Turbulence, Training and Unemployment: Do we need higher training subsidies? Pascal Belan * Arnaud Chéron † June 26, 2009 Abstract This paper develops a model where firms invest in transferable hu- man capital and workers endowed with heterogenous abilities can loose this capital during unemployment spells as a result of turbulence. We find that an increase in the probability of experiencing human capital depreciation raises unemployment and decreases training investments. However, we show that it can be optimal to reduce training subsidies in the event of higher economic turbulence. This arises in economies where the average unemployment duration of low-skilled workers is not too much greater than that of the high-skilled. Lastly, we show the robustness of our main results with respect to the wage setting process and endogenous matching technology. * Université de Nantes (LEMNA) † Université du Maine (GAINS-TEPP) and EDHEC 1
Transcript
Turbulence, Training and Unemployment:Do we need higher training subsidies?
Pascal Belan∗ Arnaud Chéron†
June 26, 2009
Abstract
This paper develops a model where firms invest in transferable hu-man capital and workers endowed with heterogenous abilities can loosethis capital during unemployment spells as a result of turbulence. Wefind that an increase in the probability of experiencing human capitaldepreciation raises unemployment and decreases training investments.However, we show that it can be optimal to reduce training subsidiesin the event of higher economic turbulence. This arises in economieswhere the average unemployment duration of low-skilled workers is nottoo much greater than that of the high-skilled. Lastly, we show therobustness of our main results with respect to the wage setting processand endogenous matching technology.
∗Université de Nantes (LEMNA)†Université du Maine (GAINS-TEPP) and EDHEC
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1 IntroductionFrom the end of the 1990’s, Ljungqvist and Sargent (LS henceforth) haveemphasized the significant role of turbulence to explain the rise of unemploy-ment in European countries with generous unemployment benefit systems(see among others Ljungqvist and Sargent [1998, 2007]). A period of tur-bulence is characterized by an increase in the probability of loosing humancapital during an unemployment spell. This analysis relies on the followingset of observations:
• Long-tenured displaced workers experienced large and enduring earn-ing losses which can be thought of as corresponding to worker’s skilldepreciation during an unemployment spell.1
• The 1980’s was accompanied by an increase in the dispersion of earningsand the intertemporal volatility of individual earnings.
LS then consider exogenous human capital sticking partly to the workerand partly to the job, and show in frictional labor market models that theinteraction of higher skill depreciation probability with generous unemploy-ment benefit system can account for the rise in European unemployment.
Nevertheless, the kind of instruments that should be implemented in orderto improve efficiency remains an open issue. This paper aims at examiningthe optimal design of training subsidies in the context of a labor market searchmodel with turbulence. Face to skill depreciation, should we implementtraining subsidies? If yes, in the context of higher turbulence, should wesubsidy more training costs? Our answers to these two questions are: yesand not necessarily.
From a first-best perspective, the usefulness of training subsidies is obvi-ous in the context of externality and inefficiencies. At the end of the 1990’s,Acemoglu [1997], Acemoglu and Pischke [1998, 1999a, 1999b] and Acemogluand Shimer [1999] put emphasis on this point by focusing on inefficient train-ing in the context of frictional labor markets. More precisely, Acemoglu [1997]points out the possibility of an externality between the worker and his futureemployers (“poaching externality”). In competitive equilibrium, the workerobtains 100% of the increase in productivity due to training in general hu-man capital, and was therefore willing to pay the cost through wage cuts
1As emphasized by Neal [1995], subsequent earnings of displaced workers are an indi-cator of human capital surviving beyond the old match.
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(see Becker [1964]).2 On the contrary, a frictional labor market may explainthe willingness of employers to bear part of the costs of general training.Wage bargaining implies that a fraction of additional productivity obtainedfrom worker’s training goes to the firm. But training investment may alsobenefit to future employers that is, with some probability, an unknown party(the future employer) is getting a proportion of the training benefit whenthe worker is displaced. This results in underinvestment because the rentsaccruing to this third party do not feature in the calculations of worker’s cur-rent employer.3 It is then obvious that the size of these externalities shouldbe related to the probability of human capital depreciation (turbulence asdefined by LS). A first contribution of our paper is to examine the interplaybetween turbulence and poaching externalities, by focusing on the optimaldesign of training subsidies.
Beyond this, our analysis considers a new externality related to the socialunemployment gain of training. Training investments may also increase theprobability of leaving unemployment and contribute to raise steady-stateemployment, hence output. To the best of our knowledge, this issue has notyet been addressed (see Leuven [2005] for a survey). Once again, firms do nottake into account this “steady-state unemployment externality”. This resultsin lower training investments than is required by the first-best allocation.
The immediate consequence is that both externalities (poaching and steady-state unemployment) make training subsidies efficient. Nevertheless, wepoint out that the way turbulence interacts with training subsidies differaccording to the externality that plays the dominant role. In particular,despite a higher turbulence leads to increase unemployment and decreasetraining investments, it can be the case that training subsidies should bereduced.
More precisely, Section 2 of this paper develops a frictional labor mar-ket model with heterogenous workers according to observable characteristics,where firms can invest in training that brings workers’ up-to-date knowledge
2By distinguishing general and specific training, Becker [1964] emphasizes that a workershould pay for any general training that may improve his productivity in future jobs.Therefore, in a competitive framework, inefficiency in training investment is the conse-quence of incomplete contracts or credit market imperfections between the worker andhis current employer (see Becker [1964] and Grout [1984]). Contractual arrangements, asexit penalties for trained workers who quit their firm, can restore efficiency. Governmentintervention should also be mostly limited to improve loan markets; training subsidies areunnecessary.
3Actually, the equilibrium could be efficient only if the bargaining power of workers is100%.
