SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Econ 401 Price Theory
Chapter 19: Profit Maximization Problem
Instructor: Hiroki Watanabe
Summer 2009
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
1
2
3 4
5 6 7
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 Overview
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Corresponds to Ch5 utility maximization problem.
( )
∗ 1
= ϕ1 (p, m)
p
ϕ(p, m)
∗ 2
= ϕ2 (p, m)
m
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Intro Overview
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
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
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
1
2
3 4
5 6 7
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
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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 xK . ¯ FC = wK xK . 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 , xK ) = ¯ pf (xC , xK ) ¯ −wC xC − wK xK
total revenue total cost ¯ = pf (xC , xK ) − wC xC − FC.
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Intro
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Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Short-Run Profit Maximization Problem
Iso-Profit Line ¯ An Iso-profit line at π contains all the production plans ¯ (xC , xK , y) that yield a profit level of π. 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 wC . p ¯ E.g., p = 1, (wC , wK ) = (1, 1) and xK = 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 8 Cheesecakes (y) 7 6
4 6 2 0 −2 −4 8 4 6 2
π=py−wCxC−FC
0
−2
5 4 3 2 1 0 0
0 −2 −4 −6 −8 0 −1 2 0 −2 −4 −6 −8
1
2
3
4 5 6 Chefs (xC)
7
8
9
10
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Intro
SPMP
Comparative Statics
LPMP
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)? ¯ E.g., y = f (xC , xK ) = 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 9 8 Cheesecakes (y) 7 6 5 4 3 2 1 0 0 1
1 0 −1 −2 −3 −5 −4 3 2 5 6 3 4 2 1 0 −1 −2 −3 1 0 −1 −2 −3 −5 −4 −6 −7 −8 −5 −4 8 7 5 6 3 4 2 1 0 −1 −2 −3
π = py − wC xC − F C √ −7 −6 f(xC , xK ) = 8xC 0 y= ¯ 1 9
−8 − −
2
3
4 5 6 Chefs (xC)
7
8
9
10
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solution to Short-Run Profit Maximization Problem
Recall:
1
2
¯ The slope of production function when xK = xK denotes the marginal product of xC . The slope of iso-profit is wC . p
¯ Kayak’s profit is maximized at (xC , xK , y) where the production function is tangent to the iso-profit curve. Tangency Condition ¯ At the optimal production plan (xC , xK , y), wC p ¯ = MPC (xC , xK ).
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Intro Example
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Example ¯ Suppose p = 1, w = (1, 1), xK = 1 and ¯ y = f (xC , xK ) = 8xC .
1 2
What is the fixed cost? Which production plan maximizes the short-run ¯ profit? (MPC (xC , xK ) = 2 ).
2xC
Tangency condition:
2 2xC
= 1.
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Intro
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Comparative Statics
LPMP
Factor Demand
Returns to Scale
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Interpretation
What does the tangency condition mean? ¯ = MPC (xC , xK ) ¯ ⇒ wC = pMPC (xC , xK ) ∆y ⇒ ∆TC = p ∆x ∆xC C ⇒ additional cost of hiring a chef = additional revenue. ¯ What if wC > pMPC (xC , xK )? ¯ What if wC < pMPC (xC , xK )? wC p
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Interpretation
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
1
2
3 4
5 6 7
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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Figure:
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 x+ and y+ .
C
Discussion
1
Does the increase in wK affect the optimal production ¯ plan (xC , xK , y)? ¯ Does the increase in xK affect the optimal production ¯ plan (xC , xK , y)?
1 2
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, w = (2, 1), xK = 1 and ¯ ¯ y = f (xC , xK ) = 8xC . MPC (xC , xK ) = ¯ optimal production plan (x+ , x , y+ )?
C K 2 2xC
). What is the
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Isoprofit 10 9 8 Cheesecakes (y) 7 6 5 4 3 2 1 0 0 1
1 0 −1 −2 −3 −5 −4 3 2 5 6 3 4 2 1 0 −1 −2 −3 1 0 −1 −2 −3 −5 −4 −6 −7 −8 −5 −4 8 7 5 6 3 4 2 1 0 −1 −2 −3
π = py − wC xC − F C √ −7 −6 y = f(xC , xK ) = 9 8xC 10 ¯
−8 − −
2
3
4 5 6 Chefs (xC)
7
8
9
10
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Isoprofit 10
5 0
Cheesecakes (y)
5
−5
−1 0
0
4
0 0 0.5
2 Chefs (xC)
−2 0
π = py − wC xC − F C √ y = f(xC , xK ) = 8xC ¯
0
−5
−1
0
−1 5
2
−5
−1 0
−1 5
10
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
1
2
3 4
5 6 7
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. ¯ xK 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 f (xC , xK ) = xC + xK .
Price of a cheesecake is p = 2 and w = (1, 1). MPC (xC , xK ) = MPK (xC , xK ) =
2 1 . xC 1 . 2 xK
+ + 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
8 4 6 4 6 4 2
4 Cheesecakes (y)
3
4 2
2 0
2
2 0
0 −2
1
0 −2
−2 π = py − wC xC − wK xK 4 − √ y = xC + 1 −4
0 0
1
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
8 6 4
4
6
4
2
Cheesecakes (y)
4
3
4 2
2
0
2
2 0
0 −2
1
0 −2
− py π = 2 − wC xC − wK x−4 K √ y = 1 + xK −4
0 0
1
2 3 Size of Kitchen (xK)
4
5
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Intro
SPMP
Comparative Statics
LPMP
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 ).
⇒
= wK MPK (xC , xK ) −wC = TRS(xC , xK ). wK
wC
MPC (xC , xK )
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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 w kitchen area by wC .
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.
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
1
2
3 4
5 6 7
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
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: MP (x , x+ ) goes
C C
down as xC increases.
K
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Intro
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Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
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Intro
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Comparative Statics
LPMP
Factor Demand
Returns to Scale
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Example + Suppose MPC (xC , xK ) = function is given by
1 . 2 xC
The factor demand 1
wC p
=
.
2
xC
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Intro
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Comparative Statics
LPMP
Factor Demand
Returns to Scale
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Factor Demand 3 p=1 p=2
Wage (w )
C
1 0.5 0 0
1 Chefs (xC)
3
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Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
1
2
3 4
5 6 7
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
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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
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Comparative Statics
LPMP
Factor Demand
Returns to Scale
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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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Factor Demand
Returns to Scale
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Intro
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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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Comparative Statics
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Factor Demand
Returns to Scale
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Intro
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Comparative Statics
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Factor Demand
Returns to Scale
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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.
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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
1
2
3 4
5 6 7
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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Figure:
Intro
SPMP
Comparative Statics
LPMP
Factor Demand
Returns to Scale
Σ
Solving profit maximization problem. Comparative statics. Factor demand. Competitive environment and compatible technology.
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