RETAINED EARNINGS DYNAMIC, INTERNAL PROMOTIONS AND WALRASIAN EQUILIBRIUM * Pablo F. Beker

Transcription

1 RETAINED EARNINGS DYNAMIC, INTERNA PROMOTIONS AND WARASIAN EQUIIBRIUM * Pablo F. Beke WP-AD Coespondence: Univesidad de Alicante, Fundamentos del Análisis Económico, Caetea San Vicente, s/n, Alicante (Spain). Tel.: / Fax: / beke@melin.fae.ua.es. Edito: Instituto Valenciano de Investigaciones Económicas, S.A. Pimea Edición Mazo Depósito egal: V IVIE woking papes offe in advance the esults of economic eseach unde way in ode to encouage a discussion pocess befoe sending them to scientific jounals fo thei final publication. * This pape is a evised vesion of chapte III of my Ph.D. dissetation submitted to the Gaduate School of Conell Univesity in Januay 2002 and supevised by Pof. David A. Easley. I would like to thank him fo comments to a pevious vesion of this pape. I am also gateful to the paticipants at the Fist Bazilian Wokshop of the Game Theoy Society held in Sao Paulo, The XIII Wold Congess of the IEA held in isbon and seminas at the Univesities of Valencia and Alicante fo useful comments. All emaining eo is mine. Suppot by the Ministeio de Ciencia y Tecnología, Gant NoBEC , as well as fom the Instituto Valenciano de Investigaciones Económicas (Ivie) is gatefully acknowledged.

2 RETAINED EARNINGS DYNAMIC, INTERNA PROMOTIONS AND WARASIAN EQUIIBRIUM Pablo F. Beke ABSTRACT In the ealy stages of the pocess of industy evolution, fims ae financially constained and pay diffeent wages because wokes have heteogeneous expectations about the pospects fo advancement offeed by each fim s job ladde. This pape agues that, nevetheless, if the output maket is competitive, the positive pedictions of the pefectly competitive model ae still a good desciption of the long un outcome. If fims maximize the discounted sum of constained pofits, financing expenditue out of etained eanings, pofits ae diven down to zeo as the pefectly competitive model pedicts. Ex ante identical fims may follow diffeent gowth paths in which wokes wok fo a lowe enty-wage in fims expected to gow moe. In the steady state, howeve, wokes pefoming the same job, in exante identical fims, eceive the same wage. I explain when the long un outcome is efficient, when it is not, and why fims that poduce inefficiently might dive the efficient ones out of the maket even when the steady state has the positive popeties of a Walasian equilibium. To some extent, it is not technological efficiency but wokes self-fulfilling expectations about thei pospects fo advancement within the fim what explains which fims have lowe unit costs, gow moe and dominate the maket. Keywods: Industy Evolution - Maket Selection ypothesis - Poduction unde Incomplete Makets - Retained Eanings Dynamic - Self-Fulfilling Expectations - Intenal abo Makets JE Classification Numbes: D21, D52, D61, D84, D92, J41 1

3 1. INTRODUCTION Conside a maket in which many fims compete to sell an homogeneous poduct. Economic theoy pedicts that, at least in the long un, pofits vanish and each fim poduces the quantity that maximizes pofits at the maket pice. Although most economists agee about this desciption of the long un outcome of the pocess of industy evolution, it is not so clea what foces lead an industy to that steady state. The theoy of industy equilibium in competitive makets elies on the existence of a pefect cedit maket and pofit maximizing fimstoexplainwhypofits ae dissipated. If thee is a complete set of pefectly competitive financial makets, each fim maximizes its maket value, the makets fo inputs ae pefectly competitive, thee ae no tunove costs and thee is eithe fee enty o the technology displays constant etun to scale, then equilibium pofits ae zeo and each active fim poduces the pofit maximizing level. In shap contast with these assumptions, howeve, the empiical evidence suggests that new fims ae financially constained and the labo maket, athe than being in a Walasian equilibium fom the stat, it is bette chaacteized by social institutions which ae not pesent in the theoy of the fim unde pefect competition. Indeed, the poblems of asymmetic infomation identified by authos like Stiglitz and Weiss [12] as the main explanation fo the failue of the cedit maket, ae paticulaly impotant at the ealy stages of the pocess of industy evolution. Theefoe, many fims finance poduction einvesting thei own funds. In moden industies, financing though etained eanings is the nom athe than the exception. To quote Allen and Gale [3]: Pehaps the most stiking point [...] is that in all counties [US, UK, Fance and Gemany] except Japan, etained eanings ae the most impotant souce of funds. Extenal finance is simply not that impotant (p. 76) The lack of access to cedit may pevent fims fom achieving its optimal size fom the stat and explains why it takes time fo pofits to be dissipated. In addition, the pesence of asymmetic infomation among fims about the ability of wokes causes wage ates to diffe fom poductivity and tunove costs ae significant. Theefoe, wokes tend to be attached to the same fim fo long peiods, fims cay out most of the taining of thei employees and pefe to pomote employees intenally athe than ecuiting new wokes. Using the tem made popula by Doeinge and Pioe [6], fims set up an intenal labo maket, with ules that ae diffeent fom the ones that pevail in a Walasian maket. As S. Rosen [11] wites: Many featues of labo makets bea little esemblance to impesonal Walasian auction makets. Chief among them is the emakable degee of obseved woke-fim attachment [...] The typical adult male woke spends twenty yeas o moe on a single job It is appaent that moden industies display many featues which ae not taken into account in the static model but ae key to undestand why industy evolution takes time and how wages evolve. Theefoe, the standad desciption of fim and industy behavio is at best the desciption of a steady state of some gowth dynamics. Economists like Alchian [1] and Fiedman [8] ecognized this long time ago. oweve, Nelson and Winte [10] wee the fist in poviding a fomal explanation on how such steady state can be attained even if no 2