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and raises their productivity. During unemployment spells, the worker mayloose its up-to-date knowledge, but firms cannot know ex ante who amongunemployed workers faced depreciation. The key variable is then the set ofobservable characteristics of the workers that firms agreed to train when thematch is formed. Turbulent times mean that depreciation of transferableknowledge during unemployment spell occurs with higher probability. Wefirst show that the partial equilibrium with exogenous contact rates is char-acterized by an increasing (decreasing) relationship between the probabilityof loosing human capital and unemployment rate (resp. share of trainedworkers).
Section 3 aims at presenting an efficient training system, and characterizesthe optimal subsidy rate of training cost. We show that the higher theturbulence, the lower the size of the poaching externality; for this reason,turbulence has a negative effect on the optimal subsidy rate of training costs.On the contrary, the higher the turbulence, the higher the size of steady-state unemployment externality; this second effect of turbulence increases theoptimal subsidy rate of training costs. Overall, we stress that the respectiveroles of both externalities depend on the gap between the contact rate of low-skilled and high-skilled workers. It turns out that in economies where theaverage unemployment duration of low-skilled and high-skilled workers areclose enough, it is optimal to reduce training subsidies when the turbulenceis rising.
Section 4 shows the robustness of these results with respect to the wagesetting process by comparing those that have been derived under the con-ventional Nash bargaining with what is obtained in the context of a strategicalternating wage bargaining game in the line of Hall and Milgrom [2006,2008].Additionally, we find that, if a hold-up problem arises, higher economic tur-bulence unambiguously leads to lower optimal training subsidies.
Section 5 generalizes the analysis by considering an endogenous matchingtechnology which leads to endogenous contact rates for the unemployed. Thisallows to show that both poaching and steady-state externalities do exist atthis general equilibrium level. We also find that the Hosios condition is nolonger sufficient to achieve efficiency and it remains unclear whether optimaltraining subsidies increase or decrease when the turbulence is rising.
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2 Turbulence, Training and Partial Equilibrium
2.1 Environment and labor market flows
Time is continuous. The population of workers is a continuum of unit mass.Workers look for jobs and are randomly matched with employers looking forworkers to fill vacant units of production. A productive unit is the associationof one worker and one firm. Workers are heterogeneous with respect toability a perfectly observed by the firms, over a set of characteristics whichis distributed on the interval [a, a] according to p.d.f. f (a). This constitutesthe ex ante observable component of the ability of the worker, hence of theproductivity of the job.
Firms can pay for a fixed training cost γF in order to provide up-to-dateknowledge to the worker. This rises worker’s productivity from a to (1+∆)a,with ∆ > 04 and brings for transferable (portable) job skills which can beused in any future occupation. During unemployment spell, with instanta-neous probability π the worker may loose the benefits of past training, andwill then need a new formation to recover its up-to-date knowledge whenmatched with a new job. This assumption introduces obsolescence of humancapital as a result of what Ljungqvist and Sargent [1998] have called turbu-lence. It embodies the possibility of substantial human capital destructionafter job loss (Jacobson & al. [1993], Farber [2005]). For the sake of simplic-ity, it is assumed that workers cannot accumulate skills according to tenureand experience: either the worker has up-to-date knowledge which improvesits efficiency on the job according to its ability (with an additional outputequal to ∆a), or his knowledge has became obsolete which precludes anyadditional output. Importantly, we consider that firms cannot know ex anteover the pool of the unemployed who are the workers that did (not) face thisdepreciation.
Training policy of a firm simply consists in determining a threshold abilitya above which the worker is trained if he faced human capital depreciation. Itfollows that any unemployed individual belongs to one of the three followingcategories: (1) type-0 individuals: unable for training (a ≤ a); (2) type-1individuals: able enough for training (a ≥ a), but with obsolete knowledgeor not previously trained; (3) type-2 individuals: able enough for training(a ≥ a), previously trained and still highly productive. In steady state, type-1 unemployed is always an individual whom general human capital becameobsolete.
At a first (partial equilibrium) stage, we consider exogenous heterogenous4In the context of Nash bargaining of wages, no matter who pay the cost of training if
no party can renege on the wage contract once the match is formed (no hold-up).
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contact rates for workers:
p(a) =
{p0 if a < ap for type-1 and type-2 individuals with a ≥ a
with p0 ≤ p. As will be stated in section 5 such ranking of probabilities willtypically arise in the context of endogenous matching probabilities with freeentry conditions.5 All jobs are assumed to separate at rate δ. We denoterespectively employment and unemployment levels of a-ability populationby e (a) and u (a), with e (a) = f (a) − u (a). In steady state, for type-0 workers (a ≤ a), inflow into unemployment δ (f (a)− u (a)) is equal tooutflow p0u (a), so that
u(a) = f(a)δ
p0 + δ∀a ≤ a
For a ≥ a, the population splits into type-1 and type-2 workers. Let u1(a)and u2(a) be unemployment levels, we have in steady state:
• inflow into type-1 unemployment πu2 (a) is equal to outflow pu1(a).
• inflow into type-2 unemployment δ (f (a)− u2 (a)− u1 (a)) is equal tooutflow (p+ π)u2(a).
This implies
u1(a) = f (a)δπ
(p+ π) (p+ δ); u2(a) = f (a)
δp
(p+ π) (p+ δ)(1)
so that u(a) = u1(a) + u2(a) ∀a ≥ a is defined by
u (a) = f (a)δ
p+ δ(2)
For workers of ability a ≥ a, the last expression shows that unemploymentincreases with respect to the turbulence parameter π.
