nicholson chapter 10 _cost functions

8/16/2019 Nicholson Chapter 10 _Cost Functions

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COST FUNCTIONS

Reference : Chapter 10 ;Nicholson and Snyder (10th Edition


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"e#nitions of Costs• It is i$portant to di%erentiate &et'een

accontin) cost and econo$ic cost –

the accontant*s +ie' of cost stresses ot,of,poc-et e.penses/ historical costs/depreciation/ and other &oo--eepin) entries

– econo$ists focs $ore on opportnity cost

–

Opportnity costs are 'hat cold &eo&tained &y sin) the inpt in its &estalternati+e se


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"e#nitions of Costs• a&or Costs

– To &oth econo$ist and accontants/

la&or costs are +ery $ch the sa$ethin): la&or costs of prodction(horly 'a)e


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"e#nitions of Costs• Capital Costs (accontants andecono$ists di%er3 – accontants se the historical price of

the capital and apply so$edepreciation rle to deter$ine crrentcosts

– the cost of the capital is 'hat so$eone

else 'old &e 'illin) to pay for its se(and this is 'hat the #r$ is for)oin) &ysin) the $achine• 'e 'ill se v to denote the rental rate for

capital


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"e#nitions of Costs• Costs of Entreprenerial Ser+ices – accontants &elie+e that the o'ner of a

#r$ is entitled to all pro#ts• re+enes or losses left o+er after payin) all

inpt costs

– econo$ists consider the opportnity

costs of ti$e and fnds that o'nersde+ote to the operation of their #r$s


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E.a$ple3

• IT pro)ra$$er/ she does a ne' soft'are on herfree ti$e and sells it &y 640007

• 8ccontin) pro#ts9640007 This see$s li-e a

)ood proect• Econo$ist pro#ts964000 minus 'hat she cold

ha+e earned 'or-in) for a #r$ in her ti$e7 i)htnot see$ sch a )ood proect any $ore3

•

part of accontin) pro#ts 'old &e consideredas entreprenerial costs &y econo$ists


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Econo$ic Cost• The econo$ic cost of any inpt is thepay$ent re=ired to -eep that inptin its present e$ploy$ent – the re$neration the inpt 'old

recei+e in its &est alternati+ee$ploy$ent


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8nother e.a$ple

• 8 shop o'ner

•

If her accontin) pro#ts ares$aller than the rental price of thephysical shop/ it $eans that she isha+in) losses as she cold o&tain$ore $oney &y no rnnin) theshop &t rentin) it


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T'o Si$plifyin) 8ss$ptions

• There are only t'o inpts – ho$o)eneos la&or (l / $easred in

la&or,hors

– ho$o)eneos capital (k / $easred in$achine,hors• entreprenerial costs are inclded in capital

costs

• Fir$s cannot in@ence the inptprices/ they are )i+en3 (they do notdepend on #r$s decisions on the

inpts to &e sed3


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Econo$ic Aro#ts• Total costs for the #r$ are )i+en &y

total costs 9 C 9 wl B vk

•

Total re+ene for the #r$ is )i+en&y

total re+ene 9 pq 9 pf (k /l

•

Econo$ic pro#ts (π are e=al to

π 9 total re+ene , total cost

π 9 pq , wl , vk

π 9 pf (k /l , wl , vk


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Econo$ic Aro#ts• Econo$ic pro#ts are a fnction of

the a$ont of capital and la&or

e$ployed – 'e cold e.a$ine ho' a #r$ 'old

choose k and l to $a.i$ie pro#t

– De 'ill do later on3


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ini$iin) costs3• For no'3

– 'e 'ill ass$e that the #r$ hasalready chosen its otpt le+el (q

0 and

'ants to $ini$ie its costs – De 'ill e.a$ine the inpts that the #r$ 'ill

choose in order to $ini$ie costs &t

prodce =0 (-ind of a co$pensatedde$and3


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Cost,ini$iin) Inpt Choices

• athe$atically/ 'e see- to$ini$ie total costs )i+en q 9

f (k /l 9 q0

• Settin) p the a)ran)ian:

L 9 wl B vk B λq0 , f (k /l

• First order conditions are∂L / ∂l 9 w , λ(∂f/ ∂l 9 0

∂L / ∂k 9 v , λ(∂f/ ∂k 9 0

∂L / ∂λ 9 q0 , f (k /l 9 0


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Cost,ini$iin) Inpt

Choices• "i+idin) the #rst t'o conditions 'e

)et)for (

/

/k RTS

k f

f

v

w l

l

=∂∂

∂∂

=

• The cost,$ini$iin) #r$ sholde=ate the RTS for the t'o inptsto the ratio of their prices

