Arts, Faculty of
Vancouver School of Economics
DSpace
UBCV
François, Patrick
2009-06-03T18:59:30Z
1995
Doctor of Philosophy - PhD
University of British Columbia
The chapters in this thesis are each concerned with problems of coordination. The coordination issues examined here each arise in distinct situations and imply the need for a
different modeling approach in each case. The first case, Chapter 2, considers gender
discrimination in contemporary, competitive labour markets. It is sh9wn there that such
discrimination can arise as an outcome of maximizing activities on the part of firms facing
the problem of worker motivation in the light of imperfect monitorability. This is shown to
lead to firms’ hiring practices (in particular discrimination) depending on the practices of
other firms and consequently to labour market equilibria of discrimination and of non
discrimination. It is shown that a policy of affirmative action can be useful in moving the
labour market away from the discrimination equilibrium. The next chapter, Chapter 3,
considers an avenue by which the structure of industries in an economy can affect the
development of new technologies through its general equilibrium impact on profits relative to
wages. It shows that a monopolistic structure in one industry, by increasing the share of
profits in aggregate income, tends to increase the relative profitability of innovative activities
elsewhere thereby leading to the creation of further monopoly rents which, in turn, feeds
back into incentives for innovation thus causing a self-perpetuating cycle. This leads to the
possibility of an economy exhibiting multiple steady states including a “Poverty trap” or
situation of zero growth. The conditions under which multiple steady states exist are
analyzed and the economy’s behaviour out of the steady state is also characterized. The role
of government intervention, in the form of subsidies, direct provision of research and patent
protection is also examined. Finally it is shown that the model can also explain the existence
of clustering of innovations and consequent sporadic growth. The final substantive chapter,
Chapter 4, centres on problems of investment coordination in the context of LDCs. These
arise when the fall in the price of one good raises the demand for complementary goods,
thereby implying that investment decisions leading to such price falls may not be privately undertaken whereas, when coordinated across sectors, such investments could be profitable.
This chapter shows that the existence of multiple equilibria hinges upon the more restrictive
Definition of complementarity between goods, namely, the Hicks definition. As a result, gross
complementarity between goods (on its own), even though causing horizontal externalities,
can not lead to the existence of multiple equilibria. A later section looks at gross
complements in the presence of knowledge spillovers and shows, in contrast, that this can
lead to multiple equilibria and coordination problems. The chapter also examines the social
optimality of coordination in the Hicks complements case, showing that it is not always
implied by the multiplicity of equilbria.
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SOME COORDiNATION PROBLEMSbyPATRICK FRANCOISB.A.(Hons) The University of Melbourne, 1988M.Com. The University of Melbourne, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR TIlE DEGREE OFDOCTOR OF PIIILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Economics)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1995© Patrick Francois, 1995In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)_______________Department of_______________The University of British ColumbiaVancouver, Canada)Date ZDE-6 (2188)AbstractThe chapters in this thesis are each concerned with problems of coordination. Thecoordination issues examined here each arise in distinct situations and imply the need for adifferent modeling approach in each case. The first case, Chapter 2, considers genderdiscrimination in contemporary, competitive labour markets. It is sh9wn there that suchdiscrimination can arise as an outcome of maximizing activities on the part of firms facingthe problem of worker motivation in the light of imperfect monitorability. This is shown tolead to firms’ hiring practices (in particular discrimination) depending on the practices ofother firms and consequently to labour market equilibria of discrimination and of nondiscrimination. It is shown that a policy of affirmative action can be useful in moving thelabour market away from the discrimination equilibrium. The next chapter, Chapter 3,considers an avenue by which the structure of industries in an economy can affect thedevelopment of new technologies through its general equilibrium impact on profits relative towages. It shows that a monopolistic structure in one industry, by increasing the share ofprofits in aggregate income, tends to increase the relative profitability of innovative activitieselsewhere thereby leading to the creation of further monopoly rents which, in turn, feedsback into incentives for innovation thus causing a self-perpetuating cycle. This leads to thepossibility of an economy exhibiting multiple steady states including a “Poverty trap” orsituation of zero growth. The conditions under which multiple steady states exist areanalyzed and the economy’s behaviour out of the steady state is also characterized. The roleof government intervention, in the form of subsidies, direct provision of research and patentprotection is also examined. Finally it is shown that the model can also explain the existenceof clustering of innovations and consequent sporadic growth. The final substantive chapter,Chapter 4, centres on problems of investment coordination in the context of LDCs. Thesearise when the fall in the price of one good raises the demand for complementary goods,thereby implying that investment decisions leading to such price falls may not be privatelyundertaken whereas, when coordinated across sectors, such investments could be profitable.This chapter shows that the existence of multiple equilibria hinges upon the more restrictivedefinition of complementarity between goods, namely, the Hicks definition. As a result, grosscomplementarity between goods (on its own), even though causing horizontal externalities,can not lead to the existence of multiple equilibria. A later section looks at grosscomplements in the presence of knowledge spillovers and shows, in contrast, that this canlead to multiple equilibria and coordination problems. The chapter also examines the socialoptimality of coordination in the Hicks complements case, showing that it is not alwaysimplied by the multiplicity of equilbria.in -Table of ContentsAbstract iiTable of Contents ivList of Figures viAcknowledgment viiChapter 1 INTRODUCTION AND OVERVIEW 1Chapter 2 GENDER DISCRIMINATION AS A COORDINATED LABOURMARKET OUTCOME 4Introduction 5The model 12Firms 14Labour market equilibrium 16Discrimination equilibrium 24Non-discrimination equilibrium 28Heterogeneous firms 31Stability of the equilibria 32Effect of affirmative action policies 34Welfare properties of equilibria 36Empirical Implications 37Conclusions 42Chapter 3 INNOVATION, MONOPOLIES AND THE POVERTY TRAP 49Introduction 50The static model 54The competitive economy equilibrium 56The monopolistic economy equilibrium 57The dynamic model 59The intermediate monopolist’s problem 61ivInstantaneous equilibrium 64Steady states 67Dynamics 73Welfare 77Policy Implications 78Publicly provided research 79Patents of finite duration 80Entrepreneurship: A reinterpretation for LDC’s 85Conclusion 88Chapter 4 COORDINATION PROBLEMS ARISING FROMCOST REDUCING INVESTMENTS IN THE PRESENCEOF HORIZONTAL EXTERNALITIES 95Introduction 96Gross complementarity and multiple equilibria 101Proposition 1 105Optimality of investment coordination 113Proposition 2 113Gross complementarities again 118Conclusion 121Chapter 5 CONCLUDING COMMENTS 124Bibliography 126Appendix to Chapter 2 132Appendix to Chapter 3 140VList of FiguresFigure 2.1 Stability of equilibria 46Figure 2.2 Stability of equilibria 47Figure A2.1 48Figure 3.1 Existence of 2 steady states 89Figure 3.2 90Figure 3.3 Middle values 91Figure 3.4 The phase diagram 92Figure 3.5 High values 93Figure 3.6 Low values 94vAcknowledgmentIf there is anything at all interesting in this thesis, it is directly attributable to theinsistence, encouragement and insights of my research supervisor, Ashok Kotwal. He gaveme the nerve to do the things I was interested in when I really doubted whether I could. As aresult, contrary to most graduate theses, undertaking this project never became tedious orboring largely because a day of work came to mean hours of discussion with him. In additionto his immense intellectual support, and perhaps of even greater value, was his guidance inmy personal life. He was there for me when I needed a friend and/or a shrink more than asupervisor, and without him there it is certain that I would never have completed thisdegree.I also thank Jean-Marie Baland, my friend and co-author on work which constitutesmuch of the third and fourth chapters of this thesis, for his patience and enthusiasm, and forsecuring funding for my visit to the University of Namur, Belgium, where much of the workwas undertaken. From September 1990 I have also been in receipt of funding from theCanadian Commonwealth Scholarship Fund, for which I am grateful. I am grateful too to themembers of my committee: Michele Piccione for not letting me get away with sloppiness, forhours spent solving my problems and for his constantly open door, and Brian Copeland fortorpedoing my bad ideas and vigorously encouraging my good ones. Though not on mycommitte, I also owe debts of gratitude to Mukesh Eswaran, especially for his earlyencouragement, and John Weymark for carefully reading and correcting an earlier version ofthis work. Finally, I thank Siwan Anderson for being a good and understanding friendthrough all this, and for providing the stimulus for the ideas contained in the work on genderdiscrimination.viiChapter 1INTRODUCTION AND OVERVIEWThe general theme running through the chapters comprising this thesis is that ofcoordination problems. In general, such problems can arise when an individual’s desire toundertake a particular activity depends positively on the number of others doing likewise. Ineconomic contexts such situations have received much attention, though not always underthe general heading of “coordination problemst’.For example many similar issues arise in theliterature on strategic complementarities, lock-in and hysteresis, clustering phenomena andpoverty and underdevelopment traps, to name a few. In particular, with respect to the lattercontext, there has been a large tradition in development economics, dating back at least asfar as Rosenstein-Rodan (1943), which has focused on explaining problems ofunderdevelopment as essentially arising due to underlying coordination problems. In thepresent thesis, Chapters 3 and 4 can be squarely located within this tradition. Chapter 2, onthe other hand, bears no direct relation to this branch of the literature as its primary focus isin explaining discrimination in contemporary labour markets. However, its relationshipstems from its focus on the same factors: complementarities, multiple equilibria andcoordination issues, in explaining the phenomenon.Chapter 2, which is concerned with theoretically explaining gender discrimination inlabour markets, may, at first sight, appear out of place in a thesis on coordination problems,given the strong traditional link between these problems and development issues. Uponcloser scrutiny, however, the apparent disparity between this chapter and the others is seento persist only at a cosmetic level. It is demonstrated that, at a theoretical level, a similarprocess of complementarity in the activities of firms, can be seen to operate when firmsemploy discriminatory practices in labour markets. The basic point of this chapter is that1firms can exploit the gender division of labour in the household by reserving good, or higherpaying jobs, for men only. However their benefits to doing so are shown to depend upon theactivities of other firms in the labour market. If all other firms discriminate then any onefirm will also find it beneficial to do so. If no others do, however, conditions are derived inwhich a non-discriminatory outcome is also a labour market equilibrium. It is shown that,much like the development economics literature on coordination problems, coordinatinggovernment policies which induce non-discriminatory practices by a sufficient number offirms, such as affirmative action, can lead to all other firms following suit.Chapter 3 explores the problem of underinvestment in quality improving technologyin the context of a multi-sector, model of endogenous growth. In this dynamic context it isshown that incentives to invest in quality improving technological change vary positivelywith the amount of quality improving investment undertaken previously. The mechanismgiving rise to this positive history dependence is shown to be the dependence of currentindustrial structure on previous levels of innovation and, in turn, the effect of industrialstructure on current returns to innovation. Thus, the initial industrial structure inherited byan economy can determine the steady state to which it ultimately converges, in particular,whether the economy converges to a long run outcome of sustained technological change andgrowth or whether technological stagnation ensues. The model developed there, characterizesthese two scenarios and analyzes the dynamic process of economy wide adjustment betweenthem. It is shown that, when in the no growth steady state, small subsidies to innovation cansometimes serve to move the economy onto a path with increasing growth and ultimately to asituation of sustained innovation and growth. This can be Pareto improving even though notbeing undertaken when the economy is in the no growth equilibrium as, in that case,individual firms find investing alone unprofitable.2 -The final substantive chapter, Chapter 4, takes a more general look at the issue ofcomplementarities in the process of implementing cost reducing technology. Much recentliterature has focused on issues arising when such cost reducing technologies are of theincreasing returns to scale variety. In particular, it has been shown that certain types ofpecuniary externalities can lead to multiple equilibria and consequent coordination problemsin implementing investments in these technologies. This paper looks at one class of thesepecuniary externalities, that is, those arising due to horizontal complementarities, and askswhat conditions are necessary for these complementarities to lead to multiple equilibria andtherefore to coordination problems. The somewhat surprising answer is that the existence ofgross complementarities is irrelevant in determining whether multiple equilibria may or maynot exist. The critical condition is shown to be the more exacting criterion of Hickscomplementarity. However, it is also shown in this chapter that, even though implyingmultiplicity of equilibria, Hicks complementarity of investments need not imply thatcoordinating investments is socially optimal. It is shownthat, when the technology usedacross sectors is symmetric, such coordination is never desired. A final part of this chapteralso shows that gross complementarity can play some part in contributing to multipleequilibria, even in the absence of Hicks complements, when it works in combination withother factors. Finally Chapter 5 provides some brief conclusions.-3Chapter 2.GENDER DISCRIMINATION AS ACOORDINATED LABOUR MARKETOUTCOME.41 IntroductionThe huge “gender gap” literature has shown that women, on average, consistently earn lessthan men.1 A difference in earnings persists even when controlling for hours worked, industry ofwork and human capital characteristics of workers, leading many to suggest that this providesevidence of wage discrimination against women, see Gunderson (1989) and Goldin (1990) forfurther discussion.This chapter explains gender discrimination as an equilibrium outcome in a world wherelabour markets are competitive, women and men are ex ante identical and neither employers noranyone else have a preference for men or exogenous propensity to discriminate. The explanationdoes, however, utilize the fact that women and men are organized into two person households inwhich both are equally well endowed with household specific human capital, that is, a capacityto provide household goods and services for themselves and each other at a cost which is lessthan the external market rate. Differences in labour market characteristics between householdmembers (in particular, wage rates or job types) combined with this household specific humancapital, give rise to the potential for mutually beneficial within household trades.In the labour market there are two types of jobs: As in Bulow and Summers (1986) thereexist a number of exogenously specified efficiency wage jobs in which workers receive a wagepremium in order to be dissuaded from shirking. There also exist non-efficiency wage jobs‘See Cain (1986 p. 750) for early references to the empirical work in this area, Gunderson and Riddell (1993)for more recent studies and O’neill and Polachek (1993) for an account of changes in gender gap through the19 80’s.5which pay no such premium. If, and only if, one member of a couple alone has an efficiencywage job, both can benefit by a contract which calls for an increase in the household work effortof the lower paid member (with corresponding decrease in the higher paid member’s effort) inreturn for a monetary payment from the high to the low wage member. That is, there existgains from trade which can be exploited within the household. Importantly, it is differencewhich gives rise to the benefits of such trades, so that they are only realized by couples whosewage rates differ. This benefit from trade acts as a benefit in kind which occurs in additionto the efficiency wage payment. Given this benefit in kind, employers can lower efficiencywage payments while still satisfying a worker’s no shirking constraint because threat of job lossentails loss of gains from trade in addition to loss of the wage premium. Herein lies a firm’sdesire to discriminate. If all other firms are discriminating and allocating the efficiency wagejobs to one sex only, then a single finn can be sure that its employees receive this benefit inkind, and thus pay them lower efficiency wages, only if this firm also employs men exclusivelyas well. Only by doing so can it ensure that its employees are in households where they aloneare the sole efficiency wage recipient, and therefore where they can enjoy the benefits of withinhousehold trades. Thus discrimination by other firms is shown to make it strictly better forany one firm to also be a discriminator.Intuitively, a culture of discrimination in the labour market (i.e. a Nash equilibrium inwhich all firms discriminate and allocate efficiency wage jobs to men only) leads a single firmto expect that men are likely to be in households where they alone have high paying jobs(and can thus trade off much of the work at home with their spouses). Women, on the otherhand, have a much higher chance of being in a household where they are not alone in having6a high paying job and cannot therefore enjoy the benefits of such trade. Thus, though ex antewomen and men are identical, in a discrimination equilibrium, all firms correctly conjecturethat women have a higher probability of shirking at a given wage and therefore rationallychoose to discriminate.This explanation differs from previous explanations of discrimination in two significantrespects. Firstly, discrimination arises here because women and men are arranged into households. Secondly, it starts from ex ante equivalence between women and men in all respects.Previous models include either exogenous gender differences or inherent preferences for onegender over the other. Becker (1971) shows how a preference for men over women, either onthe part of employers, employees or customers, can lead to a situation of women being paid lessthan men. Madden (1973) argues that the simultaneous existence of differences in labour supply elasticities between men and women, and firms with some degree of monopsonistic power,can lead to gender differences in wages, with a monopsonist being able to pay the sex with lowerelasticity less. Signalling theories, for example Rothschild and Stiglitz (1982), explain discrimination as arising due to differences in the noise of productivity signals across gender. Women,who are posited as having noisier signals, are paid a lower wage, since accurately signallingquality is more difficult for women, and a worker’s output depends upon matching their qualitytype with the correct job. Milgrom and Oster (1987) take another approach at this by positingless observability of women’s quality to outside firms. This generates strategic non- promotionof high quality women on the part of firms, in order to protect their informational advantageover outsiders, and women being worse off. A final broad category of explanations can be seenas arising from the segmented labour market theories of Bergmann (1971) and Arrow (1973).7The existence of a dual labour market, either because of employer prejudices, as in Bergmann(1971), or because of a combination of efficiency wage jobs and differences in length of workinglife, as in Goldin (1986), or efficiency wage jobs and differences in employment turnover rates,as in Bulow and Summers (1986), can lead to differences in payments across gender. Morerecently, Kuhn (1993), shows that firm specific human capital investments, and, once again, anexogenously lower labour force commitment for women than for men, can lead to involuntaryrationing of jobs to women.Despite the prevalence of explanations Cain (1986 p.781) states, in conclusion to his surveyof labour market discrimination, that:“....the theories of discrimination have been useful for providing definitions andfor suggesting measurements of discrimination but not for providing convincingexplanations of the phenomenon nor of its patterns.”2It could be argued that previous theories leave too much unexplained. Though useful in drawingout the labour market implications of preferences for men over women, theories which take suchpreferences as a starting point, as in Becker (1971) or Bergmann (1971) provide only part ofthe explanation. Madden (1973) does not explain why women should have less elastic laboursupply responses than men, which is also empiricafly hard to justify. A major drawback of theother approaches is that they treat as exogenous, factors which, themselves, could be expectedto be caused by discrimination. Differences in the noise of quality signals for men and women, if2Recently, in rationalizing their findings of discrimination in the market for lawyers, Wood, Corcoran andCourant (1993) reach a similar conclusion.8they exist at all, could be the outcome of discrimination themselves. Because of discrimination,women do not invest in obtaining such signals and therefore, because of fewer observations,are less reliable as an indicator of quality. Similarly, differences in length of working life as inGoldin (1986) or differences in turnover rates, as in Bulow and Summers (1986) can be seen asthe outcome of a situation in which women, due to discrimination and therefore lower rewards,place less importance on labour market participation and therefore are more likely to leave theircurrent job, or the labour force altogether, as suggested by Gunderson (1989 p.48).3 Thesetheories, then, are best seen as partial explanations only. A more complete explanation couldaccount for discrimination without treating as exogenous those factors which are themselves,in turn, affected by discrimination.This is the approach of the current chapter, which attempts to generate gender discrimination in a model where men and women are completely equal in all labour market characteristics. The only role that gender plays is in the organization of households, which areassumed to contain one member of each sex. The aim of this chapter is not to explain theexistence of discrimination against women per Se. It is instead to show that one outcome of acompetitive labour market where workers are identical in all respects, is discrimination againstone of the sexes. It is demonstrated that this discrimination arises precisely because workersare organized into such two person households. The importance of such an explanation whenthere already exist an abundance of theories explaining the phenomenon, is that it may implya markedly different set of policy responses. If, as in the previous literature, gender discrimi3Ixi fact, Viscusi (1980), Blau and Kahn (1981) and Osternian (1982) find that sex differences in turnoverrates all but disappear when labour market and job characteristics are controlled for, suggesting that thesequalities may reside more in women’s typical jobs than women themselves.9nation arises due to exogenous gender differences, then correcting these (say through improvedchildcare or encouragement of female labour force participation) will lead to an eventual improvement in female work and thus a narrowing of the gender gap. If, however, as the logic ofthe model presented here wifi suggest, labour market discrimination itself causes these genderdifferences (which can even persist when women have ex ante identical characteristics to men),then altering discriminatory outcomes may require direct policy responses (such as affirmativeaction). Moreover, focusing policy explicitly on the labour market, and only the labour market,may alleviate a substantial amount of discrimination even though factors sometimes consideredexternal to the labour market (differences in household responsibilities, career interruptions,turnover rates) may appear to be empirically important explanatory variables.The model developed here is very much in the tradition of the dual labour market modelsmentioned above. These models provide a useful starting point for an analysis of discrimination,because the dual labour market structure, which allows for gender segregation by type of job,accords well with our empirical understanding of the phenomenon. Empirical studies haveshown that, in contemporary labour markets, discrimination rarely takes the form of womenbeing paid less than men in the same jobs at the same establishments. Rather it is manifestin men having better access to the higher paying jobs within an occupation type, even whentraditional labour market characteristics are controlled for.4 The better paying jobs here aremodelled as those which, for reasons of imperfect monitoring (as in Shapiro and Stiglitz (1984)),pay efficiency