4 fim follows a pofit maximization ule. The key assumption in thei wok is that fims that make positive pofits expand, those that make zeo pofits do not change capacity while those that make loses contact and seach fo new decision ules, a dynamic that can be motivated by the use of etained eanings to finance investment. oweve, Blume and Easley [5] show that even though such etained eanings dynamic explains why fims that do not maximize pofits ae diven out, it may not convege to a Walasian equilibium. The wok of Nelson and Winte and Blume and Easley focuses on the ole of the etained eanings dynamic as a substitute fo maket completeness when the labo maket is pefectly competitive. In many industies, the pesence of taining costs and fim specific abilities not only implies that wages ae not closely elated to poductivity but also that they exceed wages in anothe industy. This is typically the case fo the wage of skilled intensive jobs at the top of the pogession line. Because wokes anticipate that they may pogess though the pomotion line and obtain those high wages in the futue, esevation enty-wages ae usually lowe than in othe industies. Ceteis paibus, the bette the pospects fo advancement displayed by the fim ae, the lowe the woke s esevation enty-wage is. Intuitively pospects fo advancement must be positively elated with the gowth pospects of the fim. This intoduces an additional self-fulfilling aspect in the pocess of industy evolution. Indeed, since fims ely on intenal funds, ceteis paibus, those that ae believed to have bette gowth potentials pay lowe wages, have moe evenue and end up pomoting moe wokes, fulfilling wokes expectations. This intoduces moe complexity in the pocess of industy evolution. If ex-ante identical fims follow diffeent gowth paths, does the industy convege to a steady state with zeo pofits? Which fims pay lowe wages along the tansition? What ae the efficiency popeties of the steady state? Is thee an unambiguous positive elationship between technological efficiency and gowth ates? These ae some of the questions addessed in this wok. This pape agues that when fims maximize the discounted sum of constained pofits, financing expenditue out of etained eanings and the intenal labo maket aises as a cost minimizing institution, due to fim specific abilities and costly taining, the industy conveges to a steady state whee pofits ae dissipated. My analysis coesponds to the case in which fims do not face a shotage in the supply of skilled wokes along the pocess of industy evolution. Theefoe, adjustment costs do not play any ole in this pape. Instead, I concentate on the ole of wokes expectations in shaping facto pices, an aspect that has not been addessed yet in the liteatue of industy evolution towads a Walasian equilibium. As in Waldman [13], evey fim in the industy leans something about a woke s skills by consideing his job assignment and can ty to hie him. Theefoe, the highe the taining cost is o the moe geneal the woke s skill is, the highe is the wage of pomoted wokes in a two tasks job ladde. If this wage exceeds the wage those wokes could obtain in anothe industy, thei enty-wage depends on the woke s expectations about the fim s pomotion ate. If fims ae ex-ante identical, I show that wokes who cay out equal jobs eceive the same wage in the steady state, egadless of the fim that hies them, as if the labo maket wee in a Walasian equilibium. 3

5 oweve, ex-ante identical fims can follow diffeent gowth paths towads the steady state. Ceteis paibus, fims that ae expected to gow faste hie wokes at a lowe enty-wage, which implies that technological efficiency may not hold along the tansition. oweve, it does hold in the steady state. Allocative efficiency, instead, is satisfied in the steady state if and only if wages at the uppe levels of the job ladde ae identical to those in the competing industy so that enty-wages ae identical acoss industies. Othewise, too little is poduced compaed to the efficient allocation of esouces. The failue of technological and allocative efficiency is due both to the absence of a pefect cedit maket as well as the impossibility of enfocing a wage fo old wokes equal to thei oppotunity cost in the competing industy. I also conside the case of fims with diffeent technologies. Although economists long time ago ecognized that fims with lowe costs tend to gow moe, it is usually ague that cost diffeentials ae due to technological easons. oweve, this neglects the fact that, ceteis paibus, those fims that ae believed to display bette gowth pospects can hie wokes at a lowe wage which, in tun, contibutes to lowe its costs. This evese of causality implies that even fims that poduce inefficiently may end up dominating the maket if wokes believe they display sufficiently bette pospects than the efficient ones. Indeed, the wokes willingness to wok fo a low enty-wage can moe than compensate the cost disadvantage intoduced by an inefficient technology. Can this happen in a self-fulfilling equilibium that conveges to a Walasian-like steady state? I constuct an example in which even though pofits vanish in the long un, woke s expectations ae fulfilled and inefficient fims gow moe and dominate the maket in tems of maket shae. If at the ealy stages wokes ae optimistic enough about the pospect fo advancement offeed by the fims which poduce inefficiently, almost all wokes end up employed by inefficient fims in the long un. Theefoe, almost all wokes pefoming the same job eceive the same wage, as in a Walasian equilibium. In contast with Beke [4], I do not need to assume an stochastic technology to show that inefficient fims can dominate a pefectly competitive output maket. My analysis confims the widespead intuition that in a competitive output maket, pofits ae diven down to zeo and fims do not face financial constaints in the long un. Contay to the standad static analysis, I do not need to assume the existence of a pefect capital maket o a pefectly competitive labo maket. oweve, this pape also confims Winte s [16, p. 88] skepticism about the efficiency of the equilibium in a wold of incomplete makets whee business fims play the ole of a taining institution. Indeed, he wites: We know how to go about poving the Paeto optimality of equilibia in theoetical systems in which pices povide the necessay coodinating infomation, while actos have essentially unlimited memoies and computation powe, and contacts ae costlessly enfoced. We do not know how to -and vey likely it is not tue- fo a system in which elevant economic infomation is outinely tansmitted by the daily newspape, o, indeed by any one of a lage numbe of obviously significant social institutions. The list compises, fo example, the mass media, the schools and othe educational institutions, the family, business fims (in advetising, taining pogams, etc.)... In competitive output makets, the etained eanings dynamic gives an evolutionay advantage to fims with lowe unit costs. oweve, unit costs ae detemined not only by technological efficiency but also by wages. In the pesence of intenal pomotions, unlike in Walasian makets, woke s expectations about the oppotunities 4

