production, modes of

sufficient, it is maintained that output can be expanded morethan proportionately with the labour employed in manufacture

(increasing returns to scale).Marx used these two examples to draw a distinction between

the 'heterogeneous' manufacture (exemplified by Petty'swatch-making activity) in which the final output is obtained bysimple assemblage or 'partial and independent products', and

the more sophisticated 'organic' manli;acture (exemplified bySmith's pin factory) in which a series or successive operationsgradually transforms the original raw material into the finished

product.Smith referred to three arguments in favour or the technical

superiority or an ever increasing division or labour:

first, to the increase or dexterity in every particularworkman; secondly, to the saving or the time which iscommonly lost in passing from one species or work toanother; and lastly, to the invention or a great number ormachines which facilitate and abridge labour, and enableone man to do the work or many (Smith, 1776, p. 17).

It has been observed that these arguments are not trulyconvincing. The importance attributed to increased dexterityconflicts with the relatively low level or skills required incontemporary factories (witness the common use or child

labour). Time saving does not imply specialization byindividuals: in principle, it could equally be attained by asuitable reorganization or the activity or a single artisan. Andthe introduction or machines does not seem to exhibit anynecessary relation to the increasing division or tasks.

In fact the new organization or labour associated with thefactory system did go along with the process or technicalchange associated with the industrial re\olution. But itsoriginal role was primarily to discipline the manner in whichthe work was performed and to give the capitalist the power orcontrolling the production process in every single detail.

The introduction or machinery came after labour specia!iza-tion and reinforced the need for a thorough organization orproduction. The effects or the introduction or the s:eam-engineand other complex machines were eventually studied by twoscholars who possessed the necessary technical background,Charles Babbage (1832) and William Ure (1835); their tractswere very r~pular at the time and were widely used by theeconomists (e.g. by John Stuart Mill and Marx). Theyconceived the control and management or a factory as that ora single complex machine. under the full control or thecapitalist and with manual work brought to a minimum.

It is worth noticing that these speculations about the rationalmanagement or a highly mechanized factory were easilyextended to society as a whole. At the turn or the centuf).,Mikhail Tugan-Baranovsky (1905) dreamed or an economyin which machines were automatically produced by machines,and where the labour force was paradoxically reduced to oneworker alone. In a similar vein, especially in Germany afterWorld War I, we find many suggestions for a 'rational"organization or the economy as if it were a giant Kon:ern (asan extreme example, see the 'natural economy' proposed byOtto Neurath (1921) for the ephemeral Bavarian republic).

GIORGIO GtLJBERT

s(!~ also INCREASING REn:R~S.

BIBLIOGRAPHYBabbage, C. 1832. On the Econon,)" of Machine and .\fanufacIUrt's.

London: Knight.Defoe, D. 1728. A Plan of the English Con,n,..rct'. Reprinted, Oxford:

Blackwe1l, 1928.

992

De Quinccy, T. 1844. The l.ogic of Polilical Economy. Reprinled inCollecled "'rilings, ed. D. Masson, London: Black, 1897, Vol. 9.

Marx, K. 1859. Zur Krilik der polilischen ()konomie. In Marx-EngelsGcsanllausgabe Vol. II, PI 2, Berlin: Dielz, 1980.

Marx, K. 1867. Das Kapilal, Vol. I. In Marx-Engels GesanltausgabeVol. ", PI 5, Berlin: Dielz, 1983.

Mill, J.S. 1821. Elemenls of Polilical Economy. London: Bald,,'in.Mill, J.S. 1848. Principles of Polilical Econom)'. Ed. J.M. Robson,

Toronto: University of Toronto Press, 1965.Neurath, 0. 1921. Durch die Kriegs..'irtschafl :ur Natural...irlsch,,(I.

Munich: Callway.Petty, W. 1683. AnOlher Essa)' on Political Arilhmetick. In Economic

"'rilings of Sir "'illiam Pell)', ed. C.H. Hull, Cambridge:Cambridge University Press, 1899, Vol. 2.

Quesnay, F. 1759. Tableau «ononlique. Ed. M. Kuczynski andR. Meek, London: Macmillan, 1972.

