Ex ante versus ex post optimal promotion rules: the case of internal promotion.

Waldman, Michael

I. INTRODUCTION

The literature on promotion practices within the firm identifies two distinct roles for a firm's promotion policies. Promotions serve both as a way for the firm to efficiently assign workers to tasks and as a way for the firm to reward prior performance. (1) In this article I explore the time-inconsistency problem that arises as a result of the dual nature of the promotion decision. In particular, I show that the common practice of favoring internal candidates for promotion can be understood as a response by firms to the problem of time inconsistency. (2)

Consider a firm in which promotions serve as a reward for prior expenditures of effort and/or prior investments in human capital. The optimal promotion policy for this firm depends on the time frame from which we view the problem. The ex ante optimal promotion rule is the optimal rule when the problem is viewed from a date prior to young workers deciding to work at this firm. This rule takes into account both the assignment aspects of the promotion decision and that promotions serve as a reward for prior performance. There is a second relevant rule, however, because at the date of the promotion decision the effects of promotion serving as a reward for prior performance are in the past. For example, in a world where the possibility of future promotion serves to increase the effort levels of young workers, by the time of the promotion decision those effort choices have already been made. Hence, the ex post optimal rule only takes into account the assignment aspects of the promotion decision.

The time-inconsistency problem that arises is now clear. Total profits are maximized when the firm follows the ex ante optimal promotion rule, but in the absence of commitment this is not how the firm behaves. At the time of the promotion decision the firm has an incentive to ignore the effects of promotion serving as a reward for prior performance, and instead follow the ex post optimal rule. In other words, the profits of the firm are decreased because at the time of the promotion decision the firm maximizes current profits, and the anticipation of this behavior by workers has deleterious effects on behavior in prior periods.

In this article I explore a specific example of this time-inconsistency problem. Consider a firm's decision concerning whether to fill a managerial position by promotion from within or by hiring an outsider. I show that when confronted with this decision the firm faces the exact situation described. That is, the ex post optimal rule shows no preference for promotion from within over hiring from the outside, whereas the ex ante optimal rule exhibits a preference for promoting from within. The logic here is that the possibility of receiving the higher managerial wage in the future reduces the wage required to attract young workers into the firm, but only the ex ante optimal rule takes this into account. The result is that in the absence of commitment the firm hires from the outside too often. The conclusion is that the establishment of an internal labor market in which the firm favors promotion from within can be understood as a way that the firm avoids this time-inconsistency problem.

This result extends a finding in Malcomson (1984). That article also shows that an internal labor market in which hiring from the outside is restricted can arise in a world where promotion serves as a reward for prior performance. In that work, however, there is no time-inconsistency problem. The reason is that Malcomson assumes workers are homogeneous, and thus at the time of the promotion decision it is (weakly) optimal for the firm to have all promotions be from within. In contrast, I assume that workers are heterogeneous and find that if the firm maximizes ex post profits it lowers total profits. The result is an internal labor market that not only limits hiring from the outside but also constrains the firm to behave in a manner different from that which maximizes profits at the time the promotion decision is made.

Another closely related paper is Chan (1996). He also presents an analysis in which the firm prefers to promote from within, but his argument is different both from that put forth here and from Malcomson's argument. Chan argues that to avoid the high wage associated with promotion from within, a firm will sometimes hire an outsider when the efficient contract calls for promotion of an insider. In turn, he further argues that the firm can avoid the problem by lowering the wage associated with promotion from within and only hiring an outsider when the outsider is much superior to the best internal candidate. Chan's argument is distinctly different than Malcomson's in that Chan assumes heterogeneous workers, and this is an integral part of his argument. The argument is also distinctly different than that put forth here in that there is no production in Chan's model after the promotion decision takes place. Thus, whereas the focus here is on the interplay between optimal incentives and optimal assignment, in Chan 's analysis this is not an issue because there is no assignment aspect of the promotion decision. (3)

