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Discretionary Acquisition of Firm-Specific Human Capital under Non-verifiable Performance

Akhmed Akhmedov and Anton Suvorov

Abstract

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.

Appendix

Proof of Proposition 2. (i) We need to check the SOC to show that indeed eq. [4]] gives a solution to the employee’s problem of choosing the optimal investment. The SOC is

[9]α2f(w)1+(F(w)/f(w))2α2F(w)(F(w)/f(w))′′1+(F(w)/f(w))3C′′(e)0,

where for brevity we write w instead of w(e). Logconcavity of f(ξ) implies f′′(w)f(w)(f(w))2 for any w. Then, using that F(ξ) is also logconcave (see Bergstrom and Bagnoli 2005), after some transformations we obtain that inequality [9] will be satisfied if α2f(w(e))C′′(e). This condition is satisfied if C′′(e) is large enough (or α is small enough) since the density function f(ξ) is assumed to be bounded.

(ii) Let y=αe+β and y˜=αeNC+β be the efficient productivity gain and the equilibrium gain. Furthermore, let w˜=w2NC(eNC) be the equilibrium wage and w=w2NC(e) the wage that would be paid if productivity gain y were achieved. From eq. [2] we get y˜=w˜+gw˜,y=w+gw, where g(w)=F(w)/f(w). Note that g(w) is an increasing function since f(w) is decreasing.

Let us prove that y>y˜ (or, equivalently, w>w˜). Suppose the reverse is true. The necessary conditions for y and y˜ to be decisions of the corresponding problems are

yF(y)+yξdF(ξ)C(e)y˜F(y˜)+y˜ξdF(ξ)C(e˜),
w˜F(w˜)+w˜ξdF(ξ)C(e˜)wF(w)+wξdF(ξ)C(e).

Summing these inequalities we get

[10]yF(y)wF(w)wyξdF(ξ)y˜F(y˜)w˜F(w˜)w˜y˜ξdF(ξ),

which does not hold under yy˜ because

w[(w+g(w))F(w+g(w))wF(w)ww+g(w)ξdF(ξ)]
=1+g(w)Fw+gwFw>0.

Consequently, y>y˜ and e>eNC. ∎

Example 1. Substituting F(ξ)=1exp(λξ) and its derivatives along with C(e)=ke2/2 into the SOC [9] we obtain

α2λexp(λw)1+exp(λw)2α2(1exp(λw))α2λexp(λw)1+exp(λw)3k0,

or, after transformation,

(2+exp(λw)exp(λw))1+exp(λw)3kα2λ.

The left-hand side is less than or equal to 1, since 1exp(λw)0 and 1+exp(λw)1+exp(λw). Therefore, condition kα2λ is indeed sufficient for the global SOC.

As for the derivative of the investment e with respect to productivity α, the denominator of eq. [6] is positive as long as the SOC is satisfied. Assume kα2λ. Substituting F(ξ) and its derivatives into the denominator, and then substituting

e=1exp(λw)k(1+exp(λw))

from eq. [4] we get

αλeexp(λw)1+exp(λw)2+(1exp(λw))((1+exp(λw))2αλeexp(λw))1+exp(λw)3
=(1exp(λw))1+exp(λw)4α2λk(2+exp(λw)exp(λw))+(1+exp(λw))3
(1exp(λw))1+exp(λw)4exp(λw)+(1+exp(λw))3>0.
Example 2. Although a uniform distribution does not formally fit our assumption that f(ξ)>0 for all ξ0, this does not create complications provided there is enough uncertainty about the alternative offer (A is large enough) or, equivalently, cost parameter k is high enough. Substituting F(ξ)=ξ/A for ξ[0,A] and its derivatives along with C(e)=ke2/2 into the SOC [9] we obtain
α24Ak0,

which leads to a necessary and sufficient condition for the global SOC kα2/(4A).

The denominator of eq. [6] is positive as long as the SOC is satisfied. The numerator αe4A+w2A is positive, so condition kα2/(4A) ensures that equilibrium investment is an increasing function of α.

