Contract Enforceability and the (In)e¢ ciency of Worker Turnover Nicolas L. Jacqueta, August 2016

Abstract This paper considers the e¢ ciency properties of risk-neutral workers’ mobility decisions in an equilibrium model with search frictions, but no search externalities, and where matches are experience goods. It is shown that the e¢ ciency of workers’ mobility decisions depends on the degree of enforceability of contracts: mobility decisions are e¢ cient when contracts are enforceable, while there is too little mobility when contracts are self-enforcing. I also show that in the latter case a simple …ring tax can reestablish e¢ ciency, thereby increasing mobility. Keywords: On-the-Job Search; Bargaining; Contracts. JEL codes: J30; J63. a:

School of Economics, Singapore Management University. Email: njacquetsmuedusg. Address: 90 Stamford Road, Singapore 178903.

I would like to thank for their comments, at various stages of the development of the paper, Ken Burdett, Ricardo Lagos, Guillaume Rocheteau, Tom Sargent, Serene Tan, and Randy Wright, as well as by participants of the following seminars/conferences: NUS Macro Brownbag seminar, the Far Eastern Meeting of the Econometric Society in Taipei, the University of South Australia, the Australian National University, Monash University, the Graduate Institute of Policy Studies in Tokyo, and UC Davis. The usual disclaimer applies. An earlier version of this paper was circulated under the title "Ine¢ cient Worker Turnover." This project was funded by a research grant from the Singapore Management University (Grant number: C208/MSS5E023).

1

1

Introduction

It is well documented that the level of job-to-job transitions is large.1 However, although economists have devoted considerable attention to the e¢ ciency of the labor market, they have paid limited attention to the question of the e¢ ciency of job-to-job mobility decisions.2 Given the importance of job-to-job ‡ows, understanding whether these ‡ows are e¢ cient, and if not, knowing what the source of the ine¢ ciency is, is important as it can improve our understanding of the e¤ects of labor market policies, and therefore guide policy design to increase e¢ ciency. This paper studies a model of the labor market where worker turnover is driven by both search and information frictions: workers search randomly for jobs, all jobs are ex ante alike but di¤er ex post in their productivity, and a worker and a …rm have to match before they learn their joint productivity. I show that the degree of enforceability of contracts is crucial for the e¢ ciency of job-to-job mobility decisions of workers, with worker turnover being e¢ cient when contracts are enforceable, while it is ine¢ ciently low when contracts are self-enforcing.3 This di¤erence in e¢ ciency arises even though the contract determination mechanism ensures that mobility decisions are privately e¢ cient in both contractual environments. It is also shown that the introduction of a properly calibrated …ring tax restores e¢ ciency, thereby increasing worker turnover. One might think that mobility decisions are e¢ cient when contracts are enforceable, but not when they are self-enforcing, simply because contracts are complete in the former case, whereas they are incomplete in the latter. However, it is more subtle than that. First, even though contracts are complete for a …rm-worker pair when contracts are enforceable, agents have to get around the search frictions that prevent them from contracting with other agents they have not yet met. And to do so agents end up writing contracts that imply that the trade-o¤ faced by an employed worker with a new matching opportunity is not the same as the one the planner faces whenever it is e¢ cient for the worker to quit her current match. Second, in the absence of information frictions one gets e¢ ciency in both contractual environments, even though the di¤erence in completeness of contracts holds. The di¤erence in e¢ ciency with information friction comes from what the di¤erence in contract completeness implies for the relative bargaining positions of a poaching …rm and a worker’s current employer when competing for the worker. When contracts are enforceable, a …rm wishing to poach away a worker can o¤er 1

See Fallick and Fleischman (2004) and Nagypàl (2008). See below for a discussion of the related literature. 3 The exact de…nitions of "enforceable" and "self-enforcing" are given in the setup section. 2

2

her a contract giving her the entire expected surplus created by this job change and all potential subsequent job changes until an exogenous job destruction sends the worker back into unemployment. This is possible because a …rm-worker pair can write down a contract promising the worker more than the joint value of the match if she does not quit in order to force a poaching …rm to o¤er the worker the whole value of a new match. With self-enforcing contracts, however, even though a …rm trying to poach away a worker can also o¤er her a contract giving her the whole expected surplus of the new match, this surplus contains only a fraction of the surplus created by any future job changes happening before the match she is in is exogenously destroyed. This is because in this case a …rm cannot truthfully promise a worker more than the joint value of the match if she does not quit, so a poaching …rm does not need to o¤er the worker the whole value of the new match to attract the worker. Since a …rm wishing to keep its current worker can o¤er her the whole surplus that the match creates until it is exogenously destroyed,4 we have that an existing employer and a poaching …rm are on an equal footing when contracts are enforceable, whereas a poaching …rm is at a disadvantage when contracts are self-enforcing. If the productivity of a new match is known before the match is formed, in which case the setup is essentially a version of Kiyotaki and Lagos (2005) where …rms do not have capacity constraints so there are no congestion externalities, then mobility decisions are e¢ cient in the two contractual environments: a worker quits her job for a new one whenever the new match productivity exceeds the productivity of the current match. The degree of enforceability of contracts does not matter in this modi…ed setup because a …rm trying to attract a worker never proposes a contract with a promised continuation value in excess of the joint value of the match, since it would be sub-optimal,5 implying that the ability to credibly promise more than the joint value of the match is irrelevant. As a result a …rm currently employing a worker and a poaching …rm are on equal footing: the maximum values of employment these two …rms can o¤er a worker contain the same fraction of the surpluses created by future expected job moves. The ine¢ ciency identi…ed in this paper could be empirically important for two reasons. First, in the model the lower the bargaining power of workers, the worse the ine¢ ciency with self-enforcing contracts is. Cahuc, Postel-Vinay and Robin (2005) estimate, using a French administrative data set and a variant of the Burdett-Mortensen model where wages are determined by bargaining, that 4

All new matches being alike, if the worker chooses not to quit when given the chance, she will never do

so. 5 This is because the worker would not always quit for a more productive match. It is anyway not credible when contracts are self-enforcing.

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the bargaining power of all but one group of workers is no greater than 40 per cent and can be as low as 0 per cent for the lowest-skilled groups. Although Cahuc, Postel-Vinay and Robin’s model abstracts from learning, the fact that workers appear to be far from having all the bargaining power suggests that the welfare consequences of employment contracts enforceability uncovered in this paper could be quantitatively important. Moreover, the impact of Employment Protection Legislations (EPL) like …ring taxes on worker turnover in this paper is at odds with the existing theoretical literature, which predicts unambiguously that EPL reduce worker turnover. In fact, in this paper when contracts are self-enforcing …rms can renege on a promise. But if one puts in place an appropriately calibrated …ring tax that a …rm must pay to the government each time it …res a worker, the cost of walking away from a match will exceed the expected loss of keeping a promise. A …ring tax thus works like a commitment device for …rms and leads, all else being equal, to an increase in worker turnover. This paper could therefore help explain why results from the empirical literature are ambiguous and thus provide only mixed support for the existing theoretical literature. Related Literature - This paper is related to the small literature that studies the e¢ ciency of worker turnover in excess of job turnover. The closest paper is that of Kiyotaki and Lagos (2007). They study a model without congestion externalities where both …rms and workers can search while matched. They show that, even though mobility decisions are privately e¢ cient, they are socially ine¢ cient because of a composition externality: agents do not fully internalize the impact of their match destruction and creation decisions on other agents through their impact on the composition of match qualities. Mortensen (1978) and Gautier et al. (2010) study models with congestion externalities, while in Diamond and Maskin (1979) mobility decisions are privately ine¢ cient because of a bargaining ine¢ ciency. In Stevens (2004), the ine¢ ciency is a contractual ine¢ ciency, like in this paper, but of a di¤erent nature. She shows that, in a Burdett-Mortensen framework with no congestion or composition externalities, …rms’recruitment policies are ine¢ cient if …rms are precluded from making contract postings contingent on a worker’s employment situation. Menzio and Shi (2011) consider a competitive search model with on-the-job search where jobs can be inspection and/or experience goods, and they show that the unique equilibrium is e¢ cient. The learning literature, following Jovanovic (1979), generally abstracts from search frictions, which ensures that workers get paid their expected marginal productivity and that their job mobility decisions are e¢ cient (see Jovanovic, 1979, and Felli and Harris, 1996). In learning models with search frictions with the exception of Menzio and Shi (2011), either the assumption that workers are 4

paid their marginal product is maintained (Jovanovic, 1984), or they do not consider the e¢ ciency of mobility decisions (Moscarini, 2005; and Nagypál, 2007). In models of investments that are subject to hold-up problems (e.g., Grout, 1984; Acemo¼ glu, 1997; Masters, 1998; Acemo¼ glu and Shimer, 1999, Malcomson, 1999), the ine¢ ciency in the absence of enforceability lies in the incompleteness of contract that prevents agents from maximizing the joint surplus, just like in this paper. But in these papers the ine¢ ciency is both private and social, while in this paper mobility decisions are always privately e¢ cient. Moreover, the nature of the economic decision in this paper is di¤erent, for instance, because there is no sunk cost to be paid. Finally, this paper is also related to the dynamic contract literature, for it embeds a dynamic contract problem into an equilibrium framework. In fact, there is a continuum of principals (…rms), who can commit or not depending on whether contracts are enforceable or not, who o¤er long-term contracts to agents (workers) who cannot commit, and principals compete with each other, directly or not depending on whether a worker is in contact with one or more …rms, while the value of the outside option of an agent is stochastic (it depends on whether the worker has an outside o¤er) and is endogenous. The paper is organized as follows. The setup is presented in section 2 while section 3 characterizes the e¢ cient mobility decision. Section 4 considers the equilibrium mobility decision rule when contracts are enforceable and self-enforcing. The role of labor market policies is discussed in section 5, and I conclude in section 6. All proofs not in the main text are in Appendix A, unless indicated otherwise.

