Turnover, Wage Determination, and the Formation of Human Capital Espen R Moen1 and Asa Rosen2 June 30, 2000 1 Norwegian School of Management and the Ragnar Frisch Centre for Economic Research. Address for correspondence: Norwegian School of Management, Box 580, N-1302 Sandvika, Norway. e-mail address: espen.moen@bi.no. 2 Address for correspondence: Swedish Institute for Social Research, Stockholm Uni- versity, 106 91, Stockholm, Sweden. e-mail address: asa.rosen@so .su.se. Abstract We analyse a model with human capital investments and turnover. We analyse under what conditions rms have the right incentives to train their employees themselves rather than raiding them from other rms. We show that if rms ad- vertise and commit to a "life-time" wage schedule, the levels of on-the-job human capital acquisition, of turnover, and of unemployment are socially optimal. With- out such commitment ability, there is too much turnover and too little training compared with the optimal allocation. Still, the number of training rms and the amount of human capital investments are second best optimal, and subsidising human capital investment will therefore reduce welfare. The paper also extends the competitive search equilibrium model by including on-the-job search. 1 Introduction Most workers change employers several times during their careers and a sub- stantial fraction of these changes are direct job-to-job movements. For the EU countries and Canada, Boeri (1999) reports that 45-70% of the hirings are job-to- job movements, and that the yearly job-to-job ow is 6-18%. In the U.S., 20 % of the hirings are job-to-job movements (Blanchard and Diamond (1989)), and the yearly turnover rate is around 30% (Layard et at (1991)). Furthermore, wages typically increase substantially with experience, indicating that workers become more productive throughout their career. Since wages generally increase when workers changes employers, at least part of this increase in productivity must re ect improved general skills (not only rm-speci c skills). These considerations indicate that investments in training are important. Fur- thermore, as rms may di er in their ability to train workers and to utilise their human capital, turnover is important in order to achieve e ciency. The extent to which the market provides the rms with incentives to invest in general and speci c training rather than raiding other rms for quali ed workers, and the ex- tent to which the market induces turnover are therefore important determinants of economic welfare. Furthermore, as the turnover rate in the economy in u- ence the incentives to invest in training, human capital acquisition and worker turnover are interrelated issues that must be analysed simultaneously This paper analyses the conditions under which the outcome in the labour market is e cient in a setting with endogenous human capital formation and with endogenous turnover. To this end, we develop a search model with on-the-job search and endogenous search intensity, in which worker turnover is necessary in order to obtain an e cient allocation of resources. Within a competitive search equilibrium framework, we analyse to what extent rms have the correct incentives to train their workers themselves rather than to employ workers trained in other rms, and to which extent the rms that choose to train their own workers have the incentives to invest the socially optimal amount of human capital in their employees. To be more speci c, we study a sector in which a rm may enter the market as a training rm or as a raiding rm. By working in a training rm, a worker 1 obtains an increased productivity in all rms in the sector as well as (possibly) some rm-speci c skills. In contrast, raiding rms do not o er any training, general or speci c, and only hire experienced workers. On the other hand, the productivity of an experienced worker is lower in a training rm than in a raiding rm. Thus, an experienced worker working in a training rm has an incentive to do on-the-job search in order to nd a job in a raiding rm. One determinant of the amount of general human capital is the endogenous number of training rms in the market. Within this basic structure, we examine how wage determination a ects the e ciency of the equilibrium outcome, focusing mostly on the mix of raiding and training rms and on the turnover rate in the economy. Our rst result is that the equilibrium outcome is e cient when training rms can advertise and commit to long-term wage contracts, so that the wage that a worker obtains as a novice (inexperienced and with low productivity) and when he eventually becomes experienced (which happens at a stochastic rate) is determined up-front before the employment relationship starts. E ciency implies that for both raiding