Ricardo, D. 181 5. An Essay on Ihe /nfluence of a Lo..' Price 'of Corn.In The "'orks and Correspondence of Darid Ricardo, ed.P. Sraffa, Cambridge: Cambridge University Press, 1951, Vol. 4.

Ricardo,D. 1817. Principles of Polilical Econom).. In The "'orks andCorresponden"e of Dat"id Ri"ardo, ed. P. Sraffa, Cambridge:Cambridge University Press, 1951, Vol. I.

Smilh, A. 1776. An /nquir)' inlo Ihe Nalure and Causes of the "'eal,hof Nalions. Ed. R.H. Campbell, A.S. Skinner and W.B. Todd,Oxford: Clarendon Press, 1976.

Torrens, R. 1821. An Essa). on Ihe Produclion of Wealth. London:Longman, Hurst, Rees, Orme & Brown,

Tugan-Baranovsky, M. 1905. Theorclis"he Grundlagen des Marxismus.Gem1an trans., Leipzig: Duncker & Humblot.

Ure, A. 1835. The Philosoph). of Manufaclures, London: Knight.

production, modes of. See MODES OF PRODUCTION.

production: neoclassical theories. The economic theory ofproduction is concerned with the characterization of the inputdemand and output supply functions based on a theory ofprofit maximization subject to a production function. Two setsof issues are involved: one is the technical constraint thatdescribes the range of production processes available to thefirm, and the other is the make-up of the markets where thefi~:.s transactions take place. There is a substantial literatureon the latter, which cannot be addressed here: we adopt theadmittedly unrealistic assumption of 'perfect competition' inboth commodity and factor markets. Our purpose is to discussthe properties of the production technology in the context ofthe neoclassical theory of the multiple-product and multiple-input firm, identify the specific forms of the productionfunction which are proposed in the literature, and discuss theduality principles as well as some of the new dynamic factordemand model analyses.

!\"EOCLASSICAL THEORY OF PRODUCTION. Consider a firm thatproduces n products and employs m inputs; its objective is tomaximize profits given as:

..+. ...n = L PJ'i= L Pi)'i+ L P..l'i (I)

i-i i-i i-.+ I

where)',(i = 1" , , ,n) are the outputsandp,(i = n + 1, ' ., ,m)are output prices: )'.+, = -xi (i = 1, ., , , m) are the inputs, andP. + i (i = 1, ., ., m) are the input prices. Profit, n, is maximizedsubject to the production function

f<>i,)'2'.'.')'.')'.+I"'.')'..+.)=O, (2)

I',,)'~, ., , .)'..+. are often called net outputs; they have positivesigns for outputs and negative signs for inputs,

production: neoclassical theories

Table A partial list of economic effects related to the

production function

Output level y = I{x, I)

Returns to scale II = (ti XI/; )Ii

Distributive share SI = X,/; I t xi/;I-I

Own 'price' elasticity £i = xJ.//;

{I./I: +fii//;Jj -fu/l])ElastIcIty of substItutIon l1ii = {I/xi/; + l/xJJj)

Disembodied technological change:{1) Rate of technical change{2) Acceleration or technical change{3) Rate of change of marginal

products

I: = !./f

T = (!.,//) -(!.//)2

mi/mi = /;,//;

Assuming that the production function f( .) is (I) twicedifferentiable, i.e.

ofloyl=1. and o2jloyioyj=fli.(i,j=I,...,m+n),

exist; (2) increasing in the net outputs, i.e. the derivatives. 1.are always positive; and (3) convex (subject to the conditionf( .) = 0, the function is strict/y convex); the optimal prod-uction plan of the firm can be stated using the familiarLagrangian function:

L(y.,y2 ,Y..+I1;).)=ll +).f(y..y2 'Y..+II)..+II

= L PIYi+)!(Y.'Y2 'Y..+11)' (3)i-1

where). is a Lagrange multiplier associated with the constraintf( .) = 0.