wages. For expositional simplicity this structure is useful, however it is notnecessary for the basic story to go through. The essential features of the explanation are4For more on this see, Treiman and Hartmann (1981 p.33) Johnson and Solon (1986) and an extensivediscussion and survey in Gunderson (1989).10simply that 1) termination of a contract is costly to the firm and 2) workers are less likely toterminate the contract when their spouse does not also have a similar job. These features arepresent in dual labour markets generated by efficiency wage considerations, but not uniquelyso.In considering links with the previous literature, it should be noted that the explanation inthis chapter resembles suggestions made by Marxist Feminist theorists who point to a stronglink between labour market discrimination and the position of women within the home, see DallaCosta and James (1972), Edrnond and Fleming (1975) and Zaretsky (1973, 1974). On a verydifferent tack, the analysis also draws on Becker’s (1985) insights in the modeffing of internalorganization of labour within the household, with some departures. Firstly, unlike Becker,human capital accumulation (either within the household or in the workforce) is ignored, sothat gains from the division of labour to increasing returns activities are not what drive gainsfrom trade. Though, if this were allowed, it would generally work in the same direction asthe other factors and strengthen the results. Secondly, individuals maximize their own utilityfunctions, not a household welfare function. Finally there are no exogenous differences in eitherhousehold or workforce productivity by gender which would give rise to benefits from tradewithin the household. The aim of the present chapter is to generate these differences (in theworkforce) endogenously by showing how discrimination by a number of firms (and thus lowerfemale labour productivity) affects the composition of households and can therefore lead otherfirms to discriminate.The chapter proceeds as follows. In Section 2 the model is constructed and it is established11that a Nash equilibrium in which all firms discriminate always exists. The necessary conditionsfor a non-discrimination equilibrium are also established and the stability properties of theequilibria are examined. In Section 3, the effects of an affirmative action policy are explored.It is shown there that such a policy can move the labour market from the discriminationequilibrium to the non-discrimination equilibrium. In Section 4 an attempt is made to see howwell the model’s predictions square off with our empirical understanding of male-female labourmarket differences. Section 5 concludes.2 The modelThere are N identical individuals, N/2 of each sex, living in N/2 households. Each householdhas one member of each sex. To avoid any perverse final period effects, individuals are assumedto live for an infinite number of discrete periods. The future is discounted at rate p (0 < p < 1)per period. In each period, denoted t, an individual, denoted i, cares about three things only:consumption, denoted c; effort spent at work, c; and effort allocated to household tasks H,where ‘ +H < em° which is the total amount of effort each individual has available per period.Within period preferences for individual i are denoted by a twice continuously differentiable,separable utility function, U(c) + V(e + Hz), with U’ > 0, V’ < 0, U” < 0 and V” < 0. It isalso assumed that the marginal utility of consumption is arbitrarily high as it approaches zero,i.e. limo UI(cz) = oo, that effort is equally unenjoyable whether incurred in the householdor at work; that is, e and H are perfect substitutes and that lirnei+Hi_emaz Vf(ez + Hz) = —x,5Primes denote derivatives here.12that is, if all effort is devoted to work, the marginal disutility of work is infinitely high.6The price of consumption goods is normalized to equal one, and to simplify the individual’soptimization problem, saving is not allowed. Thus if individual i receives wage w per unitof effort expended at work and works e2, thereby receiving total income w e’ for the period,then i can consume upto = we by devoting all income to consumption. Each individual isresponsible for a total of household tasks each period. Assume the simplest form of householdproduction function for these tasks where one unit of effort expended in the household performsone unit of required housework. Thus if an individual were to personally undertake all of hisor her tasks, effort expended in the household would equal H. These tasks include thingssuch as cleaning, shopping, preparing food, child care, home maintenance etc. As the signof the derivative shows, these tasks are assumed to be unpleasant. Note that, for simplicity,the utility benefit of such tasks is not modelled here, however, in order to abstract from theproblems of determining equilibrium allocations of public goods within the household, it isassumed that undertalcing one’s own housework does not create a benefit for one’s spouse.7There also exists an external market for household tasks where, by paying a wage vf perunit of labour employed, person i can reduce his or her household tasks by an amount < 1per unit of labour hired. Note the assumption of household specific human capital implicitin setting 7 strictly less than one. This human capital is assumed to be common to bothmembers of the household, thus if i’s spouse were to devote one unit of effort to undertakingi’s tasks this would be as effective as one unit of effort from i. More effort is therefore requiredfrom an outsider than from either member of the household in performing a given amount of6 separability of utility in income and effort is a standard simplifying assumption in efficiency wagemodels, see Shapiro and Stiglitz (1984).7Konrad and Lommerud (1993) analyze a situation where members of a couple non-cooperatively choose thelevel of within household public good to provide.13household tasks. This seems reasonable as, even if it is believed that householders do not haveiitherent advantages in providing for their own household services, the existence of transportcosts are sufficient to render internal provision of services cheaper. Thus, denoting HC theamount of H bought externally, it is necessary that Hey < H, as people will not employmore labour than necessary to undertalce all of their housework. It is also possible that, inreturn for a transfer from i, i’s spouse will undertake some of H2. Denote the unit pricefor work which the spouse agrees to undertake, p’1 and the maximal amount undertaken atthat price Hm(pn).s Provided p’1 < wC, one possible example of a configuration of utilitieswithin the household is U(we— p’1Hm° — eH) + V(e2 + H — Hm0— 7H) for i andU(w,e +p’1Hm)+ V(e + H +Hm)for i’s spouse j. In this case, there has been a transferfrom i to j of p’Hm and, in addition to this, i also buys H5 of household services externallyat price we. The amount of transfer and the consequent share of household tasks that eachindividual undertakes will depend on income levels, effort expended at work and the internallynegotiated price for housework. In order to specify these, let us now consider the labour marketrequirements of firms.2.1 FirmsTo avoid the complexity of determining the relative price of goods and wages in equilibrium,it is assumed that effort applied to labour outside the household, denoted e, produces goodswhich are sold on external markets at a ftxed price. Labour can be used in one of three ways.There exists a constant returns to scale production function, with labour as its only input,8As will subsequently become clear, n mnemonically denotes a negotiated price. The determinants of plZ andH”° will be analyzed further on.14which converts one unit of labour input to one unit of exported output, the price of which isp’. Anyone can set up production and produce as much output as desired without affecting theprice; from hereon this work shall be referred to as piece rate work. People can also work inproviding household services in the formal market for housework. These are provided perfectlycompetitively so that, in any equilibrium where such work is provided, workers receive p’ foreach unit of labour provided.In addition to these two types of jobs, there exist a fixed number, F, of infinitely livedfirms, each of which wishes to hire a fixed number of workers per period, n, in the productionof another good, also sold on world markets, at another exogenously determined price p’ .Once again, production involves labour input only, but is undertaken using a markedly differenttechnology. Effort inputted into these tasks is assumed to be completely unobservable. It isassumed that for e <ë (where ë> 0) output per worker is zero, however for e e output perworker is 1. If a worker provides less than ë effort, the firm does not know until it observes theworker’s output, which is assumed to be common knowledge in the next period. However, asis standard in efficiency wage models, payment must occur in the period of work and there donot exist other arrangements, such as the worker’s posting a bond, which can serve to dissuadeworkers from shirking. Thus, in order to dissuade shirking, firms must pay efficiency wagesin these jobs. The efficiency wages provide compensation greater than that required if effortwere observable, so that the threat of terminating employment, and therefore losing the highercompensation, serves to deter workers from shirking.1°i abstract from considerations of market structure, and from explicitly considering the profit maximizingproblem of each firm, by imposing F, a and p° exogenously here. F is large enough to render each firm arbitrarilysmall in comparison. The results will not be qualitatively affected if the model is extended slightly to allow anyor all of these variables to be endogenously determined.10This is a standard, if somewhat bare, characterization of an efficiency wage type problem. These models15The efficiency wage jobs will, from hereon, be called “good” jobs as they are jobs whichmore than compensate workers for the disutility of effort expended at work. It is assumed thatthere are more than enough workers of each sex to fill all of the good jobs, that is nF < N/2.Most assumptions above are made to simplify the analysis and are not critical. All resultswill hold if it is instead assumed that, in all tasks, output responds continuously to effort andthat the effort required for different tasks may differ. Furthermore, allowing each firm’s labourdemanded to vary has no substantive effect. The assumption of either perfect observabilityor complete non-observability of effort is also made for simplicity, all results still hold if it isassumed that there are tasks which differ only in the level of observability. Critical assumptionsare that there exist some tasks which, due to informational problems, require efficiency wagepayments to dissuade shirking and that there are more than enough workers of one gender toundertake all of these tasks.2.2 Labour market equilibriumAt t = 0 each firm calls a wage, where g mnemonically denotes good jobs, which it is willingto pay good workers.” Firms then hire some of the applicants for the good jobs. All individualsare equally productive as workers, and identical in characteristics, with the exception that theyare most often associated with Shapiro and Stiglitz (1984) and have been widely used, see Weiss (1990) for asurvey of the main models and extensions. These models have received some empirical support, see Dickensand Katz (1987), Katz and Summers (1989), Krueger and Summers (1987,1988) and Campbell (1993). It hasbeen demonstrated by Macleod and Malcolmson (1989) that the nature of out of equilibrium beliefs regardingthe cause of separation between employees and employers determines whether the informational asymmetriesinherent in these models lead to efficiency wage payments or instead result in the surplus from employmentaccruing to the firm. Thus, the efficiency wage structure used in this model is more accurately seen as arisingfrom restrictions on the form of beliefs, in particular, it should be believed that termination of an employmentcontract occurs due to the worker shirking, see Macleod and Malcolmson (1989) for further details.‘1Time subscripts will be suppressed, since, in equilibrium, wages do not vary.16differ in gender, which is readily observable. In principle, firms are able to dismiss and hireother workers in future periods, however most equilibria focussed on will involve wage paymentswhich dissuade shirking and thus involve no dismissal or re-hiring. Since other firms observeseparation but not the reasons for separation (for if this were observable then contracting aroundthe informational asymmetry would be possible and efficiency wages would not occur) thesefirms will strictly prefer to hire workers not previously employed in “good” jobs provided anyweight at all is attached to the possibility of the separation occurring due to shirking.’2 Thusa worker knows that if he or she shirks and is dismissed he or she can expect to never again behired.’3 Since piece rate production is always possible, an individual not employed by any firmcan always obtain a wage p’ per unit of effort spent at work. It wifi now be demonstrated thatthe wage required to dissuade shirking in a good job depends critically on the characteristicsof the household in which each individual resides. The dependence of hiring decisions acrossfirms will then be shown by demonstrating that these characteristics are affected by the labourhiring decisions of other firms.To show the dependence of efficiency wages on household characteristics I first characterize,more fully, individual labour supply decisions. For individuals not in good jobs, this involvescalculating their optimal level of e denoted ê, given a wage We= p’ for external work. Due tothe equilibrium equality between We and p’, individuals will be indifferent between supplyingeffort to either task, without loss of generality then, all workers not engaged in good jobs are2Which is, of course, consistent with the form of beliefs required to sustain efficiency wages as an outcome.‘3A11 of the equilibria to be examined involve no shirking and thus permanent employment. Introducingexogenous turnover in employment and the possibility of re-hiring for shirkers will not affect the results providedre-hiring is not instantaneous and dismissal still acts as a penalty, i.e. provided that the efficiency wage structurepersists.17referred to as piece rate workers.’4 In some households a person’s spouse may wish to buyhousehold services from them at an internally negotiated, mutually beneficial price. Gainsfrom trade within the household are possible because a piece rate worker receives only p’ fora unit of outside work and a person buying household services externally pays p’/7, denotedp, for each unit reduction in housework that they purchase. Since j5 clearly exceeds p’, anyinternally negotiated price pfl c (p’,j5) for the internal provision of household services from thepiece rate worker to a spouse with demand for housework wifi be mutually beneficial. Since thisis a situation of trade between a constrained monopolist and a monopsonist, both the value ofthe negotiated price and the quantity traded are indeterminate. It is assumed however, thatif such mutually beneficial trades exist, for some set of prices, a price from within that set, isalways found and the couple manage to exploit the benefits from trade. For the purposes ofthis chapter, it does not matter how pfl is arrived at as long as a solution is found and the gainsfrom trade exploited.’5Consider the maximization problem facing individual i, under negotiated price p’ whichwill be paid per unit of housework provided:maxU(ep’+Hfp”)+V(e+H±H) (2.1)es.t. e2 0 (2.2)0 < <Hm (2.3)14AS seems reasonable, it is assumed that aggregate labour supply is sufficient to satisfy all demand forhousehold services.‘5For example it may be the outcome of a household bargaining game, as in McElroy and Homey (1981),where the threat points are a function of employment characteristics.18where H’7(p’) < denotes the upper bound on household services which person i’s spousewifi buy at price p’. Denote the solutions to this problem H6 and ê. It is shown in theappendix that, for pfl > p’, H8’ > 0, whereas may equal zero. It is also shown that any solution obtained is unique. Denote the indirect utility function obtained from this maximizationS Hm (p’)) where the superscript S mnemonically denotes a seller of household services,note that the wage p’ is suppressed in as it will not play an important role. The appendixalso derives the indirect utility function for a person who does not have the opportunity to sellhousehold services to their spouse. In this case, the indirect utility function is denoted iS(., 0)where the first argument which denotes the negotiated price, is left blank, and the 0 shows thatone’s spouse wishes not to buy household services. It is clear that the opportunity to tradewith one’s spouse raises (weakly) the value of the indirect utility function. Furthermore, sinceHS* > 0, such trades make an individual strictly better off. That is:S(pnHmac) > (2.4)Intuitively, this is because internal trades, occurring as they do at a mutually beneficial price,ensure that the seller of household services can work at a wage which is strictly better thanthat available in the market.It is people in good jobs who potentially demand household services. Here I establish theconditions under which this demand is positive. Someone in a good job will always choosebetween two values of e at work. If they were to shirk they would set e = 0, as any othervalue less than ë yields strictly less utility and does not affect their probability of dismissal. Ifthey decide not to shirk, they then set e = ë as any higher level of effort yields strictly lower19utility and again does not affect their probability of dismissal. I characterize the indirect utilityfunction of a person in a good job under the two cases separately. Let pB,O denote the indirectutility function of a shirker and Be denote the indirect utility function of a non-shirker, whereB mnemonically shows that this is the case for a buyer of household services.For a non-shirker, i, this function is obtained by maximizing utility by choice of HB,the amount of household services bought internally, and by choice of H the amount boughtexternally:U(w9ë— PnHB — piHe) + V(e + B — HB — (2.5)s.t. HB < HLzm(pn) (2.6)HB+7e<H (2.7)pnHB +pIHe m9è (2.8)where Hl2 (pfl) is the most labour that i’s spouse wifi supply at price pfl. Solutions aredenoted HB* and Hem. The appendix formally demonstrates that if an individual wishes tobuy household services, they will always prefer to do so from their spouse than from themarket. This is simply due to the household specific human capital which allows the buyer tobuy services cheaper than that available in the market even though the seller is providing theseat a wage above the market wage. It is shown in the appendix that a sufficient condition fora person in a good job to wish to buy household services, HB* > 0, is given by the followingassumption.A 1 For a person working ë in a good job, the ezternal price of household services is less than20the marginal rate of substitution of effort for income, i.e. p’ < —_______This condition implies both that good jobs pay well enough and that these jobs demand highenough effort from workers so that these workers wish to buy household services even at theexternal market price.’6 The maximization problem for a shirker is identical except thatV(O + H — H8 — 7H5) replaces V(ë+ E — H8 — -‘H5) in the maxirnand. Both maximizationproblems are explicitly shown in the appendix.An inirnediate implication of Al is that the possibifity of household trade in services atpfl <p’ makes a non- shirker strictly better off. That is:B(pnHlim(pn)W9) > IB(.,0W) (2.9)For this indirect utility function I include the wage variable w9 as it will play an importantrole. For a shirker, however, it is shown in the appendix that the possibility of household tradesdoes not necessarily strictly raise utility. Furthermore the following Lemma shows that, forgiven p’, a shirker wifi trade no more than a non-shirker.Lemma 1 H H*.Proof: See appendix.This Lemma is intuitively obvious. If expending less effort at work, effort at home causes less16t should be noted that ensuring household difference leads to mutually beneficial trades does not require thisassumption, since ë> 0 and the marginal disutility of effort is increasing, mutually beneficial within householdtrades will often exist between a person in a good job and one working a piece rate. At the cost of somecomplexity it is possible to relax this assumption and more accurately determine the reduced limits withinwhich p’ must lie, however this adds nothing to the results. In fact, modelling the household in Becker’s (1991)classic framework (which includes all members of the household maximizing a joint welfare function) ensuresthat a household will always benefit by such a division of labour and will not alter any of the model’s conclusions.21disutility so one wifi not wish to buy more household services. Furthermore, it follows fromLemma 1 that not being able to trade household services internafly, makes a non-shirker atleast as much worse off as a shirker. That is:Lemma 2 pB(w9 pfl) — iIIB(w9,.) iBO(w9,pn) —Proof: See appendix.I now determine the value of wages which are sufficient to dissuade shirking.2.2.1 Wages sufficient to dissuade shirkingIt is necessary to define an individual’s no shirking constraint for two different situations: 1)the person in a good job has a spouse who is a piece rate worker, 2) the person in a good jobhas a spouse in a good job.In situation (1) the no shirking constraint is the following:BO(n) + p1_s(I) (w,p) (2.10)remembering that S (.) is the indirect utility function of a potential seller of household serviceswhen trades are not available. To simplify notation, the argument for the limit on the amountof H which can be traded is suppressed, since it plays no role. The left hand side is thediscounted lifetime stream of utility obtained when deciding to shirk in the current period.The right hand side is the discounted lifetime stream of utility obtained if one decides to be a22non-shirker. ‘ Note that there is no uncertainty in these calculations. A non-shirker expects tomaintain employment forever while a shirker is certain to be dismissed in the next period and tonever again receive a good job.’8 The first term on the left hand side is the one period benefitto shirking, note that, even if shirking, an individual in a good job stifi has the opportunity totrade with his or her spouse, though such trade may not be worthwhile. The second term isthe discounted infinite stream of utility obtained when employed as a piece rate worker fromthe next period on. In this case, trade within the household is not possible, as one’s spouse isalso a piece rate worker and does not demand one’s services. The right hand side is calculatedallowing for trades, since in this situation the worker has a good job and buys household servicesfrom their piece rate working spouse.In situation (2) the no shirking constraint is given by:iF1S( n’ g ‘+‘ ‘‘‘‘‘) (2.11)i—p— i—pThis equation is similar to (2.10) except that now within household trading opportunities arereversed. Here, whenever one is in a good job, i.e. the first term on the left hand side andthe term on the right hand side, household trades are unavailable since one’s spouse also hasa good job and wifi not provide household services. However when not in a good job, one cansell household services to one’s spouse and thus benefit from internal trade, i.e. the first termon the left hand side.‘TAs is standard in efficiency wage models, if one wishes to shirk, then it is optimal to do so immediately.The no shirking condition then needs to only compare returns to shirking in the current period to those fromdeciding to never shirk.