Lastly, the overall unemployment rate writes
u = F (a)δ
p0 + δ+ (1− F (a))
δ
p+ δ(3)
5In particular, one should already stress that, if firms cannot ex ante observe whetheran unemployed worker has up-to-date knowledge or not, firms cannot discriminate betweentype-1 and type-2 workers. In Section 5, we provide a general equilibrium analysis whereequilibrium contact rates depend on ability a with a point of discontinuity at a. Forsimplicity here, we first only consider this discontinuity and assume homogenous contactrates within each segment.
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where F is the c.d.f. associated to f . This shows that unemployment isnegatively related to the share of workers who are eligible for training ifp > p0. The effect of turbulence on unemployment therefore depends on therelationship between a and π.
2.2 Training decision of the firm
For a firm, the intertemporal value of a filled job depends on worker’s abilityand worker’s type. We denote this value by Ji(a), i ∈ {0, 1, 2}. If the workeris of type 0 (i.e. a < a), instantaneous production is a. If the worker is of type1 or 2 (i.e. a ≥ a), training will increase his productivity and instantaneousproduction becomes (1 + ∆) a. Further, we consider that the governmentcan subsidy training at the time of job creation, by paying a fraction s of thetraining cost γF , so that the net training cost is γF (1− s).
The Bellman equations for jobs therefore write as follows:
where r is the interest rate, and wi (a), i ∈ {0, 1, 2} , the wage rate. Thetraining policy consists in determining the ability threshold a above which itis in the interest of firms to finance the training of workers who do not haveup-to-date knowledge. Let γF ≡ γF (1 − s), the threshold ability a is thendefined by
J1(a) = J0(a) + γF (6)
which implies∆a = w1(a)− w0(a) + (r + δ)γF (7)
This condition states that a firm trains a worker if the present value of theproductivity gain ∆a/ (r + δ) is, at least, as high as the present value of thewage gap plus the training costs. Since additional output ∆a increases withworkers’ ability, it can be the case that firms do not train low-ability workers.
2.3 Nash bargaining of wages
Since Shimer [2005] and Hall and Milgrom [2008], Nash-bargaining of wagesis a somewhat disputed assumption, at least from an empirical perspective.6
6According to Shimer [2005], the standard matching model does not explain the volatil-ity in the ratio of job vacancies to unemployment. In order to reconcile the model withthe data, these authors point out the role of real wage rigidity, and challenge the Nash-bargaining assumption for wage determination. For these reasons, a number of authors
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We devote a specific attention to the robustness of our results with respectto the wage setting assumption in section 4. At this stage, we consider thestandard Nash bargaining of wages.
The respective intertemporal values of employment and unemploymentare denoted by Ei (a) and Ui(a), i ∈ {0, 1, 2}. For individuals with a < a,steady-state Bellman equations write
where b ≤ a represents home production.For individuals with ability a ≥ a, Bellman equations turn out to be
rU1 (a) = b+ p (E1 (a)− U1 (a)) (10)rU2 (a) = b+ p (E2 (a)− U2 (a))− π (U2 (a)− U1 (a)) (11)rE1 (a) = w1 (a)− δ (E1 (a)− U2 (a)) (12)rE2 (a) = w2 (a)− δ (E2 (a)− U2 (a)) (13)
Let us emphasize in particular that any type-1 employed worker (who facedhuman capital obsolescence) become a type-2 unemployed in the event of jobdestruction, because his current employer has provided him with up-to-dateknowledge at the time of job creation.
Wages are the solutions of the following Nash-sharing rules:
From the wage equations, we deduce the following property.
have adopted the strategic alternating wage bargaining game, introduced by Hall and Mil-grom (see for instance Nagypal [2007] for a recent application of this wage setting gameto a search-matching model)
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Property 1. Human capital depreciation is associated with a wage differen-tial that is characterized by:
w2(a)− w1(a) = γFβ(r + δ)(r + π + p)
(r + π + βp)> 0 ∀a ≥ a
Property 1 states that holding up-to-date knowledge always brings a wagepremium. This is consistent with Ljungqvist and Sargent’s analysis whoconsider that turbulence entails substantial wage losses for workers.
Wages actually correspond to a weighted average of worker’s net contri-bution to output and reservation wages. Importantly, the reservation wagesof type-1 and type-2 workers are negatively related to the unemploymentgap U2 (a) − U1 (a) which is due to higher wage prospects (w2(a) > w1(a)),as shown by the following equality
U2 (a)− U1 (a) =p(w2(a)− w1(a))
(r + δ)(r + π + p)
Type-1 unemployed workers expect that, by having access to up-to-dateknowledge, in the future event of job destruction at rate δ, they would enterthe pool of type-2 unemployed instead of their actual type-1 position. Hencethey could earn a higher wage. This reduces their reservation wages at thetime of bargaining. In turn, type-2 unemployed workers expect that, if thebargaining process fails, they will face a risk of human capital depreciationwith probability π which accounts for a loss U2 (a)− U1 (a); this also lowerstheir reservation wages.
2.4 The impact of turbulence on training
At this stage, the (partial) labor market equilibrium is mainly characterizedby the equilibrium value of the ability threshold a. Our goal is here to showhow turbulence affects this threshold.
Proposition 1. The equilibrium ability threshold with Nash bargaining ischaracterized by:
∆a = γF
(r +
(r + π) δ
r + π + βp
)+ (a− b)
(x− x0
1− x
)(21)
When r → 0, it collapses to
∆a = γF
(δπ
π + βp
)+ (a− b) β
(p− p0
δ + βp0
)(22)
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Proof. Combining equation (7) with (17), (18) and (20) lead to equation(21).
Property 2. Consider ∆ > x−x0
1−x. A higher turbulence reduces the share of
trained workers and raises the unemployment rate.