and training rms, the private and social gains from entry coincide, and the turnover rate is optimal. Moreover, training rms have the right incentives to provide general and rm-speci c training. As in Moen (1997), e ciency is obtained despite the presence of search frictions. This e ciency result is somewhat surprising, as it seems to be a consensus in the literature that the combination of turnover and frictions necessarily leads to underinvestments in training. This view is forcefully put forward by Ace- moglu (1997). In his model, there is too little general training even though rms and workers can write binding long-term contracts. He attributes the ine cient outcome to the workers' inability to contract with future employers. Our e - ciency result illustrates that contracts with future employers are not necessary to obtain e ciency, and that his ine ciency result arises because he let wages be determined by Nash bargaining. The fact that the equilibrium that arises when rms may advertise (and stick to) long-term wage contracts yields an e cient allocation of resources serves as a convenient starting point when introducing other imperfections than search frictions into the model, and sharpens and simplify the intuition related to various kinds of ine ciencies that may arise. We rst analyse the case where rms 2 are unable to di erentiate between wages paid to inexperienced and experienced workers. This is the situation studied in a well-known paper by Salop (1979). In this case, a (single) wage rate advertised by the training rms serve two purposes: to attract applicants in the rst place and to economise on turnover. Wages are driven up, and we get too few training rms and too many raiding rms compared to the e cient solution. Our analysis sharpens the result obtained by Salop: in his model, turnover may in uence wages only for parameter values within a certain range (which may be narrow), while in our model, wages are a ected for all parameter values. Furthermore, in contrast to our model, his framework is not suitable for welfare comparisons, because it does not specify the search technology underlying the turnover process. We show that although there are too few training rms entering the market compared to the socially optimal level, the number of training rms, and the amount they invest in each worker, is second-best optimal-Subsidising training rms will therefore actually reduce welfare. We also consider the case where training rms at the hiring stage cannot commit to the wage that a worker will earn once he is experienced. Instead the wage for experienced workers is set so as to maximise the rm's ex post pro t. Again we nd that, compared with the socially optimal level, the equilibrium turnover rate is too high and there are too few training rms and too many raiding rms. However, it now turns out that investments in rm-speci c training may be used as an e cient commitment device for the rm. When rms invest more in rm-speci c training, the optimal wage for the rm to set ex post, when the worker has become experienced, increases. That is, rm-speci c human capital is used strategically to reduce the workers' propensity to quit. Therefore, the productivity gains from rm-speci c investment are lower than the investment costs. Nonetheless, these "over-investments" are welfare improving, because high rm-speci c investment mitigates the adverse e ects of excessive turnover. The paper is organised as follows. Section 2 presents the basic model. Section 3 analyses the case where rms can commit to a "life-time" wage schedule. Section 4 considers the case where training rms cannot disentangle the wages of novices and experienced workers. Section 5 studies the case where training rms cannot commit to wages for experienced workers. 3 2 The model The main features of the model is as follows: Workers: Workers are risk neutral and identical ex ante. They enter the labour market as unemployed, and leave the market at a constant, exoge- nous rate s. New workers enter the market at the same rate so that the total number of workers is constant. Firms: There are two types of rms: training rms and raider rms. In a training rm, the productivity of a novice is yn and of an experienced t worker ye. In a raiding rm, the productivity of an experienced worker is t y r . We assume that yn < ye < y r . A novice never starts working in a t t raiding rm. Each rm hires at most one worker. Time structure: The model is set in continuous time. A worker that is hired by a training rm will stay inexperienced for a period. The natural way to model a period of time within a continuous-time framework is to let the period length be stochastic: an inexperienced worker employed in a training rm becomes experienced at a rate . Search technology: The matching technology is described more fully in the next subsection together with the equilibrium concept. For now we just state that unemployed and employed workers search in di erent submarkets, with di erent wages, and do not cause congestion for each other. Let pu and pe denote the arrival rate of job o ers to unemployed workers and to employed workers, respectively, searching with a search intensity equal to one. Employed workers choose a search intensity e, and the arrival rate of job-o ers to these workers is thus epe, We assume that the search intensity for unemployed workers and rms are constant and normalised to one. For workers doing on-the-job search, the cost of search is given by a continuous, convex function c(e), with c(0) = 0 and c0 (e) > 0, c00 (e) > 0 for e > 0. Competitive search equilibrium In the matching literature, there exist two ways to model wage competition be- 4 tween rms. In the Walrasian avoured "competitive search equilibrium", the labour market is divided into submarkets with di erent wages, and workers and rms choose which submarket to enter (Shimer, Moen (1997), and Mortensen and Wright (1998)). In the wage advertisement literature, rms announce wages and workers respond strategically when sending o applications, and the equi- librium concept thus has a game-theoretic foundation. Peters (1994) shows that the two equilibrium concepts are equivalent in a one-shot matching game with an exponential matching function. In this paper, we choose to apply the competitive search equilibrium framework. De ne a concave and constant returns to scale matching function x(u v) that maps a certain number of workers and rms searching for each other into a a ow x of new matches (search-intensities will be introduced later). If p denotes the probability rate for an unemployed worker of nding a job, it folows that p = x(u v)=u = x(1 ) = p( ). If q is the probability rate that a rm with a vacancy nds a worker, it follows that q = x(u v)=v = x(1= 1) = q( ). ~ The matching technology can thus be summarised by a function q = q( ) = ~ q (p ~ ;1 (p)) = q (p). We introduce search intensity by assuming that the worker may choose the number e of e ciency units he puts into search, at a cost c(e). The matching function is then given by x(ue v), where e denote the average number of e ciency units provided by the workers in the economy. If we let p denote the arrival rate of job o ers to a worker that provides one unit of search e ort, it still follows that q = q(p). The arrival rate of job o ers to a worker that searches with search intensity e is then pe. We will now derive the competitive search equilibrium in a general matching market, without specifying whether the market in question is a job market for employed workers or for unemployed worker. Let Y denote the expected dis- counted joint income for a worker- rm pair that is matched. If the productivity of the rm is y, and the match dissolves at an exogenous rate s, it follows that Y = y=(r + s), where r is the discount factor. Let ws denote the income to searching workers while searching, and let W s denote the expected discounted income for a searching worker. Similarly, let W e denote the expected discounted income if the worker is employed. We assume for now that the search intensity is constant and equal to one. The asset value equation for the searching worker 5 is then (r + s)W s = ws + p(W e ; W s) (1) The asset value equation for a vacancy is given by rV = q (Y ; W e ; V ) where V is the value of the vacancy. Since we assume free entry, the value of a vacancy is equal to the cost of creating a vacancy, which we denote by K . With free entry it thus follows that W e = Y ; K r+q , and thus that q (r + s)W s = ws + p(Y ; K r + q(p) ; W s) (2) q (p) In the competitive search equilibrium, all submarkets that attract searching work- ers must yield the workers exactly their equilibrium expected discounted income, which we refer to as W s: From (1) it follows that we can write W s = W s(p W e), and it thus follows that in any submarket, W s(p W e) = W s. Alternatively, we can write this as q = q(p(W e W s)). In competitive search equilibrium, rms with vacancies o er wages W s so as to maximise the value of their vacancy given that q = q(p) and that W s(p W e) = W s. De ne the maximum value of the vacancy as V (W s). Equilibrium can then be written as (omitting the upper-bar for convenience) V (W s ) = K (3) This equation uniquely determines W s, and (2) then determines p and the fact that q = q(p) then determines q. In Moen (1997), it is shown that the competitive search equilibrium also can be derived if each rm with a vacancy announces a wage W s that maximises pro t, and that the rms' beliefs (for all values of W , not just the equilibrium