Cost Minimization Problem - University of Minnesota Duluthd.umn.edu/~watanabe/econ3821fl12/docs/cmpho.pdf · 3 Keeping to Output Requirement / Isoquant 4 Cost Minimization Problem

Intro Minimizing TC Keeping to y CMP Sub/Scale Effects Affirmative Action

Econ 3821 Labor Economics

Chapter 3C:Cost Minimization Problem

Instructor: Hiroki WatanabeFall 2012

Watanabe Econ 3821 3C CMP 1 / 113


1 Introduction

2 Minimizing Total Cost / Isocost

3 Keeping to Output Requirement / Isoquant

4 Cost Minimization Problem

5 Substitution & Scale Effect

6 Affirmative Action

7 Now We Know



1 IntroductionAgendaWhen PMP Doesn’t WorkCost Minimization Problem






7 Now We KnowWatanabe Econ 3821 3C CMP 3 / 113


Agenda

Question 1.1 (Agenda)

Where does labor demand function come from?



Agenda

Question 1.2 (Robots Ate My Job)

Listen to Marketplace Audio Clip :National Robotics Initiative to improvemanufacturing and the betting is to create jobs,not take them away, as so often is the worrywith robots.

When exactly do robots eat your job and when theydon’t?



Agenda

1 To analyze relationship between C and K as inQuestion 1.2 , y needs to be pegged at some constant.

2 If not...Even when robots eat my job, if y is increasing, Iprobably still have a job.Ambiguous result.



When PMP Doesn’t Work

Profit maximization problem is intuitive, but1 it is cumbersome when you let C, K and y move at

the same time.2 we can’t use it for some technologies.




Example 1.3 (Case When PMP Doesn’t Work)

Suppose 1 chef bakes 1 cheesecake, 2 chefs bake 2cheesecakes, ... (ƒ (C) = C). If wage is C = 1 andcheesecakes sell for p = 2 apiece, how many chefs willJack hire?




Jack’s output level exceeds Duluth population atsome point.The market is no longer perfectly competitive.We can still pose a question:

Given y, what is the best combination of(∗

C, ∗

K) that can produce y most

cost-effectively?



Cost Minimization Problem

Problem 1.4 (Cost Minimization Problem)

Given y = y, Jack

minC,K TC(C, K) =CC +KKsubject to y = ƒ (C, K).

(1)

We’re asking “You got to produce y no matter what.Find the combination of (C, K) that is easiest onthe pocketbook.



Cost Minimization Problem

How do we find the solution?Split (1) in two parts:

1 Minimizing total cost TC(C, K ) =CC +KK .2 Keeping to the asked amount y = ƒ (C, K ).

Do not mix them up:1 TC(C, K ) doesn’t tell Jack how much y he can

produce.2 ƒ (C, K ) doesn’t tell Jack how much (C, K ) will

cost.



1 Introduction

2 Minimizing Total Cost / IsocostSetupTrinity on the Cost SideRelative PriceOpportunity CostComparative Statics





7 Now We Know



Setup

Definition 2.1 (Components of Total Cost)1 An input bundle = (C, K) contains C chefs

and K ft2 of kitchen.2 A factor price = (C, K) marks the wage and

rental rate each.3 Total cost of input bundle is

TC(C, K) =CC +KK .



Setup

Example 2.2 (Computing Total Cost)

If = (C,K) = (2,1),

TC(C, K) = 2C + K .

1 = (C, K ) = (4,10) will cost TC(4,10) = 18.2 = (C, K ) = (5,10) will cost TC(5,10) = 20.3 = (C, K ) = (4,12) will cost TC(4,12) = 20.



Setup

We can list the total cost for each = (C, K) allday long.Very tiring.

Definition 2.3 (Isocost Line)An isocost line at $c is a collection of all = (C, K)that costs exactly $c.

If = (C, K) is on the isocost line at c, it satisfies

CC +KK = c.



Setup

Continued from Example 2.2 :

Example 2.4 (Finding Isocost Line)

If = (C,K) = (2,1), what is the isocost at c = 20?

(C, K) = (0,20), (2,16), (10,0) etc.2C + K = 20.K = −2C + 20.