Proof. From Proposition 1, it comes that
a =γF
(r + (r+π)δ
r+π+βp
)− b
(x−x0
1−x
)∆−
(x−x0
1−x
)Then, if ∆ > x−x0
1−x, the threshold a is unambiguously increasing with respect
to π. Moreover, equation (3) implies that the unemployment is positivelyrelated to the proportion of type-0 workers, F (a). This concludes the proof.
The assumption ∆ > x−x0
1−x(or ∆ > β
(p−p0
δ+βp0
)when r → 0) is weak but
necessary to consider an interior solution (a > b); this actually correspondsto (1− x)(1 + ∆)a > (1− x0)a which means that, once the training cost hasbeen paid, the instantaneous profit value of training has to be positive. 7
To understand Property 2, it is worth emphasizing that the equilibriumtraining rule highly depends on the expected unemployment surplus relatedto training U2(a) − U1(a). The derivation of Proposition 1 indeed relies onthe fact that the equilibrium ability threshold solves:
∆a = (r + δ) γF − δ (U2(a)− U1(a)) + (a− b)
(x− x0
1− x
)(23)
where the unemployment gap U2(a)−U1(a) is defined by (20). Then, the pointis that this expected unemployed surplus related to training is decreasingwith respect to the probability of human capital depreciation. A higherturbulence reduces the relative unemployment gain associated with up-to-date knowledge U2(a)−U1(a), because workers expect to switch more quicklyfrom type-2 to type-1 unemployed position. Therefore, turbulence increasesthe reservation wage of type-1 workers and raises the ability threshold a.Otherwise stated, turbulence discourages firms to train by increasing threatpoints of type-1 workers.8 Therefore, the fraction of untrained workers F (a)
7Since the “effective” bargaining power of workers (x) is increasing with p, and since weconsider p ≥ p0, we have x ≥ x0. Then, ∆ may be too low insomuch that the instantaneousprofit of the firm decreases with training; we rule out this possibility which would lead toa corner equilibrium without training (a = a).
8It should be emphasized that this result is consistent with an average wage thatdecreases with turbulence. Indeed, the share of untrained workers who earn the lowestwage w0(a) is increasing with π.
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who face a lower probability of exiting unemployment (p0 < p) increases withπ. From equation (3), this results in a rise of the overall unemployment rate,as Property 2 states.
On the contrary, it should also be added that the expected unemployedsurplus related to training is increasing with respect to worker’s bargainingpower β. Higher bargaining power for workers means that workers internalizea higher wage cut due to the cost of training (see equation (18)). Hence, therelative value of having up-to-date knowledge (type-2 position) is higher.
3 Turbulence, Efficient Training and OptimalSubsidy
3.1 The efficient training policy
We consider that the problem of the planner consists in maximizing thesteady-state average output value net of turn over costs by choosing the op-timal ability threshold below which workers are not trained (i.e. throughoutwe consider r → 0). This problem can be stated as follows:
maxa?
∫ a?
a
[a(f(a)− u (a)) + bu (a)] da+
∫ a
a?
[(1 + ∆) a(f(a)− u (a)) + bu (a)] da
−γFp
∫ a
a?
u1 (a) da
subject to the equilibria between inflows and outflows in unemployment, orequivalently
u(a) =
{f(a) δ
δ+p0if a < a?
f(a) δδ+p
if a ≥ a? ; u1 (a) = f(a)δπ
(p+ π) (p+ δ)if a ≥ a?
Proposition 2. The efficient training ability threshold is defined by
∆a? = γFδπ
π + p− (a? − b)
δ
δ + p0
(p− p0
p
)(24)
Proof. Straightforward by taking into account of the discontinuity of u(a) ata = a?.
Corollary 1. The efficient ability threshold is increasing with turbulence.
As in equilibrium, but for a different reason, the higher the turbulence,the higher is the efficient ability threshold. From the point of view of the
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planner, it is indeed less worthwhile to train workers because turbulenceincreases the probability of loosing training investment costs.
The efficient ability threshold depends on parameters p and p0, but in away that differs from the equilibrium perspective. For that purpose, in orderto disentangle the different externalities internalized by the planner, we focusin a first step to the two following configurations: (i) p0 = p and (ii) p0 = 0.
3.2 The poaching externality
Property 3. Consider p0 = p, s = 0 and r → 0. The efficient trainingability threshold is characterized by
∆a? = γFδπ
π + p≤ ∆a = γF
δπ
π + βp
and the equilibrium unemployment is higher than its efficient value.
This result is consistent with the findings of Acemoglu [1997]: firms donot internalize the social gain for future employers related to their own train-ing decision. Training not only increases productivity of the worker in thecurrent firm but also increases productivity in his future job if his humancapital does not depreciate during unemployment spells. Unlike firms, work-ers may internalize this value of training (higher expected wages in otherfirms). Nevertheless, since β < 1, they only get a fraction of the additionalproductivity related to training when they move to another job. Thus, therelative unemployed value of having up-to-date knowledge (U2 (a) − U1 (a))is not high enough. Consequently, the reservation wage of type-1 workers istoo high and this keeps too much people out of training. Of course, if β = 1,workers would capture the total gain related to training and equilibriumefficiency would be achieved when p0 = p.
3.3 The steady-state unemployment externality
Let us now abstract from the poaching externality by assuming that theequilibrium is characterized by β = 1 and consider that workers who havenever been trained face a lower job finding rate.
Property 4. Consider p0 = 0, β = 1 and r → 0. The efficient trainingability threshold is characterized by:
∆a? = γFδπ
π + p− (a? − b) < ∆a = γF
δπ
π + p+ (a− b)
p
δ(25)
and the equilibrium unemployment is higher than its efficient value.