2

The Setup

In order to focus on the contractual ine¢ ciency presented in this paper, I have made a number of modelling choices meant to assume away congestion and composition externalities as well as bargaining ine¢ ciencies that have already been identi…ed in the literature. These choices are highlighted as they arise in the setup.

2.1

Agents

Time is continuous and the horizon is in…nite. There are two types of agents in the economy. There is a mass one of in…nitely-lived and homogeneous workers who maximize the expected discounted sum of ‡ow utility and who discount the future at rate r > 0. The ‡ow utility of being unemployed

5

for a worker is b

0, while the ‡ow utility of being employed at wage w is w.

There is also a mass n of pro…t-maximizing and homogeneous …rms. Firms do not have any capacity constraint, in that a …rm can hire as many workers as it wishes, and the productivity of a match with a worker is independent of the number and quality of other matches the …rm is engaged in.6 These two assumptions jointly ensure there are no composition externalities. Firms are assumed to have deep pockets so that they do not face any ‡ow of fund constraint.7

2.2

Search and Matching

Workers’search is random, and both unemployed and employed workers search at no cost and with the same e¢ ciency: workers contact …rms according to a Poisson process with parameter

> 0.

Hence, there are no congestion externalities. It is further assumed that the productivity of a new match x is (i) initially unknown, (ii) drawn from a known distribution with cdf F on the support X = [x; x], x

b, (iii) learnt perfectly by the worker and the …rm as soon as the matched is formed,

and (iv) …xed thereafter. There are two types of match destructions in the model. First, matches are destroyed by some exogenous shock that follows an exponential distribution with parameter

> 0. In addition,

matches are endogenously severed when an employed worker leaves her current match for a match with a new …rm. It is assumed that x > b, that there is no cost of search, and that there is no di¤erence in the e¢ ciency of search while employed and unemployed, implying it is always optimal and e¢ cient for an unemployed worker to match. The only decision in the model is thus about whether a match between a …rm and a worker contacted by another …rm should be severed so that the worker can join the new …rm. Let the mobility decision rule followed by a worker in a match of quality x be indicated by

x

2 [0; 1],8 where

x

= y indicates a worker employed in a match with productivity

x moves to a new job with probability y. Denoting the date t mass of unemployed workers and distribution of quality of existing matches respectively by ut and Gt respectively, we have that 6

In general …rms make pro…ts, but I will ignore the dividends associated with these pro…ts. This is inconsequential for the mobility decision of workers. 7 This is not unlike the Diamond-Mortensen-Pissarides search and matching model where a …rm pays for the cost of posting a vacancy before it generates revenue to cover this cost. Moreover, in equilibrium some …rms need to post a vacancy for such a long time before …lling their job that the expected value of the job net of the cost of posting the vacancy is negative. 8 Note that I consider only stationary policies, but this is because the optimal policy is stationary. In fact, the productivity of a match is …xed once the match is formed, the productivity of new matches is drawn from a constant distribution, and …rms have unlimited capacity, so a worker’s allocation does not matter for other workers. This means that if it is optimal to quit a match with a given productivity to form a new match at some point in time, then it is optimal to do so at any other time.

6

ut = (1 and Gt (x) = F (x)

ut 1

ut

+

Z

x0 2X

x0

ut )

dGt (x0 )

ut ,

(1)

Gt (x) +

Z

x

x

0 x0 dGt (x )

.

(2)

Equation (1) is the law of motion for unemployment, where ut is the time derivative of unemployment: given that there is a mass 1

ut of employed workers and a mass ut are unemployed, a mass

(1 ut ) of employed workers become unemployed because their matches are exogenously destroyed, while a mass ut of unemployed workers meet a …rm and therefore become employed. Equation (2), which characterizes the evolution of the distribution of productivity of existing matches, highlights that G is an endogenous object which depends on the mobility decisions of employed workers. The …rst two terms on the right-hand side capture the in‡ow of workers into matches with productivity no more than x: unemployed and employed workers meet …rms at rate , and if a worker forms a new match then the productivity is no more than x with probability F (x); furthermore, unemployed workers always accept to form a match, and therefore the ‡ow of unemployed worker into matches of quality no more than x is F (x)u, while workers employed in a match with productivity x0 accept with probability R is (1 u) x0 2X

x0

so that the ‡ow of employed worker into matches of quality no more than x

0 x0 dG(x ).

The last two terms capture the out‡ow: workers employed in matches

with productivity no more than x see their matches being exogenously destroyed at rate , so the ‡ow out of matches of quality no more than x caused by exogenous destructions is (1

u) Gt (x),

while the ‡ow of workers employed in matches with quality no more than x who quit their current Rx job to create a new match is (1 u) x x0 dG(x0 ). A few remarks are in order. Workers’search e¢ cacy, as captured by the parameter , is assumed

not to depend on their employment status. It is possible to introduce di¤erentiated arrival rates of meetings for unemployed and employed workers without changing the main results. It is also possible to assume that x < b, but the extra “risk” of unemployment associated with changing jobs in this environment actually clouds out the real source of ine¢ ciency. Finally, the resolution of uncertainty regarding the productivity of a new match is purposely kept as simple as possible by assuming that the productivity is revealed right after a match is formed.9 The results I obtain 9

Wright (1986) makes the same assumption. It would be posible to extend the analysis to introduce a signal of match quality before the decision to match is made. There would then exist a threshold for the signal for each productivity: workers employed in a job with productivity x would change jobs only if the signal obtained is above some value p (x), with p(x) increasing in x. See Pries and Rogerson (2005) for a model with such signals.

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would be left unchanged if instead it were assumed that either (i) all matches start with the same productivity but are eventually hit by a permanent productivity shock; or (ii) productivity were to be learnt gradually over the match tenure. But these alternative assumptions would complicate the algebra without bringing any additional insight.

2.3

Employment Contracts and Surplus Sharing Rules

Contractual Environments – I consider two types of contractual environments, where the distinction between the two lies in the degree of enforceability of contracts. It is assumed in both cases that the contract governing a match cannot be changed unless both agents agree to the change.10 In the …rst environment …rms are not allowed to unilaterally terminate a match, but workers can do so, while in the second environment both workers and …rms can terminate a match unilaterally. Contracts in the …rst environment are said to be enforceable, while it is said that in the second environment contracts must be self-enforcing.11 Contracts – Even though a …rm and a worker can renegotiate a contract at any time if both agree to, I focus my attention on contracts that do not need to be renegotiated, including when a worker gets a chance to quit her existing match for a new one. This amounts to focusing on contracts that are: (i) such that neither the worker nor the …rm have an incentive to walk away from the contract if they have the option to; and (ii) privately optimal, in that it is not possible to increase the value of the match to one agent without reducing the value of the match to the other agent. I call such contracts renegotiation-proof contracts. I follow the dynamic contract theory literature and adopt a recursive representation of the contracting problem. A worker and a …rm can bargain, according to the bargaining protocol presented below, either when they …rst meet, in which case the productivity of a match is not yet known, or when they have already been matched for some time, in which case the productivity of the match is known. In the …rst instance a contract is a list f ; (Wx )x2X g where

is a lump-sum

paid by the …rm to the worker as soon as the match is formed and Wx is the starting value of employment, net of the initial lump-sum, that the …rm promises to the worker if the quality of the match turns out to be x. In the second instance a contract simply speci…es a continuation value of 10

I present brie‡y in Appendix B how the analysis changes when agents can force their partner to renegotiate the contract constantly, very much like in the standard Diamond-Mortensen-Pissarides framework. 11 One could alternatively say that in the …rst environment there is one-sided commitment (on the …rm side) and that there is two-sided lack of commitment in the second environment. However, the lack of commitment here is about whether to stay in a match, but workers and …rms cannot unilaterally change a contract. For this reason I adopt a di¤erent terminology from the one adopted in the dynamic contract literature.