‘81t wiil be shown that in both the discrimination and non-discrimination equilibria which are consideredfurther on, no one shirks and thus no one is dismissed in any period. Thus if someone were to shirk theycalculate their return to shirking under the expectation that, if dismissed, they will never again receive a goodjob. Note however that given expectations which define a non-zero probability of termination being due toshirking, firms will never hire workers previously dismissed from good jobs. Thus (2.10) would stifi apply evenif the model were extended to allow for exogenous, involuntary turnover as in Shapiro and Stiglitz (1984).23Denote solving (2.10) with equality w and denote ziP solving (2.11) with equality w.As will soon be clear, d and n are intended to denote discrimination and non-discriminationrespectively. From the assumed separability of the utility function and the use of the envelopetheorem, it follows that for given w9, < where is the derivative of ‘P withrespect to uP, see the appendix for details.’9 Thus, since p > 1, an increase in uP serves tomake shirking less likely in both situations since it raises the right hand side in equations (2.10)and (2.11), by more than the left. That is, as would be expected in an efficiency wage model,increasing the wage raises the relative returns to not shirking. The separability can also beshown to imply that the values w and w are unique.2°2.3 Discrimination equilibriumI now establish a result which is essential in proving the existence of a discrimination equilibrium, i.e. an equilibrium in which only men receive good jobs.Proposition 1 w <w.Proof:w is given by4JBO(Wn)+p’PS(.) =4iBéE(w,pa) (2.12)i—p i—p19Note that since the constraint set in the maximization of equation (2.5) is convex, ‘I’,. is therefore welldefined except at kink points where ffB = Hztm and H8 + 7He = iT. For simplicity, the problems raised bythese two points are ignored since they are negligible in the domain of H.20Note the extreme usefulness of the separability assumption here. It is possible to replace this assumptionwith a weaker assumption which limits the complenientarity between leisure and income, and still preservethe basic efficiency wage structure. However, since this has no qualitative effects we persist with the standardassumption of separability.24w is given by= B9)(2.13)This proof proceeds by demonstrating that, when w is substituted into (2.12) the left handside [LHS] of the equation obtained is less that its right hand side [RHS]. This wifi imply thatw which solves (2.12) with equality is less than w, since, as discussed previously w9 raisesthe [RHS] by more than the left.Substitute w into (2.12). This yields[LHS] = BO(w9 pfl) + (2.14)andB,ë(wg[RHS] = ‘S ‘“ “. (2.15)i—pAdd and subtract the left hand side of (2.13) to [LHS] yielding:[LHS} JBO(w9) + [lBo(uP,pn) —+ps(pn)+[10 — j,S(n) (2.16)i—p i—p i—pDenote the first term in square brackets, which represents the one period utility gain from beingable to trade, D1 and the second term, which represents the net present value of the infiniteloss to a piece rate worker from losing trading opportunities with their spouse, D2. Similarly,by adding and subtracting the right hand side of (2.13) to [RHS] yields,+[fBë(w)Pfl) — Be(w,.)1 (2.17)i—p i—p i—pIn this expression, define the term in square brackets D3, this represents the net present valueof the gain from being able to trade in all fnture periods to a non-shirker. Since p> 0 we know,25from Lemma 2, that D3 > D1, that is, the one period benefit of trade to a shirker must be lessthan the infinite period discounted benefit of trade to a non-shirker. Also, from equation (2.4)we know that D2 < 0. Thus, since it follows from (2.13) that the remaining two terms not insquare brackets in [LHS] equal the term not in square brackets in [RHSj, it must be the casethat [LHS] < [RHS]. Since it was shown earlier, that an increase in w9 raises the left hand sideby less than the right hand side, it therefore follows that w9 solving (2.12) with equality is lessthan w. Thus w <wa.The intuition for this result is straightforward. Good jobs, though requiring higher effort,yield higher income. The income is useful to an individual both for buying goods and reducinghousework. However trading housework for income internally depends on individuals havingdifferent characteristics. If one’s spouse does not have a good job, then obtaining a good jobopens up possibilities for mutually beneficial trade, the net benefits of which for a non- shirkerare denoted by the expression D3, which is greater than that for a shirker, D1. if one’s spousehas a good job, then working at piece rate opens up possibilities for mutually beneficial trade,the benefits of which are equal to the negative of expression D2. Thus if one’s spouse has agood job, the cost of losing a good job is lower than if one’s spouse has a bad job, because inthe former case, losing a good job also entails the loss of mutually beneficial trades. Thus thewage required to ensure no shirking is higher if a worker’s spouse also has a good job.It will now be shown that a Nash equilibrium in the labour market is for all firms todiscriminate in allocating jobs. Since the model assumes no inherent differences between menand women, discrimination could involve the systematic exclusion of either men or women from26“good” jobs. For concreteness, however, discrimination wifi be taken to imply the reserving ofgood jobs for men not women.Proposition 2 If the employment status of a person’s spouse is not freely observable, and ifall other firms discriminate, then each firm finds it individually worthwhile to discriminate.Proof:Consider the decision faced by one firm, f. Suppose all other firms discriminate, that is theyonly employ males in “good” jobs. If f discriminates, wage w for good jobs will satisfy eachof f’s “good” worker’s shirking constraints, since f can be sure that none of these workers wifibe married to spouses in good jobs. This yields a total labour cost for f of nw to produceoutput of value np9.Suppose that f does not discriminate. Each of f’s employees will be either married to a spousein a good job or a piece rate worker. If an employee’s spouse does not have a good job, thenw will be sufficient to discourage shirking. If, however, an employee’s spouse has a good jobthen only a wage of at least w wifi discourage shirking. At the lower wage of w the employeewill take the job and shirk. To see this, note that Proposition 1 implies that the no shirkingconstraint is not satisfied at w and shirking yields utility j,B,0(, ., 0) + (p’,Hm°(pfl)),*IIS(p?L Hm(pn))which exceeds . Smce a firm cannot determine the employment status of anemployee’s spouse, a non- discriminating firm chooses between offering a wage w to eachemployee and guaranteeing output n or offering wage w and expecting some workers to shirk.With all other firms discriminating, (F — 1) men are hired. Thus the probability of a womanhaving a spouse in a good job is 2(F — 1)n/N > 0, denoted a. Thus the firm can expect a27proportion a of the women they hire to shirk at wage w. Thus since Proposition 1 establishesthat w > w, a firm that discriminates and offers good jobs to men only, either faces lowerlabour costs to obtain the same output or pays the same wage bill for lower output. Thus ifall other labour hirers are discriminators, f strictly prefers to be a discriminator. 0This proposition establishes that one possible Nash equilibrium in the labour market involves each finn discriminating in the allocation of good jobs and paying a wage w for theseworkers. Intuitively when all other firms discriminate against women, a single firm can ensurethat all of its “good” employees do not have spouses in “good” jobs, only if it reserves these jobsfor men exclusively. Then, by Proposition 1, discrimination ensures that the firm is guaranteedeffort ë at w which is less than the wage required to guarantee ë if women were also hired, w.Thus, in this equilibrium, the outcome is wage discrimination. Women and men with identicalcharacteristics as workers, receive different wages, since only men get the “good” jobs.Note that the existence of profits plays no part in this result. Extending the model andallowing entry in the output market and elimination of profits still requires that survivingfirms are those employing the cheapest labour hiring policy. If all other firms discriminate, thisremains a policy of discrimination. 2121j should be noted that this explanation bears some resemblance to the arguments often made by Marxistfeminists. Here, firms benefit because good workers pay their spouses to do housework at less than the marketrate. Compare this explanation with the conclusions of Dalla Costa and James (1972 p. 35):“Men appear to be the sole recipients of domestic services but, in fact, the figure of the boss isconcealed behind that of the husband.”And aiso that of Edmond and Fleming (1975 p. 8):“All the tolling power of thousands of housewives enables the possessing classes to increase theirriches, and to get the labour power of men and children in the most profitable way.”Proposition 2 provides one possible rationale for these assertions, even when firms in the labour market actcompetitively.282.4 Non-discrimination equilibriumI now derive a necessary and sufficient condition for the existence of a Nash equilibrium in whichall firms do not benefit by discriminating. It is shown that this condition depends criticallyon the costs of lost output as a result of shirking. In contrast, it is also demonstrated thatthe necessary and sufficient condition for the existence of a Nash equilibrium in which all firmsdiscriminate is always satisfied.Proposition 3 (n/N)(F — l)p > w, — w is a necessary and sufficient condition for theexistence of a Nash equilibrium in which no firm discriminates.Proof:Assuming a large number of firms, if all other firms do not discriminate then a single firm, f,expects equal numbers of men and women employed in good jobs elsewhere. If the firm setsw = w, only workers with spouses not in good jobs will not shirk. If denotes the probabilityof hiring a worker in a good job whose spouse also has a good job, i.e. a shirker at w, thenexpected profits aren[P(i—c)— wj = (2.18)recalling that p9 denotes the price of the output of good workers as defined in Section (2.1). Ifa firm sets wage at i4 it is assured that no worker will shirk. Therefore profits are given by(2.19)29Thus lll d — w — c) — w which implies— < ap9. (2.20)This condition will depend on the value of a which is determined as follows.Here we determine a as a function of D, the number of discriminating firms. For thepurposes of this proposition, however, the only relevant case is D — 0. Since N is the sizeof the workforce, there are F — 1 other firms, and D < F — 1 is the number of firms thatdiscriminate, then F — 1 — D do not discriminate. This implies that (F — 1 — D)n/2 womenare hired and Dn + (F — 1 — D)n/2 = (F — 1 + D)n/2 men are hired. Since there are N/2 ofeach gender, the probability of a man having a spouse with a good job is (n/N)(F — 1 — D)and the probability of a woman having a spouse with a good job is (n/N)(F — 1 + D). TIa firm pays w and discriminates, the firm allocates all good jobs to men, if D > 0, as thislowers the chances of hiring a shirker since (n/N)(F — 1 + 13) > (n/N)(F — 1 — D). Thereforea = (F — 1— D)n/N. However if 13 = 0 then a firm following a low wage strategy is indifferentbetween hiring all men and all women. The firm will, however, not hire a mixture of both sinceby hiring all men or all women a = (F — 1)n/N, whereas by mixing with proportion 5 femalesa = (F — 1)n/N + S/N for male workers and a = (F — 1)n/N + (1 — 5)/N for female workers,both of which exceed (F — 1)n/N. Thus if the firm chooses the low wage option it will still dobetter by discriminating even if other firms do not. This follows simply because by hiring allof one gender the firm at least ensures that it is not hiring a married couple itself. Thereforeif D = 0, we use a = (F — 1)n/N in equation (2.20), to obtain the following condition underwhich the low wage option will not be chosen:30(n/N)(F—l)p9 > w— w. (2.21)This is thus a sufficient condition for the non-discrimination equilibrium since, when itholds, if all other firms do not discriminate a single firm will not care about its employee’sgender. By reversing the inequality in (2.21) we obtain the condition under which firms preferto follow the low wage (discrimination) strategy when all others discriminate, thus (2.21) isalso a necessary condition for the existence of the non-discrimination equilibrium. DIf this condition holds, then a Nash equffibrium exists in which no firms discriminate and allpay wage w. This condition is more likely to hold when the cost to the firm of a worker shirking,p9, is large, the proportion of people with good jobs, nF/N, is high and when the differencebetween wages necessary to dissuade shirking for a non-discriminator and a discriminator,— w, are small. Intuitively, a non-discrimination Nash equilibrium may fail to exist because,if either the costs of hiring a shirker are small or the probabffity of hiring one low, firms preferto pay wage w instead of w and take the chance of hiring a shirker. They discriminate, even ifother firms do not, because, by doing so, they ensure that two people from the same householddo not work in their firm and thus slightly reduce the probability of hiring a shirker.Proposition 4 The necessary and sufficient condition for there to exist a Nash equilibrium inwhich all firms strictly prefer to discriminate is always satisfied.31This condition is given by setting D the number of firms discriminating, equal to F — 1 indetermining a, which implies a = 0, and reversing the inequality in (2.20) which gives thecondition under which a firm prefers the discrimination policy. This yields0<w—w. (2.22)Since w > w by Proposition 1, condition (2.22)is always satisfied. DNote that in both the discrimination and non-discrimination equilibria, no one ever shirks.Thus the assumptions under which equations (2.10) and (2.11) were calculated are correct.222.5 Stability of the equilibriaEven though not a dynamic model, we discuss stability here in the sense of robustness todeviations from the equilibrium strategy by small numbers of players. Unstable equilibria aretherefore those in which deviations from the equilibrium strategy by a small number of firmslead to further deviations by other players and thus to an unravelling of the equilibrium. It isoften the case with models of multiple equilibria that asymmetric equilibria tend to be stablewhile symmetric equilibria are unstable, as surveyed in Matsuyama (1993). Alternatively, wherethere exist many interior equilibria, these often alternate between being stable and unstable.Neither of these outcomes are a property of the current model where it will be shown thatalmost any configuration of stability properties may apply. This section shows the dependence22The stability properties of both equffibria are discussed subsequently. Under some conditions there may a’soexist another equilibrium in which half of the firms discriminate against women and the other half against men.However since such an equilibrium is never stable it is ignored.32of the stability results on the model’s primitives and provides an interpretation of these results.This model’s unusual stability properties arise from the existence of an alternative employment strategy for finns, the returns from which do not depend upon the activities of otherfirms in the market. Labour market policies of other finns affect a firm by determining thegender composition of the labour force and thus the probability of hiring an employee with aspouse in a good job. However by pursuing the strategy of non-discrimination and paying thehigher wage, w, no shirking is ensured irrespective of household characteristics, thus implyingthat other firms’ policies do not matter. The extra cost of pursuing such a strategy is the wagedifferential w — w. Thus, if parameter values render this a preferred strategy, deviations bya number of other firms may not matter and the equilibrium can be stable.To see this clearly consider Figure 2.1, which plots a firm’s costs on the vertical axis andD, the number of discriminating firms, on the horizontal axis. The cost of pursuing the highwage strategy is depicted by the horizontal line, w — w, showing the differential between lowand high wage jobs, which is independent of D. The cost of following the low wage strategyis the risk of hiring a shirker (someone with a spouse also in a good job), which is denotedby the downward sloping line (F — 1 — D)(n/N)p9. The line slopes downward because theprobabifity of hiring a shirker at w falls with an increase in economy wide discrimination,since fewer women participate in good jobs and thus fewer men come from households wherethey undertake substantial household tasks. At D F—i all firms discriminate, the probabilityof hiring a shirker is zero, since women never have good jobs, and the line touches the horizontalaxis.33A non-discrimination equilibrium exists when the downward sloping line intersects with thewage differential line, corresponding to condition (2.21), and touches the vertical axis at thepoint (F — l)(n/N)p9.As shown in Figure 2.1, this equilibrium will be stable to deviations byany number of firms less than D, i.e., if fewer than D’ firms discriminate, then the preferredstrategy of any other firm is to be a non-discriminator, and thus, all other firms will remainnon-discriminators. Clearly the greater the value (F— 1)(n/N)p relative to w — w, the morestable the equilibrium. As the point of intersection approaches the origin, the equilibriumbecomes less robust to deviations and eventually completely unstable where it meets the linew-w.Conversely a discrimination equilibrium, D = F, always exists and is locally stable. Thediagram thus makes clear the logic of proposition 4. With all others discriminating D = F — 1the cost of discrimination is zero whereas proposition 1 shows w — w > 0. The stability ofthe equilibrium increases with the wage differential which is determined implicitly by equations(2.12) and (2.13). Interrns of the model’s primitives, it is easy to show that the wage differentialis falling in y. As y rises, the benefits of household trades relative to market purchases ofservices fall and w must correspondingly rise to ensure incentive compatibility. With closeto 1 the difference is very small and the equilibrium becomes less stable, as in Figure 2.2, wheredeviations by more firms than F — 1 — D’ breaks down the discrimination equilibrium.Thus, though it is possible for the usual configuration of stability properties to arise in thismodel, it is also possible that both equilibria be locally stable. It will be shown in Section 3that the stability of the discrimination equilibrium can have important policy implications.342.6 Heterogeneous firmsOne implication of assuming identical finns is that stable outcomes of the model are all symmetric Nash equilibria; either all firms discriminate or none do. However, in reality, firmsmay differ in the importance they place on ensuring that workers do not shirk. For example,shirking by carelessly building the foundations of a house is probably more important thanshirking in effort put into weeding a field. A simple way of introducing heterogeneous finns isto allow the effect of shirking on output to differ by firms. In the appendix it is shown thata Nash equilibrium can exist in which finns for whom shirking is costly pay w and do notdiscriminate, while other firms continue to discriminate. Intuitively, if the costs of shirkingdiffer by firm, then low cost firms may be willing to risk hiring shirkers at a lower wage like wwhereas high cost firms will prefer to pay w to ensure that no shirking occurs.3 Effect of affirmative action policiesThe model can provide some rationale for the use of affirmative action policies in alleviatinglabour market discrimination. Affirmative action can require the setting of target quotas forthe hiring of selected groups, in this case women. The usual rationale for these policies is thatequal opportunity legislation (which is almost universally law) is insufficient to compensatefor the legacy of a cumulative history of discrimination, Gunderson and Riddell (1993 p.56fl.Thus these policies are advocated as a temporary corrective device serving to partially offsetthis history of discrimination.35In the model presented here, an affirmative action policy which requires all targeted firmsto give half of their good jobs to women, can also be shown to be useful in moving the labourmarket away from an equilibrium in which firms discriminate to one in which firms do not.Proposition 5 If a non-discrimination Nash equilibrium exists, and is stable a policy of affirmative action, if applied widely enough, can move the economy from a discrimination equilibrium to a non-discrimination equilibrium.Proof: Equation (2.21) gives the necessary and sufficient condition for the existence of a Nashequilibrium in which all firms do not discriminate. If this condition is satisfied then it is possibleto find a value of D, denoted D, (corresponding to the same point in figure 2.2) such that(n/N)(F—1— D)p = w— 4 (3.1)Suppose the labour market starts in a situation where D = F, that is, the discrimination Nashequilibrium. Now, suppose that affirmative action policies can be applied to F — D + 1 ofthe F firms. This yields the following inequality:(n/N)(F — D)p > w — w. (3.2)Which is the condition for any one firm to prefer paying the wage w and not discriminate(derived from equation (2.20), substituting (n/N)F — D* for a). Thus, from (3.2), we seethat the remaining firms, who are not directly stopped from discriminating by the affirmativeaction policy, voluntarily choose to stop discrimination and D goes to zero. DIn this situation, a policy of affirmative action is able to move the labour market from a36situation in which all firms discriminate to one in which there is no discrimination. The intuitionfor this result is straightforward. Discrimination may cease to be beneficial if the affirmativeaction policy can induce enough other firms to not discriminate because, with enough womenemployed, the probability of hiring a man with a spouse not in a good job is small enoughto make it no longer worth the risk. Firms expect a high chance of hiring a shirker at thelower wage, w, and thus pay wage w to induce ë from any worker. In this case, even firmsnot directly affected by the affirmative action policy, no longer benefit by reserving “good”jobs exclusively for men. An implication of this result is that to conclude affirmative actionpolicies do not have wide reaching effects, because they only affect a few firms directly, as inGunderson (1989 p.63), may be incorrect. This model shows that, to the extent that such apolicy contributes to increasing opportunities for women outside the household, it may serveto reduce incentives for other firms to discriminate.3.1 Welfare properties of equilibriaIt can be shown that the discrimination equilibrium is a potential Pareto improvement overthe non-discrimination equilibrium. This is because the existence of household specific humancapital makes it always more socially efficient for either one or both members of the householdto undertake their own housework instead of hiring in an outsider to do it. By restrictingaccess to good jobs, a discrimination equilibrium limits the discriminated against sex to undertaking housework or piece work and thus ensures that household specific human capital is notwasted. Since the Pareto criterion does not, of course, account for distributional considerationsthis is an unambiguous improvement. Clearly a social welfare function which values equality37across sexes in either outcomes or employment opportunities, may yield different implications.However, provided employment, per S e, is not valued and the government can levy lump sumtaxes on employees in good jobs (so that the discrimination equilibrium is not affected by thetaxes)it will, in general, be possible for a government policy of taxing men in good jobs toattain the desired distribution across sexes as well as the efficiency gains of the discriminationequilibrium. It should be noted however, that this result can be sensitive to the simplifyingassumption that all individuals are of equal ability, or, in other words, that ability does notmatter in production. In a more complex framework where ability matters, the dominance ofthe discrimination equilibrium may no longer be maintained as there is a social loss to excludingall high ability individuals of one gender from the good jobs. Although not explored here, thissuggests that perhaps changes in technology which utilize high ability workers more intensively,may precipitate a movement away from the discrimination equilibrium.4 Empirical implicationsRather than provide an empirical test of the model, this section attempts, less ambitiously, todemonstrate that the model’s conclusions are at least consistent with the results of existingempirical studies and to also show that the model suggests a tentative explanation for theobserved recent reduction in the wage gap.Occupational segregationGoldin (1986) argues using historical data, that women have been over represented in jobs38paying by the piece in comparison with men. In U.S. manufacturing, in 1890, 47% of femaleoperatives worked in piece rate jobs, defined as jobs in which monitoring was easy or direct,as compared with men among whom only 13% had such jobs. She also finds a much higherprobability of employment in team production for men, where one would expect that individualeffort is more difficult to determine. Evidence from U.S. clerical employees in the 1940’s suggestsa similar trend. It was seen there that a large influx of female employees followed technologicalchange in clerical work which lead to the “routinization” of clerical tasks. She shows that“routinization” lead to a fall in supervision requirements, a consequent lowering of wages paid,and, soon after, to a large influx of female employees. A similar pattern is echoed in Reskin andRoos (1990) survey of the “fejjzation” of 11 previously male dominated American industries.They find that women’s entry had little impact on the gender gap since women were mostlyplaced in lower paying, less prestigious occupational sub-specialties. An example fitting wellwith the theme of this chapter is the experience of adjusters in the insurance industry asdocumented by Phipps (1990). In the 1960’s, insurance adjusters scheduled their own claims(relatively few per day) and were largely self-monitoring, having their own car and conductingmuch of their work outside the office. In the 1970’s however, adjusting was standardized sothat it was able to be conducted from within the office, largely with the use of a phone andcomputer. Workers then became subject to quotas and electronic management. Just after thisdramatic fall in supervision costs, the profession saw a marked change in gender composition.U.S. census bureau data shows that from 1970 to 1980 the occupation increased by more than67,000 new jobs to total 98,407 with a rise in female adjusters to 73,744 and a correspondingreduction in male employees of 6,477.39Although not all incidences of feminization surveyed in Reskin and Roos (1990) followedsuch stark changes in the technology of work, the pattern of “feminization” in most occupationsexamined was consistent with a clustering of women in the relatively low paying, part-time,and/or less responsible and autonomous jobs which is consistent with the pattern predicted bythe model.Marriage PremiaGoldin (1990, p.102) states that:the role of marriage in enhancing the earnings of male workers is still only dimlyunderstood.”As Korenman and Neurnark (1991) note, the existence of a wage premium for married menis one of the most robust empirical findings in the social sciences. Most studies, however, findno similar premium for women and often instead that marriage is actually negatively correlatedwith female earnings. This sort of result is not unique to the U.S., in a recent analysis of Swedishlabour markets Richardson (1995) finds a similar premium for married men, even though muchof it is reduced when fixed unobservables are accounted for. It was also found in a sample of12 OECD countries by Schoeni (1990).Korenman and Nenmark (1991) attempt to better understand the existence of the malemarriage premium with a detailed analysis of company level data. They summarize the workof Medoff and Abraham (1981) who looked at company personnel data for a large U.S. manu40facturmg firm’s white male, managerial employees, the information included detailed employeeappraisals and extensive job classification. It was observed that married men were more likely tobe located in higher paying job grades, as opposed to being paid more within a grade, and thatmarriage was positively correlated with promotion once employed in a firm, the probability ofpromotion being 10.5% higher for married men, but had little effect on entry level employmentprospects. Interestingly, analysis of worker evaluations suggested that the promotion success ofmarried men stemmed from their exhibiting actual higher productivity on the job rather thanfor reputational or some other reason. They find supervisors’ performance ratings of marriedmale subordinates tend to dominate the ratings of unmarried males.The model presented in this chapter provides an explanation for why married men shouldhave higher on the job productivity. In the discrimination equilibrium, men alone have goodjobs and those who are