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From the point of view of the planner, equation (24) already shows thata lower contact rate for type-0 workers reduces the efficient ability thresh-old a?, so that it increases the fraction of workers who should be trained.This reflects the fact that firms do not internalize the impact of training on(un)employment rate. When p0 = 0, keeping a worker of ability a out oftraining leads to a social output loss that amounts to a − b. Firms do notvaluate this loss, whereas the planner does.9 Then, obviously, the size of theexternality is greater when the difference between market and home produc-tions is large, i.e. the social cost of a rise in unemployment increases witha? − b.
In addition, from the equilibrium’s perspective, it also appears that p >p0 = 0 contributes to increase the ability threshold by raising the “effective”bargaining power of workers (x instead of x0), hence wages.10
Accordingly, from both arguments, firms underinvest in training. Thentoo many workers face a low probability to exit unemployment; the equilib-rium unemployment rate is too high as compared to first-best efficiency.
3.4 Turbulence and the optimal training subsidies
The main issue is then: what should be the optimal subsidy rate of trainingand how should it evolve according to the level of turbulence? Otherwisestated, should the subsidy rate increase or decrease when the probability ofloosing human capital during unemployment spell raises? Our point is thatthere exist two offsetting effects according to the externality we take intoconsideration, either “poaching” or “steady-state unemployment”.
Proposition 3. Consider r → 0. The optimal subsidy rate of training is:
s? = (1− β)
(p
π + p
)+
1 + β(
δ+p0
δ+βp0
)pδ
1 + ∆(
δ+p0
p−p0
)pδ
[π + βp
π + p− ∆b
δγF
(π + βp
π
)]
Proof. Propositions 1 and 2 allow to determine s? as a solution of a = a?
where γF = (1− s?)γF .
In a first step, we focus on the design of the optimal policy by payingattention to the two particular parameter specifications {β < 1, p0 = p} and
9For 0 < p0 < p, the externality consists in the fact that firms do not internalizethat untrained workers face a longer unemployment spell than those who can benefit fromup-to-date knowledge.
10As stated in the sequel, this effect vanishes in the context of alternating bargaining ofwages à la Hall-Milgrom.
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{β = 1, p0 = 0}. This allows to disentangle the two externalities at work. Amore general statement is provided at the end of the section.
Corollary 2. Consider p0 = p and r → 0. In the context of Nash bargaining,the optimal subsidy rate is:
s? = (1− β)
(p
π + p
)(26)
It is decreasing with π.
For β < 1, due to the poaching externality, it is efficient to subsidythe training costs. Corollary 2 states that the optimal subsidy rate shoulddecrease with turbulence. Both equilibrium and efficient ability thresholdsare increasing with π. But, the higher the turbulence, the lower the size ofthe poaching externality. Indeed, a tends to a? as π goes to infinity. Whenthe turbulence is high, a firm would less likely benefit from training decisionof other firms; the social return to training investment converges to its privatereturn. Ultimately, if π → ∞ training investment collapses to job-specifichuman capital, so that both equilibrium training policy and unemploymentrate are optimal and no subsidy is required. From that point of view, weneed less training subsidies in the context of high turbulence.
Corollary 3. Consider p0 = 0, β = 1 and r → 0. In the context of Nashbargaining of wages, the optimal subsidy rate is:
s? =1 + p
δ
1 + ∆
[1− ∆b
δγF
(π + p
π
)]and it is increasing with π.
Assuming p0 = 0 < p and β = 1 eliminate the poaching externality.Corollary 3 only deals with what we labelled “steady-state unemployment”externality. It turns out that a higher probability of human capital deprecia-tion (turbulence) leads to implement a higher training subsidy rate in orderto restore efficiency. Indeed, both efficient and equilibrium training abilitythresholds are still increasing with economic turbulence. But the point is nowthat the planner internalizes the fact that an increase in the ability thresholda? also raises the size of the output loss a?−b since type-0 workers face higherunemployment duration (this duration is actually infinite when p0 = 0). Onthe contrary, firms no longer internalizes this incidence of an increase in abil-ity threshold. From this perspective, the higher the turbulence the largerthe gap between equilibrium and efficient thresholds. Consequently, trainingsubsidies increases with π.
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Figure 1: Turbulence and the optimal subsidy rate
π
p0
p
πmin
ds*/dπ > 0
ds*/dπ < 0
C
p0~
p0
Therefore, the overall impact of turbulence on the subsidy rate of trainingcost is ambiguous. In response to higher turbulence, the increase in steady-state unemployment externality introduces an opposite force to the reductionof inefficiencies related to higher job-specificity of human capital. It is thusunclear whether we need more or less training subsidies in the context ofhigher economic turbulence. Figure 1 reports the sign of the marginal impactof an increase in turbulence on the optimal subsidy rate of training, accordingto the value of the contact rate for type-0 workers p0 and the initial state ofhuman capital depreciation probability π.
This figure shows that three regimes typically emerge, according to thevalue of p0:11
• If p0 < p0, then the steady-state unemployment externality is so highthat an increase in turbulence requires to increase the subsidy rate oftraining.
11We only pay attention to the case a∗ > b, which is equivalent to π > πmin where πmin
satisfies γFδπmin
πmin+p = ∆b. The curve C is the set of values of π and p0 such that dsdπ = 0.
C is decreasing, passes through the point (0, p) with an asymptote p0 = p0 for π → +∞.Finally, p0 is the value of p0 where the curve C intersects the vertical line π = πmin.
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• If p0 > p0, then the steady-state unemployment externality is so lowthat an increase in turbulence requires to reduce the subsidy rate oftraining.
