Status and incentives.

* The article has argued that social recognition plays a major role in the workplace. Social aspects are all the more significant given that much of labor relations takes place outside of the market and is medium to long term. Our analysis relies on the following two premises: recognition and income are complements; and recognition is scarce because it is valued in relative terms. Our main findings are that whereras it is costly to introduce differentiation between identical coworkers in a static environment, such differentiation might prove to be a relatively powerful incentive device in a dynamic setting. In the intertemporal incentive scheme, pay is attached to job, rewards are delayed in time, and higher income is associated with greater recognition. From an empirical perspective, the proposed framework yields predictions on the shape of the compensation scheme in relation to the hierarchical structure in internal labor markets and spot markets. Stylized facts are consistent with our results.

Our theoretical analysis predicts that internal labor markets are a superior mode of work organization. If this is the case, we may wonder why firms do not resort to them more systematically. This might not always be possible. To organize an internal labor market, firms need not only commit to keep employees but also be large enough or growing fast enough to propose stimulating career paths. For firms in recession or in unstable economic environments, flexibility matters, so that commitment is not always possible. There will then be no benefit in creating a hierarchical structure for incentive purposes. In recent years, there has been a significant move toward delayering in industrial countries. For instance, Bauer and Bender (2001) examine a representative German employer-employee data set and reveal that between 1993 and 1995, 50.73% of the 251 firms sampled reduced their number of hierarchical levels. In the same spirit, using a panel of 300 U.S. firms, Rajan and Wulf (2003) find that firms' depth (i.e., the number of positions between the CEO and division heads) fell by more than 25% between 1986 and 1999. (34) According to our analysis, this might reflect a weakening of employer commitment, which could itself be explained by an anticipated rise in the job loss rate. There is indeed evidence of such an increase during the 1990s (see Farber, 1997).

Appendix

* Proof of Proposition 1. The proofs of Conditions (i) and (ii) are straightforward: first note that, for a given status level, [s.sub.j], total surplus is a strictly concave function of effort, which reaches a maximum at [e.sup.*] ([s.sub.i][DELTA]q). Thus, if [DELTA][w.sub.i] > [DELTA]q, total surplus may be increased by decreasing [DELTA][w.sub.i]. Then profit may be increased while keeping the agents' utility unchanged by increasing [[w.bar].sub.i]. By a symmetric argu [ 0, profit could be increased by increasing [DELTA][w.sub.i] and decreasing [[w.bar].sub.i].

Proof of Condition (iii). First note that in the optimal incentive scheme, we must have ([[w.bar].sub.i], [DELTA][w.sub.i]) = (0, 0) [??] [s.sub.i] = 0. Thus, if [s.sub.i] = 0, the result holds. Second, when [s.sub.j] > [s.sub.i] > 0, we prove the result by showing that if (iii) does not hold, the principal can increase her prof e

Let [phi] be the composition of [mu] and [e.sup.*], [phi] = [mu] [omicron] [e.sup.*]. The probability [mu] being increasing and concave in effort, [phi] is concave as long as [e.sup.*] is concave, which is the case under Assumptions 1 and 2. This can be seen from

[e.sup.*]"(x) = [(e').sup.3][x[mu]'"(e) - [psi]'"(e)] + 2[(e').sup.2] [mu]"(e)/[mu]'(e). (A1)

Consider a change in status for some agent i by some amount [epsilon] and consider changes in wages that keep the agent's utility constant: because effort is chosen optimally by the agent, when taking the derivative of utility with respect to [epsilon], the envelop theorem implies that only the direct impact of changes in status and wages need be considered. First suppose that [[w.bar].sub.i] > 0 so that, from (ii), [DELTA][w.sub.i] = [DELTA]q. Then let [[alpha].sub.i]([epsilon]) be the low performance wage that keeps utility constant. Thus, [[alpha].sub.i](0) = [[w.bar].sub.i] and the derivative of ([s.sub.i] + [[epsilon]])[[[alpha].sub.i]([epsilon]]) + [DELTA]q[phi] (([s.sub.i] + [[epsilon]) [DELTA]q)] with respect to [[epsilon]] must be zero so that [[alpha]'.sub.i]([epsilon]) = - [[alpha].sub.i]([epsilon])+ [DELTA]q[phi](([s.sub.i] + [epsilon])[DELTA]q/[s.sub.i]+[epsilon] (where the derivative with respect to the term inside [phi] is ignored due to the envelop condition on effort). If [[w.bar].sub.i] ([s.sub.i] + [epsilon]) [[beta].sub.i]([epsilon]) = [s.sub.i][DELTA] [w.sub.i]. Hence, [[beta].sub.i](0) = [DELTA][w.sub.i] and [[beta]'.sub.i]([epsilon]) = - [[beta].sub.i]([epsilon]])/[s.sub.i] + [epsilon]. Finally, note that if we consider the profit generated by agent i's work, its derivative with respect to [member of] evaluated at [member of] = 0 is merely the change in the expected wage bill [[alpha]'.sub.i] (0) or [phi]([s.sub.i] [DELTA][w.sub.i])[[beta]'.sub.i] (0). In the former case, because [DELTA][w.sub.i] = [DELTA]q, the effort level maximizes profit subject to the individual rationality constraint and thus the envelop theorem applies. In the latter case, there is no change in effort because ([s.sub.i] + ([epsilon])[beta]([epsilon]) is kept constant.