8

employment W the …rm promises the worker should she decide to stay. Once a …rm and a worker have decided to stay or become matched, and once the productivity x is drawn if it was not yet known, the …rm o¤ers to the worker a contract that maximizes its own value of the match given the value promised to the worker, and given the relevant participation constraints. At this stage a contract is constituted of (i) a …xed wage w the worker receives from f the …rm the …rm until she receives a new employment opportunity, and (ii) a continuation value W promises to the worker if she meets another …rm and decides not to quit.12

Since I focus on renegotiation-proof contracts, a contract for a new match is contingent only on the continuation value of the worker in her previous job,13 if any, i.e., for a worker previously emf a contingent contract is f (W); f (Wx (W)) f x2X g; ployed in a match with promised continuation value W and a contract proposed by a …rm after a match has been formed and the productivity drawn is

x when the initial value of employment promised is W depends on x and W, so that the contract

fx (W)g. can be written as fwx (W); W

Note that I do not include termination or quitting fees. This is to facilitate the comparison be-

tween the two contractual environments, because such fees are not enforceable when contracts must be self-enforcing and they are not necessary when contracts are enforceable. I could also consider tenure-dependent contracts and additional lump-sum payments, but it will be shown soon that the restrictions I impose are immaterial for mobility decisions. I moreover assume, for simplicity, that contracts only include the value of transfers within a match and therefore that they do not specify whether a worker should quit a match for a new one when she meets another …rm. However, this is also immaterial, for a mobility rule is implicitly de…ned by the value of transfers. Contract Determination Mechanism and Privately E¢ cient Mobility Rules - Any contract determination mechanism is arbitrary to some extent. Moreover, I want to make as clear as possible the role of contractual issues for the new ine¢ ciency of mobility decisions identi…ed in this paper. I therefore assume that the mechanism through which contracts are determined is a simple bargaining game which induces privately e¢ cient mobility decisions, by which I mean that a matched worker who has another matching opportunity always chooses the match with the greater private joint value. 12 Alternatively, I could model the contracting problem in a more traditional way with the …rm o¤ering at each date a spot contract, with a spot wage and a promised continuation value. However, the only potential instance when there might be a need to change a contract in the context of this paper is when the worker contacts another …rm. 13 So a contingent contract does not directly depend on the productivity of the previous match of the worker, although it can indirectly depend on x through the promised continuation value in that previous match.

9

The bargaining protocol followed by a worker, either employed or unemployed, and one or two …rms,14 is as follows. Bargaining Protocol: 1. In the …rst stage …rms propose a contract to the worker. a. If the worker accepts a contract, bargaining is over; b. If the worker turns down all o¤ ers, she keeps her existing contract, if any, and the bargaining process reaches the second stage. 2. In the second stage the worker gets to propose a contract with probability get to propose with probability 1

2 (0; 1) and …rms

. In the latter case if the worker is employed then …rms enter

a Bertrand competition for the worker.15 a. If the worker and one …rm agree on a contract, this contract agreed on governs the match between the two agents. b. If agreement is reached with two …rms, then the worker picks her preferred contract. c. Otherwise the worker keeps her previous contract, if any, or stays unemployed. Although the game is simple, I focus on the outcome of the game and I therefore do not formally describe the strategies followed by each agent. And, in order to keep the exposition as simple as possible, I assume the following tie breaking rules: (i) if the private joint value of an existing match is equal to the private joint value of a new match, the worker quits her match to start a new one; (ii) if the value of a match to a …rm is zero, then it accepts the o¤er.

2.4

Value Functions

As a …rst step I present the value functions assuming that all new contracts are equilibrium contracts, and that any existing contract is renegotiation proof. Unless speci…ed otherwise I treat an unemployed worker as being a worker employed in a match of quality b, with the joint value of the match and promised continuation value promised to the worker being both equal to the value of unemployment, U.

f denote the equilibrium expected value of a new match, as determined by the barLet W e (W)

14 Because the probability of meeting two …rms at the same time is zero, an unemployed worker will almost surely negotiate with only one …rm at a time. I therefore ignore the possibility of tripartite bargaining involving an unemployed worker. However, employed workers can be in contact with other …rms, in which case the bargaining will involve three parties, ignoring once again measure zero events where a worker is in contact with two or more new …rms at the same time. 15 The result would not change if the stochastic processes giving the right to make an o¤er were uncorrelated for the two …rms. This is because the second stage of the game is never reached in equilibrium, and thus only the expected value of going to the second stage matters for the outcome of the game.

10

gaining game and inclusive of the initial lump-sum fee, for a worker currently employed in a match R f i.e., W e (W) f f with promised continuation value W, + Wx (W)dF (x). The value of unemploy-

ment, U, is then such that

rU = b + (W e (U)

U).

(3)

f does not The value for a worker of being employed in a match of quality x with contract (w; W) f and is such that directly depend on x, is denoted W (w; W), f = w rW (w; W)

+

f (W (w; W)

U)

e f f (W)(W (W)

f + (1 W (w; W))

(4) f W f (W))(

f W (w; W)),

f is again an indicator function equal to 1 if the worker chooses to quit her job when a where (W) new match opportunity arises and 0 otherwise.16

f the value of a job of quality x for a …rm when the contract with the worker is (w; W), f Jx (w; W),

is such that

f = x rJx (w; W)

w

+ (1

( +

f x (w; W) f (W))J

f f (W))(J x (W)

(5)

f Jx (w; W)),

f denotes the optimal value for a …rm of a match with productivity x when the utility where Jx (W)

f That is, promised to the worker is W.

f Jx (W) = max Jx (w; W),

(6)

f fw;Wg

subject to the promise-keeping constraint

f =W W (w; W)

and the relevant participation constraints.

f denote the joint value of a It is also useful to de…ne the joint value of a match. Let Vx (W)

f i.e., Vx (W) f match with quality x when the contract’s continuation value for the worker is W, f + W (w; W). f By de…nition Vx (W) f is such that Jx (w; W) 16

f =x rVx (W)

f (Vx (W)

U)

e f f (W)(W (W)

f Vx (W)).

(7)

This expression again assumes that contracts are renegotiation-proof: the value of staying in her current f the expected value of a new match can depend on W, f but not directly on match for the worker is indeed W; f x, although it might depend indirectly in x through W; and the mobility decision is assumed to depend on f but not directly on x. The upshot is that the value of a match for a worker depends only on the terms W of the contract and not at all on the quality of the match, at least not directly.

11

We can similarly de…ne Vx (W) as the optimal joint value of a match with productivity x given the value promised to the worker is W, i.e., Vx (W)

Jx (W) + W.

(8)

Using the de…nitions of Jx (W) and W we have that f + W (w; W)g, f Vx (W) = max fJx (w; W) f fw;Wg

f and subject to the relevant participation constraints, which, given the expression for Jx (w; W)

f in (4) and (5), yields that W (w; W)

f Vx (W) = max Vx (W), f W

(9)

subject to the relevant participation constraints. That is, the optimal joint value of a match is independent of the initial value of the match promised to the worker and will from now on be denoted simply as Vx . We then have that the optimal expected joint value of a new match is V

2.5

e

Z

Vx dF (x).

(10)

Equilibrium De…nition

Let Wx be the set of feasible employment values for workers given the degree of enforceability of contracts. The set is restricted by the ability of agents to walk away from a match, which holds in both contractual environments for workers and for …rms when contracts are self-enforcing. More speci…cally, the value of employment for a worker must be no less than the value of being unemployed at all times and for all possible productivity levels in both contractual environments, while for …rms there are no constraints on the value of a match if contracts are enforceable, whereas the value of a match must be no less than 0 at all times and for all possible productivity levels when contracts are self-enforcing. We then have the following de…nition. De…nition: A steady-state symmetric equilibrium for a given Wx is an allocation fx )x2X ; u; Gg such that: fU; W; (Jx ; Vx )x2X ; ; ; (Wx ; Jx ; Vx ; wx ; W

fx )x2X g, U, W , and (Jx ; Vx )x2X solve (3)-(5) 1. Value Functions: Given f ; ; (Wx ; Jx ; Vx ; wx ; W

and (7);

2. Bargaining and Mobility: For each x 2 X [fbg, given Vx and V e and a promised continuation

f 2 Wx , W f and f (W); f (Wx (W)) f x2X g are subgame perfect equilibrium contract values of value W 12

f the bargaining game o¤ered to the worker by her current and poaching …rm respectively, and (W)

is the implied mobility decision rule;

3. Optimal Contracts: for each x 2 X and each initial promised employment value W 2 Wx ,

fx g solve (6); fJx (W); wx ; W

4. Demographics: u and G are as given in (1) and (2) with u = 0 and G = 0 and for

for all x 2 X.

x

fx ) = (W

Note that J captures the present expected value of pro…ts of a …rm from a match, which is all is needed given it is assumed there are no ‡ow of funds constraints. Note also that all equilibrium objects except for u and G are independent of whether we consider a steady-state.

