In this paper we develop a model of discretionary employee investment in firm-specific human capital. When a firm has full bargaining power, the non-contractibility of this investment completely undermines the employee’s incentives. However, the incumbent firm’s inability to observe competing wage offers at the interim stage prevents it from completely expropriating the surplus, thereby creating incentives for the employee to invest. We demonstrate that commitment to a wage floor for the second period destroys the worker’s incentives to acquire human capital, but makes turnover efficient. Therefore, such a commitment has value only if the return on the employee’s deliberate investment in human capital acquisition is sufficiently low. When firms are privately informed about the productivity of their human capital acquisition technology, more productive firms offer higher entry wages to separate themselves from less productive firms. Furthermore, in contrast to the case of symmetric information about human capital acquisition technology, the commitment opportunity now has value for firms with higher return on investment: commitment to a wage floor acts as a substitute for raising the entry wage.
Proof of Proposition 2. (i) We need to check the SOC to show that indeed eq. ] gives a solution to the employee’s problem of choosing the optimal investment. The SOC is
where for brevity we write w instead of . Logconcavity of implies for any w. Then, using that is also logconcave (see Bergstrom and Bagnoli 2005), after some transformations we obtain that inequality  will be satisfied if This condition is satisfied if is large enough (or is small enough) since the density function is assumed to be bounded.
(ii) Let and be the efficient productivity gain and the equilibrium gain. Furthermore, let be the equilibrium wage and the wage that would be paid if productivity gain were achieved. From eq.  we get where . Note that is an increasing function since is decreasing.
Let us prove that (or, equivalently, ). Suppose the reverse is true. The necessary conditions for and to be decisions of the corresponding problems are
Summing these inequalities we get
which does not hold under because
Consequently, and . ∎
Example 1. Substituting and its derivatives along with into the SOC  we obtain
or, after transformation,
The left-hand side is less than or equal to 1, since and Therefore, condition is indeed sufficient for the global SOC.
As for the derivative of the investment e with respect to productivity the denominator of eq.  is positive as long as the SOC is satisfied. Assume Substituting and its derivatives into the denominator, and then substituting
from eq.  we get
which leads to a necessary and sufficient condition for the global SOC
The denominator of eq.  is positive as long as the SOC is satisfied. The numerator is positive, so condition ensures that equilibrium investment is an increasing function of ∎
Proof of Proposition 3. (1) Analysis preceding the proposition demonstrated that any equilibrium with employment belongs to one of the types described in Proposition 3. It remains to fully describe the equilibrium strategies and show the there always exists a unique SPNE. The firm’s wage setting policy for the second period has been described: Let us determine the employee’s optimal investment given a contract offer If defined by
the employee is indifferent between no investment and . If she strictly prefers zero investment; if she prefers to invest
It remains to determine the optimal contract for the firm. Among the contracts that are subsequently renegotiated, the optimal one is where is determined by participation constraint  satisfied as equality. The profit under this optimal short-term (ST) contract is then the same as in the model without commitment:
If the firm offers a long-term (LT) contract that is not renegotiated afterward, its equilibrium profit is
and it is maximized when : in this case the ability to commit eliminates the inefficiency of turnover. Moreover, if with is greater than it follows that
so that and the agent indeed prefers to choose rather than deviate to a positive effort level. If the firm is indifferent between an optimal ST and an optimal LT contract, so both equilibria are possible. Thus, we have proved that depending on parameters there exists a generically unique SPNE with ST contract if with a LT contract if and no employment if ∎
Proof of Proposition 4. Let us begin with the following useful observation:
Lemma 3In a separating PBE the worker’s investment in human capital acquisition at the highly productive firmis larger than her investment at the less productive firm
Proof. The equilibrium investment levels and are determined by eq. , where is replaced by and , respectively. Taking into account that since , and that is increasing in e by statement (i) of the same proposition, we immediately get the result from eq. . ∎
To prove part (i), note that the entry wage offered by the less productive firm must be . Indeed, this wage cannot be lower than since otherwise the worker would not accept it. It also cannot be larger since the firm would increase its profit by offering defined in eq. .
