Advanced Microeconomics Petri Ch 5

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f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. 1

Fabio Petri - Microeconomics for the critical mind

CHAPTER 5. november 20

THE THE!R" !F THE F#RM$ PART#A% E&'#%#(R#A$ PERFECT C!MPET#T#!)$

A)* THE ATEMP!RA% +ACAP#TA%#,T#C E)ERA% E&'#%#(R#'M /#TH

PR!*'CT#!)

5.1. The present chapter extends the marina!ist theor" of competitive enera! e#ui!i$rium to

inc!ude production %&ithout capita! oods and &ithout a rate of interest' capita! oods and rate of

interest raise specia! pro$!ems in the marina!/neoc!assica! approach and &i!! $e discussed in

(hapters )* + and ,-. o&ever* in order to mae our treatment of production decisions sufficient!"

enera! and thus a!so usefu! for su$se#uent chapters* &e admit the presence of capita! oods &hen

discussin the sin!e firm and the sin!e industr"' &e on!" !eave them out &hen &e come* in the

third part of the chapter* to discuss the enera! e#ui!i$rium of atempora! production and exchane

as this ind of enera! e#ui!i$rium is ca!!ed %exp!anation for this termino!o" must &ait for

(hapters ) and +-.

e start &ith the necessar" notions a$out the theor" of production and of price-taking firms.

4ricetain $ehaviour* &hich &e have a!read" assumed in the stud" of consumers* means the

economic aent treats prices as iven parameters in her maximiations' hence a $u"er %respective!"*a se!!er- $e!ieves that the price at &hich she can purchase %respective!" se!!- additiona! units of a

ood is the same as the price of the previous units' therefore for a firm* revenue from sa!es or

expenditure on inputs are !inear functions of the #uantit" so!d or $ouht. A discussion of &hen the

pricetain assumption is !eitimate* and of its connection &ith the notion of competition* is

provided in 4art 777 of the chapter.

The firms &e stud" in this chapter produce undifferentiated oods8 each product is produced

$" man" firms and is so standardied %and unaccompanied $" maretin expenses- that consumersare indifferent as $et&een the severa! producers. As a resu!t the on!" pro$!em of the pricetain

firm is ho& much to produce and &ith &hat com$ination of inputs.

4A9T 7

49:;<(T7:= 4:>>7?7@7T7E> >ET> A=; 49:;<(T7:= <=(T7:=>

5.2.. 7maine a &or!d &here a variet" of consumption oods is produced throuh the use of unproduced %oriina!- factors* i.e. different t"pes of !and and of !a$our' there are no produced




means of production i.e. no capita! oods* and no need to date oods no ro!e for time* no interest

rate. e &ant to understand ho& the marina! approach determines the competitive enera!

e#ui!i$rium of such an econom".

There are t&o t"pes of aents8 consumers* &ith iven endo&ments of factors and iven

preferences' and firms. 7n the $acround* ensurin respect of private propert" and of contracts*

there must $e an institutiona! setup* somethin !ie a state &ith !a&s* po!ice and courts* and this

re#uires resources* $ut for the moment &e ne!ect this aspect.

irms have rod1ction ossibilit sets. A production possi$i!it" set B is the set of a!!

com$inations of inputs and outputs that are possi$!e for a iven firm. These com$inations are ca!!ed

rod1ction rocesses or rod1ction lans. A production process is a vector* of inputs and of

outputs* &ith n e!ements if the possi$!e outputs and inputs tota! to n.

7n modern enera! e#ui!i$rium theor"* the preferred forma!iation of a production process is

as a vector of net1ts* a vector &here neative num$ers indicate %net- inputs and positive num$ers

indicate %net- outputs of oods %or services-.

h" the specification C%net- inputsC and C%net- outputsCD =etputs are especia!!" convenient

&hen one studies intertempora! e#ui!i$ria &ith produced intermediate means of production %i.e.

capita! oods-. Then inputs and outputs are dated. A firm ma" for examp!e consider a p!an inc!udin

the production of 100 units of a circu!atin capita! ood at time t* and a!so the uti!iation of +0 of those units at time t to o$tain other products at time t1' one sa"s then that the p!anned netput of

that capita! ood $" that firm at time t is 20' its &hat the firm can se!! of that capita! ood to outside

aents accordin to that p!an. Then the inner product of a netput vector and of the vector of input

and output %discounted- prices "ie!ds the %discounted- profit of adoptin that production p!an* &ith

neative netput entries %amounts of inputs- contri$utin to cost and positive netput entries %amounts

of outputs- contri$utin to %discounted- revenue.

7n the same &a" as for consumer theor"* inputs and outputs can a!so $e distinuished $" the!ocation in &hich the" are avai!a$!e* and $" the state of nature &ith &hich the" are associatedF 1.

hen oods are dated* it &i!! $e enera!!" !eitimate to assume that if B inc!udes a productive

process &ith netputs distinuished $" their dates* %"t*..*"T-* then it a!so inc!udes the same se#uence

&ith a!! netputs dates increased $" the same num$er* indicatin that a!! that matters is the !a

$et&een inputs and outputs* and not the moment &hen the productive process is started'

furthermore it is natura! to assume that outputs cannot precede their inputs in time.

?ut no& &e !eave aside #uestions re#uirin the datin of commodities. The most accepta$!e

1 (f. footnote 12 in chapter H* and chapter 12DD.




interpretation of the mode! &e are oin to present in 4art 777 is that it depicts the e#ui!i$rium of an

econom" &here f!o&s %per time unit- of services of non-produced factors %t"pes of !and and t"pes

of !a$our- produce f!o&s %per time unit- of consumption oods* either &ith no !a %continuousf!o&

production-* or in production c"c!es of one period !enth &ith the product comin out a!! toether at

the end of the period' and that there is no intertempora! transfer of purchasin po&er %no !oans- and

thus no interest rate. This is ca!!ed the atempora! econom" $ut in fact it ma" &e!! refer to an

econom" in time* &ith production tain time* $ut &ith no interest rate and no !oans' the

e#ui!i$rium can $e conceived as the norma! situation &hich the econom" ravitates to&ard* &ith

averae prices constant throuh time or chanin sufficient!" s!o&!" for the chane to $e ne!ii$!e.

The endo&ments of the econom" consist therefore on!" of nonproduced factors* and &e assume

that the inputs consist on!" of services of these factors' the products are on!" consumption oods*

that are so!d as the" come out.

5.2.2. hen the vector of outputs to $e produced $" a firm is iven* the in1t re31irement

set is the set of input vectors that a!!o& the production of that vector of outputs. 7n the vectors of the

input re#uirement set the inputs are measured as positive #uantities. 7f one assumes that some of the

#uantities of inputs can $e !eft id!e* the input re#uirement set for a certain output vector inc!udes a!!

vectors x of inputs that a!!o& producin at !east that output vector* p!us a!! vectors xIx.irms &i!! enera!!" on!" uti!ie efficient production processes. 7n terms of netputs &e sa"

that "∈B is efficient if there is no other "C∈B such that "CJ" and "CI"' in other &ords it is not

possi$!e to produce the same outputs &ith !ess of some input* or to produce more of some output

&ith the same inputs. 7f the output vector I0 is iven* an input vector x %inputs $ein measured

no& as positive #uantities- is efficient if no vector xKx exists such that the netput vector %*x-∈B.

hen one considers production processes that produce on!" one output* it is often assumed

that the economically relevant production processes that produce that output can $e descri$ed $" arod1ction f1nction #Lf%x1**xn-Lf%4- &here output # is the maximum output o$taina$!e from the

vector of inputs 4L%x1**xn-* the !atter measured as positive #uantities. The set of input vectors that

a production function #Lf%x- associates &ith a iven output #N is ca!!ed the iso31ant associated &ith

#N' note that its e!ements need not $e a!! efficient8 if x is an efficient input vector associated &ith

output #N* and if the addition to x of some additiona! amount of an input* e.. Ox 1* is una$!e to

increase production* the ne& input vector xL%x1 Ox1**xn- is a!so part of the iso#uant associated

&ith #N. 7n common par!ance in economics &hen one speas of output o$taina$!e from certaininputs* one means the maximum output. The reason &h" the economica!!" re!evant production



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. H

processes can inc!ude some nonefficient input vectors is that if an input is free %ero price-* a firm

need not !imit the demand for it* so it miht even demand a use!ess!" !are amount of it.

7t is sometimes assumed that in the short eriod the #uantit" of some inputs cannot $e varied'

these are ca!!ed fixed inputs. The short-eriod rod1ction f1nction has t&o a!ternative

representations8 it can inc!ude amon the inputs the iven #uantities of the fixed factors* $ut for

$revit" it can a!so descri$e output as a function of the so!e variable inputs.

The e#uiva!ent of the production function for production processes that produce severa!

outputs simu!taneous!" is a transformation f1nction imp!icit!" defined $" an e#uation T%x1*

*xn'#1*...*#m-L0* &here* aain* inputs are measured as positive #uantitiesF2. or a!! inputs fixed

and a!! outputs $ut one fixed* the e#uation "ie!ds the maximum o$taina$!e output of the !ast ood*

and for a!! outputs and a!! inputs $ut one fixed* the e#uation "ie!ds the minimum necessar" amount

of the !ast input.

5.. hen the production possi$i!ities set B is a set of netput vectors* some axioms that ma"

$e postu!ated on it are8

1 0∈B* inactivit" is one possi$i!it"

2 B∩9 n L {0}* no production of outputs &ithout inputs

3 B∩ B L P* production is irreversi$!eH B is convex

5 B is $ounded a$ove

6 for an" ood i* and an" positive sca!ar #* the vector "L%0*...*0*#i*0*...0- is ∈B %free

disosal-.

Axioms 1* 2 and 3 are unpro$!ematic and are assumed in &hat fo!!o&s. Axiom H imp!ies

perfect divisi$i!it" of a!! inputs and outputs' its connection &ith returns to sca!e &i!! $e discussed

present!". Axiom 5 is convenient in a first stae of some mathematica! proofs* and it is Qustified $"referrin to !imited factor endo&ments that do not a!!o& producin more than certain maximum

#uantities* $ut this means mixin up endo&ments &ith techno!oies* so it &i!! not $e assumed in

&hat fo!!o&s. Axiom 6 postu!ates that for each ood there is avai!a$!e a process that uses that ood

a!one as an input and produces nothin* so one can a!&a"s et rid of an" amount of an" ood

&ithout an" cost' it is ca!!ed the free disosal ass1mtion. hen the free disposa! assumption is

made* then an" firm can coup!e an" production process &ith free disposa! processes* and the resu!t

2 Exercise8 :$tain the transformationfunction representation of a production function.




is a propert" or assumption sometimes ca!!ed monotonicit of the rod1ction ossibilities set+8

if "∈B* then "C such that "CK" is a!so ∈B*

$ecause "C is a process that emp!o"s at !east as much of each input as " % H-* and produces not more

of each output than "* and it is a!&a"s possi$!e to o$tain "C from " $" use of free disposa! processes.

#Lf%x-

#

: x

i. 5.1. 4roduction possi$i!ities set resu!tin from a production function #Lf%x- p!us free disposa!.

ree disposa! ma" appear a #uestiona$!e assumption in man" situations %it is often cost!" to

et rid of oods* or to prevent some output from comin out-* $ut it can $e arued to $e

fundamenta!!" harm!ess. An a$i!it" cost!ess!" to et rid of excess inputs is not needed as !on as

these inputs have a positive cost* $ecause then firms &i!! not $u" them to start &ith* so a free

disposa! assumption of excess cost!" inputs is superf!uous $ut then a!so harm!ess' and if inputs are

cost!ess and &ith a neative marina! product* the" can $e !eft id!e and a!! one needs is to

distinuish the technological from the economic production method %as &as done in R3.3.H-. As to

undesired outputs* if the" must not $e produced it can $e assumed that* &hatever disposa! process is

necessar" in order not to produce them %or in order to dispose of them-* its inputs %and costs- are

inc!uded amon the inputs %and costs- of the desired output. hen there is a choice a$out ho&

much to produce of an undesired side product* then a forma!iation can usua!!" $e found in &hich

the abatement of its production is counted as an output* and then the pro$!em $ecomes aain the

3 >ome authors ca!! monotonicit" the fo!!o&in s!iht!" different assumption8 !et # stand for a vector of

outputs and !et S%#- stand for the set of input vectors %measured as positive #uantities- that a!!o& the

production of at !east #' if an input vector x is in S%#-* and xCIx* then xC∈S%#-.H 9emem$er that inputs are neative num$ers* so a greater use of an input means a reater a$so!ute va!ue

of a neative num$er i.e.* a!e$raica!!"* a smaller num$er indicatin input use.




usua! pro$!em of cost minimiation and profit maximiation* this time &ith Qoint production. Thus

the free disposa! assumption is not rea!!" necessar" $ut it is a!so enera!!" harm!ess.

5.6. Axiom H stipu!ates that if "∈B and "∈B* then for 0a1 a!! netputs "ULa"%1a-" are

a!so ∈B' strict convexit" of B means that if " and " are efficient* then "U is not efficient. (onvexit"

re#uires divisi$i!it" of inputs and of outputs.

Exercise8 V- >uppose a sin!e output # is produced $" t&o inputs x 1 and x2 accordin to the

production function #Lax1$x2* a*$W0. ;ra& some iso#uants and prove that the production

possi$i!it" set is convex.

X- >uppose t&o outputs 1 and 2 are Qoint!" produced $" a sin!e input x &ith transformation

function 122

2 xL0. 4rove that the production possi$i!it" set is convex and that for fixed x the

!ocus of possi$!e efficient com$inations of 1 and 2 %the transformation curve- is concave.

Y- >uppose a sin!e output # is produced $" t&o inputs x 1 and x2 accordin to the productionfunction #Lx1

2x22' dra& some iso#uants and prove that the production possi$i!it" set is not convex.

An assessment of Axiom H re#uires a discussion of ret1rns to scale* a notion iven different

meanins in the histor" of economic thouht. The term oriina!!" referred to the dependence of

returns* that is of net earnins* on the sca!e of norma! output. The usua! meanin in modern

ana!"ses is technolo7ical ret1rns +in terms of o1t1t to the scale of in1t emloments * or

deree of homoeneit" of the production function &hen a!! inputs are varia$!e. A different meanin

that &e &i!! discuss !ater %R5.21DD- is firm returns in terms of output from the sca!e of tota! cost * a

notion especia!!" he!pfu! in the presence of indivisi$i!ities' this definition of returns to sca!e taes

input prices as fixed. A sti!! different meanin is industry returns in terms of output from total

industr" costs* a ver" comp!ex Marsha!!ian notion* historica!!" important in partia! e#ui!i$rium

ana!"sis* that taes into account %i- returns to sca!e at the firm !eve!* %ii- &hat happens to the prices

of the industr"Cs inputs* and %iii- externa! effects* &hen the industr"Cs output chanes%5-' this notion

of returns %in this meanin the appendae to sca!e is usua!!" a$sent- &i!! $e discussed &hen &e

arrive at the industr"Cs supp!" curve.

4roduction functions a!!o& a #uic definition of techno!oica! returns to sca!e. @et 4 $e the

vector of inputs to a production function f%x-. e sa" that f%4- exhi$its

constant returns to scale (CRS) if f%t4-Ltf%4- for tW0'

increasing returns to scale (IRS) if f%t4-Wtf%4- for tW1'

decreasing returns to scale (DRS) if f%t4-tf%4- for tW1.

7n &ords* constant returns to sca!e means that dou$!in a!! inputs dou$!es output' increasin

5 This is the notion of returns in the famous controvers" oriinated $" 4iero >raffa &ith his 1,25 and 1,26

artic!es.



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. )

returns* that dou$!in a!! inputs more than dou$!es output' decreasin returns* that dou$!in a!!

inputs "ie!ds !ess than dou$!e output. 9eturns to sca!e can $e varia$!e8 for examp!e a production

function miht exhi$it returns to sca!e that are increasin at first* then constant* then decreasin.

ocal returns to scale for a iven vector of inputs 4 are ascertained $" checin &hich one of the

three ine#ua!ities ho!ds for sma!! variations of t around 1 %cf. R5.6.1 $e!o&-.

Technica! returns to sca!e can a!so $e defined in terms of netputs and of frontier of the

production possi$i!it" set %B-. 7 ive a taste. 7f netput "∈%B- imp!ies t"∈%B-* &ith t an" positive

sca!ar in a neih$ourhood of 1* &e sa" that B exhi$its !oca! (9>. There are increasin returns to

sca!e at least locally* if there is some "∈%B- and some sca!ar tW1* such that t"∈B and t" ∉ %B-*

that is* ∃ "∈B and Jt" such that "It".

=ote that if there are increasing returns to scale at least locally! " is not convex#

4roof8 $" contradiction. ?" Axiom 1* "L%0*...*0-∈B* so convexit" re#uires that for an" "∈B a!so tC"

&ith 0tC1 is in B' then consider netputs "∈%B-* t" and "CIt" of the a$ove definition of !oca!!" increasin

returns in terms of netputs8 if B is convex* "ZLtC"C &ith 0tC1 is in B' choose tL1/t* then "ZI" and "ZJ"* so "

cannot $e in %B-* contradictin the assumptions' hence B cannot $e convex. [

This sho&s that Axiom H exc!udes increasin returns to sca!e. =o&* increasin returns to sca!e

are arued $" man" economists to $e often present in rea!it"' therefore Axiom H is a restrictive

assumption' in spite of this* &e &i!! enera!!" assume it $ecause it is necessar" for the standard

theor" of enera! e#ui!i$rium* &hose presentation is no& our main aim.

As an Exercise* "ou are ased to prove that a convex production possi$i!it" set &ith a sin!e

output imp!ies a $uasiconcave production function. %int8 prove first that the input re#uirement set

is convex. The definition of #uasiconcavit" &as iven in (h. H* RH.5-

5.5. >ome c!arification on the notions of productive process* inputs* indivisi$i!ities can $e

usefu!. A production process usua!!" starts &ith certain inputs and !asts some time* durin &hich

time the inputs $ecome transformed into a series of other thins $efore reachin the fina! form of

the sa!ea$!e product. There is therefore some ar$itrariness in the description of a production

process' one miht a!&a"s $rea it into t&o successive processes* the first one producin

intermediate products &hich are the inputs to the second one. or examp!e* the production of a sofa

from &ood p!ans* sprins and fa$ric miht $e $roen do&n into a series of staes* each one

producin an incomp!ete sofa c!oser and c!oser to comp!etion. The intermediate products miht

themse!ves $e priced* as made evident $" the fact that sometimes the process oriina!!" performed



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. +

in its entiret" $" a firm is $roen do&n into processes performed $" different firms* for examp!e

production of a car has historica!!" $een more and more $roen do&n amon separate firms &hich

produce the seats* the &indshie!d* the tires* the $raes* etc.* parts &hich are fina!!" assem$!ed into a

car $" a different firm. hich inputs are initia! inputs that appear in the production function* and

&hich are intermediate staes that are on!" imp!icit!" considered %$ut miht $ecome exp!icit if the

process &ere $roen do&n into successive su$processes performed $" different firms-* depends on

the oraniation of production and* from the theoristCs point of vie&* is !are!" ar$itrar". This maes

the notion of intermediate in1t am$iuous. !our is an intermediate input in the production of

$read from &heat* $ut it is an initia! input in the production of $read from f!our.

The term Cintermediate inputC is sometimes used as s"non"m of circ1latin7 caital 7ood*

&hich means a capita! ood that is fu!!" destro"ed %one miht sa"* that disappears into the product-

in a sin!e uti!iation. These t&o notions are $est ept distinct' it is $est to reserve the term

Cintermediate inputsC for the products produced and reuti!ied inside a production process* and

therefore not appearin amon the inputs of the production function %the" must not $e paid for-.

The inputs appearin in the production function are a!! the Cinitia!C inputs that must $e paid for% 6-'

&hen the" are capita! oods* the" can $e either circu!atin capita! oods* or d1rable caital 7oods'

in the !atter case the" reappear amon the outputs* o$vious!" &ith the a!terations caused $"

uti!iation%)

-.e &i!! enera!!" assume that a!! inputs and outputs are perfect!" divisi$!e* i.e. can $e

represented $" continuous varia$!es. This is a ood approximation for !and and for !a$our timeF +*

$ut it is o$vious!" unrea!istic for capita! oods and for man" products' ho&ever* if the ana!"sis dea!s

&ith $i #uantities the assumption ma" sti!! $e accepta$!e if the indivisi$i!ities are sma!! re!ative to

tota! input use or tota! output.