3

The E¢ cient Mobility Decision Rule

In this section I characterize the constrained socially e¢ cient allocation of workers, where the criterion used for e¢ ciency is the maximization of the discounted sum of aggregate output augmented by the discounted sum of the value of leisure for unemployed workers.17 Formally, the planner chooses a mobility decision rule ( Z

1

e

sp x )x2X rt

to maximize

(1

ut )

Z

xdGt (x) + ut b dt,

(11)

x2X

0

subject to (1) and (2), and given initial conditions u0 and G0 . Let ! u and ! x respectively denote the shadow values of an unemployed worker and of a match with productivity x, and let ! e R x2X ! x dF (x). We then have the following lemma. Lemma 1 The necessary conditions to the planner’s problem include 8 > > = 1, if ! e > ! x , > < sp 2 [0; 1], if ! e = ! x , x > > > : =0 otherwise.

(12)

and the Euler equations

r! x

!x = x +

sp e x (!

for all x 2 X, and r! u 17

!u = b +

!x)

(! e

(! x

!u) ,

!u) .

Workers are risk neutral and therefore this is what a utilitarian planner would do.

13

(13)

(14)

The e¢ cient mobility policy requires that a worker employed in a match with productivity x quits her job for sure to form a new match for sure if and only if the expected shadow value of a new match ! e strictly exceeds the shadow value of the current match ! x . The social planner is indi¤erent between

sp x

equal to zero or one for ! e = ! x , and I will, in order to facilitate the

comparison with the equilibrium mobility decisions, use as a tie-breaking rule that says the worker should accept a new match if indi¤erent. It is clear that ! x is strictly increasing in x, and therefore the e¢ cient mobility decision is characterized by a reservation strategy with cut-o¤ productivity xsp and mobility decision such that 8 < = 1 , for x xsp ; sp x : = 0 , for x > xsp .

(15)

The expressions for the shadow values are intuitive. Consider an unemployed worker. She enjoys an instantaneous ‡ow value b, she encounters a …rm at rate , and the expected capital gains arising from the formation of a new match are ! e

! u , and therefore r! u , the ‡ow shadow value of an

unemployed worker, is given by (14). Consider now a worker matched with productivity x. The ‡ow product is x as long as the match lasts. When the match is destroyed, the gross capital loss is ! x . If the match destruction is exogenous, then the worker becomes unemployed and the net capital loss is ! x

! u . If, however, the match has been destroyed because the worker leaves for another …rm,

the expected net capital gain is then ! e ! x . Hence, the shadow value of a match with productivity x is given by (13). Note that none of the shadow values depend on the endogenous distribution of match quality G, which con…rms that there are no composition externalities of the kind found in Kiyotaki and Lagos (2007), and any deviation of the shadow values from their steady-state values would not be driven by intrinsic changes to the labor market. Given that the equilibrium mobility decision rule is always stationary, I focus my attention on the steady-state shadow values and the associated stationary e¢ cient mobility decision rule to facilitate the comparison between the equilibrium and e¢ cient mobility decision rules. Equation (13) and the mobility policy (15) yield that the steady-state shadow value of a match with productivity x is !x = Let xe

R

x2X

8 > > < > > :

x+ ! u + ! e r+ +

,

if x

x+ ! u r+

,

otherwise.

xsp , and

xdF (x). Since the e¢ cient mobility cut-o¤ is such that ! xsp = ! e we have the

following proposition. 14

Proposition 1 There exists a unique steady-state e¢ cient mobility cut-o¤ xsp and it is such that Z x sp e (x xsp ) dF (x). (16) x =x + r+ xsp We thus obtain that the e¢ cient mobility productivity cuto¤ xsp exceeds the expected productivity of a new match xe . The di¤erence between xe and xsp in this setup comes from an option value of search that appears when the quality of a new match is uncertain. To see this, consider …rst a worker in a match of quality xe . If she never quits she generates xe per unit of time until the match is destroyed exogenously. If the planner instead instructs the worker to quit all matches with productivity no more than, say, xe , the new expected productivity of the next match is xe , so at this point there is no gain. However, if the next draw falls below xe , the mobility rule instructs the worker to change job again, and so on until the worker draws a productivity greater than xe . Hence, if a worker quits all matches with productivity no more than xe , then her e¤ective expected productivity over the time spell until her match is exogenously destroyed is strictly greater than xe . However, the cuto¤ xsp is by de…nition such that the present value (before the next unemployment spell) of the output in a match with productivity xsp is equal to the expected present value of output the worker generates by changing job whenever her match productivity is no more than xsp , so we obtain that xsp > xe . For the sake of completeness we can use (1) and (2), the mobility decision rule (15) for xsp as given in (16), and the tie-breaking rule mentioned above to derive the steady-state demographic characteristics of the model. We have that the steady-state unemployment rate is u=

+

,

and the steady-state distribution of match productivities among employed worker is 8 < , for x 2 [x; xsp ], and + (1 F (x)) F (x) G(x) = G(xsp ) : F (x) (1 F (x)) , for x 2 (xsp ; x].

(17)

(18)

Note that G depends only on primitives since xsp itself depends only on primitives of the model.

4 4.1

Equilibrium Mobility Decision Rules Some Preliminary Results on Contracts

Since I focus on renegotiation-proof contracts, a contract must be such that the participation constraints are satis…ed. That is, if the worker is promised an employment value W and a continuation 15

f and the productivity of the match is x, then it must be that W and W f are no less than value W

f U , and when contracts are self-enforcing the contract must also be such that Jx (W) and Jx (W) are no less than 0.

Before proceeding to the determination of the outcome of the bargaining game and the characterization of optimal contracts, it is useful to establish a few results. f maximizes Jx (w; W) f subject Lemma 2 Consider a match with productivity x. A contract fw; Wg

f = W for some W (W (w; W) f subject to Jx (w; W) f = J for some J ) and the relevant to W (w; W)

f maximizes Vx (W) f subject to the relevant participation participation constraints if and only if W constraints.

The intuition behind this lemma is simple: the greater the joint value of a match is, the greater is the value of the match to the …rm (worker) given the …xed value that has to be o¤ered to the worker (…rm). This leads to another result.18 Lemma 3 Consider a match whose productivity is known. The contract governing the match is renegotiation proof only if it is such that (i) the relevant participation constraints are satis…ed and (ii) the joint value of the match is maximized (given the relevant participation constraints). The intuition for this lemma is again simple. If a contract is such that either the …rm or the worker is better o¤ walking away when it is possible, then the contract will be renegotiated. And if the joint value of the match is not maximized, then one can design a contract that makes one party strictly better o¤ without making the other worse o¤. An important consequence of lemma 3 is that when I consider a match I can focus my attention on contracts that maximize the joint value of the match. This in turn implies that when a …rm and a worker bargain before a match is formed, they bargain assuming that the joint value of the match to be split is maximum for each possible productivity. This matters because the value that the worker and the …rm can each obtain from a match depends on the joint value created by the match. f and a renegotiationLemma 4 Consider a match of quality x, a renegotiation-proof contract fw; Wg,

proof contract f 0 ; (w0 )

f0 g such that

0; W

0

+ Jx ((w0 )

lump-sum and w0 is the wage paid at tenure . Then 18

0

f0 ) = Jx (w; W), f where

0; W

+ Wx ((w0 )

f0 ) = Wx (w; W). f

0

is the

0; W

The result is not an equivalence. This is because renegotiation proofness also means that the value promised to a worker is what is delivered, and this aspect is ignored in lemma 3.

16

This result establishes that focusing on contracts with a …xed wage without an initial lump-sum is without loss of generality, for any renegotiation-proof contract with an initial fee and a tenuredependent wage can be replicated by a contract with a …xed wage and without a lump-sum, and vice versa. We will soon also establish that, whenever the continuation value promised to the worker matters for the joint value of a match, there is a unique optimal continuation value and a unique f and f 0 ; (w0 ) associated mobility decision rule. This implies that the contracts fw; Wg,

f0 g

0; W

f=W f0 , and therefore the only di¤erence between must have the same continuation value, i.e., W the two contracts is in the timing of payments before the worker receives an outside o¤er.