6 fo advancement within the fim ae key to detemine wages. Theefoe, the fitness of a fim depends not only on its technological efficiency but also on the self-fulfilling beliefs of the wokes. I conclude that, at least in the long un, the etained eanings dynamic justifies the use of the standad static analysis of competitive makets to make positive pedictions but does not always justifies its efficiency popeties. Unlike in Blume and Easley s model, even the steady state of the etained eanings dynamic may fail to be efficient in the pesence of intenal pomotions. As in Athu [2], what happens at the oigin of the industy can have a decisive ole on the technology that dominates the maket. oweve, it is not a netwok extenality o the pesence of inceasing etuns what dives the esult in this model but the self-fulfilling beliefs of the young wokes about the pospects fo advancement offeed by the fims. 1.1 Oveview In section 2, I descibe a patial equilibium model of industy evolution in which etained eanings detemine the scale of opeation, fims ae long lived and evey peiod a new geneation of wokes, who live fo two peiods, entes the labo foce. The desciption of the labo maket is stongly influenced by Waldman s fomalization of the aguments in Doeinge and Pioe. 1 In section 3, I define an Industy Equilibium (IE). In an IE, each fim and the wokes it contacts play a subgame pefect Nash equilibium (SPNE) and the output and labo makets clea. Fims may follow diffeent stategies eithe because they ae endowed with diffeent technologies o because of the existence of multiple SPNE of the game played between each fim and the wokes. Since taining wokes is costly, fims have an incentive to hie wokes tained by a competito. oweve, these wokes ae not as poductive as those pomoted intenally. Theefoe, the highe the taining cost is o the moe geneal the taining is, the highe the equilibium wage of a pomoted woke is. In section 4, I show that if the wage of a pomoted woke exceeds what those wokes would eceive in anothe industy, the game played by the wokes and the fim has two SPNE. In one SPNE, evey geneation of young wokes believes the next geneation will accept employment at wages low enough to induce the fim to pomote a lage faction of its cuent employees the following peiod. Anticipating this, they accept employment at a low enty-wage. In anothe SPNE, evey geneation of young wokes believes the next geneation will accept employment only at a wage so high that the fim will pomote a small faction of wokes. Theefoe, they accept employment only at a high enty-wage. Instead of looking fo a futhe efinement of the notion of ationality, I analyze how the maket shae of fims which face diffeent labo maket conditions evolve along time. In sections 5 and 6, I analyze the dynamic and efficiency popeties of the industy equilibium fo the case of ex-ante identical fims and heteogeneous fims, espectively. Conclusions ae in section 7. All the poofs ae in the Appendix. 1 Thee ae some slight diffeences between the two models of the labo maket. In Waldman s model, wokes ability takes values in a continuum while in mine it can take only two values but a law of lage numbes holds at the fim level. e assumes that fims ae not financially constained but instead the technology is such that they hie only one woke each peiod. 5

7 2. TE MODE At date zeo, the industy adopts a new technology to poduce a final good. et q denote the fim s output level. The technology to poduce this good equies only labo and the poduction pocess can be descibed as a function of two tasks. The level at which task 1 and task 2 ae pefomed ae denoted by q 1 and q 2, espectively, and the poduction function takes the following functional fom: q = q α 1 q 1 α 2 0 < α < 1. Task 1 equies a skill that is not industy specific. If l is the numbe of wokes employed in task 1 then 2 q 1 (l) =l Evey woke develops a new ability while pefoming the fist task. Ability is a andom vaiable that takes only two values: high o low. Ability tuns out to be high with pobability λ (0, 1). In ode to be able to pefom the second task, a woke needs not only to have high ability, but also to eceive some additional taining to develop the industy specific skill. Then a necessay condition to be able to cay out the second task is to have pefomed task 1 in the past. In pinciple, thee ae thee diffeent ways in which a fim can lean whethe an old woke has the necessay ability to develop the industy specific skill: 1. Since ability is evealed while pefoming task 1, fims lean which of thei employees have developed high ability. Doeinge and Pioe emphasize this point [6, p. 31]: The efficiency of intenal ecuitment and sceening deives fom the fact that existing employees constitute a eadily accessible and knowledgeable souce of supply whose skill and behavioal chaacteistics ae well known to management. Infomation about intenal candidates is geneated as a by-poduct of thei wok histoy in the entepise. At the beginning of peiod t +1, each woke bon at t who developed high ability can be tained, at a unit cost of c, to pefom task 2 duing t +1.Ifthefim hies those wokes to pefom task 2 at date t +1,thefim is said to pomote wokes intenally. et s i t be the numbe of wokes pomoted intenally by fim i at date t. If all wokes pefoming the second task have been hied intenally then it is said that the fim has a closed intenal labo maket with one enty pot. 2. Obseving who ae the employees that pefom the second task in othe fims in the industy, a fim can lean who ae those that developed high ability. A fim can make an offe to any of those wokes. If the woke accepts the offe, he does not need additional taining to be able to pefom the second task in his new job. The fim that employs him is said to hie wokes extenally. oweve, that employee is not as poductive as one that also has the skill but woked in the same fim when young. In paticula, I assume that e skilled wokes 2 I assume that the numbe of wokes takes values in < + so it would be moe appopiate to say that l is the measue of wokes hied by the fim. The same applies to all othe types of labo in this pape. 6

8 that change fims ae equivalent to e 1+θ, with θ > 0, skilled employees who ae pomoted intenally. et ei t be the numbe of wokes that have been tained by anothe fim and ae hied by fim i at date t. 3. Fims could also hie a woke who pefomed the fist task in anothe industy when young and sceen him in ode to lean whethe he has high ability o not. oweve, as Doeinge and Pioe [6, p. 31] note: In contast, potentially inteested outsides must fist be located and then sceened [...] The poblem of identifying the vaiables which will completely pedict a new hie s wok pefomance, howeve, is geneally viewed as eithe insoluble o soluble only at a pohibitive cost. Accodingly, I ule out this possibility and fo the est of the pape I assume that the second task is pefomed eithe by intenally pomoted wokes o by extenally hied employees. If s and e denote the numbe of the two types of wokes employed to cay out the second task, then q 2 (s, e) =s + e 1+θ denotes the level of activity of the second task. The paamete θ measues the degee of fim s specificity of the taining pocess. Geate values of θ coesponds to geate fim specificity of the skill obtained duing the taining pocess. The technology to poduce q can be witten as a function of labo in the following way: q (l, s, e; α) =q 1 (l) α q 2 (s, e) 1 α Since poduction takes time, a fim that employs (l, s, e) wokes at date t, obtains q (l, s, e; α) units of output at t +1. Finally, the demand fo the good is D(p). I assume that D has standad popeties. Assumption AD: The function D : < + < + is continuous and stictly deceasing, limd (p) = 0and p α D (p) =0 p α w1 1 α v +c 1 α >. whee the last condition ensues that demand is zeo only at pices high enough so that fims can make positive pofits hiing young and old wokes at wages w 1 and v = Max c θ, w 2ª, espectively. This assumption will ensue that the equilibium output level is not zeo. 2.1 Wokes Evey peiod t 0, a new geneation of wokes, who live fo two peiods, entes the labo foce. Wokes do not consume the good poduced by this industy. They only face uncetainty about thei ability and, theefoe, about thei wage (and consumption) when old. Wokes ae isk neutal and have pefeences ove andom bundles of the numeaie that have a discounted expected utility epesentation with discount ate 0 < β < 1. A woke who does not wok in this industy can wok in anothe industy, o at home, and obtain expected lifetime utility u = w 1 +β w 2, when young, and w 2 w 1, when old. Without loss of geneality, one can think 7