Let us now write the incentive compatibility condition for the L-type firm: it does not want to imitate the equilibrium behavior of its H-type counterpart if
Let be the value of which turns constraint  into equality. Similarly, the condition that the H-type firm does not want to mimic the equilibrium strategy of the L-type firm is
Let be the value of which makes constraint  binding. Thus, both incentive compatibility conditions will be satisfied iff Let us check that Substituting to inequalities  and , we obtain that is equivalent to
which is satisfied by Lemmas 1 and 3.
It is easy to see that any can indeed be an equilibrium wage if we specify “pessimistic” out-of-equilibrium beliefs: the worker interprets any wage as indicative of the L-type firm.
Finally, since condition  implies that therefore, the employee’s participation constraint is satisfied by the H-type firm’s offer (note that we assumed that agent would be hired under full information for both values of ).
First, let us analyze the separating equilibria. If a separating equilibrium satisfies the Intuitive criterion, for any out-of-equilibrium wage offer the agent should believe that the firm has the high type (the L-type firm would not want to offer such a wage even if it induced the agent to believe that her technology is efficient). Hence, no can be offered in a separating equilibrium that satisfies the Intuitive criterion (otherwise, the H-type firm would prefer to deviate to ). Finally, the lowest-wage separating equilibrium does satisfy the Intuitive criterion.
Lemma 41. There existsuch that for anythere is a pooling PBE in which both types of firm offer wage. There are no other pooling equilibria.
2. No pooling equilibrium satisfies the Intuitive criterion.
Proof. (i) In a pooling equilibrium the firm offers the same first-period wage This wage should be sufficient to induce the worker to accept the offer:
, where is the worker’s equilibrium effort level:
. On the other hand, the equilibrium wage cannot be too high – otherwise the L-type firm would prefer to pay a lower wage (although the cost of such deviation would be the disclosure of the firm’s low type):
where is defined in eq. . From Lemma 1 we can easily obtain that if satisfies inequality  the H-type firm also does not gain by deviating to wage Let be the values of which make constraints  and , respectively, be satisfied as equalities. Note that from  and  it can be seen that and from  so that Clearly, there is no pooling equilibrium with On the other hand, by specifying “pessimistic” out-of-equilibrium beliefs (i.e. such beliefs that the worker interprets any wage as being given by the L-type firm), any wage can be supported as equilibrium wage.
Consider a pooling equilibrium with wage Let be the first-period wage that makes the L-type firm indifferent between its equilibrium payoff and the payoff it gets if offers , assuming the agent reacts to by choosing effort defined in eq. :
By Lemma 1 the H-type firm would strictly prefer to deviate to (and, by continuity, to a wage for small enough ) were the agent to react by choosing . But then, for the equilibrium to satisfy the Intuitive criterion the agent should interpret as indicative of the H-type firm, which would indeed create incentives for the H-type firm to deviate to thus destroying the equilibrium. ∎
It remains to prove that the hybrid equilibria do not satisfy the Intuitive criterion. In the first type of hybrid equilibria the H-type firm randomizes between and and the low type offers Since the H-type firm is indifferent between and the L-type firm strictly prefers the lower wage to any wage with small enough. But then the Intuitive criterion implies that the worker should believe that wage is offered by the H-type firm, thus destroying the equilibrium.
