Malin Vaud Chap 3

8/3/2019 Malin Vaud Chap 3

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44 The consumer

(I)general

preference relation also is perfectly known. Contrary to Samuelson'ssuggestion, therefore, it is possible to determine this relation from directobservation of consumer behaviour wi thout having to confr ont theconsumer with a series of binary choices.3

The producer

L DefinitionsWe come now to the activity of producers, also called 'firms'. This will be

investigated in two successive stages. First of all we shall study the representa-tion of the technical constraints which limit the range of feasible productiveprocesses. We must then formalise the decisions of the firm which must actwithin a certain institutional context. Our discussion will be carried onmainly in the context of 'perfect competition', which cannot pretend to bean always valid description of real s itua tions. But i t is the ideal model onwhich the study of the problems of general equilibrium arising in marketeconomies has been based so far.As in our discussion of consumptiQn theory, we shal l omit the index jrelating to the particular agent considered. So ah, bh and Yh will simplydenote input, output and net production of the good h in the firm in question.

For the purposes of economic theory, a detailed description of technicalprocesses is as pointless as knowledge of consumers' motivations. All thatmat te rs in this chapter is that we should formalise the constraints whichtechnology imposes on the producer . These can be summarised in a verysimple way: certain vectors Y correspond to technically possible transforma-tions of inputs into outputs; other vectors correspond to transformationswhich are not allowed by the technology at the disposal of the firm.To take account of this, we need only define in Rl the production set Yasthat set containing the net product ion vectors which are feasible for theproducer. Thus the demands of technology are represented by the simpleconstraint

Y E Y.(We must not forget that Y relates to a particula r producer; inequilibrium theory, each producer j has his own set Yj .)Of course, all the technically feasible transformations are not of interest


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a pnon; some may require greater inputs and yield smalfer outputs thanothers. The firm's technical experts must eliminate the former in favour ofthe lat te r. Thi s is why we can often confine ourselves a priori to technicallyefficient net productions. By thi s we mean any transformation which cannotbe a lt ered so as to yield larger net production of one good without thisr esul ting in smaller net production of some other good. Relative t o s uch atransformation, therefore, output of one good cannot be increased withoutincreasing input or reducing output of another good.

Formally, the vector yl is said to be technically efficient if it belongs to the

llset Y of feasible net productions and if there exists no other vector y2 of Ysuch that

y / :? y/; for h = 1,2, .. . , I.So the technical ly efficient vectors y belong to a subset, or possibly to the

whole, of the boundary of Y in the commodity space.tIn the construction of optimum and equilibrium theories we could impose

on ourselves to use the production set Yas the sole representation of technicalconstraints. This is the method adopted in the most modern approaches tothe subject. Following a tradition of almost a century, however, mathematicaleconoml'sts often introduce another more restrictive concept, that of the'production function', which formalises in particular the idea that marginalsubstitutions between inputs are feasible.

Actually, in their approach to the problems of general equilibrium economists have alternatively used two types of formalisations, which stress twoopposing features of product ion. One fea tu re is the existence of 'proportionalities' or 'coefficients of production': some input s mus t be combined ingiven proportions, like iron ore and coal in the process of producing pigiron. Another feature is the possibility of substituting an input for another:machines can replace men, one fuel can be substituted for another, more orless fertilizer can be pu t in a given piece of agricultural land and more or lesslabour can be spent on it, hence the same crop may be achieved with a l it tl eless fertilizer and a l it tl e more labour.

Economists such as K. Marx or L. Walras in the first editions of histreatise constructed their systems assuming fixed proportionalities, i.e. complementarity between inputs. Others like V. Pareto have used formalisationsimplying that substitutabilities are everywhere prevalent. The great advantageof the modern set theoretic approach is to cover both complementari t ies

t Rigorously, we can confine ourselves to technically efficient vectors only if, corresponding to every y of Y, we can find an efficient y* such that ) ' / ~ :> )'h for all h. This will bethe case if Y is a c losed set and if, without leaving Y, we cannot increase onecomponent ofy indefinitely without reducing another. It docs not restrict the validity of the theory toassume this.

(3)

(2)

47Dejinitiol1sand substitutions. The definition of Y can take into accoun.t s i m . u l t a n e o u s : ~the substitutability of machi ne s f or men an d the proportIOnality ~ e . t w e h '. and coa l Hence t he theory built directly on Y IS fully genera III t ISIron ore, .respect . fWhe;1 we want to build models that lend themselves to c o m p u ~ a t l O n ordea ling with quest ions of applied economics, we have ~ h cholce

dt o ~ a y

between two types of more specific f o r m a ! i s ~ ~ i ~ n : either pro u C ~ l O nfunct ions, usual ly allowing fo.r large s ~ ~ s t i t u t a b I ! J t l e s , or fixed coeffiCIentprocesses combined into 'actIVity analySIS models. d .

L e ~ t u r e s such as the pr es ent ones should no t ignore th.e pro uctlOnfunction concept. In fact it will b e used extensively with ~ h aIm of m ~ k l l 1 g. . .' d to a llow the f ree use of differential calculus. omecxposltlon easier an . Y bessel1tJal proofs will be g iven under the assumption that the se.ts .i can e, . h h' tlOn IS no t rerepresented by production functIOns, even th.oug t I assump h' f b'd for the validity of the result. ProductIOn functIOns must t ~ I ore. eqUIre h 11 . t t 111 passll1gdeflned and discussed with some care. Later on we s a P01l1. ou

t h ~ s e places where the use of such functions conceals some difficulty.!:::.J?D2..duction {unction (for a particular finn is, by definit ion, a real function

defined on R 1 such that:I t ] ' ! , Y2, ..., ) '/) = 0

if and only if Y is an efficient vector, and such that/(1'[,)'2, .,Y/) 0

if and only if) ' belongs to Y. . ". . fi d bFor the moment we shall not inquire Into the c o n d l ~ l O n s to be ~ a : l s e .yY if weare to be able to define such a function. ThiS wIiI be dIscussed 111Section 2. . . esentAccording to this definit ion, we can use (I ) or (3) eq.Ulvalentl y todrepr ththe technical constraints on productiont (the functIOn / depen s on eparticular producer j, as does Y). . .Geometric illustrations of the production set and the productIOn f u ~ c ~ ~ o nare often fruitful. Suppose, for example, that there are four commo lIes,the first two of which are outputs of the firm and the las t two Il1puts. t g ~ e s o l .and 2 represent two intersections ,of Y, t l ~ : ~ t bY= \ ~ r . p ~ : I ~ ~ ~ t [ ~ e ~ i a ~ ~, _ ;0 ) the second by a hyperplane (YI - J ! , Y2 . 2 .,

) 4 - ) 4 t' tl set of tJv' productions that are feasible from the quantllJesrcpresen s 1e .-t'- We may point out that, like tile Ulility] ffunccttiOonn" w t h l e t h P ; ~ : ~ ~ : ~ 1 ~ n S ; ~ ~ c ~ ~ o i ~ s l : : : u ~ ; ~ t ~. I F ,Ie If IS a rea un Idefined unique y. or examl , . then (f) corresponds to the same se t as f.and which is zero when Its argumcndt IS ~ r o , tl .n consumption theory, we shall not laySince this has already been dlscusse sU 1Clen y Ifurther stress on it.

The producer6


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48 The producer Definitions 49The function g* will generally be increasing with the ahand thefunction g willconsequently be decreasing with respect to the Yh' or at least non-increasing.

Later on we shall often assume that the function f is twice differentiable.Let yO and yO + dy be two neighbouring technically efficient vectors. We canwrite

a = - Y and a2 = - y2 of the two inputs; the second represents the set ofinputs allowing the quantit ies b? = Y? and bg = yg of the two outputs to beobt ai ne d. The poi nt s satisfying (2) are represented by the North-Easiboundary on Figure 1 and the South-West boundary on Figure 2. ryve note

'. in passing that a s et which, like the curve in Figure 2, represents the technically

IIe f f i ~ i e n t combinations of inputs yielding given quantities of outputs is calledan zsoquant.)