values) about the relationship between wages and the arrival rate of workers is given by q = q(p(W e W s)). In this paper, we will characterise the competitive search model in a somewhat di erent way. Note that the competitive search equilibrium allocation is such that V is maximised given W s , while free entry ensures that V = K . The dual problem to this maximisation problem is to maximise W s given that V = K : Lemma 1 In the competitive search equilibrium, W s is maximised given that V =K 6 Let us now derive the optimal allocation of resources. Let N s denote the number of searching worker, and let b denote an exogenous in ow of searching workers. Now suppose the productivity of the searching worker is equal to his wage ws. Then the planner wants to maximise Z 1 t r + q (p) R(N s ) = pY N s + wsN s ; pNe K ]e;rt dt 0 q (p) with respect to p, given the constraint _ Ns = b ; (s + p)N s The associated Bellman equation is given by r+q rR(N s ) = max pY N s ; N s ws + pN s K + R0 (N s )(b ; (s + p)N s )] (4) p q It follows that (r + s)R0 (N s) = ws + p(Y ; r+q K ; R0 (N s)), which is independent q of N s . Furthermore, by comparing (2) and (4) it follows that the expressions for R0 and for W s are equivalent. Now the maximisation problem in (4) can be written as max p(Y ; r + q K ; R0) p q which is equivalent to maximising R0 . But it then follows that the planner max- imises W s given by (2), that is, maximises W s given that V = K , just as in the competitive search equilibrium. Proposition 1 The following holds: a) The socially optimal allocation maximises W s given that V = K . b) The competitive search equilibrium allocation is socially e cient c) In the competitive search equilibrium, the social and private value of an additional worker entering the search market coincide 3 Equilibrium with commitment In this section, we derive and evaluate the equilibrium of the model where training rms, when advertising their vacancy, also advertise (and commit to) wages for the worker as a novice and as an experienced worker, denoted by wn and we, t t 7 respectively. We can then write the expected income of a worker employed in a rm o ering a wage schedule (wn we) as Wn(wn we), given by t t t t t (r + s)Wn = wn + (Wet ; Wn) t t t (5) where Wet denotes the expected discounted income to the worker when experi- enced, and the rate at which he becomes experienced (exogenous for now). Wn t is given by (for later reference we write it as a function of we) t (r + s)Wet(we) = we + max epe (W r ; Wet(we)) ; c(e)] t t e t (6) where W r is the expected income for a worker in a raiding rm, e the search intensity of the worker and c(e) the search cost. It follows that the worker's choice of e is given by c0(e) = p(W r ; Wet). Let Jn denote the value of a training t rm with a novice, and Jet the value of a training rm with an experienced worker. Then (r + s)Jn = yn ; wn + (Jet ; Jn) t t t t (7) For a training rm with an experienced worker we have that (r + s)Jet = ye ; we ; epeJet t t (8) We rst consider the choice of we given that Wn(wn we) Wn for some Wn. It t t t t t t follows from (5) that wn = (r + s + )Wn ; Wet. Inserted into (7) this gives t t (r + s + )Jn = yn ; (r + s + )Wn + (Jet + Wet ) t t t (9) The rm's only choice variable is we. Only the last term depends on we, and it t t follows that the rm chooses we so as to maximise Jet + Wet , the joint expected t discounted income for the worker and the rm when the worker becomes expe- rienced. The point is that we can be used to govern the search intensity of the t worker. In the appendix we show the following lemma: Lemma 2 For any Wnt , the optimal wage for experienced workers is given by t we = ye t 8 The proof is given in the appendix. However, the intuition is simple: Workers choose e so as to maximise Wet. If we 6= ye, the worker does not internalise the t t e ect of quitting on the rm's pro ts. If we < ye, on-the-job search gives rise to t t a negative externality for the current employer, and there is too much on-the-job search and turnover. On the other hand, if we > ye, on-the-job search gives rise t t to a positive externality for the rm, and there will thus be too little on-the-job search and turnover. Or, put di erently, given the overall compensation Wn to t the worker, it is optimal for the rm to induce the worker to exert the on-the-job e ort level that maximises the workers' and the rms' joint surplus, and where the gain from successful on-the-job search for the worker is a part of this surplus. The worker is induced to do this exactly when there is no externality from his on-the-job search on the rm, that is, when ye = we. t t Let Y t denote the joint expected discounted income for a training rm and its novice employee. It follows that we can write it as (r + s)Y