Much more de$ata$!e is the assumption of differentia$i!it" of the production function* and

here &e come to a ver" important issue. 7n most industries a different productive process re#uires*not different proportions amon the same capita! oods* $ut different capita! oods* and for each

ensem$!e of capita! oods a rather riid !a$our input. The amount of !a$our services needed to

assem$!e a car* for examp!e* is rather strict!" determined $" ho& mechanied the production

6 7n the case of intertempora! production functions* Cinita!C inputs are not necessari!" enterin the process

at the same date* the term Cinitia!C must $e interpreted as meanin not produced $" the production process

itse!f.) 7t is a!so possi$!e to imaine cases in &hich a dura$!e capita! ood is an intermediate ood* $ecause the

production process considered !asts a num$er of periods* and produces itse!f a dura$!e capita! ood &hich is

then entire!" uti!ied durin the remainder of the process.+ @a$our time is* ph"sica!!" speain* perfect!" divisi$!e* $ut reu!ations often !imit this divisi$i!it"' for

examp!e a firm ma" $e o$!ied to hire fu!!time !a$our on!". Even then* if the firm emp!o"s 1000 !a$ourers a

perfect divisi$i!it" assumption ma" $e an accepta$!e approximation.



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. ,

process is' and there is no su$stituta$i!it" $et&een the parts to $e assem$!ed* a car needs exact!" one

enine* five t"res* etc.* each one of these parts needin in turn riid #uantities of materia! inputs and

riid amounts of !a$our determined $" the machiner" used. There enera!!" is no* or near!" no*

varia$i!it" of proportions amon the same inputs. ;ifferentia$i!it" is a very unrea!istic assumption

&hen a!! inputs are ph"sica!!" specified and production uses capita! oods. ?ut then ho& come the

differentia$i!it" assumption is so &ide!" acceptedD The reason &ou!d appear to $e a historica! one*

name!" the Marsha!!ian ha$it* shared $" a maQorit" of economists of his eneration and of the next

one* and sti!! &idespread toda"* to descri$e production functions as com$inations of !a$our and

!and &ith capita! treated as a single factor \* measured as an amount of value' imp!icit!"* the prices

of the severa! capita! oods are treated as iven* and the production function is determined as

fo!!o&s8 the firm is assumed to determine* for each iven vector of noncapita! inputs and each

iven \* the vector of capita! oods of va!ue \ that maximies production' thus* iven the amounts

of !a$our and !and* sma!! increases of \ can &e!! imp!" a tota!!" different vector of capita! oods'

a!on a tota! factor productivit" curve* capita! so conceived chanes not on!" in #uantit" %an amount

of exchane va!ue- $ut a!so in form %ph"sica! composition-F ,. ith such a specification of the

capita! input* the assumption of smooth varia$i!it" of proportions $et&een the inputs ac#uires much

reater p!ausi$i!it"8 increasin on!" the num$er of t"res certain!" does not increase the output of

cars' on the contrar" if &hat can $e increased is the value of the capita! oods used* &hich can $eassociated &ith a chane in the capita! oods used* then it is !ie!" that a &a" can $e found to use

the capita! increase so as to increase the num$er of cars produced $" a iven num$er of &orers.

<nfortunate!" in more recent times this oriin of the use of smooth production functions and of

nice!" decreasin marina! product curves appears to have $een !ost siht of* &ith the resu!t that in

modern microeconomic theor" and enera! e#ui!i$rium theor" this treatment of capita! has one out

of fashionF10* inputs are a!! measured in technica! units* the severa! capita! oods are each treated as

a separate factor' $ut continuit" and differentia$i!it" of production functions are sti!! common!"assumed. 7 must fo!!o& no& this usua! practice in order to introduce the readers to this !iterature'

$ut it is important to rea!ie that &e are encounterin here an instance of surviva!* in a context no

!oner Qustif"in them* of assumptions that &ere oriina!!" Qustified $" the treatment of capita! as a

sin!e va!ue factor of varia$!e form. %More on this in chapter ).-

, Ana!oous!" an iso#uant in terms of* sa"* !a$our and \ indicates* for each !eve! of !a$our* the minimum

va!ue of capita! re#uired to produce the iven output' aain* sma!! movements a!on the iso#uant can &e!!

mean a passae to a production method re#uirin ver" different capita! oods.10 7t is not difficu!t to understand &h". The treatment of capita! as a #uantit" of va!ue in the firms

production function is on!" !eitimate if the prices of capita! oods are iven. ?ut these prices cannot $e

taen as iven &hen the purpose of the ana!"sis is not a partia!e#ui!i$rium one $ut rather the determination

of income distri$ution &hich affects re!ative prices.




>o un!ess different!" specified* &e assume production functions to $e continuous and

differentia$!e functions of perfect!" divisi$!e inputs.

5.8. Returns to scale#

5.6.1. 7n (h. H &e met homoeneous functions %cf. (h. H fn. 35DD-. (9> imp!ies that the

production function is homoeneous of deree 1 F11. 7f the production function is homoeneous of

deree W1* it has increasin returns to sca!e' if homoeneous of deree 1 it has decreasin

returns to sca!e. Even &hen a function f%x- is not homoeneous* &e can determine its local degree

of homogeneity $" increasin a!! independent varia$!es $" a common sma!! percentae* sa"* .1]*

and o$servin &hether f%x- increases $" more or !ess than .1]. This indicates the local

technolo7ical ret1rns to scale %of course* under perfect divisi$i!it" of a!! inputs-. Their most

&ide!" used measure is the scale elasticit of o1t1t %or simp!" elasticit of scale- &ith respect to

inputs. @et 4 $e the vector of inputs in an initia! situation &ith #Lf%4-* and consider f%t4- &ith tW0'

the sca!ar t measures sca!e* and the sca!e e!asticit" of output in #Lf%4- is defined as

e % &df(t x )'f(t x )'(dt't) L %$'t) * (t'$) % ln $ ' ln t eva!uated in t%+! ,ith x fixed .

=ote that this definition does not re#uire differentia$i!it" of f%x-* it on!" re#uires

differentia$i!it" of f%t4- &ith respect to t* i.e. &ith respect to proportiona! variations of a!! inputs*

therefore it is app!ica$!e to production functions &here inputs* or some of them* are perfectcomp!ements. ?ut if f%x- is differentia$!e* then $" the derivative ru!e of a function of function it is e

% ∑ =

n

+i ) x( f

+&xi*f'xi . ?" Eu!erCs theorem on homoeneous functions* if the production function

has constant returns to sca!e then i(xi*f'xi )%f( x ) so eL1. Accordin as e is e#ua! to* more than* or

!ess than* 1* the production function exhi$its locall constant$ locall increasin7$ or locall

11 T&o properties of homoeneous functions %a!read" $rief!" indicated in footnote 35 of ch. H- are of reat

re!evance for (9> production functions. A continuous function f%x1*...*xn- is homogeneous of degree k if*

&ith t a positive sca!ar* f%tx1*...*txn- L t

f%x1*...*xn-. The first propert" is that if a function homoeneous of deree is differentia$!e* then its partia! derivatives are homoeneous of deree 1' the proof is $"

differentiatin $oth sides of f%tx1*...*txn- L t f%x1*...*xn- &ith respect to xi* and indicatin &ith ^f%tx-/^%tx i- the

partia! derivative of f re!ative to the ith independent varia$!e* ca!cu!ated in tx8 one o$tains

i

k

i x

x f t

tx

tx f t

∂

∂=

∂

∂ -%

-%

-%&hich imp!ies that the partia! derivative ca!cu!ated in tx is the partia! derivative

ca!cu!ated in x mu!tip!ied $" t1' thus if f is a (9> production function its marina! products are

homoeneous of deree ero i.e. depend on!" on factor ratios %hence the expansion path* the !ocus of

tanencies $et&een iso#uants and isocosts* is a ra" from the oriin-. or the second propert"* differentiate

$oth sides of f%tx1*...*txn-Lt f%x1*...*xn- &ith respect to t* o$tainin -%--%

-%% 1

1

x f kt xtx

tx f k n

i

i

i

−

=

=⋅∂

∂∑ * and then

set tL18 one o$tains _i%xi`^f/^xi-Lf%x-* a resu!t sometimes ca!!ed .uler/s theorem for homogeneous

functions' for (9> production functions it is L1* hence if each factor is paid its ph"sica! marina! product

the pa"ment to factors exhausts the product* a resu!t a!so ca!!ed the product exhaustion theorem.




decreasin7 ret1rns to scale.%12-

7f all re!evant inputs are taen into account in the specification of the production function*

then an exact dou$!in of a!! inputs a!!o&s the exact rep!ication of the same p!ant t&ice and

therefore it shou!d permit at least a dou$!in of production8 for inteer tW1* f%t4-Itf%4-* i.e. there

must $e at least constant returns to sca!e. h" do 7 sa" Cat !eastCD ?ecause there ma" $e some

indivisi$i!it" of the techno!oica! process* &hich &as not fu!!" exp!oited at the oriina! sca!e* and

can $e $etter exp!oited at a $ier sca!e* a!!o&in for increasin returns to sca!e. Thus consider the

production function of a firm that extracts and transports oi! and a!so produces a!! the pipes for the

pipe!ine. The pipes are intermediate products in the overa!! production process and appear neither

amon the inputs nor amon the outputs of the oi! productionandtransport function. The inputs

inc!ude* for examp!e* the stee! needed to mae the pipes. =o&* up to a point the carr"in capacit" of

pipes increases more than proportiona!!" &ith the increase in the stee! uti!ied to mae pipes*

$ecause the stee! uti!ied is &ithin certain !imits rouh!" proportiona! to the diameter of the pipe

$ut the carr"in capacit" is proportiona! to the s#uare of the diameter. ;ou$!in the amount of

produced and transported oi! ma" then re#uire pipes of dou$!e diameter that use !ess than dou$!e the

stee!* &ith !ess than dou$!e the cost. A simi!ar issue arises &ith tans. This examp!e sho&s that*

&hen the production function ref!ects vertica!!" interated production processes &hich inc!ude the

production and uti!iation of intermediate oods* a dou$!in of a!! inputs need not correspond to arep!ication of the same production method t&ice* and a dou$!in of output need not re#uire a

dou$!in of inputs%13-. >ti!!* the rep!ication of the same production method t&ice %the $ui!din of a

second p!ant identica! to the first one- is a!&a"s possi$!e and therefore returns to sca!e for inteer

tW1 are at !east constant. hat a!!o&s the existence of increasin technica! returns to sca!e is the

existence of indivisi$i!ities %either of oods* or of processes- &hich are not fu!!" taen advantae of

at sma!! sca!es of production.

The resu!t f%t4-Itf%4- need not ho!d for fractiona! increases in sca!e* aain o&in toindivisi$i!ities. 7t ma" $e impossi$!e to increase a!! inputs $"* sa"* 30] if some inputs are

indivisi$!e' or the indivisi$i!ities can $e in the production process* e.. the production process miht

$e vertica!!" interated and inc!ude the interna! production and uti!iation of a !are indivisi$!e

12 The extension of these definitions to transformation functions is !eft to the reader % rays of outputs &i!!

rep!ace the sin!e output' the so!e comp!ication is that* since &ith transformation functions enera!!" a iven

vector of inputs does not uni#ue!" determine the vector of outputs* it is no& possi$!e to imaine cases &here

returns to sca!e differ accordin to &hich output ra" one considers-.13 The same can happen if capita! oods are areated into the sin!e factor %va!ue of- capita!' then

dou$!e the capita! and dou$!e the noncapita! inputs need not correspond to purchasin t&ice as man" of the

same capita! oods* it ma" for examp!e mean the use of a different fixed p!ant that costs t&ice as much $ut

a!!o&s more than dou$!e the production. (f. $e!o&* R5.20* the notion of sca!e returns to cost.




capita! ood. ?e!o& 7 &i!! assume that this pro$!em is of minor importance* &e &i!! see that &hen

one considers an entire industr" &ith free entr" it is enera!!" p!ausi$!e to treat the industr" output

as comin from a (9> production function even &hen at the firm !eve! there are re!evant

indivisi$i!ities

5.6.2. 4!ausi$!"* even if indivisi$i!ities cause increasin returns at sma!! sca!es of production*

at each stae of technica! no&!ede there is a finite production sca!e $e"ond &hich returns to sca!e

are no !oner increasin8 &e can ca!! it minimum optima! sca!e. ?e"ond a certain dimension !arer

pipes and tans are no !oner convenient $ecause the" re#uire specia! reinforcin structures. 7n a

perfect!" competitive industr" &ith free entr" firms must produce at minimum averae cost and

therefore &i!! tend to adopt p!ants of at !east minimum optima! sca!e* possi$!" severa! of them. As

!on as this efficient sca!e of production is sma!! re!ative to tota! industr" output* it &i!! $e

approximate!" true that the areate of firms composin an industr" can $e seen as havin a

production function exhi$itin constant returns to sca!e &here the constanc" is enerated $"

variations in the num$er of identica! efficient p!ants. or this reason* $e!o& &e enera!!" assume

constant technica! returns to sca!e for industries. This assumption ma" appear va!id for on!" a ver"

restricted set of industries* iven the o$serva$!e tendenc" of firms in man" industries to ro& as

!are as the" can. ?ut the advantaes of sie can $e due to man" other reasons $esides increasintechnica! returns to sca!e* reasons that do not concern us no&%1H-. An"&a" !o$a!iation has

increased competition in man" industries &here minimum efficient p!ant sie is ver" !are' for

examp!e* one can $u" cars produced a!! over the &or!d* \orean cars compete &ith German and

<>A cars* thus even for industries &here minimum p!ant sie is ver" !are there often appears to $e

sufficient competition for the assumption of price e#ua! to averae cost to $e $road!" accepta$!e for

man" ana!"ses.

5.6.3. hat a$out decreasing technica! returns to sca!eD The" are difficu!t to defend if the

inputs appearin in the production function are rea!!" all re!evant inputs* $ecause then* as arued

a$ove* identica! dup!ication of p!ant and process shou!d "ie!d dou$!e output. ;ecreasin returns to

sca!e can $e admitted on!" if some re!evant inputs are fixed in #uantit" and do not appear in the

production function. This is the case in shortperiod ana!"sis &hen on!" varia$!e inputs are made to

appear in the production function' $ut for !onperiod ana!"sis* the so!e &a" &hich appears

1H e on!" mention at this stae the possi$!e advantaes in terms of funds for maretin expenditures* or

for research and deve!opment %9;- expenditures* and the possi$i!it" to o$tain discounts on some input

prices .




accepta$!e to arrive at decreasin returns to sca!e is to postu!ate a reater difficulty of control and

co-ordination over the performance of su$ordinatesF15. 7t miht $e ans&ered that if one dup!icates

not on!" the p!ant $ut a!so the manaers* there is no reason &h" there shou!d $e a reater difficu!t"

of contro!. The o$Qection to this is that manaers must $e contro!!ed too* and the o&ner of the firm

or top manaer ma" have reater difficu!t" in avoidin su$ordinate manaers shirin or

em$e!in &hen she must contro! man" manaers' furthermore* the information that the top

manaer must process increases* and its re!ia$i!it" decreases* as it must pass throuh a reater

num$er of $ureaucratic !a"ers* main correct decisions more difficu!t. hether this o$Qection is

sufficient to Qustif" an optima! sie of pricetain firms is sti!! an o$Qect of disareement amon

theoreticiansF16. ?ut for the purposes of va!ue theor" &hat is important is the $ehaviour of

industries* and then the moment one admits free entr" the difficu!ties of contro! ma" exp!ain &h"

individua! firms are !imited in sie* $ut industr" output can $e varied $" variation in the num$er of

firms' so at the industr" !eve! there &i!! $e (9> an"&a" in !onperiod ana!"sis as !on as one can

assume that minimum efficient sie is sma!! re!ative to tota! demand.

5.6.H. ina!!" a &ord on the difference $et&een production process and production method

&hen there are (9>. An" netput vector in the production possi$i!it" set is a production process.

ith (9>* if % 4*#- is the netput representation of a sin!eoutput production process then %t4*t#-&ith tW0 is a production process too* and if the first one is efficient so is the second. :ne means then

$" rod1ction method a set of ratios $et&een inputs and outputs8 a vector % 4*#-∈B and a vector

%t4*t#-* tJ1* are considered t&o production processes representin the same method activated at

t&o different activity levels.

15 ;ecreasin economic returns to sca!e %i.e. profits that increase !ess than proportionate!" &ith output-

can arise $ecause an output increase raises the renta!s of some inputs' $ut this is $est ept separate from the

issue of technica! returns to sca!e.16 or examp!e* >cherer and 9oss p. 106 aree &ith the traditiona! position &e!! represented $" EAG

9o$inson DDref on the reater difficu!t" of contro! as a cause of u!timate!" decreasin returns to sca!e* and

their aruments on the difficu!t" of contro! increasin &ith sie are prima facie convincin. Edith 4enrose on

the contrar" &rote8 be do not no& ho& effective the decentra!iation of authorit" can $e as a means of

eepin costs per unit of output from risin as a firm expands. 9e!ia$!e empirica! evidence does not exist and

a!! studies of the matter are inconc!usive* $ut there is no evidence that a !are decentra!ied concern re#uires

supermen to run it....=either is there sinificant evidence that the a$i!it" to fi!! the hiher administrative

positions is excessive!" rare or that the demands on the men occup"in these positions exceed their a$i!it" to

cope &ith them effective!".U E. 4enrose* b@imits to the ro&th and sie of firmsU* AE9 1,55* vo!. H5 %2-*

Ma"* 531H3* p. 5H2. (ases supportin 4enrose* for examp!e Mac;ona!ds or (oca(o!a or <nited ruits*

easi!" come to mind. 4erhaps in man" cases the advantaes of increasin sie are so reat %especia!!" &hen

one considers the sca!e economies in maretin* 9;* transport costs* emp!o"ee trainin* etc.-* as to more

than counter$a!ance the increasin difficu!ties of coordination' a!so* t"in decentra!ied manaers pa" to

resu!ts ma" $e often sufficient to motivate them.






!and method 1

method 2

isocost A

?C

method 3

?

(

: !a$our

i. 5.1$is %i. 3.15-. Activit"ana!"sis iso#uant &ith three a!ternative methods. The red

$roen !ine is the iso#uant associated &ith method 2 a!one.




5.8. Marginal product, isoquant, transformation curve

5.+.1. e assume no& that the firm produces a sin!e output and its efficient production

possi$i!ities are represented $" a production function f%4- &hich is t&ice continuous!" differentia$!e

%that is* it has partia! derivatives that are differentia$!e* and their partia! derivatives are continuous-.

Then &e can define the mar7inal rod1ct of factor xi as M4i^f/^xi' and the !ocus of input

com$inations &hich "ie!d an assined !eve! of output* the iso31ant* is a differentia$!e surface %in 9 n

if there are n inputs- $" the imp!icit function theorem.

7f some factors are fixed* the !ocus of com$inations of the remainin inputs &hich "ie!d the

iven output is ca!!ed a restricted iso31ant or short-eriod iso31ant %note that it ma" $e empt"-.

x2 #2

M9T

iso#uant

transformation curve

T9> x1 #1

i. 5.2 >tandard iso#uant and standard transformation curve in 9 2.

>uppose &e consider an iso#uant restricted to the t&o inputs i and Q. The condition that

production $e e#ua! to an assined #uantit" #g and that a!! inputs $e iven except for xi and x Q8

f(x+ !###!xi !###!x 0 !###!xn ) % $1 ! ,ith xh given except for h%i!0

imp!icit!" maes x Q a function of xi. The s!ope of this function x QLx Q%xi-* i.e. the s!ope of the

restricted iso#uant &hen xi is treated as the independent and x Q as the dependent varia$!e* is the

e#uiva!ent in production theor" of the consumerCs marina! rate of su$stitution' it is ca!!ed the

technical rate of s1bstit1tion of factor Q for factor i* to $e indicated as T9> Qi' $" the derivative ru!e

for differentia$!e imp!icit functions the T9> Qi is iven $" T9> Qi ≡ dx Q/dxi L %^f/^xi-/%^f/^x Q- L



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. 1)

M4i/M4 Q* the &e!!no&n resu!t of first"ear text$oos. The T9> is a negative #uantit" if $oth

marina! products are positive' economists sometimes spea of the T9> as a positive #uantit"* then

the" are imp!icit!" referrin to its a$so!ute va!ue.

7n (hapter 3 &e sa& that marina! products need not $e positive* and therefore technological

iso#uants need not $e do&n&ards!opin. %The reader is invited to reread in (hapter 3 the

distinction $et&een techno!oica! and economic iso#uants.- e supp!" no& a numerica! examp!e.