/L f dYh = 0h= 1 (8)

(9)

(11)

(12)

(10)

s =f. I

s, r =f. 1.

for

for

- g

dYr g,;- - ==dy', g;

and

The ratio (II) measures the inc rease in production resulting from anincrease of one unit in the input of s (note that Ys is equal to minus the input).It is often called the marginal productivity of s. The rat io (12) defines, apartfrom sign, the additional quantity of input of r which is necessary to compensate in output for a reduction of one unit in the input of s. This is, in fact, amarginal rate of substitution.We note also that the first derivativesfh of the production functionfmust

take non-negative values at every technically efficient point yO. Consider asmall variation dy all of whose component s a re zero except dYk' which isassumed positive. Since yO is technically efficient, yO + dy is no t technicallypossible, that is, f(yO + dy) is pos it ive. But , s ince f(yO) is zero, f(yO + dy)can be posit ive only if fk is no t negative.

where.fh denotes the value at yO of the derivative of f with respect to Yh'In particular, if all the dYh except two, dYr and dys' are zero, then (8) reducesto

dYr f dy', 7 ~

The ra ti o on t he r ig ht hand side of (10) can be called theJnarginal rate ofsubstitution b . ~ L i h L g Q Q d s - L ~ l l i L L . f u r the producer in gueiliml, Thisexpression is similar to that encountered in consumption theory. To avoidconfusion, we shall sometimes speak instead of the !!!:.arginal r{lte of t r a ~formation.

I n the par ti cu la r case where f t ak es t he f orm (4), equalities of the type(10) become

or

Fig . 2c : o - r - - - - - - - - - - a 3 - = - _ - ~

Fig. 1

*YI g(Y2' ..., Yf) (5)and the expression 'production function' is a ls o used for the function gwhich defines the output resulting from given quantities of inputs. Thereshould be no real possibility of confusion from this ambiguity.Note tha t we could show inputs and outputs explicitly in (5). Thus

b i g(- a2, - aJ, ... , - aa (6)or, after an obvious change in notation,

b l g*(a2, aJ, ..., az) (7)

The mos t general form of a production function is that in (2). Slightlymore particular expressions are often used. Thus i t is often assumed that thefirm h as onl y one output, the good I, to fix ideas ; the product ion funct ionis then given the form:t

~ f ( Y I ' Y z , ",Yf) = YI - g(Y2' ... ,y/). (4)The technical constraint is

b Y2 2

t Obviously this part icular form is no longer affected by the indeterminacy alreadymentioned in relat ion to the general form (2), Here the function g representing a givenset Y is determined uniquely. In fact , even i f these are several outputs , in most cases wecan solve the equali ty fCy) = 0 fo r YI and so revert to (5).


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50 The producer The validity ofproduction functions 512. The validity of production functions

We must now investigate the conditions to be sat isfied by the productionset Y in order that, first of all, there exists a product ion funct ion/ , and in thesecond place, that this function is differentiable. These conditions arecertainly more restrictive than i t would appear at first glance.

Differentiability implies that f is continuous and consequently that Y is a, closed set in RI. This property is not restrictive; if the vectors {y\, y 2 , . . ,}of a convergent sequence each define a feasible production then the limitingvector certainly corresponds in reality to a feasible production.

2

efficient vector o f Yand thatf(y) > s satisfied for every vector y outside Y.(At a point such as N,/(y) should be equal to a n e g a t i v ~ num?er, b ~ s h o u l ~Ie positive ~ o e v ~ r y point n ~ a r !y whose second coordll1ate IS pOSItIve; thISis incompatIble WIth the contll1Ulty o f f at N.)However, we cantake account of these limitations by altering the definitionof the production function and explicitly adding inequalities to the formalrepresentations of the set Y and the set of technically efficient vectors. Forexample, to characterise Y we rep lace (3) by

{ [(Y\,YZ' ... ,y/) 0, (13)Yh for a specified list of goods h.To characterise the se t of technically efficient vectors, (2) is replaced by

{ Yl + o:Y2 = (16)yz 0.This complication will no t be taken into account in ou r discussion of the

general theories. That is, we shall proceed as if the limits on the domains J2fvariation of t h e . . . . l ' ~ r e neveriil force. Aswe-sawli1-consumption theory,~ e r t a i n new particular features are revealed if we take account of constraintsexpressed by inequalities, bu t this does not alter basically the nature of theresults. We shall presently return to this point.

(ii) In the second place, in some productive operations the different goodswhich constitute input s mus t be combined in fixed proportions. This isparticularly the case for most of the raw materials used i n many industrialprocesses.When such proportionality ratios exist, the i soquan ts do not have the

same form as in F igu re 2. If there is free disposal of surplus, they look likethe isoquant in Figure 4. Apart f rom the surplus of one of t he two inputs,a3 and a4 must take values whose ratio corresponds to that defined by thehalf-line GA. Except at the point A, the half-lines AN and AM correspond tonon-technically efficient productions. At the point A, the first derivatives o f fwith respect to Y3 and Y4 are not continuous. (The situation is similar t o t ha tin Chapter 2, with the utility function (8) illustrated in Figure 7.)The real situation is sometimes less clear-cut than Figure 4 aS5umes, since

M

Fig. 3But the continuity o f f implies also that every point y* on the boundary ofYsatisfiesf(y*) = 0 since i t can be approached both by a sequence of vectors

y such thatf(y) 0 and by a sequence of vectors such thatf(y) > 0. So t he 1\definition o f f implies that every point y* on the boundary of Y is technicallyefficient. Moreover, differentiability assumes that, with respec t to anytechnically efficient vector, the marginal rates of substitution are all welldefined. Taken literally, these consequences are ditDcult to accept.(i) In the first place, the domains of variation of all, or some, of the y" maybe limited. For example, technology may demand that some good r occursonly as input and some other good s only as output. So the inequalitiesYr nd Ys 0 appear in the def in it ion of Y. ( In fact, t he second inequality can be eliminated if we assume that the firm can always dispose ofits surplus without cost, s ince this assumption is naturally expressed as :yO E Y and y" y for all h implies y E Y.) Because of the limits on thedomains of variation of SOll,e y", the set Y has boundaries corresponding tonon-technically efficient )roductions ( for examp le, the half -li ne GN inFigure 3).The existence of such boundaries is incompatible with the cOlltinuity of ftogether with the conditions thatf(y) < s satisfied for every non-technically

{ f(Yl' Y2, ..., y/) = 0,y" for the same list of goods h.Thus, for Figure 3, (13) and (14, become

{ y\ + o:yz 0,yz 0,and

(14)

(15)


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52 The producer Assumptions about production sets 53

t It is the aim of a new branch of economic science, 'activity analysis', to integrate intothe theory formalisations which describe technical constraints more accurately than doproduct ion funct ions. A very good account of thc resulting modifications is given inDorfman, Application of Linear Programming to the Theory of the Firm, University ofCalifornia Press, Berkeley 1951. See also Dorfman, Samuelson and Solo'.", Linear programming alld activity analysis, McGraw-Hili, New York, 1958.

3. Assumptions about production setsWe must n ow discuss cer ta in assumpt ions which are frequently adopted

about production sets or production functions.ADDITIVITY. If the two vec tors y 1 and y 2 define feasible productions

(y l E Y and y2 E Y or f( y 1) 0 andf(y2) 0), thenthe vector y = )'1 -+ y2defines a feasible production (therefore y E Y or fey) 0).

This appears a natural assumption. For; it s eems that we can alwaysrealise y by realising independently yl and y2. Additivity fails to hold onlyif yl and y2 cannot be applied simultaneously. A pr io ri there seelTl$ noreason for this to be the case.

However, it may happen that t he model does not identify all t he commodities which in fact occur as inputs in production operations. Fo r example,if t he l and i n t he possession of an agricultural undertaking does not appearamong the commodities, then additivity does not apply to its production set,since, if the available land is totally used by yl on the one ha nd and by y2on the other , realisation of yl -+ y2 requires double the actually availablequantity of land. Similarly, if the capacity for work of the head of anindustrial firm does no t appear among the commodities, a nd i f his capacitylimits production, then additivity no longer strictly applies.

encountered in their rigorous exposition. The changes in product ion theoryintroduced by their presence will be described briefly. tFinally, we see that the above-mentioned difficulties can be avoided if we

base ou r reasoning directly on the set Y of feasible productions and on theset of technically efficient productions rather than on the production function.This is the approach adopted in the most modern treatments of the theorieswith which we are concerned here.As when a uti li ty function is substituted for a preordering of consumerchoices, the substitution of a production function for a production set makesexposition easier since it allows the use of the differential calculus and offairly standard types of mathematical reasoning. Moreover, this approachalone leads to certain results which every economist must know. Knowledgeof these results is essential for the student, even if their application is somewhat restricted by the simplifications r equi red to justify the product ionfunction.