There are m I n first-order conditions that can be interpretedas equality between the marginal profitability of each netoutput and its revenue or cost. The Lagrange multiplier is thechange in profit made by the firm with respect to a change inits production plan. Manipulating these equalities, we obtainfamiliar expressions such as the marginal transformationamong commodities and inputs. the marginal rate of technicalsubstitution among inputs and the expansion path of inputs. Itfollows that the profit-maximizing output and input levels,jil(i = I. ..., m + n). and the Lagrange multiplier. I, arefunctions of the prices PI(i = I, ..., m + n). That is:

I=I(PI'...'P..+II)

and

ji;=SI(PI'...,P..+I1)' (i=I,...,m+n). (4)

'-(.) is homogeneous of degree one, while Si(.) is homogeneousof degree zero. jii are the net supp/y functions. For outputs, theequations jil are the usual supply functions; for inputs, theyarethe negative of the demand functions. Thus net supplyfunctions exist provided that the marginal profitabilityconditions are satisfied and that the production function hasthe appropriate properties.

PROPERnES A~"D FORM OF rnE PRODUCTION FUNCTIONS. Thecharacterization of the input demand and output supplyfunctions depends on the specific properties of the productionfunction. A number of studies have tried to specify theseproperties and discover more flexible functional forms toaccommodate various economic effects often imbedded in theproduction process. Some economic concepts of interest arelisted below. Though the concepts shown in Table I aredefined in terms of a single-output production function. theycan easily be extended to multiple-output production func-tions. Given the production function Y = f(x, t), where x is avector of inputs and t the index of technological change, itis possible to deduce expressions shown in Table I for returnsto scale, shares of factors of production, price elasticity andelasticity of substitution, as well as various indices ofdisembodied technical change. Other effects such as indices ofembodied technical challge can also be derived. By imposingspecific restrictions across these effects. different functionalforms of the production function can be obtained. Of thisarray of economic effects, those associated ,,;th returns toscale, degree of substitution among inputs and the type andnature of technological change, have received prominentattention in the literature.

These economic effects arise from the inherent nature ofthe underlying production process, and the specific formof the production function is therefore critical in determining

993

Source: adapted from Fuss and McFadden (eds), 1978, p. 231.

the existence and magnitude of these effects. These propertiesof the production function -homogeneity, additivity andseparability -have played an important role in the derivationof input demand and output supply functions. A homogeneousproduction function of degree k is defined as:

f().xl,. ..,).x.)=J.y(xl' ...x.); J. >0

and a monotonic transformation of a homogeneous prod-uction function yields a homothetic production function iny = g[f(x., ..., x.)]. This family of production functions is

characterized by straight-line expansion paths through theorigin. Additivity may take the form:

fl().y.)+ ...f.().y.) =fl(yl)'. ..,f.(y.) =0 for any J. > 0

where }'i represents net output of ith commodity, some ofwhich are inputs to the production process. If the function f iis either homogeneous of some degree, or logarithmic, theadditivity condition holds.

Most of the theoretical formulations of the productionfunctions described in the literature implicitly assume thatseparability conditions prevail. The f(x) is M'eakly separablewith respect to partition R when the marginal rate ofsubstitution (MRS) between any two inputs xi and Xi from anysubset N" s = I, ..., " is independent of the quantities outside

N, (Leontief, 1947; Green, 1964; Bemdt and Christensen, 1973)or a(J;/Jj)/ax. = 0. Strong separability, on the other hand,exists when MRS between any two inputs inside N, and N,does not depend on the quantities outside N, and N, orJjfik -J;f.,. = 0.

Functional separability plays an important role in aggregat-ing heterogeneous inputs and outputs, deriving value-addedfunctions and estimating production functions. It also opensup the possibility of consistent multi-stage estimation, whichmay be the only feasible procedure when large numbers ofinputs and outputs are involved in the production activities of

highly complex organizations.A major preoccupation in the literature for empirical

estimation of production functions has been to find flexiblefunctional forms. Well-known functions (e.g. the Leontief and

Cobb-Douglas production functions) impose restrictions ofzero and one, respectively, on the elasticity of substitution, (7,while for C:ES production functions, (7 is an arbitrary constantto be estimated. Attempts to relax this stringent requirementhave led to the development of the variable elasticity of

production: neoclassical theories

only contributes important insights of its own but also offersmore immediate empirical applications. A mapping of thecharacteristics of the transformation function and its dual costfunction is indicated in Table 2.The cost formulation is used extensively in econometricstudies. This approach has two main advantages: (I) demandand supply functions can be derived as explicit functions ofrelative price and output without imposing arbitrary con-straints on production patterns required in the traditionalmethodology; (2) cost and profit -functions arecomputationally simple and permit testing of a wider class ofhypotheses by utilizing economic variables (Nadiri, 1982).

