An Introduction toCost and Production Functions

ALSO BY DAVID F. HEATHFIELD

The Econometric Study of the UK (editor)

Production Functions

Topics in Applied Macroe conomics (editor)

The Economics of Co-determination (editor)

Perspectives on Inflation : Models and Policy (editor)

AN INTRODUCTIONTO COST ANDPRODUCTION

FUNCTIONS

David F. Heathfieldand

Soren Wibe

MMACMILLANEDUCATION

© David F. Heathfield and Soren Wibe 1987Softcover reprint of the hardcover Ist edition 1987

All rights reserved. No reproduction, copy or transmissionof this publication may be made without written permission.

No paragraph of this publication may be reproduced, copiedor transmitted save with written permission or in accordancewith the provisions of the Copyright Act 1956 (as amended).

Any person who does any unauthorised act in relation tothis publication may be liable to criminal prosecution andcivil claims for damages .

First published 1987

Published byMACMILLAN EDUCAnON LTDHoundmills, Basingstoke , Hampshire RG21 2XSand LondonCompan ies and representativesthroughout the world

British Library Cataloguing in Publication DataHeathfield, David F.An introduction to cost and production functions .I. Production functions (Economic theory)I. Title II. Wibe, Soren338'.001 HB241ISBN 978-0-333-41607-5 ISBN 978-1-349-18721-8 (eBook)DOI 10.1007/978-1-349-349-18721-8

Contents

Preface ix

Symbols xii

Abbreviations xiv

1

2

Basic Concepts 11.1 . Production I1.2 Factors of Production 21.3 Factor Rewards 41.4 Aggregation 51.5 Varying Inputs and Outputs 101.6 The Production Function 11I. 7 The Isoquant 131.8 Homothetic Production Function 141.9 Technological Progress 151.10 Returns to Scale 171.11 Factor Substitution 181.12 Total, Average and Marginal Products 221.13 Diminishing Returns 26Selected Reading 27

Cost Functions and the Theory of the Firm 282.1 Optimisation 282.2 The Isocost line 28

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vi Contents

2.3 The Cost-minimising Choice of Technology 302.4 Cost-minimising Choice of Technology in the Short

Run 332.5 Cost Functions in the Long Run 342.6 Cost Functions in the Short Run 382.7 The Relation Between Long- and Short-run Cost

Curves 402.8 Factor Demand in the Long and Short Run 412.9 Determination of Output Level 422.10 Output Determination in Perfect Competition 432.11 Output Determination in a Monopolistic

Market 47Appendix: Returns to Scale, Imperfect Markets and

the Second Order Conditions for a Maximum 48Selected Reading 52

3

4

Elasticity, Efficiency and the Theory of the Firm 533.1 Introduction 533.2 Output and Substitution Effects 533.3 The Elasticity of Scale 553.4 The Elasticity of Scale and Shape of the Cost

Functions 563.5 The Elasticity of Substitution 583.6 Factor Demand and Elasticity of Substitution 603.7 Putty-Clay Technology 633.8 Putty-Clay and Frontier Production Functions 643.9 Long- and Short-run Industrial Supply Curves 69Appendix 70

The Elasticity of Substitution in the General n-factorcase 70The relation between the elast icity of substitution andthe price elasticities in the general case 72The elasticity of substitution and the cost functions 73Expressing the elasticity of substitution in terms of thepartial derivatives of the production function 73

Selected Reading 74

The Cobb-Douglas Function 764.1 Introduction 764.2 Cobb-Douglas Isoquants 77

5

6

7

Contents vii

4.3 Short-run Total, Average and Marginal Product Curvesfor Cobb-Douglas 78

4.4 The Cobb-Douglas Elasticity of Substitution 804.5 Returns to Scale and the Cobb-Douglas 814.6 Cobb-Douglas Factor Demand Functions 824.7 Cobb-Douglas Cost Functions 844.8 The Adding-up Problem 864.9 The Aggregate Cobb-Douglas Function 89Selected Reading 91

The CES Function 925.1 Generalising the Cobb-Douglas 925.2 The Elasticity of Substitution of the CES 945.3 The CES Isoquants 955.4 Returns to Scale 975.5 Short-run Total, Average and Marginal Products for

the CES 985.6 The CES Factor Demand Functions 1005.7 CES Cost Functions 102Selected Reading 104