Assume

# L %x1x2 .+x12 .2x2

2-1/2

The reader can chec that this production function has (9>. Marina! products are iven $"

^#/^x1L 2/12

1

$%x2 1.6x1-* ^#/^x2 L 2/12

1

$% x1 .Hx2-' provided it is #W0* &hich &i!! $e the case as

!on as 1x2/x1H* the marina! product of factor 1 is positive as !on as x 2W1.6x1* and the marina!

product of factor 2 is positive as !on as x22.5x1. ence $oth marina! products are positive and

iso#uants are neative!" s!oped on!" as !on as 1.6x2/x12.5. :utside this fair!" restricted rane of

factor proportions* one of the t&o marina! products is ero or neative* imp!"in an up&ard

s!opin iso#uant.

or a differentia$!e transformation function 2(x+ !3!xn4$+ !###!$m )%5* since there is more than

one output* an input has severa! marina! products* one for each output % Exercise8 &rite the

expression for them from the ru!e of derivation of imp!icit functions-. The determination of the T9>

$et&een t&o inputs re#uires that a!! other inputs and a!! outputs $e fixed. 7f a!! inputs* and a!!

outputs $ut t&o* are fixed* the !ocus of efficient com$inations of the t&o remainin outputs %&here

efficienc" means that one output is maximied &hen the other one is iven- is ca!!ed a

transformation c1rve+9 and its s!ope is ca!!ed the mar7inal rate of transformation* M9T* and is

iven $" M9T Qi d# Q/d#i L %^T/^#i-/%^T/^# Q-.

ith constant returns to sca!e* marina! products on!" depend on factor proportions and not

on the sca!e of production. This is $ecause the partia! derivatives of a function homoeneous of

deree one are homoeneous of deree ero* i.e. do not var" for e#uiproportiona! variations of a!!

independent varia$!es. Thus a!on a ra" from the oriin a!! iso#uants have the same T9>.

5.+.2. The reader &i!! have noticed the resem$!ance $et&een iso#uants and indifference

curves* and $et&een marina! products and marina! uti!ities. Thus* for examp!e* convex iso#uants

imp!" that the production function is #uasiconcave. o&ever* there are some differences.

1) 7t is a!so ca!!ed a production possi$i!it" frontier for the firm %in order to distinuish it from the

production possi$i!it" frontier for the entire econom"-.



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. 1+

A first difference ref!ects the cardinal character of production possi$i!ities versus the ordinal

nature of preferences. An" strict!" increasin transformation of an increasin uti!it" function

represents the same preferences' on the contrar"* an" transformation of a production function a!ters

the production possi$i!ities. Therefore* #uasiconcavit" is not as important a notion as in uti!it"

theor"8 in production theor" &e a!so &ant to no& returns to sca!e* &hich are ar$itrar" in ordina!

uti!it" theor".

A!so* as long as CRS and perfect divisibility are assumed! an iso$uant is necessarily convex .

This is $ecause under these assumptions if x and xC are t&o input vectors $e!onin to the iso#uant

associated &ith output #* then the firmCs production possi$i!ities set B inc!udes a!! netput vectors %

tx*t#- and %txC*t#- for an" tW0* so the firm can produce # $" usin an" convex com$ination of the

t&o processes %ax*a#-%%1a-xC*%1a-#-* &here 0 ≤ a ≤ 1* and therefore the iso#uant cannot o

a$ove the sement connectin an" t&o points of it* cf. i. 5.3$. :n the contrar"* in uti!it" theor" it

&ou!d $e a most specia! case if a!! !inear com$inations of t&o consumption $und!es on the same

indifference curve "ie!ded the same uti!it" %the consumption oods &ou!d $e !oca!!" perfect

su$stitutes-.

x2 A

x2C ?

(

x1C x1

iure 5.3a >hortperiod restricted iso#uant &ith i. 5.3$. ith divisi$i!it" and (9>* iso#uants

incompati$i!it" $et&een the t&o varia$!e factors. cannot $e strict!" concave* point ? of input use

a!!o&s the same production as A or ( throuh a

!inear com$ination of the activities %processes- correspondin to A and (.

or the same reason* under divisi$i!it" and (9>* production functions are concave $ecause*

iven an" t&o efficient vectors %xC*f%xC-- and %xZ*f%xZ--* it is possi$!e to produce at !east an" !inear

com$ination of these vectors and therefore f%axC%1a-xZ-Iaf%xC-%1a-f%xZ-.

This is no !oner necessari!" the case &ith restricted* or shortperiod* iso#uants. or examp!e*

&ith a iven machine for chemica! reactions* it miht $e possi$!e to produce a iven output $"

usin a certain chemica! process* or another chemica! process $ased on different components* $ut



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. 1,

an" Qoint use %or even a!ternate use- of the t&o processes on the same machine miht $e impossi$!e

$ecause the residues of the components of one process &ou!d react &ith the components of the

other process and destro" the machine. 7n this case* assumin free disposa!* the restricted iso#uant

&ou!d have the shape sho&n in iure 5.3a. The meanin of such an iso#uant is that* iven the

other inputs* the desired output can $e produced either &ith the #uantit" x 1C of input 1* or &ith the

#uantit" x2C of input 2* $ut not &ith $oth. o&ever* cases such as this one can $e considered hih!"

unusua! and therefore &e !eave them aside.

5.9. Profit maximization and APM

The most common assumption a$out the $ehaviour of firms is that the" aim to maximie

profit . The meanin of CprofitC here and in the entire chapter is the marina!ist one* i.e. in this

chapter CprofitC stands for &hat is !eft to the entrepreneur %the o&ner of the firm- after pa"in a!!

costs including interest on capita! advances%1+-. Even &hen the apparent aim of the firm is another

one* e.. sa!es maximiation* a case can usua!!" $e made that this does not entai! sinificant!"

different choices from the ones aimed at maximiin !onrun profit. More re!evant is the

possi$i!it" of inefficienc"* $ut as !on as manaement strives for profit maximiation the fact that

the oa! is on!" imperfect!" rea!ied does not a!ter the $road pattern of industr" $ehaviour. or

examp!e* the tendenc" to invest more in the industries that offer $etter profita$i!it" prospects &i!!sti!! exist even if on averae manaement is not ver" ood at minimiin costs. The first"ear

text$oo shortperiod supp!" curve of the firm* coincidin &ith the marina! cost curve* most

pro$a$!" remains up&ards!opin and therefore suests an increase in output if the product price

rises* even if marina! cost ref!ects inefficiencies. And the occasiona! episodes of manaers

pursuin strateies of persona! enrichment at the expense of the profita$i!it" of their firm usua!!"

end up rather #uic!" in the disappearance of the firm* &hose maret shares are a$sor$ed $" $etter

run firms. e accept profit maximiation as $road!" va!id as a surviva! condition* especia!!" incompetitive environmentsF1,. A monopo!ist entrepreneur not threatened $" taeovers miht indu!e

in other aims* e.. to have po!itica! inf!uence* or p!a" o!f* or $e enerous to&ard emp!o"ees' firms

in competitive environments and under threat of taeovers end up $ein taen over or oin

1+ And inc!udin an a!!o&ance for ris too' $ut &e are not considerin ris for the moment. %The reader

ma" $e surprised $" our mentionin interest here' $ut as &e said* the treatment of firms aims to $e enera!*

the assumption that there are no capita! oods and no interest &i!! on!" $e made &hen &e come to the

enera! e#ui!i$rium mode! to $e studied at the end of this chapter.-1, brom the vo!uminous and often inconsistent evidence* it appears that the profit maximiation assumption

at !east provides a ood first approximation in descri$in $usiness $ehavior. ;eviations* $oth intended and

inadvertent* undou$ted!" exist in a$undance* $ut the" are ept &ithin more or !ess narro& $ounds $"

competitive pressures* the se!finterest of stoco&nin manaers* and the threat of manaeria! disp!acement

$" important outside shareho!ders or taeovers.U >cherer and 9oss p. 52.




$anrupt if the" do not stru!e to survive* and this re#uires profit maximiation.

7n &hat fo!!o&s the prices of outputs are indicated as p Q* QL1*...*m' the renta!s %prices of the

servicesF20- of inputs are vi* iL1*...*n. The vectors L%p1*...*pm- and vL%v1*...*vn- are to $e conceived

as ro, vectors un!ess other&ise indicated* and the vectors of inputs 4 and of outputs 3 of a

production process as column vectors. At iven prices %$v-* for each production process &ith netput

% 4*3- tota! cost is v4 and profit is j8L3 v4. ith the netput notation L% 4*3-* one can put a!!

prices of inputs as &e!! as of outputs into a sin!e vector' if one uses the s"m$o! P:+v$ for this

encompassin ro& vector* then profit can $e represented more compact!" as j8LP. 7f inputs and

outputs are paid at different dates* the severa! prices of the previous expression are to $e intended as

discounted or capita!ied to a common date8 for the moment* the date at &hich output is so!d. 7f* as 7

&i!! most!" assume in &hat fo!!o&s* the firm produces a sin!e output* then vector 3 has on!" one

nonero e!ement.

A resu!t* some&hat ana!oous to the &ea axiom of revea!ed preferences* that derives

immediate!" from profit maximiation is the fo!!o&in8 if at prices P;L%v;$;- the firm chooses a

netput vector ;* the profit must $e at !east as reat as &ith an" other netput in B* i.e. P;;IP;* ∀'

This is sometimes ca!!ed the 6eak 7xiom of 8rofit 9aximi:ation* /APM. 7t has the fo!!o&in

imp!ication8 if netput "N is chosen at prices 4N and netput "k is chosen at prices 4k* it must $e

P;;IP;< and P<<IP<;W' re&rite these ine#ua!ities as P;%; "k-I0* P<+;=<I0 and addthem to et %P; P<-%; <-I0* &hich is often &ritten

OPÒI0.

7n &ords8 the dot product of the vector of price chanes and the vector of netput chanes is

nonneative and enera!!" positive' eometrica!!"* the t&o vectors form an acute an!e. The

interpretation re#uires to remem$er that inputs are neative num$ers in the netput representation.

Thus if price i is the on!" one to chane and it increases* the A4M imp!ies that netput i must

increase or at !east not decrease8 if netput i is an input* an a!e$raic increase of its neative #uantit"means a sma!!er uti!iation of that input.

5.!". #ptimal employment of a factor

@et us first consider the decision of ho& much to emp!o" of a sin!e input* &hen the

#uantities of the other inputs are iven. The firm is competitive* i.e. price taer* and produces a

sin!e output. The production function is #Lf%4-* differentia$!e. 4rofit is πLp# v4Lpf%4- v4 &here #

20 As pointed out in (h. 3* it is prefera$!e to spea of input rentals to mean price of the services of inputs'

$ut input price is so often used to mean input renta!* that in this chapter &e use the t&o terms

interchanea$!".




is output %a sca!ar-* p its price* 4 the vector of inputs %positive #uantities-. 7nterior maximiation of

π &ith respect to xi under xiW0 re#uires the firstorder condition p`^f/^xi vi L 0* i.e. the e#ua!it"

$et&een mar7inal reven1e rod1ct of the factor and CpriceC %i.e. renta!- of the factor* &here the

marina! revenue product of a factor is the derivative of revenue p# &ith respect to the emp!o"ment

of the factor* i.e. %under pricetain- p`^f/^xi. ith the usua! s"m$o!s8

p*98 i%vi#

This can a!so $e expressed as e#ua!it" $et&een marina! product of the factor* and real renta!

of the factor measured in terms of the product* M4 iLvi/p.

The secondorder condition is that the marina! revenue product must $e decreasin in x i. The

increase in profit if the firm emp!o"s one more sma!! unit of factor i is p`M4 i vi* and if it is positive*

or if it $ecomes positive for further increases of the factor* the firm finds it convenient to expand the

use of the factor* so the optima! !eve! of factor emp!o"ment must $e &here one more unit of the

factor no !oner increases profit and further units &ou!d on!" mae thins &orse.

(arefu!8 there ma" $e no positive so!ution to this maximiation pro$!em' in other &ords* it

ma" happen that no positive va!ue of x i* ho&ever sma!!* avoids a marina! revenue product inferior

to the iven renta!. 7n this case* since it must $e xi I 0* the firm reaches a Ccorner so!utionC &ith xiL0

and p`M4i K vi. 7t is possi$!e* a!thouh a f!ue* that at x iL0 it is p`M4iLvi.

5.!!. $ost minimization

5.11.1. 7f t&o varia$!e factors i and Q are $oth demanded in positive amounts* then p`M4 iLvi

and p`M4 QLv Q imp!" the &e!!no&n condition M4i/M4 QLvi/v Q' ho&ever* this !ast e#ua!it" can $e

satisfied &hen the t&o other ones are not* and this &i!! mean* as &e no& sho&* that a different

pro$!em is $ein so!ved8 cost minimiation.

A necessar" condition for profits to $e maximied is that the tota! cost of producin the profit

maximiin output $e minimied. 4rofit maximiation can $e achieved in t&o steps8 first* for each!eve! of output* minimie cost* and find ho& this minimied cost varies &ith output* i.e. find the

cost function' second* maximie profit $" findin the !eve! of output that maximies the difference

$et&een revenue* and the minimied cost.

The cost f1nction is the minimumva!ue function

c%v*#- L min v4 s.t. f%4-I# .

This &i!! $e a short-r1n cost function if some factors cannot $e varied' usua!!" the other

ones* ca!!ed variable factors* are the so!e factors that are made to appear in 4.Assume there is a uni#ue costminimiin so!ution 4 for each iven %v*#-. @et 4L%v*#- $e the




vector function indicatin ho& the so!ution chanes &ith v and #' %v*#- is a vector function ca!!ed

the conditional in1t demand f1nction. Then c%v*#-Lv%v*#-. The &ord Cconditiona!C derives from

the fact that these are the input demands conditional on the !eve! of output.

There is a strict simi!arit" $et&een the cost function in production theor" and the expenditure

function in consumer theor"F21* and $et&een the firmCs conditiona! input demand function and the

consumerCs compensated %or icsian- demand function8 mathematica!!" the" are Qust the same

thin %this is &h" &e use the same s"m$o! to indicate the conditiona! factor demand function

too-. Therefore &e need not prove the properties &e no& !ist $ecause the proofs are the same as for

the expenditure function* cf. chapter H.

As !on as f%4- is continuous and 4 is such that f%4- is strict!" increasin in a neih$ourhood

of 4* the cost function has the fo!!o&in properties8

%1- c%v*#- is nondecreasin in vi

%2- c%v*#- is homoeneous of deree 1 in v

%3- c%v*#- is continuous in vi* for vWW0.

%H- c%v*#- is strict!" increasin in # as !on as #WW0

%5- c%v*#- is concave in vi.

7f the production function is continuous &e can rep!ace the constraint f%4-I# &ith the

constraint f%4-L#' if it is a!so differentia$!e* for interior so!utions %4WW0- &e can use the @aranianapproach &ith e#ua!it" constraint %the \uhnTucer conditions are the more enera! necessar" first

order conditions-. ormu!atin the pro$!em as one of maximiation of c%v*#-* the @aranian

function is v4 λ%#f%4-- &here # and v are iven%22-. The firstorder conditions for an interior

so!ution "ie!d

vi L λ^f/^xi* iL1*...*n

from &hich one derives the &e!!no&n condition

vi/v Q L M4i/M4 Q .This is interpreta$!e eometrica!!". Assume a!! input !eve!s apart from those of inputs i and Q

to $e iven and to cause a cost ?L_sJi*Q%vsxs-. or each iven tota! cost (* the expression

vixiv Qx QL(? maes x Q a !inear function of xi. This function is ca!!ed a restricted %t&odimensiona!-

21 7ndeed a num$er of economists ca!! cost function the consumers expenditure function.22 7n the consumer maximiation pro$!em &e &rite the constraint in the @aranian function as λ%pxm-*

here &e &rite it as λ%#f%x--8 the difference derives from the fact that* in order to o$tain a nonneative

va!ue for the @arane mu!tip!iers in \uhnTucer theor"* the constraint %the expression in parenthesis- must

$e &ritten in such a &a" that it is constrained to $e nonpositive %if there is a minus sin $efore the l-8 in the

case of the consumer* the constraint is mIpx* in the case of the firm it is f%x-I#. Another &a" of puttin the

thin is* that in the consumer pro$!em a re!axation of the constraint re#uires an increase of m* in the cost

minimiation pro$!em a re!axation of the constraint re#uires a decrease of #.




isocost. 7ts eometrica! representation is a do&n&ards!opin straiht !ine in %x i*x Q-space* &ith

s!ope e#ua! to vi/v Q and intercepts %(?-/vi on the a$scissa and %(?-/v Q on the ordinate axis. 7t is

the !ocus of #uantities emp!o"ed of the t&o factors that cause the same tota! cost (. 7ncreases in (

induce a para!!e! out&ard shift of the isocost !ine. (ost minimiation re#uires that the isocost $e as

c!ose as possi$!e to the oriin under the condition that it has a point in common &ith the iven

iso#uant correspondin to the desired output. 7f the iso#uant is smooth* the condition v i/v Q L

M4i/M4 Q imposes that the isocost $e tanent to the i*Qiso#uant associated &ith #. 7n order for this

tanenc" actua!!" to indicate a point of minimum cost* an" isocost c!oser to the oriin must have no

point in common &ith the iso#uant. This is ensured if the iso#uant is convex.

5.11.2. The partia! ana!o" &ith the uti!it" maximiation pro$!em is raphica!!" c!ear in the

t&ofactors case8 in $oth cases &e have a map of straiht !ines and a map of curves' the difference

is that in order to maximie uti!it" &e !oo for the point* on a iven straiht !ine %the $udet !ine-*

that touches the curve %the indifference curve- farthest from the oriin' &hi!e in order to minimie

cost &e !oo for the point* on a iven curve %the iso#uant-* that touches the straiht !ine %the isocost-

c!osest to the oriin.

x2

isocosts iso#uant

x1 iure 5.H

ith n factors* the iso#uant is a surface of dimension n1 in 9 n* the isocost is a h"perp!ane'

the firstorder conditions for an interior so!ution imp!" tanenc" $et&een iso#uant and isocost* and

the secondorder sufficient condition is that the iso#uant surface $e convex. The thin is raphica!!"

evident &ith t&o factors. As in the <M4* convexit" of the iso#uants ensures that* &hen the first

order conditions are satisfied* the so!ution is a global maximum of (* that is to sa"* a !o$a!minimum of (. Mathematica!!"* this secondorder condition can $e expressed* as in the <M4



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. 2H

pro$!em* as the condition that at the so!ution xk it is d2%(-W0 for disp!acements from xk that satisf"

the constraint f%x-L#N* and it can $e aain sho&n that this corresponds to the condition that the

!eadin* or natura!!" ordered* principa! minors %startin from the third one- of the $ordered

essianF23 a!ternate in sin startin from positive %remem$er that the pro$!em is formu!ated as the

maximi:ation of (F2H-.

or the reasons exp!ained ear!ier* convexit" of the iso#uant surface a!&a"s o$tains &hen (9>

and divisi$i!it" are assumed and a!! factors are varia$!e.

The condition vi/v Q L M4i/M4 Q can $e rea!ied &ithout the conditions M4 iLvi/p* M4 QLv Q/p

$ein rea!ied' &hen so* factor emp!o"ments are not optima!* the output !eve! is not profit

maximiin. (ost minimiation is on!" one part of &hat is necessar" for profit maximiation8 one

must a!so choose the optima! output !eve!. This !atter choice can a!so $e examined in terms of cost

function and revenue function* see $e!o&. ?ut $efore* &e exp!ore the cost function a !itt!e more.

5.!%. A$m& 'u(n)*uc+er conditions and cost minimization& (ep(ard-s lemma.

5.12.1. (ost minimiation has an imp!ication ana!oous to the A4M. 7f for a iven output #

and iven input renta!s v the firm finds it optima! to uti!ie an input vector 4;:%v*#-* it must mean

that an" other input vector capa$!e of producin # %or more- must cost at !east v4;* in other &ords*

v4;Kv4 for a!! 4 such that f%4-I#N* a resu!t sometimes ca!!ed ,eak axiom of cost minimi:ation*/ACm for short. 7f at input prices v; the firm chooses input vector 4; and at input prices v< the

firm chooses input vector 4< to produce the same output * proceedin in the same &a" as for the

A4M one reaches the conc!usion %v; v<-%4; 4<-K0* more often expressed as

OvÒ4K0.

This imp!ies* for examp!e* that if on!" one input price chanes* the demand for that input must

chane in the opposite direction. %=ote that if inputs &ere measured as neative #uantities the

ine#ua!it" sin &ou!d $e reversed* sti!! this is not the same resu!t as &as derived from the A4M* $ecause here output is ept fixed.-

7n enera!* not a!! inputs &i!! $e used in positive amounts $" a firm' the condition v i/v Q L

M4i/M4 Q must ho!d for inputs $oth used in positive #uantities' the more enera! necessar" first

order conditions for cost minimiation are deriva$!e from the \uhnTucer theorem. 7n this case*

&ith #N the iven output* the function to $e maximied is v4 and the constraints are #Nf%4-K0*

and xiK0* iL1*...*n. The firstorder conditions are therefore %mu!tip!"in $oth sides $" 1-8

23 Exercise8 derive the $ordered essian in the t&ofactors case and sho& that its determinant is positive

if and on!" if the ana!oous $ordered essian of the <M4 in the t&ooods case has a positive determinant.2H 7f the pro$!em is formu!ated as the minimiation of (* then the !eadin principa! minors of the $ordered

essian must $e a!! neative.




vi % λ 5f'xi ; λ i ! i%+!###!n! ,ith the complementary slackness condition that if λ i<5!

that is to say! if vi<λ 5f'xi* then xi%5#

The @arane mu!tip!ier λ0 in this pro$!em can $e interpreted throuh an app!ication of the

Enve!ope Theorem. e define the mar7inal cost function M(%v*#- as the derivative of the cost

function &ith respect to output8 M(%v*#-L^c%v*#-/^# . 7t te!!s us $" ho& much tota! cost increases if

the firm increases output $" one %sma!!- unit. =o& !et M $e the va!ue function of the maximiation

pro$!em &ith constraint

( ) $ x f #t # s )vx( xam x

=− '

hence M L c%v*#-' the @aranian function of the maximiation pro$!em is

=vx= λ 5($=f(x))*

and the Enve!ope Theorem imp!ies

9'$% = λ 5* i.e. c'$%λ 58

the agrange multiplier >5 in the cost minimi:ation problem is the marginal cost .