Setup

0 2 4 6 8 100

4

8

12

16

20

Chefs xC

Kitc

hen

x K (

ft2 )

Isocost at (wC, w

T, c)=(2,1,20)

0 2 4 6 8 100

4

8

12

16

20

Chefs xC

Kitc

hen

x K (

ft2 )



Setup

Fact 2.5 (Intercepts & Isocost)

The y-intercept denotes the kitchen size whenJack doesn’t hire anyone ( c

K).

The x-intercept denotes the number of chefswhen Jack doesn’t rent a kitchen ( c

C).



Setup

00

Chef xC

Kitc

hen

x K (

ft2 )

Under BudgetOver BudgetIsocostc/w

Cc/w

K



Setup

0 2 4 6 8 100

4

8

12

16

20

4

8

12

16

20

24

28

32

36

Chefs xC

Kitc

hen

x K (

ft2 )

(xC, x

K): TC(x

C, x

K)=c

0 2 4 6 8 100

4

8

12

16

20

4

8

12

16

20

24

28

32

36

Chefs xC

Kitc

hen

x K (

ft2 )



Trinity on the Cost Side

Question 2.6 (Slope of the Isocost Line)

What does the slope of the isocost line imply?

Answer 2.7 (Trinity on the Cost Side)

The following are the same:1 The slope of the isocost line.2 Relative price.3 Opportunity cost.



Relative Price

One chef buysC

Kunits of kitchen.

1 If you layoff one chef, Jack will save C dollars.

2 With C dollars, Jack can buyC

Kft2 of kitchen at

the market.3 The slope tells Jack the market rate of exchange

between C and K a.k.a. relative price of akitchen in terms of chefs. 1

1Note that the relative price is predetermined in the factormarket and Jack’s technology has no bearing on it.



Opportunity Cost

Definition 2.8 (Opportunity Cost)

is the best alternative that Jack forgoes or gives up,when he makes a choice or decision.

When Jack hires one more chef, Jack has to give upC

Kft2 of kitchen. 2

2Once again, we do not care why Jack would trade a chef in forkitchen here. It is simply a market matter but not Jack’s technologymatter.



Opportunity Cost

To sum up, the slope of the isocost line representsthe relative price, andthe opportunity cost of buying (hiring) one unit of C(measured in terms of kitchen size).

Note we do not care which bakery we analyze atthis point. Factor prices are given in advance andthe bakery is a price taker.



Comparative Statics

Question 2.9 (Change in Parameters and BudgetLine)How does the insocost line change when factor pricesand total cost change?

1 Stay put2 Parallel shift3 Rotation (pivotting)4 Combination of parallel shift and rotation



Comparative Statics

Suppose the allotted expense increased from 1 to100:

CC +KK = 1↓

CC +KK = 100.



Comparative Statics

The intercept changes because ...The slope remains constant because ...

K =−C

pKC +

1

pK→ K =

−C

pKC +

100

pK.

A change in total cost results in a parallel shift.



Comparative Statics

00

Chef xC

Kitc

hen

x K (

ft2 )

00

Chef xC

Kitc

hen

x K (

ft2 )



Comparative Statics

Suppose C increased from 2 to 100:

2C +KK = c↓

100C +KK = c.

The slope changes because ...The horizontal intercept changes because ...The vertical intercept remains constant (why?)A change in price results in rotation.

K =−2pK

C +c

pK→ K =

−100pK

C +c

pK.



Comparative Statics

00

Chef xC

Kitc

hen

x K (

ft2 )

00

Chef xC

Kitc

hen

x K (

ft2 )



Comparative Statics

Exercise 2.10 (Change in Price)

Jack spends $12 on chef (C) and kitchen (K). Theprice is given by (C,K) = (3,2).

1 Draw Jack’s isocost line.2 What happens to the isocost line if C drops to 1?

In particular, consider the change in1 maximum number of chefs he can hire2 maximum size of kitchen he can rent3 his opportunity cost of a chef in terms of kitchen

3 Can we say the rotation of isocost line will result inmore cheesecakes? If not, what additionalinformation do we need?



Comparative Statics

0 2 4 6 8 10 120

1

2

3

4

5

6

Chef xC

Kitc

hen

x K (

ft2 )

(wC, w

T, c)=(3, 2, 12)

(wC, w

T, c)=(1, 2, 12)

0 2 4 6 8 10 120

1

2

3

4

5

6

Chef xC

Kitc

hen

x K (

ft2 )



Comparative Statics

While reduced wage expands Jack’s choice set, itdoes not necessarily increase his output level.He may not care for the chef in the first place.