In the second type of hybrid equilibrium the L-type firm randomizes between and and the H-type firm offers Then, the same reasoning as in the proof of part 2 of Lemma 4 shows that such an equilibrium does not pass the Intuitive criterion. ∎
Proof of Proposition 5. The first claim is obvious and it is proved in the text. We need to prove the second claim, so assume Let us first show that separating equilibria exist in this case. If is also larger than separating equilibria constructed in Section 3.1 still exist (e.g. with pessimistic beliefs for all out-of-equilibrium contracts). It remains to analyze the case
Let us prove that there exists a separating equilibrium where the H-type firm offers for some and the L-type firm offers the optimal long-term contract where . There are three constraints to be satisfied by for this pair of contracts to form a separating equilibrium21:
The first two constraints are incentive compatibility conditions for the H- and the L-type firms, the third is the participation constraint for the employee when she is offered the H-type firm’s contract. Let be the value of that makes constraint  binding. Substituting into  we get
which is true since
where the first inequality follows from and the second from the optimality of
The assumption that under symmetric information the H-type firm prefers short-term contracting implies that constraint  will be satisfied when In fact, any with where satisfies eqs  and , may be a part of separating equilibrium; to support it, the employee should hold the belief that she is facing the L-type firm when offered any other contract.
The proof that equilibria with some pooling (when they exist) do not pass the Intuitive criterion follows the same steps as the proof that pooling equilibria are not “Intuitive” in the absence of commitment opportunities (see the proof of Lemma 4). ∎
Proof of Lemma 2. By Proposition 5 the more productive firm offers either a short-term or a renegotiated long-term contract in any separating equilibrium. If the less productive firm also offers a short-term contract, Lemma 3 applies. Otherwise, which verifies the claim. ∎
We would like to thank Andrei Bremzen, Guido Friebel, Sergei Guriev, Sergey Stepanov and Jean Tirole for their helpful comments. Anton Suvorov is grateful to the Institute of Advanced Studies in Toulouse for a travel grant and hospitality while he was working on the paper, and to CAS at HSE for a travel grant.
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Regarding the definition of firm-specific human capital, Lazear (2009) argues that it might be better interpreted as firm-specific combinations of general skills rather than specific skills per se. Indeed, giving examples of purely specific skills that play important role in the production process is often problematic.
A related idea that the firm can use promotions as a mechanism to commit to reward the acquisition of specific skills is considered in Prendergast (1993).
Moore (1985) studies labor contracts that emerge when workers have privately observed reservation wages, but his model does not include discretionary employee investment.
Indeed, Since can be checked to be positive under our assumptions, the sign of the mixed derivative is not clear. However, it will be positive if the marginal cost of effort is high enough so that the equilibrium effort is low.
See, for example, Fudenberg and Tirole (1991).
A sufficient condition for this is that the marginal cost is high enough and is increasing fast enough; see Footnote 8.
Indeed, suppose are the entry wages offered by the less and the more productive firms, and and are the investment levels subsequently chosen by their employees. Then, necessarily and a revealed preference argument shows that which contradicts Lemma 1.
In the hybrid equilibrium of the first type the more productive firm randomizes between and and the less productive firm offers in the second the less productive firm randomizes between and and the more productive offers
In our model the worker does not benefit from that asymmetry since the firm has the full bargaining power ex ante. If the worker had some bargaining power ex ante (e.g. due to competition between potential employers), she would also gain.
There would exist a range of pooling equilibria with where the lowest wage leaves no rent to the employee, while the highest wage just allows the less productive firm to break even. Any wage in this range can be supported by pessimistic beliefs, and all these equilibria satisfy the Intuitive criterion.
To be precise, there exists a function such that symmetric information contracts form a separating equilibrium if and Function is decreasing since the larger is the more tempting it is to deviate for the less productive firm; as .
In the model without commitment the less productive firm’s incentive constraint implied that the more productive firm’s offer meets the worker’s participation constraint. This need not be true in the model with commitment when the less productive firm chooses to offer a long-term contract.
In some of the cases we consider here the Intuitive criterion leaves more than one equilibrium. This happens because we do not explicitly model the cost of commitment. However, the outcomes in all these equilibria are essentially the same, up to an infinitesimal difference in payoffs.
Unfortunately, we have been unable to find simple conditions on parameters that would discriminate between the different types of equilibria we considered.
Sharing future profits is a substitute mechanism, but its scope is limited when the number of employees grows.
Recall that in the optimal LT contract so that the firm’s expected profit from such contract is
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