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: w2R(e)=maxwˆ2,w2NCe. Let us determine the employee’s optimal investment given a contract offer (w1,wˆ2). If wˆ2=wˉ, defined by

[11]F(w¯)w¯+w¯ξdF(ξ)=F(w2NC(eNC))w2NC(eNC)+w2NC(eNC)ξdF(ξ)c(eNC),

the employee is indifferent between no investment and e=eNC. If wˆ2>wˉ, she strictly prefers zero investment; if wˆ2<wˉ, she prefers to invest eNC.

It remains to determine the optimal contract for the firm. Among the contracts that are subsequently renegotiated, the optimal one is (w1,0), where w1 is determined by participation constraint [5] satisfied as equality. The profit under this optimal short-term (ST) contract is then the same as in the model without commitment:

[12]πNC=F(w2NC(eNC))(αeNC+β)+w2NC(eNC)ξdF(ξ)C(eNC)u.

If the firm offers a long-term (LT) contract that is not renegotiated afterward, its equilibrium profit is

[13]πLT=F(w^2)β+w^2ξdF(ξ)u,

and it is maximized when wˆ2=β: in this case the ability to commit eliminates the inefficiency of turnover. Moreover, if πLT with wˆ2=β is greater than πNC, it follows that

F(β)β+w^2ξdF(ξ)F(w2NC(eNC))(αeNC+β)+w2NC(eNC)ξdF(ξ)C(eNC)
>F(w2NC(eNC))(w2NC(eNC))+w2NC(eNC)ξdF(ξ)C(eNC)
=F(w¯)w¯+w¯ξdF(ξ)

so that wˆ2=β>wˉ and the agent indeed prefers to choose e=0 rather than deviate to a positive effort level. If πLT=πNC 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 πNC>max{πLT,0}, with a LT contract if πLT>max{πNC,0} and no employment if max{πLT,πNC}<0.

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 firmeHis larger than her investment at the less productive firmeL.

Proof. The equilibrium investment levels eH and eL are determined by eq. [4], where w2NC(e) is replaced by w2H(e) and w2L(e), respectively. Taking into account that ddew2H(e)>ddew2L(e) since d2w2NCdαde>0, and that w2i(e) is increasing in e by statement (i) of the same proposition, we immediately get the result from eq. [4]. ∎

To prove part (i), note that the entry wage offered by the less productive firm must be w1L. Indeed, this wage cannot be lower than w1L, since otherwise the worker would not accept it. It also cannot be larger since the firm would increase its profit by offering w1L defined in eq. [8].

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

[14]F(w2L(eL))(αLeL+βw2L(eL))w1LF(w2L(eH))(αLeH+βw2L(eH))w1H.

Let w_1H be the value of w1H which turns constraint [14] into equality. Similarly, the condition that the H-type firm does not want to mimic the equilibrium strategy of the L-type firm is

[15]F(w2H(eH))(αHeH+βw2H(eH))w1HF(w2H(eL))(αHeL+βw2H(eL))w1L.

Let wˉ1H be the value of w1H which makes constraint [15] binding. Thus, both incentive compatibility conditions will be satisfied iff w_1Hw1Hwˉ1H. Let us check that w_1H<wˉ1H. Substituting w1L to inequalities [14] and [15], we obtain that w_1H<wˉ1H is equivalent to

Δ(eH,eL,αH)>Δ(eH,eL,αL),

which is satisfied by Lemmas 1 and 3.

It is easy to see that any w1Hw_1H,wˉ1H can indeed be an equilibrium wage if we specify “pessimistic” out-of-equilibrium beliefs: the worker interprets any wage w1w1H as indicative of the L-type firm.

Finally, since eH>eL, condition [14] implies that w1H>w1L; 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 α).

  • (ii)

    First, let us analyze the separating equilibria. If a separating equilibrium satisfies the Intuitive criterion, for any out-of-equilibrium wage offer w1w_1H,wˉ1H 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 w1H>w_1H can be offered in a separating equilibrium that satisfies the Intuitive criterion (otherwise, the H-type firm would prefer to deviate to w1Hε). Finally, the lowest-wage separating equilibrium does satisfy the Intuitive criterion.