• For intermediate values p0 ∈ {p0, p0}, there exists a threshold initialvalue for π (defined by C) below which a marginal increase in turbu-lence require to reduce the subsidy rate of training.
This suggests, that the gap p − p0 is crucial. Therefore, in economieswhere the average unemployment durations of low-skilled workers (1/p0) andhigh-skilled workers (1/p) are close, it is optimal to reduce the subsidy rate oftraining when economic turbulence is rising. On the contrary, in economieswhere the discrimination against low-skilled workers is high, we need moretraining subsidies in the context of higher turbulence.
4 Robustness to the Wage Setting ProcessBefore turning to the general equilibrium analysis with endogenous contactrate, it seems worthwhile to consider robustness of these former results withrespect to the wage setting rules.
4.1 Strategic alternating bargaining
In our framework, the main difference between the standard Nash bargainingassumption and the strategic alternating bargaining concept lies in the threatpoints. Following Hall and Milgrom, we now consider that during the bar-gaining process the threat point of employed workers is not the unemployedposition but the so-called delay Di (i ∈ {0, 1, 2}), which satisfies
rDi (a) = b+ δ (Ui (a)−Di (a)) ∀i = 0, 1, 2
Importantly, for type-1 workers, we assume that bargaining takes place beforethe worker’s training. Therefore, if bargaining fails, the worker goes back tothe type-1 unemployed position.
Accordingly, the sharing rules can be re-stated as follows
βJ0 (a) = (1− β) (E0 (a)−D0 (a))
β (J1 (a)− γF ) = (1− β) (E1 (a)−D1 (a))
βJ2 (a) = (1− β) (E2 (a)−D2 (a))
16
so that it is straightforward to derive the following wage equations
This deserves further discussion. Unlike the conventional Nash-bargaining,weights obtained with the strategic alternating wage setting game do not de-pend on the contact rates. The share of the surplus that goes to the workeris now lower (β < x).
Proposition 4. Consider r → 0. The equilibrium ability threshold withstrategic alternating bargaining is characterized by:
∆˜a = γFδπ
π + βp≤ ∆a (31)
Proof. Combining equation (7) with (27), (28) and (30) lead to equation(31).
Comparing equations (22) and (31) shows that the equilibrium abilitythresholds are the same under the restriction p0 = p; the optimal subsidyrate of training is then given by equation (26). Otherwise (for p > p0), thethreshold with strategic bargaining is lower than with Nash bargaining. Thisis indeed due to the fact that, under strategic bargaining, wages no longerdepend on the contact rates of the unemployed; the wages gap related to theswitch from type-0 to type-1 status is lower).
Interestingly, the ambiguity of the relationship between the subsidy rateof training and turbulence remains however unchanged. On the one hand,for p0 = p, the optimal subsidy rate, still given by (26), is decreasing withπ. On the other hand, if we revisit the optimal subsidy rate of trainingwhen only the unemployment externality exists, the subsidy rate of trainingis lower than in the case with Nash bargaining, but is still positively relatedto turbulence. Corollary 4 states this point.
17
Corollary 4. Consider p0 = 0, β = 1 and r → 0. In the context of strategicalternating bargaining the optimal rate of training subsidy is:
s?? =1
1 + ∆
[1− ∆b
δγF
(π + p
π
)]=
s?
1 + pδ
and it is increasing with π.
4.2 Bargaining with hold-up
From both Nash bargaining and strategic alternating wage setting processes,it is obvious that workers have ex-post a clear incentive to renege on theinitial wage agreement. Without enforceable contractual arrangement of thetype developed by Malcomson [1997], a hold-up problem may then arise.12In other words, we consider here that type-1 workers are able to bargain thesame earnings as type-2 workers. Since we assume homogenous contact ratep for type-1 and type-2 unemployed workers, it comes that U1 = U2.
Then, if we let denote wh(a) the wage equation for workers with abilitya ≥ a, it is now straightforward to see that:
wh(a) =
{x(1 + ∆)a+ (1− x)b if Nash bargainingβ(1 + ∆)a+ (1− β)b if strategic alternating
Proposition 5. Consider r → 0. In the context of unenforceable wage con-tracts, the equilibrium training ability threshold is characterized by
∆ah =
{γF
δ+βp1−β
+ (ah − b)β(
p−p0
δ+βp0
)if Nash bargaining
γFδ
1−βif strategic alternating
In the context of contractual incompleteness, each type-1 worker has ex-post incentives to renege on the wage contract and claims for type-2 wagelevel. Accordingly, firms face an additional wage cost that moves upward thetraining threshold: either ah > a or ah > ˜a according to the wage settingprocess we consider.
Property 5. If wage contracts are unenforceable, the share of trained workersdoes not depend on turbulence.
12This point has already been emphasized in a context where firms invest in physicalcapital by Acemoglu and Shimer [1999]. Chéron [2005] also considers the case of trainingcosts.
18
Otherwise stated, since wages are (re)negotiated after training, turbulencehas no longer impact on wages and equilibrium training rules. Furthermore,it also appears that the sign of the relationship between the optimal subsidyrate of training and turbulence is now clear cut even though p > p0.
Proposition 6. Consider r → 0 and p0 = 0. The optimal rate of trainingsubsidy is
s? =
1− ∆1+∆
(1− β)[
ππ+p
(δ
δ+βp
) (1− βp
∆δ
)+ b
δγF
]if Nash bargaining
1− ∆1+∆
(1− β)(
ππ+p
+ bδγF
)if strategic alternating
and it is decreasing with π.
An increase of the probability of human capital depreciation unambigu-ously involves a decrease of the optimal subsidy rate of training, whateverthe wage setting process we consider, and despite we are looking at a param-eter configuration where the steady-state unemployment externality is at itshighest value (with p0 = 0).
