Managerial economics cap 4

8/9/2019 Managerial economics cap 4

1/17

CH PTER

PRODU TION THEOR

Once

managers

determine the

demand

for the firm s product

or

service,

their

is far from over. Now

they

must choose

the

optimal method

to

need

to

be

as efficient

hallmark

of

good

managers.

Efficiency

tion

process.

Simply

staled, a

production

process

explains

or

service (output). The

production

precisely specifies the

relationship

between inputs

and

outputs.

Production issues are not confined

to

the physical transformation of

into outputs.

In business,

goods

and services, such as employment

distribution.

managers are

concerned

with rtflnenttv

intellec-

THr

PRODUCTION FUNCTION

WITH ONE VARIABtr INPUl

THE

PRODUCTION

FUNCTION

WIT ONE

VARIABLE INPUT

lEARN1NG OBJECTIVeS

The

Production Function with

One Vanahle

The Law 1 Dlfmnishlng

I ~ a r g i f l a

Returns

The

PlOuuction Function

with

Two

Variable

lsoquants

The Marginal Rate

of

Technical

Substitution

The

Optimal

Combination 1

Inpuls

Corner

Solutions

Relurns 1 Scale

The

Output

Elasticity

Eslimattons 1

Produclion

Funclions

Appendix. Lagrangian

Moltipliers and Oplimallnput

Combinations

tual

resources.

The

production function

is a table, a grapn,

or

an

product

output achieved

from any specified set

of

inputs. The function

summa

nto

rizes

the

characteristics of

existing

tedlilology

at

a

time; it shows

the

tech-

emand,

is required

for managers to optimize

process.

constraints managers face. Any manager should want

to

use the most

efficient process known. So we

assume

managers presuppose technical efficiency.

annot

understand

their

firm s cost structure unless they

understand

Unfflrtnnotptv

ll1any

managers view processes as static.

Production is

dynamic:

he

production

process.


2/17

Output of Metal Parts When Various

Amounts of labor

Are Applied to

FIve

MachIne Tools.

Thomas

Machine

Amount of

Amount

1 Cdpital [Number

of

Labor Machines]

o

?

3

4

5

5 5

6

5

6.67

7

8

10

5

11

12

13

5

14

5

15

CHAPitR I.: PRODliCTION [HEORY

a process

uses

two

level of the second mput, the

is the level of the first

fUBction s

Q

where

Q

s

the firm s

output

rate.

Cognitivel) , the simplest case has one input whos\:

whose

qnantity

is

variable.

Excd inputs

cannot be

l be sure,

economists

assume the

time

needed to

ning of what

is

called

the

long term. Fixed invuts

often

machinery,

Variable inputs can

be

example. In

run, all inputs are variable.

John

Thomas

is

an

entrepreneur who

He woils

as

a

contractor

in the airplane

TABlE4.1

THE

PilODUCTION

FUNCTION WITH ONE

VARIABLE I I ~ P U j

Olltput

if he were

to hire various

numbers

of machinists.

(Please note

the following output l1tllllhers

are

in hundreds.) Tbomas estimates

one machinist

produces 49 piirts

per year. Thomas Lall produce more parts

Jllore

workers, as we see

in Table

4.1.

This table

represents a

ThollYas Machine when

five machine

tools are used. More

the

curve

ill

i g u r ~

4.1 presents exactly

the

same

results.

In fact, the

between Total Output and Amount of

Labor

Used

Machine

Tools,

Thomas Machine Company

oulput

increases as labor

increases at an

increasing rate lup

106.67

ulliis

labor!

and

Hlcreascs at adecreasing rate luntil

slightly

more than 11 units of

lahorl. Thereafter, output decreases

as.more

units

of

labor

are deployed. Managers

.

Will

never williully deploy labor

in

the latter circumstance. The

production function

shows

the

relationship between o \ l t ~ u l l i n this

case nUl1\ber

1 parts

oroducedl

and

input

[in lhis case unils

01 labor!'

otparts

'

1 ~ O O

--

Tolatoutpul

1.400 .