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Fig . 5o~ I C - - . _

Fig . 4

fA M~ I ~ - - - 8 _!/3 3

Y4 = aY3'I n the case of two techniques, as in Figure 5, the supplementary constraints

may be

there may be available t o t he firm two or more production techniques eachrequiring fixed proportions of inputs, the proportions differing for thedifferent techniques. Figure 5 relates to an example of two techniques, thefirst represented by the point A, the second by the point B. The firm canemploy the two t ~ c h n i q u e s simultaneously to produce the same quantit ies ofoutputs. Fo r example, if each technique can be employed on a scale reducedbyone half relative to that represented by A or B (the assumption of constantreturns to scale, to be defined presently) then the same output can be obtainedby simultaneous use of the two techniques on this new scale; the poi nt onFigure 5 corresponding to this method of production is the midpoint of AB.

Similarly, each point on AB def ines a possible combination of the twotechniques yielding the same output as A or B. In this case, the first derivatives offare in fact continuous at each point within AB, but not a t A nor a t B.In order formally to represent such situations as those of Figures 4 and 5,

we can add other constraints to the equation fey) = 0 to characterise the setof technically efficient vectors. Fo r example , if, as in Figure 4, there must bea fixed proportion between Y3 and Y4' we write:

- f3YJ - Y4 - aYJ. (18)The theory becomes very complicated if such const ra in ts are taken into

account. Fo r this reason, they a re bet te r i gnored in a course of lectureswhose a im is to provide the student with a sound grasp of the general logicof the theories to be discussed rather than the difficulties which are


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54 The producer Assumptions about production sets 55

DIVISIBILITY. If the vector yl defines a feasible product ion (yl E Y or(fyl) :s;; 0) and if 0 < a < 1, t hen the vec tor ayl also defines a feasibleproduction (therefore ayl E Y and f(ayl) :s;; 0).

This assumption is much less general ly sat isfied than the previous one.It assumes that every productive operation can be split up and realised on areduced scale without changing the proport ions of inputs and outputs.Taken l ite ra lly , i t can be said to be rarely satis fied . For every productiveopera tion there is cer ta in ly a level below which i t cannot be carried out inunaltered conditions. But this indivisibility may vary in its degree of effectiveness and in many industrial operations it appears negligible.

CONSTANT RETURNS TO SCALE.t If the vector yl defines a feasible production(yl E Y or f(yl) :s;; 0) and if f3 is a, posit ive number, then the vector f3y l alsodefines a feasible production (therefore f3y l E Yandf(f3y l) :s;; 0).

,Obviously the constant ~ u r n s defined by this assumption imply divisibility..#. C o n ~ ~ d i t i v i t y and divisibility imply constant returns to scale. F ; ; ~le t k be the integra l part -of f3; we (;ill apply the-property oTadditivit;repeatedly, taking the vectors yl , 2y l, ..., (k - I)yl successively for yZ andthus proving that 2y\ 3y\ ..., ky l are feasible; divisibil ity shows that(f3 - k)yl is feasible; finally, additivity shows that f3y l = (f3 - k)yl + ky lis feasible.

In practice, we shall consider that returns to scale are constant preciselywhen addit ivity and divisibility ca n be considered to hold, alth oughrigorously, additivity is not necessary.

Consider the particular case where the technical constraints are expressedin the form (5). I f the function 9 is homogeneous of the first degree, then theassumption of constant returns to scale is clearly satisfied.

Conversely, constant returns to scale imply thatg(f3Yz, ..., f3y/) = f3g(yz, ..., y/)

for every vector Y and every positive number fJ. Indeed, on the one handthe hypothesis implies, by definition,g(f3yz, ..., f3y/) > f3Yl = f3g(yz, ..., y/),

since f3y is feasible whenever Y is feasible, On the other hand, t he samehypothesis implies:g(Y2, .. ,Ya > Yl = g(f3Y2' .. . , [JY!)lfJ

since Y = :::1f3 is feasible whenever::: (= f3y) is feasible. The two precedinginequalities do imply pos itive homogeneit y, as was to be p roved .t The expression 'constant returns to scale' is explained as follows: if the first good isthesole output, the return with respect to the inputI in the productive transformationy ' is,

by definition, the ratioyi!e - yn. This assumption specifies that the volumeof output canbe changed without changing the return with respect to anyof the inputs.

To characterise the second of the above assumptions, we often speak of'non-increasing returns to scale ' rather than of divisibility. The relationshipwith the assumption of constant returns is obvious from the above formulations . However, there must not be any confusi on of the assunlQ!ion ofdivisibility, or non-increasing returns to s c a i e ~ with the ~ - t i ( ; n o j n c r e a s i D K m a r q i ! ! g L r e t u r n - s ; ~ ~ i 1 . 1 . L ' 0 : ' b j c h _ V j _ ~ _ shilJLs h Q ! ~ be concerned.-W e 'also speak of non-decreasing returns to scale when f(yl) :s;; 0 (ory 1E Y) and a > 1 imply f( ay l) :s;; O.Figure 6 illustrates the three situations for the case of a single input and asingle output. The production set bounded by r I relates to constant returnsto scale, that bounded by r Z to decreasing returns and that bounded by r 3to increasing returns (of course, a given production set may come into noneof these three categories).

b =YI I

oFig, 6

CONVEXITY. If the vectors yl and yZ define two feasible productions andif '0 < a < I: then the vector ayl + (I - a)yZ defines a feasible production.

In short , there is convexity if the set Y contains every segmentjoining twoof its points. Figures J and 2 correspond to the intersections of a convex setYof R4. Similarly, the sets in Figures 3, 4 and 5 satisfy the assumption ofconvexity. Finally, in Figure 6, the set bounded by r 3 is not convex, and theother two sets are.Obviously divisibility and additivity imply convexity. Since the null vectornatural ly belongs to Y, convexity implies divisibility i l 1 - . . E . ~ e . (To show

- - . . . . ~ ~ - - - - - < - - _ . _ - _ . -


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56 The producer Equilibrium for the firm in perfect competition 57

h = 2, ..., I,

this, we need on ly app ly t he property of convexity, taking the null vectorfor y 2 .)Convexity has consequences for the second derivatives of the production

function. To investigate these consequences, we shall deal with the case of afunction of the form

Here gj, is t he value at yO of the first derivative of g with respect to y".Similarly gi:k is the value at yO of the second derivativeof g with respect toy" and J'k. The two numbers e and 1] are infinitely small with the dy".

Subtracting (22) multiplied by CI. from (23), and taking account of the factthat 0 < CI. < I, we have

1 1

I I g ~ dYh dYk :::; 0 (24)h=2 k=2

4. Equilibrium for the firm in perfect competitionWhen dealing with the consumer, we reduced the problem of choosing

the best consumption complex to that of maximising a utility function. Weshall nowassume that the firm tries to maximise thenet value of its production:

I I IPY = I PhI'" = I Ph b" - I Ph a". (25),,= I ,,= I ,,= I

To conclude our discussion, we return to the two reasons mentionedearlier for departures from additivity and divisibility.

The fact that certain factors available in limited quantities have not beentaken into account explicitly in the formulation of the model obviously doesnot affect the marginal returns to the other factors. On the other hand, thisfact may expla in why we choose . f unct ions for wh ich r etu rns to scale ar ediminishing, while additivity implies constant returns.

The presence of considerable indivisibilities may explain the appearanceof production functions with increasing returns to scale for which theassumption of non-increasing marginal returns is not satisfied.

M. Allais suggests that we distinguish two situations. In some branches ofproduction, divisibility can be considered to be approximately satisfied toa sufficient extent. In this situation we usually find that production is carriedon by a relatively large number of technical units functioning in similarconditions. The technology of this branch satisfies the assumption of constantreturns to scale. M. Al lai s uses the term 'di fferentiated sector' to cover allproductive activity of this ,kind.

In other fields, considerable indivisibilities exist. The market for each ofthe goods produced is then served by a very small number of very largetechnical units. To represent this situation, M. Allais assumes that a singlefirm exists in each such field, all of which const it ut e what he calls the'undifferentiated sector'.