The intuition behind this result is as follows. On the one hand, theequilibrium ability threshold ah is no longer related to the probability ofhuman capital loss. This occurs because in the context of hold-up workersdo not expect wage losses in the event of human capital depreciation. Fromequation (7), the impact of turbulence on the training policy of the firm isrelated to variation in the wage gap w1(a) − w0(a). Due to hold-up, thereservation wages of type-1 workers no longer depends on the probability ofhuman capital depreciation (because workers do not expect any wage cuts),and the labor cost premium for the firm is then always at its highest value,whatever the value of π. Then, whereas in an economy without hold-up,turbulence reduces the relative value of training and increases reservationwages of type-1 workers, this result does not apply in an economy wherecontracts are unenforceable and the equilibrium ability threshold is leavedunchanged when turbulence is rising.
On the other hand, the efficient ability threshold still increases with tur-bulence, because it is less worthwhile from the planner’s point view to paythe training costs which are more likely to become unproductive. As a con-sequence, the gap between efficiency and equilibrium thresholds unambigu-ously decrease when the probability of human capital depreciation raises.The higher the turbulence, the lower the optimal subsidy rate of training.
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5 General Equilibrium with Matching and theOptimal Training Subsidy
Until now results have been derived under the assumption of exogenousmatching probabilities. Our main purpose is now to show that the rank-ing of contact probabilities we have assumed can be an equilibrium outcome.
5.1 Labor market equilibrium with Nash bargaining andendogenous matching
As a preliminary step, we should devote some time to discuss our assumptionsconcerning the segmentation of the labor market in the context of firms’ free-entry:
• First, a can be thought of as corresponding to ex ante observable het-erogeneity among workers, according for instance to diploma, which ishere assumed to be constant over time. Firms can therefore direct theirsearch according to the observable component of worker’s ability.
• Second, we consider on the contrary that firms cannot know who are theunemployed workers that has been hit by the human capital deprecia-tion shock. Otherwise stated, we assume that firms cannot discriminateex ante between workers with either up-to-date knowledge or not.
Let θ(a) denote the labor market tightness associated with ability a, andq(θ(a)) be the probability for a vacant job directed toward workers withability a to become a filled position. Expected values of a vacancies aredefined by:
where c stands for the instantaneous flow of recruitment cost, expected re-turns of a filled job Ji(a) are characterized by equations (4)-(5), and thecut-off point in ability space is still given by (6).
Free-entry conditions for each ability-segment of the labor market, V (a) =0, then allows to determine the equilibrium labor market tightness and there-fore the endogenous contact probability for the workers. Under the assump-tion of a matching function with constant returns to scale, the latter is defined
20
by p(a) ≡ θ(a)q(θ(a)). We should substitute out for this definition of thecontact probability in both expressions of the unemployment rates (equations(1)-(2)). As will be stated hereafter, one feature of this equilibrium is thatthere exists a discontinuity of p(a) around a, which is consistent with ourprevious assumption of a lower contact rate for the low-ability workers whonever been trained.
Proposition 7. Consider r → 0. The labor market equilibrium is a functionθ(a) and a threshold a characterized by:
cδ
q(θ(a))= (1− β)(a− b)− βcθ(a), ∀a ∈ [a, a[ (34)
cδ
q(θ(a))= (1− β)((1 + ∆)a− b)− βcθ(a) (35)
− γF (1− β)
(π
π + p(a)
) [δ − (p (a) + δ) βp(a)
π + βp (a)
], ∀a ∈ [a, a]
∆a = γFδπ
π + βp(a)+ (a− b) β
(p(a)− p(a−)
δ + βp(a−)
)(36)
where p(a) = θ(a)q(θ(a)).
Proof. The definition of the labor market equilibrium with endogenous match-ing probabilities is obtained from the training condition (7), Bellman equa-tions (32) and (33), and free entry conditions V (a) = 0, for any a.
Property 6. The labor market equilibrium is characterized by a discontinuityof the contact probability p(a) at a, with p(a) > p(a−) ≡ lima→a, a<a p(a).
Proof. Let us define
Ψ(θ(a)) ≡ cδ
q(θ(a))− (1− β)(a− b) + βcθ(a)
Ψ is an increasing function of θ. From (34) and (35), it comes that
limε→0
[Ψ(θ(a))−Ψ(θ(a− ε))] = (1− β)
(∆a− γF
(δπ
π + p(a)
))+(1− β)γF
(δ + p (a)
π + p(a)
) (πβp(a)
π + βp (a)
)which is positive from (36). Hence, there exists a discontinuity at a: θ(a) >θ(a−) and p(a) > p(a−). This concludes the proof.
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By posting a vacant position directed towards workers with ability a, orhigher than a, firm might be matched with a type-2 worker with up-to-dateknowledge, who allows to reach a higher productivity of the job without anyadditional cost. The possibility of hiring a type-2 worker leads to highertightness θ(a). Therefore, this provides general equilibrium foundations toour former assumption p > p0 and gives rise to the steady-state unemploy-ment externality related to the training decision of the firms.
From Proposition 7, it is clear that there is no impact of turbulence onlabor market tightness for workers with ability a < a. For workers withability a ≥ a, one can suspect that the higher the turbulence the lowerthe labor market tightness.13 To understand this, it should be first noticedthat the impact of turbulence on equilibrium labor market tightness relieson wage costs variations. The point is that an increase in turbulence raisesreservation wages of type-1 and type-2 workers by reducing the relative valueof having up-to-date knowledge when unemployed (the gap U2(a) − U1(a)).Hence, this entails an increase in w1(a) and w2(a), which in turn implies arise of the expected wage costs for the firms so that fewer firms choose topost vacancies directed toward workers with ability high enough to be trained(a ≥ a). Interestingly, this suggests that this additional mechanism (withrespect to the exogenous probability case) accounts for a decrease of the gapin probabilities p(a)− p(a−) (previously p− p0) when turbulence is rising.