Another &a" to prove this resu!t is the fo!!o&in* &here f i?f(x)'xi8

5

nn++

+n5++5

nn++

+n++

dx f ###dx f

dx f ###dx f )conditionsorder first the from(

dx f ###dx f

dxv###dxv

$

)$ !v( cλ

λ λ =

++

++−=

++

++=

∂

∂.

5.12.2. @et us no& consider aain the function = c( v !$) as the va!ue function of the

maximiation pro$!em &ith constraint* ( ) $ f t s x xam x

=− x v ..-% . @et us app!" the Enve!ope

Theorem to the derivative of this va!ue function &ith respect to factor renta!s' &e o$tain %no& &e

indicate the @arane mu!tip!ier as simp!" λ-8

( ) ( )

i

i

i

i

xv

f x

v

$c−=

∂

∂+−=

∂

∂−

x v

λ *

%$ecause 0-%=

∂

∂

iv

f x

-'

the #uantit" xi that appears in this expression is in fact the conditiona! demand for input i* $ecause it

is the #uantit" of this input demanded &hen output is ept fixed at the !eve! $' so &e o$tain8

,hehard>s %emma for the firm8 )$ !v( xv

)$ !v( ci

i

=∂

∂ .

7n &ords8 2he conditional factor demands are the partial derivatives of the cost function ,ith

respect to factor rentals#

This is the oriina! >hephardCs @emma* &hich &as !ater extended to consumer theor".

5.!. *(e profit function and /otelling0s 1emma




5.13.1. The rofit f1nction π%p*v- is defined as the va!ue function of the %unconstrained-

maximiation pro$!em

max x pf( x )= vx *

&hich ass to find the input vector that maximies profit* as a function of output price and input

renta!s. %The function j%#-Lp#(%#-* profit as a function of output under cost minimiation* is not

ca!!ed profit function.- 7n shortperiod ana!"sis the inputs are the varia$!e ones.

The profit function is on!" defined &hen the condition pLM( "ie!ds a definite optima! output.

This is not the case if the production function has constant returns to sca!e %or if there is perfect

rep!ica$i!it" of p!ants* cf. R5.DD-8 then averae and marina! cost coincide and are constant* and for

a pricetain firm* if pWA(* profits ro& end!ess!" &ith increases in output* so there is no optima!

output* &hi!e if pLA( there is an infinit" of so!utions a!! "ie!din ero profit. Thus the profit

function re#uires sufficiently decreasin returns to the sca!e of varia$!e inputsF 25' this can $e

Qustified in shortperiod ana!"ses $" tain some inputs as fixed' on the contrar"* a long-period

profit function re#uires assumptions on returns to sca!e %a <shaped @A( curve- that not a!!

economists find p!ausi$!e. ?ut precise!" in !onperiod ana!"ses the profit function is irre!evant

even &hen it can $e defined* $ecause entr" &i!! an"&a" maintain profits e#ua! to ero* so the

determination of e#ui!i$rium industr" output does not re#uire consideration of the profit function* as

&e exp!ain !ater. As to the short-period profit function* it is $ased on a rather mis!eadin definitionof profit as revenue minus varia$!e cost* ne!ectin the need to inc!ude amon costs the #uasirents

of fixed factors as opportunit" costs. These #uasirents* if inc!uded in tota! cost and if the

entrepreneur is neither $etter nor &orse than other entrepreneurs at maximiin profit* &ou!d

a!&a"s $rin profit to ero even in shortperiod ana!"sis $ecause the" are the opportunit" cost of

usin the fixed p!ant instead of rentin it out to other entrepreneurs %these &ou!d $e read" to pa" for

the use of the fixed p!ant a maximum amount e#ua! precise!" to &hat &ou!d $rin do&n their profit

to ero-. Therefore the profit of the shortperiod profit function is the sum of true profit and of the#uasirent to $e attri$uted to the fixed factors. As a conse#uence* it ma" $e the case that this profit

is positive $ut the entrepreneur &ou!d do $etter to se!! the firm $ecause other entrepreneurs &ou!d

et more out of that set of fixed factors and so the" va!ue the fixed factors more than she does.

o&ever* the profit function is &ide!" used in microeconometric practice* so &e !ist its main

properties8

25 Exercise 5.%2 h" the stress on sufficiently/ D 7s decreasin returns to sca!e &ithout #ua!ifications a

sufficient condition for the existence of a profit functionD Tr" exp!orin the case in &hich (%#-

as"mptotica!!" approaches from $e!o& a function a$#* &ith a*$W0.




Properties of t(e profit function. Suppose that the production function f@ Rn;AR; is

continuous! strictly increasing and strictly $uasiconcave and that the profit function π (p!v)! i#e# the

value function of the problem max x pf(x)=vx! is ,ell defined for given (pB!vB) and continuous in (p!v)

in a neighbourhood of (pB!vB)4 then in that neighbourhood π (p!v) is

(+) increasing in p

() decreasing in v

() homogeneous of degree one in (p!v)

(E) convex in (p!v)

(F) differentiable in (p!v)<<5#

49::8 e on!" prove the !east intuitive of these properties* convexit"* i.e. that for an" sca!ar a such

that 5GaG+ it is aπ (p!v);(+=a)π (pB!vB)H π ≈(+=a)pB!av;(+=a)vB. ;efine paap%1a-pC and vaav%1a-vC.

@et x* xC* xa and #Lf%x-* #CLf%xC-* #aLf%xa- $e the so!ution inputs and outputs associated respective!" &ith

π%p*v-* π%pC*vC- and π%pa*va-. Then it is

π%p*v- L p#vx I p#a vxa

π%pC*vC- L pC#CvCxC I pC#a vCxa

&hich imp!" aπ%p*v-%1a-π%pC*vC- I a%p#a vxa-%1a-%pC#a vCxa- L pa#a vaxa L π%pa*va-. [

5.13.2. =o& !et us app!" the Enve!ope Theorem to the profit function. The !atter is the va!ue

function of a maximiation pro$!em &ithout constraints* so &e o$tain8

π 'p % f(x)%$ i#e# the partial derivative of the profit function ,ith respect to the output price

yields the optimal output4

π 'vi % =xi i#e# the partial derivative of the profit function ,ith respect to input rental vi

yields the (unconditional) demand for input i measured as a negative number (i#e# in accord ,ith

the netput notation)#

These t&o resu!ts constitute Hotellin7>s %emma %sometimes ca!!ed the ;erivative 4ropert"-.

Thus the partia! derivatives of the profit function* &hen the !atter exists* ive us the output

supp!" function and %the neative of- the input demand functionsF26. >ince the profit function is

convex* &e reach the resu!t that* &hen the profit function is &e!! defined* $oth π 'p and π 'vi are

nondecreasin and enera!!" increasin in p* respective!" in vi* that is to sa"8

supp!" is a nondecreasin* and enera!!" an increasin* function of output price %$ecause

convexit" of imp!ies π 'p is nondecreasin and enera!!" increasin in p* $ut π 'p%$-'

the unconditiona! demand for an input* &hen measured as a positive #uantit"* is a non

increasin* and enera!!" a decreasin* function of the input o&n renta!8 there are no Giffen inputs.

26 Exercise8 4rove that if the firm is a mu!tiproduct one and profit depends on inputs and outputs* then

ote!!ins @emma* aain derived from the Enve!ope Theorem* enera!ies to e#ua!it" $et&een the partia!

derivative of the profit function re!ative to an" one output price* and supp!" function of that output.




5.!3. $onditional and unconditional factor demands, inferior inputs, rival inputs,

su4stitution effect and output effect.

5.1H.1. hen the profit function is &e!! defined* for each output price p and vector of factor

renta!s v there is an optima! output and an associated vector 4 of optima! factor uti!iations %the

!atter need not $e uni#ue* $ut 7 &i!! assume it is-. 7n this case &e can define the s1l f1nction +of

the individ1al firm >%v*p- that indicates ho& optima! output chanes &ith factor prices v and the

output price p* and the %vectoria!- 1nconditional factor demand f1nction 4%v*p- that indicates the

associated optima! factor emp!o"ments' it is

4%v*p-L%v*>%v*p--.

7f the profit function is defined for the short period* i.e. &ith some inputs fixed* then on!" the

varia$!e inputs appear in 4 and in v. hen the profit function does not exist* the output supp!"

function and the input demand functions do not exist either8 no uni#ue profitmaximiin output

exists. hen these functions can $e defined* &hat a$out the sin of their partia! derivativesD

7f profit is considered a function of #* its maximiation re#uires so!vin the pro$!em

max$ p$=C( v !$)*

&hose firstorder necessar" condition is the e#ua!it" of product price p* and marina! cost

M(%v*#-8L^(/^#' the secondorder sufficient condition is that M( must $e risin at the optima! #.ence

^>%v*p-/^pW0

the %inverse- s1l c1rve %optima! # as a function of p* &ith p in ordinate and # in a$scissa- is

increasin %as !on as to increase output is possi$!e-* a resu!t actua!!" a!read" reached via

ote!!ins @emma p!us the convexit" of the profit function' $ut it can $e usefu! to see the same

resu!t from different perspectives.

e cannot reach a resu!t on the sin of ^x i%v*p-/^vi direct!" from the condition v iLp`M4i* $ecause xi is not the on!" input use that &i!! chane &hen v i chanes' $ut ote!!ins @emma and

the convexit" of the profit function imp!" ^xi%v*p-/^viK08 the o&nrenta! effect is neative.

5.1H.2. ;oes the a$ove resu!t on input use imp!" ^>%v*p-/^vi K 0 D in other &ords* is it a!&a"s

the case that the optima! output* &hen it exists* decreases if the price of a factor %in positive use-

risesD 4erhaps surprisin!"* not a!&a"s. The reason is that the factor miht $e an inferior input *

defined as an input &hose conditiona! demand fa!!s as output increases* that is* such that^xi%v*#-/^#0 %at !east at the iven input renta!s and in a neih$ourhood of the initia! #-. 7t is indeed




possi$!e that an increase of # is $est achieved $" increasin some input and decreasing some other

input.

p#

x Q tota! cost*

expansion path revenue

c

c

xi #

i. 5.H$is%a- i. 5.H$is%$-

This possi$i!it" is sho&n in i. 5.H$is%a-* &here* assumin t&o varia$!e factors &ith iven

prices* severa! iso#uants are dra&n and for each one the tanenc" &ith an isocost is sho&n' the

!ocus of tanenc" points is ca!!ed the expansion path of the firm for iven input prices' this

expansion path can exhi$it a decrease of the uti!iation of one factor as hiher iso#uants are

reached. An examp!e of inferior input can $e !a$our in some aricu!tura! productions on a iven

specia!ied !and &here* as !on as the re#uired output is not !are* it can $e optima! to achieve it

&ith a$undant use of !a$our near!" unassisted $" machiner"* $ut &hen output $ecomes !are* it

$ecomes convenient to mechanie production &hich reduces the needed amount of !a$our. Another

examp!e can $e the use of fue! in a chemica! production process %carried out in a fixed p!ant- that

re#uires heat* such that &hen the process is run on a sma!! sca!e* heat must $e provided $" f!ames

under the cau!dron* &hich means consumption of fue!* $ut the reater the sca!e on &hich the

process is run* the more heat is produced $" the chemica! reaction itse!f* and the need for fue!

consumption decreases8 fue! is an inferior inputF2)

.The re!evance of inferior inputs for the sin of ^>%v*p-/^vi comes from the fact that !oca!!" the

increase in the rental of an inferior input shifts the marginal cost curve do,n,ards* so the pLM(

condition is achieved for a greater $ a!thouh at a sma!!er profit* as in i. 5.H$is%$- &here the

increase in the price of an inferior factor shifts the cost curve from c to c. This means that

S( v !p)'vi G 5 only if factor i is not inferior .

The proof that the marina! cost curve shifts do&n&ards* that is* ^M(/^v i0 if input i is

2) Exercise8 suppose the chemica! process in the text produces output # &ith chemica! input x and as "*

accordin to the production function #Lf%x*"- &ith #Lx1/2 if "I1/x* #L0 if "1/x. Assume pxL1/3* p"L3* and

fixed cost %due to the presence of fixed machiner"- is 2. (onfirm that " is an inferior input' derive the cost

function and prove that minimum average cost is reached for #L3.




inferior* is an interestin app!ication of the notions studied thus far. The cost function is assumed

t&ice continuous!" differentia$!e* so secondorder cross partia! derivatives coincide.

8roof . @et input i $e inferior' differentiate $oth sides of x i%v*p-Li%v*>%v*p-- &ith respect to p* and

app!" >hephardCs @emma and the coincidence of cross partia!s to o$tain8

.-*%-*%-*%-*%-*%-*%-*%-*%

p

pvS

v

9C

p

pvS

$

$vc

v p

pvS

v

$vc

$ p

pvS

$

$v J

p

p x

iii

ii

∂

∂⋅

∂

∂=

∂

∂⋅

∂

∂

∂

∂=

∂

∂⋅

∂

∂

∂

∂=

∂

∂⋅

∂

∂=

∂

∂ v

>ince p

- p*v%>

∂

∂W 0 and &e are assumin that xi is inferior* the conditiona! demand for factor i decreases

&hen output increases &ith iven input prices* so it is p

- p*%x i

∂

∂ v

0* hence it must $eiv

M(

∂

∂ 0. :r

a!so* the a$ove sho&s thati

i

v

9C

$

$v J

∂

∂=

∂

∂ -*%* and the !efthand side is 0 $" the definition of inferior

input.[

7 &i!! not o into the intricacies of inferior inputs except to prove that &hen the production

function is differentia$!e a necessar" %$ut not sufficient- condition for an input xi to $e inferior is

that it $e rival of some other input x Q* &hich means that an increase in x Q decreases the marina!

product of xiF2+' this a$ove a!! as a &a" to introduce riva! inputs* that have some riht to attention

on their o&n and are $rief!" discussed in the next pararaph.

8roof . @et ? L 1/p ` v $e the vector of rea! factor renta!s in terms of the firmCs output' f% 4-L# is the

firmCs production function. Assume that marina! cost is !oca!!" strict!" increasin so the profit function j% ?-

is &e!! defined at the optima! #. Maximiation of j%?- determines the unconditiona! factor demands 4%?-.

These satisf" the &e!!no&n condition &i L ^f%4-/^xi* a!! i* that for $revit" 7 indicate as

;4f%4- L ?*

&here ;4 means the vector of partia! derivatives of the su$se#uent function &ith respect to the varia$!es in 4.

@et us assume that 4%?- is inverti$!e* i.e. to each vector 4 there corresponds a uni#ue vector ?. @et ?%4%?--

$e this inverse functionF2,. >ince ? L ;4f%4-* it is ;4%?%4%?-- L ;42f%4%?--. And $" the derivative ru!e for

inverse vectoria! functions* the aco$ian ;4%?%4%?-- is the inverse of the aco$ian ;?4%?-8

;4%?%4%?-- L F;?4%?- 1

L ;4

2

f%4%?--.?ut then* invertin ever"thin8

;?4%?- L F;4%?%4%?-- 1 L F;42f%4%?-- 1.

=o& &e use a resu!t in the theor" of matrices %cf. e.. Taa"ama* 9athematical .conomics* 1,)3* p.

3,3* Theorem H.;.3- that states that if the offdiaona! e!ements of a s#uare nonsinu!ar matrix are a!! non

2+ 9iva!r" can arise* for examp!e* if there is some third input x h* &hose services cooperate &ith either xi

or x Q* and such that &hen x Q increases it is convenient to a!!ocate x hs services to cooperate main!" &ith x Q' or*

one input can have direct neative side effects on the efficienc" of another input* for examp!e o&in to

chemica! interactions ferti!iers miht decrease the marina! productivit" of pesticides in fruit production. :f

course &hen the production function is t&ice continuous!" differentia$!e the cross partia! derivatives

coincide so riva!r" is reciproca!.2, As 7 have used 4%?- to represent the function that maes 4 depend on ?* its inverse shou!d $e actua!!"

represented as ?L4 1%4-.




neative then its inverse is nonpositive. 7f no inputs are riva!* the marina! product of no input decreases

&hen some other input is increased* so the essian of the production function has nonneative offdiaona!

e!ements* hence its inverse is nonpositive* and therefore ;?4%?- is nonpositive.

=o& fix the product price p so &e can return from ? to v. 7t is ^>%v*p-/^viL ∑ ∂

∂

∂

∂

0 i

0

0 v

pv x

x

x f

-

-*%-%

%

0 $ecause &e have sho&n that the second terms in the parentheses are nonpositive. Therefore a$sence of

riva! inputs imp!ies that supp!" cannot increase &hen an input price increases. [

5.1H.3. The possi$i!it" of rivalry amon inputs has some re!evance for the marina!ist or

neoc!assica! approach to income distri$ution. As exp!ained in (hapter 3* accordin to this approach

each factor tends to earn* in the !on period* a renta! e#ua! to %the va!ue of- its fu!!emp!o"ment

marina! product. 7n the !on period* competitive industries $ehave !ie firms &ith (9> production

functions* and this prevents riva!r" if factors are on!" t&o* as 7 prove $e!o&' $ut if factors are more

than t&o* an increase in the e#ui!i$rium use of a factor $ecause of an increase in its supp!" can

cause a decrease of the fu!!emp!o"ment marina! product of a riva! factor in fixed supp!"8 thus the

marina! approach does not exc!ude the possi$i!it" that an increase in the supp!" of a factor causes a

decrease of the e#ui!i$rium renta! of another factor. o&ever* cases of riva!r" appear rare and

specific to certain industries* so marina!ist/neoc!assica! economists unanimous!" consider the

!ie!ihood of the decrease of e#ui!i$rium renta! Qust i!!ustrated to $e ero for factors* !ie most t"pes

of !a$our* the demand for &hich comes from ver" man" industries.

8roof that ,ith t,o factors and CRS there cannot be rivalry# 7f factors are on!" t&o* ca!! them x and "*

since marina! products on!" depend on the proportion x/" o&in to (9>F 30* if an increase of x &ith " fixed

causes a decrease of M4" it must mean that an increase of " &ith x fixed causes an increase of M4 "* and this

means a nonconvexit" of output as a function of " &ith x fixed' $ut such a nonconvexit" is exc!uded $"

profit maximiation p!us (9>8 in i. DD* &here the curve represents output as a function of " &ith factor x

fixed* a!! points on the sement A? can $e reached $" a !inear com$ination of the factor vectors %" A*x- and

%"?*x-' this enera!ies &hat &as exp!ained in (hapter 3 on the difference $et&een techno!oica! and

economic tota! productivit" functions or iso#uants* and &e conc!ude that &ith (9> and t&o factors*

increasin marina! products are impossi$!e. This exc!udes riva!r". [

output

?

A

30 (f. footnote 11.




"A "?

i. DD

The possi$i!it" Qust mentioned of a decrease of the e#ui!i$rium renta! of a factor &hen supp!"

of another factor increases is due to riva!r"* not inferiorit"8 an input is inferior if the demand for it

decreases &hen output increases at given factor renta!s* $ut in this case &ith (9> a!! inputs increase

in the same proportion as output* and 7 have arued that constant returns to sca!e industries is the

on!" !eitimate assumption for the !onperiod $ehaviour of competitive industries &ith possi$i!it"

of p!ant rep!ication* &hi!e inferior inputs re#uire nonconstant returns to sca!eF31. =o&* the

determination of e#ui!i$rium factor renta!s is a !onperiod issue* o&in to the comp!ex* time

consumin adQustments %chanes in outputs* shifts of !a$ourers across firms* etc.- re#uired for

e#ui!i$rium to $e approached on factor marets' therefore input inferiorit"* different!" from riva!r"*

is irre!evant for the marina!ist theor" of income distri$utionF32.

7nput riva!r" can a!so cause perverse effects of shifts in the composition of demand for

consumption oods on e#ui!i$rium factor renta!s. As i!!ustrated in (h. 3* if one !eaves aside possi$!e

perverse income effects then a shift in the composition of demand in favour of oods that use a

factor in a hiherthanaverae proportion tends to raise that factors e#ui!i$rium renta! if technica!

coefficients are fixed* and if there is technica! su$stituta$i!it" the effect on the factor renta! is

norma!!" considered to $e of the same sin* on!" &eaer. ?ut if a factor is specia!ied and used on!"