We also need to know Jack’s technology as well ashis cost structure to understand his hiring decision(to be discussed in the next section).



1 Introduction


3 Keeping to Output Requirement / IsoquantIsoquantCobb-Douglas Production FunctionTrinity on the Technology SideComparison to Relative Prices




7 Now We Know



Isoquant

Recall Problem 1.4 :

minC,K TC(C, K) =CC +KKsubject to y = ƒ (C, K).

Liz has to produce y.Which combination = (C, K) will produce y mostcost-effectively?



Isoquant

Definition 3.1 (Production Function)ƒ () describes the relationship between quantity ofinputs = (C, K , · · · ) and quantity y that canproduce.



Isoquant

Definition 3.2 (Isoquant)

The isoquant at an input bundle = (C, K) is acollection of all input bundles that producey = ƒ (C, K).

If an input bundle y = (yC, yK) and z = (zC, zK) areon the same isoquant, then they produce the samenumber of cheesecakes.



Isoquant

Liz hires left-handed chefs L and right-handedchefs R to produce cheesecake y = ƒ (L, R).Each chef (lefty or righty) bakes 1 cheesecake.Her production function is

y = ƒ (L, R) = L + R.



Isoquant

Question 3.3 (Isoquant)

Trace isoquant at (L, R) = (0,4) and (2,6) of aproduction function

y = ƒ (L, R) = L + R.



Isoquant

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

2

4

6

8

10

12

14

16

18

Left−Handed Chefs xL

Rig

ht−

Han

ded

Che

f xR

(xL, x

R): f(x

L, x

R)=y

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

2

4

6

8

10

12

14

16

18


Rig

ht−

Han

ded

Che

f xR



Cobb-Douglas Production Function

The most commonly used production function isCobb-Douglas production function:

ƒ (C, K) = CbK,

where > 0 and b > 0.E.g., ƒ (C, K) = CK .Isoquant at (C, K) = (1,10)?




10 0 10 20 30 40 50 60 70 80 90 1009 0 9 18 27 36 45 54 63 72 81 908 0 8 16 24 32 40 48 56 64 72 807 0 7 14 21 28 35 42 49 56 63 706 0 6 12 18 24 30 36 42 48 54 605 0 5 10 15 20 25 30 35 40 45 504 0 4 8 12 16 20 24 28 32 36 403 0 3 6 9 12 15 18 21 24 27 302 0 2 4 6 8 10 12 14 16 18 201 0 1 2 3 4 5 6 7 8 9 100 0 0 0 0 0 0 0 0 0 0 0

K \C 0 1 2 3 4 5 6 7 8 9 10




0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

10

20

30

40

50

60

70

80

90

Chefs xC

Kitc

hen

x T (

ft2 )

Isoquants {x: f(x)=y}

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

10

20

30

40

50

60

70

80

90

Chefs xC

Kitc

hen

x T (

ft2 )



Trinity on the Technology Side

Just like the slope of the isocost line has themeaning (recall trinity),the slope of the isoquant carries two importantmeanings.

Fact 3.4 (Trinity on the Technology Side)

The following are the same:1 The slope of isoquant curve.

2 Marginal rate of technical substitution.

3 Ratio between MPC and MPK .a

aYou can forget this one. Proof is left at Appendix .




Definition 3.5 (Marginal Rate of TechnicalSubstitution)If Liz hires 1 chef, she can maintain the currentproduction level y even if she reduces K by ft2. This is called her marginal rate of technicalsubstitution (MRTS).

1 chef replaces MRTS ft2 kitchen without takingany toll on the ongoing production level y.




E.g., if ƒ (4,10) = ƒ (5,6), then Liz can give up 4 ft2in exchange for 1 chef.

She will still get the same number of cheesecakes.Her MRTS at (4,10) is 4.