married are able to trade with their spouses to undertake the provisionof household services. These people will tend to outperform unmarried men since, for a givenwage, they are less likely to be shirkers. To married men, loss of the job involves loss of thebenefits from within household trades as well as loss of the wage premium, whereas to unmarriedmen the only cost of job loss is the loss of the wage premium. Unmarried women, on the otherhand, should fare no worse than married women since, in the discrimination equilibrium, nowomen at all receive good jobs.2323Note that if, for some reason, a firm were forced to employ some women in efficiency wage jobs it maystrictly prefer unmarried women, depending on the number of men employed in efficiency wage jobs. If a highproportion of such men have good jobs, then married women are very likely to be married to men in good jobsand thus more likely to be shirkers than unmarried women (since losing the job opens up the possibility of withinhousehold trades and thus raises the possibility of shirking). This may explain the policy of marriage bars, whichwere prevalent in the U.S., see Goldin (1990 cli. 6, and 1991), which explicitly discriminate against the hiringof married women and call for the dismissal of female employees upon marriage.41On a final and more speculative note, the model also suggests a direction of explanationfor observed recent declines in the gender gap. O’neffl and Polachek (1993) show that theunadjusted female male wage differential in the U.S. rose from .602 in 1976 (approximatelywhere it had been since the end of World War II) to .716 in 1990. They show that this trendreflects improved female training (both on and off the job) but cannot determine whetherthis arises due to a decrease in employer discrimination, or increase in women’s educationefforts and/or work attachment. Whichever explanation holds, and it seems likely that all playsome part, the current model would suggest a link between those factors and the decline inprominence of the traditional nuclear family.This decline has been well documented by sociologists. It appears that almost all of thetraditional characteristics of the family have seen a marked change since 1960. Fewer personsare marrying, they are marrying later, more marriages are broken by divorce and those who aremarrying are having fewer children, see the survey article by Popenoe (1993) for a discussionof evidence from U.S. census data. According to Popenoe p.528, since the 1960’sThey [families] have grown smaller in size, less stable, and shorter in life span.People have become less willing to invest time, money and energy in family life,turning instead to investments in themselves.Since the model presented here does not include investment in human capital decisions, itcan only suggest a link between weakening family structure, better female training and reduceddiscrimination by employers. It would suggest firstly, that the presence of more single people42of either gender reduces employers’ benefits from discrimination. Secondly, since the size offamilies is declining, the benef. s of within household trade due to specific human capital maybe smaller than previously, thus again reducing incentives for discrimination. Finally, thoughthis is not explicitly explored in the current model, if household human capital were explicitlyconsidered, lower expected marriage duration, would imply that the benefits to male employeesin good jobs from trade with their spouses may not be available, this would then imply thatfirm’s benefits to reserving good jobs for men exclusively would also be lower and incentivesfor discrimination weaker.5 ConclusionsIn reality, it is not because firms understand the impact of different jobs on the allocation ofhousehold tasks that they favour men over women in the allocation of “good” jobs. Firmsprobably discriminate because they follow a rule of thumb which states that women are not asgood in those jobs or not as reliable as men and should therefore not be hired. This chapterprovides no explanation or historical analysis for why such a situation may have evolved to thedisadvantage of women and not men. The model presented above does, however, provide oneexplanation of why rational, profit seeking firms following that rule of thumb may outperformfirms who have a more egalitarian hiring policy. By constructing equilibria in which a firmhiring women in good jobs, does worse than a firm who does not, it provides one explanationfor the persistence of such a rule. It shows how a hiring policy of discrimination is reinforcedand maintained even in a competitive labour market setting.43This explanation differs from traditional explanations of discrimination in that 1) it does notrely on any inherent asymmetry between men and women and 2) it is based on the interactionbetween men and women within a household. The model also shows, however, that there existconditions under which an equilibrium in which firms do not wish to discriminate can exist andbe a stable outcome. When a non- discrimination equilibrium is stable, Section 3 shows that apolicy of affirmative action can move the labour market from the discrimination equilibrium tothe non-discrimination equilibrium. That is, non-discrimination can become preferred even byfirms not directly affected by affirmative action policies. This suggests that, even if a policy ofaffirmative action can not target many establishments, it may still have wide ranging effects.Empirically, the model implies that the existence of wage discrimination will be greatestacross jobs where effort is unobservable. It predicts that men will occupy jobs which rely mostheavily on difficult to monitor individual effort. This seems reasonable, as it would include jobsrequiring management or supervision, jobs which have been traditionally considered part of themale realm. It suggests that males will also be disproportionately represented in other jobswhich require an efficiency wage payment, a supposition which is supported by the empiricalanalysis of Goldin (1986). This model can also provide one explanation for why, within occupation types, men seem to be allocated the good, or higher paying, jobs. (For a recent example ofthis in a competitive labour market see Wood Corcoran and Courant (1993)). It also explainsthe existence of a wage premium for married men, without any similar premium to marriedwomen, and ilnics the recently seen improvements in women’s labour market performance tochanges in the stability of the traditional family structure.44In conclusion, this chapter shows, perhaps unsurprisingly to Marxist feminist economists,that labour market discrimination can be linked to the allocation of tasks and resources withinthe household. Perhaps less obviously, the model developed here also demonstrates the self-sustaining nature of a discrimination equffibrium. Not only does discrimination stem from anunequal allocation of household tasks, but, in the other direction, such an unequal allocationwithin households, is shown to be reinforced when firms exclude women from the high payingjobs.45Figure 2.1(F-) 946Figure 2.2(F- rD** D=F473IChapter 3.INNOVATION, MONOPOLIES AND THEPOVERTY TRAP.49INNOVATION, MONOPOLIES AND THE POVERTY TRAP241. IntroductionThis chapter examines a situation in which an economy may simultaneously possessa technologically stagnant, no growth, steady state and a steady state involving sustainedtechnological progress and growth. Here, multiple steady states arise principally because ofself-reinforcing effects between technological change and market structure, that is, becausetechnological change affects market structure in a way which increases future technologicalchange, thereby once again altering market structure and so on. An advantage of thisexplanation is that multiplicity here arises through the effect of technological change onmarket structure (i.e. from the process of technological change itself) , rather than fromspillovers or production non-convexities, or due to complementarities in demand orproduction (i.e. through the exogenously posited characteristics of the technology).At an industry level, the impact of market structure on innovation has received muchattention in partial equilibrium analyses (as expounded upon by Schumpeter (1951) andsurveyed in Kamien and Schwartz (1982)). We, however, focus on the general equilibriumeffect of industrial structure on innovation, through its impact on factor returns, and thesubsequent effect, in turn, of innovation on industrial structure. The crucial factor we wish tohighlight is that returns to a factor employed in research relative to production are lowerwhen all sectors are competitive than when there exist sectors which are monopolistic. Thusa monopolistic industrial structure, i.e. one in which other industries tend to be dominatedby monopolists rather than being competitive, raises relative returns to innovation in non24’phjs chapter represents a revised version of Baland and Francois (1995a).50 -directly related industries and is, therefore, more conducive to technological progress andgrowth. Furthermore, since industrial structure is itself the outcome of past innovations (i.e.there are more likely to be monopolists in industries which have devoted resources toinnovative activities than in industries which have not), current returns to innovationdepend upon previous levels of innovation through the inherited industrial structure and canthus make innovation a self-sustaining process.Recently there have been many models focusing on the process of growth throughthe purposeful development of new technology. In contrast to the previous literature in whichaggregate productivity increased almost incidentally, often as a byproduct of investment inanother activity (e.g. Arrow (1962), Romer (1986)) this generation of research focuses on theintended creation of new knowledge. Aghion and Howitt (1992) and Grossman andHelpman’s (1991 Ch. 4) quality ladders model explore the Schumpeterian process of growththrough innovation and creative destruction.25In these models, creative destruction takesthe following form: innovation leads to the displacement of an incumbent monopolist by asuccesful innovator and thus to the transfer of monopoly rents from the former to the latter.The lure of monopoly rents leads profit seeking agents to devote resources to innovativeactivities. As research activities are carried out in a decentralized manner, the size ofpotential rents is an important determinant of the intensity of research activity andtherefore of growth. Importantly, these models implicitly start with incumbent monopolistsalready holding infinite patents on the use of their technology, so that when a newinnovation occurs in an industry, one incumbent is simply replaced by another and theunderlying industrial structure remains basically unchanged.The point of these models is to capture, in a tractable way, the effects of knowledgespillovers and casacading technological progress in explaining how an economy can exhibit25See also Shell (1967), Romer (1990) and Sergestrom, Anant and Dinopoulos (1990)).- 51sustained growth. In contrast, this chapter aims to understand the process leading to thepoint where these previous analyses take over. That is, by generalizing these previousapproaches slightly and allowing some industries to not initially be dominated bymonopolists, we firstly demonstrate that an economy otherwise capable of sustainedtechnological progress and growth may be caught in a no growth steady state or “PovertyTrap”. We determine, in terms of exogenous parameters, the conditions under which apoverty trap can emerge, and show the initial conditions under which an economy will becaught there. We then seek to understand the process of transition from stagnation to self-sustained growth, and the role of government therein, through an analysis of the model’stransitional dynamics.This chapter is also related to another branch of the endogenous growthliterature which has been concerned with explaining the diversity of observed cross countrygrowth experiences through models exhibiting multiple steady states (Azariadis and Drazen(1990), Durlauf (1993), Matsuyama (1991)). Their analyses draw on a venerable tradition indevelopment economics dating back to Rosenstein-Rodan’s (1943) theory of the Big Push,Nurke’s (1953) discussion of balanced growth and Rostow’s (1956) emphasis on stages ofgrowth and the take-off. This literature shared in common the belief that some form ofcomplementarity between individuals (often investors) could explain underdevelopment as alow level equilibrium trap, and that the process of development therefore implied a drasticmove of the economy from a low level equilibrium to a higher one. Although, retrospectively,the insights provided by the early contributors may not always appear theoretically wellfounded, they offered historical evidence of economies experiencing the sorts of dramaticchanges which were consistent with their evocative explanations of complementarities andcoordination problems. More recently, Murphy, Shleifer and Vishny (1989) and, before them,new Keynesian approaches (as synthesized in Cooper and John (1988)), provide a theoreticalfoundation for the possibility of Poverty Traps by explaining plausible mechanisms which-52give rise to these complementarities. In contrast, the endogenous growth literature onmultiple steady states is not primarily concerned with the mechanism leading tocomplementarities, per Se, but rather with developing the implications of suchcomplementarities in an endogenous growth framework, or, as in Matsuyama (1991),understanding the conditions under which coordination can affect a movement from adominated steady state. Despite this being a model of endogenous growth, we think of ourmodel as more closely related to the work of Murphy, Shleifer and Vishny (1989) and the newKeynesian approaches in its focus on the mechanism giving rise to multiple steady states.The terms “technological progress” and “innovation” used above have differentinterpretations depending on the context in which they are used. In the modern creativedestruction literature, concerned with already developed economies, attention is focused onone particular aspect of innovative activities, namely, the creation of new or better productsor inputs. Innovative activity there implies the use of resources in research and developmentwith an aim to both generating new ideas (development of new products, technologicalimprovements...) and seizing rents on them through exclusion (patents, confidentiality,contracts or licences). In the context of LDCs however, a different interpretation ofinnovation, emphasizing entrepreneurship rather than R&D, seems more relevant. It is notso much lack of R&D which appears to constrain LDCs to the use of less productivetechnology but rather insufficient entrepreneurial incentive to adopt new methods byborrowing from ideas already developed elsewhere. Thus, in our view, a theory of growthemphasizing the role of entrepreneurship may yield useful insights into the causes of lowproductivity in LDCs. In a later section, therefore, we emphasize the entrepreneurialinterpretation of innovation, with a view to using the insights developed in the basic model toexplain stagnation in LDCs.53Before developing the main model, Section 2 of the chapter presents a stylizedversion in a static framework in order to provide some basic intuition of the generalequilibrium effect causing the complementarity which motivates the chapter. Section 3 thenpresents a dynamic model, based on Grossman and Helpman (1991), in which multiplesteady states are shown to exist and determines the conditions under which they do. Thissection also examines the welfare properties of the steady states and describes the dynamicsof the system between them Section 4 discusses a possible role for government researchsupport, either through subsidies or publicly provided research, in moving the economy awayfrom Pareto dominated situations. It also explores some implications of introducing patentsof finite duration. In Section 5 the entrepreneurial interpretation of innovation is developed,with emphasis placed on the self-generating nature of the entrepreneurial process. A briefconclusion is provided in Section 6.2. The Static ModelThere is one final good denoted y which is produced competitively using a continuumof intermediate inputs distributed along a unit interval according to the following productionfunction:f lnx1diy=e° , (1)where Xj stands for the intermediate good i. The final good is taken as the numeraire: Py=l.There are L consumers with well-behaved preferences over the final good. Individuals are allidentical and each is endowed with one unit of inelastically supplied labour.54Labour is used in production of intermediate inputs and in research activities. Aslabour is homogeneous across all sectors and activities, there is a unique wage rate denotedw.26 Intermediate inputs are assumed to be produced according to a linear productiontechnology with labour as the only input as follows:x=41,, (2)where A denotes labour productivity and £, the amount of labour used in the production ofintermediate good i. The level of labour productivity depends upon the arrival of aninnovation in that sector. Innovations take the form of a new technology in which labourproductivity is ytimes higher than under the previous technology (with y >1), so that theproduction function for the intermediate good i after innovation becomes:x, = ‘yAi,. (3)If £Rfunits of labour are devoted to research in sector i, then the probability of aninnovation in that sector is denoted ‘-t(R). Similar to Aghion and Howitt (1992), I(R)denotes the aggregate probability of an innovation arriving in sector i with (F (J?R) denotingthe increase in probability due to a marginal increase in research activity: ct(j) is twicecontinuously differentiable, (flj) > Oand I’(R1) 0.27 An advantage of this26Uniqueness of wage rates is a simplifying assumption only, the model could easily beextended to allow for heterogeneous labour (for example with private investment in humancapital) and multiple equilibrium wage levels without affecting the results provided labour issubstitutable between production and research.271n Aghion and Howitt (1992) the concavity of cI is a useful modelling device whichintroduces diminishing returns to research labour without requiring explicit modelling of theresearch sector. They also include another type of labour, specialized individuals denoted R,who can only work in research. Thus, strictly, our function (J) corresponds more accurately totheir function p which implicitly supresses the notation for specialized labour. If the readerwishes, our model can equivalently be thought of as including specialized labour only able toundertake research, with the notation similarly suppresed in the function (I). It is alwayspossible, however, for CJ) to be a linear function, as in Grossman and Helpman (1991), whichhas the more desirable property of independence between an individual’s research55specification is that it abstracts from explicitly considering the structure of research firms. Itis assumed that within each sector at most one innovation can occur.28 A succesful innovatorbecomes a monopolist in the production of that intermediate good using the new technology.Since the final good sector is competitive, it follows from the final good productionfunction that final good producers minimize costs by allocating expenditures evenly over allinputs. Thus, as the final good is the numeraire and it is a competitive sector, the unit cost ofproducing the final good is equal to 1: 1 is therefore also equal to expenditure on each input i(since there is a unit interval of inputs). The monopolist’s decision problem is therefore verysimple; profit is maximized by limit pricing at the marginal cost of the firm’s best competitor.Profits for a successful innovator in sector i are therefore denoted by:(1 1 (i 1 y (i 1 1 yA. (y—i’H=I——— Iwx.=I——— Iw—=I——— Iw_!=I ly. (4)LA1 yA1) ‘ A1 ‘yA,) p, LA, yA,) w )We normalize L to 1, so that the labour-market clearing condition can be written as:$‘.e1di +J0Rid1 = 1. (5)The comretitive economy equilibriumLet us first consider a situation where all intermediate firms share a commontechnology and, therefore, produce all intermediate goods competitively. We proceed by firstassuming that no research is undertaken in this economy, and then establish the conditionsproductivity and the level of R & D in the sector. This will, in fact, be the interpretationfollowed in analyzing the model’s dynamics.28TMs is a strong assumption which is made for expositional convenience here. However inthe dynamic version of the model, with continuous time, the analogous assumption is thatinnovations do not occur simultaneously.56 -under which this assumption is valid. The production function of each firm is given byequation (2) and the price of any intermediate good is therefore given by --. From equation(1), the amount of each input used to produce output y equals or equivalentlypi wwhich corresponds to units of labour. Therefore, labour market clearing implies:1=$--di = y=w. (6)Let us now consider the profits which would accrue to a successful innovator. Fromequations (4) and (6), these are equal to 2’ 1 w. As long as the costs of undertaking2’)research exceed the expected benefits, the above described equilibrium is the uniqueequilibrium of this economy. This is given by the following equation:I or I2’’’(O)<1. (7)kY} Y)If the above inequality does not hold, then any equilibrium must involve a positive amount ofresearch.The monoyolistic economy equilibriumWe now consider the situation in which, within each industry, one firm only hasaccess to the technology described in equation (2). All other firms have in common thefollowing inferior technology:=eA,.1?, (8)57 -where E < 1. Thus, within each industry, the privileged firm limit prices its potentialcompetitors out of the market by setting its price at their marginal cost of production whichequals —-. Once again, competition in the final good market implies demand of -s-, whichp1here corresponds to -- eA, units of each input. For each input, labour demanded thereforeequals . Equilibrium in the labour market implies:wl=j —di=y=— (9)ow eA comparison of equations (6) and (9) shows the main point of this section, namelythat the ratio of w toy is lower with the existence of monopolies. Note, however that the totalamount of output in the economy does not differ across the two situations; all labour isemployed and the technology used in production of intermediate inputs is still given byequation (2). What differs is that the existence of monopolists in each sector reduces labour’sshare of total income and, since the real wage is lower, the cost of undertaking research isthereby reduced. Furthermore since the expected benefits of research depend on the size ofsales and therefore income, y, they remain unaltered across situations. As a result, thecondition for there to be no research in equilibrium, corresponding to that given by equation(7) above, is weaker when industries are monopolistic:or [Y_l(o)<E (10)A comparison of (10) with (7) shows that the existence of a monopolist in other intermediatesectors raises the incentives for research in any one sector. This arises, even though theCobb-Douglas specification of final good production ensures no direct cémplementaritybetween intermediate sectors, as a result of the general equilibrium impact of monopolies onrelative returns. It is therefore clear that there exists a range of values for y such that58condition(10)isnotsatisfiedwhilecondition(7)is.Inotherwords,forsucharangeofparameters,acompetitiveindustrialstructureinvolvesnoresearchwhile,inamonopolisticstructure,researchwilloccur 29.Thisholdseventhough,consideringanyonesectoralone,returnstoinnovationinthatsectordonotdependuponwhetherthecurrentbesttechnologyisusedexclusivelybyanincumbentmonopolistoravailabletoallproducers.3.TheDynamicModelTheabovemodelshowedthatindustrialstructureaffectsincentivesforresearchbyinfluencinglabour’sshareoftotalincome,oralternatively,byalteringtheproportionofaggregateincomeaccruingintheformofmonopolyrents.However,properlyconsidered,theprocessofresearchandinnovationisinherentlydynamic:currentresourcesareinvestedinexpectationoffutureprofits.InthissectionwethereforedevelopthebasicinsightofthemodelpresentedaboveinanintertemporalframeworksimilartothatofbothAghionandHowitt(1992)andGrossmanandHelpman(1991).ThoughsimilartoGrossmanandHelpman(1991)inparticular,ourapproachdiffersinthatweallowforendogenouschangeinindustrialstructure,thatisforthepossibilityofindustrieswhichare,initiallyatleast,notmonopolistic.WealsopursueadifferentmethodofsolutionandnormalizationtothatusedinGrossmanandHelpman(1991)soastobestmakecleartheimpactofchangesinindustrialstructureonincentivesforinnovationandthereforegrowth.Itwillbeshownthat,inthissituation,itispossiblethattheeconomyconvergestowardsdifferentsteadystates.Theconsumer’sproblem29Notethat,intheabovestaticframework,thenoresearchoutcomeissociallyoptimal.Weignorethisaspectoftheissueheresince,aswillbeclearlater,thisneednotbetrueinadynamiccontext.59There exist L infinitely lived agents with the following intertemporal preferences:U= f 1nC(r) d’r (11)where C( r) denotes the amount of final good consumed at time ‘C and p is the rate of timepreference. As in Section 2 the production function for final goods is given by:lnC(’r)=f’ lnx1(’r)di, (12)where x, (r) denotes the amount of industry i’s input used at time ‘C in production of thefinal good. Each individual’s intertemporal budget constraint is given by:eR p(’r) C(’r) di W(’r) + e__t w(’C) d’r with(13)R(r)= s: r(s) dswhere R(’r) represents the discount factor from time ‘c to time 0 and W(t) denotes the netpresent value of the household’s wealth at time t (see equation (2.7) in Grossman andHelpman (1991)). The left hand side of equation (13) represents the net present value ofexpenditure and the right hand side, the net present value of wage and asset income. In theaggregate, the right hand side of the budget constraint consists of total wage income and netassets held by consumers.Standard dynamic optimization of the utility function, equation (12), subject to theintertemporal constraint, equation (13), yields:C(’r) p (‘r)+ y = —p+r(’r), (14)CQr) p()60which describes an agent’s optimal consumption path. Taking the price of the final good, p,as the numeraire, equation (14) reduces to:C( r)= —p+r(’r). (15)C( r)The intermediate monopolist’s problemAs in Section 2, in each sector, intermediate goods are produced linearly with labouras the only input. At time t, the production function for firm j in sector i is defined as:= (16)where A,.’ (t) denotes labour productivity of firm j in sector i at time t. Since production islinear and equation (12) implies unit elastic demand for intermediate goods, the firm withhighest labour productivity within each sector, if unique, acts as a constrained monopolistand limit prices at the marginal cost of its closest competitor. Thus if A7’ (t) denotes themost productive firm’s productivity in sector i and A,c (t) denotes the productivity of theirclosest rival, then instantaneous profits for the sector i monopolist are given by:fl.(t) =w(t)- w(t)y(t)A1C(t)(17)A1C(t) A1M(t)) w(t)where y and w again denote output of the final good and wages respectively.61If the highest labour productivity technology is shared by more than one firm, it isassumed that these firms act as Bertrand competitors3°and, by competition, all profits aredissipated with price equalling marginal costs of production, that is:w(t)p1(t)= 4(t) (18)Note that we interpret innovations as increases in the productivity of labour inintermediate production. There are many alternative interpretations which yield an identicalstructure. Grossman and Helpman (1991, Ch. 4) interpret innovations as either raising theproductivity of inputs used in production of the fmal good (as do Aghion and Howitt (1992)),or, assuming away intermediate industries, as increasing the quality of final goods measuredby an index of utility.ResearchIt is assumed that, if a firm succesfully innovates, its labour productivity rises to alevel y times greater than that of the previous incumbent monopolist (where 2’> 1).Successful innovations within an industry are assumed not to arrive simultaneously.31Notethat, as in Aghion and Howitt (1992) and Grossman and Helpman (1991), a research successyields spillovers: it raises the initial level of all future research to that of the succesfulinnovator. Nevertheless, even though the knowledge embodied in new technology iscompletely public, the use of that technology in production is protected by a patent of infiniteduration.30Modelling interaction between firms with access to similar technology in some other waywhich does not imply the complete dissipation of profits will lower the magnitude of impactassociated with industrial structure. However, the qualitative nature of the results presentedhere will be completely unaffected.3-Or, since this is such a low probability event in a continuous time world, simply ignored asin Grossman and Helpman (1991).62Within an industry, innovations arrive at the instantaneous rate t(1Rl(t)) where£RI(t) denotes the total amount of labour devoted to research in industry i at time t. Thefunction is identical to that used in Section 2. A succesful research firm obtains a flow ofinstantaneous profits as defined in equation (17), until the arrival of a superior technologyreduces the value of its patent to zero. We can think of research firms as financing theirinvestments by selling claims to the future stream of potential profits. As in both Aghion andHowitt (1992) and Grossman and Helpman (1991), incumbent monopolists do not undertakeresearch because the net present value of a succesful innovation is lower than for a potentialentrant. Let (t) denote the present value of this flow of profits in sector i. Then, as there isfree entry into research, in equilibrium the marginal expected value of research cannotexceed its marginal costs:w(t) ‘I” (,(t)) (t) with equality f £R1(t)> 0.32 (19)Since 2’ is identical across sectors and since there is equal expenditure onintermediate inputs in the production of the final good, profit to a monopolist within industryi is given by:H1Q) = y(t) (20)2’C Mby substituting y17 for A1 and y for A. in equation (17), where n1(t) is the numberof innovations that have taken place in industry i upto time t. As a result:32Since firms finance investment by the sale of equity shares in the future stream ofpotential profits, and since the arrival of innovations is governed by independentthstributions across sectors, equity shareholders will diversify and receive riskiess returns.Firms thus only consider the expected value of returns to research, and not their variance,when deciding upon investment.631(t)=V(t)=V(t) ‘v’i,j. (21)Moreover, since ‘ is also identical across sectors, research effort will be distributedevenly:33£RI(t) = £RJ(t) = £R(t) V i,j. (22)As a result, in equilibrium, equation (19) can be re- written as:w(t)‘((t)) V(t) with equality if £R(t)> 0. (23)Since the arrival of research success is independent across industries and since thereexist a continuum of industries, the economy wide level of innovation success is completelydetermined by the level of research. Also, as capital markets are perfect, researchinvestments can be completely diversified so that returns to research are non-stochastic. Inequilibrium, the economy wide value of wealth simply equals the value of the aggregateclaims on research firms.Instantaneous equilibriumHere we closely follow the procedure used in Section 2. By appropriate normalizationwe assume that L = 1 and 4(0) = 1 for all i. Labour market clearing then implies that:=£,?(t)di+ J £1(t)di (24)33Where is not strictly concave (the linear case) this is only true weakly. Theinstantaneous allocation of labour across research sectors is not then uniquely determined,however we shall proceed by only considering the symmetric allocation.641 = £R(t) + a(t) y(t) + (1 — a(t)) (25)2w(t) w(t)where cx(t) defines the number of sectors in which production is undertaken by atechnologically dominant monopolist. This can be rearranged to yield:y(t) (1—eR(t))W(t) 2’ (26)It is worth noting that, as in the static model of Section 2, the relationship between y(t) andw(t) depends upon industrial structure. Equation (26) shows that the general equilibriumconsequence of an increase in the number of sectors dominated by a unique monopolist (anincrease in a) is a lowering of wages relative to output.We now derive an expression for the expected present value of the flow of profits to amonopolist. We first define the real wage rate at time t in terms of the state variables, a(t)and n1 (t), the number of innovations having occured in industry i upto time t. Fromy(t)equation (12) and by substituting in the demand for each intermediate input, , thep1(t)amount of final good produced is given by:r1 y(t)lny(t)= j in di (27)° p.(t)which implies:o=in di =—inw(t) di — f lndi (28)° p.(t) 2huu1 a yfland therefore,65lnw(t) = (1n7)J0t(n(t)_ 1)di = (my)1t(eR(’t))d’t— x(t)(1n y)(29)This expression defines the wage rate in two equivalent ways. The middle expression is aslight manipulation of (28) where the integration over sectors 0 to aQ) involves monopolisticsectors, while integration from a(t) to 1 is over competitive sectors which disappear since noinnovations have arrived in these sectors and the denominator equals 1. The last part of (29)makes use of the deterministic nature of innovation successes at the aggregate level and thesymmetry of labour allocation to research. This implies that integrating across sectors withresearch successes to determine the aggregate number of successes is equivalent tointegrating over the instantaneous probability of a research success arriving from 0 to thepresent date.The flow of instantaneous profits defined in (17) simplifies down to — 1 y ( t)7)which is then discounted by the interest rate and the probability of a new research successdriving future profits to zero, to determine the net present value of the flow of profits to amonopolist, V(t) as:V(t) e_t)_tD er_t8 •‘ 1 y(’t) dt (30)Thus, using (26) and w(t) from (29), we obtain:34There are a few technical issues glossed over here. Firstly, note that for the integral in themiddle expression to be well defined we must be able to order sectors by their number ofresearch successes, and, since these occur discretely, to integrate over the pieces. This isalways possible since the arrangement of sectors along the continuum is arbitrary and can beadjusted to ensure integrability. A further issue is the integrability of (‘ (t)) throughtime. This does not matter in steady states where £ is fixed, nor on saddle paths where £will be shown to vary continuously, but is liable to be important when £ changes discretely(as will be seen to occur when policy variables change). The integral is again only welldefined upto the point of discontinuity so we must again proceed by integrating the pieces.66V(t)= $ e__R(t etRS 2’ 1 (1— £R(r))y ye_i)di dry a(’r)+(1—a(r))y(31)The first term within the first integral, , is the discount rate, the secondJ—cI’(tR (s))dsterm, e’ , is the probability of at least one innovation having occured in the sectorupto time t , i.e. the probability of being displaced before time t, the third, 2’ , is the2’monopolist’s markup, the fourth, (1— £R(r)), is the amount of labour directly used inproduction, the fifth term, 2’ , the inverse of labour’s factor share ofa(T)+(l— a(’z))yf (n(r)—i)diincome, and finally the last term, ye 0 , denotes the wage rate. An instantaneousequilibrium is then defined by a value of £R(t) such that, after substituting in for V(t)fromequation (31), the inequality in equation (23) is satisfied.Steady statesIn this framework, a steady state is defined as an allocation of labour, betweenresearch and production which does not change through time. We restrict attention to perfectforesight equilibria only. Here we will first establish the conditions under which there existsa steady state in which there is a positive amount of research, secondly we derive theconditions under which there is no labour allocated to research and no growth in the steadystate, the Poverty Trap, and finally we show that there exist a range of parameter valuessuch that the two steady states coexist.A steady state with positive research67In any steady state with positive £R(t) it must be the case that the proportion ofindustries in which there exists a monopolist with a technological advantage, a, equals one.This is because equation (20) shows that a leader’s profit flow does not vary with the numberof innovations in a sector and the concavity of R) ensures that research efforts aretargetted evenly across all sectors with equal profit flows.35 Thus with £R (t)> 0, if a < 1 itwill rise through time, and since equations (23) and (31) show that £R(t) increases with a,this cannot be a steady state.Define the steady state level of £R(t), £R, which, as previously shown, will be equalacross industries. By defmition of 4(A?R), the number of innovations that have taken placeupto time t is given by:J’n1(t)di = (32)In the steady state,y(t) = (I—eR) w(t) y (33)Using equation (29) and (32), w(t)can be expressed as follows:1w(t) = —e ° . (34)The present value of holding a monopoly is then given by:Recall that when is linear this will not necessarily be true, but that we arefocusing only on symmetric outcomes so that a will again rise with positive £R (t).68- ‘‘( ( )d 1 1 (1ny)J(R(s))dsV(t)= f e_(R(t)_R(t)) eJt R S S ‘ (1—R)y — e ° dt.(35)From equation, (33) and (34), it is clear that the wage rate and the output, at the stationaryequilibrium, grow at a constant rate lnO’)(F(R). As a result, from equation (14) andremembering that the numeraire is the final good, the stationary equilibrium interest rate isthen given by i=p +1(7)’R) Since, by definition, R(s) = s, integrating theexpression in (35) and using equation (23) yields the following steady state value for £R.‘C€R)__ £ (y— 1) = 1 (36)It is easy to verify that any solution to (36), where it exists, is unique36.The existence of asolution to (36) between 0 and 1 depends upon the values of exogenous parameters 7 and p,as well as the function (I). We explicitly characterize these conditions when (I) is linear i.e.where I(er) = &r implying ‘J’ (er) ö. must be less than one to ensure that theprobability of an innovation arriving in a sector always falls below one. The equality in (36)now becomes ö(1—.)(7— 1) = R +p, implying:(36*)It will therefore always be the case that £R < 1 and £R will exceed zero provided:(‘y—1)—p>O. (37)36Note that by substituting for and 8 for in equation (36) we obtain ana 7expression which is identical to that obtained in equation (4.18) of Grossman and Helpman(1991).69The linear case thus makes clear the intuition for the existence of a steady state with apositive allocation of labour to research and therefore sustained growth. It will be more likelyto exist the higher is the productivity advantage of new products, y, the higher theproductivity of labour in research, 6, and the lower the discounting of the future p, as onewould expect.A steady state with no researchIn a steady state with no research, technology does not change and there is nogrowth in per capita incomes, this may be thought of as corresponding to a Poverty Trap.Without research in the steady state, the initial level of a, ã, persists through time and it isalso easy to show that:(38)Cy wAs a result, r=p. One can also express the equilibrium level of output in the steady state asfollows:y= w (39)Thus, the present value of a patent equals:V(t)=et ( — — dt. (40)y70 -Finally, there should be no incentive to allocate labour to research in the steady state. Thisholds as long as the inequality defined in equation (23) is satisfied, that is:(y—l)< 1. (41)p ia+(1—a)2’)For given 7, we define & as the value of a which solves (41) with equality, in the linearcase above this becomes jC = — + . Provided < 1, it can be seen by comparingp y—lequation (36) to equation (41), that there always exist values of y and p such that theeconomy could be caught in a poverty trap, even though it is also capable of experiencingsustained growth. For = 0 we can thus determine the conditions under which both steadystates exist. These are given by 7 and p such that:I’(0)(y—1)and eR>o. (42)p(+(1—)-y)We can interpret this condition in terms of the model’s primitives most easily if we againconsider the case of a linear (i). In this case, equation (41) reduces to ( y — 1) < py which,when combined with the inequality in (37), yields the following range of parameter values forwhich both a steady state with sustained growth and a growth trap exist:(42*)Figure 3.1 plots this parameter range for the case of = 1 in p, y space. Bothsteady states exist for p, y pairs lying within the region between the two curves. Recall thatp is not constrained from above and that, as p rises, the range of values for 7 in which twosteady states exist increases. For p greater than 1 (or delta in the general case) this range isunbounded from above, so that with high p, for arbitrarily high 7, both steady states exist.Intuitively this is because, with a high value of p, even a large value for 7 will not induce- 71research when all sectors are competitive and most of aggregate income is paid to labour inthe form of wages, since high future discounting makes future instantaneous profits lessworthwhile. However, a high value of y also implies that when many sectors aremonopolistic much of instantaneous aggregate income accrues to holders of patents in theform of profits. Wages in production are then relatively low and, not withstanding highdiscount rates, allocating labour away from production and towards research will beworthwhile.With a unique steady state, any initial industrial structure and hence any initialratio of aggregate income to wages ends up not mattering in the long run, as the economywill attain its steady state. When both steady states exist, history matters, in the sensethat the initial industrial structure may determine which of the steady states is eventuallyattained. Figure 3.1 demonstrates that history is more likely to matter the higher the rate atwhich the future is discounted, this resembles the result presented in Matsuyama (1991)though for entirely different reasons. In Matsuyama (1991), low future discounting impliedthat from any given initial starting point (i.e. pattern of labour allocation betweenagriculture and the increasing returns activity, industry) there was more likely to be atransition path leading from the unindustrialized (or high agriculture employment steadystate) and converging on the industrialized or high manufacturing steady state, to which theeconomy could jump given the correct coordination of individuals’ expectations. Historyhowever matters with higher discounting as then a given initial position implies a uniqueperfect foresight path along which the economy will evolve. History thus matters more withhigher discounting because, given an initial position, the steady state to which the economyin his model will converge is pre-determined. Here, however, the possibility of more than onesteady state existing rises with the discount rate. History thus matters here, with higher371n the following section, where the model’s out of steady state behaviour is examined, thisassertion will be demonstrated more formally.-72discounting, for the reason that the initial conditions eventually determine whether theeconomy ends up in a steady state with sustained growth or a growth trap.It is, of course, not strictly correct to make inferences about the time path of theeconomy out of steady state without undertaking a complete analysis of the model’stransitional dynamics. It could, for instance, be the case that though the growth trap existsas a steady state, all paths away from the initial steady state imply convergence to thesteady state with sustained growth. We therefore now turn to explicit consideration of themodel’s dynamics.DynamicsThe behaviour of the system outside the steady states is defined by equations ofmotion for a and £ (t). In this section we show that when in the neighbourhood of thesteady state with growth, the economy converges, along a saddle path, to that steady state.However, it will be seen that the equations of motion describing the economy’s evolution arehighly non-linear and thus do not admit the possibility of a qualitative global analysis. Wetherefore revert to simulation methods in order to describe the economy’s transition betweenthe steady states.The equation of motion for a(t) is simply given by the level of research in period taccording to:(43)so that a(t) is unchanged if and only if either there is no research in period t or a(t)zd.73In the neighbourhood of the steady state with growth, or with any positive allocationof labour to research, equation (23) holds with equality, so that:()with time variables suppressed from here on. The appendix shows that using equations (15),(20), (26) and (43) we obtain, after some manipulation, an expression for £R:=I(e )Ii++ (-1)(1-R) 1R Ra(1—y)+y) p a+(1—a)7 )1—”(R)+ 1R) i—eR(45)Figure 3.2 depicts this equation for £R =0, which is easily verified to be upward sloping inx space and to cut the horizontal axis at a = o The equation of motion for a alsoshows that a = 0 only where either a=1 or R = 0. The diagram thus shows the modelstwo possible types of steady state. If the initial value of a lies between 0 and &, incentivesto invest in research are too low to induce research and therefore a remains at its initialvalue. However if the economy starts with a=1, £R = £R, sustained research levels of £ aremaintained each instant.We now show the existence of a saddle path, locally, to the steady state withsustained growth, i.e. where £ = £R. By linearizing the system around the point wherea=1 we obtain:rai E—( ) 0 [czlB I+K (46)LRJ L z R)7—(R)(1—eR)(Y1) [j74where K denotes residual terms. Since the product of the diagnol is always negative in thecase of own effects and zero in the case of cross effects, the determinant of the square matrixis negative and there thus exists a saddle path to the steady state with sustained growth.The saddle path is depicted in Figure 3.1. For a given value of a, levels of research above thesaddle path value imply an increase in labour allocated to research until eventually £R = 1.However, in this case the instantaneous value of a leading firm’s profits is infinite whichviolates the transversality condition in the consumer’s optimization problem. On thecontrary, too low a level of research implies that research eventually stops. However, witha> ac and £R =0, this implies that the value of labour in research exceeds that inproduction which is inconsistent with the labour market clearing each instant. Thus in theneighbourhood of a = 1, £R = £R, the economy moves along the saddle path until a 1 andconverges to a steady state with growth and constant allocation of labour to research.Linearizing the model in the neighbourhood of the steady state with growth,unfortunately, yields no insight into the economy’s performance a discrete distance awayfrom that steady state. However the complexity of equation (45) in particular, implies that aqualitative analysis cannot be undertaken. We thus provide simulations of economy’s pathwhen ae(acl)We simulate for the simplest possible case, i.e. cJ ( £R) = £R. Values of 7 and p arechosen to satislr equation (42) evaluated with = 0, i.e. to ensure that both steady statesexist. We firstly present calculations for the case with 7=3, p =.8.38 These values implyo=025 and £R = 0.4. By running the system backwards through time from a point close to38Simulations were also undertaken for a wide variety of values satisfying (42). In all casesthe outcomes were qualitatively identical to that reported for the representative values andwe also include two other plots of simulations for values chosen from extremes of theparameter range.75the growth steady state, we trace out one possible path of evolution for the economy. It willbe shown that this is the unique saddle path. Figure 3.3b shows a plot of the simulated pathrunning between (a=Oc=0.25, £R = 0) to (a = 1, £ = = 0.4. ). If for a given values ofa, £R is above the saddle path then £R explodes upwards until it equals one. By identicalreasoning to that above, such a path is not admissable. The simulations also show that forvalues of £R below the saddle path, the economy follows a trajectory in which research goesto zero, however with a> XD this is again inadmissable. Thus for starting values of a> ctCthe economy converges along the saddle path to the steady state with sustained growth.Figure 3.3a shows the monotonically increasing time path of £R starting from a point on thesaddle path arbitrarily close to ctc. For values of a less than (Xc the economy is in a nogrowth steady state and thus remains there. Figure 3.4 thus extends the plot in figure 3.3band depicts the complete behaviour of the system in a, £R space. To the left of the dashedline the economy is below & and in a steady state with no growth, thus both and £R = 0.Small shocks to the system in the interior of this range (i.e. an exogenous increase in or£) has no permanent effect, the economy stays in the Poverty trap. At any point to the rightof (Xc the analysis above applies and the economy is on a saddle path converging to thesteady state with sustained growth. Figures 5 and 6 are plots derived from a similar processof simulation for parameter values in a different region of the feasible range. Note thequalitative similarity between these and the above plots.Analysis of the models dynamics thus shows that when both steady states exist, forvalues of < cx the economy stagnates with no research and no growth. Shocks to theeconomy will not move it from this situation unless it is close to ac. However for an economystarting with a> & there is monotonic convergence to the steady state with sustainedgrowth.76WelfareThe social optimum is calculated by maximizing an agent’s intertemporal utilityfunction by choice of £Ri (t), the amount of labour devoted to research in each industry attime t. Due to the concavity of both and the production function, labour devoted toresearch or production will be allocated evenly across industries. Thus we re-specifyvariables in terms of the aggregate allocations of labour to research and production, £R.Define n(t)=I(R (t))dt, then ñ(t) = I(R (t)), the present value Hamiltonian canbe written as:H = ln(1— £R (t)) +(lny)n(’t) + O(t)1(eR (t)). (47)Dynamic optimization, the details of which are in the appendix, yields:4(t)=1— p (48)(lny)c1’(4(t))Where £ denotes the solution. In terms of the linear example:p (48*)in yA comparison of 4 with £R from equation (36) shows that the optimal level of investmentmay diverge from that which occurs in a decentralized economy experiencing growth. Inparticular, for the linear example, when the following inequality holds, the socially optimallevel of research exceeds that which would be undertaken in the steady state with growth:(49*)-Y4. Policy Implications77The potential for divergence between the socially optimal level of research and thatwhich obtains in either steady state suggests the possibility of a role for governmentintervention. In this section, we consider the impact of two distinct forms of governmentintervention in the development of new technologies: government subsidies (or taxes) toprivate research and publicly provided research. We also modify the basic model in order toconsider the effect of changes in the duration of patent protection.Pareto imDroving government subsidiesIn this section we show that an economy in a zero growth steady state , be movedto a path with sustained growth by a government subsidy to research, and that such a movecan be a Pareto improvement. For the purpose of this example we consider the linear casewhere equation (42*) is satisfied (both steady states exist) equation (49*) is also satisfied (thesteady state with growth invioves less than socially optimal research) and we assume that Ctis arbitrarily less than o so that:i=-1’_ 1_ +e (50)where s is arbitrarily small and strictly positive, so that the economy is currently in the zerogrowth steady state. Our previous example where , =1, p=.8, y=3, satisfies theseconditions for Ct arbitrarily below 0.25. The government now sets .S> E by an arbitrarilysmall amount at time t39, implying that (41) is no longer satisfied. Since C is arbitrarilysmall, once investment has been undertaken in all industries, the increase in x, as describedby (43), increases relative returns to innovation in subsequent periods so that some research391t can be assumed that the government finances this subsidy by a once off, lump sum taxon all individuals. By specifying c, and therefore s, as arbitrarily small, we are able to ignorethe implications of this tax on individual decisions.78 -is now always worthwhile.4°The economy now moves onto the saddle path, depicted forsimulation values in Figure 3.3b, research increases monotonically in all periods after thesubsidy and the economy tends towards the steady state with growth.Since the optimal control problem of maximizing social welfare involves a strictlyconcave objective function with a convex constraint set, all £R such that 0 < £R 4 will besocially preferred to £R = 0. Also, since (49*) holds (R <4) and convergence to £R alongthe saddle path is monotonic, we know that all subsequent values of £R will lie between £Rand 4, thus, in all future instants, labour market equilibrium will yield a socially preferredlevel of research. Finally, since by construction s is arbitrarily small, its current period costcan be ignored and it follows that the subsidy constitutes a Pareto improvement for thiseconomy.In general, it is possible to specify the optimal level of subsidy for moving theeconomy from the low growth steady state, however considering discrete subsidy levels addsthe complication of having to explicitly consider the government’s method of finance and itswelfare impact. Note finally that (49*) is sufficient though not necessary for such subsithes tobe Pareto improving, it may still be the case that moving from the zero growth steady state isPareto preferred even if it eventually implies that the economy converges to a level ofresearch which is greater than optimal.Publicly provided research401f were larger, then a once off subsidy may not raise a sufficiently to move the economyfrom the zero growth steady state. In that case, a subsidy sustained over an interval may berequired, it is, however, always possible to effect a move from the zero state by this methodprovided (42) is satisfied.79 -In addition to granting research subsidies, the government also has the option ofincreasing innovation and growth by undertaking research of its own. In most models this isqualitatively equivalent in effect to directly subsidizing research. However, the implicationsof this research effort for growth in our framework depend critically upon the way in whichthe government uses it. In the case where the government acts as a profit maximizingmonopolist, such research can be beneficial if it draws the economy out of the zero growthtrap 41•It will have this positive effect if the research allows the proportion of industries withmonopolists to increase past the critical level of O, (Xc. However, starting from a steadystate with growth, the effect of public research is markedly different if, instead ofappropriating the returns to that research, the government makes it publicly available.There are two distinct effects here. Firstly, a displacement effect: government researchcrowds out private research. More importantly, a disincentive effect: since access togovernment research is public, the proportion of competitive industries in the economy rises(X falls) and therefore wages rise relative to private returns from research. As a result, theaggregate level of research falls, lowering the rate of growth (as long as the level of publicresearch is less than the initial steady state level). Furthermore, if government research islarge enough to reduce the proportion of monopolistic sectors below the critical level (XC,private research disappears, i.e. the economy can move to the zero growth steady state.Patents of finite durationAnother government policy tool is the length of patent protection. In the model so far,patents have been assumed to last forever, as in both Aghion and Howitt (1992) and41Note that, if the economy is currently in a steady state with growth, such research simplydisplaces private research with no aggregate effect.