MicroEconomics 5e by besanko braeutigam chapter 07

Copyright (c)2014 John Wiley & Sons, Inc.

Chapter 7

Costs and Cost Minimization

1


Chapter Seven Overview

1.1.What
Whatare
areCosts?
Costs?
2.2.Long
LongRun
RunCost
CostMinimization
Minimization

••
••
••

The constraint minimization problem
The constraint minimization problem
Comparative statics
Comparative statics
Input demands
Input demands

Chapter Seven

Copyright (c)2014 John Wiley & Sons, Inc.

3.
3. Short
ShortRun
RunCost
CostMinimization
Minimization

2


Explicit Costs and Implicit Costs

Explicit
ExplicitCosts
Costs––Costs
Coststhat
thatinvolve
involveaadirect
directmonetary
monetaryoutlay.
outlay.

Copyright (c)2014 John Wiley & Sons, Inc.

Implicit

ImplicitCosts
Costs––Costs
Coststhat
thatdo
donot
notinvolve
involveoutlays
outlaysofofcash.
cash.

Chapter Seven

3


Opportunity Cost

The
Therelevant
relevantconcept
conceptofofcost
costisisopportunity
opportunitycost:
cost:the
thevalue
valueofofaa
resource
resourceininits
itsbest
bestalternative

alternativeuse.
use.

Copyright (c)2014 John Wiley & Sons, Inc.

•• The
Theonly
onlyalternative
alternativewe
weconsider
considerisisthe
thebest
bestalternative
alternative

Chapter Seven

4


Economic Costs and Accounting Costs

Economic
EconomicCosts
Costs––Sum
Sumofofaafirm’s
firm’sexplicit
explicitcosts
costsand
andimplicit

implicitCosts.
Costs.

Copyright (c)2014 John Wiley & Sons, Inc.

Accounting
AccountingCosts
Costs––Total
Totalofofaafirm’s
firm’sexplicit
explicitcosts.
costs.

Chapter Seven

5


Sunk Costs
Sunk
SunkCosts
Costsare
arecosts
coststhat
thatmust
mustbe
beincurred
incurredno
nomatter
matterwhat

whatthe
thedecision.
decision. These
Thesecosts
costsare
arenot
notpart
partofof
opportunity
opportunitycosts.
costs.

•

$5M is not sunk cost for the decision of whether or not to build the factory

•

$5M is sunk cost for the decision of whether to operate or shut down the factory
Copyright (c)2014 John Wiley & Sons, Inc.

Example: Bowling Ball Factory
• It costs $5M to build and has no alternative uses

Non-Sunk
Non-SunkCosts
Costsare
arecosts
coststhat
thatmust

mustbe
beincurred
incurredonly
onlyififaaparticular
particulardecision
decisionisismade.
made.

Chapter Seven

6


Cost Minimization

Cost
Costminimization
minimizationproblem:
problem: Finding
Findingthe
theinput
inputcombination
combinationthat
thatminimizes
minimizesaafirm’s
firm’stotal
totalcost
costofof
producing
producingaaparticular

particularlevel
levelofofoutput.
output.

Cost
Costminimization
minimizationfirm:
firm:AAfirm
firmthat
thatseeks
seekstotominimize
minimizethe
thecost
costofofproducing
producingaagiven
givenamount
amountofofoutput.
output.

Long
Longrun:
run: AAperiod
periodofoftime
timewhen
whenthe
thequantities
quantitiesofofall
allofofthe
thefirm’s
firm’sinput

inputcan
canvary.
vary.

Copyright (c)2014 John Wiley & Sons, Inc.

Short
Shortrun:
run: AAperiod
periodofoftime
timewhen
whenatatleast
leastone
oneofofits
itsinputs’
inputs’quantities
quantitiesisisfixed.
fixed.

Chapter Seven

7


Long-Run Cost Minimization

Minimize
Minimizethe
thefirm’s
firm’scosts,

costs,subject
subjecttotoaafirm
firmproducing
producingaagiven
givenamount
amountofofoutput.
output.