t = yn + (Wet(ye) ; Y t ) t t (10) Here we use that rms set the wage to the experienced worker equal to his pro- ductivity ye. Similarly, we can write the joint expected income for a raiding rm t and an experienced worker employed in that rm as Y r = yr =(r + s). Let V r denote the value of a raiding vacancy, and let V t denote the value of a training vacancy. From the last section we know that in competitive search equilibrium, the income to the searching worker is maximised given that the rms break even. From (2) it thus follows that the equilibrium in the on-the-job search market solves the problem max(r + s)Wet (ye) = maxfye ; c(e ) + e pe(Y r ; r + q(p ) K ; Wet(ye))g (11) e t t t pe pe q (pe ) where pe is the arrival rate of job o ers (per unit of search intensity) in the on- the-job search market. Analogously, the equilibrium in the unemployed-search market is given by (with Wet given by (11)) max(r + s)Wet(ye) = maxfye + pu(Y t ; r + q(p ) K ; Wet (ye))g (12) u t t t pu pu q (pu ) where pu is the arrival rate of jobs in the unemployment search market. It is now easy to show the following 9 Proposition 2 Suppose yn + (Y r ; K ) > (r + s + )K . Then the equilibrium t exists It is now also easy to show that the equilibrium is e cient. First, consider the on-the-job search market. By introducing the choice of search intensity into the Bellman equation (4), it follows that the choice of search intensity in equilibrium is optimal. Since ye = we it follows that proposition 1 applies and the on-the-job t t search market is e cient. Furthermore, from the same proposition it follows that the social value of one more experienced worker is equal to the private value, Wet (ye ). But proposition 1 then tells us that the unemployed-search market is t e cient as well, and thus that the equilibrium as a whole is e cient. Proposition 3 The equilibrium de ned by (11) and (12) is socially e cient Note that wage pro le for a worker is steeper than his productivity pro le. This seems to be consistent with empirical ndings. Lazear argues that rms use steep wage pro les as an incentive device. In this paper, rms use a steep wage pro le in order to economise on turnover. Firm-speci c human-capital investments Suppose now that rms can invest in rm-speci c human capital in their workers. We assume that the rms undertake the investments when workers are novices. Investing k dollar in a novice increases the productivity of an experienced worker, and we write ye = ye(k). As the human capital is rm-speci c, yr is independent t t of k. It follows that we can write Wet = Wet (k). For simplicity, we assume that the investments in rm-speci c human capital are undertaken just before the worker becomes experienced. Since the entire return from the investments accrues to the worker, the socially optimal investment level is such that Wet0(k) = 1. Proposition 4 When rms can advertise and commit to wages, the investments in rm-speci c human capital are socially e cient The rst thing to note is that since the social and private value of getting one more experienced worker coincide, it is su cient to show that the rm will 10 set k so that Wet0 (k) = 1. The second thing to note is that for a given wage wet for experienced workers, the entire gain from rm-speci c investments accrues to the rm, thus the rm has no commitment problem when it comes to rm- speci c investments. For a given Wn, a rm chooses wages and investment in t rm-speci c capital such that the pro t is maximised given that Wn = Wn. For t t any given ye, we know that we = ye is optimal. The rm thus minimises w+++ k t t t t n r s given that Wn = Wn. Using (5) to eliminate wn then gives that the rm will t t t minimise Wn + (kr;W+(k)) , which obviously implies that Wet0 (k) = 1. t t e +s It is worth noting that the worker in e ect receives the entire gain from the rm-speci c human capital through higher wages when experienced, and therefore also in e ect pays the entire cost of the investment as a novice through a lower wage. This seems to contradict conventional wisdom that rm-speci c human capital should be (at least partially) nanced by the employer. General human capital investments Until now, all general human capital acquisition has been associated with learning by doing, in the sense that the rate at which the workers obtain human capital is exogenous. In this subsection we endogenise the rate at which the worker acquire general skills (i.e., the rate at which a novice becomes experienced).1 To this end, we assume that the rate at which a given worker becomes ex- perienced, , can be written as a function of the ow h of investments in the worker. We also assume that the rm can advertise and commit to h. The social gain associated with having one more experienced worker is given by Wet (ye ) ; (Wn + Jn ) = Wet (ye) ; Yn . The socially optimal value of h thus t t t t t solves maxh (h) Wet (ye) ; Ynt ] ; h with the rst order condition t 0 (h) W t (y t ) ; Y t ] = 1 e e n The rm, on the other hand will chose an optimal mix of h and wages for expe- rienced workers and inexperienced workers in order to minimise wage costs given that Wn = Wn. Since the rm's pro t is given by Ynt ; Wn , it follows that the rm t t t 1 We do not assume that the agents can manipulate the general human capital level for experienced workers, as this will take us into huge technical di culties. 