The Translog Function 1056.1 Introduction 1056.2 The Elasticity of Scale and the Translog 1076.3 The Translog Isoquant 1096.4 Short-run Product Curves for the Translog 1096.5 The Translog Cost Function 1106.6 The Elasticity of Substitution of the Translog 112Appendix: Some other functional forms 112Selected Reading 117

Technological Progress 1187.1 Introduction 1187.2 Product Innovation and Process Innovation 1187.3 Embodied and Disembodied Technological

Progress 1197.4 Neutrality of Technological Progress 1207.5 Some Functional Forms 1227.6 Learning by Doing 1277.7 Research and Development 128

viii Contents

7.8 Patents 1297.9 Technological Progress and Economic Growth 1307.10 Summary and Conclusions 133

8

9

From Firms to Industry: the Johansen ProductionModel 1348.1 Introduction 1348.2 Ex Ante Functions at Micro and Industry Levels 1358.3 The Ex Post Micro Function 1378.4 The Ex Post Industry Production Function 1388.5 Competitive Markets and the Bang-Bang Solution 1398.6 The Short-run Industry Function Obtained Through a

Competitive Market 1418.7 Construction of an Isoquant -an Example 1428.8 The Elasticity of Scale 1478.9 The Elasticity of Substitution 149Appendix: The Houthakker Model 150Selected Reading 152

Empirical Work on Production Functions 1539.1 Introduction 1539.2 The Econometric Approach 1549.3 Stochastic Equations 1559.4 The Error Structure 1569.5 Simultaneous Equation Systems 1599.6 Estimating the Translog Production Function 1679.7 The Engineering Approach 1709.8 The Formal Structure of Engineering Analysis 1709.9 An Illustrative Example of Engineering

Functions 1729.10 Statistical Estimation from Engineering Data 1769.11 Merits and Drawbacks of the Engineering

Approach 1779.12 Empirical Engineering Production Functions 1789.13 Empirical Best Practice Studies 1799.14 Empirical Short-run Macro Functions 180Selected Reading 182

Bibliography 183

Index 190

Authors' Preface

There can be little doubt that production functions, and theirassociated cost functions , form an integral part of an enormous rangeof economic theory. In microeconomics , production functions under­lie the supply side of markets, generate production possibilityfrontiers , offer an explanation of income distribution and yield factordemand functions. Production functions are also central to the theoryof economic growth and to investigations into the rate of tech­nological progress . In international trade, production functions areused to provide a rationale for product and factor movements acrossnat ional boundaries. In macroeconomics , production functions liebehind aggregate supply functions , aggregate labour demand func­tions and form the link between output and the consequent employ­ment. Even the demand side can make use of product ion functions.The functional forms used for utility functions are often 'borrowed'from production theory. Indeed the production model has beentransferred wholesale to model 'demand'.

It is clear from this somewhat impressive list that a knowledge ofproduction functions is a useful if not essential part of being aneconomist.

The kind of production function most widely used is the so-called'neoclassical' production function. Neoclassical functions can take ahost of forms (as we shall see in the following chapters) but they allhave three fundamental characteristics. First they represent ways inwhich labour, capital and land can be combined to produce goods.Second, they assume that capital is a separate, independent input

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x Authors' Preface

directly comparable with labour and land. And third, they focusattention on the production possibilities and decisions within pro­cesses, firms or industries.

Many of the results which spring from these functions, whenapplied to the various aspects of economics listed above, rely on theseimplicit assumptions. They are not without their critics.

A slightly different approach derives from general equilibriummodels and focuses attention on inter-firm relationships of produc­tion. The Leontief production model is perhaps the best knownexample of this approach. Each sector uses capital and labour but alsoneeds inputs from other sectors. The 'Food, Drink and Tobacco'sector, for example, buys materials from Agriculture, and Agriculturebuys from Chemicals and so on. There is no substitution amonginputs and so the inputs into each sector are usually assumed to bedetermined simply by the output of that sector. This approach isknown as input-output analysis and concentrates on the interdepen­dence of industries, firms and processes rather than choice oftechniques within an industry, firm or process.

The input-output approach is widely used in planning modelswhere the 'balance' among industries is important. And input-outputis largely regarded as an alternative to neoclassical functions ratherthan a contradiction of them.