The analysis in this article builds on an insight found in Milgrom and Roberts (1988). That study was the first to recognize the time-inconsistency problem that arises as a result of the dual nature of the promotion decision. Milgrom and Roberts (1988) consider a world characterized by influence activities, where influence activities refer to activities that a worker can engage in that are not directly productive but that can influence a firm's beliefs concerning the worker's ability. Their focus is a time-inconsistency problem that arises due to the unproductive nature of these activities. In particular, they show that from an ex post standpoint the firm has an incentive to promote the workers it believes are most productive in the higher-level task, but this creates an incentive for workers in prior periods to expend effort in unproductive influence activities that affect the firm's perceptions concerning who would be best at the higher-level task. As a result, in their analysis the firm sometimes commits t o promote the workers most productive in the lower-level task, rather than the ones it thinks will be most productive in the higher-level task. In other words, as in the current analysis, the firm commits to an assignment of workers to tasks that is ex post inefficient. (4)

As a final introductory point, it is worth noting that there are alternative explanations for why higher levels of a firm's job ladder might be staffed entirely or almost entirely with insiders. For example, this could be due to the presence of specific human capital as first explored by Becker (1962). Or it could be because the firm has better information about the abilities of its own workers and is thus hesitant to fill a high-level position with an outsider for whom it has less precise information. (5) One aspect of the argument presented that is distinctly different than those previous explanations is that, as discussed, the firm does not only refuse to hire outsiders for higher-level positions. The firm also needs to constrain its own behavior to achieve this goal. I come back to this issue in section IV, where I discuss methods of commitment.

The outline for the article is as follows. Section II presents an overlapping-generations model in which the possibility of future promotion serves as an incentive for young workers to provide effort. Section III analyzes the model and shows that time inconsistency can cause the formation of an internal labor market in which the firm always promotes the best internal candidate. Section IV first discusses using a promotion standard for internal promotion (rather than a total prohibition on hiring from the outside), and then discusses methods a firm may employ to bind its own future actions. Section V presents concluding remarks.

II. THE MODEL

Consider an overlapping-generations setting in which each worker lives two periods. In each labor cohort there is a pool of Z risk-neutral workers who discount the future by a factor [beta]. Each worker has associated with him or her a value for a variable that will be called ability and will be denoted [[theta].sub.i] for worker i. It is assumed that [[theta].sub.i] is a random draw from a distribution described by a probability density function h([theta]), where h([theta]) > 0 for all [[theta].sub.L] [less than or equal to] [theta] [less than or equal to] [[theta].sub.H], and h([theta]) = 0 for all [theta] outside this interval. At the beginning of a worker's career, everyone in the economy (including the worker) knows h([theta]) but does not know the worker's actual realization of ability. Each worker must also choose an effort level each period. [e.sub.it] is the effort level choice of worker i in period t, where effort can be either low or high, that is, [e.sub.it] [member of] {[e.sup.l], [e.sup.h]}, and it is assumed [e.sup.h] - [e.sup.l] < [[theta].sub.H] - [[theta].sub.L]. The assumption [e.sup.h] - [e.sup.l] < [[theta].sub.H] - [[theta].sub.L] ensures that there are realizations for an entry-level worker's output such that the firm is uncertain whether the worker chose high or low effort. g([e.sub.it]) is the disutility an individual receives from effort, where g([e.sup.h]) > g([e.sup.l]) = 0 and g([e.sup.h]) < [e.sup.h] - [e.sup.l]. The assumption g([e.sup.h]) < [e.sup.h] - [e.sup.l] ensures that high effort is efficient.