4.2

Bargaining Outcomes

4.2.1

Enforceable Contracts

The following lemma characterizes the mobility decision and the value of employment for the worker in the match of her choice as a function of the promised continuation value for her current match. Lemma 5 Assume contracts are enforceable, and consider a worker in a match of quality x 2

f and who gets an opportunity to form X [ fbg with a contract with promised continuation value W a new match. If V e

f Vx g, then the worker quits her job to form a new match, i.e., maxfW;

f = 1, and

x (W)

the contract agreed to by the worker and the new …rm is such that the expected value promised to f = V e + (1 the worker is W e (W)

f Vx g. ) maxfW;

f Vx g, then the worker stays in her job, i.e., If instead V e < maxfW;

f = 0, and the

x (W)

f if V e new contract is such that the value of employment promised to the worker is W V e + (Vx

V e ) otherwise.

f and W,

This is intuitive. If the maximum expected joint value of a new match exceeds strictly both the maximum joint value of a worker’s current match and the continuation value promised to her (which can be greater because contracts are enforceable), then the newly met …rm is able to poach away the worker. In fact, the current employer has no incentive to retain the worker for it would lead to a negative value of the match (since V e > Vx ). And the value the newly met …rm has to o¤er the worker depends on the continuation value promised by her current employer if and only if the continuation value is less than the joint value of the match. If instead the joint value of a new match is less than the maximum joint value of a worker’s current match and the continuation value promised to her, then the newly met …rm is unable to 17

poach away the worker. In fact, either the promised continuation value is enough for the current f employer to retain the worker (W

V e ), in which case the …rm can turn down any new o¤er from

the worker; or it is forced to bargain with the worker over an improved contract to retain her, in which case the worker obtains a new value of employment V e + (Vx a fraction

V e ) such that she captures

of the surplus over her alternative value of employment.

f = U the expected value of employment Note that for a currently unemployed worker, since W

is W e (U) = U + (V e

U). Finally, it is important to note for future reference that in cases where

an employed worker quits her current match, her expected value of a new match is increasing in the value of the continuation value, strictly when the quality of the match is such that the promised f exceeds the joint value of the match Vx . continuation value W

4.2.2

Self-Enforcing Contracts

The following lemma characterizes the mobility decision and value of employment for the worker in the match of her choice as a function of the promised continuation value for her current match. Lemma 6 Assume contracts must be self-enforcing, and consider a worker in a match of quality f and who gets an opportunity to x 2 X [ fbg with a contract with promised continuation value W form a new match. If V e

Vx , then the worker quits her job to form a new match, i.e.,

f = 1, and the

x (W)

contract agreed to by the worker and the new …rm is such that the expected value promised to the f = Vx + (V e worker is W e (W)

Vx );

If instead V e < Vx , then the worker stays in her job, i.e.,

f = 0, and the new contract is

x (W)

f if V e such that the value of employment promised to the worker is W otherwise.

f and V e + (Vx W,

V e)

It is worth nothing that when contract must be self-enforcing the mobility decision of the worker, as well as the value of a new match for the worker, are independent of the promised continuation f value. In fact, either W

Vx , in which case the current employer improves its o¤er to Vx to try to

f > Vx , in which case the current …rm will not keep its promise and therefore retain the worker, or W the continuation value promised is irrelevant.

Furthermore, the promised continuation value matters for payo¤s if and only if the joint value of the current match is such that the worker does not quit and the promised continuation value is at least as good as the best o¤er the newly met …rm can make, i.e., V e 18

f Finally, note that, W.

just like in the case of enforceable contracts, the value of employment for a currently unemployed worker is W e (U) = U + (V e

4.3

U).

Optimal Contracts

Now that the outcome of the bargaining game has been determined for the di¤erent possible cases I can turn my attention to the characterization of renegotiation-proof optimal contracts for matches whose productivity is known, that is contracts that maximize the joint value of a match given the constraints imposed by the contractual environment.19 Lemma 7 Consider a match of quality x. Let x

rV e + (V e

U).

(i) Assume contracts are enforceable. If x > x , then a contract is an optimal renegotiation-proof f contract if and only if W

f = V e. if and only if W

V e . If x

x , then a contract is an optimal renegotiation-proof contract

(ii) Assume contracts must be self-enforcing. If x > x , then a contract is an optimal renegotiationf 2 [V e ; Vx ]. If x proof contract if and only if W

x , then a contract is an optimal renegotiation-

f = Vx . proof contract if and only if W

When contracts are enforceable and the productivity of the match is such that the joint value

of the match exceeds the expected joint value of a new match, then the joint value of the match is maximized if and only if the worker stays. But either the promised continuation value exceeds the joint value of a new match, which implies the …rm can get the worker to stay with the promised value, or it is not in which case the …rm must improve its o¤er to retain the worker, as was established in lemma 2. However, in the latter case the contract is not renegotiation-proof. If instead the productivity of a match is such that the joint value of the match is less than the joint value of a new match, then a contract is optimal if and only if the worker quits and the value of the new match for the worker is as large as possible given the worker should move, which is achieved when the promised value is equal to the entire expected value of a new match. When contracts are self-enforcing, the value of the promised value of continuing a match does not matter in general. In fact, either it is no more than the joint value of the match, in which case the …rm will be able to up its o¤er if it is needed and desired to retain the worker, or it is more than 19

When a match productivity is unknown the contract only speci…es the value o¤ered by the …rm to the worker for all possible draws, under the assumption that the …rm will deliver this promised utility by o¤ering a contract which maximizes the joint value of the match since such contracts maximize its own value of the match.

19

the joint value of the match in which case the …rm will not deliver on its promise. However, only the contracts with the promised continuation value indicated in lemma 7 are renegotiation proof.

4.4 4.4.1

Equilibrium Mobility Decision Enforceable Contracts

The equilibrium joint private value of a match is given by (7), with the promised continuation value f mobility decision (W), f and expected value of a new match W e (W) f given by lemmata (5) and W, (7). It follows that

Vx =

8 > > < > > :

x+ U+ V e r+ +

, for x

x+ U r+

,

x , and

otherwise.

The interpretation of the private joint value of the match Vx is similar to that for the social value, ! x and the expression for the equilibrium joint value of a match of quality x is identical to that of the shadow value of a match of same quality in the planner’s problem if we set U = ! u . We then have the following proposition. Proposition 2 Assume contracts are enforceable. The equilibrium mobility cut-o¤ x is equal to the e¢ cient mobility cut-o¤ xsp for all values of

2 (0; 1].

It is useful to compare the way mobility decisions are taken in equilibrium and in the planner’s problem to understand why mobility decisions are e¢ cient when contracts are enforceable. In the planner’s problem, if the worker never quits a match with productivity x the shadow value of the match ! x includes the whole product of the match for as long as the match is not exogenously terminated, while the expected shadow value of a new match ! e includes the expected product of the new match for as long as the match exists, as well as the whole expected product of any subsequent match until the worker becomes unemployed. And as long as x

xsp , the expected

shadow value of a new match exceeds the shadow value of the existing match, implying that it is e¢ cient for the worker to quit for a new job when an opportunity arises. When contracts are enforceable, the trade-o¤ that a worker faces is di¤erent from that of a planner but yields the same mobility decisions: the value of not quitting an existing match is V e for all x

xsp , which is as if the worker were capturing the whole product of a match with productivity

xsp ; and the expected value a new match is also V e , which is as if she captures the whole product of all her future matches until she is sent back into unemployment. 20

The equilibrium surplus of not quitting V e quitting ! x

U therefore exceeds the shadow surplus of not

! u for all x < xsp , but it does not a¤ect the mobility decisions since the expected

value of a new match V e is no less than the value of an existing match for the worker V e for all these productivity levels. Moreover, if it were not the case the worker would be in a weakened bargaining position when negotiating with a poaching …rm, and it would lead to ine¢ ciently low mobility. This feature of the equilibrium with enforceable contracts highlights how agents go around the fact that they cannot contract with other agents they have not yet met. It is also clear from the nature of optimal contracts that enforceability of contracts is crucial, for …rms would prefer to walk away from a match than honor its promise when the productivity is below the cuto¤. Demographics - The steady-state equilibrium unemployment rate u is given by (17), and the distribution of match productivity G is as given in (18) with x = xsp . 4.4.2

Self-Enforcing Contracts

In this case the equilibrium joint private value of a match is given by (7) with the promised f mobility decision (W), f and expected value of a new match W e (W) f given continuation value W, by lemmata (6) and (7). It follows that 8 x+ U+ > > < r+ + Vx = > > x+ U : r+

Ve

, for x ,

x , and

otherwise.

We then have the following result.

Proposition 3 (i) The equilibrium mobility cut-o¤ with self-enforcing contracts, x , is such that Z x e x =x + (x x ) dF (x), (19) r+ x and it is unique. (ii) For all

2 (0; 1], x

xsp , with strict inequality for

< 1, and @x =@ > 0.

Part (i) of proposition 3 says that in steady-state the equilibrium mobility cut-o¤ x exists and is unique, while part (ii) says that the equilibrium mobility cuto¤ is strictly lower than the e¢ cient one as long as workers do not always capture the entire surplus of a match, i.e., as long as

< 1,

and that ine¢ ciency increases as the bargaining power of workers decreases. The issue with self-enforcing contracts is that the value of a new match is depressed relative to the case with enforceable contracts, because the lack of enforceability generates an endogenous 21

incompleteness of contracts. When contracts are enforceable there is no limit to the value a …rm can truthfully promise a worker, thereby implying a …rm-worker pair can write a contract that guarantees the worker captures the whole value of the next match if it is optimal for her to quit for a new job. When contracts must be self-enforcing, however, contracts must be such that neither the …rm nor the worker are better o¤ unmatched than matched, thereby implying contracts enable the worker to capture only a fraction of the value of the next match she will join if it is optimal to quit for a new job. The depressed value of a new match with self-enforcing contracts makes new matches relatively less attractive and thus leads to ine¢ ciently low turnover. Before moving on I would like to go over a modi…cation of the setup to illustrate the fact that the issue with self-enforcing contracts is that the value of a new match does not incorporate the whole value of all future matches. Assume that when an unemployed worker becomes employed she is restricted to changing employer only once before her match is exogenously destroyed. In this example it is clear that the e¢ cient mobility decision is such that a worker quits her current match for a new match if her current match productivity is x < xe , while she should remain in her current match if x > xe , and if x = xe then it does not matter whether she moves or not. That is, xsp = xe . It can be shown20 that in this case the equilibrium mobility decision with self-enforcing contracts is given by

x

= 1 for x

x = xe , and

x

= 0 otherwise, that is, the equilibrium mobility

cuto¤ is equal to the e¢ cient mobility cuto¤. In this example the equilibrium mobility decision is e¢ cient even when contracts are not enforceable, for no matter whether a worker decides to quit her current job, she does not have to bargain with another employer before she is forced into an unemployment spell. Hence, when a worker is able to bargain for the entire expected joint value of a new match V e , the worker de facto captures the entire surplus of the match she is in until she becomes unemployed. Demographics - The steady-state equilibrium unemployment rate u is given by (17), and the distribution of match productivity G is as given in (18) with x as given in (19).

