problem
Solution to Short-Run Profit Maximization Problem
Example
Interpretation
Comparative Statics
Long-Run Profit Maximization Problem
Solution to Long-Run Profit Maximization Problem
Tangency Condition & Technical Rate of
Substitution
Factor Demand
Returns to Scale and Profit Mazimization Problem
Summary
34 / 49
Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Change in wC affects xC as well as π.
Tangency condition:
pMPC (xC , xK ) = wC .
At each wC , Kayak’s sets xC at which the additional
increase in revenue equates wC (factor demand
function).
Diminishing marginal product: MPC (xC , xK ) goes
down as xC increases.
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
36 / 49
Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Example
Suppose MPC (xC , xK ) =
function is given by
1
p .
2 xC
wC
p
=
The factor demand
1
p .
2 xC
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Factor Demand
3
C
Wage (w )
p=1
p=2
1
0.5
0
0
1
3
Chefs (xC)
38 / 49
Intro
SPMP
1
2
3
4
5
6
7
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Introduction
Overview
Short-Run Profit Maximization Problem
Definitions
Short-Run Profit Maximization Problem
Solution to Short-Run Profit Maximization Problem
Example
Interpretation
Comparative Statics
Long-Run Profit Maximization Problem
Solution to Long-Run Profit Maximization Problem
Tangency Condition & Technical Rate of
Substitution
Factor Demand
Returns to Scale and Profit Mazimization Problem
Summary
39 / 49
Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
If a competitive firm’s technology exhibits
decreasing returns to scale then the firm has a
single long-run profit-maximizing production plan.
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
41 / 49
Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Figure:
If a competitive firm’s technology exhibits exhibits
increasing returns to scale then the firm does not
have a profit-maximizing plan.
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Figure:
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?
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Figure:
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.
When a firm’s technology exhibits constant returns
to scale, earning a positive economic profit is
inconsistent with firms being perfectly competitive.
A CRS firm is compatible with perfect competition
only when firm earns zero profit.
46 / 49
Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
47 / 49
Intro
SPMP
1
2
3
4
5
6
7
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Figure:
Introduction
Overview
Short-Run Profit Maximization Problem
Definitions
Short-Run Profit Maximization Problem
Solution to Short-Run Profit Maximization Problem
Example
Interpretation
Comparative Statics
Long-Run Profit Maximization Problem
Solution to Long-Run Profit Maximization Problem
Tangency Condition & Technical Rate of
Substitution
Factor Demand
Returns to Scale and Profit Mazimization Problem
Summary
48 / 49
Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solving profit maximization problem.
Comparative statics.
Factor demand.
Competitive environment and compatible
technology.
49 / 49