$" one industr" and is riva! of other inputs in that industr"* then &hen demand for the industr"s

product rises the industr" increases the use of other inputs in order to satisf" the increased demand

and this can cause a decrease of the marina! product of the specia!ied factor and hence a decreaseof the demand for it if its renta! remains the same8 the excess supp!" of the factor &i!! then cause the

renta! of the factor to decrease' so it is not impossi$!e that a rise in the e#ui!i$rium output of the

industr"* a!thouh associated &ith a hiher output price* $e associated &ith a !o&er e#ui!i$rium

renta! of the specia!ied factor. This possi$i!it" !oos exceptiona!* $ut it cannot $e exc!uded.

31 7n the Exercise in footnote 2)DD the industr" &ou!d increase output $" increasin the num$er of optima!

p!ants %o&ned $" the same firms* or $" ne&!" formed firms-* each one producin 3 units* thus at the industr"

!eve! the inferiorit" of factor " disappears. 7ndeed constant returns to sca!e imp!" that expansion paths are

ra"s from the oriin.32 Except possi$!" for specia!ied inputs to $e associated &ith other specia!ied inputs* e.. specia! ferti!iers

to $e used on ver" specia! !ands for the production of specific products.




5.1H.H. ina!!"* &e consider ^xi%v*p-/^v Q. 7t shou!d $e intuitive that its sin is am$iuous8 if

factor Q is not inferior* optima! output decreases &hen v Q increases* and this &ou!d reduce the

demand for xi if factor proportions &ere fixed* on the contrar" these proportions chane %and in

directions that depend on the specific production function- $oth $ecause optima! factor proportions

chane &ith # even &ith fixed re!ative factor prices* cf. i. 5.H$is%a-* and $ecause re!ative factor

prices chane and this a!ters optima! factor proportions for each !eve! of #. The possi$i!it" that

factor Q $e inferior adds further uncertaint" $" renderin the sin of the chane in optima! #

uncertain. A!! this can $e ana!"ed more forma!!" %$ut &ithout reat ains in c!arification-.

;ifferentiatin 4%v*p-L%v*#%v*p-- &ith respect to v Q* one ets

^xi%v*p-/^v Q L î%v*#-/^v Q Q

i

v

- p*v%>

#

-#*v%.

∂

∂⋅∂

∂L %cross substitution effect - %output

effect -.

This sho&s that the direction of chane of xi is the composition of t&o directions of chane8

a!on the oriina! iso#uant as indicated $" the first term on the rihthand side* that indicates the

chane in xi if output &ere ept fixed' and a!on the expansion path o&in to the chane in # at

iven factor renta!s* as indicated $" the second term. ?oth directions are of uncertain sin. The sin

of ^>/^v Q depends on &hether input Q is inferior' the sin of#

-#*v%.i

∂

∂ depends on &hether input i

is inferior. ?ut even assumin that neither input is inferior %so the sin of Q

i

v

- p*v%>

#

-#*v%.

∂

∂⋅

∂

∂ is

neative-* sti!! the sin of ^x i%v*p-/^v Q for QJi is uncertain* $ecause the sin of the cross su$stitution

effect î%v*#-/^v Q is uncertain as !on as there are more than t&o factors. The reason is that &hen v Q

rises and hence x Q%v*p- decreases* the chanes in other inputs are affected $" t&o forces8 first* it is

necessar" to restore # to its iven !eve!* and this &ou!d tend to increase the demand for the inputs

other than x Q' $ut second* the optima! proportions amon the inputs other than x Q &i!! chane in

favour of the factors &hose marina! product re!ative to the other marina! products rises &hen x Q

decreases8 no&* it is possi$!e that the decrease of x Q affects the marina! product of x i stron!" and

neative!"* so much so that the optima! proportion amon the factors other than x Q chanes aainst

xi so much that the compensated demand for xi decreases.

5.!5. Elasticity of su4stitution.

(ost minimiation re#uires that firms &hich intend to produce a iven output !ocate




themse!ves on the point of the correspondin %convex- iso#uant &here the a$so!ute s!ope e#ua!s the

ratio $et&een factor renta!s. A chane in the re!ative renta! of t&o factors ma" induce no

su$stitution* !itt!e su$stitution* extensive su$stitution. or examp!e* if factors are perfect

comp!ements then iso#uants %&ith t&o factors- are @shaped and costminimiin firms &i!! a!&a"s

!ocate themse!ves at the in8 chanes in re!ative factor prices induce no su$stitution. 7n order to

measure the effect of chanes in re!ative factor renta!s on the proportions in &hich firms find it

optima! to com$ine t&o factors* economists use the elasticit of s1bstit1tion. e !imit ourse!ves to

the t&ofactors case. e have a!read" met this notion in chapter H. hen app!ied to production* this

e!asticit" is the a$so!ute va!ue of the ratio of the percentae chane in the factor proportion x 1/x2

%a!on a iven iso#uant- to the percentae chane of their T9>* i.e. M41/M428

21

21

21

21

21

/M4M4

-/M4M4%d

x/x

-x/x%d

=σ L 21

21

21

21

x/x

v/v

-v/v%d

-x/x%d⋅ .

The second of the a$ove t&o expressions for the e!asticit" of su$stitution is $ased on the

assumption that the firm chooses the factor proportion that satisfies T9> 21Lv1/v2' thus the e!asticit"

of su$stitution measures the sensitivit"* of the proportion in &hich factors are demanded* to re!ative

factor renta!s' its usefu!ness !ies a$ove a!! in that it ives an indication of &hat happens to the

relative shares of factors in tota! cost as re!ative factor renta!s var". The re!ative share of factor 1 in

tota! cost is iven $" %v1x1-/%v2x2-* &hich can $e re&ritten as %x1/x2-`%v1/v2- or %x1/x2-`T9>21. hen

v1/v2 increases* x1/x2 decreases' an e!asticit" of su$stitution e#ua! to 1 means that the t&o chanes

neutra!ie each other and re!ative factor shares in tota! cost do not chane. An e!asticit" of

su$stitution less than + means that &hen factor 1 $ecomes re!ative!" more expensive* x1/x2

decreases !ess than in proportion* so the re!ative share of factor 1 in cost increases.

This resu!t is used in oneood* t&ofactor enera! e#ui!i$rium neoc!assica! mode!s to derive

predictions on factor shares from the e!asticit" of su$stitution. 7n these mode!s the econom"

produces &ith a (9> production function* and income distri$ution is determined $" fu!!

emp!o"ment marina! products* hence the product exhaustion theorem ho!ds %cf. footnote 11-* and

tota! factor cost is a!so tota! revenue. (hanes in factor supp!ies &i!! then a!ter factor shares in a

direction that depends on the e!asticit" of su$stitution. Thus assume the factors are !a$our and !and*

riid!" supp!ies and fu!!" emp!o"ed. >uppose !a$our immiration raises !a$our supp!"* causin the

rea! &ae to decrease re!ative to !and rent. (hec "our understandin of the issues &ith the

fo!!o&in #uestions. 7f the e!asticit" of su$stitution is !ess than one* the percentae decrease of the




ratio of &ae to rent* needed to ensure the fu!! emp!o"ment of the increased supp!" of !a$our* &i!!

have to $e reater or sma!!er than the percentae increase in !a$our supp!"D and the share of &aes

in nationa! income &i!! decrease or increaseD The correct ans&ers are in a footnote on the next pae.

%The popu!arit" of the (onstantE!asticit"of>u$stitution %(E>- production function amon

macroeconometricians derived from the c!aim that the share of !a$our in nationa! income did not

chane much for man" decades in the <>A. The empirica! evidence &as ho&ever immediate!"

disputed* and certain!" !oos much !ess convincin no&* $ecause after the 1,+0s the share of &aes

has decreased considera$!"' furthermore there are formida$!e areation pro$!ems $ehind an"

attempt to stud" an econom" as if it &ere producin a sin!e output' a!so* it is tota!!" unc!ear &h"

technica! proress shou!d not a!ter the e!asticit" of su$stitution over the decades' and !ast $ut not

!east* the va!idit" of the marina!/neoc!assica! approach to income distri$ution can $e disputed &ith

stron aruments* as &i!! $e exp!ained in !ater chapters.-

5.!. 6ntegra4ility of conditional factor demands

e touch ver" $rief!" on the dua!it" $et&een some of the notions exp!ained in this chapter.

e have seen that cost function and conditiona! factor demands stand to the production function in

exact!" the same re!ationship as expenditure function and icsian %or compensated- consumer

demands stand to the uti!it" function. Therefore the resu!t reached in consumer theor"* that theexpenditure function or the icsian consumer demands a!!o& the reconstruction of the uti!it"

function %more precise!"* of its convexification-* a!so ho!ds for production theor"8 the cost function

contains the same economica!!" re!evant information as the production function* and from it one

can recover the %convexified- iso#uants. :f course this is on!" possi$!e if the chosen function rea!!"

is a cost function* i.e. if there exists a production function that enerates it' the conditions

uaranteein it are !isted in the fo!!o&in proposition %&e omit the proof* cf. Sarian* 1,,2* p. +5-8

et c(v!$) be a differentiable function ,hich is

(i) non-negative if (v!$) is non-negative!

(ii) non-decreasing in (v!$)!

(iii) concave in v! and

(iv) satisfying homogeneity of degree + in v 4

then c(v!$) is the cost function of a production function#

7t can $e convenient* in app!ied &or* to start direct!" from a cost function rather than from a




production functionF33.

As to conditiona! factor demands* the" can $e CinteratedC to "ie!d the production function that

enerated them* &ith the same procedure that derives the uti!it" function from a s"stem of

compensated demands for consumption oods.

5.!7. unctional separa4ility2 1eontief separa4ility and :ea+ly separa4le utility;. >uppose

&e can separate inputs into t&o su$vectors* x %(x+ !###!xm ) and y%(y+ !###!yn )* respective!" &ith prices v

and :* and that the production function satisfies the fo!!o&in condition8 f% 4*-Lf%%4-*-* &here %`-

has the characteristics of a production function8 it is as if inputs 4 produced a sin!e intermediate

ood &hich then produces the fina! output in com$ination &ith inputs . This can ref!ect a true

production of an intermediate ood* or $e simp!" a propert" of the production function.

7f %4- is differentia$!e and f%*- is a!so differentia$!e* then f'xi%(f':)*(:'xi )' as a resu!t*

the %eontief ?ea@ searabilit condition ho!ds8 the marina! rate of su$stitution $et&een an" t&o

xoods is independent of the amounts of yoods8

9RS x0!xi % = (f'xi )'(f'x 0 ) % = (:'xi )'(:'x 0 ).

Siceversa if the @eontief &ea separa$i!it" condition ho!ds* then a differentia$!e f( x ! y ) can $e

&ritten as f(:( x )! y ) &here :( x ) is a sca!ar function. %e omit the proof.-

<nder &ea separa$i!it"* the firm can adopt a t&ostae costminimiation procedure8 it canfirst determine the costminimiin input com$ination of the xinputs for each !eve! of * and the

resu!tin cost of ' and then it can determine the costminimiin input com$ination of %* - for

each !eve! of output. 7f f%`- has constant returns to sca!e* so does %`-' then the cost function for the

ood can $e &ritten as %v-* &ith %v- representin the unit price of .

The production function must $e additively separa$!e if one &ants that not on!" the marina!

rate of substitution $et&een t&o inputs* $ut a!so the marina! product of an input* $e independent of

the amounts of other inputs. 7t is not eas" to thin of rea!istic examp!es to &hich such anassumption miht app!"* except &hen a firm uses ph"sica!!" separate processes that produce the

same ood usin inputs of different #ua!it"* and one treats the inputs of each process as a sin!e

input $ecause in each process the" are com$ined in fixed proportions8 this miht perhaps $e the

case for some aricu!tura! or minera! product produced on !ands* or $" mines* of different #ua!it".

5.!8. upply curves2 (ort)period Mars(allian analysis.

33 Ans&ers to the #uestions posed on the previous pae8 reater* decrease. =ote that the decrease in the share

of &aes in nationa! income does not imp!" a decrease of the total ,age bill * $ecause the increased supp!" of

!a$our raises tota! output* it is therefore possi$!e that tota! &ae pa"ments increase* $ut $" a !o&er

percentae than the increase of tota! output so that the share of &aes decreases.






production function is ca!!ed the short-eriod$ or restricted$ rod1ction f1nction. (ost

minimiation can on!" operate on the varia$!e inputs* and the shortperiod cost function resu!ts from

the minimiation of the cost of varia$!e inputs.

(orrespondin!"* there &i!! $e fixed costs and varia$!e costs. ariable cost is the tota! cost of

varia$!e inputs. The costs that are fixed %in the sense of not dependin on the #uantit" produced- $ut

on!" exist as !on as the firm exists* i.e. disappear if the firm is c!osed do&n* are ca!!ed 31asi-fi4ed

costs. Fi4ed costs proper are those costs independent of the #uantit" produced* that must $e $orne

$" the o&ners of the firm even if production is discontinued and the firm is c!osed do&n. ixed

costs are due to irrevoca$!e contracts that o$!ie the o&ners of the firm to pa" them in a!! instances*

e.. the repa"ment of de$ts. ixed costs proper do not inc!ude* for examp!e* those overhead !a$our

costs %manaerCs secretar" and ana!oous accountin !a$our etc.- independent of the #uantit"

produced $ut &hich can $e e!iminated $" c!osin do&n the firm and firin a!! &orers. ixed costs

do not necessari!" coincide &ith the cost of fixed factors. A firm miht have a de$t to $e repaid* that

causes a fixed cost $ut is due to past expenses and has no connection &ith the firms present fixed

p!ant.

or the #uestion &hether the firm shou!d c!ose do&n &hen profit is neative* fixed cost

proper shou!d not mae a difference since the o&ners of the firm must sti!! $ear it even if the firm

c!oses do&n' #uasifixed cost on the contrar" does mae a difference and therefore it must $einc!uded in the varia$!e cost. ?ut for simp!icit" in &hat fo!!o&s there are no #uasifixed costs.

The %short-eriod variable cost f1nction* to $e indicated as S(%v*#-* resu!ts from the

choice of varia$!e inputs that minimies varia$!e cost for each assined !eve! of output.

7t is p!ausi$!e that* since there are fixed factors* the shortperiod production function &i!!

exhi$it decreasin returns to sca!e at !east after a certain !eve! of output' as a resu!t* at !east $e"ond

a certain !eve! of output S(%v*#- &i!! increase more than in proportion &ith output. This is

forma!ied $" assumin that the short-eriod mar7inal cost* i.e. the derivative of varia$!e cost&ith respect to output*

M(%v*#-8L^S(/^#

is an increasin function of # at !east $e"ond a certain !eve! of output. 7n &hat fo!!o&s 7 tae v as

iven and for $revit" 7 often drop the indication of the functiona! dependence on #. @et us no&

define +short-eriod avera7e variable cost as

AS(%v*#- S(/#'

it is possi$!e that initia!!" AS( is a decreasin function of output* indicatin that the fixed p!ant &as p!anned to $e optima! for a certain !eve! of production and up to that !eve! varia$!e cost increases




!ess than in proportion &ith output. 7f a!! these manitudes are continuous functions of otuput &e

can stud" the re!ationship amon them. or #L0 it is AS(LM( $ecause for the first sma!! unit of

output averae varia$!e cost coincides &ith the increase in cost. Then if AS( is initia!!" a

decreasin function of #* M( is a decreasin function of # too and is !ess than AS( %in order for the

averae cost to decrease* the additiona! units of output must cause an additiona! cost !o&er than the

averae-. AS( remains a decreasin function of output as !on as M(AS(. ?ut if M( $ecomes an

increasin function of # at !east from a certain !eve! of output on&ards* then sooner or !ater it

$ecomes e#ua! to AS(* and from that !eve! of output on&ards M(WAS( and AS( $ecomes an

increasin function of # %$ecause the additiona! units of output cause an additiona! cost reater than

the averae-. 7t fo!!o&s that AS( reaches its minimum &here its curve crosses the M( curve' if one

no&s the t&o functions* this minimum can $e determined simp!" $" so!vin M(LAS( for #W0.

o&ever* if M( is increasin from the ver" start* then the minimum AS( is reached for #L0. %The

mathematica! proof of these statements is eas" and !eft to the reader as Exercise' $e sure to chec

the secondorder conditions.-

=o& define short-eriod avera7e +total cost as A(L%(S(-/#LA(AS(. A( is

avera7e fi4ed cost* defined as (/#. 7f (W0 then A(WAS(' the vertica! distance $et&een the A(

curve and the AS( curve decreases as # increases* $ecause it measures A(. or the same reason as

for AS(* A( is a decreasin function of # as !on as it is reater than M(* and an increasinfunction of # as !on as it is sma!!er than M(* and as a resu!t it too reaches a minimum &here it

crosses the M( curve. A!! these re!ationships are sho&n in i. 5.58 a particu!ar!" important point is

* &here the A( and the M( curve cross each other' this point determines the minimum average

cost * MinA(* associated &ith the iven fixed p!ant and the iven factor renta!s* and the

correspondin #uantit" of output #g. As !on as the price at &hich the firm se!!s its output is reater

than MinA(* the firm maes a positive profit.

o& does the firm maximie profit in the short periodD ?" e#ua!iin marina! cost andoutput price. This can $e sho&n as fo!!o&s. @et 9Lp# stand for the firmCs revenue* and >(%#- for its

shortperiod tota! cost function* &hose derivative &ith respect to # is the marina! cost M(%#-. Then

πL9>(%#-Lp#>(%#-

and the firstorder condition for a maximum is pM(%#-L0. The secondorder condition is

dM(%#-/d#0*

i.e. M( must $e increasin &here it e#ua!s the iven output price.

The condition pLM(%#- imp!ies a s1l c1rve of the firm &hich coincides &ith part of theM( curve %except that no& the independent varia$!e is the one on the vertica! axis- if on the vertica!



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. H0

axis &e a!so measure the output price. The part of the M( curve &hich coincides &ith the supp!"

curve is the part a$ove the AS( curve. The reason is that the firm finds it convenient to o on

producin even if pA(* as !on as pWAS(* $ecause in this &a" it can at !east minimie its !oss8 the

excess of revenue over varia$!e cost compensates at !east partia!!" for the fixed cost &hich must $e

$orne an"&a". ?ut if pAS( then the firm minimies its !oss $" not producin at a!!%35-.

35 7n concrete situations* a firm ma" decide to o on producin even &hen the price is $e!o& minimum

averae varia$!e cost* if it esteems that this is a temporar" situation and that the interruption &ou!d damae

profit more than continuin to produce at a !oss for a time.




p

M(LMS(

A(

AS(

pkLMinA(

A(

: #g #

i. 5.5. Averae %shortperiod- cost A( and averae varia$!e cost AS( &hen marina! cost M( isinitia!!" decreasin' averae fixed cost A( is a rectanu!ar h"per$o!a and e#ua!s A(AS(. =ote that if

AS( inc!uded some #uasifixed costs then the AS( curve &ou!d not start at the same !eve! as the M( curve*

$ut &ou!d have initia!!" a shape simi!ar to that of the A( curve.




5.1+.3. The idea that a firm &i!! continue to operate even &hen main a !oss* as !on as

revenue covers at !east the varia$!e costs* can derive from considerin fixed cost proper a sun cost*

to $e paid &hether one continues to produce or not* for examp!e interest on the de$t contracted to

start production' $ut this ind of exp!anation is endanered $" the varia$!e confines of the firm as a

!ea! entit"* for examp!e the uncertain existence of fixed costs proper in the presence of !imited

!ia$i!it". A c!earer theor" is o$tained if one reformu!ates the #uestion as reardin* not the surviva!

of the firm* $ut rather the surviva! of a fixed p!ant. The e" is to consider the cost of usin a fixed

p!ant a residua!!" determined $uasirent . The ana!o" &ith !and is c!arificator". 7maine that

someone $u"s a !and &ith $orro&ed mone" in order to rent it to firms* and then discovers that the

rent she can earn is insufficient to repa" the interest on the de$t' this is no reason not to rent the !and

out to firms8 as !on as rent is positive* it is not convenient to !eave the !and id!e' perhaps our o&ner

&i!! o $anrupt* $ut then the !and &i!! $e $ouht %for a price appropriate to its rentearnin

capacit"-* and uti!ied or rented out to firms* $" someone e!se. ixed p!ants are !ie !ands in that*

once created* it is $est to uti!ie them as !on as the" can earn a positive renta!. The renta! earned $"

the fixed p!ant is not made exp!icit in the usua! forma!iation of shortperiod firm cost and profit*

$ut it can $e derived from it $ecause it is the difference $et&een revenue and cost of varia$!e

factors. 7ndeed* a firm miht !ease its fixed p!ants to other entrepreneurs' &hat maximum renta! &i!!an entrepreneur $e read" to pa" for the riht to use a fixed p!ant she does not o&nD A renta! e#ua! to

the maximum residua! o$taina$!e after su$tractin a!! other varia$!e costsF 36 from the revenue one

can earn $" operatin the fixed p!ant' such a renta! &ou!d reduce profit to ero. 7f the renta! is !ess

than that* the profit of the !easee is positive* and entrepreneurs must $e expected to compete for the

riht to use the fixed p!ant* therefore the renta! &i!! rise to the eroprofit residua! Qust discussed. 7f

the entrepreneur is a!so the o&ner of the p!ant* she shou!d inc!ude in the costs the oort1nit cost

of the use of the fixed p!ant the revenue the o&ner ives up &hen decidin not to !ease the fixed p!ant to other entrepreneurs * and this opportunit" cost is the maximum renta! thus determined*

ca!!ed 31asirent $" A!fred Marsha!! $ecause of its ana!o" &ith the rent of !and %the difference is

that fixed p!ants deteriorate-. The entrepreneur &ho first purchases the fixed p!ant is in the same

position as the person &ho purchases a !and' she ma" o $anrupt if the p!antCs #uasirent fa!!s $e!o&

the !eve! expected at the time of purchase renderin it impossi$!e to repa" the de$t incurred to

purchase the p!ant* $ut as !on as #uasirent is positive the p!ant &i!! not $e shut do&n* it &i!! $e

$ouht $" some other entrepreneur at its ne& va!ue %the present va!ue of its ne& expected #uasirents

36 7nc!usive of #uasifixed costs if these are positive.




for its remainin economic !ife%3)-- and &i!! $e ept in operation.