5

Labor Market Policy and E¢ ciency with Self-Enforcing Contracts

The mobility cuto¤ is too low when contracts are not enforceable, implying there is too little mobility. However, it is possible to design labor market policies that yield e¢ ciency without 20

See Appendix A for the proof.

22

relying on the productivity of a match being observable. This last feature is important because it is in practice often di¢ cult to determine accurately the productivity of a worker on a job. Moreover, a labor market policy that would require the productivity of a match to be observable for it to be e¤ective would run into the same issues of enforceability as private contracts. In fact, the impossibility of accurately observing the productivity of a match is one of the reasons why enforceability of contracts by a third party like a court of law or a court of arbitration might not be possible. There are more than one possible policies that can deliver e¢ ciency, and I focus on a …ring tax and show that a ‡at …ring tax does the job. I make this choice because it is a simple policy, and it shows that an appropriately chosen …ring tax increases mobility, which is in contradiction to the results obtained in the literature,21 and does not hold when the quality of a match is known before the match is formed. Since …rms are better o¤ severing a match whose value is negative to them, one way to get around the absence of enforceability is to make it costly for …rms to sever a match: if a …rm’s value to a match is negative but the cost of destroying the match exceeds the loss associated with continuing the match, then …rms will choose not to severe the match. Hence, in the context of this paper a …ring tax can play the role of a commitment device for …rms to not severe a low productivity match, thereby protecting workers against the risk of a bad productivity draw. There are many ways to implement the equilibrium match values derived earlier, but there is one way which is very useful because it permits implementing a …ring tax based on the wage paid. When a …rm-worker pair meet they can write down a Fixed-Wage Contract (FWC) (wx )x2X where there are no lump-sums and the …rm promises to pay the worker a …xed wage wx until the match is severed if they match and the productivity draw is x. When contracts are enforceable a …rm and a worker will agree to an optimal FWC such that for all productivity below the cuto¤ x the worker receives a wage wx = x + rV e + (V e

U) which yields a value of the match to the worker of V e .

Hence, when the worker contacts a new …rm she moves if and only if the …rm o¤ers her the entire expected value of the match V e with an optimal FWC. Consider now a …ring tax such that when a …rm lays o¤ a worker the …rm is liable to pay to a third-party, say the government, a tax, possibly contingent on the wage the worker was paid before 21 See, for instance, Section V-A in Kiyotaki and Lagos (2007) for the impact of a …ring tax in a model with on-the-job search. And see Boeri and van Ours (2008) for a review of the theoretical results in the absence of on-the-job search.

23

she got laid o¤.22 Let

= ( (w))w2R+ be the …ring tax schedule. Clearly, as long as (w)

Jx (w)

a …rm is better o¤ continuing the match than breaking it. Since the e¢ cient mobility decision is obtained if workers receive the wage xsp for all productivities x

xsp and that in such a case the

value of the match to the …rm is Jx (xsp ) =

x xsp , r+ +

which is negative, with strict inequality for all x < xsp , a …ring tax yields the e¢ cient mobility decision if it is such that (xsp ) In fact, if (20) holds, then Jx (xsp )

xsp x . r+ +

(xsp ) for all x

pay workers a wage xsp for productivities x

(20) xsp , i.e., …rms which have agreed to

xsp are better o¤ keeping the worker until she …nds

another match than …ring her. Hence, the optimal FWCs with enforceability become self-enforcing because the penalty that the government can impose on …rms for breaking a match is severe enough to deter …rms from doing so. The rest of the …ring tax rate schedule is indeterminate and thus a ‡at …ring tax such that (w) = (xsp ) for all w works. This means that if …rms face a large enough …xed administrative or legal cost to …ring a worker, then they would retain her.

6

Concluding Comments

It has now been well documented that worker turnover is much lower in continental Europe than in the U.S. or the U.K.23 At the same time, countries from continental Europe tend to have much stronger employment protection legislations (EPL) than the U.S. or the U.K. This seems at odds with the impact that a …ring tax has on worker ‡ows in this paper. However, I have ignored the e¤ects of EPL on labor turnover already identi…ed in the literature, which all reduce worker turnover - EPL imply (i) reduced job creation (because creating a job is less pro…table); (ii) a longer search period before a …rm …lls a vacancy (because …rms are more careful when hiring when they are uncertain about the quality of a match); as well as (iii) lower ‡ows into unemployment (because it is costlier to terminate a match). Furthermore, as Pries and Rogerson (2005) highlight in a model without on-the-job search, EPL can also have a signi…cant downward impact on worker turnover through its interaction with other legislations like a binding minimum wage, unemployment bene…ts, and labor income tax. Hence, whether EPL in practice increase or decrease turnover depends on 22

If the payment made by the …rm were to be to the worker, then it would have no e¤ect. This is reminiscent of Lazear (1990). 23 See Pries and Rogerson (2005) for a review of the evidence.

24

the strength of the already known negative e¤ects of EPL relative to the positive e¤ects identi…ed in this paper. Lastly, the existence of these two opposite types of e¤ects might explain why empirical studies have failed to reach unambiguous conclusions about the impact of EPL on labor turnover.24 24

See Boeri and van Ours (2008) for a review of the evidence.

25

Appendix A: Proofs of Results in the Main Text Proof of Lemma (1): The Hamiltonian for the planner’s problem is H =

Z

x (1

x2X

+

Z

!x

Z (1 u) g(x)dx u u) g(x)dx + ub + ! u x2X Z ( + x ) (1 u) g(x0 )dx0 f (x) u + x0 (1

u) g(x) ,

x0 2X

x2X

where g is the pdf associated with the cdf G where at the point of discontinuity x I use the convention g(x ) and (1

lim

!0+

[G(x )

G(x

1.

)]

The …rst-order conditions with respect to u

u)g(x) are respectively r! u

!u = b

Z

!u +

! x f (x)dx, and

x2X

r! x

!x = x + !u +

Z

x

! x0 f (x0 )dx0

x0 2X

( +

x) !x.

These two expressions simplify to r! u

!u = b +

r! x

!x = x +

And clearly the planner optimally chooses

(! e x (!

x

! u ) , and e

!x)

(! x

= 1 if ! e > ! x ,

x

!u) .

= 0 if ! e < ! x , and is indi¤erent

if ! e = ! x : Proof of Proposition 1: Solving for ! e yields !e =

!u + r+

(xsp ; ),

where (xsp ; )

r+ +

1 (1

xe +

F (xsp ))

r+

Z

x

xdF (x) .

xsp

Hence we obtain that

!x =

8 > > < > > :

x+

x r+

(xsp ; ) r+ +

+

+

!u r+

!u r+

xsp , and

,

if x

,

otherwise.