9 that w 1 and w 2 ae the expected wages of a young and old woke, espectively, in anothe industy. Wokes cannot boow fom futue wages. Theefoe, each woke consumes out of his wage and decides whee to wok to maximize his expected utility. Each fim in this industy faces an infinite supply of ex-ante identical young wokes. 2.2 Efficient Allocations Since this is a patial equilibium model, to make efficiency consideations one has to make some additional assumptions. In paticula, I assume that the consume suplus is an adequate measue of welfae and that the social oppotunity cost of woking in this industy when young and old, in tems of the numeaie, is given by w 1 and w 2, espectively, and 1 is the socially optimal discount ate. As usual, the set of efficient allocations can be chaacteized as the solution to the following Social Planne s poblem: max X 1 t 1 CS (q t) w 1 l t (w 2 + c) s t ½ qt = l s.t. t α s 1 α t s t+1 λ l t l t, s t 0 t=0 whee CS (q) R q 0 D 1 (x) dx is the Mashallian Consume Suplus. At any date t 0, thee ae only two elevant types of labo fo the planne: the young wokes who pefom task 1 and the old wokes who pefomed the fist task in this industy when young. An industy poduces efficiently if moe output cannot be poduced using at most the same amount of evey input and stictly less of one of them. Allocative efficiency holds if the aggegate suplus is maximized. et p µ α w1 α µ w2 + c 1 α 1 α and Q = D (p ). The following lemma chaacteizes the set of efficient allocations fo those paametes such that the second constaint in the Social Planne s poblem is not binding. This set of paametes gives the appopiate benchmak because in all the equilibia I analyze late the constaint does not bind eithe. w emma 2.1 If α > 1 w 1 +λ (w 2 +c) then Q is the efficient level of output while the efficient allocation of labo is l t = l and s t = s whee: l = s = µ α 1 α w2 + c 1 α Q w 1 µ 1 α α w α 1 Q w 2 + c 8

10 2.3 Fims Fims eceive a name j in the unit inteval and take the output pice sequence {p t } t=0 as given. Each fim is endowed with a 0 > 0 units of the numeaie and l 1 1 α w a 2+c 0 tainees. The lowe bound chosen fo l 1 ensues that thee is no shotage of skilled wokes at date zeo. 3 One can think that the fims have been opeating fo a while, pehaps using anothe technology based only in task 1, and know the ability of those wokes that wee employed befoe. I assume that the wokes distibution acoss fims is such that a law of lage numbes holds at each date: if fim i employs l t wokes in task 1 at date t, exactly a faction λ of these wokes develops high ability. 4 Theefoe, since taining is costly, at most λ l t wokes eceive taining at date t and ae eady to pefom task 2 at date t +1. I assume fims cannot boow in the capital maket. This may be because these fims ae ationed in the cedit maket but I do not explicitly model this phenomena. At evey date t 0, each fim chooses how much of its assets to use as financial capital to hie inputs, 0 m t a t, andwhatpattoinvestinanaltenative activity, b t = a t m t, with goss ate of etun >1. Fo the est of the pape, I take this altenative activity as lending at the inteest ate. Ifafim hies (l t,s t,e t ) wokes and invests b t in bonds at date t, then its assets at date t +1ae a t+1 = p t q t + b t whee q t q (l t,s t,e t ; α i ). At evey date t 0, each fim collects evenue and leans who ae the employees that developed high ability. In that infomation set, the fim decides how much of its assets to allocate as financial capital and how to spend it. That is, the fim chooses how many wokes to contact and what wages to offe so its expenditue does not exceed m t. The hiing pocess is descibed in detail below. Once the hiing phase has ended, poduction is caied out. Figue 1 illustates the timing of decisions. t t+1 Collects evenue fom sales made at t-1 and Obseves ability Chooses m t Spends m t to pay fo wages and taining costs Sells output iing phase Poduction phase Payments phase Figue 1. Timing of decisions 3 4 Seebelowfomoediscussionontheassumptionthatthe intenal labo maket constaint is neve binding. Since independence has no ole in this model, the agument in Feldman and Gilles [7] implies that thee exists a distibution of wokes fo which the law of lage numbes holds in evey Boel set. 9