The considerations Qust deve!oped have an interestin imp!ication. 7f one ne!ects those fixed

costs &hich are* in the short period* ine!imina$!e and therefore not an opportunit" cost %e..

pa"ments for de$ts contracted in the past-* then a correct imputation of costs* inc!usive of

opportunit" costs and hence of #uasirents of fixed p!ants* renders the shortperiod profit al,ays

e#ua! to ero* even &hen it appears positive &ith the usua! forma!iation8 the reason &h" it appears

positive is that shortperiod tota! cost as usua!!" defined does not inc!ude the #uasirent earned $" the

fixed factors.

5.1+.H. The a$ove considerations a!so sho& that in order to determine the shortperiod supp!"

curve of an industr" &hat is necessar" is the supp!" curves of the severa! fixed p!ants in the

industr"' ho& their propert" is su$divided amon firms is not re!evant %as !on as efficienc" is

independent of o&nership-. The shortperiod supp!" curve of the industr" is iven $" the horionta!

sum of the parts* of the shortperiod marina! cost curves of the sin!e p!ants* &hich !ie a$ove the

respective AS( curves.

price

p a $ c d e f

#1 #2 #1#2

i. 5.6. orionta! sum of the shortperiod supp!" curves of t&o pricetain firms havin different

minimum AS(. Areate supp!" at price p* the sement ef* is the horionta! sum of the supp!ies of the t&ofirms* the t&o sements a$ and cd.

%7n order to determine the shortperiod supp!" curve of the pricetain firm the consideration

of fixed costs* as &e!! as of the A( curve* is not necessar"* $ut these notions $ecome necessar" for

!onperiod ana!"sis* and this is &h" the" have $een introduced.-

3) e are here introducin a rate of interest. This part of the chapter is concerned &ith the theor" of the

firm* and &e &ant to present this theor" so that it is app!ica$!e a!so to economies &here there is a rate of

interest.



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. HH

5.!9. rom s(ort)period to long)period supply

5.1,.1. The shortperiod A( and M( curves &ere derived for a iven fixed p!ant and iven

fixed and #uasifixed costs. e can no& extend the ana!"sis to the lon7 eriod i.e. to the ana!"tica!

situation &here &e treat all inputs as varia$!e* $" imainin that the firm can choose amon a set of

fixed p!ants* and for each one of them it can derive the A( and M( curves and find the minimum

averae cost and the associated output !eve!. e need not assume perfect divisi$i!it" of the e!ements

&hich o to form a fixed p!ant8 the firm can $e confronted &ith a finite num$er of a!ternative fixed

p!ants* for each one of &hich it can determine the A( and M( curve. The sma!!est of the minimum

averae costs associated &ith the severa! a!ternative fixed p!ants is the true minimum averae cost*

to $e indicated as Min@A(' !et us indicate the associated output as #k.

The !onperiod averae cost curve @A( is the inferior enve!ope of the shortperiod averae

cost curves* cf. i. 5.). Thus each point of the @A( curve is associated &ith a t"pe of fixed p!ant*

$ut not enera!!" &ith that fixed p!ants minimumaveraecost output. 7f the fixed p!ant consists of

a sin!e factor &hose amount can $e varied continuous!" %no indivisi$i!ities-* then each shortperiod

supp!" curve has a point in common &ith the !onperiod supp!" curve* &here the t&o curves have

the same s!ope.

@A(

#

i. 5.)

4roof. Assume the fixed factor is factor n' consider the unconditiona! !onperiod demand x n%v*#- and

assume v is iven' for a iven !eve! #N of output* cost minimiation determines a demand xn%v*#N- for factor

n' if its fixed amount is Qust x nkLxn%v*#N-* then for # different from #N !onperiod cost cannot $e reater than

shortperiod cost >(* $ecause the additiona! shortperiod constraint %the fixed amount of xn- cannot possi$!"

permit a !o&er cost and &i!! enera!!" imp!" a hiher cost* hence c%#-K>(%#*xnk-' and for #L#N !onperiod

and shortperiod cost coincide' hence the !onperiod averae cost curve @A( coincides at #N &ith the short

period averae cost curve $ased on xnLxnk* and is not a$ove it %and enera!!" $e!o& it- for # different from

#N. or each #N* there &i!! $e a xnkLxn%v*#N- for &hich one can repeat the a$ove reasonin' if $oth inds of

averae cost curves are smooth* at the #N for &hich the iven x nk is optima! the t&o curves must $e tanent

to each other. 9epeatin the reasonin for a!! #N comp!etes the proof. [




=ote that some of the shortperiod A( curves in i. DD have $een dra&n as havin no point

in common &ith the @A( curve. The reason is that if the fixed p!ant consists of several factors

com$ined in fixed amounts* the com$ination ma" $e a su$optima! one for a!! output !eve!s' then the

A( curve correspondin to that fixed p!ant &i!! $e ever"&"ere strict!" a$ove the @A( curve.

7f there is perfect divisi$i!it" and constant returns to sca!e* then Min@A( can $e reached for

an" #' if there are indivisi$i!ities and rep!ica$i!it" of p!ant* then there is a minimum efficient sca!e

of output #k that a!!o&s the firm to achieve an averae cost e#ua! to Min@A(* and the firm can

reach the same minimum averae cost $" producin 2#k &ith t&o fixed p!ants identica! to the one

&hich produced #k* or $" producin 3#k &ith three fixed p!ants* etc. The A( and M( curves &ith

t&o fixed p!ants are the curves &ith one fixed p!ant* CstretchedC riht&ards so as to reach the same

va!ue on the ordinate for a dou$!e va!ue on the a$scissa. 7n i. 5.)$ &e see the A( curves and M(

curves &ith one* t&o and three fixed p!ants of the same t"pe. A pricetain firm considers that it

can se!! an" amount of product at the iven price* therefore as !on as the output price p is reater

than Min@A(* the firm finds it convenient to ro& &ithout !imits $" rep!icatin infinite times the

p!ant associated &ith Min@A(' if pLMin@A(* then the firmCs maximum profit is ero and the firm

is indifferent $et&een producin #k* 2#k* 3#k etcetera' if pMin@A(* in the !on period the firm

does not produce.

A(

M(1 A(1 M(2 A(2

M(3 A(3

Min@A(

#k 2#k 3#ki. 5.)$. Averae and marina! cost curves &ith one* t&o or three identica! p!ants.

5.1,.2. @et us see ho& these considerations connect &ith profit maximiation.

Mathematica!!"* the pro$!em is

max$ π ($)%R($)=C($)%p$=C($)

&here (%#- is the !onperiod cost function* and 9Lp# is revenue. or !eve!s of production re#uirinthe use of a hih num$er of fixed p!ants* if rep!ication of p!ants does not cause a decrease in




efficienc" it is !eitimate to treat (%#- as proportiona! to #* $ecause averae cost remains near!"

constant as # varies o&in to the fact that the varia$i!it" in the num$er of p!ants ensures that* even

&hen # is not an inteer mu!tip!e of #k* each p!ant produces a #uantit" ver" c!ose to #k' for examp!e

in $et&eeen #L100#k and #UL101#k the production of each p!ant differs from #k $" at most 1]

and therefore remem$erin the <shape of the averae cost curve of a p!ant averae cost is ver"

near!" e#ua! to Min@A(' in i. 5.) this is evident a!read" &ith three p!ants. Therefore in this case

it is !eitimate to treat the !onperiod averae cost as constant* e#ua! to Min@A(. 7t is then a!so

e#ua! to the !onperiod marginal cost @M(* the derivative of the !onperiod cost function c%#-.

Therefore if pW@M(LMin@A(* no # satisfies the condition pL@M( for a pricetain firm' the

profit π%#-L%pMin@A(-# increases &ithout !imit $" increasin #' the pro$!em max # πLp#c%#- has

no so!ution. 7f pLMin@A(* supp!" is indeterminate $ecause jL0 for an" output !eve!' on!"

pMin@A( "ie!ds a determinate so!ution* #L0. Therefore in this case there is no supp!" function of

the firm* the firms supp!" is either ero* or infinite* or indeterminate.

?ut this fact does not create pro$!ems to the theor". A!! one needs to assume is that if

pWMin@A( the firm &i!! p!an to expand productive capacit" %i.e. costminimiin output- $"

expandin or rep!icatin p!ant* and if pMin@A( the firm &i!! p!an to reduce productive capacit"'

$ut variations of productive capacit" tae time* and since the firm &i!! not $e the on!" one to tae

such decisions* and since it is un!ie!" that there $e perfect s"nchroniation of the decisions of thesevera! firms %possi$!" inc!udin ne& entrants-* it is !eitimate to assume that enera!!" the

expansion or contraction of industr" productive capacit" &i!! $e radua!' thus the shortperiod

supp!" curve shifts radua!!"* and the shortperiod e#ui!i$rium price* determined $" the intersection

of the demand curve &ith the shortperiod supp!" curve* tends to&ard Min@A(F3+. 7f the minimum

averae cost is not the same for different firms* the !ess efficient firms &i!! $e e!iminated $"

competition* and on!" the firms &ith the !east Min@A( &i!! survive8 in the !on period* competition

enforces productive efficienc" in the sense of minimiation of averae cost. The tota! num$er of p!ants &i!! $e such as to $rin output price as c!ose as possi$!e to Min@A( &ithout fa!!in $e!o& it.

The industr" supp!" curve is derived as fo!!o&s. @et @M( n%#- stand for the horionta! sum of

the !onperiod marina! cost curves of n identica! efficient p!ants* &ith # their tota! output and #k

the sin!ep!ant minimumaveraecost output' the supp!" curve consists of a discontinuous series

of up&ards!opin sements "ie!din a sa&!ie shape* the nth sement $ein the portion of the

@M(n%#- curve correspondin to the semiopen interva! Fn#k* %n1-#k-. A!! sements start at a

3+ >"nchronied decisions to a!ter productive capacit" miht cause phenomena ain to co$&e$ c"c!es %cf.

(h. 6 RDD- $ut for !onperiod decisions such phenomena are !ess !ie!" $ecause there is time to revise

decisions in the !iht of information of &hat other producers in the industr" are doin.



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. H)

heiht e#ua! to Min@A(' as n increases the sements $ecome f!atter* the supp!" curve approaches a

continuous horionta! !ine at a !eve! e#ua! to Min@A(. 7f the do&n&ards!opin demand function

for this ood crosses this supp!" curve more than once* the e#ui!i$rium intersection is the one

correspondin to the reatest output at a price not !ess than Min@A(* #E in i. 5.+.

@M(1 demand curve

@M(2

Min@A(

#k 2#k 3#k etc. #E #

i. 5.+. @onperiod industr" supp!" curve &ith p!ants that reach minimum

averae cost at output !eve! #k* and a demand curve crossin the supp!" curve t&ice. 7n

this case in e#ui!i$rium there is room for 6 p!ants. The @M( curves are dra&n as

straiht!ine sements on!" for simp!icit".

@et us for examp!e suppose that* at pLMin@A(* the demand for the industr"Cs output fa!!s

$et&een 100#k and 101#k. There is therefore room in the industr" for 100 p!ants* each one

producin s!iht!" more than #k* and therefore havin a !onperiod marina! cost Qust s!iht!"

a$ove Min@A(. The e#ui!i$rium price &i!! $e so c!ose to pLMin@A(* that to assume that the

e#ui!i$rium price is e$ual to Min@A( is an exce!!ent approximation. There is no room in the !on

run for 101 p!ants $ecause @M(101%#- crosses the do&n&ards!opin demand curve at a price $e!o&

Min@A(* and firms &ou!d mae !osses. The conc!usion is that* &ith perfect rep!ica$i!it" of p!ants*

as !on as the output of a sin!e p!ant is on!" a sma!! fraction of tota! output* the !onperiod

industr" supp!" curve can $e treated* to a!! re!evant purposes* as a horionta! straiht !ine %if input

prices are iven- at a !eve! e#ua! to minimum averae cost.

o&ever* one ma" fee! uneas" &ith the fact that the theor" does not determine the sie of

individua! firms. e pass to discuss this issue.

5.%". *(e size and t(e num4er of firms

Economists sti!! discuss on the issue of the !onperiod e#ui!i$rium sie of competitive firms.

hen there is product differentiation* the sie of a firm is !imited $" the maret for its product%s-*

the #uestion $ecomes &hat determines the sie of that maret* and ans&ers are not difficu!t to find8

the #ua!it" of the product re!ative to the tastes of consumers* if it is a consumption ood' the



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. H+

num$er of firms that use techno!oies needin that product* if it is a productive input' the extent of

competition from riva! products* and the maretin strateies of the firm and of its riva!s. =o&* it is

important to understand that perfect product homoeneit" is a ver" rare occurrence' usua!!"

!ocation* or the importance for customers of past tradins that have $ui!t confidence in their usua!

supp!iers* are sufficient to render it cost!" for a firm to extend its sa!es $" su$tractin customers

from other firms* a!thouh often the diverence from perfect!" competitive $ehaviour remains sma!!

enouh that one ma" continue to app!" the !onperiod theor" of the perfect!" competitive industr".

7n other cases the sie of the firm is determined $" the nature of the product8 roc $ands are firms

too* and their product re#uires a certain sie of the &orforce' there is no possi$i!it" of rep!icatin

the p!ant identica!!" &ithin the same firm. >imi!ar considerations app!" to a!! firms $ased on a

strict* creative interaction amon fe& persons. Apart from these cases* one finds the disareement

amon economists mentioned in R5.6.3* &ith some %e.. Edith 4enrose- aruin that in man"

instances firms are a$!e to ro& to enormous sies &ithout an" increase in averae cost and

therefore the !imits to sie must $e found either on the demand side or on the need for o&n capita!

or co!!atera!* and others aruin that the enera! case is <shaped @A( curves $ecause of co

ordination difficu!ties that increase &ith sie.

7n the discussion &hether @A( curves are <shaped or not* &e find here the second meanin

of returns to sca!e mentioned in R5.H* returns to the scale of total cost or* $rief!"* (scale) returns tocost * o$vious!" a notion that assumes iven input prices. These returns are defined $" the elasticit

of o1t1t to total cost* and need for their definition neither that a!! inputs $e increased in the same

proportion* nor differentia$i!it" of the production function* nor divisi$i!it" of a!! inputs' as tota! cost

increases* there ma" &e!! $e discontinuous chanes in the #uantities emp!o"ed of some inputs* e..

some capita! oods ma" $e rep!aced $" capita! oods of a different t"pe* the fixed p!ant ma"

chane* or fixed p!ants ma" $e indivisi$!e and $e discrete!" increased from one to t&o* three etc.'

there ma" then $e some discontinuities in maximum output as tota! cost increases* and the pointe!asticit" of maximum output to tota! cost &i!! not $e defined at those points' $ut ever"&here e!se*

and ever"&here for discrete chanes in tota! cost* the e!asticit" of output to cost &i!! $e &e!!

defined* and therefore returns to cost is a more enera! notion than technical returns to scale. hen

returns to cost are constant* output increases in the same proportion as tota! cost* and therefore

averae cost is constant' &hen returns to cost are increasin* successive increases in tota! cost "ie!d

increasin returns i.e. $ier and $ier increases in output* so averae cost is a decreasin function

of output' &hen returns to cost are decreasin* averae cost is an increasin function of output.(onstant returns is a!so used for the case &hen Min@A( is reached on!" for the optima! outputs



f petri Adv Micro chapter 5 firms and productionGE 10/03/2016 p. H,

correspondin to rep!ication of indivisi$!e p!ants. >ca!e economies is another term used to indicate

increasin returns to cost.

Assumin no& that maximum output is a continuous function of tota! cost and that the @A(

curve is <shaped* Min@A( is reached &here the returns to costs thus defined* in passin from

!oca!!" increasin to !oca!!" decreasin* are !oca!!" constant. 7f the production function is

differentia$!e &ith respect to a!! inputs* then the !oca!!" constant returns to costs at Min@A(* that is*

the e#ua!it" of averae and marina! cost* imp!" !oca!!" constant technica! returns to sca!eF 3, and

therefore imp!" that the pa"ment to each factor of its marina! revenue product exhausts revenue if

pLMin@A(. Thus the fact that the !onperiod cost curve is <shaped entai!s no contradiction

$et&een assumin ero profits of competitive firms in e#ui!i$rium* and assumin that each factor is

paid its marina! revenue product.

7f the firmCs !onperiod A( curve is <shaped* the firmCs dimension is no !oner

indeterminate8 profit is maximied &hen @M(%#-Lp. hen this is the case* the !onperiod industr"

supp!" curve is derived in the &a" a!read" sho&n* &ith firms rep!acin p!ants in the reasonin8

the num$er of firms is endoenous* $ecause competition a!so means free entr"* and in the !on

period there is time for entr". The conc!usion is aain that* as !on as the minimum optima!

dimension of firms is sma!! re!ative to tota! industr" output* if factor prices %factor renta!s- are iven

then to a!! practica! effects the !onperiod supp!" curve of the industr" is horionta! at a price e#ua!to minimum averae cost.

?ut even &hen the <shaped cost curve is not accepted* the supp!" curve of the industr"

remains horionta! at the Min@A( !eve!' the sie of the firms composin the industr" is then simp!"

irre!evant. :ne can for examp!e tae it as determined $" historica! accidents* or $" !imits to the

ro&th of individua! firms derivin from !imits to possi$!e inde$tedness.

7n conc!usion $oth &hen there are* and &hen there arent* constant returns %to cost- $" firms*

the assumption of competition &ith free entr" imp!ies an essentia!!" horionta! !onperiod industr"supp!" curve once input prices are iven* as !on as the minimum #uantit" that a!!o&s a firm to

minimie averae cost is sma!! re!ative to the tota! demand forthcomin at a price e#ua! to that

averae cost. hen this is the case* since in the !on period competition e!iminates inefficient firms*

one can assume a common techno!o" &ithin the industr". e can therefore treat the industryBs

3, e prove this for the t&ofactors case. At the point of minimum averae cost it is 9C % 1

1

98

vL L

-*% 21

2211

2

2

x x f

xv xv 7C 98

v +

== ' this can $e re&ritten f(x+ !x ) % 98 + x+;98 + 2

1

v

v

x % 98 + x+;98 x * &hich

imp!ies that the production function is !oca!!" homoeneous of deree 1.




!onperiod areate production function as exhi$itin constant technical returns to sca!e even

&hen &e cannot do so for sin!e firms.

This conc!usion &i!! a!!o& us* &hen in 4art 777 of the chapter &e formu!ate the neoc!assica!

competitive enera! e#ui!i$rium &ith production* to assume product prices e#ua! to minimum

averae costs* and that at those prices supp!" adapts to the demand forthcomin at those product

prices %and at the factor prices that determine them-.

>o far in this !onperiod ana!"sis &e have taen a!! input prices as given. 7n this case* except

for f!ues on!" one production method &i!! $e the costminimiin one for the production of each

product. o&ever* &hen &e come to determinin factor renta!s endogenously* it ma" &e!! happen

that t&o %or more- methods &i!! coexist in the production of the same product8 the different factor

emp!o"ments need not imp!" differences in averae cost if factor renta!s* for some of the factors*

adapt so as to ensure the same averae cost &ith $oth methods. :ne t"pica! such case is that of

extensive differentia! !and rent8 &hen the same product is produced on !ands of different ferti!it"* a

differentia! rent &i!! arise on more ferti!e !ands* that &i!! mae the uti!iation of different !ands

e#ua!!" convenient. ere &e need not add to &hat &as said on this topic in (hapters 1 and 3.