This dis tinc tion will be taken again later , notably in Chapter 7 when weshall consider economies involving a large number of agents.

that is,0(- g ~ :::; O.o( - Yh)

The marginal return to h(ogjoah = - gj,), also calledthe marginal productivity,is therefore a decreasing function of the quantity of input h used (ah = - Yh)'We should point out that diminishing marginal returns and constant

returns to scale are not contradictory, as can be verif ied from the functionYI = -JY2 Y3- A l s o , _ ~ ~ _ ~ _ i ~ i ~ i . ! r ! n d _ ~ ~ v i s i b i l i t y imply both c o n s t a n t ~ t u r n s . ! ?scale and convexity, therefore non-increasing m a r g i n a ~ . r e t ~ r n s . '*_ . _ - ~ _ . _ - - - ~ - ~ - _ . _ . _ - - _ . _ . _ - - ~ ~ < ~ - - - - - - - , - - -

(23)

YI = g(Y2, ..., YI)' (5)Consider two infinitely close vectors yO and yO + dy which satisfy (5):y? = g ( y ~ , ..., Y?) (19)

andY? + dYI = g ( y ~ + dY2' ..., Y? + dYI)' (20)

If 0 < CI. < I, then yO + (X dy i a possible vector; it therefore satisfiesY? + (X dYl :::; g ( y ~ + CI. dyz, ..., Y? + CI. dYI)' (21)

Let us assume that the second derivatives of g are continuous. Expandingthe right hand sides of (20) and (21) up to the second order , and takingaccount of (19), we obtain

1 (1 + e) 1 IdYI = I 0;, d.l\ + -2'- I I g ~ dy" dYk (22)"=2 "=2 k=2and

(the multiplier CI.(CI. - I + Cl.1] - e) IS certainly negative if the dy" aresufficiently small).Since a priori the dy" can have any values, convexity implies that the

IJIatrix G" of the second derivatives ghk i.)' negative definite or negative semi-d . ! f i n i ~ ~ : - - - - - - " - ' - - " - - - - - ' - - - ' ' ' - ' ' ' - - - - - - - - - - - - - - - - ...Conversely, it can be shown that, if G" is negative definite for any system of

values given to Y2, Y3, ..., YI, then the assumption of convexity holds.The condition on G", which we have just established, is a general form of

the assumption of non-increasing marginqlrc!urns . In particular, this conditionimplies


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58 The producer Equilibrium fo r the firm in perfect competition 59

t Concerning the difficulties faced by the theory of management of f irms and the m a n ~references dea ling with it, see H. Leibens te in , 'The Mlssmg Lmk: Micro-MIcro Theory,Jot/mal oj Economic Literature, June 1979.

inadequate for bui ld ing a ' theory of the firm' that could serve as a gene.ralconceptual framework for the discussion of the many problems concernmgdec is ions to be taken by business managers. We must remember that themicroeconomic representation considered here aims at a theory of pricesand resources aIlocation not a t a theory of the management of the f i r ~ ~ . t

Adopting the assumptions of profit maximisation and perfect competitIOn,and us ing a production function representing the technical constraints, wecan easily determine equilibrium for the firm. We need on ly maXImIse pysubject to the constraint

f(.1'I' .1'2' .. , )'J) = O. (26)(In what follows, we assume that no price p" is negative, so t h ~ t the firm. losesnothing by limiting itself to technically efficient net productIOns. ObvIOuslywe also assume that the price vec to r is not identically zero.)

If we follow the same approach as for consumption theory, we should no.winvestigate the existence and uniqueness of equilibrium. We shall not do thIS,which in any case raises some difficulties of principle (see the footnote at th estart of Section 6). So we shall go straight on to conside r the margmalequalities satisfied in the equilibrium. . ,Maximisation of (25) subject to the constraint (26) IS a sImple case ot theclassical problem of constrained maximisation. The necessary first orderconditions for a vector yO to be a solution imply the existence of a Lagrangemultiplier A such that

P h = A j ~ h=I ,2 , ... ,1 (27)where f(, is the value at yO of the derivative of f with respect to Yh' Fo r theapplication of theorem VI of the Appendix, it is a ssumed here that the fhare not all simultaneously zero. It foI lows from the remark at the end ofSection 1 that the ff< are not negative and consequently that A is positive.

Conditions (27) imply

This expression, which is the amount by which the value of outputsexceeds the value of inputs a lso def ines the 'prof it ' that the firm derivesfrom production. In fact , t he mic ro economic theory wi th which we areconcerned considers the behaviour of the firm to be motivated by its desire tor eali se the g re at es t possibl e p ro fi t subject to the con st ra in ts imposed bytechnology and the institutional environment. This assumption, adopted inall theories of general equilibrium, has been subject to criticism. However,no alternative has so far been suggested which stands up to examination andcan provide the basis for a general theory.t Also, some criticisms arise frommisunderstanding of the wide generality of the model under study. In orderto avoid the same errors, we shaIl later discuss the definition of 'profit' whentime and uncertainty are taken into account. Fo r our present purposes it issufficient that the assumption of profit maximisation seems to afford the bestway for a simple systematisation of the behaviour of firms.Again, we cons id er the firm to be in a s itua tion of perfect competition if:- the price of each good is perfectly defined and exogenous for the firm,

and therefore independent of its production decisions;- and if, at this p rice , the f irm can acquire any quantity it requires of a

good, or dispose of any quantity it has produced.Of course, this is an abstract model of real situations. BasicaIly, it assumes

that the firm is small relative to t he market , s o that i ts act ions have noinfluence on prices. Moreover, it assumes that the demands and suppliesemanating from other agents are completely flexible so that they can reactinstant ly to any supply or any demand emanating f rom the particular firm.This model is clearly inappropriate to the 'undifferentiated sector' . At theend of this chapter we sha ll d iscuss the case of t he firm in a monopol is ti csituation and in Chapter 6 we shall briefly consider the formulations proposedfor other situations of imperfect competition. When in Chapters 10 and11, we shal l h ave expl ici tl y introduced time and uncertainties, we shallalso understand that strictly speaking perfect competition implies a muchricher market system than the one actually prevailing.Thus, the hypotheses of profit maximisation and perfect competition

have the advantage of being simple, but t hey lead to an idealisation thatmay look s trong with respect to an essentially complex reality . I repeatthat these hypotheses are introduced here in order to permi t the buildingof a general equ il ib rium theory and that, for this pur pos e, t hey mayprovide an admissible first approximation. They would on the contrary bet r mus t, however , men ti on here the exi stence o f a general e qu il ib ri um theo ry foreconomies with labour managed firms. The objec tive of the fi,nn is then said to bemaximisation of value added per worker rather than maximisation of profit. On this subject

see 1. Dreze, 'Some Theo ry of Labor Management and Participation', Econometrica,November 1976.

f Psf; PrIn the equil ibrium, the margina l rate of substitution between the. ~ w modities rand s must equal the ratio of the prices of these commodltles.

In particular, if the production function isYI = g(Y2' Y3' ..., YI),

conditions (27) becomePI = A and Ph = - A g for h f:. 1,

(28)com-II

(29)


9/18

60 The producer The case of additional constraints 61

(32) (34)- dt)yO

and so_ g ~ = P h h=2,3 , ... ,l . (30)P1

The marginal productivity of commodity h must equal the ratio between itsprice and that of the output.As in consumption theory, we can find the necessary. second order conditions for a profitmaximum.With the general form of the production function,

(26) say, these conditions require/I f ~ dYh dYk 0 (31)h,k= 1

for every set of dYIl such that/I f dYh = 0,

11=1where, of course, J,;'k denotes the value at yO of the secon.d derivative of fwith respect to YII and Yk (see theorem VIII in the Appendix).

In the particular case of the production function (29), the second orderconditions imply more simply that/I g ~ dYIl dYk 0 (33)

lI.k = 2for every set of dYh's (where h = 2, 3, ..., I). For, we can always associate

Iwith these dYh's a number dY1 such that (32) is satisfied; (33) then follows1from ( ~ 1 ) . So we come back to the assumptio.n ofnon-increasingmarginal returns,whzch zs therefore s a t i ~ j i e d at an equilibrium for the firm.