DYNAMIC FACTOR DEMAND MODELS. These types of productionfunctions emphasize the intertemporal aspect of the produc-tion process by focusing on the movement from oneequilibrium state to another. The models incorporate costs ofadjustment that are incurred in order to change the level ofquasi-fixed inputs, costs which can take two forms. The firsttype is external: as the firm adjusts its quasi-fixed factors itmust face either a higher purchase price for these factors(Lucas, 1967; Gould, 1968) or a higher financing cost for theaccumulation of these inputs (Steigum, 1983). The second typeis internal and reflects the fact that firms must make thetrade-off between producing current output and divertingsome of the resources from current production to accumulatecapital for future production (Treadway, 1974).

Suppose the firm maximizes its present value:

V =fo~ {Py -WL -rK-Gk.} e-P/dt

subject to the production function f(y, L, k., K) = ° and theinitial condition K(O) = Ko. p is the price of output, y is thelevel of output, W is the nominal ".age, r is the user cost ofcapital, G is the purchase price of investment, K is a vectorof capital inputs, L is labour, and k. is net investment. /( isintroduced in production on the assumption that .firmsproduce essentially two types of outputs: y, to ~ell, and K, theinternally accumulated capital which ",ill be used in futureproduction. K is assumed to be neither perfectly fixed norperfectly variable. Suppose, in addition, that the productionfunction is characterized by the relation y + C(/() -g(K, L) = 0, where C and 9 are continuous and the marginalproducts of f L and f K are positive and diminishing.

From the necessary conditions, it follows that for perfectlyvariable inputs its marginal product must equal its price, "'.hilefor the quasi-fixed inputs the discounted sum of future netvalues of its marginal product must equal the sum of thepurchase price of investment and the marginal value of realproduct foregone as a consequence of expansion at the rate k.

substitution functions (VES) where 11 is dependent oneconomic variables such as input mix (Liu and Hildebrand,1965; Kadiyala, 1972). Efforts to relax the homogeneityproperty have led to the development of a number ofhomothetic production functions that make the returns toscale depend on output and/or input mix (Zellner andRevankar. 1969; Fare, Jansson and Knox Lovell. 1978). Amajor advance has been the formulation of non-homotheticfunctions by Christensen, Jorgenson and Lau (1973). whoformulated the translog production function, which does not apriori impose restrictive constraints such as homotheticity,constancy of 11, additivity. and so on.

TECHNICAL PROGRESS. Technical progress deals with the processand consequences of shifts in the production function due tothe adoption of new techniques which either have a neutraleffect on the production process or change the input-{)utputrelationships. Neutrality of technical progress can be measuredby its effect on cerlain economic variables such ascapital-{)utput, output-labour and capital-Iabour ratios,which should remain invariant under technical change. Severaldefinitions of technical progress have been proposed, such as(1) product-augmenting, (2) labour- or capital-augmenting,and (3) input-decreasing and factor-augmenting, amongstothers (Beckmann, Sato and Schupack. 1972). However. themost familiar definitions are the Hicks, Harrod, and Solowforms of technical progress.

Part of technical change can be endogenous and would bedetermined by the firm to maximize its long-run profit.Technical knowledge is expensive to produce but, onceproduced, its transmission cost is almost zero, giving rise tothe 'indivisibility' and 'inappropriability' characteristics ofinventions. Attempts have been made to incorporate R&D asan input in the neoclassical production and cost functions, toestimate its contributions to the firm's productivity growth andcost behaviour, ard to measure its spillover effects on otherfirms or industries (Nordhaus, 1969; Griliches, 1979.) Theresults indicate substantial private and social rates of return toR&D (Mansfield, 1969). Changes in relative prices and outputnot only affect endogenous technical change but also the rateof factor productivity and the bias of technical change, whichwill in turn alter the structure of the production process(Jorgenson and Fraumeni, 1981).