5.2 Efficient allocation and policy revisited with endoge-nous matching
The problem of the planner now consists in maximizing the steady-stateoutput net of turnover costs by choosing, on the one hand, the labor markettightness for each ability segment and, on the other hand, the ability cut-off.
• For ability level a where workers are type-0, the optimal tightness θ?(a)maximizes
a
(1− δ
δ + p
)+ (b− cθ)
δ
δ + p
with respect to θ, where p = θq(θ).13A formal proof of this latter statement is not trivial, because equation (35) which can
be written as f(a, θ) = 0 does not imply in general a clear cut sign of the derivative ∂f(a,θ)∂θ .
This is actually due to the fact that the gap U2(a)−U1(a) is increasing with p, hence θ. Asa consequence, reservation wages for workers with ability a ≥ a are decreasing with labormarket tightness and this introduces an opposition force to the conventional positive wageimpact of an increase of search costs (cθ). During our discussion, we consider that thispositive wage impact of labor market tightness via search costs still dominates its negativeimpact via reservation wages.
22
• For ability level a where workers are type 1 or 2, the optimal tightnessθ?(a) maximizes
(1 + ∆)a
(1− δ
δ + p
)+
(b− cθ − γFp
π
π + p
)δ
δ + p
with respect to θ.
• The optimal cutoff a? delimits the range of ability where output ishigher with training.
Proposition 8. Let us denote ψ = 1 − θp′(θ(a))p(θ(a))
. The efficient allocation ischaracterized by:
cδ
q(θ?(a))= (1− ψ)(a− b)− ψcθ?(a) ∀a ∈ [a, a?[ (37)
cδ
q(θ?(a))= (1− ψ)((1 + ∆)a− b)− ψcθ?(a) (38)
− (1− ψ)γFπ
π + p?(a)
[δ −
(δ + p?(a)
π + p?(a)
)p?(a)
], ∀a ∈ [a?, a]
∆a? = γFδπ
π + p?(a?)− (a? − b) δ
δ + p?(a?−))
(p?(a?)− p?(a?
−)
p?(a?)
)(39)
Property 7. The Hosios condition ψ = β does not achieve efficiency.
Proof. The proof is straightforward when we consider ψ = β in Propositions7 and 8.
First, it is obvious that the equilibrium ability threshold does not cor-respond to its efficient counterpart (a 6= a?); this is due to poaching andsteady-state externalities. But importantly, even though training would beoptimal (for instance, a = a?), efficiency would still not be achieved. Indeed,with positive investment costs γF , equation (35) and equation (38) are stilldifferent. This is due to the fact that the valuation of the gap U2−U1, whichis related to training costs paid through wage cuts by type-1 unemployedwhen they come back to employment, remains lower than the correspond-ing valuation by the planner. Accordingly, reservation wages of type-1 andtype-2 workers are too high and this pushes down labor market tightness forworkers with ability a ≥ a.
Two policy tools are then required to restore equilibrium efficiency. Thetraining subsidy, at rate s, should be completed by another instrument such
23
as an employment subsidy which can take the form of a transfer to workers,µ(a), ∀a ∈ [a, a]. The transfer µ(a) is actually shared with the firm accordingto the wage bargaining process. This modifies the equilibrium allocation asfollows:
cδ
q(θ(a))= (1− β)((1 + ∆)a− b+ µ(a))− βcθ(a)
− (1− s?)γF (1− β)π
π + p(a)
[δ − (p (a) + δ) βp(a)
π + βp (a)
], ∀a ∈ [a, a]
∆a =(1− s?)γF δπ
π + βp(a)+ (a− b) β
(p(a)− p(a−)
δ + βp(a−)
)− µ(a) (40)
Then, the optimal policy {s?;µ?(a), a ∈ [a, a]} can be derived from the equal-ities a = a? and θ(a) = θ?(a). Therefore, the general equilibrium analysisdoes not change our main message: it is not clear cut whether it is optimalto increase or to reduce the subsidy rate of training when the turbulence isrising. This still holds here because the two externalities we discussed earlierdo exist in the general equilibrium framework.14
6 ConclusionThe main goal of this paper was to examine the role of training subsidies inan economy where workers face a risk of human capital depreciation. Theidea that an increase of this risk (turbulence) was the main driving forcebehind the rise of European unemployment due to its interplay with generousunemployment benefits system was recently developed by Ljungqvist andSargent. Our main result is that despite training and employment are loweredin a context of high turbulence, this does not necessarily involve that a highersubsidy rate of human capital investments (training) should be implemented.The optimal subsidy rate indeed depends on the relative size of the standardpoaching externality and what we have called the steady-state unemploymentexternality.
This work should be extended in further directions. On the one hand,one would like to embody this approach into a calibrated realistic frameworkas it has been developed by Ljungqvist and Sargent. This could allow toquantitatively determine whether it is relevant or not to increase training
14One should also have noticed that a positive optimal level of the employment subsidy(µ?(a) > 0), which arises for instance when β = ψ (the Hosios condition), contributes toreduce the optimal subsidy rate of training s? because this employment subsidy reduceswages of type-1 workers, and therefore tends to decrease the equilibrium ability threshold(see equation (40)).
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subsidies. On the other hand, we should extend our work to account for asecond-best analysis. Taking into account distortions related to welfare stateis also an important issue and it should be interesting to examine the designof training subsidies from that point of view.
25
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