T h ~ e c o n d order conditions reveal aE important point: the firm cannotb e ~ ~ l ' ~ i t i v e _ ~ . ! : ! i E 1 : > ! i l , l l ! 1 l ! t ~ - P Q i I ! ! ~ ( I ! ~he PIQcLuction st:Lwhere reLl.![!}"!o scale are locally increasing. Let us take the case of the production function(29) and a s s u m e t h a : t l r o m ~ y O , inputs are increased by the quanti ties y do:,..., Y? do:. Let dY1 be the corresponding increase in output. We can say thatthe returns to scale are locally increasing if dyddo: is an increasing functionof do:. If we consider a limited expansion of dY1 and ignore the case wherethe second order term is zero, we see that the multiplier of do: in the expressionfor dyddo: is

/" , ,0 0L , gllkYhYk'h,k=Z

It cannot bepositivewithout contradicting the necessarysecond order condition.Thus competitive equilibrium is incompatible with such increasing returns

to scale, which are often characteristic of t he secto r i n which very largeproduction uni ts predominate . The maintenance of equil ibrium for thi ss ec to r demands forms of institutional organisation other than perfect

competi tion (see, for example, the case of monopoly in Section 9 below, orthe management rule for certain public services given in Chapter 6, Section 6).We can also now consider the inverse problem and p r o v ~ h a t t " ~ _ n : ! ~ i I ! < l . . !conditions (27) are sufficient for an e"quilil:lrTum-or"the-ftrD; ifilie assumption

"aTeonvexlty issatiSfiecl.TIieT6110Wiii"gpropertYlliemorema-tcnespropos ition2"ffi


10/18

62 The producer The case oj additional constraints 63After introduction of a second Lagrange multiplier, the first order conditionsbecome:

t The introductionof such a composite good raises no difficultywhen we are consideringthe firm in i so la ti on ; but it is usually inappropriate for the discussion of general equilibrium, since goods 3 and 4 may be produced by two distinct firms, or consumed by otheragents in a proportion other than Yz, Y3, Y4) 0- Y4 0Y4 - aY3 O.----

Ph = A j + ! . U p ~ , h = 1,2, ..., I, (37)which replaces (27).Does such a substi tut ion have much effect on ou r results? No t necessarily.

A relat ively simple alterat ion in the propert ies is sufficient in some cases.Let us return to the example of four goods an d the additional constraint

Y4 = aY3, (38)which expresses s tr ic t proport ionali ty between two inputs. System (37)becomes

(41)

(43)

(42)

B

h ,= 1, 2

h = 1,2.

Fig. 7

/IIIIIIII

II II II IIIIIo

Let A, fl.l an d fl.2 be the corresponding Kuhn-Tucker multipliers. Th e necessaryconditions for a maximum are

for{Ph = A i P3 = Ai; + afl.2P4 = A j + fl.l - fl.2

where each of the multipliers A, fl.l and fl2 must be non-negative, and must bezero when the corresponding constraint is a strict inequality.

If PI or P2 is positive, as we shall assume, the multiplier Amust be positiveand the equilibrium yO must strictly satisfy f(yO) = O. We can then distinguish three cases:(i) If the equil ibrium is such that 0 < - y < - ayg (the point M on

Figure 7), the multipliers fl.l an d fl2 are zero. System (41) reduces to system(27) exactly as if the constraints (40) did not exist.(ii) If the equilibrium is such that y = 0 an d yg < 0 (point B on Figure 7),

/(2 = 0 and fl.l O. After elimination of fl.l, system (27) is replaced by{ Ph = A i h = 1,2,3.P4 A j

In particular, if the product ion funct ion takes the form (5), the marginalproductivity - 94 of good 4 is less than or at most equal to the price rat ioP4/Pl'(iii) If the equil ibrium is such that y = a y < 0 (point A in Figure 7),/( 1 = 0 and fl z O. System (27) becomes

{Ph = A j P3 + ap4 = AU; + a j ~ )P3 Aj ; an d P4 A j

(39)

(40)

h = 1,2.

h = 1,2.or{Ph = Aj;,P3 = Aj ; - fl aP4 = A j + fl

Eliminating fl, we obtain{ Ph = A j P3 + ap4 = AU; + a j ~ )This new system has the same form as (27) provided that goods 3 and 4 are

replaced by a composi te good one uni t of which consists of one unit of good3 an d a times one unit of good 4;/3 + af4 is then the partial derivative o f fwith respect to the composite good.tSimilarly, no insurmountable problem arises if we take account of con

s traint s expressed by inequal it ies. Suppose , for example , that t he re a reaga in fou r goods and , apart f rom the production function, the two constraintso - Y4 - aY3'(Goods 3 an d 4 a re i nputs , an d the proportion of 4 with respect to 3 isbounded above; see Figure 7.)Here we have a case for app li ca ti on of theorem XI of the Appendix.Th e function to be maximised is


11/18

64 The producer Supply and demand laws for the firm 65

(47)

(45)

(44)

h = 1,2, ..., l,

h, k = 1,2, ..., t.

[F" I'J_1[f']' 0while the right hand side is the element on the kth ~ o .and the hth colum,n.Now, the matrix (47), which we assume here to eXIst, IS clearly symmetnc,which proves the equality. dThis property shows that we can say unambiguously whe th er two goo sare substitutes or complements for the particular firm .. We n ~ e d only ~ o O k atthe sign of the partial derivative Ol]h/OPk' More p r e ~ l s e l y : w s a ta t . :wo

t ' ts h al1d k are complements if thiS denvatlve IS 'pOSItIve,outputs or wo mpu .and are substitutes if it is negatIve.

p r M e ~ ; S ~ e t su I unction is homo eneous 0 de ree zero with res ec.t .toP and for any multiplication of these prices by the same pOSItIvePuPz, ... , I h t' t E Y orf(y) 0hIS is an 0 vious property since t e cons ram, y ,' ~ involve p and the function to be maximised is ~ o ~ o g e n e o u s II1 p.

If yO maximises py subject to the' constraint, it also maXimIses rJ.py when rJ.is positive. . . f I f ctionsJust as in consumption theory, this homogeneity 0 . ~ e t . supp Y ~ n shows that the choice of numeraire doe.s no.t a ~ e c t eqUIhbnum. Agam It canbe described as 'the absence of money IllusIOn. . .o The substitution eU!ct of h for k is equal to the s U b s t l t u t ~fpr h Consider the increase in the supply of h when the pnce 0 Iml.I1IS 1 e ~ .

t he net supply functions are d i f f e r e n t i ~ b l e : we can :haractense thiS'substitution effect' of h for k by the pa:tial d e n ~ a t l v e of I] h With respect to Pk'Property (ii) then expresses the followmg equalIty:

Ol]h Ol]k0Pk 0Ph

To establish this property, we differentiate the system consisting of (27)and (26) and obtain

\

Art f ~ dYk + f dA = dphI f ~ d Y h = 0,h= 1

which can be wri tten in matrix form:

[AF" I'J [dYJ = [dPJ, (46)[fT 0 dA 0with the obvious notation. This equality shows that the left hand side of (44)is the element on t he hth row and kth column of

Thi s br ing s us back to (39); we can introduce a composite good f or t heinterpretation of the las t equality; but we can now identify the irtdividualmarginal productivities of inputs 3 and 4 with r espec t t o output l, namelyfi/f; and f;.!({. We see that the marginal productivity of input 3 is at mostP3/P1> and tha t of input 4 is at least P4/Pl' In fac t, to incrc:ase the input offactor 3 without changing the input of fac tor 4 is possible but n ot wor thwhile, whereas to increase the input of factor 4 without changing the inputof fac tor 3 might be worth while but is impossible.

In short, consideration of additional constraints entails some modificationin the equilibrium conditions but makes no basic change in their nature.6. Supply and demand laws for the firm

The theory of the fi rm must lead to some general properties of supplyand demand functions, as happened with the theory of the consumer. In thecontext of the per fect competition model, the suppl y functi on for c ommodity h defines how the firm's output of this good var ies as the prices ofall goods vary. Similarly, the demand function for commodity h defines howthe firm's input of this commodity varies. We shall deal with these twofunct ions s imul taneously by considering net supply, which , by def in it ion,is equal to supply for an output and to demand with a change of sign for aninput.

