# Profit Maximization

**Topics:**Economics, Economics of production, Microeconomics

**Pages:**38 (2684 words)

**Published:**February 26, 2013

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 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

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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.

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.

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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

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...

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