11 Although fims ae pefectly competitive in the output maket, they ae not so in the labo maket. This is because each fim has pivate infomation about the ability of the wokes that it employed the pevious peiod. oweve, as in Waldman [13], when a fim makes an offe to a fome employee, it ealizes that othe fims in the industy may lean something about that woke s ability by obseving his job assignment and can ty to hie him. et v e t w 2 be the equilibium outside value of a woke who pefoms task 2. Any woke who pefoms the second task at date t can move to anothe fim and obtain utility v e t. 5 To simplify the discussion, I do not model the game of simultaneous offes played by the fims and those wokes that ae pomoted by its fist peiod employe. oweve, I do equie v e t to be compatible with the fims stategies in equilibium. The inteaction between each fim and the successive geneations of wokes is descibed as a game whee fims take as given both the output pice sequence as well as the outside value of a pomoted woke. In pinciple, thee is a lage set of labo contacts that a fim could offe to the young wokes. Fo example, one could imagine a contact in which a fim assigns a young woke to task 1, pays him a cetain wage at date t and pomises futue wages contingent on being pomoted o not. One could even think of a contact whee the fim details the faction of wokes that it will pomote at t+1, as in Malcomsom [9]. oweve, many contacts like these ae not implementable because eithe the fim cannot commit to take actions that ae not sequentially ational o the woke cannot commit to stay in the fim in case of eceiving a bette offe in the futue. In this wok, I estict myself to spot contacts. Assumption AC: Whenafim hies a young woke, it can neithe commit to a wage in the event that such woke is pomoted when old no to a pomotion pobability. Each fim takes as given both the sequence of output pices P = {p t } t=0 as well as the esevation utility levels V = {vt e } t=0. At evey date, the game between the fim and the successive geneations of wokes has two stages: 1 st stage: Each fim decides how much of its assets (a t ) to spend as financial capital, 0 m a t.it also decides the numbe (l, s, e) < 3 + of wokes it wants to hie and makes wage offes fo young and old wokes, (w, v) < 2 +, such that its expenditue does not exceed its financial capital. w l +(v + c) s + v e = m (1) Implicit in the financial constaint (1) is the assumption that the fim offes the same wage to all employees pefoming task 2, independently of thei past employment histoy. In pinciple, one could allow the fim to make diffeent wage offes to those pomoted intenally and those hied in the maket. As I show below, since wokes pefom task 2 only in the last peiod of thei life, then no fim has an incentive to pay to that woke 5 Notice that I defined the outside value of a woke that pefoms task 2 to be independent of his employment histoy. This seems easonable because all high ability wokes ae equally poductive when woking in any othe fim diffeent fom the one that tained them. 10

12 moe than what the maket would pay. Thus, given the assumption that all pomoted wokes ae equally poductive in a fim diffeent fom the one that tained them, the assumption of equal wage offes within the fim is made without loss of geneality to simplify notation. Each young woke is appoached by just one fim. Fo simplicity, I assume that fims adopt an up o out pomotion system: old wokes who ae not pomoted ae fied. This assumption is also made without loss of geneality because, as it will become clea late in the pape, in equilibium, no fim could make a pofit by hiing an old woke to pefom task one. Since the numbe of intenal pomotions cannot exceed the numbe of employees that developed high ability, then the fim faces the following intenal labo maket constaint: 0 s t λ l t 1 (2) If s t < λ l t 1,thenthefim decides at andom who eceives taining because, fom the fim s point of view, high ability wokes ae homogeneous. It follows that each woke hied at t 1 has an ex-ante objective pobability st l t 1 of being pomoted at t. 2 nd stage: Each young woke contacted by fim i obseves the wage offe, w i t, and decides whethe to accept (A) o eject (R) it. Those old wokes that went though the taining pocess decide whethe to stay in the fim that tained them (A) o to move to anothe one (R) whee they obtain utility v e t. Moe fomally, let d t =(l, s, e, m, b, w, v) be the quantity demanded of each facto, the financial decisions and the wages offeed by a fim at date t. etd w t {A, R} {A, R} be the date t esponses of young and old wokes and let h t =(d t,d w t ) be the actions of the playes at date t 0. 6 et h 0 =(l 1,a 0 ) be the histoy at the stat of play, h t =(h 0,h 1,...,h t 1 ) denotes the patial histoy of play up to date t 1 and h t τ the patial histoy whee the fist τ t elements ae omitted. The set of actions that a fim can choose afte histoy h t is given by: whee A h t = d <7 + : m + b = a t w l +(v + c) s + v e = m s λ l t 1 ½ pt 1 q (l a t = t 1,s t 1,e t 1 ; α)+ b t 1 if d w t 1 =(A, A) b t 1 + w t 1 l t 1 1 σ1,t 1 =R + v t 1 s t 1 1 σ2,t 1 =R othewise and 1 σk,t=r is the function that takes value 1 if σ k,t = R and zeo othewise. Theefoe, the set of all histoies up to date t is 6 t = {(h 0,...,h t 1 ):d τ A (h τ ) & d w τ {A, R} {A, R} fo all 0 τ t 1} Implicit in the desciption of the actions played at date t, h t, is the assumption that all wokes of the same geneation take the same decision. This assumption is made without loss of geneality because I only conside stationay equilibia whee wokes of the same geneation play the same histoy independent stategy against a given fim. 11

13 and the set of teminal histoies is = (h 0,h 1,...):(h 0,h 1,..., h t 1 ) t fo all t 0 ª. At date t, each young woke obseves histoy h t and decides whethe to accept o eject the wage offe he eceived. If he ejects, he woks in anothe industy with lifetime utility u. The payoff that a young woke obtains at date t is ½ w u 1 (x, w) = w 1 if x = A if x = R Each old woke who woked in the fim when young and eceived taining can stay in the fim that tained him o leave. If he stays, he obtains utility v t. oweve, he can obtain utility v e t by leaving to anothe fim. It follows that the date t payoff of an old woke who undewent taining is u 2 (x, v t,v e t )= ½ vt v e t if x = A if x = R et Γ (P, V, α) be the extensive fom game played between a fim with technology α and the infinite geneations of wokes. A stategy fo fim j specifies the numbe of wage offes it makes fo each task at date t, (l t,s t,e t ), the wages it offes, (w t,v t ) and the financial decisions, (m t,b t ), as a function of the histoy. Fomally, a pue stategy fo the fim is a sequence f = {f t } t=0 whee f t : t A h t.etf be the fim s set of pue stategies. The stategy of a woke bon at date t specifies whethe he accepts o ejects the offe made by a fim at date t and whethe he stays o moves to anothe fim at t +1afte eceiving taining. That is, the stategy of a woke bon at t is a pai σ t =(σ 1,t, σ 2,t ) whee σ 1,t : t < + {A, R} is the decision of the young woke who eceives an offe to pefom task 1 and σ 2,t : t+1 < + {A, R} is his esponse at t +1afte going though the taining pocess and being offeed pomotion by his fist peiod employe. et W t be the set of pue stategies fo the wokes bon at date t. I assume that all wokes of the same geneation play the same stategy against a given fim. Theefoe, the sequence σ = {(σ 1,t, σ 2,t )} t=0 W W 0... W t... is the collection of stategies that the infinite geneations of wokes play against a fim. Fo any (f,σ) F W,let h 0 = h 0 and h h h h t+1 = t, f t h i t, σ t t denote the actions chosen by the playes befoe date t 0, i.e.thepathofplayuptodatet. et l f t = t, s t, e t, m t, b t, w t, h v t = f t t and h σ t = σ t t be the actions chosen at date t by the fimandwokesonthepathofplayof(f,σ). Each woke decides whethe to accept o eject an offe in ode to maximize his payoff. I define the set of wages that induce wokes to accept a job at date t as: Θ σ h t (w, v) < 2 + : σ 1,t h t,w = σ 2,t 1 h t,v = A ª 12