11 will choose h so as to maximise Ynt . Now (r + s)Ynt = yn + (h)(Wet ; Ynt ) ; h. t Obviously, the rst order conditions for maximum is given by 0 (h) Wet ; Ynt ] = 1. We have thus shown the following proposition: Proposition 5 Suppose rms can advertise and commit to wages and to the amount of general human capital investments. Then the equilibrium allocation is socially e cient 4 One wage rate In the full commitment case, e ciency is obtained because rms can be compen- sated for a high wage for experienced workers by a low wage for inexperienced workers. In equilibrium, the costs K of opening a training rm is capitalised during the period where the worker is a novice. However, for several reasons this compensation form may be impossible. This may be because there is a lower bound on wages because there exists a minimum wage law, or that having a low wage to novices is costly because the workers are credit constrained. A lower bound on wages is most likely to bind if the training period is relatively short. Firms may also be unable to commit to a wage for experienced workers that is higher (or su ciently higher) than the wage for novices. We choose the latter interpretation, and thus assume that rms cannot commit to pay a higher wage for novices than for experienced workers. In this section we also assume that the parameters are such that rms, ex post (when the worker is experienced) do not want to increase the wage for experienced workers in order to economise on turnover (this assumption will be removed in the next section). When a high wage rate for experienced workers cannot be compensated for by a low wage to novices, the same wage rate (the experienced wage rate) serves two purposes, to share income between workers and rms and to adjust the incentives to do on-the-job search. However, as a single wage rate cannot serve two purposes, the equilibrium wage turns out to be an ine cient compromise between these two ends. 12 Let wt denoted the wage in the training rms, which now is independent of whether the worker is an experienced worker or a novice. It follows that we can write e = e(w), with e0 (w) < 0. Using (7) and (8) yields + pe0 wt)J t (r + s + ) @wnt = ;1 ; 1 r + s (+ ep e @J t From equations (5), (6), and the envelope theorem it follows that t (r + s + ) @Wtn = 1 + r + s + ep @w Now t t t @Jn @Wnt = @Wnt=@w t @J =@w n It thus follows that t @Jn pe0 (wt )Je t = ;1 ; (13) @Wnt r + s + ep + Since e0 (wt) < 0, the last term is positive, and it follows that @Wnn > ;1. Thus, @J t t giving one unit more to the worker in terms of wages reduces the rm's pro t with less than one unit. Training rms choose w so as to maximise V . Taking derivatives with respect to w and setting it equal to 0 thus gives Jn ; K t pe0 (wt )Je t Wn ; W u t =1+ r + s + ep + <1 (14) It thus follows that the cost to the rm of increasing expected discounted wages with one unit is less than one unit. Thus, it is cheaper to increase wages in this case than in the full-commitment case, and for given values of Jn and W u, the t equilibrium wage is higher than in the full commitment case. However, wages obviously will be set below ye,otherwise rms will never cap- t italise on K . First, it will (cet :par ) increase the workers' incentives to do on-the- job search. Second, more raiding rms will ( cet :par ) enter the market: According to lemma 1, the equilibrium in the on-the-job search market max- imises Wet (wt). The equilibrium thus solves the maximisation problem max wt ; c(e) + peee(Y r ; Wet ; r +(q(p ) K ) e e pe q pe) (15) 13 It follows that pe is decresing in wt. To see this, note that from the envelope theorem, Wet0(wt) = 1=(r + s + pe). From (15) it follows that the equilibrium value of p maximises pe(Y r ; Wet ; r+(qp(ep) ) K ) with respect to pe, and when Wet e q falls, the optimal value of pe increases as well. It is also possible to show that the search intensity is decreasing in wt. To see this, rst note that we can write Wet (wt) as r + q (pe ) (r + s) ( ) = maxfw ; c(e) + e max p (Y ; Wet t yee t pe e e r Wet ; q (p e) K ) ; Wet (ye)]g t From the envelope theorem it follows that the derivative of maxpe pe(Y r ; Wet ; r+(qp(ep) ) K ) with