Since xsp is such that ! xsp = ! e , we have xsp !u !u + = + r+ r+ r+ 26

(xsp ; ),

(21)

or sp

x

r+ = r + + (1

e

F (xsp ))

x +

r+

Z

x

xdF (x) ,

xsp

which can be rewritten as sp

x De…ning )

R

x2X

(x

xe

(z) = z

x) dF (x) < 0 and

e

=x +

=(r + ) (x) = x

r+

Rx z

Z

x

(x

xsp ) dF (x).

xsp

(x

(x) = x

z) dF (x), we have that

xe > 0 and therefore since

xe

=(r +

is continuous on X we have

proven existence by the Intermediate Value Theorem. Uniqueness is obtained from the fact that @ (z) (1 F (z)) =1+ > 0: @z r+

Proof of Lemma 2: I present the proof for the case of a …rm maximizing the value of the match subject to providing the worker with a minimum value. The other case is perfectly symmetric. Suppose …rst that f maximizes Jx (w; W) f subject to W (w; W) f = W for some W and other relevant participation fw; Wg

f does not maximize Vx (W) f subject to the relevant participation constraints. constraints, but that W f0 6= W f such that Vx (W f0 ) > Vx (W) f and such that all relevant participation Then, there exists W constraints are satis…ed. There are three cases to consider: (i) f = 1 and

x (W)

f0 ) = 0, and (iii)

x (W

f =

f = 0 and

x (W)

f0 ) = 1. In fact, if

x (W)

x (W

f =

x (W)

the joint value of the match would be left unchanged.

f0 ) = 1, (ii)

x (W

f0 ) = 0, then

x (W

f0 ) Case (i) - For the mobility decision rules to be what they are, it must be that W e (W

f0 W

f > W e (W). f And the only way this is consistent with Vx (W f0 ) > Vx (W) f is if W e (W f0 ) > and W f We then have that W (w; W f0 ) > W (w; W) f Vx (W).

f0 ) = W (w; W) f and W (w0 ; W

f0 ) = Vx (W f0 ) Jx (w0 ; W

W, and therefore one can w0 < w such that

f0 ) W (w0 ; W

f = Vx (W) f > Jx (w; W)

f W (w; W).

f maximizes Jx (w; W) f subject We therefore obtain a contradiction to the assumption that fw; Wg

f = W. to W (w; W)

f0 ) < W f0 Case (ii) - For the mobility decision rules to be what they are it must be this time that W e (W f and W

f And the only way this is consistent with Vx (W f0 ) > Vx (W) f is if W e (W) f < W e (W). 27

f0 ) = Vx independent of W f0 . Since the worker does not quit the match, the continuation value Vx (W

f0 can be as low as W e (W), f in which case W (w; W f0 ) = W (w; W). f Since Vx (W f0 ) > Vx (W), f it W

follows that

f0 ) = Vx (W f0 ) Jx (w; W

f0 ) W (w; W

f = Vx (W) f > Jx (w; W)

f W (w; W).

f maximizes Jx (w; W) f subject We once again obtain a contradiction to the assumption that fw; Wg

f = W. to W (w; W)

f0 ) > Vx (W) f and Case (iii) - Since Vx (W

f =

x (W)

f0 ) > W (w; W) f However, this implies that W (w; W

f0 ) = 1, it must be that W e (W f0 ) > W e (W). f

x (W

f0 ) = Jx (w; W)), f and therefore W (and Jx (w; W

f0 ) = W (w; W) f and thus such that one can …nd w0 < w such that W (w0 ; W f0 ) = Vx (W f0 ) Jx (w0 ; W

f0 ) W (w0 ; W

f = Vx (W) f > Jx (w; W)

f W (w; W).

f maximizes Jx (w; W) f subject Hence, we get another contradiction to the assumption that fw; Wg f = W, which completes the proof of the "if" part. to W (w; W)

f maximizes Vx (W) f subject to the relevant constraints, but that there Suppose now that W

f maximizes Jx (w; W) f subject to W (w; W) f is no w such that fw; Wg

W for some W and the

same relevant participation constraints. This implies that Jx is maximized for some other contract f0 g with W f0 6= W. f However, for w such that W (w; W) f = W, then since W (w0 ; W f0 ) fw0 ; W

W, we

have that

f0 ) = Jx (w0 ; W f0 ) + W (w0 ; W f0 ) > Vx (w; W) f = Jx (w; W) f + W, Vx (W

f > W, then there exists w0 such that W (w0 ; W) f =W a contradiction. And for w such that W (w; W)

f > Jx (w; W), f another contradiction. This proves the "only if" part and completes and Jx (w0 ; W)

the proof of the lemma.

Proof of Lemma 3: (i) It is clear that if a participation constraint is violated, say for the …rm, then the …rm would prefer to walk away from the match than to continue with the current contract. Since all matches generate a surplus over the pair being unmatched, the worker will agree to renegotiate the contract as it is possible to ensure both the worker and the …rm enjoy a non-negative surplus by staying matched over being unmatched. Formally, assume the match is of quality x and the current contract 28

delivers a value of the match of W and Jx to the worker and the …rm respectively, and assume the participation constraints is violated for the …rm, i.e., Jx < 0 for the case of self-enforcing contracts (since with enforceable contracts there are no ex post participation constraints). If the worker and the …rm renegotiate the contract, in the second round when the worker proposes she proposes a contract such that the value of the match to her and the …rm are (Vx ; 0), and the …rm accepts the o¤er. In fact, the …rm’s options are to stay matched with previous contract that gives a strictly negative value or walk away which give a zero value, so it is strictly better accepting than turning it down. Note that the worker is strictly better o¤ following the renegotiation, for otherwise the …rm would walk away and the worker would then become unemployed and receive U < Vx . If the …rm is making the o¤er in the second round, it proposes a contract such that the value of the match to worker and itself are (U; Vx

U), and the worker accepts the o¤er since it is

indi¤erent. Note this time it is the …rm that is strictly better o¤. We therefore obtain that in the …rst stage of bargaining the …rm o¤ers to the worker a contract such that the value of the match to her and itself are (U + (V x

U); (1

)(V x

U)), the worker accepts the o¤er, and both the

worker and the …rm are strictly better than walking away from the match. (ii) If the joint value of the match is not maximized, then it is possible to increase the value of the match for one of the parties without decreasing the value for the other party, and therefore both are willing to renegotiate the contract, and at least one of them strictly wants to. Formally, assume the match is of quality x and the current contract delivers a value of the match of W and Jx to the worker and the …rm respectively, and it is such that W + J x < V x . If the worker and the …rm renegotiate the contract, in the second round when the worker proposes she proposes a contract such that the value of the match to her and the …rm are (Vx

Jx ; Jx ), and the …rm accepts

the o¤er since it is indi¤erent. Note that the worker is strictly better o¤ since Vx > W + J x . If in the second round it is the …rm making the o¤er, it proposes a contract such that the value of the match to worker and itself are (W; Vx

W), and the worker accepts the o¤er since it is indi¤erent.

Note this time it is the …rm that is strictly better o¤. We therefore obtain that in the …rst stage of bargaining the …rm o¤ers to the worker a contract such that the value of the match to her and itself are (W + (V x

W

J x ); Jx + (1

)(V x

W

J x )), and the worker accepts the o¤er.

Proof of Lemma 4: We know from lemma 3 that any renegotiation-proof contract maximizes the joint value of a match V . Note that this holds for contracts with tenure-dependent wages, for the proof of part (ii) of lemma 3 is silent over the wages being paid before a worker receives an

29

outside o¤er. Hence, if f 0 ; (w0 )

f0 g is such that

0; W

that V

h

0

+ Jx ((w0 )

0

+ Wx ((w0 )

0

+ Jx ((w )

i 0 f ; W ) = V 0

f0 ) = Jx (w; W), f we have

0; W

f or Jx (w; W),

f0 ) = Wx (w; W). f

0; W

Proof of Lemma 5: I reason by backward induction. Second Round - (i) If the worker is the one making the o¤ers, she o¤ers to the newly contacted …rm a contract such that she captures the entire expected value of the new match, i.e., such that her value of the match and that of the …rm are V e and 0. The values of the match corresponding to the o¤er the worker makes to her current employer depends on how the promised f compares with the joint value of the match V e . If W f > V e , then the …rm will continuation value W

f since the worker turn down any o¤er that gives the worker a value of employment more than W f which leaves Vx will stay anyway, and therefore the worker asks for W

f to her current employer. W

f < V e , then the worker will quit her current match if the …rm does not accept an If, however, W

improved contract. The worker takes advantage of this and o¤ers a contract which gives her a value of employment of Vx and leaves nothing to her employer. If V e

f Vx g the worker can get a value of employment as least as large with the new maxfW;

…rm than what she can get in her current match, in which case she quits her current job to start a

new match. Otherwise she stays in her current match. Note that I assume that if current contract cannot be improved, then the worker asks for the same value even though she could ask for anything and just keep current contract. (ii) If it is the …rms making the o¤ers, given that the …rms enter a Bertrand competition for the worker the o¤ers are such that the split of the joint value of the match between the worker and the f minfVx ; V e gg; minfVx …rm are (maxfW;

f Vx W;

minfVx ; V e gg) for the current’s …rm o¤er and

f Vx g); maxf0; V e maxfW; f Vx g) for the poaching …rm’s o¤er. If V e (minfV e ; maxfW;

f Vx g maxfW;

the workers elects to quit her current job to start a new match, while otherwise she stays. First round - Since the worker gets to propose with probability propose with probability 1

and the …rms get to

, when the …rms propose they know the worker will turn down any

o¤er such that her value of the match is strictly less than the expected value of going to the next round, which is the maximum of the expected values she can obtain from her current and poaching 30

…rm. Hence, if V e

f Vx g the current …rm o¤ers to the worker a contract such that her maxfW;

f Vx g, while the newly contacted …rm o¤ers the worker a contract value of employment is maxfW; with value of employment V e + (1

f Vx g. And the worker quits her current job to take ) maxfW;

f Vx g, then the newly contacted …rm up the newly contacted …rm’s o¤er. If instead V e < maxfW;

o¤ers the worker a contract with value of employment V e , while her current employer o¤ers her a value of employment match.