• This is s&ect to the sa$ereser+ations as 'ith tility (if RTS

is strictly decreasin)/ no corner


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Cost,ini$iin) Inpt

Choices• Cross,$ltiplyin)/ 'e )et

w

f

v

f k l

=• For costs to &e $ini$ied/ the $ar)inal prodcti+ity per dollar

spent shold &e the sa$e for all inpts

• The soltion to this $ini$iation pro&le$ for an ar&itrary le+el

of otpt =0

)i+e s the contin)ent de$and fnctions forinpts:

lc9lc('/+/=0; -c9-c('/+/=0


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Contin)ent "e$and for

Inpts• The contin)ent de$and for inpt 'old &e

analo)os to the co$pensated de$andfnction in cons$er theory

• Notice that the contin)ent de$and for inpt is&ased on the le+el of #r$*s otpt7 So/ it is aderi+ed de$and7


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=0

De #. the iso=ant of otpt =07

C1

C!

C

Costs are represented &yparallel lines 'ith a slopeof ,'G+

Hraphically37Cost,ini$iin) Inpt Choices

l per period

- per period

C1 C! C


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C1

C!

C

=0

The $ini$$ cost of prodcin) =0 is C!

Hraphically37 Cost,ini$iin) Inpt Choices

l per period

- per period

k*

l*

The opti$alchoice is lJ/ -J

This occrs at the

tan)ency &et'een theiso=ant and the totalcost cr+e


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Total Cost Fnction• The total cost fnction )i+es the

$ini$$ cost incrred &y the #r$

to prodce any otpt le+el 'ith)i+en inpt prices7 C 9 C(v /w/q

• De can co$pte it as:

• C 9 C(v /w/q9'Jlc(v /w/qB +J-c(v /w/q


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8+era)e Cost Fnction• The a+era)e cost fnction ( AC is

fond &y co$ptin) total costs pernit of otpt

q

qw v C qw v AC

),,(),,(costaverage ==


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ar)inal Cost Fnction• The $ar)inal cost fnction (MC is

fond &y co$ptin) the chan)e intotal costs for a chan)e in otptprodced

q

qw v C qw v MC

∂∂

==),,(

),,(costmarginal


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Hraphical 8nalysis of

Total Costs

• Sppose that k 1 nits of capital and

l 1 nits of la&or inpt are re=ired to

prodce one nit of otpt

C(q91 9 vk 1 B wl 1

•

To prodce m nits of otpt(ass$in) constant retrns to scale

C(q9m 9 vmk 1 B wml 1 9 m(vk 1 B wl 1

C(q9m 9 m ⋅ C(q91


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Hraphical 8nalysis of Total Costs

Otpt

Totalcosts

C

Dith constant retrns to scale/ totalcostsare proportional to otpt

8C 9 C

Koth 8C and

C 'ill &econstant


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• Sppose instead that total costs startot as conca+e and then &eco$escon+e. as otpt increases – one possi&le e.planation for this is that

there is a third factor of prodction thatis #.ed as capital and la&or sa)e

e.pands – total costs &e)in risin) rapidly after

di$inishin) retrns set in


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Otpt

Totalcosts

C

Total costs risedra$atically asotpt increasesafter di$inishin)retrns set in


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Otpt

8+era)e and

$ar)inal

costs C

C is the slope of the C cr+e

8C

If 8C L C/

8C $st &efallin)

If 8C C/

8C $st &erisin)

$in 8C


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Shifts in Cost Cr+es

• The cost cr+es are dra'n nder theass$ption that inpt prices and thele+el of technolo)y are held constant – any chan)e in these factors 'ill case

the cost cr+es to shift


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Aroperties of Cost Fnctions

17 Mo$o)eneity:

If the inpt prices are $ltiplied &y

an a$ont t, the total cost is$ltiplied &y the sa$e a$ont

• cost $ini$iation re=ires that the ratio ofinpt prices &e set e=al to RTS/ a

do&lin) of all inpt prices 'ill not chan)ethe le+els of inpts prchased

!7 Nondecreasin) in q/ v / and w


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7 Conca+e in inpt prices – The cost fnction increases less than

proportionally 'hen one inpt priceincreases &ecase the #r$ cans&stitte it &y other inpts

– 8s the e.penditre fnction incons$er theory


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C(+/'/=1

TC is conca+eity of Cost

Fnction

'

Costs

The cost fnction C(+/'1/=1 is conca+e

in inpt prices7

C(v,w1,q1)

w1

Asedo Cost


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Reaction to inpt prices increases

• If the price of an inpt increases/ the cost'ill increase

• If #r$s can easily s&stitte another inptfor the one that has risen in price/ there

$ay &e little increase in costs• It is i$portant to $easre the s&stittion

of inpts in order to predict ho' $chcosts 'ill &e a%ected &y an increase in the

price of an inpt (possi&ly de to a ta.increase


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Inpt S&stittion

• 8 chan)e in the price of an inpt 'illcase the #r$ to alter its inpt $i.