There is a single risk-neutral firm in this economy that also has a discount factor [beta] and is in operation through period T (in section IV I also discuss what happens when the firm is infinitely lived). Each period the firm is in operation it has 2N job slots to fill, 2N < Z. N slots are entry-level positions filled by young workers. If young worker i is in an entry-level position in period t, then the worker's output equals [[theta].sub.i] + [e.sub.it] There are N - 1 skilled positions filled by old workers (a young worker placed in a skilled position produces zero). If old worker i is in a skilled position in period t, then output equals [lambda][[theta].sub.i] + [e.sub.it], where [lambda] > 1. There is a single managerial position, which is also filled by an old worker (as with the skilled positions, a young worker placed in the managerial position produces zero). If old worker i is in the managerial position in period t, then output equals [delta][[theta].sub.i] + [e.sub.it], where [delta] > [lambda]. Finally, the firm does not observe a worker's effort level, and a worker's output is privately observed by the firm as opposed to being publicly observed and verifiable. Note that because a worker's output is privately observed by the firm, contracts where payment is directly contingent on output are not feasible. As a result, old workers employed by the firm choose effort [e.sup.l] rather than [e.sup.h].

A worker who does not work at the firm is self-employed, and a self-employed worker receives utility U. It is assumed that [[theta].sub.L] + [e.sup.l] U. This assumption guarantees that productivity at the firm is sufficiently high that the firm can attract workers and earn positive profits. Note in this specification the firm acquires no information about those who were self-employed in the previous period. As a result, if the firm hires an outsider for the managerial position in period t, then it simply chooses randomly from the pool of old workers who were self-employed in period t - 1. I impose this assumption for tractability reasons. Clearly, a more realistic model would allow the firm to acquire some information about the ability levels of external candidates for the managerial position. My sense is that as long as the firm is limited in terms of the number of outside candidates it can evaluate, allowing the firm to acquire some information about the ability levels of external candidates would not affe ct the qualitative nature of the results.

In the analysis that follows it is assumed that in each period t, 1 [less than or equal to] t [less than or equal to] T - 1, the firm hires young workers by offering a contract that specifies [W.sup.E.sub.t], [W.sup.S.sub.t+1], and [W.sup.M.sub.t+1], where [W.sup.E.sub.t] is the entry-level wage in period t, [W.sup.S.sub.t+1] is the skilled wage in period t + 1, and [W.sup.M.sub.t+1] is the managerial wage in t + 1. Note because it is assumed that workers cannot bind themselves to the firm, a worker always has the option of choosing self-employment when he or she is old. Hence, in equilibrium the firm always specifies wages [W.sup.S.sub.t+1] [greater than or equal to] U and [W.sup.M.sub.t+1] [greater than or equal to]U. (6)

Finally, in addition to specifying wages, the contract offered to young workers in period t also indicates the extent to which the firm can hire outsiders for the managerial position in the following period. I consider two different possibilities for the extent to which the firm can commit its future promotion decisions. In section III it is assumed that commitment is all or nothing. That is, in each period t the firm either makes no commitment or commits to fill the managerial position in period t + 1 with a worker who was in an entry-level position in period t. (7) In section IV I consider the alternative assumption that the firm can commit to a promotion standard. (8)

III. ANALYSIS

The literature on labor market tournaments typically assumes that a firm can bind itself to a set of prizes that it will pay in future periods. The point of the following analysis is that when workers vary in terms of ability, it may also be important for the firm to commit to the manner in which workers will be assigned to jobs in the following period. In particular, extending the analysis of Malcomson (1984), I show that the firm sometimes has an incentive to commit to an assignment rule that limits its own ability to hire an outsider for positions that serve as a reward for prior performance.