If ƒ (8,4) = ƒ (9,3),Her MRTS at (8,4) is 1.




The slope of isoquant represents themarginal rate of technical substitution.Take Question 3.3 .Slope of isoquant is −1.How many right-handed chefs R can Liz lay off ifshe adds one left-handed chef to payroll withoutinterfering production?It’s 1 as well.




0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

2

4

6

8

10

12

14

16

18


Rig

ht−

Han

ded

Che

f xR

(xL, x

R): f(x

L, x

R)=y

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

2

4

6

8

10

12

14

16

18


Rig

ht−

Han

ded

Che

f xR




Question3.3 is a rather extreme case.MRTS usually varies depending on the input bundle.Consider two bundles:

= (C, K) = (1,29384720396)y = (yC, yK) = (3240894603,1).

MRTS at is pretty large (why?)MRTS at y is pretty small (why?)




I.e., Liz prefers a balanced combination of twoinputs rather than something extreme.This tendency is called convex technology.



Comparison to Relative Prices

Notice the difference:Relative price is ft2 that Liz has to sell (give up) tohire one more chef to keep to her budget c.Marginal rate of technical substitution is ft2 Lizcan give up to hire one more chef whilemaintaining output level y.



1 Introduction



4 Cost Minimization ProblemIsocost Curves Meet IsoquantTangencyTo Find the Exact Solution





Isocost Curves Meet Isoquant

Definition 4.1 (Long-Run Cost MinimizationProblem)Liz chooses the input bundle = (C, K) that costs theleast while maintaining the production level, i.e.,

minC,K TC(C, K) =CC +KKsubject to ƒ (C, K) = y.




How does Liz make her hiring and investmentdecision?i.e., how do we find Liz’s optimal number of chefsand size of kitchen ∗ = (∗

C, ∗

K)?

Consider a case whenOutput requirement ƒ (C, K ) = CK = 150.Factor price (C,K ) = (3,2).




0 5 10 15 200

5

10

15

20

25

30

10

20

30

40

50

60

70

80

90

100

110

Chef xC

Kitc

hen

x K (

ft2 )

Isocost Lines

0 5 10 15 200

5

10

15

20

25

30

10

20

30

40

50

60

70

80

90

100

110

Chef xC

Kitc

hen

x K (

ft2 )




0 5 10 15 200

5

10

15

20

25

30

Chef xC

Kitc

hen

x K (

ft2 )

Isoquant at y=150

0 5 10 15 200

5

10

15

20

25

30

Chef xC

Kitc

hen

x K (

ft2 )




0 5 10 15 200

5

10

15

20

25

30

10

20

30

40

50

60

70

80

90

100

110

Chef xC

Kitc

hen

x K (

ft2 )

Isocost LinesIsoquant at y=150

0 5 10 15 200

5

10

15

20

25

30

10

20

30

40

50

60

70

80

90

100

110

Chef xC

Kitc

hen

x K (

ft2 )



Tangency

Condition 4.2 (Tangency Condition)

For standard technologies, the isocost line is tangent toisoquant at the optimal bundle, i.e., they share thesame slope at ∗.



Tangency

What does tangency condition imply?The slope of isoquant denotes MRTS:

ft2 of kitchen Liz can give up to hire one morechef.

The slope of isocost line denotes relative price(opportunity cost):

ft2 of kitchen Liz has to give up to hire onemore chef.

Liz’s idea of the chef’s worth coincides withlabor market’s idea of the chef’s worth whenshe chooses the right amount.



Tangency

What if tangency condition is not met?



Tangency

0 5 10 15 200

5

10

15

20

25

30

10

20

30

40

50

60

70

80

90

100

110

Chef xC

Kitc

hen

x K (

ft2 )


0 5 10 15 200

5

10

15

20

25

30

10

20

30

40

50

60

70

80

90

100

110

Chef xC

Kitc

hen

x K (

ft2 )



To Find the Exact Solution

Where exactly is the isocost line tangent toisoquant?We need two conditions to find the point oftangency:

Condition 4.3 (Finding the Solution)

For standard production functions, the solution can befound using

1 tangency condition and

2 output requirement




Problem 4.4 (CMP)

minC,K TC(C, K) = 3C + 2Ksubject to ƒ (C, K) = CK = 150.