14 Then, the payoff to the fim in the subgame that begins afte histoy h t is whee f t A h t and R f t, σ; α ht Π f,σ; α h t = X β k+1 R f t, σ; α h t a t k=0 p t q l t, m t wt l t vt e t,e v t +c t;α + b t a t if (w t,v t ) Θ σ h t bt a t + wt lt 1σ 1,t =R+vt st 1σ 2,t =R a t othewise The payoff to the young woke bon at t is U σ t,f,bσ h t,w t ( u 1 [σ 1,t,w t ]+β st+1 l t u 2 σ2,t,v t+1,vt+1 e w2 + w2 u 1 [σ 1,t,w t ]+β w 2 if bσ 1,t ( ) =A othewise whee the second line eflects that if a geneation of wokes eject woking in the fim,thenthatfim closes. Finally, I define the equilibium concept fo the game Γ (P, V, α). Since both young and old wokes take thei decisions at date t knowing only h t,w t and h t,v t, espectively, the game Γ (P, V, α) is one of impefect infomation. Theefoe, subgame pefection does not exclude the possibility that wokes follow a stategy that pescibes a suboptimal action on some infomation set out of the path of play. In paticula, it does not eliminate the possibility that fo some ε > 0, old wokes eject any wage offe below vt e + ε even though they would be stictly bette off accepting it. If fims make a pofit by hiing wokes at a wage vt e + ε, thei best esponse would be to offe that wage to the old wokes even though no othe fim is willing to pay that sum. To eliminate these equilibia, I conside only those SPNE in which no woke chooses a stictly dominated action in o out of the equilibium path. Definition 2.1 A Subgame Pefect Nash Equilibium (*SPNE) of the game Γ (P, V, α) is a pofile of stategies bf,bσ F W such that fo all t 0 and h t t 1. u 2 bσ2,t 1,v,vt e h t,v u 2 x, v, v e t h t,v fo all v 0 and x {A, R} 2. U bσ t, f,bσ b h t,w U σ t, f,bσ b h t,w fo all w 0 and σ t W t 3. Π bf,bσ; α ht Π f, bσ; α ht fo all f F. 3. INDUSTRY EQUIIBRIUM In this section, I define an Industy Equilibium. In an Industy Equilibium, fims take both the output pices as well as the esevation values of a skilled woke as given, the stategies of fims and wokes constitute a 13

15 *SPNE of Γ (P, V, α) and all elevant makets clea when fims and wokes behave accoding to the equilibium path of the *SPNE they play. In section 3.1, I intoduce the notion of pospects fo advancement and show that in any IE, ceteis paibus, one fim displays bette pospects fo advancement than anothe if and only if it pomotes a lage faction of its wokes than its competito. In section 3.2, I discuss what detemines the outside value of a pomoted woke. In the pevious section, I descibed the behavio of wokes and fims fo exogenous sequences of the output pice and the outside-value of pomoted wokes. This analysis is appopiate because each fim is competitive in the output maket and once a woke is pomoted the fim loses any monopoly powe ove him. oweve, both the output pice sequence as well as the outside value of the pomoted wokes actually depend on the aggegate behavio of the fims though the coesponding maket cleaing condition. On the one hand, the output pice, p t, evolves such that the output maket cleas evey peiod. On the othe hand, the utility that a pomoted woke can obtain by moving to anothe fim, vt e, must be consistent with the fims actions on the equilibium path of the *SPNE of the game Γ (P, V, α i ). At date zeo, afte the fims announce thei names, evey woke who is contacted by a fim updates his common pio about the stategy of that fim afte obseving the ealization of a binay sunspot vaiable that assigns pobability µ to the stategy f and 1 µ to the stategy f. To be moe pecise, the decision ule of a woke bon at date t is a mapping fom the set of fims, [0, 1],totheset σ t, σ ª t Wt W t.inanie,a measue µ (0, 1) of the fims follow stategy f while the est of the fims follow stategy f. At date zeo, the assets in hands of those fims that follow stategies f and f ae a 0 = µ a 0 and a 0 = 1 µ a 0, espectively. If µ =0o µ =1,thenallfims follow the same stategy. Fo any i {, }, qt i denotes the output poduced at date t, on the equilibium path of the *SPNE f i, σ i,byafim that follows stategy f i. Definition 3.1 An Industy Equilibium (IE) is (P, V ) < + < + togethe with stategies f i, σ i F W fo i {, } and µ [0, 1] such that P, V, f, σ, f, σ,µ ª satisfies: 1. Fo each i {, }, f i, σ i is a *SPNE of Γ (P, V, α j ) fo some j [0, 1]. 2. qt 1 µ + qt µ = D (p t ), fo all t e i t =0fo all t 0 and i {, } 4. If f i, σ i is a *SPNE of Γ (P, V, α j ) fo some i {, } and j [0, 1], then Π(f i,σ i ;α j h t ) e t all t 0 (with equality fo some i if v e t > w 2.) 0 fo If evey fim follows the same stategy, I denote the IE simply by P, V, f, σ ª. This equilibium concept does not impose the estiction that ex-ante identical fims must follow the same stategy in the game Γ (P, V, α). The behavio of fims and wokes may be heteogenous eithe because fims 14