I

1.300

1.200

1,1DO

1,000

gOO

800

700

600

500

400 -

300

200 -

Output 01 Parts

10.

hundreds

~ e r year]

1,9

132

243

376

52'J

684

792. 39

847

1,008

1.161

1,300

1.419

1,512

1.573

1,596

1 :175


3/17

Average product iAPI Common

measunng

device

lor

estlmat nQ

the

uOlls 1

output, on

average

p r mpul

Metric

lor

eSllmallog the efficiellcy 01 each

rnpullo which the

"'pufs MP

is

(qual 10 Ihe

Incremental change

rn output created by a small

change in the input

CHAPrEll '

PRODUCTION

1HFORY

:l IIf,;rL

THE

PRODUCTION

[,UNCI

ION

WITH O I ~ E VARIAI3LE INPUT

when

the

latter reaches a maximum; thal

is

MP AP 130

when

10

numbers in Table

4.1

(and

Table 4.2) are derived from the

to Q 30L + 20U P. Lis

We

can think

of the production

lunctlOn

as

technology use. Thomas is clearly

interested in

the

number of machinists varies. One common measure used by m lIlY m a n a g e r ~ i I I

is

01ltput

Jl9T wOIker

This measure

is W h ~ l i tconOinim

call

aVefdge product

Because

we

are

v a r v i n ~

machinists,

this

is

output

per

worker or

Q

AP

.

holding X

conslanl

tells

Thomas

how many units of output, on

average,

each

ist is respollsible

for.

If

he

wants a better metric to estimate the

worker,

he should

use what economists

call

the

marginal

product

IMP],

MP

is

eaual

to the incremental change

ill

output created by a small

MP = IlQ

f X

I

x

constanl

Por machinists, the marginal product represents the impact on output of a

in machinists. If

Thomas adds a

machinist,

the

more units

did

we produce

because

I hired

this last

machinist?" If he mllst let

go,

it

is,

"How many

fewer units

did

we

produce because

I

let this

machinist

The

marginal

of Parts Q,

tA..verage

Marginal

. n; -.I

5,

average

parts per machinist between the fllst and second hires.

Results

for

hires are

shown

in

Table 1.2.

QUANT OPTION

More

precisely.

the product 1

an

Input the derivative

of

output

Average and Marginal

Products of

Labor,

Thomas

Machine Company

Amount Output of

1'1011j IOl

{Number

of

Hundreds Product Product Product

Machines of Parts

D/L1

IllOIM

I

Ida/dUo

0

5

49 49

49 67

5 1:l2

66

83

98

3

5

21,3

81

111

123

4

5 376

9t,

133

142

5

5 525

105

149

6

5 684

114

159

162

6.67

5

792

59 118.89

162.89

16133

7

5

847

121

163

163

8 5

1,008

126

161

158

5

1,161

129

153

1 7

10

5

1,300

1:jQ

139

130

11

5

1,419

129

119

107

12

5 1,512

126

93

78

13

5 '1,573

121

61

43

14

5

1,596 114

23

2

15

5

IS/5 105 -19

-45

ligures

In Ihe

I lO/M

column pertain to the ,nterval belween Ihe indicated amounl 01

tabor

and

one

Ilnitlesslhan

Ihe mdicaled

amount 01

tabor The ligures

inlhe dO/dt

column


4/17

CHAPTER I r O D U c r I O I ~ THEORY

FIGURE 4.2

Average and Marginal

Product

Curves for

Labor

Mal

[Jlnal

praGue: eHcelb average

product when the lallel is incrEaSing

dnd

is less

:hdfi JVeragf

product

when the iatler 15 IOutput

~ e r

unil ollabar is

mea

sur(ld if

I hundrp.d::

Oilipul

\

I I I I I I I

3 4 5 B

10

11

12 13 14

1:)

Amounlo1

rullor

The second dQI

elL assumes that

Thomas can employ

labor continuously, as

J.25

workers

or J,33

workers.