MRTS at (C, K) is−KC

.3

The relative price is −32 .

3In what follows, I’ll get you MRTS unless it is too obvious.Watanabe Econ 3821 3C CMP 63 / 113



Tangency condition:

−KC

=−32⇒ K =

3

2C. (2)

There are lots of bundles (C, K) satisfying (2).




0 5 10 15 200

5

10

15

20

25

30

10

20

30

40

50

60

70

80

90

100

110

Chef xC

Kitc

hen

x K (

ft2 )


0 5 10 15 200

5

10

15

20

25

30

10

20

30

40

50

60

70

80

90

100

110

Chef xC

Kitc

hen

x K (

ft2 )




To pin down the solution, use output requirement:

CK = 150.

Conclude ∗ = (10,15).




Exercise 4.5 (UMP)Liz is asked to produce y = 3 according to herproduction function ƒ (C, K) = C +

pK . Factor price is

given by (C,K) = (2,1). His preferences arerepresented by (C, K) = C +

pK .

1 Write Liz’s cost minimization problem.2 What is the tangency condition in this example?

(MRTS at (C, K) is −2pK).

3 Which bundle does Liz choose?




0 0.5 1 1.5 2 2.50

1

2

3

4

5

1.0

2.0

3.0

4.0

5.0

5.0

6.0

7.0

8.0

9.0

Chef xC

Kitc

hen

x K (

ft2 )


0 0.5 1 1.5 2 2.50

1

2

3

4

5

1.0

2.0

3.0

4.0

5.0

5.0

6.0

7.0

8.0

9.0

Chef xC

Kitc

hen

x K (

ft2 )



1 Introduction




5 Substitution & Scale EffectSubstitution EffectScale EffectGross Substitutes & Complements




Substitution Effect

Getting back to Question 1.2 .

Example 5.1 (Substitution & Scale Effect)

Liz produces cheesecake y with a production function

ƒ (C, K) = CK ,

where C is a number of chefs and K measures thekitchen equipment. The marginal rate of technical

substitution at (C, K) is MRTS(C, K) =−KC

. She was

asked to produce y = 100 cheesecakes. Factor price is(C,K) = (4,1).

1 Suppose C = 4↘1. Should she invest more or less inK?



Substitution Effect

Split the price change in three progressive stages:1 Original: O2 Transitional: T3 Final: F



Substitution Effect

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

2040

60 80

100

120

140

160180

Chef xC

Kitc

hen

Equ

ipm

ent x

K


0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

2040

60 80

100

120

140

160180

Chef xC

Kitc

hen

Equ

ipm

ent x

K



Substitution Effect

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

10

20

30

40

50

60

70

Chef xC

Kitc

hen

Equ

ipm

ent x

K


0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

10

20

30

40

50

60

70

Chef xC

Kitc

hen

Equ

ipm

ent x

K



Substitution Effect

Liz used to choose O =

�

OC

OK

�

=�

520

�

.

Now she chooses T =�

1010

�

.

She traded in 10 kitchen appliances for 5 chefs totake advantage of cheaper labor:

ΔS := T − O =�

1010

�

−�

520

�

=�

5−10

�

.



Substitution Effect

Definition 5.2 (Substitution Effect)

Substitution effect ΔS is a difference between O

(the original choice) and T (the transitional choice),where

1 O is a solution to

minC,K TC(C, K) =OCC +KK

subject to ƒ () = y.

2 T is a solution to

minC,K TC(C, K) =FCC +KK

subject to ƒ () = y.



Substitution Effect

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chef xC

Kitc

hen

Equ

ipm

ent x

K

Isocost at 40Isocost at 20Isoquant at y=100

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chef xC

Kitc

hen

Equ

ipm

ent x

K



Scale Effect

Liz used to spend $40 (=OCOC+KOK )

Now she spends only $20 (=FCTC+KTK)

With the extra $20, she can produce more.



Scale Effect

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

10

20

30

40

50

60

70

Chef xC

Kitc

hen

Equ

ipm

ent x

K

Isocost LinesIsoquant at y=100Isoquant at y=400

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

10

20

30

40

50

60

70

Chef xC

Kitc

hen

Equ

ipm

ent x

K



Scale Effect

With extra $20, she can produce y = 400cheesecakes.