16 have diffeent technologies o due to a coodination poblem among the infinite geneations of wokes. Fo example, if fo some sequences (P, V ) < + < + the game Γ (P, V, α) has multiple *SPNE equilibia, ex-ante identical fims may follow diffeent gowth paths. Conditions (2) - (4) efe to the quantities hied and poduced by the fims on the equilibium path of the *SPNE. This implies that unilateal deviations not only ae not pofitable but also they do not affect the equilibium pices fo output and pomoted wokes, which is consistent with the competitive hypothesis. Conditions (2) - (3) state that in an IE both the output as well as the skilled labo maket cleas. In the maket of skilled labo, the supply is given by the sum of those wokes who ae offeed a pomotion but chose to leave to anothe fim. Since taining is costly, in any *SPNE, those wokes that ae offeed pomotion eceive a wage offe that induce them to stay with thei pevious employe. ence, in an IE, the supply of extenally tained wokes is zeo. Condition (3) says that the quantity demanded of extenally tained wokes, e i t, equals the quantity supplied. Finally, the last condition guaantees that when v e t > w 2, some fim in this industy is willing to pay v e t to hie a woke pomoted by anothe fim. Notice that the deivative in condition (4) takes into account that because the fim faces a financial capital constaint, a maginal incease in e t implies a eduction on the use of some input at date t. Since wokes pefom the second task only when old, the equiement that thei stategy is pat of a *SPNE of Γ (P, V, α) implies that they accept any wage offe which is geate o equal than vt e, and eject any offe below that level. In case of indiffeence, I assume that an old woke pefes to stay in the fimwheehewoked when young. Theefoe, in any *SPNE the old wokes stategy is: bσ 2,t = ½ A if vt v e t R if v t <v e t and fo the est of the pape, I assume that {bσ 2,t } t=0 descibes the behavio of the old wokes in a *SPNE. The set of stationay, o histoy independent, stategies fo the wokes is: W = n σ W : k =1, 2 and x 0, σ k,t h t,x = σ k,t eh t,x t 0 and h t, e h t t o 3.1 Pospects fo advancement Since in the ealy stages of the evolution of an industy fims ae financially constained, those that pay lowe wages can poduce moe and obtain moe evenues to finance expansion. Theefoe, in ode to explain the outcome of industy evolution, it is impotant to identify what enables one fim to hie wokes at lowe wages than anothe. Insofa woke s abilities ae, at least to some degee, fim specific and developed by on-the-job taining, one would expect that his esevation wage depends not only on his oppotunity cost and futue wages, but also on othe factos such as his expectations about the oppotunities fo pomotion within the fim. Fo the moment, I will be athe vague and call all those elevant factos the pospects fo advancement 15

17 displayed by the fim. Although intuition suggests that pospects fo advancement depends on many factos, I believe that in this model the following definition captues the main idea: Definition 3.2 A woke believes that fim i displays bette pospect fo advancement than fim j if he is willingtowokinfim i at a lowe wage than in fim j. The elevant question is: what aspects of the fims stategies make wokes believe that one fim displays bette pospects fo advancement than anothe? I show that in an IE, ceteis paibus, onefim displays bette pospects fo advancement than anothe at date t if and only if wokes believe that the fome will pomote a lage faction of its employees than the latte at t +1. To see why, let s conside the case of a young woke bon at date t who believes that he will be pomoted at date t +1with pobability π t.etv t+1 > w 2 be the wage, o utility, he anticipates in case of being pomoted. If he eceives a wage offe w t at date t and he accepts to join the fim, his lifetime expected utility is w t + β [π t (v t+1 w 2 )+w 2 ]. Othewise, his lifetime utility is w 1 + β w 2. It is not difficult to obtain the wage offe, w (π t,v t+1 ), which makes the woke indiffeent between accepting a job at date t o not, i.e the esevation enty-wage. Clealy, w (π t,v t+1 ) must be the unique solution to the following equation in w: w + β [π t (v t+1 w 2 )+w 2 ]=w 1 + β w 2 and it follows that the esevation enty-wage is: w (π t,v t+1 )=w 1 β π t (v t+1 w 2 ) As one could expect, ceteis paibus, wokes ae willing to wok at a lowe enty-wage in fims that ae expected to pomote a lage faction of thei wokes. Fomally, emma 3.1 et P, V, f, σ, f, σ,µ ª be an IE. If v t+1 = v t+1 > w 2,thefim that follows stategy f displays bette pospects fo advancement at date t than the fim that follows f does iff π t > π t. 3.2 The outside value of a pomoted woke InanIE,anywokewhoisoffeedpomotionisfeetomovetoanothefim whee he obtains utility vt e.if young wokes follow a stategy that do not depend on the histoy of play, fims have no incentive to pay to a pomoted woke moe than his esevation utility, vt e. Theefoe, the esevation utility of a pomoted woke is detemined eithe by his wage in anothe industy o by what the fims in this industy ae willing to pay to a high ability woke tained by anothe fim. 16