The net supply law for commodity h is therefore that law which defines Yha s a funct ion of the P1> P2, ...,PI' the set Y of feasible productions, or theproduction function f, being fixed. We sha ll wri te thi s law rth(P1, Pz, ..., PI),assuming that yO exists, and is unique, for every vector P belonging to anI-dimensional domain of R'.t We can easily establish the following threet In fact, this assumption is more restrictive than appears at first sight . For example,if the production function satisfies the assumption of constant returns to sca le and isexpressed in the form (5) or (29), the derivatives g{, are homogeneous of degree zero andcan therefore be expressed' as functions of the 1 - 2 variables YZ/Y" .. " ,Y,-I!Y,. Now,there are I - 1 equations (30), necessary for equilibriumand also sufficient in the case ofconvexity. If the Ph are chosen freely, these equations will not generally have a solution.In the part icular case where thePh are such that a solution exists, yO say, then every proportional vector exyO will also be a solution (ex > 0).In economic terms these formal difficulties have the following significance. The decisionto produce can be split into two stages: (i) the choiceof the technical coefficientsYl /Y" .. "Y,-l/Y" (ii) the determination of the volume of production, In the case of constant returnsto scale, the two stagesare independentof each other and, oncethe best technical coefficientsarechosen, profit is proportional to thevolumeof production. I f it is positive, no equilibriumexists since it is always advantageous to increase production. If it is negative, only zeroproduction gives an equilibrium which does not obey the marginal equalities (30).I f profit

is zero, then any level of production is optimal.The most modern versions of microeconomic theory take account of these difficulties:net SUR-pIx. functions can be defined only for a subset of the values that area priori p o s ~T o r - ; - ~ ~ , S o the t e r m ' s u p p l y C ~ p p l yfunctions' is used.


12/18

66 The producer Cost functions 67

This is the general form of the relation of comparative statics, which mustbe obeyed in the comparison of two dif ferent equil Ibna for the samefirm.

In particular, if pI and pZ are identical except where price P" is concerned,the inequality becomes:( p - p t ) ( y ~ - yt ) O.

This establishes property (iii).

When the price of a good increases, the net supply of this good cannot'lldiminish. For the proof of this property we can use the second order conditionfor an equilibrium and establish that the partial derivative of/lh with respect toPh is not negative. The reasoning is similar to that used for consumer demand(cf. property 3 in Chapter 2, Section 9). We can also proceed directly on thebasis of finite differences, which makes the result clearer and more general.

Consider two price vectors, pI and pZ say, and two corresponding equilibria,yl and yZ. Since yl maximises pl y in the set of the feasible y's and since yZ isfeasible, we can write

the cost function changes when these prices change. The production set orproduction function are more fundamental since they represent the technicalconstraints independently of the price system.

In the second place , a production theory based on the analysis of costs isout of place in a general equilibrium theory which treats prices as endogenousand not determined apriori. Since our aim is to lead up to the study of generalequilibrium, we must start with production sets or functions.However, an examination of cost functions reveals certain useful classicalpropert ies which are simple to establish at this point and may be neededlater . We assume here that the markets for inputs are competi tive so thatthe Ph are given for the firm (h = 2,3, ..., I).Since we restrict ourselves to the case of only one output, we can take theproduction function as

YI = g(yz, Y3, ..., Yl)' (29)Before defining the cost function, we must first f ind the combination of

inputs which allows production of a given quant ity Y1 of commodity 1 atminimum cost, so we must maximise profit subject to the constraint thatYl = YI' Thi s is a par ti cu la r case of the problem discussed at the start ofSection 5 where (y) = YI - YI' Here the sys tem of first order conditions(37) becomes

{ PI = A +Ph = - Agh for h = 2,3, ..., [.The first equation allows us to find /1 and is of no f ur ther use. If, as we

assume here, the first order conditions are sufficient for cost minimisation, thesolution isobtained by determiningvaluesof Aand of yz, Y3' ...,YI which satisfy

(52){ g(y:: Y3' ...,' yz) = ~ Ph - - Agh h - 2,3, ..., l.

(48)

(49)

(50)

plyZ plyland also

pZyl pZyzor equivalently,

_ pZyz _ pZyl.Adding (48) and (49), we obtain(pi _ pZ)yZ (pI _ pZ)yl

or:* (pI _ pZ)(yl _ yZ) O.

We need only replace the Yh in this expression by their values in thesolution of (52) when we want to determine the cost function, which relatesthe value of the minimum of C with the production level YI (the Ph being

When the firm minimises its cost of production, the marginal rates ofsubstitution of inputs are equal to the ratios of their prices; but the marginalproductivity of an input, hfor example, is not necessarily equal to Ph/Pl' It isequal to P,.!PI ifYI is the optimal production for the firm selling on a competitive market. But for freely chosenYI' in most cases i t is not equal to this ratio.

Cos t C is def ined as

7. Cost functionsSuppose that the prices Ph of the different commodities are given and that

the firm produces only one good, the good I to fix ideas. The cost functionrelates to the quantity producedYI, the minimum value of the input mix whichyields this production.The theory of t he firm is o ft en bui lt up on the initial basis of t he cos t

function. This greatly simplifies the analysis, but is subject to criticism ontwo counts.In the first place, the relationship between the value of input complex andthe quantity produced depends on the pricesPh of the different inputs, so that

I IC = L Phah = - L PhYh'h=Z h=Z

(53)


13/18

68 The producer Cost functions 69

hence, taking account of the definition of C and the marginal equalities (52),C = AYl'

This equation, together with (55) shows that A, which a priori i s a funct ion ofYb is i n fact a constant (always assuming that the p" are fixed).tt We saw that the assumption of constant returns to sca le would usual ly not hold i fall the factors of production were not accounted for in the model. When defining marginalcost, we assumed that the quantit ies of al l the fac tors could be freely fixed. This lat te r

assumption is inappropriate to factors such as the work capacityof the managing director.So the case of constant marginal cost is not necessari ly frequent in relat ion to a firm someof whose factors cannot vary. (See below the distinction between long-term and short-termcosts.)

(56)

(57)

(55)

h = 2,3, ... , I.

de = A L g ~ d y " = AdYl', , ~ z

This equation establishes that ). equals marginal cost.We can also verify that the assumption of non-increasing marginal returns If

i m p l i ~ s t h a ~ marginal cost is increasing or constant. Let us differentiate (52), \1keepmg pnces constant:

r t g dy" = dYl, , ~ zld A g ~ + A kt z g ~ dYk = 0Multiply the hth equation by dy,,; s um for h = 2,3, . . . , I; take account ofthe first equation: we obtain

IdA dYl + A I g ~ dy" dYk = O.

h . k ~ zSince marginal cost A is positive, the assumption of non-increasing marginalreturns implies

dAdA ' dYl 0 or - 0, (58)dYlwhich is the required result.So a cost curve derived from a production function with non-increasing

marginal returns is concave upwards. The classical curve of the. cost function, I.as exhibited in Figure 8, is concave downwards at the start: thIS correspondsto the range of values of output for which indivisibilities are significant andmarginal returns are increasing.

We note also that marginal cost is rigorously constant when the p r o d ~ c t i o n 1\function satisfies the assumption of constant returns to scale. The functIOn gis then homogeneous of the first degree, and so

I

I g;,Yh = YI ;h ~

considered as given).t This function is often assumed to have the form of thecurve C in Figure 8.

t The term 'cost function' is somet imes a lso used for the funct ion tha t relates C to11and to Pl , PJ .. ,PI'

Fig. 8When looking for the equilibrium of the firm, we can work in two stages:(i) Define t he cos t function, that is, determine for each value of YI the

Yz, Y3, ..., Yl which minimise cost and find the value C corresponding to thisminimum cost.

(ii) Choose YI s o as to maximise profit (pd l - C(YI))'The solution of stage (ii) is obvious. The first order condition requires

PI = C'(jil)' (54)C' measures the increase in cost resulting from a small increase in production,and is therefore the 'marginal cost'. Equation (54) shows that, in competitive equilibrium, marginal cost is equal to price of the output. The second ordercondition requires that the second derivative of the profit is negative or zero,that is, that marginal cost is increasing or constant.

We shall verify tha t, in (52), A equals the marginal cost. When marginalcost is equated to price PI ' the first order conditions for cost minimisation,equations (52), are transformed into first order conditions for profit maximisation, equations (29) and (30).

Let us differentiate (53), the expression for cost, keeping prices p" constant:I

dC = - L p" dy", , ~ z

or, taking account of (52) and, in particular, differentiating the first equation,


14/18

70 The producer Short and long-run decisions 71

Fig. 9(iii) the firm should increase production i n d ~ f i n i t e . l y ( h i g ~ ..price PI)' .As we said previously, the existence of situatIOns (I) and . ( I ~ I ~ , . together WIththe multiplicity of equilibria in (ii), are sufficiently real possIbIlItIes to ~ a k e us

avoid trying to prove for producer equilibrium a general property ~ e x ~ s t e n ~ eand uniqueness corresponding to that stated for consumer eqUIlIbnum 111proposition I of Chapter 2.8. Short and long-run decisions

Cost minimisation hasjust been presented as a stage in profit m a x i m i s a ~ i o n .In fact abandoning the s tr ic t model of perfect competition, we sometImesc o n s i d ~ r that some firms actually behave so as to provide an exogenouslydetermined output and minimise their production cost. System (52) thenapplies directly to the equilibrium for the firm.Similarly, in some contexts, the firm does not choose all, but only some ofits inputs, the others being predetermined. Thus for the s.ame firm. we. oftendistinguish between long-run decisions relating to the entIre orgamsatlOn ofproduction (choice of equipment and manufacturing processes) and sh?rt-rundecisions relating to the use of an already existing productive capacIty. Sofor short-run decisions, the inputs relating to capital equipment are fixed.