f Vx g + (1 maxfW;

f V e g, and the worker stays in her current ) maxfW;

Proof of Lemma 6: I again reason by backward induction. Second Round - (i) If the worker is the one making the o¤ers, she o¤ers to the newly contacted …rm a contract such that she captures the entire expected value of the new match, i.e., such that her value of the match and that of the …rm are V e and 0. The values of the match corresponding to the o¤er the worker makes to her current employer depends on how the promised f compares with the joint value of the match V e . If W f > V e , then the …rm will continuation value W

f and therefore the turn down any o¤er that gives the worker a value of employment more than W, f which leaves Vx worker asks for W,

f for her current employer. If, however, W f < V e , then the W

worker will quit her current match if the …rm does not accept an improved contract. The worker takes advantage of this and o¤ers a contract which gives her a value of employment of Vx and leaves

nothing for her employer. If V e

Vx the worker can get a value of employment as least as large with the new …rm than

what she can get in her current match, in which case she quits her current job to start a new match. Otherwise she stays in her current match. Note that I assume that if current contract cannot be improved, then the worker asks for the same value even though she could ask for anything and just keep current contract. (ii) If it is the …rms making the o¤ers, given that the …rms enter a Bertrand competition for the worker the o¤ers are such that the split of the joint value of the match between the worker f V e gg; minf0; Vx and the …rm are (minfVx ; maxfW; (minfV e ; Vx ); maxf0; V e

f V e gg) for the current’s …rm o¤er and maxfW;

Vx ) for the poaching …rm’s o¤er. If V e

Vx the workers elects to quit

her current job to start a new match, while otherwise she stays. First round - Since the worker gets to propose with probability propose with probability 1

and the …rms get to

, when the …rms propose they know the worker will turn down

31

any o¤er such that her value of the match is strictly less than the expected value of going to the next round, which is the maximum of the expected values she can obtain from her current and poaching …rm. Hence, if V e

Vx the current …rm o¤ers to the worker a contract such that

her value of employment is Vx , while the newly contacted …rm o¤ers the worker a contract with value of employment V e + (1

)Vx . And the worker quits her current job to take up the newly

contacted …rm’s o¤er. If instead V e < Vx , then the newly contacted …rm o¤ers the worker a contract with value of employment V e , while her current employer o¤ers her a value of employment Vx + (1

f V e g, and the worker stays in her current match. ) maxfW;

Proof of proposition 2: Solving for V e yields

where

Ve =

r+

U+

(x ; ),

(x ; ) is de…ned as in (21), and hence we obtain that 8 x+ (x ; ) > > + r+U , for x x , and < r+ + Vx = > > x U : , otherwise. r+ + r+

Therefore the equilibrium mobility cut-o¤ in this case is such that x = xe +

r+

Z

x

(x

x )dF (x).

(22)

x

Equation (22) is identical to (16) which characterizes the e¢ cient mobility productivity cut-o¤, and since this equation has a unique solution, we obtain that x = xsp . Proof of proposition 3: Solving for V e yields

where

(x ;

Ve =

U + r+

(x ;

),

) is as de…ned in (21), and hence we obtain that 8 x+ (x ; ) > > + r+U , if x x , and < r+ + Vx = > > : x + U , otherwise. r+ r+

(i) Since x is such that Vx = V e we have x r+

+

U U = + r+ r+ 32

(x ;

),

or r+ x = r+ + (1

Z

e

x +

F (x ))

which can be rewritten as

Z

x = xe +

r+

x

xdF (x) ,

x

x

r+ x Rx =(r + ) z (x

(x

De…ning e (z; ) = z xe Rx =(r + ) x (x x) dF (x) < 0 and e (x; ) = x

x ) dF (x).

z) dF (x), we have that e (x; ) = x

xe

xe > 0 and therefore since e is continuous on

X we have proven existence by the Intermediate Value Theorem. The fact that @ e (z; ) =1+ @z

(1 F (z)) >0 r+

yields uniqueness. (x; ), @ e (z; ) =@z

(ii) The results follows from the facts that e (x; ) strict inequality for all

< 1, and that

@ (z) =@z, with

@ e (x ; ) =@ > 0. @ e (x ; ) =z

dx = d

Proof that x = xe in the example of Section 4.4.2 - Consider the equilibrium mobility decision: when an employed worker considers whether to leave her current match to form a new one she knows this will be her last job change before her match is hit by an exogenous destruction cx (w) be the value of employment for a worker in a match with productivity x when shock.25 Let W

the wage is w and she is no longer allowed to change job, and let Vbx be the associated joint value cx (w) and Vbx are such that of the match. W which is independent of x, and

cx (w) = w rW

cx (w) (W (Vbx

rVbx = x

U ),

U ).

Hence, the expected joint value of the new match is

25

xe + U . Vb e = r+

If she decides not to accept to form a new match once, she will never accept a new job. And since the value of

employment is always no less than the value of unemployment a worker will never quit a job unless she has another job lined up.

33

One can then follow the analysis in the main text to show that the outcome of the bargaining game is such that the mobility decision of the worker is given by 8 < 1, for x x , and x = : 0, otherwise,

(A7)

where x is such that Vx = Vb e . We then have that for x > x the value of staying with the current f the promised continuation employer when we consider renegotiation-proof contracts is simply W,

value. While for x

x the expected value of changing job is given by f = Vx + (Vb e W e (W)

Vx ).

(A8)

The optimal value Vx of a match with productivity x in which the worker was recruited while unemployed is thus such that be

rVx = x + with

x

x (V

Vx )

(Vx

U) ,

given by (A7). (A9) and (A7) together yield that 8 x+ U+ Vb e > > , for x x , and < r+ + Vx = > > x+ U : , otherwise. r+

From there, it is straightforward to show that x = xe .

34

(A9)

Appendix B: Continuous Renegotiations In the main text it was assumed that when contracts are self-enforcing workers and …rms can walk away from a match, but if they wish to stay matched they either have to abide by the terms of the contract or obtain the agreement of their partner to change the contract. In this appendix I present very brie‡y how the analysis changes with a di¤erent de…nition of self-enforcing contract where instead agents can force their partner to renegotiate the contract constantly, very much like in the standard Diamond-Mortensen-Pissarides framework. For the sake of brevity I will only sketch the arguments. Consider …rst a matched …rm-worker pair whose joint productivity is x 2 X. Since both the …rm and the worker can force their partner into renegotiating the terms of the contract at any time, it is straightforward to show that the rules of the bargaining game imply that a renegotiation-proof contract is such that for any such match Wx , the value of being matched for the worker (outside contact times), and Jx , the value for the …rm, are such that Wx = U + (Vx Jx = (1

U ), and

)Vx .

Given W e , the expected value of a new match, we also know that rWx = w

(Wx

+

e x (Wx

U) Wx ) + (1

and rJx = x

w

( +

x )Jx

(1

f

x )(W

e

x )(J

Wx ),

Jx ),

f and Je are the continuation values for the worker and …rm respectively. Note that these where W values have to be determined by the bargaining game. And as before we can de…ne the joint value

of a match with productivity x, Vx as the sum of the value for the worker and the …rm, and it is such that rVx = x

(Vx

U) +

e x (Wx

Vx ).

Denote by V e the expected joint value of a new match, and consider the situation where a worker employed in a match with productivity x contacts another …rm. Assume …rst that x for x such that Vx = V e , that is x = rV e + (V e in the …rst round Vx and Vx + (V e

U ). Then, since Vx

x ,

V e , the worker is o¤ered

Vx ) by her current and potential new employer respectively, 35

and the worker accepts the o¤er of the poaching …rm so that x = 1. Note that the worker can R obtain a value of starting in a new match greater than W e = Wx dF (x) = U + (V e U ) because

the poaching …rm can pay a lump-sum (a sign-on bonus) of (1

)(Vx

U ) when the worker agrees

to join. But the value of employment for the worker goes down to Wx after the lump-sum payment has been made. Assume instead that x > x . Then Vx > V e , and the worker is o¤ered in the …rst round V e + (Vx

V e ) and V e by her current and potential new employer, and the worker accepts the

o¤er of her current employer so that

x

= 0. Since V e + (Vx

V e ) > Wx = U + (Vx

…rm needs to make a lump-sum payment (pay a retention bonus) of (1 It follows from the above analysis that 8 < x+ U r+ Vx = : x+ U +

)(V e

U ), the

U ).

, for x > x , and V

e

r+ +

, for x

x .

Clearly, the expressions of the value of a match are identical to the case in the main body of the paper with self-enforcing contracts where contracts are not continuously renegotiated. This also implies that the mobility decisions are the same in the two cases, and therefore mobility is ine¢ ciently low with continuous renegotiations. The intuition as to why the expressions are the same in the two cases is simple: the starting values of employment for a worker can be the same with continuous renegotiations than in the case in the main text because …rms can use lump-sum payments at the start of a match or when a worker gets an outside o¤er to compensate for the fact that the worker will not be able to obtain as high a wage later on. In other words, the only di¤erence between the two cases lies in the timing of payments.

36

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