18 Since any old woke can wok in anothe industy when old and obtain w 2 then v e t w 2 fo all t 0. oweve, the best option of a pomoted woke need not be to move out of the industy but to wok fo a competito of the fim that tained him. In that case, condition (3) in the definition of an IE implies that vt e must be equal to the competitos value of an extenally tained woke. If those fims pay vt e to thei intenally pomoted wokes, and wokes follow histoy independent stategies, the value of an intenally pomoted woke is at least vt e + c. Since a woke pomoted intenally is, oughly speaking, as poductive as 1+θ wokes tained by anothe fim then the value of the latte is at least ve+c t 1+θ. Theefoe, ve t ve+c t 1+θ. In the case in which vt e > w 2, condition (4) in the definition of an IE implies that vt e = ve t +c 1+θ o, equivalently, ve t = θ c. One concludes that in any IE in which wokes follow stationay stategies, vt e =max ª c θ, w 2 v.etv denotes the sequence with elements vt e = v fo all t SUBGAME PERFECT NAS EQUIIBRIUM In this section, I conside the game which descibes the inteaction between a fim with technology α and the infinite geneations of wokes, Γ (P, V, α), in isolation. I divide the analysis in two cases accoding to the value of v. Fo each case, I estict the analysis to a set of pice sequences Σ that is the natual candidate to contain an IE pice sequence and analyze the existence of a *SPNE of the game Γ (P, V, α) fo those P Σ. Fo the est of the pape I assume that young wokes follow a stationay stategy. Theefoe, fims have no incentive to offe a pomoted woke moe than what its competitos would pay. As I agued in the pevious section, in an IE the outside value of a pomoted woke must be given by the sequence V. Anticipating this, the esevation enty-wage of a young woke becomes w π t,vt+1 e = w (πt,v ) and depends on π t if and only if v > w 2. Wheneve v = w 2, the optimal stategy of the fim is the solution to a one agent poblem and, theefoe, easie to analyze than the case in which v > w 2. Since v = w 2 ifandonlyif c θ w 2,I conside fist the simplest case in which θ c w 2 and late the case θ c > w Case I: c θ w 2 = v Since young wokes eceive w 2 when old, egadless whee they wok in the futue, thei esevation entywage is w 1 no matte what the fim s pomotion policy is. Conside the following stategy fo the wokes ½ σ s A if wt w t = 1 R if w t < w 1 Given the young wokes stategy, the fim has no incentive to offe its wokes moe than w 1 when young and w 2 when old, as in a Walasian equilibium. The detemination of the optimal financial capital and the numbe of employees becomes a one agent poblem. At evey date t and patial histoy h t, {(l k,s k,e k,m k,b k )} k=t 17

19 must solve X Max β k+1 R k a k k=t w 1 l k +(w 2 + c) s k + w 2 e k = m k s.t. R k = pk q(lk,sk,ek;α)+ bk a k m k + b k = a k, a k+1 = R k a k (l k,s k,e k,m k,b k ) < 5 +, s k λ l k 1 The solution to this poblem depends, among othe things, on the sequence P. Instead of solving the poblem fo each possible sequence P, I estict myself to a set whose elements shae some natual popeties that makes them a candidate fo an IE pice sequence. Wheneve pofits ae positive, the economic intuition suggests that fims fully einvest eanings to hie inputs. Although the behavio of each fim in isolation does not affect the output pice, the decision of fully einvesting eanings, taken by all of them togethe, eventually dives the pice down. If this is so, the gowth ate of the aggegate financial capital is necessaily lage than one, which suggests that pofits must be diven down to zeo in finite time. Since the pupose of this pape is to analyze the convegence to a Walasian-like equilibia, it seems natual to conside those pice sequences in the set Σ = {P < : T such that p t <p t 1 t T and p t = p t T } of deceasing pice sequences that convege in finite time to p, the socially optimal maginal cost of the good. Fom date T on, the fim can make at most zeo pofits. Since the fim can make zeo pofits by allocating all assets to the bond, it follows that the poblem above has value 0 t T 1, {(l k,s k,e k,m k,b k )} T 1 k=t T 1 Max must solve X β k+1 R k a k + β T +1 k=t β 1 β a T fom date T on. Theefoe, fo any β 1 β a T w 1 l k +(w 2 + c) s k + w 2 e k = m k s.t. R k = pk q(lk,sk,ek;α)+ bk a k m k + b k = a k, a k+1 = R k a k (l k,s k,e k,m k,b k ) < 5 +, s k λ l k 1 (3) In geneal, the optimal stategy of the fim depends on the histoy of play not only though a t but also though l t 1, making it difficult to find a closed fom solution. oweve, the case in which c θ = w 2 is easy to analyze because the cost of poducing one unit of task 2 does not depend on whethe the fim employs wokes 18

20 pomoted intenally o wokes tained by anothe fim. This is because c θ = w 2 w 2 + c = w 2 (1 + θ). Theefoe, the fim s payoff depends only on the numbe of wokes pefoming task 2, s t + e t n 1+θ. ence, one can solve the poblem above fo l k,s t + et 1+θ,m k,b k o ignoing the intenal labo maket constaint k=t andthensets t and e t so that the constaint is satisfied. But once the labo maket constaint is not taken into account, the poblem of maximizing the intetempoal sum of discounted pofits is equivalent to a sequence of one peiod poblems. Indeed, the fim must hie wokes to maximize one peiod pofits subject to the financial capital constaint and fully allocate its assets as financial capital up to date T 1. One optimal stategy is f (T,δ; α,p), which consists in offeing wages w t = w 1 to the young wokes, v t = w 2 to the old wokes and allocates the fim s assets in the following way: s t + e t 1+θ m t = whee δ [0, 1] and b t = a t m t. a t l t = α w 1 m t δ a t Min{m T,a t } if t<t if t = T if t>t = 1 α 1 α w 2 + c m t and s t = Min w 2 + c m t, λ l t 1 Poposition 4.1 et p Σ and c θ = w 2. Fo any δ [0, 1] and T T s, [f (T,δ; α,p), σ s ] is a *SPNE of Γ (P, V, α). Ifp 0 p α 1 α w2+c w 1 λ,then ;α e t =0, Π(f,σs h t ) e t =0, q t = p m t fo all t 0, m 0 = a 0 and m t = p t p m t 1 if 1 t<t δ pt 1 p m T 1 if t = T m T if t>t (4) The uppe bound on the date zeo pice in Poposition 4.1 ensues that the intenal labo maket constaint is not binding at date zeo. Fom Poposition 4.1, it is clea that the game Γ (P, V, α) does not have a unique *SPNE. Indeed, the game Γ (P, V, α) has two souces of multiplicity. oweve, neithe of them esult in an IE whee identical fims follow diffeent gowth paths towads the steady state. I show in section 5 that maket cleaing in the output and labo maket as well as the equiement that financial capital stays constant afte date T helps to eliminate all but one of those *SPNE. Fist, since the elative cost of a pomoted woke and a woke tained by anothe fim equals thei constant maginal ate of technical substitution, then the fim is indiffeent between these two inputs. Stategy i i f (T,δ, α,p) assumes that s t = Minh 1 α w 2 +c m t, λ l t 1 but actually any s t 0,Minh 1 α w 2 +c m t, λ l t 1 is also a best esponse to the wokes stategy and implies that e t > 0. oweve, this multiplicity is only elevant when one analyzes a single fim in isolation. In any IE, instead, maket cleaing implies that no fim hies 19