Managerial economics cap 4
Transcript of Managerial economics cap 4
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8/9/2019 Managerial economics cap 4
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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.
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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
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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
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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
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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'\
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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
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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,
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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
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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?
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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
-
8/9/2019 Managerial economics cap 4
11/17
CHAP TR 4 PRODUCTION
mEORY
MP
L
5
K
ilL
L
MPK ilK ._\