respect to wt is equal to ;pe =(r + s + pe) < 0. Since the e q rst order condition for e is given by c0(e) = maxpe pe(Y r ; Wet ; r+(qp(ep) ) K ), it e q thus follows that e0 (wt) < 0. We have thus showed the following proposition: Proposition 6 Suppose rms advertise one wage only, as described above. Then, relative to the socially e cient equilibrium, the following holds: 1. There is too much on-the-job search (e is too high) 2. Given the number of training rms in the market, there are too many raider rms entering the market (p is too high) It also follows that Ynt is lower in this case than with commitment, as the joint expected income for the worker and the rm falls for experienced workers. To see this, rst recall that when w = ye, J is zero, and maximising Wet is equvialent to t maximising the joint income for the experienced worker and his employer. For wt < ye it follows that t r + q (pe) (r + s) We + Je ] = w + (y ; w ) + max ;c(e) + e p (Y ; We ; q(pe) K )] ; eepeJet t t t t t e pe e e r t r + q (pe ) < y t + max ;c(e) + ee pe (Y r ; Wet ; K ; Je )] t e pe q (p e) = We (ye) t t 14 It is straight-forward to show that in the competitive search equilibrium, a fall in productivity leads to increased unemployment. In our model, the unemployment rate therefore exceeds its rst best level for two reasons: rst, there is too much turnover, which reduces the incentives for training rms to create jobs. Second, given the level of turnover, unemployment increases because a larger proportion of the joint income is allocated to the worker, thus reducing the incentives to open up vacancies in training rms. Proposition 7 In equilibrium, too few training rms enter the market Still, although the market is ine cient compared to the rst best solution, it is rst best in the following sense: suppose the planner was setting the wage in training rms (not being allowed to discriminate between experienced and inexperienced workers), while all other decisions were carried out by the market. Then the planner would set the wage equal to the equilibrium wage in training rms. To see this, note that the social value of an experienced worker entering the on-the-job search market, leaving out the interests of training rms, is given by Wet(wt). When the interests of the training rms is included, it thus follows that the social value of an experienced worker is equal to Wet(wt)+ Jet (wt). It thus follows that the equilibrium wage wt is still set so as to maximise W u given that V t = K . Thus, the training rms maximise welfare given the search behaviour of experienced workers and the entry decisions of raiding rms. Lemma 3 Suppose the planner determines the wage level in training rms (not being able to discriminate between experienced workers and novices), while all other decisions were taken by the market participants. Then the planner would set the wage in training rms equal to the equilibrium wage in training rms. By using exactly the same argument, it follows that the number of training rms entering the market, and the investment levels in general and in rm-speci c human capital (given that they can be truthfully advertised) are second-best e cient: Proposition 8 The unemployment search market is second-best e cient. Thus, the government cannot improve welfare by subsidising training rms or invest- ments in general or rm-speci c human capital. 15 It is interesting to note the di erence between our result and the result in Salop (1979). Salop derives the wage that is optimal for the rms in order to economise on turnover. If this insider wage is higher than the outside wage (the wage necessary to attract applicants), job rationing occurs in equilibrium. However, if the outside wage is higher than the insider wage, the insider wage has no in uence on the wage rate in the economy. In this paper, by contrast, the market-clearing wage has no natural de nition. When setting wages, rms trade o search costs and wage costs. If a high wage, in addition to speeding up the hiring process, also has a positive impact on worker turnover, this tilts this trade-o in the direction of higher wages. This is true independently of the costs of turnover to the rm, although the e ect in numerical terms of course is larger the more important are the costs to the rms associated with turnover. Since Salop does not model the turnover explicitly, this prevents welfare anal- ysis of his model. We model turnover explicitly, and show that the high wages paid in training rms, as well as the reduced number of training rms and the reduced human capital investments in each training rm, may be considered as an optimal respons to the imperfections caused by the assumption that rms cannot make wages contingent on experience. Finally, note that welfare can be improved by policy measures that deal di- rectly with the externalities caused by turnover. For instance, welfare can be improved if the government taxes wages in excess of wages earned in training rms (and thus only paid by workers employed in raiding rms). This will re- duce the incentives to do on-the-job search for experienced workers and for raiding rms to enter the market instead of training their own workers. 