F =

�

FC

FK

�

=�

2020

�

.

She increased the scale of production withcash savings from reduced labor cost.



Scale Effect

Definition 5.3 (Scale Effect)

Scale effect Δ is a difference between F (the finalchoice) and T (the transitional choice), whichmeasures the change in hiring and capital investmentplan due to cash savings from reduced labor cost.



Scale Effect

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chef xC

Kitc

hen

Equ

ipm

ent x

K

Isocost at $40Isocost at $20Isoquant at y=100Isoquant at y=400

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chef xC

Kitc

hen

Equ

ipm

ent x

K



Scale Effect

Total effect Δ:

Δ = ΔS + Δ .

In Example 5.1

Δ =�

5−10

�

+�

1010

�

=�

150

�

.



Scale Effect

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chef xC

Kitc

hen

Equ

ipm

ent x

K

Isocost at $40 (wC=4)



Isoquant at y=100Isoquant at y=400

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chef xC

Kitc

hen

Equ

ipm

ent x

K



Gross Substitutes & Complements

How does wage reduction affect investmentdecision?

1 ΔSK< 0.

2 ΔK> 0.

Which one exceeds which?




Definition 5.4 (Gross Substitutes & Complements)

If C È leads to1 K È , chefs and kitchen equipment are gross

substitutes.2 K Î , chefs and kitchen equipment are gross

complements.

Remark1 If ΔS

Kis larger than Δ

Kin magnitude, inputs are

gross substitutes.2 If Δ

Kis larger than ΔS

Kin magnitude, inputs are

gross complements.




In Example 5.1 , kitchen equipment is not a grosssubstitute or gross complements of chefs:

ΔSK+ Δ

K= −10+ 10 = 0.

In general, cross-price effect is absent forCobb-Douglas production function.




0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chefs xC

Kitc

hen

Equ

ipm

ent x

K

IsocostO

IsocostT

IsocostF

Isoquant

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chefs xC

Kitc

hen

Equ

ipm

ent x

K




0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chefs xC

Kitc

hen

Equ

ipm

ent x

K

IsocostO

IsocostT

IsocostF

Isoquant

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chefs xC

Kitc

hen

Equ

ipm

ent x

K




0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chefs xC

Kitc

hen

Equ

ipm

ent x

K

IsocostO

IsocostT

IsocostF

Isoquant

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Chefs xC

Kitc

hen

Equ

ipm

ent x

K



1 Introduction





6 Affirmative ActionBackgroundHiring Decision & Affirmative ActionCMP for Color-Blind LizCMP for Discriminatory LizPotential Backfiring

7 Now We Know



Background

Discussion 6.1 (Affirmative Action & HiringDecision)Take a listen to Marketplace Audio Clip .

Irani says private industry would be better offvoluntarily hiring lower-caste employees.

1 What are the economic benefits of making hiringdecision based on the caste?

2 Can affirmative action plans backfire?



Background

Affirmative action programs are intended toencourage firms to hire underrepresenteddemographics in terms of

1 race2 ethnicity3 gender.

E.g., requiring Jack to hire one black worker forevery two workers hired.



Background

Definition 6.2 (Types of Firm)1 Liz is color-blind if race does not enter the hiring

decision2 Liz is discriminatory otherwise.



Hiring Decision & Affirmative Action

Example 6.3 (Affirmative Action & CMP)

Liz hires white labor W and black labor B for theproduction of 30 Rock. Factor price (wage) is(W,B) = (1,1). Liz has to produce 100 episodes.

1 How does color-blind Liz make her hiring decision?2 How does discriminatory Liz make her hiring

decision?3 How does affirmative action affect two Liz’s profit?



Hiring Decision & Affirmative Action

Assume W and B are not perfect substitutes.They may contribute to the production differently:

Different educational attainmentDifferent previous occupations

Suppose that Liz has a convex technology:

ƒ (W, B) = WB.

MRTS(W, B) =−BW

.