S ~ c h situations can easily be analysed using the principles applied above.Suppose, to fix ideas, that capital equipment is represented by a. s.ingle gO?d,the /th. LetYI bethe predetermined value ofYI' The short-run deCISIOn conSIstsof profit maximisation subject ~ the constraint YI = YI' The s ~ o r t - r u n costfunction relates cost C to the ~ a l . u e . Yl of output :vhen Y.I = YI' t ~ _o t ~ e rinputs Y being fixed so as to m1l1ImISe cost. Let thiS functIon be C (y I, YI).

As before, we see that inputs Y2' Y3' ... , YI_I> cost C* and marginal cost),*obey the system

c

,7,,- - - - - - - - - - - - - - - - ~ '" 1./ i, 1

" Iem - - - - - - ~ - ~ - ~ - ~ - - - , ~ ' - -"" ....- - - - ~ - - . " . . "

In addition to total cost C and marginal cost C' we often consider averagecost per unit of output, namely c = C/h. If we differentiate c with respectto YI, it is immediately obvious t'hat average cost is increasing or decreasingaccording as it is greater or less than marginal cost (a typical curve c appearsin Figure 8).

It is sometimes convenient to give a diagram representing the last stage inprofit maximisation. Let the curves c and y represent respectively variationsin average cost and marginal cost as a funct ion of YI for given values ofPl, P3, ..., Pi- The equilibrium point yO is determined by the abscissa Y? ofthe po int on the curve }' whose ordinate is PI ' The profit is then Y? timesthe difference in the ordinates of the points on y and c with abscissa Y?Examination of the figure rounds off the preceding analysis, which waslimited to finding necessary conditions for a profit maximum at a point yOfor which constra ints other than the product ion funct ion do not operate .Are these conditions also sufficient, as we assumed earlier when we said thaty? corresponds to the equilibrium?Ambiguity may exist if several points on y have PI as ordinate. In practice,

this is likely to arise only in two ways. In the first place, there may be twosuch points, one on the decreasing part and the other on the increasing partof the marginal cost curve; the first point cannot correspond to an equilibriumsince it does not satisfy the second order condition, so that the ambiguitydisappears. Also, at the ordinate PI the curve y may be flat (in particular,we saw that marginal cost is constant if the productwn function satisfies theassumption of constant returns); all the points on this flat section give thesame profit; if one of them corresponds to an equilibrium, then the othersalso correspond to equilibria.The point or points with ordinate PI and lying on the non-decreasingpart of y may not correspond to an equilibrium if it is to the interest of the

firm to have zero outputY I' This situation arises ifPI is lessthan the minimumaverage cost Cm and if Y l = 0 implies zero profit, since the points consideredthen give negative profit.Finally, if the whole curve y lies below the ordinate corresponding to PI>there is no limit on the increase of profit and i t is to the interes t of the firmto go on increasing production indefinitely. (Of course, in practice it wouldcome up against a limit sooner or later, but the chosen cost function ignoresthis fact.)To sum up, for given values of Pl , P3, " "PI ' the value of PI may be suchthat:(i) the firm should choose YI = 0 (low price PI);(ii) the firm should choose a finite output Y?, which mayor may not bedefined uniquely;


15/18

long-run cost, gives the value YI for J'l. For, the solution of (52) then satisfies(59) with c* = C. Let yy be this particular value of YI' At yy, the equalityPI = - A*g; is satisfied, so that dC* = A* dYI = dC. At this point , long andshort-run marginal cos ts are equal , long and short-run average costs aretangential. A priori, this may seem an obvious result, since if existing equipment coincides with what the firm would choose in the long run in the same

73

(61 )

Monopolyprice situation, then short and long-run equilibria must naturally coincide.

Hence, the long-run average cost curve is the envelope of short-runaverage cost curves (obviously the same property holds for total cost curves).In any case, the short -run cos t cannot be lower than the long-run cost smcethe minimisat ion which defines the former is subject to one more constramtthan that which defines the latter.

t The assumption of independence of demand with respect to prices Pz, ... , p, is madehere for the s ake of simplicity. It can obviously be eliminated if prices Pz, ... , p, areindependent of the decisions of the firm, that is, i f the markets for a ll goods except the fi rs tare competitive.

9. MonopolyThe formal approach developed so far is more or less easily transposed to

institutional situations that differ from perfect competi tion. We may brieflyexamine here the classical theory of monopoly, leaving for Chapters 6 and 8the analysis of other situations.

In the applied study of market structures a firm is said to have a monopolyposition on the market for commodity h if i t supplies alone this commodityand if demand comes from many agents who are individually small and ac tindependently of one another. Classical monopoly theory represents thissituation starting from the hypothesis that the same price Ph will apply to theexchange of all units of commodity h but that this price will depend on thequantity )'h that the seller will supply. Thus the monopoly faces a demandwhose quant ity var ies wi th the price of his product but is otherwise independent of his decision.

The firm facing such a situation necessarily takes account of the fact thatthe price at which it will dispose of its output depends on the quantity whichit puts on the market. We c an no longer analyse its behaviour on theassumption that i t considers price as exogenous. We have to adopt a formalmodel other than t ha t o f perfect competition.

Suppose, for example, that the firm produces good I and sells it on a marketwhere there are many buyers whose demand depends on price PI and not onother prices.t We can represent this demand by a relat ion between PI and YI :where n I is the funct ion defining the price at which the monopol is t candispose of the volume of production YI '

lt may also happen that a firm is the onlyone to use a fac tor h (for example,when it is the only employer of labour in a town). lt is said to be in a s itua tionof 'monopsony'. It knows that price Ph depends on the quantity ah = - Yh

(59)

eL

yO yC yL, I ,Fig. 10

o

The producerI(Yz, ..., YI-I, YI) = hPh = - A * g ~ h = 2, 3, . .. , I - 1,1- 1IC* = - L PhYh - PIYI'h= 272

Differentiating the first and last equations for given Ph and taking accountof the intermediate equalities, we obtaindC* = A* dh - (A*g; + PI) dYI,

which replaces (55). The short-run marg inal cos t is aga in equal to theequilibrium value of the Lagrange multiplier A*. We could also verify that,to determine the value of YI which maximises profit subject to the constraintYI = YI' we must addto (59) the condition that the marginal cost A* equalsPl'Let us illustrate this theory by a d iag ram in which th e different costfunctions are represented as a function of YI ' Let cL and yL be the long-runaverage and marginal cost curves. The long-run equilibrium value ofproduction for price PI is determined as the abscissa yf of the point on yL whoseordinate isPl' Also let cCand yC be the short-run average and marginal costcurves. The short-run equil ibrium is determined by the abscissa yC of the. Ipomt on yC whose ordinate is Pl '

The long and short-run average cost curves generally have a common pointcorresponding to the value of YI for which the solut ion of (52), defining the

TC neC I .' /, .

