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# Profit Maximization

By hammerbang Feb 26, 2013 2684 Words
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

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 2 / 49

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

3 / 49

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.

4 / 49

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 5 / 49

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 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.

8 / 49

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

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 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. 10 / 49

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
11 / 49

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
13 / 49

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

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

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 18 / 49

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 ↓.

22 / 49

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
25 / 49

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 26 / 49

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
30 / 49

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.

33 / 49

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 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: MP (x , x+ ) goes C C

down as xC increases.

K

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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

Example + Suppose MPC (xC , xK ) = function is given by
1 . 2 xC

The factor demand 1

wC p

=

.

2

xC

37 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Factor Demand 3 p=1 p=2

Wage (w )

C

1 0.5 0 0

1 Chefs (xC)

3
38 / 49

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 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

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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.

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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 48 / 49

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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