- 80Grossman and Helpman (1991). However allowing for patents of finite duration cansignificantly alter the results obtained as it may affect the stability of the high growthequilibrium. Once again this follows from the effect of industrial structure upon factorreturns in the different activities. If patents lapse in a number of industries and monopolyrents are dissipated, returns to research relative to wages fall and research may be reducedor stop altogether. Since the arrival of innovations is stochastic and independentlydistributed across industries, in some periods many industries experience success,conversely, in others, a relatively large number of industries do not innovate and, dependingon patent length and age of invention, patents lapse. If the corresponding reduction in theproportion of monopolistic sectors is large enough, a falls below c, condition (41) issatisfied, and research stops forever.The level of aggregate variation across periods will, of course, depend inversely uponthe number of industries. However, in the model presented so far, such variation cannotoccur. With a continuum of industries there is no aggregate uncertainty and, therefore, themodel does not allow for variation in the number of innovations across periods, thus thenumber of patents lapsing per period is known in advance. With a smaller number ofindustries, however, the economy may be subject to aggregate level shocks. There then existsthe possibility that, in a single period, relatively few research successes occur, many patentslapse, and returns to research relative to wages fall by so much that inventive activity stops.We now demonstrate this in a simple discrete version of the above model.Suppose now there are only two intermediate industries, time is discrete and patentslast only one period. 42 The two industry version of the production function in equation (1) isgiven by:42Since this example involves only two firms we must assume here that firms do not exercisemonopsonistic power in the labour market, nor act strategically. With a larger number offirms this is guaranteed but with n large, the calculations become increasingly complex.81lny(t) = --lnx1(t)+--1nx2(t) - (51)which can be re-written as:1 y(t) 1 y(t)In y(t) = — in + — in (52)2 2p1 2 2pWe establish the possibility that investment in research proceeds provided both industrieshave experienced research sucess and terminates if either one experiences a period withoutinnovation. Starting in period n, the productivity of the leading monopolist in each industryw(n)equals y” thus a monopolist will price at n-I while m competitive sectors the price will be2’w(n)Substituting into equation (52) we then obtain that, in the case where both industriesare monopolistic:in y(n) = iny(n)‘+ iny(n)yfl(53)2 2w(n) 2 2w(n)which yields the wage rate:(54)2However when the patent has lapsed in one industry, using (52) we obtain:in y(n) = -- in y(n)yfl1 +-- in y(n)yfl(55)2 2w(n) 2 2w(n)which upon re-arrangement yields the wage rate:(n—I)+nw1(n)= 2’2(56)Similarly the wage rate when afi sectors are competitive, i.e. both patents have lapsed, isgiven by:wc(n)=3,fl• (57)As in the previous section, investment in innovative activity depends upon a comparison ofexpected returns with the wage rate. Here, since monpoly rents accrue for one period only,the marginal benefit of investing one unit of labour is given by:82(n+J)i-nE(V(n + 1))=(y— l)’ (R)[(l — £R(n+ l))LR(fl + 1))+1 2 (i—+ 1)))](58)a( ()‘r())For simplicity, we assume that 0, implying that the expected valueaRfunction is decreasing in £R. As a result the expected return to innovation is highest whenthe current level of reseach is lowest. A sufficient set of conditions for positive growth whenboth industries are monopolistic but zero growth when one patent has lapsed is given by:1 ()and,l>2t(0)2Fl. (60)If these conditions are satisfied, the economy will continue to experience positive growthonly when innovations continue to arrive in both industries. Following a period in whicheither one or both industries has not realized an innovation, wages rise sufficiently relativeto aggregate income to deter any investment in innovative activity. From then on growthstops. At the cost of increased complexity, this result will generalize to the case of multi-period patents and a finite number of industries, the only pre-requisites are that patents beof limited duration and there exist aggregate uncertainty with respect to the number ofinnovations.More generally still, even if the zero research condition is not satisfied, it will alwaysbe the case that the level of innovative activity falls with the lapsing of patents. Thissuggests the possibility of cyclical investment levels in the economy and clustering ofinnovations; after bad periods in which few inventions arrive, patents lapse and researchfalls, thus decreasing the likelihood of innovations in the next period. In the currentexample it is possible to characterize three distinct levels of £R, with associated conditionalprobabilities, between which the economy can move. As w’(n)> w’(n)> Wm(fl), the lowest83equilibrium research level corresponds to that in which both industries are competitive, it isgiven by‘R solving.wc(n)= E(V(n+1)), (61)which we denote £ (the uniqueness of a solution is guaranteed by the independence of £ Racross periods). The state in which most research obtains is given by:wm(n) = E(V(n + 1)), (62)which we denote £. Similarly there exists a level of research between the two which occursin situations where one patent has lapsed, given by:w’(n) = E(V(n+l)), (63)which we denote 4. In any one period there exists a probability that the economy will be inany one of the three states, with the probability conditioned upon the state in the previousperiod. Interestingly, the probability of a particular state is highest if that state prevailed inthe previous period, suggesting the possibility of clustering in innovations and thereforediscontinuous growth.The existence of clustering in the pattern of innovations was a factor noted bySchumpeter (1936) and has recently been explained in a partial equilibrium setting by Stein(1994). He shows that clustering may occur when incumbent firms enjoy an advantage overrivals in terms of “customer base” or long standing relationships which give them anadvantage over new entrants. These can act as a barrier to entry so that when one firmsuccesfully enters and thus destroys an existing customer base, there is more likely to be asuccession of entrants in turn before the new incumbent has a chance to establish its ownbase. An empirical difference between the clustering predicted in our model and Stein’s isthat within Stein’s framework clustering emerges as a within industry phenomenon due tothe externality a new entrant creates for those that follow. However our model predicts theclustering of innovations across sectors due to complementarities which operate at theaggregate level.845. Entrepreneurship: A Reintepretation for LDCsAs mentioned earlier, establishing the existence of multiple steady states has beenmotivated, in the previous literature, by the desire to provide an explanation for observeddifferences in cross-country growth experiences. Thus the stagnation of some LDCs has beeninterpreted as corresponding to their being trapped in a low level steady state. However, it ishard to ascribe poor growth perfomances in LDCs to insufficient investment in R&D and thewant of new knowledge, as would be suggested by the present model. The results establishedabove may therefore seem of little relevance to this issue.However, a broader interpretation of the model can provide some insights. The mainthrust of the model is that the process of innovation in one sector provides a spillover topotential future innovators by increasing the relative profitability of research activities. Inthe context of LDCs, it is likely to be the implementation of new and superior productionmethods, rather than their invention, which provides the engine of growth. In this case, theimportant actors are entrepreneurs taking the risk of departing from existing or traditionalproduction methods, rather than researchers. If, however, such entrepreneurship does notrequire unique or specialized skills, then we can simply interpret £R as the amount of labourdevoted to entrepreneurial activity (instead of R&D), with (1- I(.R)) then representing therisk of a new venture failing. By employing a superior technology, the successfulentrepreneur displaces traditional producers from her sector by limit pricing at theirmarginal cost and receiving a markup on each unit sold until she, in turn, is displaced by anew entrant. Note that there is still a knowledge creation effect of such entrepreneurialactivity: assuming that some part of the method of production used by the entrepreneur isobservable, it will be useful to new entrants in that sector seeking to improve upon the first85 -entrepreneurs methods. Moreover, as it is not protected by patent, it can also be readilyimitated. However, as long as such imitation carries a positive probability of failure or apositive cost (no matter how small), it will not be undertaken in a perfect equilibrium sincethe ensuing Bertrand competition between imitator and incumbent yields zero profit.Under this interpretation, the basic results developed in the previous sections carryover. Two steady states may exist. In the no growth steady state, all resources are used intraditional production and, since production techniques are commonly known, there are zeroprofits. Disequilibrating entrepreneurial activity displaces traditional producers to othersectors, thereby lowering relative returns to traditional production there. This displacementeffect makes entrepreneurship there relatively more worthwhile. Entrepreneurial activity inone segment of the economy can thus launch a surge of such activities throughout alltraditional sectors and move the economy towards a path of self-sustained growth.The main difficulty with the interpretation expounded above is that it assumesentrepreneurship can be undertaken by anyone whereas, more realistically,entrepreneurship requires particular skills which may not be widespread. In the appendixwe demonstrate that a simple modification of the existing model can accomodate a nonuniform distribution of entrepreneurial skills without altering the basic results.The aim of this section was to show that the model, when suitably interpreted, couldprovide a realistic description of sectoral linkages giving rise to multiple steady states. Inparticular, the appendix demonstrates that having factors which are not uniformlydistributed does not affect the configuration of steady states. What really matters is that thefactor (or combination of factors) used in entrepreneurship is also used in traditionalproduction and that this factor is mobile. Under such conditions, successful entrepreneurshipin sector i induces a reallocation of factors previously used in sector i to other sectors,86thereby lowering returns to those factors in traditional production relative to their returnsfrom entrepreneurship. The existence of two steady states again suggests that the status quoin traditional production may be self sustaining in economies not having experiencedentrepreneurship, even though, once some critical proportion of sectors have beenmodernized by successful entrepreneurs, traditional production is no longer attractive sincereturns to the remaining traditional producers are now so low that they find entrepreneurialrisks worth bearing.-876. ConclusionSchumpeter (1951) argued that monopolies were an important element in the processof capitalist innovation and growth. His principal concern, and that of most subsequentliterature, has been with the incentive effect provided by monopoly rents. In this chapter weconsidered another avenue through which the existence of monopolies can affect thedevelopment of new technologies by investigating the general equilibrium effect of industrialstructure on relative prices.The main result of our analysis is that a monopolistic structure, by increasing theshare of profits in aggregate income, tends to increase the profitability of innovativeactivities. An important implication is that in a multi-sector model of growth throughcreative destruction there may exist more than one steady state, with the initial industrialstructure determining the state eventually attained: thus, the more competitive the initialindustrial structure, the less likely that the positive growth steady state is attained. Inparticular, in LDCs, an interpretation is that the avenue through which this occurs isentrepreneurial activity. This suggests that, though perhaps difficult to set in motion, thereare forces at work, even in the absence of direct externalities, which make entrepreneurship,once started, self-sustaining. This allows for the possiblity of Pareto improving governmentintervention. However, though government subsidies to private research are shown to bebeneficial, in contrast, government development of new technologies, if made publiclyavailable, may be detrimental to growth. A further implication is that with finite livedpatents, in the presence of aggregate uncertainty, new innovations will have a tendency toemerge in clusters, thus suggesting a discontinuous time path of growth.88Figure 3.1Existence of 2 steady states.tBetween curves both exist1086420.2 0.4 0.889sç.I; 0CD 00 1’1•AFigure 3.3Middle values of rho and gamma.= 3,,O=.8, ç=.4. Iç’= .25Time path of labour devoted to research from vicinity of growth trap to steady statewith sustained growth.ocThe saddle pathIc-100—so—60—400.0.4 0.5 0.6 0.7 0.8 0.9cC91Figure 3.4The phase diagram.(z)I IIThe saddle path0o(Q0 0.3 0.4 0.5 0.6 0.7 0.8 0.9c-5—100 —80 —60‘‘z3/,O=.8. LR=.4—40.25—2092Figure 3.5High rho and gamma.“3” =30,,,O =.99, R.9336 0< .O24381Time path of adjustment from vicinity of growth trap to steady statewith sustained growth.Plot of saddle path in (horizontal), (vertical) space.0.00.0.0.893Figure 3.6Low rho and gamma..LR=.OOO29411. (. =.2385Time path of adjustment of labour devoted to research from vicinity ofgrowth trap to steady state with sustained growth.- -p_npp/ 0.00025/ 0.0002/ 0.00015J 0.00010.00005-100000 -80000 -60000 -40000 -20000Plot of saddle path in tx (horizontal), ,f . (vertical) space.o:6osi94Chapter 4COORDINATION PROBLEMS ARISINGFROM COST REDUCING INVESTMENTS INTHE PRESENCE OF HORIZONTALEXTERNALITIES95COORDINATION PROBLEMS ARISING FROM COST REDUCING INVESTMENTSIN THE PRESENCE OF HORIZONTAL EXTERNALITIES.43IntroductionRosenstein-Rodan (1943), in arguing for the importance of external investment in there-construction of Europe, was the first to enunciate a doctrine which would later be knownas that of balanced growth or the “Big Push” . The main conclusion of this doctrine was thateconomies could be trapped in low growth equilibria where, due to complementarities in costreducing investment across industries, single firms would not find investment profitableeven though coordinated investment was worthwhile. A central question implied by thisdoctrine, the answer to which is stifi not fully understood today, is under what conditions canan economy support multiple, Pareto rankable equilibria; some in which there is large scaleinvestment and continual improvement and others in which investment does not come forth?Externalities to investment were certainly a central component of discussion at the time ofRosenstein-Rodan, and there was much debate as to the relative importance of differingtypes (see also Nurkse (1953), Scitovsky (1954) Fleming (1955) for a discussion of the issue).More recently interest has been renewed by development economists seeking to explain theseeming stagnation of economies at low levels of productivity and welfare (for example seeBasu (1984), Murphy, Shleifer and Vishny (1989), Matsuyama (1992a, b), Ciccone andMatsuyama (1993), Gans (1994), and a recent survey in Matsuyama (1994)). In explainingsub-optimal stagnation, traditional theory suggests a number of market imperfections which,43This chapter presents a revised version of Baland and Francois (1995b).96by now, are quite familiar: public goods, spillovers in knowledge or technology, incompletemarkets, information problems, to mention a few. These factors, whether implying multipleequilibria or not, usually suggest a divergence between market outcomes and those which aresocially preferred and there can be little doubt that in fully understanding problems ofunderdevelopment these factors need to be taken into account.However, economists at the time of Rosenstein-Rodan also emphasized a differentmechanism. They focused instead on processes internal to the market, so called pecuniaryexternalities, to explain underdevelopment. Pecuniary externalities arise when agentsimpact on each other through market mediated effects. For example, in contrast totechnological externalities (such as the famous bees and honey example), pecuniaryexternalities could be said to arise when a firm invests and raises aggregate income so thatother firm& investments become more worthwhile. Another example occurs when a firm, bylowering its price, raises demand for producers of other (complementary) goods and thusraises their profits. Scitovsky (1954) who also emphasized the importance of pecuniaryexternalities, provides a useful taxonomy. He breaks these up into those benefits arising to(1) vertically linked firms, (2) horizontally linked firms such as makers of complementaryproducts, and, (3) demand benefits arising from an increase in income, each of which, heargues, could serve as an avenue giving rise to coordination problems. Recenly the role ofpecuniary externalities has been emphasized for another reason. Matsuyama (1994) arguesthat economists find the macroeconomic implications of technological externalities acrosssectors unconvincing, since such externalities provide strong incentives for firms tointernalize benefits through integration (as pointed out by Milgrom and Roberts (1992)). Onthe other hand, pecuniary externalities, which arise as an outcome of the market mechanism,do not easily admit individual level internalization of external benefits, since they arediffused more broadly and can, perhaps, provide a more compelling explanation of macrolevel stagnation.97To understand the role of pecuniary externalities, models based on the standardcompetitive equilibrium framework, in which all agents treat prices as parametric, are oflittle use. Therefore, in order to analyze these issues Murphy Shleifer and Vishny (1989)employ a model which departs from the standard assumption of price taking behaviour. Intheir model, a unique potential monopolist in each sector has access to a superior increasingreturns to scale technology, which, when invested in, lowers unit costs below that of acompetitive fringe using an inferior technology. The monopolist then sets price taking intoaccount the pattern of demand and the marginal costs of the competitive fringe. This model,which allows for non-price taking behaviour and the existence of profit, provides a frameworkin which pecuniary externalities can have real effects. However, a surprising result obtainedby Murphy Shleifer and Vishny (1989) is that pecuniary externalities through income, type(3) in Scitovsky’s schema, cannot, on their own, lead to the existence of a coordinationproblem. Though undertaken in a stylized framework their model demonstrates a very basic,general point: This being that pecuniary externalities through income only provide a positivebenefit when investment itself is profitable. Unprofitable investment actually makes otherinvestors worse off and, contrary to the doctrine of balanced growth, coordinated investmentonly compounds the losses of investing firms.An essential insight provided by their analysis is that for the existence of multipleequilibria, investing firms, though themselves becoming worse off by investing, must actuallyincrease the profitability of investment to other firms. This can lead to a threshold effectwhereby, if enough others invest, any one firm will find it profitable to do so as well, andinvestment can become self-sustaining, i.e. an equilibrium. Murphy Shleifer and Vishnythus develop some extensions of the basic framework to demonstrate this possibility. Forexample one version of their model includes a factory wage premium, so that workers ininvesting firms which switch from traditional production to factory production receive higher98wages than others. This can lead to demand increases for other firms from the increasedpurchases of the factory workers even if investing firms incur losses, and thus to thepossibility of multiple equilibria. Another modification of their model, explored inMatsuyama (1994, sect. 2A), shows that their basic framework can support multipleequilibria if the fixed costs incurred in the increasing returns to scale technology, involve theuse of final goods instead of a primary input, such as labour in the Murphy, Shleifer andVishny model. The basic reason for multiplicity there is again that, with fixed costs in termsof final goods, investments which incur the fixed cost actually raise demand for other firms’output (and thus returns to investment) even when making a loss themselves.These previous analyses leaves unexplored the other possible avenues through whichpecuniary externalities may operate. In Scitovsky’s taxonomy these are: verticalexternalities, type (1) above, or what we shall refer to as horizontal externalities, type (2).Horizontal externalities occur when, due to complementarity between goods, a fall in onegood’s price increases the demand for goods in another sector, so that the investing firm’sprofits underestimate the social benefit created by its investment. As Fleming (1955) notes,this was an avenue stressed in the balanced growth debate. Of course, goods must, at least,be gross complements for this positive externality to occur. (If goods were substitutes, then afall in one good’s price would actually make the other’s demand lower, i.e. a negativeexternality.) Such horizontal externalities actually underlie many contemporary models ofcoordination problems with underinvestment. For example, Matsuyama (1992b) shows thatcomplementarity can arise even when goods are directly, highly substitutable. In a modelserving to explain geographic agglomeration, he shows that with sufficiently stronglocational complementarities, goods which are, in terms of characteristics, highlysubstitutable, can end up being complementary. In another environment, the same author,Matsuyama (1994 sect. 4A), demonstrates that specialized inputs which, in a partial sense,99are highly substitutable, can become complementary (and lead to multiple equilibria) ifinvestment and supply of one of these inputs induces producers to change final goodproduction towards methods which more fully utilize specialized inputs, so that, indirectly,demand for all specialized inputs increases. Finally Rodriguez (1993) shows that withdifferent tradeable final goods utilizing industrial inputs in different intensity, investment byenough industrial input producers, may lead final good producers to specialize in theindustrial input intensive final good, and thus increase demand and profits for otherindustrial input producing sectors, i.e. that investment in these sectors can again becomecomplementary.In this chapter we develop a simple model which abstracts from the situationalrichness of these previous approaches in order to focus on a more basic question. We askwhat factors constitute necessary conditions for multiplicity of equilibria to arise throughhorizontal externalities. In particular, since this sort of externality seems to be closelyrelated to whether goods are complements, we seek to understand which, if any, definition ofcomplementarity (gross or Hicksian, which will be clearly defined further on) is relevant fordetermining whether horizontal externalities can and cannot give rise to multiple equilibriaand a need for coordination of investments. In the next section it is shown that, whereinvestment involves a fixed input cost leading to a lowering of unit costs, (as in Murphy,Shleifer and Vishny (1989)), and for any specification of preferences consistent withmaximizing behaviour, investment in horizontally related goods (which is not individuallyworthwhile) can never be made worthwhile when investments are coordinated unless thesegoods are Hicksian complements. In other words, multiple equilibria cannot exist if goods arenot Hicks complements. Thus, somewhat surprisingly, gross complementarity is irrelevantfor determining the existence of multiple equilibria through the horizontal avenue. This is asomewhat unintuitive result which arises from the fact that the real income effect of a pricefall (which alone generates complementary demand increases in the gross complements case)- 100is just offset by the direct fall in income due to unprofitable investment (a business stealingeffect) so that income effects disappear and all that remain are pure substitution effects (thusrequiring Hicks complementarity). In Section 3, we then ask whether the existence ofmultiplicity implies a need for coordination. That is, we examine the social welfare propertiesof the equilibria. This leads to another surprising result, namely that when the technology ofproduction is symmetric across sectors, Hicks complementarity, though possibly leading tomultiple equilibria, never implies the need for coordination to improve social welfare. That is,the no-investment, uncoordinated outcome is socially preferred. We also discuss why thisresult is not true in a non-symmetric world and thereby better understand situations inwhich coordination may lead to improvements. In section 4, by relaxing a central conditionrequired in section 2, we discuss a possible avenue through which horizontal externalitiesbetween gross complements can give rise to multiple equilibria and investment coordinationproblems. Section 5 concludes.2. Gross complementarity and multiple equilibriaThe model presented below is a modified version of the framework provided byMurphy, Shleifer and Vishny (1989) to analyze the problem of pecuniary externalities. Asthey were exclusively concerned with the effects of income externalities, they employed aCobb-Douglas utility function, for which the cross price elasticity of demand is zero, thusensuring that horizontal externalities did not arise. Positive horizontal spillovers (i.e. whatScitovsky (1954) calls a pecuniary external economy) can only exist if goods are gross- 101complements, that is, if the fall in price of one good raises demand for other goods. To allowfor this possibility we will draw upon a more general preference structure.44Demand theory distinguishes between two main concepts of horizontalcomplementarity between goods. Two goods are defined as gross complements if a rise in theprice of one good decreases the demand for the other good, keeping income fixed. They areHicks complements if a rise in the price of one good decreases the demand for the other good,holding constant the level of utility. These two concepts of complementarity are relatedtogether by the Slutsky equation as follows:dx. a. ax.