5.%!. Aggregation

5.21.1. hen the num$er of firms in an industr" is given* and for each firm a profit function*and therefore a supp!" function* exists* then the industr"s competitive supp!" can $e determined as

if forthcomin from a sin!e mu!tip!ant pricetain firm that operates a!! the individua! production

functionsFH0. orma!!"* the area$i!it" condition is that* iven the individua! production

possi$i!it" sets %&hose e!ements are netput vectors- B1* ... * B Q* ... *B of the individua! firms %&here

is the num$er of firms-* the aggregate production possi$i!it" set $e

B L B1 ... B L "∈9 n8 " L _ Q " Q for some " Q∈B Q* QL1*...*.

7n &ords* the areate firm must have no additiona! production possi$i!ities at its disposa! $e"ond a simu!taneous activation of the production processes avai!a$!e to the individua! firms* at

most one per firm. 7n this case* the areate firm can do no $etter in terms of profits than the sum

of the individua! firmsC profits* $ecause it can do no $etter than cop" &hat the individua! firms

H0 This is true as !on as the possi$i!it" is exc!uded that a sin!e manaement of a!! the factors of the

individua! firms %inc!udin the fixed factors &hich need not exp!icit!" appear in the shortperiod production

functions- &ou!d achieve cost reductions. or examp!e* the fusion of five sma!! farms into a sin!e $i farm

miht permit the uti!iation of $i aricu!tura! machiner" &hich &as uneconomica! for each individua! farm'

or there miht $e unnecessar" dup!ication of some indivisi$!e factors %for examp!e* each separate farm miht

need to $u" its o&n tractor if no sharin is a!!o&ed* &hi!e four shared tractors &ou!d suffice for the five

farms-. :n!" &hen one exc!udes such phenomena can one conc!ude that the industr" $ehaves in the same

&a" as if a sin!e firm &ere to operate a!! the individua! production functions.




&ou!d choose autonomous!".

hen this is the case then* converse!"* the firms in the industr" maximie areate profits'

since profit maximiation re#uires cost minimiation* this a!so sho&s that the a!!ocation of the

industr" output amon the severa! firms in the industr" is costminimiin. 7ndeed* since for each

firm output price e#ua!s marina! cost* marginal cost is the same in all active firms * &hich is an

o$vious efficienc" condition8 if marina! cost &ere different in t&o %active- firms* a sma!! transfer

of production to the firm &ith the !o&er marina! cost &ou!d decrease areate cost. Ana!oous!"*

for each varia$!e %and hence* transfera$!e- factor the marina! product &i!! $e the same in a!! firms

&here the factor is used* $ecause e#ua! to the factor renta! divided $" the output price%H1-* aain an

o$vious efficienc" condition8 if the marina! product of a factor &ere not the same in a!! firms %and

!o&er in the firms not usin it-* transferrin a sma!! amount of the factor to the firm &here it has the

reater marina! product &ou!d increase tota! production. These considerations sho& that the

areate firm o$e"s the conditions for profit maximiation &hen each individua! firm does.

5.21.2. 7n !onperiod ana!"sis &ith free entr"* aain the industr"s $ehaviour can $e derived

as comin from the decisions of a sin!e firm* a constantreturnstosca!e firm &hich* for each

vector of factor renta!s and each output !eve!* adopts the factor emp!o"ments correspondin to

minimum averae costFH2

. (ompetition e!iminates !ess efficient firms and thus causes the production function to $ecome the same for a!! firms. 7f the sin!e firms have a (9> production

function* then the industr" acts !ie a iant firm &ith that same production function. 7f the

individua! firms production function "ie!ds <shaped averae cost curves* then %assumin a

sufficient!" sma!! minimum efficient sie re!ative to areate output- $ecause of the possi$i!it" of

rep!ication of p!ants or firms the industr" acts !ie a sin!e (9> firm* &ith a production function

&hich* for each vector of re!ative factor renta!s* "ie!ds the same optima! factor emp!o"ments per

unit of output as the averaecostminimiin choice of the individua! firms.e i!!ustrate &ith a numerica! examp!e. The firms production function is # L 2%1x 1

1x2 1- 1 .

7f $oth factors are mu!tip!ied $" a sca!ar t* &e o$tain #%t- L

21

2

11

2

x xt + ' it is convenient to put

x1x2L1/A' then #%t- L 2t2/%t2A-' and the sca!e e!asticit" of output %remem$er that it is eva!uated at

tL1- is eL2A/%1A- &hich is ⋛1 accordin as A⋛1. Thus for each iven factor proportion this

H1 Assumin that marina! products can $e defined. The reader is reminded that factor renta!s must e#ua!

marina! revenue products.H2 ere as e!se&here in this chapter it is assumed that the averaecostminimiin factor proportions are

uni#ue!" determined.




function produces a <shaped cost curve* $ecause it exhi$its !oca!!" increasin returns to sca!e

&hen x1x21* !oca!!" decreasin returns to sca!e &hen x1x2W1* and !oca!!" (9> %and minimum

averae cost- &hen x1x2L1 i.e. &hen #L1. Therefore in the !on period each firm produces one unit

of output* and the iso#uant map of the industry/s !onperiod production function is the radia!

expansion of the iso#uant x2L1/x1 associated &ith #L1' in order to no& ho& much an industr"

input vector %x1*x2- produces* &e must vie& it as t times the vector %x1k*x2k- that produces 1 unit of

output &ith the same factor proportion as %x1*x2-* &here t is determined $" x1Ltx1k* x2Ltx2k such that

x1kx2kL1' in other &ords* inputs %x1*x2- produce t units of output &here t2 L x1x2. ence the

industr"s production function is q L 21 x x . The reader can chec that it has (9>FH3.

e can use this examp!e to c!arif" a forma! pro$!em arisin &ith <shaped cost curves of

individua! firms in !onperiod ana!"sis &ith free entr". >uppose that factor renta!s are v 1Lv2L1'

then in this examp!e Min@A(L1. >uppose that the demand curve is decreasin* and at the product

price pL1 demand is 100.5 units. There is room for 100 firms' if there &ere 101 firms* price &ou!d

o $e!o& 1 and a!! firms &ou!d mae neative profits. ?ut &ith 100 firms the e#ui!i$rium price is

s!iht!" a$ove 1 and profits are positive' hence* if &e assume that firms enter as !on as

pWMin@A(* there &i!! $e entr"' no e#ui!i$rium exists if &e define it as simu!taneous!" re#uirin

riorous!" demandLsupp!" and profitsL0. o&ever* if the aim is to determine a !onperiod

e#ui!i$rium %the averae situation around &hich the econom" osci!!ates-* this is not a pro$!em $ecause even if firms do enter and $rin the tota! num$er of firms a$ove 100* the num$er &i!!

su$se#uent!" decrease* and &e can sti!! assume that the averae around &hich the price osci!!ates is

1. @onperiod e#ui!i$rium on!" aims at determinin the averae around &hich actua! maret

varia$!es osci!!ate. %urthermore it is p!ausi$!e that potentia! entrants tr" to mae an estimate of the

effect of their entr"* and enera!!" the" &i!! rea!ie that it is !ie!" that the sma!! profits associated

&ith 100 firms &i!! disappear if the" enter* and hence &i!! not enter.-

H3 To render exp!icit the industr"s production function startin from the firms production function is not

a!&a"s possi$!e* $ut* as the examp!e indicates* one procedure to find the amount produced from the amounts

of factors emp!o"ed $" the industr" is as fo!!o&s. or simp!icit" assume on!" t&o factors. @et f%x 1*x2- $e the

individua! firms production function "ie!din a <shaped @A( curve' !et x1*x2 $e the iven industr" inputs'

!et ?x2/x1' find x1k such that f%x1k*?x1k- minimies averae cost' !et #k f%x1k*?x1k-' !et tx1/x1k' then the

industr" output is qLt#k. Exercise 5.8 assume that the individua! firms production function is # L x 1x2

%x13/2x2

3/2-/3 and ?L,' find x1k* #k* and q if the industr"s emp!o"ment of input 1 is x1L100.




PART ##

5.%%. PA<*6A1 E=>616?<6>M

5.22.1. As a first step to&ard the stud" of the marina!ist/neoc!assica! competitive enera!

e#ui!i$rium of production and exchane* &e discuss in reater detai! the construction that* more or

!ess exp!icit!"* &e have $een usin in this chapter %for examp!e in i. 5.+DD and in the text

discussin it-8 the determination* via the intersection of a supp!" curve and a demand curve* of the

soca!!ed particular e#ui!i$rium* as it &as oriina!!" ca!!ed* or %its current usua! denomination-

partial e#ui!i$rium* of a sin!e maret studied in iso!ation. The prices and #uantities on other

marets are taen as iven and are considered essentia!!" unaffected $" chanes in the maret under

stud"8 this is ca!!ed the assumption of coeteris paribus %!atin for Cother thins remainin the sameC-.

A famous 1,26 artic!e descri$es this approach as fo!!o&s8

This point of vie& assumes that the conditions of production and the demand for a

commodit" can $e considered* in respect to sma!! variations* as $ein practica!!"

independent* $oth in reard to each other and in re!ation to the supp!" and demand of a!!

other commodities. 7t is &e!! no&n that such an assumption &ou!d not $e i!!eitimate

mere!" $ecause the independence ma" not $e a$so!ute!" perfect* as* in fact* it never can $e'

and a s!iht deree of interdependence ma" $e over!ooed &ithout disadvantae if it

app!ies to #uantities of the second order of sma!!s* as &ou!d $e the case if the effect %for

examp!e* an increase of cost- of a variation in the industr" &hich &e propose to iso!ate

&ere to react partia!!" on the price of the products of other industries* and this !atter effect

&ere to inf!uence the demand for the product of the first industr". %>raffa 1,26* p. 53+-

The partia! e#ui!i$rium approach can a!so $e used for the stud" of imperfect!" competitive

marets* for examp!e a monopo!istic maret.

>ometimes it ma" $e !eitimate to iso!ate not one* $ut t&o %or perhaps even more-interdependent marets8 one examp!e is the stud" of the effects of chanes of the supp!" conditions

and hence of the price of one product on the demand and hence on the e#ui!i$rium price of a

comp!ementar" or of a su$stitute product' another examp!e is the determination of the supp!" curve

of an industr" that uses a specia!ied factor* &hose renta! rises &hen* o&in to a rise in the demand

for the industr"s product* the industr"s output rises.

5.22.2. The partia!e#ui!i$rium supply curve of a competitive industr" can $e a !onperiod or




a shortperiod one. The !onperiod supp!" curve is horionta! if factor renta!s are ivenF HH. The

shortperiod supp!" curve is up&ards!opin even &ith iven factor renta!s* o&in to the iven

amounts of some factors. Their derivation has $een i!!ustrated a!read"' &e add some considerations

on their !eitimac" and on the ana!o" $et&een them.

?esides needin the pricetain assumption* the use of a partia!e#ui!i$rium !onperiod

supp!" curve is !eitimate in t&o cases. The first one is &hen the industr" uses on!" a sma!! fraction

of the tota! supp!" of each one of the factors of production it uti!ies' then a variation in the #uantit"

produced $" that industr" &i!! not exert an apprecia$!e inf!uence on its factors renta!s* $ecause it

&i!! cause a ver" sma!! percentae variation of the demand for them. As a resu!t* factor renta!s can

$e taen as iven. 7f the inf!uence on the renta! of some factor &ere sinificant* this &ou!d a!ter the

cost conditions of other products too and then their prices &ou!d chane* renderin the coeteris

pari$us assumption i!!eitimate.

The second case is &hen there is a specia!ied factor demanded on!" $" the industr" one is

iso!atin. This miht $e for examp!e a specia! t"pe of !and indispensa$!e to %and on!" demanded

for- the production of one product* sa" a famous &ine. 7n this case the specia!ied factors renta! is

determined endoenous!"' it &i!! rise as demand for the product rises* so as to maintain the

producers profit at ero' the marina! product of the remainin factors %&hose renta!s are iven-

decreases as output increases' the supp!" curve is up&ards!opin. The independence $et&eensupp!" curve and demand curve* necessar" for partia! e#ui!i$rium ana!"sis* additiona!!" re#uires

that the chanes in the incomes of the o&ners of the specia!ied factor do not apprecia$!" inf!uence

the demand for the product of the industr" under ana!"sis.

Marsha!! enera!ied this second case to inc!ude cases &here the supp!" to the industr" of

some specia!ied factors* different!" from the supp!" of specia!ied !and* is varia$!e in the !on

period $ecause those factors are produced factors* $ut it varies sufficient!" s!o&!" re!ative to the

supp!" of the other factors for it to $e treated as iven in shorterperiod e#ui!i$ration processes.>ome t"pes of fixed p!ants ma" indeed tae a !on time to $e $ui!t' this authories treatin their

supp!" as fixed for e#ui!i$ration processes on time horions of* sa"* a fe& months. %As pointed out

ear!ier* &hat is needed for the determination of the shortperiod supp!" curve and hence for short

period partia! e#ui!i$rium ana!"sis is that the supp!" of fixed factors to the industry* not to each

firm* $e iven.-

The supp!" curve is va!id on!" for comparative!" sma!! variations of the #uantit" produced*

HH This assumes that if there are profits to $e made* there &i!! a!&a"s $e someone %existin or ne& firms-

read" to add further p!ants in the industr". This assumption appears stron!" confirmed $" experience &hen

ade#uate account is taen of barriers to entry* that &e are assumin a$sent here and &i!! $e discussed in

(hapter 11.




$ecause for considera$!e variations the coeteris pari$us condition $ecomes dou$tfu!FH5. 4articu!ar

difficu!ties arise &hen one attempts comparative statics of partia! e#ui!i$ria of a capita! ood8 a

chane in the supp!" conditions of a capita! ood %due for examp!e to the discover" of a ne&*

cheaper production method* or to a tax- a!ters the costs and hence the prices of a!! oods for &hose

production that capita! ood is used* and man" of these ma" $e in turn inputs to the production of

that capita! ood* so the supp!" curve of the capita! ood shifts for more reasons than the direct

effect of the first chaneFH6' and the derivation of the demand curve is not eas" to conceive.

5.22.3. A!fred Marsha!! attempted to arue that the !onperiod partia!e#ui!i$rium supp!"

curve of a product can a!so $e do&n&ards!opin* o&in to t&o effects of increases of the

dimension of an industr"8 first* the possi$i!it" $etter to exp!oit sca!e economies' second* an increase

in external effects or externalities. e &as thus tr"in to mae room in the theor" of partia!

e#ui!i$rium for a phenomenon no dou$t often o$served* an association $et&een increase in

production and decrease in price of a product produced $" an industr" &here it &as difficu!t to den"

the existence of competition amon producers. ?ut in t&o artic!es* in 1,25 and 1,26* 4iero >raffa

sho&ed that the decreasin supp!" curve is incompati$!e &ith competitive partia! e#ui!i$rium. e

remem$ered that the existence of unexp!oited sca!e economies is incompati$!e &ith competition

&ith undifferentiated products* $ecause competition re#uires firms to $e rather sma!! re!ative to tota!industr" demand* and the perfect su$stituta$i!it" for the $u"er amon the products of the different

firms in the industr" imp!ies that an" sma!! price reduction $" a firm &i!! attract to the firm enouh

$u"ers to mae it a$!e to se!! the increased output that a!!o&s the exp!oitation of sca!e

economiesFH). As to externa!ities* he noticed that the positive externa! effects due to increases of

economic activit"* e.. reater ease in findin repairmen or transportation firms or si!!ed &orers*

H5 Marsha!! 8rinciples p. 3+H fn. of the oriina! +th edition %1,20' p. 31+ fn. in the after1,H, reset

editions-8 bthe ordinar" demand and supp!" curves have no practica! va!ue except in the immediateneih$ourhood of the point of e#ui!i$rium.U The reason is part!" different for demand curves* cf. $e!o& in

the text and a!so RDD%consumer surp!us-.H6 (hanes in input use can $e sometimes ver" surprisin &hen these interre!ations are taen into account'

their exp!oration has started on!" recent!" and is sti!! proceedin* cf. :pocher and >teedman DDH) Marsha!! had arued that unexp!oited sca!e economies exist* $ut re#uire time to $e exp!oited $ecause firms

are s!o&ed do&n in their expansion $" the need for co!!atera! and therefore for accumu!ated profits* and this

prevents firms from $ecomin indefinite!" !are $ecause the founders of successfu! firms pass the firm to

their chi!dren &ho are much !ess competent and cause the firm to dec!ine and die' he compared the firms in

an industr" to trees in a forest* some of &hich are ro&in* &hi!e others are d"in. This picture is

occasiona!!" confirmed $" facts* $ut no&ada"s more and more it is the case that firms are o&ned $" man"

shareho!ders and run $" hired manaers* and a dec!ine due to incompetent o&nerentrepreneur heirs is rare'

therefore a dec!ine $efore sca!e economies can $e exp!oited is enera!!" imp!ausi$!e' the more so* $ecause

there are more and more iant firms* con!omerates &hich have the financia! potentia! to set up ver" !are

firms from the start.




are ver" se!dom interna! to an industr"* the" enera!!" concern !are roups of firms $e!onin to

different industries $ut connected $" common !ocation or $" simi!ar needed si!!s' these effects

cannot $e admitted in the partia! e#ui!i$rium ana!"sis of one industr" $ecause the" extend to other

products* a!terin their prices* and therefore vio!atin the coeteris pari$us condition. >raffa

conc!uded* and su$se#uent economists have admired the coenc" of his criti#ue* that competitive

!onperiod partia! e#ui!i$rium theor" can admit on!" constantcost industries* or increasincost

industries in the so!e case of an industr" $ein the so!e demander of a specia!ied factor. %hen the

expansion of an industr" affects the renta! of a factor a!so used $" other industries* then the costs of

these other industries are affected as much as in the first industr"* the prices of the products of those

other industries are re!evant!" affected* and aain the coeteris pari$us condition does not ho!d.- This

does not mean that unexp!oited sca!e economies do not exist* it on!" means that their causes and

effects re#uire a different approach8 >raffa suested to a$andon the assumption of perfect

competition and admit that the enera! case is rather one of differentiated products* &hich* if

coup!ed &ith free entr"* ensures nonethe!ess a $road tendenc" of prices to&ard averae costsFH+.

5.22.H. The demand curve is a function that specifies the #uantit" demanded of the product

under investiation as a function of its price. The !eitimac" of assumin the existence of a partia!

e#ui!i$rium demand curve re#uires81- Given prices of other oodsH,' this imp!ies a iven income distri$ution. %This in turn

imp!ies that partia! e#ui!i$rium ana!"ses cannot stud" chanes in the price and #uantit" of a product

induced $" chanes in income distri$ution.-

2- Given incomes of consumers. This re#uires* in addition to a iven income distri$ution* a

iven !eve! of uti!iation of resources %in particu!ar* a iven !eve! of !a$our emp!o"ment-8 this &as

traditiona!!" Qustified $" an assumption of fu!! uti!iation of resources* the tendentia! resu!t of the

&orin of maret economies accordin to the marina! approach* as &e no& from (hapter 3.3- Given preferences* unaffected $" actua! none#ui!i$rium consumptions.

These three sets of conditions too are su$sumed under the expression coeteris paribus. These

ivens are* and must $e* assumed not to chane %or more precise!"* to chane on!" ne!ii$!"-

H+ The readin of $oth >raffas artic!es* the 1,25 and the 1,26 oneDDrefs in En!ish* is stron!" recommended

as the" are exce!!ent examp!es of penetratin reasonin attentive to the economic Qustifications of theoretica!

constructs.H, 7t is a!so possi$!e to consider the demand curve as derived under an assumption that the prices of some

stron!" interconnected oods chane &hen the #uantit" demanded of the first ood chanes' for examp!e if

one considered pro$a$!e that a $i increase of taxes on aso!ine &ou!d considera$!" decrease the demand for

cars* an" estimate of the demand curve for aso!ine not restricted to the ver" short period shou!d tr" to tae

into account the effect on the car maret* inc!udin the possi$!e effect on the price of cars. ortunate!"* for

most industria!!" produced oods the price is rather insensitive to demand* cf. (hapter 11.




durin the adQustment processes to&ard the partia! e#ui!i$rium* other&ise the e#ui!i$rium &ou!d

!ac the persistence necessar" to ive a ood indication of the averae $ehaviour of the maret' it

&ou!d $e impossi$!e to assume* for examp!e* that the demand curve has the persistence that a!!o&s

a monopo!istic firm to form a reasona$!" correct idea of its position and s!ope %a necessar" premise

to the derivation of the marina! revenue curve-.

