Status and incentives.

The expression in (A5) is strictly negative if the expression in the bracket is negative. The expression in the bracket is the derivative of [gamma](x) = x [e.sup.*'](x)[mu]'([e.sup.*](x)) with respect to x = [s.sub.i] [DELTA][w.sup.*.sub.i]. Using the first-order conditions for optimal effort, we obtain that [gamma](x) = [e.sup.*]'(x)[psi]'([e.sup.*](x)). Thus, using (A1), [gamma]'(x) = [e.sup.*]" [psi]'([e.sup.*]) + [([e.sup.*]').sup.2] [psi]"([e.sup.*]) = [([e.sup.*]').sup.2][psi]'([e.sup.*])[[e.sup.*]'[x[mu'"([e.sup.*]) - [psi]'"([e.sup.*])]/[mu]'([e.sup.*]) + 2[mu]"([e.sup.*])/[mu]'([e.sup.*]) + [psi]"([e.sup.*])/[psi]'([e.sup.*])], which is negative by Assumptions 1, 2, and 3. Hence, the partial derivatives in (A4) and (A5) have the same sign, so that [DELTA][w.sup.*]' ([s.sub.i]) < 0. Proposition 1 (iii), combined with the fact that [DELTA][w.sup.*] (s) is strictly decreasing in s, implies that all agents with zero low performance wage and for whom (IR) is not binding must have identical status levels.

Finally, suppose that there are two agents i and j with = [[w.bar].sub.j] = 0 and such that (IR) is binding for i only. Then, from Proposition 1(iii), this can only be possible if [s.sub.i] < [s.sub.j]. We have shown above that [[partial derivative].sup.2][PI]/[partial derivative][w.sup.*.sub.i] < 0 so that profit is concave in [DELTA][w.sup.*]. Hence, because the (IR) constraint for j is not binding, we must have [DELTA][w.sup.j] = [DELTA][w.sup.*]([s.sub.j]), which is optimal if the (IR) constraint is ignored. Similarly, the (IR) constraint being binding for i implies that [DELTA][w.sup.i] > [DELTA][w.sup.*i]([s.sub.i]) so that [DELTA][w.sup.*]([s.sub.i]) < [DELTA][w.sup.*]([s.sub.j]), which contradicts our result above that [DELTA][w.sup.*] is decreasing in status. Thus, this situation cannot be part of any optimal solution.

Proof of Proposition 3. Consider a steady state. There then exists ([c.sub.1], [c.sub.l], [c.sub.h]) such that ([c.sub.1t], [c.sub.lt], [c.sub.ht]) = ([c.sub.1], [c.sub.l], [c.sub.h]) for all t. The proof proceeds in three steps.

Step 1. [c.sub.1] = (0, 0, 0). If [s.sub.1] = 0, then it is optimal to set [[w.bar].sub.1] = [DELTA][[w.bar].sub.1] = 0. Thus, the proof of the result amounts to showing that [s.sub.1] = 0. Suppose to the contrary that [s.sub.1] > 0. At some date t, the principal may switch to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A6)

If each generation from t on is offered these contracts, the young's expected intertemporal utility is held constant. Basically, the young's wages are transferred from the first to the second period while being divided by the ratio of the original period 1 status to the new second period status [s.sub.1]/[s.sub.1]+[s.sub.p], p [member of] {l, h}, so that the increase in status exactly compensates for the decrease in income. The new intertemporal utility is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A7)

which is the intertemporal utility in the original contract. On the other hand, the utility of an old agent is increased (by [s.sub.1][[w.bar].sub.1]/[delta] for the l type and [s.sub.1]([[w.bar].sub.1] + [DELTA][w.sub.1])/[delta] for the h type). Furthermore, all effort levels are maintained. Finally, the intertemporal wage bill for each generation is lower: that is, ([mu]([e.sub.1])[s.sub.1]/[s.sub.h] + [s.sub.1] + (1 - [mu]) ([w.sub.1))[s.sub.1])/[s.sub.l]+[s.sub.1])E[w.sub.1] + [delta][mu]([e.sub.1]) [s.sub.h]/[s.sub.h]+[s.sub.1] E[w.sub.h] + [delta](1 - [mu]([e.sub.1])) [s.sub.l]/ [s.sub.l]+[s.sub.1] E[w.sub.l] < E[w.sub.1] + [delta][mu]([e.sub.1])E[w.sub.h] + [delta](1 - [mu]([e.sub.1]))E[w.sub.l]. Hence, a steady state with [s.sub.1] > 0 cannot be part of any optimal solution.