This could be achieved by

or workers who work more or

Jess

time than in a

standard

day s work.

MP equals

AP

when AP is maximized. A

llltuitive frame may help. Assume your

IJrofessOT

is

average

of test

scores by

score

is

must decrease. This is a

natural

law of

mathern


5/17

CHAPTER /

PllODUCTION

THeORY

ISOQUANTS

TABLE

4,3

Production

Two Variable

Amount

of Labor

Unitsl

3

1

2

11

3

22

I.

30

J

35

choices, the process is similar to that of the onevariable input

case.

tional eNC

machines.

Engineers

XI is the

amount

of the first input and X

function is

where Q

is

the firm's

output

rate.

The

AQ/AX,; the

marginal

product of

the

Thomas Machine Company

Quantity 01 Machine Tools

IHundreds

of Parts Produced per Year

5

11

18

30

50

60

80

Bl 115

84

1/,0

exira

input

combinations to

consider.

Though Thomas will have to consider m

To illustrate, suppose

Thomas is considering whether to

machines and

derive

Table 4.3,

The

average product of

either

machirie tools

machinists is computed by dividing the

total

output

by

the amount

of cit

machine

tools

or machinists

used, The

marginal

product

of

each input

is obla'

by holding the

other

input

constant.

For example, the

marginal product 0

additional machine tool when using four machinists and three machine

t,

5,1 00 parts

per

machine tool;

the marginal product

of an

additional

macrun

when using

three machinists and four CNC machines

is 2,100 parts

per unit.

is the amount of

the

second

o

f X ~ ,

X

2

QUANT

OPIION

of

the first

input

is AQ/6.X

~ X ,

z

bundle.

For

exam

MP

iJO

ilX

2

surface, OAQB

shows

Ihe amollnl

o lold

output thai (on be

Tolal I

oulpul'

o

B

B B

Amellol 01

labor

function by a surface, as shown

in

We

measure output

for any

down from a point

machinists

and

machine tools.

Conversely,

we

can take any

amounts of machine

tools

say OA

l

machine tools

and

OB

l

machinists, and find thcir output

the

height of the production

surface

at D', the point where

'machinists is OE

l

and machine tool

input

is OAr According

to

Figure 4.3, the

equals

D'D, Input bundles that

produce

identical

output

have

the

same

bundles capable

of

surface

of

Figure

4.3.

Suppose

we

want to

find

to an

output of G' G. All we need

to

do is cut the

sur-

the result

beiM EGF and

Isoquanl Curve showing all i::'

Sible iejfieientimpul bundle

capable

nf producing

a

9 VlO

the iaslldio1l5,

we

have

Qulput

level.

L This

surfacf

is no

meilnt

to

MP

resent the

numerical

values

in

1;

.

4.3

but

a general reprcsen at>:;;

how aproduction

surface

of lh:.J .

is likely to appear.

1 ?

HI'\


6/17

CHAPTER 4. PRODlJC110N lHEORY

Nucor

is one

of tile largest steel firms In

the

United

Slates, although it did

not

focus

on

steel production until

Ihe

't960s.ln 2U08,15 first qUilrler sale';.

were approach

ing

5

billion;

the

t,rm achieveurecord first quarter

net

earnings

tor

the fifth consecutive year.

Managers

hali ?

past

implemented to achieve this

,

One difference is thai Nucor is a

minim I , hot an

integratedsteel firm.

MinimiUs have

a

different prOduc

tion function than do integrated mills.

They

use electric

arc

furnaces

to make stcel producls from scrap

metal.

In 2007

Nucorwas

the nation's recyder, repro

How

db

Nucor managers

keep employees focused on

efficient production?

Nucor

uses

the

following

multiprongedapproilch:

It maintains

a

slre.amlined

tlonal structure

t h ~ t encourages decentralized decision

Most divisions use only three of man

agement

Each diviSion is

treated

as

a cenler and

IS

exoecled

to

earn

a25

percent

return on total assets.