DUALITY .A major advance in the economic theory ofproduction has been the dual formulation of productiontheory (Shephard, 1953; Diewert. 1974; Fuss and McFadden,1978). The main features of this approach is to recoverthrough indirect functions- that is. by means of a dualrepresentation such as profit or cost functions -the propertiesof the underlying production function. The dual approach not

Table 2. Comparison of the properties on the transformation function and its dual cost function

Property Aon :he transformation fu:Jction FU", x)

Property B

on the cost function C(y:p)

I Non-increasing in )'2 Uniformly decreasing in )'3 Strongly upper semi-continuous in U', x )4 Strongly lower semi-continuous in U', x )5 Strongly continuous in U', x)6 Strictly quasi-concave from below in .\:7 Continuously differentiable in positive .\:8 Twice continuously differentiable strictly

differentiably quasi-concave from below in x

Non-decreasing in )'Uniformly increasing in )'Strongly lower semi-continuous in (),. p )Strongly upper semi-continuous in ()'.p)Strongly continuous in ()'. p )Continuously differentiable in positive pStrictly quasi-concave from below in pTwice continuously differentiable and strictly~ifferentiably quasi-concave from below in p

994

production and cost functions

The basic problem in this type of model is to deal withexpectations about future prices of inputs and outputs. Asimple and often-used approach to the problem is to assumestatic expectations, but that begs the question. Uncertaintyabout future prices are handled in two ways, either byapproximating optimization under uncertainty with certaintyequivalence, which requires a quadratic objective function andlinear constraint (Hansen and Sargent, 1981) or by makingadjustment costs a function of the level of the quasi-fixedinputs, which exploits the expectations of future prices that arecontained at the quasi-fixed input levels.

There is a fairly large and growing theoretical and empiricalliterature using the dynamic production function or its duals,dynamic profit or cost functions. The main result of thesemodels is that, because of the existence of adjustment costs,substitution possibilities and technological biases may belimited in the short run, and the effects of prices and taxchanges on factor demands may be quite differept from theireffects in the long run.

ECONOMIES OF SCALE A~D SCOPE. An important extension of thetheory of the firm has been the production and pricingbehaviour of a multi-product firm when economies of scaleprevail. To derive the net supply functions, the necessaryconditions for equilibrium noted earlier break down whenincreasing returns or declining long-run average costs prevail.In such cases, monopolistic organization of an industry mayoffer cost advantages over production by a multiplicity offirms. An interesting and important question is what are thenecessary and sufficient conditions for a multi-product firm tobe a natural monopoly and for it to be sustainable againstentry (Baumol, 1977). The condition for natural monopoly isthat a cost function be stricti}' and globally subadditive in theset ofcomrnodities C(yl + ...+ ym) <C(yl)+ .00 + C(ym),

which means that it is always cheaper to have a single firmproduce whatever combinations of output is supplied to themarket. If the output vectors are restricted to be orthogonal,then the production function exhibits economies of scope. Thenatural monopoly is a sustainable set of products set at pricesthat do not attract rivals into the industry (Baumol, Bailey andWillig, 1977). Even if rivals are attracted, the monopoly maybe able to protect itself from entry by changing its prices. But,by definition, only a sustainable vector of prices can prevententry and yet remain stationary. The conditions necessary forsustainability are (1) the products are weak gross substitutes;(2) the cost function exhibits strictly decreasing ray averagecosts; and (3) the cost function is also transray convex.Ramsey prices often ensure sustain ability under specifiedcircumstances.

M. IsHAQ NADIRI

See also COBB-DOUGLAs FUNCTJO~s; COST A"D SUPPLY CL'II\'ES; COST

FUNCTIO"s; HUMBUG PRODUCTION FUNCl1ON; JOINT PRODUCl1O~; st;PPL Y

FUNCTJO"s.

Christensen, LR., Jorgenson, D.W. and Lau, L.J. 1973. Transcenden-tal logarithmic production frontiers. Revie,,' of Economics andStatistics 55(1), 28-45.

Diewert, W.E. 1974. Applications of duality theory. In Fron1iers ofQuantitatil'e Economics, ed. MoO. Intriligator andD.A. Kendrick, Amsterdam: North-Holland, Vo!, ".