Cost
Costto
tothe
theFirm:
Firm:

TC
TC==Total
TotalCost
Cost
ww==wage
wagerate
rate
LL==Quantity
QuantityofofLabor
Labor
Copyright (c)2014 John Wiley & Sons, Inc.

r r==price
priceper
perunit
unitofofcapital

capitalservices
services
KK==Quantity
QuantityofofCapital
Capital

TC = wL + rK
Chapter Seven

8


Isocost Line

The
Theset
setof
ofcombinations
combinationsof
oflabor
laborand
andcapital
capitalthat
thatyield
yieldthe
thesame
sametotal
totalcost
cost

Copyright (c)2014 John Wiley & Sons, Inc.

for
forthe
thefirm.
firm.

Chapter Seven

9


Isocost Line

w = $10/hour
r = $20/hour
TC = $1 million

Or more generally:

TC
K=
r − (w / r ) L

Chapter Seven

10

Copyright (c)2014 John Wiley & Sons, Inc.

⇒$1 mil = $10L + $20K
⇒K = $1 mil/20-(10/20)L


Isocost Lines
K
Direction of increase
Direction of increase

TC2/r

in total cost
in total cost

TC1/r

Slope
Slope==-w/r
-w/r

Combinations of labor and capital
that yields the same total cost for the
firm
Copyright (c)2014 John Wiley & Sons, Inc.

TC0/r

L
TC0/w TC1/w TC2/w
Chapter Seven

11


Long-Run Cost Minimization
Suppose that a firm’s owners wish to minimize costs

Let the desired output be Q0

Technology: Q = f(L,K)

TC
TC==rKrK++wL
wL…or…
…or…
KK==TC/r
TC/r––(w/r)L
(w/r)L

Owner’s problem: min TC = rK + wL

isisthe
theisocost
isocostline
line

K,L
Subject to Q0 = f(L,K)

Copyright (c)2014 John Wiley & Sons, Inc.

•
•

Chapter Seven

12


Long-Run Cost Minimization

•• Cost
Costminimization
minimizationsubject
subjecttotosatisfaction
satisfactionofofthe
theisoquant
isoquantequation:
equation:QQ00==f(L,K)
f(L,K)
•• Note:
Note:analogous
analogoustotoexpenditure
expenditureminimization
minimizationfor
forthe
theconsumer
consumer
Tangency
TangencyCondition:

Condition:

Copyright (c)2014 John Wiley & Sons, Inc.

•• MRTS
= -MP /MP = -w/r (or) MP /w = MP /r
MRTSL,K
L,K = -MPLL/MPKK = -w/r (or) MPLL/w = MPKK/r
•• Constraint:
Constraint:QQ00==f(K,L)
f(K,L)

Chapter Seven

13


Long-Run Cost Minimization

Solution to cost minimization:

•

Point where isoquant is just tangent to
isocost line (A)

•
•

G – Technically Inefficient

E & F – Technically Efficient but do not

Copyright (c)2014 John Wiley & Sons, Inc.

minimize cost

Chapter Seven

14


Long-Run Cost Minimization
Solution to cost minimization:

Slope of isoquant = slope of isocost line

− MRTS
(or) L , K
•

w
=
r

MPL w
=
MPK r

Ratio of marginal products = ratio of input prices

Copyright (c)2014 John Wiley & Sons, Inc.