Step 2. If [U.sub.h] > [U.sub.l], then ([s.sub.h] > [s.sub.l]) and ([[w.bar].sub.h] [greater than or equal to] [[w.bar].sub.l] and ([DELTA][w.sub.h] > [DELTA][w.sub.l]) must hold. First note that the arguments used to prove Proposition 1(iii) may be applied to the old population at each period so that ([s.sub.h] > [s.sub.l]) implies ([[w.bar].sub.h] [greater than or equal to] [[w.bar].sub.l]) and ([DELTA][[w.sub.h] [greater than or equal to] [DELTA][w.sub.l]). Furthermore, if [U.sub.h] > [U.sub.l], we cannot have [s.sub.l] > [s.sub.h], because this would imply that wages for type l old workers should be at least as high as those of type h old workers, which contradicts [U.sub.h] > [U.sub.l].

Step 3. [U.sub.h] > [U.sub.l]. Because young agents have no status, proving the result amounts to showing that a steady state in which the young's effort is zero cannot be part of an optimal solution. In such a steady state, at each date, only the old exert effort. Now suppose that at some date t, the principal commits to giving only half of the status to the old at date t + 1. Then she is in a position to implement the egalitarian solution of Proposition 2, which is optimal in the static problem. That is, all agents can be awarded identical status and wages and they all exert the same effort: in particular, young agents are not induced to exert additional effort by the prospect of future utility differentials because there are none. Because the solution in which only the old (i.e., one fraction of the agents) exert effort is also feasible in the static problem, this yields a strictly lower per-period profit than the egalitarian solution. Thus, the young's effort must be strictly positive in the steady state of an optimal solution. Because the young exert effort in spite of zero status, we must have [U.sub.h] > [U.sub.l].

Proof of Proposition 4. Consider a steady state. An agent may face four possible states of nature depending on her performance in each of the two periods (i.e., ll, lh, hi, hh). To simplify the notation, the reference to the state of nature is dropped in the remainder of the proof. For one such state of nature, let st and w t denote the agent's status and wage when young, and [s.sub.2] and [w.sub.2] the agent's status and wage when old. Let v = g([s.sub.i])h([w.sub.i]) + [delta]g ([s.sub.2])h([w.sub.2]). Now suppose that [s.sub.l] > 0. If the principal switches to a solution ([s'.sub.1], [w'.sub.1], [s'.sub.2], [w'.sub.2]), with [s'.sub.1] = [w'.sub.1] = 0 and [s'.sub.2] = [s.sub.1] + [s.sub.2], v is unchanged as long as

h([w'.sub.2]) = g([s.sub.1])h([w.sub.1]) + [delta]g([s.sub.2])h([w.sub.2])/[delta]g([s.sub.1] + [s.sub.2]) (A8)

It can easily be shown that if this is done for all states of nature, effort levels and intertemporal expected utility are unchanged whereas the agent's utility when old increases. Suppose that h(w) = w. Then (A8) becomes to [w'.sub.2] = g([s.sub.1])h([w.sub.1]) + [delta]g([s.sub.2])h([w.sub.2]) /[delta]g([s.sub.1] + [s.sub.2]) Because g is strictly increasing, the discounted wage bill [delta][w'.sub.2] is lower than [w.sub.1~] + [delta] [w.sub.2]. Thus, the principal is better off. Suppose that g(s) is linear. Then (A8) can be written as

h([w'.sub.2]) = 1/[delta] [s.sub.1]/[s.sub.1] + [s.sub.2] h([w.sub.1]) + [s.sub.2]/ [s.sub.1]/[s.sub.1] + [s.sub.2] h([w.sub.2]). (A9)

Strict monotonicity and concavity of h imply

h([w.sub.1] + [w.sub.2]) > h[s.sub.1][w.sub.1] + [s.sub.2][w.sub.2]/[s.sub.1] + [s.sub.2] [greater than or equal to] [s.sub.1]h([w.sub.1]) + [s.sub.2]h([w.sub.2])/([s.sub.1] + [s.sub.2]). (A10)

Thus, for [delta] close to 1, because h is strictly increasing, if [w'.sub.2] satisfies (A9), then [delta][w'.sub.2] < [w.sub.1] + [delta][w.sub.2].

We would like to thank the editor and an anonymous referee for their fruitful comments and suggestions. We are also indebted to seminar participants at Fourgeot seminar (Pads), Institut d'Analisi Economica (Barcelona), Universit6 d'Aix-Marseille 2, Universit6 Catholique de Louvain, Universit6 de Caen, Universite des Sciences Sociales de Toulouse, University of Virginia, Stockholm School of Economics, Erasmus University, and the participants in the workshop Social Interaction and Economic Behavior in Paris December 1999 for stimulating criticism and comments on an early version of the article, participants at the conference Organizational Behaviour, Structure and Change; The Economics of Personnel and Organizations in Toulouse May 2003, and especially Lucy White, for their comments and fruitful discussions. We are thankful to Thomas Madotti and Made-Christine Henninger for their help and suggestions. Finally, we are also grateful to Andrew Clark for proofreading the entire manuscript and improving the English. All remaining errors are ours.

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