Fare, R., Jansson, L. and Knox Lovell, C,A. 1978. On

ray-homothetic production functions, In The Importance ofTechnologJ' and the Permanence of Structure in Indus1rial Gro,,'1h,ed. B, Carlsson, G. Elliasson, and M.I, Nadiri, Stockholm:Industrial Institute for Economic and Social Research, Vol. ",228-37,

Fuss, M. and McFadden, D. (eds) 1978. Production Economics: aDual Approach to TheorJ' and Application. Amsterdam: North-Holland, Vol. I.

Gould, J.P. 1968. Adjustment costs in the theory of investment of thefinn. Rer'ie,,' of Economic Studies 35, 47-55.

Green, H.A.J. 1964. Aggregation in Economic Anai}'sis: an In1roduc-tory Sun'eJ'. Princeton: Princeton University Press.

Griliches, Z. 1979. Issues in assessing the contribution of researchand development to productivity growth. Bell Journal of Econom-ics 10(1),92-116.

Hansen, L.P. and Sargent, T.J. 1981. Linear rational expectationsmodels for dynamically interrelated variables. In Rational Expec-to lions and Econometric Practice, ed. R.E. Lucas andT.J. Sargent, Minnearolis: University of Minnesota Press, Vol. I,127-56.

Jorgenson, D.W. and Fraumeni, B.M, 1981. Relative prices andtechnical change. In Modeling and Measuring Natural ResourceSubstitution, ed. E.R. Berndt and B.C. Fields, Cambridge, Mass.:MIT Press, 17-47.

Kadiyala, K.R. 1972. Production functions and elasticity of substitu-tion, Southern Economic Journal 38(3), 281-4,

Leontief, W.W. 1947. Introduction to a theory of the internalstructure of functional relationships. Econometrica 15,361-73.

Liu, T,C. and Hildebrand, G,H. 1965. Manufacturing ProductionFunctions in the United States, /957. Ithaca: Cornell UniversityPress.

Lucas, R,E. 1967. Adjustment costs and the theory of supply. Journalof Political EconomJ' 74(4), 321-34.

Mansfield, E. 1969. Industrial research and development: characteris-tics, costs and diffusion of results. American Economic RCl.ie".59(2), 65-71.

Nadiri, M.I. 1982. Producers theory. In Handbook of Ma1hematicalEconomics, ed. K,J. Arrow and MoO. Intriligator, Amsterdam:North-Holland, Vol,ll, 431-90.

Nordhaus, W. 1969. Invention. Gro,,'th and Welfare: a TheoreticalTreatment of Technological Change. Cambridge, Mass.: MITPress.

Shephard, R.W. 1953. Cost and Production Functions. Princeton:Princelon University Press.

Steigum, E. 1983. A financial theory of investment beha\'ior.Econometrica 51(3).637-45.

Treadway, A. 1974. The rational multivariale flexible acceleralor.Journal of Economic TheorJ. 7(1), 17-39.

Zellner, A. and Revankar, N. 1969. Generalized prOduclion funclions.Rt'I'ie..' of EcJnomic Studies 36(2), 241-50.

production, prices of. See PRIC~ OF PRoDucnoN.

production and cost functions. The traditional starting pointof production theory is a set of physical technologicalpossibilities, often represented by a production or transfonna-tion function. The development of the theory parallels thefinn's objective (cost minimization or profit maximization) andleads to input demands (and output supplies in the case ofprofit maximization) constructed from an explicit consider-ation of the underl)'ing technology (i.e. derived directly fromthe production function).

BIBLIOGRAPHYBaumol, W.J. 1977. 0!1 the proper cost tests for natural monopoly in

a multi-product industry. Amc,ica/1 Eco/1omic RCl.ic...67(5),809-22.

Baumol, W.J., Bailey, E.E. and Willig, R.D. 1977. Weak invisiblehand theorems on the sustainability of multi-product naturalmonopoly. Amcrica/1 Eco/1omic RCIic... 67(3), 350--65.

Beckmann, M., Sato, R. and Schupack, M. 1972. Alternative ap-proaches to the estimation of production functions and oflechnical change. 1/1lc'/1atio/1al Eco/1omic RCIic... 13, 33-52.

Berndt, E.R. and Christensen, L.R. 1973. The internal slructure offunctional relationships: separabilily subslitution and aggregation.

RCIi"... of Eco/1omic Studics 40(3), 403-10.

995