Now assume [s.sub.j] > [s.sub.i] > 0. We show that profit may be increased by an [epsilon] > 0 transfer of status from j to i along with an adjustment in wages so that both agents' utility levels remain unchanged. From (i) and (ii), if (iii) does not hold, three cases may arise.

Case 1. [[w.bar].sub.i] > [[w.bar].sub.j] > 0 (and [DELTA][w.sub.i], = [DELTA][w.sub.j] = [DELTA]q). The derivative of profit with respect to [epsilon] evaluated at [epsilon] = 0 is [[alpha]'.sub.j](0) - [[alpha]'.sub.i](0) = [[w.bar].sub.i]/[s.sub.i] - [[w.bar].sub.j]/[s.sub.j] + [DELTA]q([phi]) ([s.sub.i][DELTA]q)/[s.sub.i] - [phi]([s.sub.j][DELTA]q)/[s.sub.j])). This derivative is strictly positive because, [phi](s [DELTA]q) being concave and equal to 0 when s = 0, [psi] (s [DELTA]q)/s is decreasing in s.

Case 2. [[w.bar].sub.i] > [[w.bar].sub.j] = 0 and 0 [DELTA][w.sub.j] < [DELTA][w.sub.i] = [DELTA]q. The derivative of profit with respect to [member of] at [epsilon] = 0 is [[beta]'.sub.i](0) - [[alpha]'.sub.i](0) = [[w.bar].sub.j]/[s.sub.i] + [[phi]([s.sub.i][DELTA]q)[DELTA]q/[s.sub.i] - [[phi]([s.sub.i][DELTA]q)[DELTA]q/[s.sub.j]] + [[phi] ([s.sub.j][DELTA][w.sub.j])[DELTA]q/[w.sub.j]/[s.sub.j]], which is positive because [phi](s[DELTA]q/s is decreasing in s (see Case 1) and [phi](s [DELTA]w) [DELTA]w is increasing in [DELTA]w.

Case 3. 0 < [DELTA][w.sub.j] < [DELTA][w.sub.i] [less than or equal to] [DELTA]q. The derivative of profit with respect to [epsilon] for [epsilon] = 0 is [[beta]'.sub.j](0) - [[beta]'.sub.i](0) = [[phi]([s.sub.i][DELTA][w.sub.i])[DELTA][w.sub.i] - [phi]([s.sub.i] [DELTA][w.sub.j])[DELTA][w.sub.j]]/[s.sub.i] + [[phi]([s.sub.i][DELTA][w.sub.j])[DELTA][w.sub.j]/[s.sub.i] - [phi]([s.sub.j][DELTA][w.sub.j])[DELTA]/[w.sub.j]/[s.sub.j]], which is strictly positive because [phi](s[DELTA]q)/s is decreasing in s (see Case 1) and [phi](s[DELTA]w) [DELTA]w is increasing in [DELTA]w.

Finally, the "if" part of Condition (iii) does hold since if [s.sub.i] = [s.sub.j], the monetary incentive for the two agents will be the same.

Proof of Proposition 2. We prove Proposition 2 under the following assumption.

Assumption 3. The functions [mu] and [psi] satisfy

[psi]"(e)/[psi]'(e) [less than or equal to] - 2[mu]"(e)/[mu]'(e) (A2)

We have shown in the text, after Proposition 2, that if [[w.bar].sub.i] > 0 for some i, then all agents in the organization must have equal status, and thus by virtue of Proposition l(iii), the same contract.

Now consider agents for whom [[w.bar].sub.i] = 0 and (IR) does not bind. Setting the first derivative of expected profit with respect to [DELTA][w.sub.i] equal to 0, the optimal solution [DELTA][w.sub.*]([s.sub.i]) must satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A3)

Applying the inverse function theorem, we have [DELTA][w.sub.*]' ([s.sub.i]) = - [[partial derivative].sup.2]E[PI]/ [partial derivative][DELTA][w.sub.i][partial derivative][s.sub.i] / [[partial derivative].sup.2]E[PI]/[partial derivative][DELTA][w.sup.2.sub.i] The second partial with respect to [DELTA][w.sub.i] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A4)

This is strictly negative if [e.sup.*] is concave, which is true by Assumptions I and 2. The cross partial is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A5)