The second type of PBE is a pooling equilibrium. We analyze such equilibria in the following Lemma.

Lemma 41. There existw_1P<wˉ1Psuch that for anyw1P[w_1P,wˉ1P]there is a pooling PBE in which both types of firm offer wagew1P. 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 w1P. This wage should be sufficient to induce the worker to accept the offer:

[16]w1Puλ[F(w2H(eP))w2H(eP)+w2H(eP)ξdF(ξ)]
(1λ)[F(w2L(eP))w2L(eP)+w2L(eP)ξdF(ξ)]+C(eP)

, where eP is the worker’s equilibrium effort level:

eP=argmaxeλ[F(w2H(e))w2H(e)+w2H(e)ξdF(ξ)]
+(1λ)[F(w2L(e))w2L(e)+w2L(e)ξdF(ξ)]C(e)

. On the other hand, the equilibrium wage w1P cannot be too high – otherwise the L-type firm would prefer to pay a lower wage w1L (although the cost of such deviation would be the disclosure of the firm’s low type):

[17]w1Pw1L+F(w2L(eP))(αLeP+βw2L(eP))F(w2L(eL))(αLeL+βw2L(eL)),

where eL is defined in eq. [7]. From Lemma 1 we can easily obtain that if w1P satisfies inequality [17] the H-type firm also does not gain by deviating to wage w1L. Let w_1P,wˉ1P be the values of w1P which make constraints [16] and [17], respectively, be satisfied as equalities. Note that from [8] and [16] it can be seen that w_1P<w1L, and from [17]wˉ1P>w1L, so that w_1P<wˉ1P. Clearly, there is no pooling equilibrium with w1Pw_1P,wˉ1P. On the other hand, by specifying “pessimistic” out-of-equilibrium beliefs (i.e. such beliefs that the worker interprets any wage w1w1P as being given by the L-type firm), any wage w1Pw_1P,wˉ1P can be supported as equilibrium wage.

  • (ii)

    Consider a pooling equilibrium with wage w1P. Let w˜1L be the first-period wage that makes the L-type firm indifferent between its equilibrium payoff and the payoff it gets if offers w˜1L, assuming the agent reacts to w˜1L by choosing effort eH defined in eq. [7]:

F(w2L(eP))(αLeP+βw2L(eP))w1P=F(w2L(eH))(αLeH+βw2L(eH))w˜1L.

By Lemma 1 the H-type firm would strictly prefer to deviate to w˜1L (and, by continuity, to a wage w˜1L+ε for small enough ε>0) were the agent to react by choosing eH. But then, for the equilibrium to satisfy the Intuitive criterion the agent should interpret w˜1L+ε as indicative of the H-type firm, which would indeed create incentives for the H-type firm to deviate to w˜1L+ε 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 w1P and w1H>w1P and the low type offers w1P. Since the H-type firm is indifferent between w1H and w1P, the L-type firm strictly prefers the lower wage w1P to any wage w1Hε with ε>0 small enough. But then the Intuitive criterion implies that the worker should believe that wage w1Hε is offered by the H-type firm, thus destroying the equilibrium.

In the second type of hybrid equilibrium the L-type firm randomizes between w1L and w1P>w1L and the H-type firm offers w1P. 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 αH>αˉ. Let us first show that separating equilibria exist in this case. If αLis also larger than αˉ, 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 αL<αˉ.

Let us prove that there exists a separating equilibrium where the H-type firm offers (w1H,wˆ2H)=(w1H,0) for some w1H and the L-type firm offers the optimal long-term contract (w1L,wˆ2L)=(w1LT,β), where w1LT=uF(β)ββξdF(ξ). There are three constraints to be satisfied by w1H for this pair of contracts to form a separating equilibrium21:

[18](ICH):F(w2H(eH*))(αHeH*+βw2H(eH*))w1Hu+F(β)β+βξdF(ξ),
[19](ICL):F(w2L(eH*))(αLeH*+βw2L(eH*))w1Hu+F(β)β+βξdF(ξ),
[20](IRH):F(w2H(eH*))w2H(eH*)+w2H(eH*)ξdF(ξ)+w1HC(eH*)u.