=— x. __L, (1)dp, ap, ‘dywhere Xj denotes the Marshallian demand and the Hicks demand for good j. Two goods idx. dh.and j are gross complements if 0, they are Hicks complements if —i- 0. Withdp1normal goods, it can easily be checked that, if two goods are Hicks complements, they arealso gross complements, but the reverse is not true. This distinction will prove crucial in thedetermination of our results.Assume that labour is the only factor of production, wage is the numeraire and thetotal amount of labour in the economy is normalized to 1. There are n> 1 sectors in thiseconomy, each of which produces a different good. Within each industry, subscripted by i,there exists a unique firm with access to a superior technology, called the incumbent. Theconstant marginal cost of this firm is denoted 2’? and its potential competitors only haveaccess to a technology which enables production of an identical good at higher constant44We also remove the assumption of symmetry across sectors, to show ihat this is notnecessary for our result.45Note that the Hicks definition of complementarity is symmetric, so that if i is a Hickscomplement of j,j is a Hicks complement of i. For more on complementarities in demandsystems see Phlips (1983) and Deaton and Muellbauer (1980).102marginal cost y , i.e. y> y.46 For the sake of analytical simplicity, we only considergoods which are normal and gross complements here.47 (The introduction of other sectorswhere this is not true does not affect the basic results but unnecessarily complicates theanalysis of price setting behaviour within each sector.) Since goods are gross complements,the elasticity of substitution between goods produced in each sector is less than zero, theoptimal strategy for the incumbent is to limit price at the marginal cost of competitors in its0 c48own sector, so that p1 =As in Murphy, Shleifer and Vishny (1989), we consider a representative agenteconomy, where the agent inelastically supplies one unit of labour and owns all profits. Thewell behaved utility function of this individual (defined over all n goods) is denoted u(x)where x is the vector of all goods in the economy. The maximization of this utility functionsubject to the aggregate income constraint yields the following (Marshallian) demand for anygood k produced in sector k:XkXk(P,Y), (2)where p is the price vector andy denotes aggregate income, defined as:y= 1+Hk(p,y), (3)where, since wage is numeraire and total labour endowment is normalized to unity, 1represents labour income and the second term is obtained by summing profits across allsectors of the economy.460ne can alternatively interpret the technology as improving the quality of the goodproduced, for example as in Grossman and Helpman’s (1992) quality ladders model. In such a1case — denotes the quality of the good produced, with quality clearly increasing in y.47Note that the definition of gross complementarity requires that when the price of one goodrises the quantity of its complement falls. This implies that, for all those goods which aregross complements without being Hicksian complements, income effects outweighsubstitution effects in the Slutsky equation, and that therefore they cannot be inferior goods.48Note that we also depart from Murphy, Shleifer and Vishny’s analysis by assuming thatpotential entrants displace an incumbent monopolist rather than a competitive fringe. Sincewe are concerned with the effects of price falls, this is the only case of interest since with acompetitive fringe, no price fall occurs (see Murphy, Shleifer and Vishny (1989)).103Consider the decision problem faced by a potential entrant in one industry who alonehas the opportunity to invest in a cost reducing technology, and then compete with theincumbent, or not invest.49 Investment in industry i involves fixed cost I which reduces theentrant’s marginal cost from 7 to y1, where y> y. This entrant maximizes profit bydisplacing the former incumbent through pricing at the latter’s marginal cost. Price inindustry i thus falls from y to 7. Note that after entry, or investment, price falls by adiscrete amount, however for the purposes of our analysis we shall suppose that this discretefall is small enough to be well approximated by a derivative.In this framework, a horizontal externality would imply that investment in oneindustry makes demand in other industries rise, thus perhaps boosting the profitability ofsimilar types of cost reducing investment there. Where goods are gross complements, theseexternalities exist, as argued by Scitovsky (1954) and Rosenstein-Rodan (1943). Intuitively,by lowering price in its own sector, an investor in cost reducing technology, even whenmaking a loss itself, could precipitate an increase in demand for others. If this were the case,multiple equilibria might exist and a coordination problem arise. The following propositionshows that, in general, this reasoning is not valid and that Hicks complementarity betweengoods is required for an unprofitable, cost-reducing investment in one industry to increasedemand and profits in another. Before considering the proposition, note that if onlyprofitable investment raises profit elsewhere in the economy, then there can only be oneequilibrium. In this case, though creating positive externalities forj, profitable investmentin industry i is always undertaken, thus ruling out the existence of a no investmentequilibrium. Hence, a necessary condition for the existence of multiple equilibria in thisframework is that unprofitable investment serves to raise profit elsewhere. We thus49To avoid unnecessary complication in determining returns to investment when more thanone firm invests, it is assumed that only one firm has the potential to invest in the costreducing technology (as in Murphy, Shleifer and Vishny( 1989)).104examine the impact of investment in i onj’s demand under the most favourable case, that iswhere an investment in i leads the investor to just break even, i.e. zero profits net ofinvestment cost. If it can be shown that, even in this case, f’s demand does not rise, then itwill also not rise when the initial investment makes a loss since, in that case, for the samechange in relative prices, aggregate income, and hence demand, is even lower.Proposition 1: If a firm i just breaks even when investing in cost reducing technology, theensuing pecuniary externality through price fall does not increase profits and returns to costreducing investments in another sector if there are no Hicks complements to good i in theeconomy.Proof:Denote the investing firm by i, and suppose that this firm, upon investing alone,breaks even. The break-even assumption implies:(4)where x’ stands for the demand for industry i’s product and 11 denotes profit afterinvestment has taken place in industry i (in the following, superscript 1 denotes post-investment values and superscript 0 pre-investment values). From equation (3), preinvestment income, y°, is given by:y°=l+H, (5)After investment in one sector, hereafter denoted industry i, aggregate income is given by:y’=l+H—1, (6)which, at the break-even point, by substituting from (4) yields:y’=l+H. (7)ki105The net effect on the demand for good j resulting from an investment in industry iwill be denoted Ax (sector i will always denote the investing sector, unless otherwisespecified, so that the notation need not explicitly refer to the investing sector). It can bedecomposed into two effects: the change in demand for j due to the fall in i’s price and thechange in demand for j due to the change in aggregate income:dx. dx.&, —1-Ap +—-Ay. (8)Using the Slutsky equation, we obtain an approximation to the change in demand as follows:dh. dx. dx.Ax —‘-Ap1x,—’-Ap+--—-tXy, (9)dp1 dy dywhere h denotes the Hicks demand for good j. The discrete nature of the changes meansthat this equation only approximates the real change, however, for small enough changesthis approximation will be reasonably accurate. In the equations above and the rest of thechapter we define Ap1 = p — p10, as the discrete price fall in sector i due to investment insector i. We similarly use the notation Ax1 to denote the total change in demand for sector fsgood due to investment in i, Ay to denote the total change in income due to investment in iand All1 to denote the change in j’s profits due to an investment in i. (note again that thereference to the investing sector, sector i is omitted).1 0 0 cIt follows from the discussion above that p1—p, =—y. But, note that—= H (this follows from Bertrand competition which will be discussedsubsequently) so that the nrice fall just eauals the initial markup, where markup is thedifference between price and marginal cost. This is a critically important condition whichimplies that (9) becomes:dli. dx.Ax1 1Ap1 + —i-(Ay + ny). (10)dp1 dyOnce again, by definition, we know that Ay = y’ — y° and from equations (5) and (7)y’—y° = (ll — IT) — ll. Equation (10) then becomes:kiAx -Ap1+ (11)106Profits in sectors ki at time 1, i.e. after investment, are given by:= (y — y)x. (12)As a result, equation (11) can be written as:dh. Bx.”Ax1 —- Ap, + —- (‘y —y )Ax,< J. (13)Pg kiEquation (13) will hold for all sectors j. We have therefore a system of n-i equations, whichcan be written in the following matrix form:[&j(fll)X1— [] Ap +[] [yc —(n—1)xl (n—1)xl(14)or, equivalently,[i_[..][yc — yol] [Ax](fl_l)Xl Ap1 . (15)(n—1)x(n-1) (n—1)xlWe can apply Cramer’s rule to determine the sign of each term of [AxJ(fl_l)Xl. Thedenominator is the following determinant:a.1 o FAx1 FAx11_(y—)--—(y2r)-- ...—(y—) i—(y-y)0a 0dxIn evaluating this determinant many terms cancel, this can be most easily seen whenconsidering a three good example, i.e. three goods other than i, so that it becomes:i—(y—y) —(y—y) —(y—y)o FAx2 FAx2 o FAx2—(—rj-- 1_(y2_y)-- —(y3)--o FAx3 FAx3 FAx3—(—)-- —(72—Y2)-- 1—(y3_y)-.-expanding along any row or column, for example the first row, yields:107C 0ax2(yc 0 dx3 a2 o dX3dyJ Y2Y2)J - 32’3)J - )(-)--jdx3 dx3 ( dx2-- -{1-(Y3- 73)J(Yi - y)--(y - Y3)JJo dx2( o—(a— yi—(r —1)--—(r — — —dy JJwhich simplifies to:1 ( c dX1V ( c o dx2— Yi2’i)TJ — 2’2Y2)(Y3Y3)iJdx1 ( dx2 “ (c 0) dx1 o dx3— Y2)L_(2h1 — — -‘ —yielding, after expansion once more:1 (c dx o dx2 (rc o)dx3— y)--—(y2r-—which can be more simply expressed as:(y;—y) cqc Yqq=1 ‘YqThus for the general case of n goods the solution is:n (r—r)c Yq (16)q=1 YqqidxThe derivative of the budget constraint with respect to y yields 1 = y sinceq=1dxPq’ which implies that the term y is strictly less than 1 (since the ith term isqinot included in the summation and goods are normal). Thus, since each markup rate (the108first term in the summation in (16)) is clearly less than 1, the summation in (16) is less than1. As a result, (16) is positive.Now consider sector 1, the numerator of Cramer’s ratio is:dh1 / dx o dx17 72) — Yn )dh2dh 1 c o\dXa 7n7nJ(17)expanding this determinant leads to a similar sequence of cancellations as for (16) yielding:(18)As for the denominator, the first bracketed term is positive. So both terms will be positiveprovided there are no Hicks complements, since, in that case 0, Vi,q. This will alsoap,be true for the numerator evaluated for sectors other than sector 1, and implies that sinceAp, is negative by construction, each term of [Axj 1)x 1 is strictly negative and, as a result,All1 isnegative Vji, Q.E.D.Importantly, the Proposition says nothing about gross complementarity of the goods.With unprofitable investment, the possibility of pecuniary horizontal externalities arises onlywith the existence of Hicks complements, so that Hicks complementarity is necessary for theexistence of multiple equilibria through such an effect. This result relates to the earlierdebate, in particular Fleming (1955) who argued that horizontal pecuniary externalities wereunlikely to be important, though in a slightly different context. He suggested that, withinelastically supplied factors, horizontal externalities never implied the need for109need for coordination. The proposition above shows his conclusion to be true for the case ofgross complements (i.e., these do not imply multiple equilibria), however with the possibilityof Hicks complements it is possible that multiple equilibria exist.The intuition behind the unimportance of gross complementarity can be most easilyunderstood in a two good world. Note that, in such a world, Hick’s complementarity isexcluded.50Investment in Sector 1 now has two effects. On the one hand, the price of good 1falls from y to y. On the other, aggregate income falls: indeed, since the investing firmjust breaks even and displaces the previous incumbent, the latter’s profit disappears andaggregate income falls by the amount of that profit, i.e. H (a business stealing effect).Consider the net effect of these changes on the profits of sector 2. Since neither price normarginal cost in Sector 2 have changed, the only avenue of these effects is through a changein the demand for Sector 2’s good. From (8) the total change in demand for sector 2’s good isgiven bydx2 ax—zp + —- A)?. (19)dp1 dyThe second term on the right hand side is evaluated using the fact that Ay = —H. Theeffect of a fall in l’s price on 2’s demand, the first term, can be decomposed using the Slutskyequation into a pure substitution effect and an income effect:dX2_dh2 dX2—Ap1—41 —x1 —Ap1 (20)dp1 dp1The second term on the right hand side of (20) can be re-expressed as —x1--(y—which equals .-H. As a result, putting this into (20) and substituting the expressionobtained into (19) yields:(21)50This shall be shown more clearly a little further on.110which is always non-positive (since l’s price has fallen and the two goods are not Hickscomplements —- 0). Notice that in obtaining (20) the income terms, Ay, cancel out, thisap,is because the effect on 2s demand due to direct income fall associated with the profit lossjust equals the effect of the positive real income rise due to the price fall. Thus grosscomplements are not important because the income loss due to the investment is just thatamount required to place the consumer at her initial utility level, i.e. the income loss justoffsets the real income effect of the price fall. Thus, since income effects cancel out, the onlyeffect remaining is the pure Hicks substitution effect. But without goods that are Hickscomplements, this is negative. Hicks complementarity is therefore necessary to generatemultiple equilibria in this economy.It should be noted that Proposition 1 is robust to the model’s simplifying assumptionabout the form of interaction between firms. It could be argued that more realistic modelingof market interaction between firms would not imply that the producer’s price necessarilyfalls as low as the marginal cost of the previous incumbent. However the assumed Bertrandcompetition between incumbents and entrants which allows for the largest possible pricefalls, biases the case most strongly against proposition 1 holding. If price falls were not asgreat as those under Bertrand competition, the benefits to complementary sectors would besmaller, ceteris paribus, thus weakening the case for coordination. The case of price fallsbeing greater than that occurring under Bertrand competition seems unlikely since Bertrandcompetition seems to ensure the most intense rivalry, however, in Section 3, we will explorethis possibility.Before examining expression (18) in more detail, we digress slightly here to consideran important property of Hicks complements. It can be shown that, for all goods, aweightedsum of the derivatives of the Hicks demand with respect to the price of a particular good, saygood i, is equal to zero:ilLp1--=O. (22)This useful property will be used below. Note that, since a good is always a Hickscomplement of itself, the own price derivatives of the Hicks demand are always negative. Asa consequence, in an economy with two goods, there are no Hicks complements.More detailed consideration of expression (18) shows the different avenues throughwhich Hicks complementarity gives rise to multiple equilibria. Firstly the straightforwardcase is that by investing and lowering price in one’s own sector one increases demands in theHicks complementary sector and can thus lead to profitable mutual investment even wheninvestment alone is not so. This is captured in the first term, i.e. if negative a fall in papiincreases demand for good 1. But there is alsO an indirect avenue through the second term.Suppose that in the expression given in (18) good 1 is independent of good i: 0. Thedp1first term disappears. The second term of the expression,q=2 Yq P1}(9YqiE a 1remains and may turn out to be positive. The vector of Hicks substitution effects,— , canaJJ, Jbe such that the fall in the price of good i, diverts demand away from sectors with very lowmark-up rates (that is for which ) }s very low) to sectors with high mark-up rates‘Yq(the latter sectors producing Hicks complements to good i), and, thereby, lead to an increasein profits and in income in this economy. Clearly, when sectors are identical in technology,this second avenue is impossible, since shifts in demand across sectors following a price falldo not affect mark ups.- 1123. Optimality of investment coordinationThe two avenues mentioned above have very different implications for the socialoptimality of investment coordination in the presence of Hicks complements. This is seen inthe next proposition in which it is shown that with enough symmetry of technologies acrosssectors (which thereby rules out the indirect avenue through which Hicks complements giverise to multiple equilibria) coordinated investment is not socially worthwhile when notworthwhile investing alone. For this proposition we thus assume that, in all sectors, q, thec 0markup rates, q q , are the same. We also assume that, in each investing sector, the‘Yqnew markup is identical to the pre-investment level markup, i.e.:— Yq0 = — Yq1• Denotethese two assumptions Al.Proposition 2: Under Al Hicks complements, though leading to the possibility of multipleequilibria, always imply that the no investment equilibrium is socially preferred to theinvestment equilibrium.Let us assume that only industries i and j coordinate and invest simultaneously. (The proofpresented below can be easily extended to the case of more than two investing industries.)Consider first the change in income such a coordinated move leads to. From (5) this is:Ay =-H -F +x(y-qi,j-fly -F +4(r--)+(-r)÷(r J)]APi(23)q= 1qi,j113where, if we consider the top line which relates to i’s investment; the first term representsthe loss of the previous incumbent’s profits, the second is the fixed cost, the third is the profitof the new investor and, between brackets we have the impact of falling prices on the othersectors profits as well as on i and j through changes in demand after investment has beenmade in those two sectors. Consider now the most favourable case to social optimality ofcoordinated investment, which arises when, if they were to invest alone, each firm i and jwould just break even: each of them needs as little incentive to invest as possible:F, = x ( y — + (y — y,’ and,FEquation (23) can then be written as:& &.Ay=—H+ (y;y i+_...L(yy,1) Ap.q=l P Pqi,j—n+ Ap1q=1 P Pqz,jsubtracting —i- (y—from the first term in the first large bracket and adding it to thesecond term in that bracket (doing the same for y — y in the second large bracket) yields:Ay=+(y.&.Iq=1 Ixpi Lpqj J(24a)(24b)(25)q=lqi (26)C 0 0 1 .. . . .From Al—Yq = — 21q for q=,j. This implies that the second term in eachbracket cancels. Also since pre-investment technologies are the same across sectors, wedenote the common mark-up rate t as:Yk2’k= Vk (27)114We can now re-express equation (26) as:AxAy=—fl+ y_-- Ap1_fl5+ (28)q=l P q=1qi qjUsing expressions (16) and (18) to express in terms of the derivatives of the HicksAp. Apdemand functions and derivatives with respect to income yields:fl 1’ ( c 0) a aAy—H—Fl+ k k r,y----- Ap1ki zq P1 YziNow consider the term in brackets which is multiplied by Ap1. Ignoring the term raised tothe power -1 momentarily, consider the first few terms obtained by expanding the mainsummation (i.e. the summation over q), these are:ah1 ah1 a2 dh1 a)32’ 12 Yi Yaap, dy ap, dy+ +aia1+‘1’2+dp, dy dp, dyNotice that many of the terms in these expressions cancel out; the second term in the firstbracket cancels with the first term on the second line of the next bracket, the first term onthe second line of the first bracket cancels with the second term of the second bracket. Asimilar pattern follows for the additional terms generated by further expansion so thatfinally all that remains is the first term in each bracket, i.e. the —‘i- terms. This impliesthat equation (28) becomes:a’ ai a1 ai ax32’2 2’2 2’iap, dp, dy a, dy115Ay-H-fl+ 1_ ‘kc yL 2’—k1 Yk qiki(29)kjEquation (21) now allows us to simplify further the terms under the summation sign asfollows:(30)+utilizing the fact that Yq’ Pq° As for equation (16), the term raised to power -1 in eachbracket is positive and since the own price derivative of the Hicks demand is always negativeeach expression in brackets is positive. We replace the mark-up rates by their expressionsgiven in (27) and we simplifr expression (30) by making use of the derivative of the budgetconstraint with respect to income, [i = to obtain:--+[[ (31)+[_r[i_ r(yJ]1y L9hJ}J2Using the envelope theorem, the net impact of investment in sectors j and i on the(indirect) utility of the representative agent is given by:dU=My — A$dp, — A4dp, (32)116where A. stands for the Lagrange multiplier of the budget constraint (see, for example,Varian (1978 p. 127)). Introducing the expression for the income change from (31) implies:AU -A.xAp—A.H? —A.xz\j -A.fl +A.[_r1_+A.[_r[1_[a]ahor, remembering that, in each investing sector, the extent of the price fall just equals theinitial mark-up, leads to the canceling of the first four terms so that:AU = A.