CMP for Color-Blind Liz

Problem 6.4 (CMP for Color-Blind Liz)Color-Blind Liz solves:

minW ,B TC(W, B) =WW +BBsubject to ƒ (W, B) = WB = 100.




Color-blind Liz uses Condition 4.3 .




Color-blind Liz will hire according to Condition 4.2 :

MRTS(W, B) =−BW

Relative Price =−W

B=−11.

Tangency condition:

−BW

= −1.




Color-blind Liz chooses CB = (W, B) = (10,10).TC(CB) = TC(10,10) = 20.




0 5 10 15 20 25 300

5

10

15

20

25

30

5

10

15

20

25

30

35

40

45

50

55

White Labor xW

Bla

ck L

abor

xB


0 5 10 15 20 25 300

5

10

15

20

25

30

5

10

15

20

25

30

35

40

45

50

55

White Labor xW

Bla

ck L

abor

xB



CMP for Discriminatory Liz

Let’s say discriminatory Liz prefers to hire whiteand black labor in 4 to 1 ratio.TC(D) = TC(20,5) = 25>TC(CB)




Discriminatory Liz is not profitable.

Discussion 6.5 (Discriminatory Firm’s Profit)

Why?




0 5 10 15 20 25 300

5

10

15

20

25

30

5

10

15

20

25

30

35

40

45

50

55

White Labor xW

Bla

ck L

abor

xB


0 5 10 15 20 25 300

5

10

15

20

25

30

5

10

15

20

25

30

35

40

45

50

55

White Labor xW

Bla

ck L

abor

xB




Discriminatory Liz does not follow Condition 4.3 .Instead, her hiring decision is based on

1 Output requirement2 Feelings




Discriminatory Liz’s MRTS(D) doesn’t match therelative price:

MRTS(D) =−14, relative price = −1.

Discriminatory Liz could have taken advantage ofthe labor market.She will be better off if she trade white workers infor black workers.




As a policy maker, we can ask discriminatory Liz tochoose 1 to 1 ratio.



Potential Backfiring

but then...Do policy makers know MRTS?What if a policy maker imposed 1 to 4 ratio instead?




0 5 10 15 20 25 300

5

10

15

20

25

30

5

10

15

20

25

30

35

40

45

50

55

White Labor xW

Bla

ck L

abor

xB


0 5 10 15 20 25 300

5

10

15

20

25

30

5

10

15

20

25

30

35

40

45

50

55

White Labor xW

Bla

ck L

abor

xB




Color-blind Liz will incur a loss of $5 because ofwrongfully executed affirmative action policy.



1 Introduction






7 Now We Know



CMPTrinity on the cost sideTrinity on the production sideSubstitution and scale effectsRelationship between capital investment and labordemandAffirmative action



Appendix

Proof.On the isoquant, the output level remains the same,i.e., dƒ () = 0. Taking the total derivative,

dƒ () = ƒC()dC + ƒK()dK = 0,

which leads to

dK

dC=−ƒC()ƒK()

�

=:−MPC()MPK()

�

on the isoquant. �


affirmative action, 91black labor, 94Cobb-Douglas productionfunction, 41, 86color-blind employer, 93convex technology, 50, 95cross-price effect, 86Δ, see total effectΔ, see scale effectΔS, see substitutioneffectdiscriminatory employer,93factor price, 13gross complements, 85gross substitutes, 85input bundle, 13isocost line, 15isoquant, 37marginal rate of technicalsubstitution, 44, 45, 49,59, 70market rate of exchange,22MRTS, see marginal rate oftechnical substitution

opportunity cost, 21, 23,59original bundle, 71output requirement, 62parallel shift, 27price taker, 24production function, 36relative price, 21, 22, 59rotation, 29scale effect, 80shift, see parallel shiftsubstitution effect, 75tangency condition, 58,60, 62total cost, 13total effect, 82trinity on the cost side, 21trinity on the technologyside, 44white labor, 94W, see black laborF, see final bundleO, see original bundleT , see transitional bundleW, see white labor

Documents

Cost Minimization Problem - University of Minnesota Duluthd.umn.edu/~watanabe/econ3821fl12/docs/cmpho.pdf · 3 Keeping to Output Requirement / Isoquant 4 Cost Minimization Problem