2.

The company ;lcts

as

the

general

contractor

rural

areas where

land is

cheap land

unions

are

weak ,

Also,

each planl is

located near water

ond

is served

by at leasllwo

railroad

work

habits of individUals

.Those with

good work habits

are

recruited

to

work

in the

plant

when

it

opens. II also

brings

workers

from other plants

[who

have already built

plants] to jOin

the constuction

team.

Using

these

meth

ees

monitor

each

olher. Bonuses

are

based on

the

c a . p a b i l i t i ~ s o l

the

~ q u i p m e n t

ahqav ;'rage 80percent

output atepm

produces.

the higher are its

bonuses

. 4 ~ Nucor

treats all

employees equally.

leadel.lt

was

the

first firm

to

produce

thin-slab casilng

at a i T 1 i n i i l ) i l l ~ l d 5 e a r c h e s worldwide for

new

develop

ments.ih

sted

m d u ~ t i o h This.emphasis.on

innovation

is

reinforced by

the

firm's

flat

Decisions can be

made

and

'ISO

9000

is a

set of

quality standards.

To

receive

ISO 9000

certification, nanagers musl

luUill

various quatily assurance

requireinen s

anti

be

audrted

by

an external registrar. If a

firlll's qualily assurance syslem

IS

approved

by

Ihls

regislrar,

Ihe lirrrl is awarded an ISO 9000 cerliircalion and is allowed to

adverlise

Ihis lacl to all cuslomers.

ISOQUANl S

this

results

in a

curve

that includes

G'G

to

J different output

rate,

are shown ill

figure 4.4. The two axes Jlleasure

the

qll


7/17

I HE

MARGINAL

RAr[ Of'

TECHNICAL SUBS1ITU110N

CHIIPT[R 4:

PRODUCtiON

1HEORY

Marginal rale of technical 5ub

stitution iMRTSi

MRT:,

shoVis Ihe

rale t

which

one mpui

IS sub

sliluled lor Jnolhel IWlih

remaining

constant]

THE MARGINAL RATE OF TECHNICAL SUBSTITUTION

oquants in the Case of Fixed Proportions

inputs musl

be

used

In ilxed

proportions, the isoauilnts are

angles,

input

is sllbstituted for

another

the output

I

o f(X

X

2

MRTSis

~ - , - = , , , , v , , ' C , 300

MRTS ilX1

200

of technical substitution

is

1times the

of

the

isoquant.

This

makes sense because

6 X/6 X

measures

the slope,

which

downward or negative (so X is on the yaxis and Xl is on the x

100

It is useful for

managers

to

think

of MRTS

as

the ratio of

for inputs I and 2, Managers need to

shows the incremental

effect

on output of the ,last unit

managers

want to

increase the use of

inputs

with

relatively

o

Labor

ucts,

though they must

also

consider

the

costs of inputs,

The rate of

substitutabilitv

processes,

one type

of labor is

are

Jines connecting the

ized

sible; to

produce a unit of output, a

fixed

amount of

each

segments or

bend

back

inputs must be used in fixed proportions. Figure

4.5

shows

the firm's isoquants

4.6,

Above

au and below av the

such

acase; as you can see,

they are right

angles,

Few production

processes

that increases

in

both capital and labor

are

required

no

substitution

among inputs, but

in some,

output

rate,

f his

is

the

case,

the marginal plOduct

of

one

Above

au

the marginal

product of capital is

nega

output increases if less capital is

used

while

the

Jevel of

labor

is held

Below OV the

QUANlOPllON

lime for some fun!

manager

will operate at a point outside the ridge

lines

dO

the same output

with

less

of both inputs, This

choice

is

aXJ )

dX

+ aX 1dX

2

0

2

Consider

point

H in Figure 4,6,

This

point is located on a posi-

Therefore,

segment of the isoquant (and so outside the lines),

It will always

of both labor

and

caoital

than

(4,3)