7t is then c!ear that the partia! e#ui!i$rium method re#uires that income distri$ution and

areate demand can $e considered iven %a!thouh not necessari!" determined accordin to the

marina!/neoc!assica! approach-* and that preferences can $e considered sufficient!" unaffected $"

the dise#ui!i$rium adQustments. :n this !ast issue* &e have remem$ered at the $einnin of chapter

H Marsha!!s o&n admission that experience irreversi$!" affects tastes. e ma" add here that &hen a

price chanes sinificant!"* o$!iin consumers to a re!evant chane in consumption ha$its* it seems

p!ausi$!e that peop!e &i!! not no& in advance ho& their o&n $ehaviour is oin to chane' the"

&i!! experiment and discover and deve!op ne& consumption ha$its that cou!d not $e predicted from

past evidence* and that often can on!" $e descri$ed as due to a formation* or discover"* of

preferences unti! then undefinedF50. As a conse#uence* the assumption of a &e!!defined demand

curve for prices different from the prevai!in one ma" $e #uestioned* as aain admitted $" Marsha!!

himse!f %cf. a$ove ch. H RH.22-8 hence the assumption %that &i!! $e met fre#uent!" in (hapter 11-

that firms kno, the demand curve facin them must $e treated &ith suspicionF51

.?ut &e must introduce the reader to the dominant ana!"ses* so &e do not further #uestion the

notion of partia!e#ui!i$rium demand curve' ho&ever* !et us not foret that this notion can $e

considered reasona$!" &e!! defined on!" for consumption oods %and perhaps for some non$asic

capita! oods-* on!" for rather sma!! departures from the unti! then prevai!in price* and on!" as !on

as the incomes of consumers %hence income distri$ution and the areate !eve! of activit" of the

econom"- are iven.

5.%. ta4ility of partial equili4ria.

50 The dependence of preferences on experience can $e used for a further criticism of the marina!

approach. 7t is not on!" that if consumption of a certain ood has never $een experienced* the preference for

it cannot $ut $e vaue* and open to modification $" experience. There is a!so the fact that repeated

experience can permanent!" a!ter preferences %e.. peop!e can develop a taste for !istenin to certain inds of

music* for drinin ood &ine* for practicin certain sport activities-. =o&* &hether and ho& man" times a

ood is experienced can depend on prices. This #uestions the assumption of preferences independent of

prices* on &hich the marina!ist/neoc!assica! determination of e#ui!i$rium is $ased.51 irms must have had the possi$i!it" to exp!ore ho& demand depends on price in an economic situation

underoin ver" !itt!e chane in the varia$!es impounded in the ceteris pari$us c!ause. This ma" $e an

accepta$!e assumption in some cases* $ut not in man" other ones' for the !atter cases* one &i!! need theories

exp!ainin firm $ehaviour &ithout an assumption that there is a &e!!defined and no&n demand curve.




5.23.1. 7f the ood is homoeneous %undifferentiated- then on averae a!! units of the ood

must se!! at the same price. :f course this can on!" $e approximate!" true $ut* as a!read" pointed

out severa! times* the e#ui!i$rium can on!" aim at descri$in the averae resu!tin from the tria!

anderror hi!in of the maret. e can spea therefore of the price of the ood.

Given a supp!" curve and a demand curve* that represent supply price and demand price as

functions of the #uantit" of the oodF52* e#ui!i$rium o$tains &here areate demand for the ood

e#ua!s areate supp!"* or e#uiva!ent!" &here supp!" price and demand price coincide.

@et us examine the sta$i!it" of e#ui!i$rium. Apart from the imp!ausi$!e case of Giffen oods

the demand curve for a consumption ood can $e assumed to $e do&n&ards!opin. %or capita!

oods the issue is more comp!ex* and as arued ear!ier the partia! e#ui!i$rium method is se!dom

accepta$!e* $ut an arument can sti!! $e put for&ard to the effect that an increase in the price of a

capita! ood* &ith other factor prices iven* &i!! tend to reduce the demand for that capita! ood*

$oth $ecause of technica! su$stitution* and $ecause of the rise in cost and hence in re!ative price of

the consumption oods usin that capita! ood as an input.- The supp!" curve is either horionta!* or

up&ards!opinF53. 7n the !atter case the sta$i!it" of e#ui!i$rium is c!ear* under the assumption that

price tends to rise if demand exceeds supp!"* and tends to decrease if supp!" exceeds demand. hen

the supp!" curve is horionta! it is a !onperiod supp!" curve* and the adQustment oes on in a

succession of shortperiod situations* in each one of &hich the num$er of p!ants is iven and thesupp!" curve is a shortperiod* up&ards!opin one. The shortperiod e#ui!i$rium price is sta$!e* and

if hiher than min@A( it induces in the !on period an increase in the num$er of p!ants in the

industr"* that is* a shift of the shortperiod supp!" curve to the riht that causes the shortperiod

52 The Marsha!!ian preference for considerin price the dependent varia$!e thus supp!" price is the price

necessar" to induce a iven supp!" to $e forthcomin* and demand price is the price that induces a iven

demand has the advantae that one can sti!! spea of a supp!" function even &hen the supp!" curve is

horionta!.53 Actua!!"* Marsha!! considered at !enth the possi$i!it" that the !onperiod supp!" curve of a product $e

decreasin* o&in to economies of sca!e achieved $" an averae increase of firm dimension &ith the ro&th

of industr" sie* or o&in to cost reductions due to economies of sca!e externa! to the firm $ut interna! to the

industr"* a specia! case of externa!ities. The first cause &as soon Quded incompati$!e &ith the perfect

competition assumption* that must assume that in the !on period firms are at the min@A( sie' &hat

Marsha!! &as imp!icit!" admittin &as demand !imits to the expansion of individua! firms* and this can on!"

$e discussed $" a$andonin pricetain and turnin to theories of imperfect competition. An examp!e of the

second cause miht $e the increasin averae si!! of specia!ied si!!ed !a$our &hen the dimension of an

industr" ro&s and &ith it ro&s the num$er of &orers &ho have ac#uired hih si!!s o&in to &or

experience* and the resu!t is a communit" &here expertise is reater* &ith a reduction in costs. ?ut such

externa!ities can on!" act ver" s!o&!"* on a time sca!e superior to that of the adQustments contemp!ated in

shortperiod or !onperiod e#ui!i$ration8 the time sca!e one considers &hen one discusses economic ro&th.

urthermore externa!ities externa! to firms $ut interna! to an industr" are ver" rare* enera!!" positive

externa!ities do not reduce costs on!" in one industr" and are therefore incompati$!e &ith partia! e#ui!i$rium.




e#ui!i$rium price to decrease and thus to tend to&ard min@A(F5H' the reverse process &i!! o on if

the shortperiod e#ui!i$rium price is !o&er than min@A(. Thus &e o$tain sta$i!it" of the !on

period e#ui!i$rium too.

5.23.2. The ana!"sis suests that it seems !eitimate to assume a tendenc" of the #uantit"

produced of the severa! products to adapt to the demand for them* at prices that tend to e#ua!

minimum averae cost. 7n (hapter 6 it &i!! $e seen that the consideration of adQustment !as can

raise dou$ts on this conc!usion* $ut it &i!! $e arued there that the difficu!ties are not ver" serious.

>o &e have some Qustification for $e!ievin that the assumption that &i!! $e made in the formu!ation

of the enera! e#ui!i$rium e#uations in 4art 777 of this chapter* of e#ui!i$rium product prices e#ua!

to minimum averae costs and of #uantities produced e#ua! to the demand for them at those prices*

ref!ects actua! tendencies. o&ever* the sta$i!it" of product marets thus assumed rests on iven

factor prices and iven demand curves' therefore it does not prove the sta$i!it" of the enera!

e#ui!i$rium of production and exchane* &hich re#uires in addition the sta$i!it" of factor marets

%and* to such an end* cannot assume iven demand curves for products $ecause chanes in factor

renta!s chane incomes and demands-. This &i!! $e discussed in chapter 6.

5.%3. elfare analysis5.2H.1. @et us no& prove that in a partia!e#ui!i$rium frame&or the competitive e#ui!i$rium

of a sin!e maret* determined $" the intersection of demand curve and supp!" curve* is a 4areto

efficient a!!ocation. ?ut &e must c!arif"* a!!ocation of &hatD :f the #uantit" produced* x* amon

consumers* and of income amon consumers and producers* under a tradeoff %ana!oous to a

production function- $et&een income %Lcost- and x. e consider t&o roups of maximiers8

consumers maximie uti!it"* producers maximie their income i.e. profit. The iven prices of a!!

other oods a!!o& their treatment as a Jicksian composite commodity* &hose price can $e madee#ua! to 1 and &hose #uantit" therefore can $e !eitimate!" identified &ith expenditure on oods

other than x* or residua! income "' thus consumer hs uti!it" depends on xh and "h* and &hen a

consumer pa"s p for a unit of xood she is ivin up p units of "ood' the producer ives a&a"

amounts of "ood as cost to produce the unit of xood' thus it is as if x &ere produced $" usin

5H =ote that for the process to push the price to&ard min@A( it is not necessar" that a!! firms have optima!

p!ants' it suffices that there $e entr" of optima! p!ants unti! supp!" o$!ies the price to e#ua! min@A(' this

process &i!! enera!!" $e faster than the process of c!osure and rep!acement of o!der p!ants %the economic !ife

of p!ants and dura$!e capita! oods is enera!!" much !oner than the time re#uired $" their production-* so

one must consider it norma! that a price ver" c!ose to min@A( coexists &ith p!ants that are not optima! and

earn residua! #uasirents.




#uantities of " as input. 7n e#ui!i$rium* consumers have the same M9> $et&een xood and "ood'

throuh an opportune choice of units for uti!it" the e#ui!i$rium marina! uti!it" of income can $e

rendered e#ua! to 1 for each consumer. This means pk* the e#ui!i$rium price of x* e#ua!s the

marina! uti!it" of x for a!! consumers active on the maret' producers are interested in maximiin

their income i.e. profit* so the" e#ua!ie price and marina! cost' hence M(%xk-LM<%xk- &here xk

is e#ui!i$rium output.

4roof of the 4areto efficienc" of the perfect!" competitive partia! e#ui!i$rium. e first prove that

production of a #uantit" different from the e#ui!i$rium #uantit" xk cannot $e 4areto efficient. 8areto

efficiency means that a 4areto improvement %a chane that maes some$od" $etter off &ithout main

an"$od" &orse off- is impossi$!e. 7f xxk* there &i!! $e some consumer read" to pa" for one more unit of x

a demand price pd reater than the e#ui!i$rium price pk* and there &i!! $e some producer &ho can produce

one extra unit at a marina! cost not reater than pk %&e admit the possi$i!it" of (9> and constant M(- and

&ou!d $e therefore read" to se!! it at a supp!" price p sKpk* hence these t&o aents can strie a mutua!!"

advantaeous $arain %to produce and exchane one extra unit at an" p intermediate $et&een p d and ps-

&ithout main an"$od" e!se &orse off. 7f xWxk* there &i!! $e some producer &ith a marina! cost not

sma!!er than pk &ho is read" to pa" a sum reater than or e#ua! to pk for the riht to reduce production $"

one unit* and there &i!! $e some consumer read" to renounce one unit of x for a recompense !ess than pk*

hence aain these t&o aents can strie a mutua!!" advantaeous $arain. Thus xLxk is a necessar" condition

for 4areto efficienc". @et us no& prove that no different a!!ocation of the production of xk amon producers can $e a 4areto

improvement* and no different a!!ocation of xk amon the consumers can $e 4areto efficient. A different

a!!ocation of the production of xk amon firms either !eaves marina! costs unchaned* or causes an increase

in marina! cost in at !east one firm* &hose profit decreases8 in either case there isnt a 4areto improvement'

in the first of these t&o cases 4areto efficienc" does not uni#ue!" determine the a!!ocation of the production

of xk amon firms. A different a!!ocation of xk amon consumers means that at !east one has more xood

and at !east one has !ess xood than in e#ui!i$rium' this means that their M9>s differ so the" can strie a

mutua!!" advantaeous exchane. [

$onsumer and producer surplus and :elfare c(anges.

R5.2H.2. e have seen in chapter H the definition of consumer surp!us in an industr".

8roducer surplus is ana!oous!" defined as the area a$ove the supp!" curve up to the horionta!

price !ine* cf. the trian!e A;( in i. 5.10. 7t is a more comp!ex notion than consumer surp!us. 7t

intends to measure the maximum amount of mone" that producers* that is* the ensem$!e of

entrepreneurs and factor suppliers invo!ved in producin the ood* &ou!d $e read" to pa" in the




areate rather than foro the possi$i!it" to produce and se!! the ood at the iven price. 7t too

assumes a constant marina! uti!it" of income.

rice

(

#nd1str s1l c1rve

* C

cometitive rice <

A demand c1rve

31antit

i. 5.10. Marsha!!ian tota! surp!us is the area of trian!e A?(* the sum of consumer surp!us %the area of

trian!e ;?(- and producer surp!us %the area of trian!e A;(-.

M( AS(

p* M(* AS(

p ;

A ?

:

( E output

i. 5.11. Graphica! proof that for a firm the area a$ove the supp!" curve up to the price !ine %trapee

A?;p- e#ua!s revenue minus varia$!e cost. hen price is p and therefore the supp!" of the firm is :E*

revenue is rectan!e :E;p* and the area under the supp!" curve :A?; of the firm e#ua!s tota! varia$!e cost* $ecause the area of rectan!e :A?( is varia$!e cost up to output :(* and* $" interation* the area under the

marina! cost curve from ? to ; is the addition to varia$!e cost caused $" increasin output from :( to :E.




7n the short period* the producer surp!us of a sin!e firm at a iven product price and

#uantit" supp!ied is defined as tota! revenue minus tota! varia$!e cost or e#uiva!ent!"* as pure profit

p!us tota! fixed cost. %9emem$er that fixed cost is $" definition a cost that the producer must $ear

&hether she produces or not. Thus in the short period the entrepreneur finds it convenient to

produce as !on as she is a$!e to more than cover variable cost.- That this is e#uiva!ent to the area

a$ove the shortperiod supp!" curve up to the horionta! price !ine is easi!" proved $" remem$erin

that tota! varia$!e cost is the intera! of marina! cost and therefore it is the area under the supp!"

curve* cf. i. 5.DD. (onsiderin fixed cost to $e a sun cost &hich cannot $e avoided* the

entrepreneur* rather than $e exc!uded from the maret* &i!! $e read" to pa" up to the amount that

&ou!d !eave her &ith &hat is Qust sufficient to cover varia$!e cost* amount &hich is measured $"

that area. The sum of these areas is the area a$ove the industr"s supp!" curve.

7n !onperiod ana!"sis a!! cost is varia$!e' if the industr"s supp!" curve is horionta!

$ecause a!! factor renta!s are iven* producer surp!us is ero. ?ut if the industr" uses a specia!ied

input &hose renta! rises &ith industr" supp!"* the !onperiod industr" supp!" curve is up&ard

s!opin* so producer surp!us is positive. o&ever* profits are ero a!! the same* $ecause at each

point of the supp!" curve the specia!ied inputs price is iven and each firm treats input prices as

iven and produces the #uantit" that minimies averae cost* i.e. that causes averae and marina!

cost to coincideF55. o& is the positive producer surp!us reconci!ed &ith the ero profitsD The point

is that the rise in the renta! of the specia!ied factor due to the expansion of production imp!ies an

income ain for the supp!iers of that factor* so they &ou!d $e read" to pa" rather than see production

of the ood for$idden. The maximum amount the" &ou!d $e read" to pa"* aain mis!eadin!" ca!!ed

producer surp!us %it is in fact a consumer surp!us of the consumers &ho supp!" the factor-* is

actua!!" measured $" the area a$ove the factors supp!" curve %in the raph &ith factor supp!" and

factor renta! on the axes- for reasons simi!ar to those definin the consumer surp!us on the demand

sideF56' for examp!e if the supp!" of the factor is riid %imp!"in a ero reservation price of the

factor o&ners-* the producer surp!us is the area of the rectan!e formed $" the axes* the vertica!

supp!" !ine* and the factor renta! horionta! !ine. This area coincides &ith the area a$ove the

industr"s !onperiod supp!" curve $ecause the area belo, the !atter curve is the pa"ment to the

other factors8 this is made c!ear $" noticin that* if tota! production came from a sin!e firm &ith the

55 Thus a !onperiod industr" supp!" curve has averae and marina! cost coincide at a!! points even &hen it

is up&ards!opin.56 Aain* under the assumption of constant marina! uti!it" of mone" !ess easi!" Qustifia$!e in this case 7t is

ver" unusua! that income from o&nership of a specia!ied factor of production $e a ver" sma!! part of oneCs

overa!! income.




specia!ied input as fixed factor* the supp!" curve &ou!d $e the marina! cost curve of this firm* and

the area under it &ou!d measure its varia$!e cost.

4roof. e prove it for a simp!e case. Assume that the ood is produced $" unspecia!ied !a$our @*

&hose &ae & is constant and for simp!icit" e#ua! to 1* over a specia!ied !and T in riid supp!"* &hose

surface* aain for simp!icit"* is measured in such units as to mae it e#ua! to 1' the (9> differentia$!e

production function is #Lf%@*T-' each marina! product is a decreasin function of the ratio of the factor to

the other factor* and $ecomes neative a$ove a sufficient!" hih ratio* in &hich case as exp!ained in ch. 3 the

factor &i!! not $e fu!!" uti!ied* on!" the amount "ie!din a ero marina! product &i!! $e uti!ied. @and is

fu!!" emp!o"ed if possi$!e* so &e consider # a function of @ on!"* #Lf%@- &hich &e assume inverti$!e* @%#-Lf

1%#- &hose derivative is 1/M4@%#-. :ptima! !a$our emp!o"ment re#uires &LM4@`p &here p is the productCs

price' since &L1 is constant* it must $e pL1/M4@* and an output increase re#uires p to rise if M4@ is

decreasin' this causes the !and renta! X to rise too* $ecause !and receives its marina! revenue product too*

and M4T rises as !a$our emp!o"ment rises* except initia!!" &hen !and is not fu!!" uti!ied and its marina!

product is ero. Thus X is a function of #* XLX%#-* and initia!!" it is ero8 the opportunit" cost of !and T is

ero* the entire !and revenue is CrentC or producer surp!us. At each #* supp!" price is defined $" e#ua!it" &ith

averae cost* p%#-L%X%#-@%#--/#* and the function p%#- thus defined is the supp!" curve. e &ant to sho&

that* iven the #uantit" produced #k* tota! industr" revenue p%#k-#k minus the area under the supp!" curve

e#ua!s X%#k-. >o &hat &e need to sho& is that the area under the supp!" curve is @%#k-. =o&* p a!so satisfies

p%#-L1/M4@%#-' therefore the area under the supp!" curve is the definite intera! ∫ k

0 -%1

$

$ 98 d$

L@%#k-

@%0-L@%#k-. [

Exercise8 Assume &ine S is produced $" !a$our @ over 1 unit of specia!ied !and in fixed supp!"

accordin to the production function SLT1/2@1/2. @a$ours &ae in terms of other oods is fixed and e#ua! to

&. 4rove that the industr"s !onperiod inverse supp!" curve is pL2&S and that the area $e!o& it for an"

iven S e#ua!s the &ae pa"ments to the !a$our emp!o"ed to produce that S.

Actua!!"* shortperiod producer surp!us in a competitive industr" is determined in the same

&a"* $ecause as arued in R5.1H* the entrepreneursC shortperiod profits shou!d $e seen as earnins

accruin to the fixed factors.

7f for some reason %e.. riid factor contracts stipu!ated in the past under different

conditions- there are profits* then this is a su$traction of part of the surp!us from the factor

supp!iers* $ut the area a$ove the marina! cost curve sti!! measures producer surp!us* on!" no&

consistin part!" of entrepreneuria! profits and part!" of factor supp!iersC surp!us.

9arshallian aggregate producer surplus* or simp!" producer surplus for $revit"* is the sum

of the individua! producer surp!uses in a maret. And the sum of consumer and producer surp!us is

ca!!ed the 9arshallian aggregate total surplus. 7t is the area of the trian!e formed $" demand

curve* supp!" curve* and ordinate axis.




7t is eas" to see raphica!!" that the competitive partia! e#ui!i$rium maximies tota! surp!us.

An" price or #uantit" different from the e#ui!i$rium ones &ou!d ration either $u"ers or se!!ers*

$ecause there &ou!d not $e enouh production or enouh demand.

7t is important to $e a&are of the !imits of the a$ove conc!usion8 the arument $rinin to it

has ne!ected externa!ities* has taen incomes and factor propert" as iven* and concerns the maret

for a consumption ood.




c EE9(7>E>

5.11 The production function is (o$$;ou!as &ith (9>* #LAxV"1V. @et factor prices $e vx

and v". ind the cost function (%vx*v"*#- and confirm >hephards @emma $" sho&in that indeed

^(/^vxLx.

5.12 Exp!ain riorous!" &h" constant returns to sca!e and divisi$i!it" imp!" that iso#uants

are convex.

Advanced Microeconomics Petri Ch 5

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