- - - - - - ~ - - - - - - - - - t ~ 1.'/'II ,y;) ;

' . I/1 II ,I II II ,I II i


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Maximisation of, t h i ~ expression sub ject to the constra in t expressed by theproductIOn functIOn Implies the following first order conditions:

75

(67)

Monopoly

We could apply the same reasoning to the case of pure monopoly where allthe 8h except 8 1 are zero. However we shall adopt a rather different approachfor an alternative presentation of the analysis, which is thus reinforced.As in the case of perfect competition, we can maximise profit by means of atwo-stage procedure involving first cost minimisation and determination ofthe cost function. Fo r a pure monopoly, cost minimisation is carried out inexactly the same way as for a perfectly competitive firm and the cost ft:nctionis exact ly t he same. So we can confine ourselves to the second stage, andfind the value of Y1 which maximises

1t1(Y1) . Y1 - C(Y1)'We can write this expression in i ts usual form

R(Y1) - C(Y1),

provided that 81 i= - 1 in the equilibrium, which we assume for simplicity.The marginal productivity of the factor h is no longer equal to the ratio ofprices but to t hi s r at io mul tipl ied by a term depending on the elasticitiesrelating t o t he factor h and to output.Consider first the case of a monopsony for which al l the 8" are zero except

that relating to a particular input k. Equations (66) then reduce to the perfectcompetition equations except for the kth, where - gk must equal pdp1multiplied by the term (1 + 8k) which is usually greater than 1. The equilib rium is the re fo re the same as in a situa tion of perfect competition involvingthe same prices for al l the goods excep t k, whose price is greater than thatactua lly asked by suppl ie rs . Since, in the compe ti tive situa tion , the f irm'sdemand 1] k can only decrease, the firm in a position of monopsony usuallyemploys a smaller quantity of the factor k than i t would employ in competition. For this reason it may be said to be in the interest of the monopsonist toadopt a 'Malthusian policy'.

where R(Yl) denotes the firm's receipts from output Y1'Profit maximisation implies that Y1 is so chosen thatR'(Y1) = C'(Y1) (68)

andR"(Y1) C"(Y1)' (69)

Equation (68) generalises condition (54) obtained for the case of perfectcompetition.

We can easily compare monopoly equilibrium with equilibrium for thefirm in perfect competition. Figure 11 shows the average cost and marginalcost curves c and y, as wel l a s the curve d representing the demand function1t1(Y1), that is, average revenue, and the curve (j representing marginalrevenue, that is, the function 1tl + Yl1ti. Suppose that 1ti is negative, as will

(63)

(29)

(66)

The producer

h = 2, ,.., I

h = 1,2, ..., I,

- g

74which it uses as input. If i t takes no account of the possible interdependenceof Ph and the pnces of other goods, the firm will fix its decisions as a functionof a supply law

p" = n,,(y,,) (62)representing t h behaviour of the agents supplying the factor h and indicatingthe pnce Ph whIch the firm must pay t o acqui re a quant it y - y" of h.. We, note that the case of perfect competition corresponds to the particularsItuatIOn where n [ and nh are constant funct ions . There fo re we can dea lsImultaneously with monopoly and with monopsonies concerning one ormore fac to rs by treat ing the case where the f irm tries to maximise i ts p ro fi tand takes account of functions nh relating the price of each good h t o its n etproduction Yh (h = I, 2, .,., I).As a funct ion of y the profit, or net value of production, is

II n,,(y,,) , y",h= [

where nh is the derivative of nil and }, is a Lagrange multiplier.For w.hat follows, we shall consider the case where prices are non-zero andshall WrIte the above condi tions in the form

p,,(1 + 8,,) = A f h = 1,2, ..., I, (64)taking account of the fact that Ph is the value of the function n and defining81, as the inverse of the elasticity of demand (or supply) whichhoccurs in them a r ~ e t for. the good h because of agents other than the particular firm underconsIderatIOn:

n d log nh8" = y";,, = d log /y,J (65)

In,the case of perfect competition, market demand and supply are perfectlyelastIC from the standpoint ? t he firm; the 8h a re zero. Condi tions (64)reduce to the f irs t order condItIOns (27) obtained earlier.

In ?rder to investigate (64), we shall consider the case where the productionfunctIOn takes the formY1 = g(YZ'Y3' ""Yl),

the good 1 being the firm's output. Equations (64) implyp,,(1 + 8h)Pl(1 + 8 1)


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76 The producer Monopoly 77necessarily be the case except perhaps for an inferior good; (j then lies belowd. According to (68), monopoly equilibrium is determined by the abscissa y!of the point of intersection of y and (j. If the firm behaves as in perfectcompetition, that is, if it takes no account of the reaction of price PI to itssupply YI, the equilibrium point is determined by the abscissa y? of the pointof intersection of y and d.

The study of monopoly has taken us outside the field of perfectcompetition. We shall not pursue this line for the moment, bu t shall takeit up again in Chapters 6 and 7. However, two remarks may usefully bemade already at this stage.

In the first p lace, i t is clear that situations of imperfect competition mayinvolve consumers as well as firms. For example, it is conceivable that a

t We shou ld also n ot e t ha t, for t he definiti on of the cos t funct ion, second orderconditions implying concavity of the isoquants in the neighbourhood of the equilibriummust be satisfied. When this is not so, no equil ibrium exists as long as the markets for thefactors are competitive: but a monopsony for the firm may allow equilibrium to be realised.

c

Fig. 12

' ............ "

..........._ "I....... _- ',!- - - _ _, I~ , - - - - TI ...i ........

IIIo

particularly wealthy consumer may have such influence on a market that hehas a position of near-monopsony.In the second place, the theory of imperfect competition cannot dependentirely on the constrained maximum techniques which we have used uptill now.Of course, situations other than those we have considered can be dea ltwith by constrained maximum techniques, for example, the case of a firm

that has a monopoly on each of the two or more independent marke ts inwhich its output can be sold . In most cases, profi t maximisat ion leads toprice diffe rentiat ion, the firm releasing to each marke t a quant ity of itsproduct such tha t margina l revenue from each marke t equa ls its margina lcost over all its output.

Generally we can say that constrained maximisation is appropriate to theextent that all agents except at most one adopt a passive attitude, taking thedecisions of other agents as given. This is just the situation for a monopoly,since those who demand the proc;iuct accept as given the price which resultsfrom the firm's decision on production. They have no other possible attitudeif their number is. large and they are all of the same relative importance, andif they are unable to band together in opposition to the monopolist.But imperfect competition is not limited to such s ituations . On some

, t 'III/IIIC

I,I ,I ": "b1I

y . yO1Fig. 11

oAt the point of intersection of y and d, the margina l cost must be nondecreasing for y? to correspond to a true competitive equilibrium. I t followsfrom the fact that d is decreasing and from the ~ e s p e c t i v e positions of d and (j

that y! is necessarily smaller than y? The firm produces less in a position ofmonopoly than in a situation of perfect competition involving the same pricesfor it ; this result is similar to that encountered earlier for monopsony.We can consider R" as negative in the interpretation of (69) defining thesecond order condition for a maximum. In particular it will be negative ifthere is constant elasticity of demand, since then [;;1 is a fixed number, R' isequal to n1(1 + [;;1) and R" to n'l(1 + [;;1)' The second order condition istherefore satisfied for any situation where marginal cost is increasing.But we should point out t ha t this condi tion may also be satisfied insituations where marginal cost is decreasing. More generally, monopoly may

sometimes allow an equilibrium to.be realised which is not possible in perfectcompetition. Figure 12 shows an example for a firm with continually decreasing marginal cost, which is possible in the "undifferentiated sector".t

p.1pO1


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78

markets there are relatively few buyers and sellers; on others, coalitions takeplace. Other methods of analysis are necessary to deal with such cases.We shall return to imperfect competition in Chapters 6 and 7 in orderto clarify problems of general economic equil ib rium. We sha ll then seehow it relates to the theory of games.

4Optimum theory

Up till now we have been considering the behav iour of a single agent .With the theory of the optimum we approach the study of a whole society.We therefore change our perspective and attack the problems raised by theorganisation of the simultaneous actions of all agents.The classical approach wou ld be first to discuss competitive equilibrium,

keeping to the positive standpoint of the previous lectures, and then to go onto the normative standpoint of t he search f or t he optimum. However, weshall reverse the order of these two questions.Optimum theory involves a rather simpler and more general model than

t he model on which competitive equilibrium theory is based . It seemsplausible that the relationship of the two t heor ies will be mor e clearlyunderstood if those assumptions which are not involved in optimum theoryare introduced in the later discussion of competitive equilibrium.We are interested, therefore, in the problem of the best possible choice ofproduction and consumption in a given society. Clearly it may appear veryambitious to attempt to deal with this. But i t is one of the ultimate objectivesof economic science. Preoccupation with the optimum underlies manypropositions briefly stated by economists. Byproviding an initial formalisationand by rigorously establishing conditions for the validity of classical proposi-tions, optimum theory provides the logical foundation for a whole branchof economics.We mus t first find out what is meant by the 'best choice' for the ~ o c i e t yand go on to study the characteristics of situations resulting from this choice.1. Definition of optimal statesBefore fixing a principle of choice, we must again define what are 'feasible'

states.For our present inves tiga tion , a ' state of the economy' consists of mconsumption vectors Xi and n net production vectors Y j'

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