MP2

Ridge

lines The

tines IhJI

profil maxlmillng fir ms

wilh,n, because oulside olltlcl,

marginal products

01 inpu\s

df::

negative,
http:///reader/full/6.X/6.X1http:///reader/full/6.X/6.X1http:///reader/full/6.X/6.X1


8/17

CHAPlER 4: PRO[JUCTION THEORY

FIGUHE 4.6

No prolit-maxtmlzlnq firm operales at a point out'.lde the

OUafld

V

'""pit81

I

v

100

P

o

Labor

THE

OPTIMAL COMBINATION OF INPUTS

analysis did

not

include the costs

costs

because the inputs are

scarce.

Amanager

who wants

to maximize profit

try

to

minimize

the (Ost of producing

a given output or

maximize the

bination

derived from the

level

of

cost.

l

Suppose

a

manager

takes

capital and labor,

that

vary in

the

relevant

and

labor should

the

manager choose

to

maximize

the

level of

cost?

First

we dctermme the various input combinations that can

be obtained

cost.

f

capital and labor are he

inputs

and the of labor

is

PI per

and the price of capital is P

per

unit,

theinpl1t combinations that are obtained

of Mare slIch that

M

where Lis the

level

of labor and

is

the level

of

follows

that

M

= PI(

PI(

THE

OPTIMAL

COMBINATION

OF

INPUTS

isocost curve shows the combinations of

aiM.

can be

ohtained

Jor a

Amounl01

capilalused

MIP

Amounl

1 labor uSP O

(pel

unit 01 time)

and labor that can be purchased,

are

represented by the straight line shown in Figure

4.7.

(Capital is

plotted

on

vertical

axis,

and labor is plotted on the horizontal axis.) This line, which

has

on

the vertical

axis equal

to M P

am] aslope

of PP\ is

called

all

curve.

t

shows all

the

input bundles that

can be

purchased

at

a

specified

sUF,cnmposethe

relevant isocost curve on the isoquant map, we see the

bundle

hat

maximizes

output

for a

cost.

An efficient

manager

should

p t h t ; ) n l ' \ ~ t \ t A

combination

to

choose

an

mput bundle where

the

margi.nal prod-

per

dollar

spent of

Jabor

and

capital are

identical. f

they are ]]ot,

the

manager

increase

the

use

of

the

immt

with

the higher marginal

per

dollar value.

the manager maximizes output

by

distrib

so

the maminal oroduct

of adollar's

worth

150c05t

curve Curve

showIng

rnput

bundles thai

can be

P

chased at a specified cost

2. The conditions

(or

minimizing

the

cost

01 producing,

given output

are the same ab those (or maximizing

the

OU1put

(rom

agiven (OSlo T h j ~

is

~ h o w n in

the present section. There

fore, we

(an vlfwlhe

firm)s

problem

in either

way


9/17

CHAPTER

k

PRODUCTION THEORY

FIGURE

4.8

Maximization of

Output

for aGiven Cost

maXIn)lze

the

output for

a given cost. the firm should ci1oo;,e the Input com

tlOn at

pOint

R

Amount j

of capital.

of one

used.

In

where

o

MP'F'"

MP

a

:' P

IJ

p. P

b

of

a

dollar's

worth

bundle

such

that

P"

products

a b

11.

Arnollnt

of labor

bundle that minimizes production costs, we

along the isoquant of the stipulated

that

lies

on

the lowest isocost curve for example, S

Input bundles on isocost curves like Co that iie

below Sare

cheaper

the desired output.

like

z

that

lie above

S

is obvious

that the

optimal bundle

S s a

point where

the

isocost curve

is

tafi \cnJijl,

to the isoquant. Therefore, to minimize the cost of producing a

to maximize the

output

from a given

cost

outlay, the finn must

equa

te

and Pi>

this

means

that

needed, the manager must

l1n

RETURNS T SCALE

C ' M n ~ ; n f l to Ihis

150-

An10tJni

I

of ("pitat

fsoquent

WMlS

o

Amount

of labO

ORNER SOLUTIONS

an

isocost

curve. In the

two-

input

case,

this

means that just

one

input

is used

in

the

least expensive

way to produce

the

most

output

4.6 will now

be an inequality

reading

MPKIP

K

>

cases

where

just capilal is used and MPKIPK MP,lP

L

for

cases where

labor

is used.