5 Ex post determination of wages In this section, we assume that the wages for experienced workers are determined ex post. This may for instance be because rms want to set a higher wage in period one than in period two in order to economise on turnover. When setting the wage for experienced workers, a rm has to take into account that lowering wages implies higher search intensity and thereby a higher quit probability rate. Let us rst derive the value of we that maximises Jet . Taking the derivative t 16 of (8) with respect to we gives t t t dJ de dJ (r + s) dwet = ;1 ; dwt pJet ; pe dwet e e e The rst order conditions for maximum is thus given by de ; dwt pJet = 1 (16) e Due to the envelope theorem, we know from (6) that dWet=dwe = ;1=(r + s + pe). t Taking the derivative of the rst order condition for e with respect to we thus t gives de c00 (e) dwet = ; r + sp+ pe which inserted into (16) gives p2 Jt = c00(e) (17) r + s + pe e It is now possible to show the following lemma: Lemma 4 For all values of p, rms set wet < yet . For su ciently low values of pe, the optimal ex post wage is below the equilibrium wage in the one-wage case, while for su ciently high Jet (su ciently large frictions in the unemployed-search market), the optimal ex post wage is above the equilibrium wage in the one-wage case As wages for experienced workers are below ye, proposition 6 still holds, and t there is excess turnover in the market. Now consider investments in rm-speci c human capital. Higher rm-speci c human capital implies that it is optimal for the rm to set a higher wage for experienced workers. From (17) it follows that we is increasing in ye. Thus, t t investments in rm-speci c human capital may be considered as a partial com- mitment device. Thus, although the rm sets the wage in the next period, the worker is willing to renege on wages in the training period, and thereby pay parts of the training costs, as the wage that the rm will set when the worker becomes experienced will increase. 17 More speci cally, from (17) it follows that the wage the rm sets can be written as we = we(k). From the envelope theorem, it follows that the return t t on H.C. investments to the rm is given by Jet0 (k) = r+es(+pe . Let k be de ned yt k) 0 by the equation Jet0 (k ) = 1, that is, the point at which the gain in terms of expected increased production value from the investments is equal to the marginal investment cost. It follows that rms set k above k in order to economise on the turnover. However, given the ine ciencies caused by rms not being able to commit to a wage contract, the rm-speci c human capital investments are optimal (since the proof is almost identical to the proofs in the last section, it is omitted): Proposition 9 Suppose the planner can determine k, but no other variables in the economy. Then the planner will set k at the same level as generated by the market. Similar e ects are identi ed by other authors: however, these authors have missed that over-investments may be an optimal response to the problem that rms are unable to commit to a wage contract. 6 Appendix Proof of lemma 2 Write Jet as a function of we and e Jet = Jet (we e). Similarly, we write Wet as a function t t of we , Wet = Wet (we ). Since e is a choice variable for the worker, Wet does not depend t t on e. By the envelope theorem it follows that @wee = ; dWee . Thus, @J t t t dwt d t @Je t de t t (J + We ) dwe e t t = @wet + @Je dwt @e + dWte dw e e t de @Je = @e dwe t From (8) it follows that (r + s) @Je = ; r+ps+e e , thus t Je t @e ep d e t dwet (Jet + Wet ) = ; r +ps Je epe e0 (we ) + t (18) 18 The rst order conditions for maximum is thus that Jet = 0, i.e., that we = ye . Since t t e0 (we ) < 0 it follows that dwt (Je + We ) > 0 for we < ye , while dwt (Je + We ) < 0 for t d t t t t d t t e e we > ye . This shows that we = ye is the optimal wage to set for experienced workers. t t t t Since the worker's search e ort is decreasing in we , it follows that the sign of the t derivative is the opposite. We know that the worker sets e such that c0 (e) = p(W r ;Wet), which inserted into (9) gives @J t pJ e t (r + s + ) @wn = ; r + s +eepe e0 (we ) t t e Since e0 (we ) < 0, it follows that Jn is increasing in we as long as Jet > 0 and decreasing t t t in we when Jet < 0. Obviously, it is optimal to set we = ye . 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