•

MPL MPK
=
w
r
Chapter Seven

15


Long-Run Cost Minimization

MPL w
>
MPK r

•

At point E

•

This implies the firm could spend an additional

MPL MPK
(or )
>

w
r

dollar on labor and save more than a dollar by
reducing its employment of capital and keep
Copyright (c)2014 John Wiley & Sons, Inc.

output constant

Chapter Seven

16


Long-Run Cost Minimization

MPL w
<
MPK r

•

At point F

•

This implies the firm could spend an additional

MPL MPK
(or )

<
w
r

dollar on capital and save more than a dollar by
reducing its employment of labor and keep
Copyright (c)2014 John Wiley & Sons, Inc.

output constant

Chapter Seven

17


Interior Solution

1/2
1/2 1/2
1/2
QQ==50L
50L KK
-1/2
-1/2 1/2
1/2
MP
MPLL==25L
25L KK
1/2
1/2 -1/2

-1/2
MP
MPKK==25L
25L KK
ww==$5
$5
r r==$20
$20

Copyright (c)2014 John Wiley & Sons, Inc.

QQ0 ==1000
0 1000
MPL/MPK = K/L => K/L = 5/20…or…L=4K
1/2 1/2
1000 = 50L K

K = 10; L = 40

Chapter Seven

18


Corner Solution
The cost-minimizing input combination for
producing Q0 units of output occurs at point A
where the firms uses no capital. At this corner
point the isocost line is flatter than the isoquant.

⇒

Copyright (c)2014 John Wiley & Sons, Inc.

MPL
w
−(
) < −( )
MPK
r

MPL MPK
>
w
r
Chapter Seven

19


Corner Solution
Q = 10L + 2K
Q = 10L + 2K
MPL = 10
MPL = 10
MPK = 2
MPK = 2
w = $5
w = $5
r = $2

r = $2
Q0 = 200
Q0 = 200
MPL/MPK = 10/2 > w/r = 5/2

Copyright (c)2014 John Wiley & Sons, Inc.

But… the “bang for the buck” in labor larger than the “bang for the buck” in capital…

MPL/w = 10/5 > MPK/r = 2/2
K = 0; L = 20

Chapter Seven

20


Comparative Statics
A change in the relative price of inputs changes the slope of the isocost line.

All else equal, an increase in w must decrease the cost minimizing quantity of labor and increase
the cost minimizing quantity of capital with diminishing MRTSL,K.

Copyright (c)2014 John Wiley & Sons, Inc.

All else equal, an increase in r must decrease the cost minimizing quantity of capital and
increase the cost minimizing quantity of labor.

Chapter Seven

21


Change in Relative Prices of Inputs

•
•
•

Price of capital r = 1
Quantity of output Q0 is constant.
When price of labor w = 1 the isocost line is
C1, optimal point A

•

When price of labor w = 2 isocost line is C2,

Copyright (c)2014 John Wiley & Sons, Inc.

optimal point B

Chapter Seven

22


Some Key Definitions

An

Anincrease
increaseininQQ00moves
movesthe
theisoquant
isoquantNortheast.
Northeast.

•• Expansion
ExpansionPath:
Path:AAline
linethat
thatconnects
connectsthe
thecost-minimizing
cost-minimizinginput
inputcombinations
combinationsasasthe
thequantity
quantityofofoutput,
output,Q,
Q,varies,
varies,
holding
holdinginput
inputprices
pricesconstant.
constant.

•• Inferior
InferiorInput:

Input:An
Aninput
inputwhose
whosecost-minimizing
cost-minimizingquantity
quantitydecreases
decreasesas
asthe
thefirm
firmproduces
producesmore
moreoutput.
output.

Chapter Seven

23

Copyright (c)2014 John Wiley & Sons, Inc.

•• Normal
NormalInputs:
Inputs:An
Aninput
inputwhose
whosecost-minimizing
cost-minimizingquantity
quantityincreases
increasesasasthe
thefirm

firmproduces
producesmore
moreoutput.
output.


An Expansion Path

As output increases, the cost minimization path moves from point A to B to C when

Copyright (c)2014 John Wiley & Sons, Inc.

inputs are normal

Chapter Seven

24


An Expansion Path

As output increases, the cost minimization path moves from point A to B to C when

Copyright (c)2014 John Wiley & Sons, Inc.

labor is an inferior input

Chapter Seven

25