The former case

is

shown

in Figure 4.1 O.

We have

seen

how managers can represent

technology

as

a

production fUIlction

and

average

product

to

operate more

want to continue this theme and

examine

some

long-term

considerations manag

crs

face.

These fOCllS ll scale. Basically, what is the

incremental

change

to

output

as managers increase their

use of capital

and labor?


10/17

CHAPTER4 PRODUCTIONTHEORY

flETurlNSTO SCALE

may

double

oulput. Thisis thecaseofconstant

Constantreturns toscale

Whe,

FIGURE4.10

ouiputmcreasesbyexactty the

salnpproportion a5 Inputs

ACorner

SolutionWhere

At firstglance,

some

managers

maybelieve

thaiproduction

functions neces

Wllh

outlay

1

M,

the most thatcan be produced IS

exhibitconstant

returns

to scale.After

all,

if

amanagercan build

twofa(

1

only

laborwere

used,

the

IIrm couldproduce

only

size and Iypes ofworkers,

can't

sheachieve the same

cheapestway10

produce0:

unll

twice

the

size'Blit thingsarcnot

this

simple. jf man

couldbeprodUCedWith

aft

iJuHoy

oi

M > rvi

may

employ

tecimiqucs

that are

econbmi

thai

would

be

lne/fiCient.

infeasible

at t?C smaller

scale.

Some inputsare not

available

in

small

units;

Amount

we

cannot

install

half

a robot.

Because

ofindivisibilitiesof thissort,

olr.apltal

M'I 'K

!!.Q .

MP, .T

12

T

~ ~ ; O I f f i ;

.

l U > ~ I : ~ 0

3

related

in

the followihg way to

the

number of

Ihese expressions

lor

lEI

and

technicians

used IT]

equatton (481

and noting

thatP

E

2,OUO,

il follows

that

Q

=

20

f2 +12T 0.5P

(4.7)

tywage

ofan engineeris$.\,000,

and

the

20 - 2E

12

-'.

T

of

a technician

is$2,000.

Ifthepresl-

4,000 2,000

per month for

the

combined

2,000 20

-2E

12

T

/

0,

4,000

10 E 12 T

is to

maximizeoutput I/or

his

T E-l 2

he

rhust choose a

bundle

of

engl-

Because Belswanger allocates

$28,000

per month

MlP

M'IP, Amount

oftabor

for the

totalwages 01 engineers

and

technicians,

we

(4.8)

have

Pc

P

7

4,000E

+

2,ooOT 28,000

Supposeweconsidera

IE +

2

lor

Tgives

us

managers

increase

the level

,

and PI is the wage of

a

lech

Increasing

return toscaleWhen

to

the change in equation

(UI with

4,000E + 2,000(+ 2) 28,000

output

increases

hy

a largerpro

toEand T, wefind

thai

portion than Inputs.

doubleoutput. This is the caseofincreasing

returns

toscale. Or

output

ThiS

meansthat

E

land

T

= 61. So to

rnaximize

increase

bya smaller

proportionthan

inputs;

forexample,

doubling

all

!!.Q

output from the $28,000

outlay

onwages, thepresi

MP

E

=

.E =

20

2E (4.9a)

Decreasing

returns

to

scale

dentshould

hire

4engineers

and

6 technicians.

toless

than

adoublingof

outout.

This

is

the

caseof

decreasing

When

ou!put incre()se

C

,

by

a

smaller proportion

than

inputs

toscale.

output

may

increaseby

the

same

proportion

as

112


11/17

CHAP TR 4 PRODUCTION

mEORY

MP

L

5

K

ilL

L

MPK ilK ._\

Managerial economics cap 4

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