•

De 'ish to see ho' k Gl chan)es inresponse to a chan)e in wGv / 'hile

holdin) q constant

∂

∂

v

w

k l


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Inpt S&stittion• Rather than the deri+ati+e/ 'e 'ill se the elasticity:

l

l

/

/

)/(

)/(

k

vw

vw

k s ⋅

∂

∂=

)i+es an alternati+e de#nition of the elasticity of s&stittion

– in the t'o,inpt case/ s $st &e nonne)ati+e

– lar)e +ales of s indicate that #r$s chan)e their inpt $i.si)ni#cantly if inpt prices chan)e7 Mence/ costs 'ill not

chan)e so $ch – It can $e esti$ated sin) econo$etrics


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Shephard*s e$$a• Shephard*s le$$a (a tric- to o&tainthe contin)ent de$and fnctionsfro$ the cost fnction

•

the contin)ent de$and fnction for anyinpt is )i+en &y the partial deri+ati+e of thetotal,cost fnction 'ith respect to thatinpt*s price

• See e.a$ple 1072 in the &oo-

• 8s 'e o&tained the co$pensated de$andfro$ the e.penditre fnction in cons$ertheory


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Short,Rn/ on),Rn "istinction

• Econo$ic actors $i)ht not &eco$pletely free to chan)e the a$ontof inpts

•

Econo$ist sally ass$e that it ta-esti$e to chan)e capital le+els/ 'hilela&or can &e chan)ed =ic-ly

• Econo$ist say that the short rn is

the period of ti$e in 'hich so$e of theinpts cannot &e chan)ed

• on),rn is 'hen the period of ti$e'hen all the inpts can &e chan)ed


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Short,Rn/ on),Rn "istinction

• 8ss$e that the capital inpt is

held constant at k 1 and the #r$ isfree to +ary only its la&or inpt

• The prodction fnction in the

short rn &eco$esq 9 f (k 1/l


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Short,Rn Total Costs

• Short,rn total cost for the #r$ is

SC 9 vk 1 B wl

•

There are t'o types of short,rncosts: – short,rn #.ed costs are costs

associated 'ith #.ed inpts (vk 1

– short,rn +aria&le costs are costsassociated 'ith +aria&le inpts (wl


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Short,Rn Total Costs• In the Short,rn:

– Cannot decide the a$ont of #.ed inpts7 The#r$ does not ha+e the @e.i&ility of inptchoice

– to +ary its otpt in the short rn/ the #r$$st se nonopti$al inpt co$&inations

– the RTS 'ill not &e e=al to the ratio of inptprices

– Conse=ently/ short,rn costs 'ill &e e=al toor lar)er than lon),rn costs


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Short,Rn Total Costs

l per period

- per period

=0

=1

=!

k 1

l1 l2 l3

Kecase capital is #.ed at -1/

the #r$ cannot e=ate RTS'ith the ratio of inpt prices7Notice that short rn costs 'ill &e

E=al or lar)er than thelon) rn cost


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Short,Rn and on),Rn costs

• The short rn cost depend on the a$ontof #.ed capital a+aila&le

• There is no a ni=e short rn cost cr+e7

There 'ill &e as $any as possi&le le+els ofcapitals are

• The short rn cost of prodcin) =1 'ill &e

e=al to the lon) rn cost 'hen thea+aila&le capital in the short rn is thesa$e as the opti$al le+el of capital for thelon) rn (see =1 and -1 in the pre+ios

)raph7


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Short,Rn ar)inal and

8+era)e Costs• Re$e$&er that short rn costs are )i+en

&y:

–SC 9 vk 1 B wl

• The short,rn a+era)e total cost (SACfnction is

SAC 9 total costsGtotal otpt 9 SCGq

• The short,rn $ar)inal cost (SMC fnctionis

SMC 9 chan)e in SCGchan)e in otpt 9 ∂SCG∂q


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Relationship &et'een Short,

Rn and on),Rn Costs

Otpt

Totalcosts

SC (-0

SC (-1

SC (-!

The lon),rnC cr+e isthe $ini$$ oShort,rn ones

q0 q1 q2

C


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So$e Illstrati+e Cost Fnctions

nicholson chapter 10 _cost functions

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