A third approach, the 'classical' production model, does howeverdirectly contradict the 'neoclassical' model. This approach is like theinput-output approach in that attention is focused on inter-industryrelations. In this classical model, however, one of the sectors producescapital goods-capital is explicitly recognised as a produced input.Whereas land and labour are 'original' inputs, capital is also anoutput. This modification is sufficient to render many of the'standard' results of neoclassical production theory invalid . It is nolonger unambiguously true, for example, that increasing the interestrate vis-a-vis the wage rate will induce capital saving. Sraffa (1972)hasshown that it is possible for a particular man/machine combination tobe used at low rates of interest, fall into disuse as interest rates rise andthen, as they rise still further, switch back into operation again. Thisresult would not be possible in a neoclassical model of production. Asinterest rate rises less and less capital is used and more and morelabour so that the capital-labour ratio continues to fall for allincreases in interest rate.

According to this classical model it is simply not possible simul-

Authors' Preface xi

taneously to determine factor and product prices as It IS In theneoclassical world. It is necessary first to specify the wage/interestratio and from this the choice of technique and product prices arefound .

The so-called 'capital controversy' has been widely covered in theliterature (Harcourt (1972), Kregel (1976) and cf. Bliss (1975» and israther beyond the scope of an introductory text such as this. Whatevermerits or flaws the neoclassical production model may have it isindisputably the dominant model and economists require to havesome understanding of it.

It is not our intention here to stress the controversies or to otTer acomprehensive account of the various applications of productiontheory. Our aim is simply to bring together in one volume theprincipal neoclassical approaches to production and to compare andcontrast their properties.

Production functions imply particular cost functions, often the'self-dual' of the production function . These cost functions aresometimes of interest in their own right but are sometimes used asmore tractable alternatives to production functions .For these reasonswe introduce, where appropriate, the cost functions associated witheach production function. We have tried to keep the inevitablemathematics to a fairly simple level and have consigned the moreesoteric points to appendices which can be ignored by the generalreader . Each chapter begins with fairly simple concepts and becomesprogressively more difficult. Some students may find that the earlyparts of the chapters are all that is required.

There are two general chapters: one on estimating productionfunctions and one on technological progress. These are includedmerely to indicate to the student some of the difficulties and somepossible solutions which have been discussed in the literature .

We have, in short, tried to provide a rigorous yet accessibleintroduction to the principal aspects of cost and productionfunctions.

David F. HeathfieldSoren Wibe

Symbols

The following symbols are used throughout this book:

q = Quantity of firm output.Q= Quantity of industry output.Vi> •• • , Vn = Firm inputs No 1, .. . , n.Vb . . . , Vn = Industry input.K = Quantity capital (same for firm and industry) .L = Quantity labour (same for firm and industry).E = Quantity energy (same for firm and industry)t = time.P = Prices of output (same for firm and industry) .Ph . . . , P; = Prices of inputs.1t = Profits.s = Elasticity of scale.se = Elasticity of cost (with respect to production level).a = Elasticity of substitution (in a two-factor production model)a., = Allen partial elasticity of substitution between inputs rand s.E, = Price elasticity of demand for factor j .S, = Cost-share for factor j.Eij = Cross-price elasticity of demand for factor i.C = (Total) cost of factors./; = partial derivative of function f ( ) w.r.t. factor i~j = Input coefficient for factor j (i.e. input per unit of output)U = Error termA, B, a, b, (x , p, y, A, 8, {) are used as parameters.

xii

Symbols xiii

A dot (-) above a symbol (e.g. K) indicates rate of growth.A line above a symbol (e.g, ij) indicates any arbitrarily fixed level.A hat or a star above a symbol (e.g.qor q) indicates some speciallevel (e.g. the cost-minimising level).

Abbreviations

The following abbreviations are used throughout thi s book.

SR = Short RunLR = Long Run

TC = Total CostFC = Fixed CostVC = Variable CostMC = Marginal Co stAC = Average CostATC = Average Total Cost ( = AC)AVC = Average Variable CostAFC = Average Fixed Co stLRAC = Long Run Average CostSRAC = Short Run Average Co st

TR = Total RevenueMR = Marginal RevenueAR = Average Revenue

TP = Total ProductAP = Average ProductMP = Marginal Product

RTS = Returns to ScaleCRTS = Constant Returns to Scale

xiv

Abbreviations xv

DRTS = Decreasing Returns to ScaleIRTS = Increasing Returns to Scale

CD = Cobb-Douglas (Production Function)CES = Constant Elasticity of Substitution (Production

Function)