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 wˉ1H be the value of w1H that makes constraint [18] binding. Substituting wˉ1H into [19] we get

F(w2L(eH))(αLeH+βw2L(eH))F(w2H(eH))(αHeH+βw2H(eH))

which is true since

F(w2L(eH))(αLeH+βw2L(eH))<F(w2L(eH))(αHeH+βw2L(eH))
F(w2H(eH))(αHeH+βw2H(eH)),

where the first inequality follows from αL<αH and the second from the optimality of w2H(eH).

The assumption that under symmetric information the H-type firm prefers short-term contracting implies that constraint [20] will be satisfied when w1H=wˉ1H. In fact, any (w1H,0) with w1H[w˜1H,wˉ1H], where w˜1H=min{w1H|w1H satisfies eqs [19] and [20]}, 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, eH>0,eL=0 which verifies the claim. ∎

Acknowledgments

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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  1. 1
  2. 2
  3. 3

    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.

  4. 4

    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).

  5. 5

    Waldman (1990) shows that results of Kahn and Huberman (1988) extend to the case when human capital is general, but information about it is privately observed by the incumbent employer.

  6. 6

    Moore (1985) studies labor contracts that emerge when workers have privately observed reservation wages, but his model does not include discretionary employee investment.

  7. 7

    The firm’s expected profit if it hires the worker at her reservation wage equals πNC=F(w2NC(eNC))(αeNC+β)+w2NC(eNC)ξdF(ξ)C(eNC)u, where eNC and w2NC(eNC) are characterized by eqs [4] and [2]. Hence, the firm chooses to hire the worker if uuˉ, where uˉ is determined by πNC=0.

  8. 8

    Indeed, 2wαe=(1+g(w))2αeg(w)(1+g(w))3 Since g(w) 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.

  9. 9

    See Kahn and Huberman (1988) and Waldman (1990) for the general analysis of up-or-out contracts.

  10. 10

    See, for example, Fudenberg and Tirole (1991).

  11. 11

    A sufficient condition for this is that the marginal cost is high enough and is increasing fast enough; see Footnote 8.

  12. 12

    Indeed, suppose w1L>w1H are the entry wages offered by the less and the more productive firms, and e1L and e1H are the investment levels subsequently chosen by their employees. Then, necessarily e1L>e1H and a revealed preference argument shows that Δ(e1L,e1H,αH)(w1Lw1H)Δ(e1L,e1H,αL), which contradicts Lemma 1.

  13. 13

    In the hybrid equilibrium of the first type the more productive firm randomizes between w1P and w1H>w1P and the less productive firm offers w1P; in the second the less productive firm randomizes between w1L and w1P>w1L and the more productive offers w1P.

  14. 14

    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.

  15. 15

    There would exist a range of pooling equilibria with w1[w_1,wˉ1], where the lowest wage w_1 leaves no rent to the employee, while the highest wage wˉ1 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.

  16. 16

    To be precise, there exists a function γ(αL) such that symmetric information contracts form a separating equilibrium if αL<αˉ and αˉ<αH<γ(αL). Function γ(αL) is decreasing since the larger is αL, the more tempting it is to deviate for the less productive firm; γ(αL)αˉ as αLαˉ.

  17. 17

    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.

  18. 18

    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.

  19. 19

    Unfortunately, we have been unable to find simple conditions on parameters that would discriminate between the different types of equilibria we considered.

  20. 20

    Sharing future profits is a substitute mechanism, but its scope is limited when the number of employees grows.

  21. 21

    Recall that in the optimal LT contract wˆ2=β so that the firm’s expected profit from such contract is u+F(β)β+βξdF(ξ)

Published Online: 2014-2-4
Published in Print: 2014-1-1

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