[[1_(33)( I I dx.i’ dh.Y±JAPi.Since the two terms under brackets are both positive, and the price changes are negative, forthe representative agent, the change in utility brought about by a coordinated move ininvestments is negative. Investment coordination lowers social welfare.Q.E.D.It is possible to provide some intuition for this result: If some goods are Hickscomplements of one good, then, necessarily, others are Hicks substitutes (see equation (21)).As a result, if a cost-reducing investment in sector i increases the demand for the outputproduced by sector j, then, necessarily, the demand for output of the sectors producing theHicks substitutes to i, which must exist, is correspondingly reduced. Thus two firms can bemade better off by coordinating investments and price reductions only if such price fallsinduce a reallocation of demand towards the investing sectors and away from the sectors notinvesting, i.e. only if these other sectors are made worse off. Thus with symmetrictechnologies across sectors, it cannot be the case that coordinated investment improves socialwelfare over the no investment equilibrium. However, this conclusion critically depends upon117the symmetry of technology across sectors as can be seen the proof makes heavy use of thisassumption. Thus coordinated investment may be beneficial in the non-symmetric case.Intuitively, this is because differences in technology across sectors imply the possibility ofdifferences in markups which in turn gives rise to the possibility of a demand reallocationfrom sectors with low markups to those with high markups following a price fall. Thisdemand reallocation can thus lead to real income effects. For a given unit of expenditure, ademand reallocation to sectors with high markups generates a larger multiplier thandemand to sectors with low markups and can thus serve to increase the demand faced byother sectors.4. Gross complementarities againIn this section we revisit the question of multiple equilibria in the presence of grosscomplementarity and show that gross complementarity alone may lead to multiple equilibriawhen the relationship between price fall due to investment and profit loss in the Bertrandsituation above does not hold. The condition used above was that the extent of the price fallin the investing sector just equals the markup and is therefore proportional to the profit loss.If the extent of the price fall in the investing sector exceeds the mark-up of the incumbentmonopolist, the price effect due to investment outweighs the income effect and, in suchcircumstances, multiple equilibria with gross complements can be shown to exist. Uponinitial consideration, however, it does seem unlikely that the price fall should ever be greaterthan the markup of the former incumbent. Bertrand pricing would seem to imply the largestpossible fall in prices, as the new price is set at the cost of the monopolist, yet only leads toequality between price fall and mark-up. However in a different context, Young (1993) hassuggested a situation where this can occur. In particular, if the investing entrant is unableto prevent some of the benefits arising from their new technology spilling over to competitors(for example knowledge spillovers) then the incumbent’s marginal costs will fall as well (see118Young 1993, p. 782, footnote 7). A similar effect is explored in the model presented inCiccone and Matsuyama (1993, Section 7). Thus, in such cases, the incumbent’s marginalcost falls to a value denoted y which will always exceed the marginal cost of the entrantprovided that some benefits are excluded but which, nonetheless, will be less than theirI S 0nutial margmal costs, i.e. 7 <7, <2’,.We now demonstrate, by use of a two-good example with CES preferences, that insituations where the price fall exceeds the mark-up, there are always conditions under whichmultiple equilibria exist.51 (Bear in mind that in an economy with two goods, there are noHicks complements so that Hicks complementarity cannot be driving the result). Considerthe demands implied by symmetric, CES preferences over 2 goods:,.(34)pi+piv-1-]pwhere r = ‘° and p is the elasticity of substitution between the two goods.p—lAs in the general model, profits at time zero can be written:H? = (yc — y)x, (35)wherex is obtained from equation (34) using y, y and y°. The aggregate income at time0, y°, is given by (see equation (3) above):y° = l+H +H. (36)To show the existence of a coordination problem we have now to establish that a cost-reducing investment is unprofitable if one sector invests alone but profitable when bothinvest. To do so we proceed by characterizing the conditions under which investment justbreaks even when (i) the investor invests alone and (ii) when both sectors invest. We thenshow that condition (i) is stronger than condition (ii) which implies that, in some cases, (ii)5-Due to the positive externality of the knowledge spillover, when both equilibria exist, thecoordinated investment equilibrium Pareto dominates the no investment equilibrium.119will hold when (i) does not and that therefore multiple equilibria may exist. Denote F1 (1)the fixed cost which would lead an investor in sector 1 to break even if only sector 1 investsand F1(2) the fixed cost at which an investor would break even if both sectors invest.If investment takes place exclusively in sector 1, the investor’s marginal cost is 2’and the incumbent’s marginal cost becomes 7. The entrant’s gross profits are thereforegiven by:(37)yiDue to the break-even assumption, profits net of fixed costs in sector 1 equal zero so that:= i + = 1 + (y—.(38)Solving for y1 yields:y1= +.(39)Profits in sector 2 are then given by:(40)Finally, in sector 1, in order for the break even assumption to be satisfied, F1 (1) mustsatisfr:r1(1) ==(r — 1 r (41a)7+Y[J +fJ120Let us now consider the case where the two invest simultaneously. In sector 2,technological spillovers reduce the incumbent’s marginal cost to y. If the entrant in eachsector just breaks-even, then aggregate income in the second period equals 1. As a result,the break-even condition in sector 1 now becomes:== (y — r (41b)A comparison between the two conditions (41a) and (41b) yields, after some manipulation:(42)l’iI O’\ IL2’) iyWhen condition (42) is satisfied then, for all values of fixed cost, F; satisfyingF; (2)> F; > F; (1), the fixed cost is high enough to dissuade investing when there is noinvestment in the other sector, since the fall in price actually transfers part of the benefit ofthe investment to the other sector’s incumbent. Simultaneously F; is low enough so that,when both invest, the mutual fall in each sector’s price and the complementary externalitiesgenerated make investment worthwhile. With respect to condition (42), note that CESpreferences only imply gross complementarity between the two goods when 1 r 0 (withr = 0 implying Cobb-Douglas, and r = 1, Leontief preferences). With r 0, the two goodsare substitutes, there are no ‘horizontal’ price externalities , so that (42) can not be satisfied.For any positive values of y, y, y satisfying 7> y> one can always find athreshold value of r between zero and one such that, for all r greater than this value, (42)holds. In other words, no matter how small the spillover accruing to the incumbentmonopolist, one can always find a degree of (gross) complementarity between the two goodssuch that multiple equilibria exist.1215. ConclusionIn understanding the mechanics which give rise to multiple equilibria, recentattention has focused on the role of pecuniary externalities, rather than those of thetechnological variety. This has been because pecuniary externalities, which act through themechanism of the market, are more difficult to internalize and are therefore more likely tohave important macroeconomic implications, such as multiple equilibria. This paper isconcerned with the effects of horizontal price externalities which arise when the fall in priceof one good raises the demand for complementary goods thereby implying that investmentsmay not be privately undertaken even though, when coordinated, they could be profitable.This avenue of horizontal externalities has been shown to lead to multiple equilibria in avariety of frameworks. Section 3 of this chapter proves that such horizontal externalities,though arising in the case of gross complements, cannot, unless goods are also Hickscomplements, lead to multiple equilibria. The analysis thus demonstrates that Hickscomplementarity, and not gross complementarity, is the relevant concept of complementaritygiving rise to multiple equilibria. Section 3 then examined the optimality of suchcoordination which is implied when Hicks complementarity leads to multiple equilibria. Itwas shown there that with symmetry of technology across sectors (but not demand), thoughcoordinated investment can be worthwhile for the investing firms, it is never socially optimal.Investment coordination could, however, be beneficial with non-symmetric technologiesacross sectors. Finally, in section 4 we suggest an avenue through which grosscomplementarity may nonetheless lead to multiple equilibria and to a need for investmentcoordination. It is shown that when investors also lower incumbent’s costs (as in the case ofknowledge spillovers), causing price falls to exceed mark-ups, the condition established in122Section 2 is violated and multiple equilibria may exist, even in the absence of Hickscomplementarity, implying a benefit to coordination.123Chapter 5.CONCLUDING COMMENTSThe three substantive chapters of this thesis have analyzed distinct areas in whichproblems of coordination give rise to important economic phenomena. Although primarily anexercise in positive economics, i.e., seeking to explain elements of the real world rather thannecessarily suggest improvements, the models developed in each chapter do give insights intopossible effects of government policies and can therefore yield normative suggestions. In thisconcluding chapter I will summarize the main insights of each chapter in light of the relevantpolicy implications which each of the chapters imply.Chapter 2 focused on explaining the existence of discrimination in competitive labourmarkets as arising due to household interaction between workers and labour market hiringpractices of other firms. It was seen there that both the need to motivate effort from workers(i.e. a problem of non-observability) and the dependence of that effort on householdcharacteristics, lead to a dependence in firms’ labour market hiring decisions. In particular,if all other firms were following a policy of discrimination against women in hiring to goodjobs, then any one firm would also find it beneficial to do so. In contrast, with all other firmsnot discriminating, it was seen that discrimination lead to no substantive benefits. Thereforethe labour market could be seen to converge on either one of two distinct outcomes, one ofdiscrimination and one of non-discrimination, with government policy having a role incoordinating the labour market onto the non-discriminatory outcome. Conditions wereestablished under which a policy of affirmative action could act to achieve such coordination.With enough firms prohibited from discriminating, the remaining firms would then find itindividually rational to do so as well and thereby comply with the policy, even in the absenceof direct enforcement.124The model presented in Chapter 3 also yielded important policy implications. Thechapter developed a model where growth stemmed from innovative activities, with the levelof such activities seen to critically depend on the inherited industrial structure. In particularit was shown in that chapter that, depending on initial industrial structure, an economycould be trapped in one of two steady states with potentially widely different welfareproperties. A government policy of subsidizing innovative activities, if applied widely enough,could move the economy from the no growth steady state onto a path of continued innovationand ultimately to a steady state of sustained growth. Depending on parameters, such a policywas shown to be potentially Pareto improving. The model developed in that chapter however,suggested a possibly detrimental effect of state funded research which is made publiclyavailable. The relevant implication was that state funding, by altering industrial structure,could serve to more than crowd out private research and be, ultimately, detrimental togrowth. The chapter concluded with an examination of finite lived patents where it wasshown that length of patent protection, a government mandated variable, could effect thelikelthood of the economy ending up in a no growth situation.In Chapter 4 the focus was much more directly upon the actual mechanisms givingrise to multiple equilibria and hence coordination problems. The important policy implicationof this chapter was that Hicks complementarity, though giving rise to multiple equilibria in aframework of multi-sector investment in cost reducing technology, did not necessarily implythat coordinating investments would be socially optimal. 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(1993) Substitution and Complementarity in Endogenous Innovation, QuarterlyJournal ofEconomics, 108, 3, pp. 775-807.Zaretsky, E. (1973) Captialism, the family and Personal life, it Socialist Revolution 3(1-2)January-April pp. 69-125.Zaretaky, E. (1974) Socialism and feminism III: Socialist politics and the family, SocialistRevolution 4(1), January-March pp.83-98.130APPENDICES1311 Appendix to Chapter 2• Maximization problem of a potential seller of household services when trading with a spousebuying, at most, Hm° at internal price pfl, which determines (pfl,Hm°(p’)). Firstly, notethat both Hl and Hm° are less than , since it is impossible to provide or purchase morehousehold services than exist.The Kuhn..Tucker conditions for this problem, derived from equations (2.1) to (2.3), are:e : p’U’(.) + V) 0, = 0 if e > 0. (1.1)p”U’(.) + V’(.) — A < 0, = 0 if Ht > 0 (1 2)— H:) = 0 (1.3)where A is the multiplier on the constraint H’ Hm.Since the objective function is continuous, and the feasible set is non-empty and compact asolution to this set of conditions must exist. Since U”(.) and V”() <0 the function is globallyconcave, and thus the uniqueness of the solution is guaranteed.By the assumption that lim U’(c) = oo, it follows that ë and HS cannot both equalzero as neither conditions (1.1) nor (1.2) can be satisfied.Suppose 0 < H9 < HmZ. From (1.3) this implies that A = 0. (1.2) then holds withequality, i.e.pU1(.) = —Vt) (1.4)132Since > 0 implies that (1.1) needs to hold with equality and since p’ <p1 and (1.4) violatesequality in (1.1), it is only possible that for Hm > H’ > 0, ê = 0.Since for any H5 <Hm, = 0, and since ê and HS. cannot both equal zero, this impliesthat HS. > 0 always.Suppose H5 Hm. This implies that \> 0 from equation (1.3). Thus (1.2) implies:p’U’(.) + V’(.) 0 (1.5)ê > 0 implies p’U’(•)+V’(.) = 0 which is consistent with (1.5), and ê = 0 implies p’U’(.)+V’(.) <0 which is also consistent with (1.5). So HS = Hm° and both ê = 0 and ê > 0 are possiblesolutions.• Maximization problem of a potential seller of household services when spouse does notwish to purchase services, i.e. 0. This problem is similar to the one above exceptthat Hf = 0 by equation (2.3) and the individual chooses é only. In this case ê > 0 aslimoU’(c) = t implies that (1.1) cannot hold with equaiity.• Maximization problem of a potential buyer of household services who is a non-shirker,buying, at most from spouse at internally negotiated price p?Z, the solution of whichdetermines 1IsBe(pn):The Kuhn Tucker conditions from equations (2.5) to (2.8) are:HB : —pnU’(.)— V’(.) — A — A2—0, = 0 if Hb > 0. (1.6)—p’U’(.) — 7V’(.) — 7A2 — 1JA3 < 0,= 0 if HC > 0. (1.7)133— HB)= 0 (1.8)A2(11 — — H) = 0 (1.9).\3(uPe — pnHB — piHe) = 0 (1.10)Since 0 < ffB < H, and the maximand is continuous, a solution exists. Again, it followsfrom the strict concavity of the objective function and the convexity of the constraint set thatany solution obtained is unique. As seems realistic, I proceed by assuming that the budgetconstraint for a worker in a good job is never binding with respect to the purchase of householdservices, thus )3 will always equal zero at any solution. A solution with this constraint bindingdoes not change any of the results.Firstly, it can be seen directly from assumption Al, that even though a person in a goodjob is not precluded from also gaining piece rate work at wage p’ he or she does not wish to.This is because, to work at wage p’ a necessary condition is:p’U’(w9ë)+ V’(e+ B) ci. (1.11)But this condition violates assumption Al. Thus ignoring a non-shirker’s choice of e is notcritical.Suppose 0 < HB. < HIt and H > 0. This implies that A = 0 from equation (1.8).Equation (1.6) then implies:— pU’() — V’() = (1.12)134Equation (1.7) implies:— p’U’(.)— 7V’(.) = (1.13)Re-arranging (1.13) yieldspU’(.) — V’(.) = (1.14)recalling that p = p’/ from the paragraph before equation (2.1) in the text. Since pfl <j5 it isclear that these equations are inconsistent. This implies that for 0 H < H1”, H = 0.Suppose HB = 0, we know this implies H = 0 from above. Thus (1.6) implies:—pThU’(.) — V’(.) < 0. (1.15)This in turn implies that pr’> — which directly violates assumption Al. Thus HB > ofor a non-shirker.Suppose H = H1, it is straightforward to show in this situation that can be either0 or greater than zero.Maximization problem of a potential buyer of household services who is a shirker, buying,at most H’” from spouse at internally negotiated price pfl, the solution of which determines111B,°(p):As with the non-shirker, for a shirker it is the case that if 0 < HB < H then H = 0. Somedifferences are that, in this situation, the worker in good job may not wish to purchase householdservices from their spouse, i.e. it is possible that HB = 0. But if they wish to buy servicesat all they will wish to buy as much from their spouse as possible before considering buying135externally. It may also be the case that they will supplement their income with employmentin the piece rate job, i.e. ê may exceed zero. The condition for this is:p’U’(w9ë)+ v’(o + 1) 0. (1.16)It can be seen that this condition need not violate assumption Al because of the 0 term in V’• Proof of Lemma 1, i.e that HB is weakly lower for a worker in a good job who sets e = 0than for one who sets e =In this section denote H the solution for a non-shirker and H0 the solution for a shirker.Clearly if H = H1 for a non-shirker then the assertion is true. Thus since it has beenshown above that a non-shirker never sets H 0, the only case that needs to be consideredis the one in which 0 < H < H1’”. In this situation it has been shown above that H = 0.The first order condition for this case from equation (1.6) is:—pU’(uPë + p’H’) — V’(ë + H — H’) = 0 (1.17)since ) disappears due to equation (1.8) and )2 = 0 since HB < H1’ < H. For a shirkerthis condition is almost identical, the only difference being that in V’, ë is replaced by a zeroyielding:— p”U’(w9ë+p”H’) — V’(O + H — H’) < 0 (1.18)It is possible that the solution to this is H0 = 0, if this is the case, then a shirker clearly buysfewer household services. However Hr may be greater than zero. In this case, the conditionfor determining it is:— p’U’(w9ë+ pH’) — V’(O + H — H’) = 0 (1.19)136Since V’ < 0 and V” < 0 the solution to this must strictly be less than the solution to (1.17).Thus if a positive amount but less than the maximum is purchased by both a shirker and a non-shirker, the non-shirker purchases strictly more. Thus it has been shown that, in all possiblecases, a shirker buys less than or the same amount of household services than a non-shirker.• Proof of Lemma 2: The left hand side is the extra gain in utifity to a non-shirker by beingable to trade household services with their spouse. The right hand side is a similar expressionfor a shirker. These amounts can be most easily compared by comparing the consumer surplusgenerated by trade in each of the two cases. Consider Figure A2.1 which sketches the marginalutility of leisure and consumption as a function of total household services purchased H. Totalhousehold services purchased comprise those purchased internally upto H” since this is alwaysexhausted first, and the remaining amount purchased externally. Since a non-shirker starts offapplying ë more units of effort, for a given H’, her marginal utility of leisure is higher. Thus theshirker’s marginal utility of leisure function (MUL°) lies uniformly below that of a non-shirker(MULe). The marginal utility of consumption function will be identical for both, since bothstart with the same income and leisure does not interact with consumption due to the separableutility function. There will be two cases of this function, one with trades available (MUCt) andone without trades (MUC). The former lying below the latter since, with trades available,services are purchased at a lower price, implying that remaining consumption is greater (forgiven HP) and thus marginal utility of consumption lower. The consumer surplus from tradingfor a non-shirker is given by the area between the two curves MUG and MULE. Similarly for ashirker with the area between the curve MUL° and MUG. The left hand side of the expressionin the lemma is given by the difference between the area between MUCt and MULe and the137area between MUC and MULe, and can be thought of as the consumer surplus availablefrom trading with one’s spouse rather than on the market for a non-shirker. This is given bythe total shaded area in Figure A2.l. Compare this with the consumer surplus from tradingwith one’s spouse rather than on the market for a shirker which is given by the cross hatchshaded area only representing the difference between MUCt and MUL° and the area betweenMUC and MUL°. The diagram shows clearly that since MUL° always lies below MULe theintersection point for a non-shirker is always to the right of that for a shirker, thus leading tolarger gains from trade and therefore, in terms of the lemma implying that the right hand sideof the expression, always exceeds the left.• Proof that ‘I’’= U(w9 — pH— p’H) + V(H — HB,e — 7H — e) and B,0 = U(w9 — pnHB0p’HeO) + V(H — HB,0 — 7He0 — 0), where HB,e, HB,o Hee and H’° denote the maximizedvalues of the choice variables. The envelope theorem implies that we can ignore induced changesin the maximized variables when considering a change in Also, the separability of utilityin U and V implies that differences in e do not change the effect of changes in w9. However, itfollows from Lemma 1 that H + H : HB,0 + H0 and since U”(.) < 0 it follows directlythat > JiB0 Intuitively, since a non-shirker starts of with a lower level of consumption(because he or she buys more household services) his or her marginal valuation of extra incomeis greater.• Equilibrium in which only some firms discriminate.Assume that there are now two types of firms. In one type of finn, denoted type a, when one138worker shirks the output of all workers is zero, i.e. ej = 0 = output = 0 /j where j is anemployee of the firm in which i works. The other type of firm is the same as before;. if e = 0then output = 0 for i only. Let denote the proportion of F that are type a firms. I look foran equilibrium in which the type a firms pay wg and do not discriminate, and the other firmspay w and do discriminate. In this situation, if all type a firms pay w to avoid shirkers, theyalso hire, simply by chance since they do not care about gender, noF/2 women. Consider theexpected profits to a type a firm from paying w and discriminating. The probability of anyone person being a shirker is noF/N. So for a firm hiring n workers, assuming that ii is smallrelative to N, the number of shirkers hired is given by a binomial distribution with i trials.Thus the probability of hiring no shirkers, remembering that this is the only case in whichoutput is non zero, is (1— noF/N). Thus expected profits are p9(l — noF/N) — wm. Profitsif paying w are given by (p9 — wjn. For type a firms to prefer to pay w it is sufficient that(p9 — w)n > p9(l — naF/N)’ — wn. For the other firms to prefer to pay w it is sufficientthat p9 — w < noFp9/N — w. The reason for the difference is that type a firms care greatlyabout hiring even one shirker as then all output is lost. Thus they calculate expected outputby considering only the probability of hiring at least one shirker. Non-type a firms, however,lose only the output of the shirker, so that for them the costs of hiring a shirker are muchsmaller. They will therefore be more likely to take a chance of hiring a shirker and thus bealso more likely to follow a policy of discrimination. More exactly, if both of these conditionsare simultaneously satisfied then there exists a Nash equilibrium in which type a firms donot discriminate and all other firms do. It should be noted that wg and w are not given byequations (2.10) and (2.11) from Chapter 2, in this case because, in equilibrium, some workersshirk, but will stifi be computed in the same way.139Appendix to Chapter 3Derivation of equation (45).Taking a time derivative yields:v w_____V w—(Al)From the no-arbitrage condition in the asset market, or alternatively by taking a timederivative of equation (31) the following condition must hold at each instant:(A2)The solution to the consumer’s optimization problem, equation (15), and the non-storabiity ofconsumption goods also imply that:r=+p.y(A3)Then, taking the natural logarithm of (26), a time derivative through the results andsubstituting for & from (43) we obtain an expression for the rate of change in aggregateincome:Y RY 1R w(A4)140Substituting the result into (A3) to obtain an expression for r , which is then used in (A2),yields the following expression for the rate of change in the value of the firm:V — w ( (y—i)(1—c—= R+—+(R)I 1+V 1R w i c41—y)+y) V(A5)This expression depends on all future levels of research through V in the last term, howeverwe can obtain an expression for - in terms of present variables by using (20) and (26) for 2Zand the labour market clearing condition (23) for V from which w cancels yielding:_= R_______________________V w i—eR cv(1—y)+y) (X+(1—CL)y(A6)Substituting this expression into (Al) yields (45).Dynamic optimization of social welfare functionDynamic optimization yields the following first order conditions:1—‘Le (t)--1R(t) R )(A7)p0—0 = in y,(A8)as well as the transversality condition:Time’0(t)n(t) = 0.(A9)141(A9) and (A8) imply p0 = in y which when substituted into (A7) yields (48).Innovation as EntrepreneurshipHere, we define another factor of production, h, used in all intermediate sectors, so that thelinear production function of equation (16) is replaced by:X = A(t)f(h/,), (A16’)with f(h/ , U) homogeneous of degree one in both inputs and h/ denotes firm fs input of hin industry i. h can be used either in production or research with a possible interpretationbeing as a unit of human capital.53 Unlike U, we assume that h need not be distributeduniformly. Profit maximization implies the following cost function:c, (t) C(w(t), r(t), x,’ (t), Af (t)), and, using the homogeneity of the production function,c(w(t), r(t))average cost is equal to . This will imply that c(w(t), r(t)) actually replaces theA,’(t)w(t) term everywhere in the previous section. As a result, equation (17) can now be writtenas:11(t) =___________—y(t) A[(t) (AlT)A,c(t) A(t) ) c(w(t),r(t))Equation (18) similarly becomes:c(w(t) r(t))p.(t)= A,(t)(A18’)We model entrepreneurial activity analogously with research in the previous section, so thatt(e(t)) denotes the sector wide probability of an entrepreneurial success. We assume thatthe technology used to produce entrepreneurial effort is analogous to the one used in theproductive sectors, so that e(t) = f(he(t),Ue(t)),With f(.,.) identical to that appearing in53A more complete explanation would describe the process of human capital formationinstead of treating it as fixed. However, allowing for this in the present model yields nochanges since relative returns to £ and h are the same in all equilibria.142equation (A16’). This simplifies calculation without affecting the qualitative results of ouranalysis, and, in fact, yields a slightly modified expression for equation (36):(é) 1—e (y—1)=1(A36’)with e=f(he,e)and equation (41) remains unchanged. Thus it can be seen that the distribution of h does notaffect the existence of two steady states.143
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