The analysis that follows considers two cases. In the first case the managerial wage is the same whether the manager is promoted from within or is hired from the outside. Specifically, in any period t + 1 each type of manager is paid the managerial wage specified in the contract offered to young workers in period t, [W.sup.M.sub.t+1]. This case is useful to analyze because the problem associated with staffing the managerial position with an outsider is very clear in this case. However, in many or possibly even most cases, assuming that the firm is restricted to paying the same wage to each type of manager is probably not a valid assumption. I thus also analyze a second case in which the firm is allowed to pay a manager hired from the outside a different wage than that paid to managers who are promoted from within. (9)

I begin by considering the ex post optimal promotion rule for period t + 1 given that both types of managers are paid the same wage (remember, the ex post optimal promotion rule ignores the effects of the promotion rule on behavior in prior periods). Suppose N = 2, and let [[theta].sup.*] be the expected value for [theta] and [y.sub.jt] denote the output of the worker in the jth entry-level position in period t. At the end of period t the firm observes [y.sub.1t] and [y.sub.2t], and from these two values the firm draws inferences concerning the ability levels of its entry-level workers in period t. Let [[theta].sup.E.sub.jt] denote the expected ability of the jth entry-level worker in period t, that is, [[theta].sup.E.sub.jt] = E([[theta].sub.j]\[y.sub.jt]). We also know that for the young workers who were self-employed in period t, the firm learns nothing concerning the workers' abilities, and thus [[theta].sup.*] is the expected ability in period t + 1 for each of these workers.

Given this information, the ex post optimal promotion rule is simple. There are three cases. First, suppose [[theta].sup.E.sub.1t] > [[theta].sup.*] and [[theta].sup.E.sup.2t] > [[theta].sup.*]. Then in period t + 1 both entry-level workers are retained by the firm, where the managerial position is staffed by the worker for whom expected ability is higher. Second, suppose [[theta].sup.E.sub.1t] > [[theta].sup.*] and [[theta].sup.E.sub.2t] < [[theta].sup.*] (or [[theta].sup.E.sub.1t] < [[theta].sup.*] and [[theta].sup.E.sub.2t] > [[theta].sup.*]). Then in period t + 1 the entry-level worker with the higher expected ability is retained and placed in the managerial position, and the skilled position is filled by a worker who was self-employed in period t. Third, suppose [[theta].sup.E.sub.1t] < [[theta].sup.*] and [[theta].sup.E.sub.2t] < [[theta].sup.*]. Then in period t + 1 both positions are filled with workers who were self-employed in period t. To summarize, from the ex post standpoint profits are maximized w hen the firm hires the workers with the highest expected abilities.

The analysis easily extends to the general case of N entry-level positions. In period t + 1 the firm will retain all the period t entry-level workers whose values for [[theta].sup.E.sub.jt] are above [[theta].sup.*], whereas the firm's remaining old workers will be drawn from the ranks of the period t self-employed. Furthermore, of the old workers it employs in period t + 1, the one assigned to the managerial position is the individual with the highest expected ability. This means that, if none of the entry-level workers in period t have a value for [[theta].sup.E.sub.jt] above [[theta].sup.*], then the managerial position in period t + 1 is filled by an individual who was self-employed in period t. (10)

In the preceding discussion I showed that from an ex post perspective the firm has an incentive to sometimes staff the managerial position with an outsider. I now consider equilibrium behavior, in particular whether the firm ever commits to only staff the managerial position with an insider. The reason the firm might commit to such a rule is that in equilibrium the firm will take an ex ante rather than an ex post perspective. That is, in equilibrium the firm will be concerned with how the promotion rule in one period affects behavior in prior periods.

PROPOSITION 1. Holding everything (including [[theta].sub.L] and [[theta].sub.H]) fixed except h(*), if [[theta].sup.*] - [[theta].sub.L] is sufficiently small, then (i)-(v) characterize equilibrium behavior in periods t and t + 1 for all 1 [less than or equal to] t [less than or equal to] T - 1. (11)

(i) [W.sup.M.sub.t+1] > [W.sup.S.sub.t+1] = U.

(ii) [W.sup.E.sub.t] - g([e.sup.h]) + [beta][(1/N)[W.sup.m.sub.t+1] + ([N - 1]/N)U] = (1+[beta])U.

(iii) Entry-level workers in period t provide high effort, whereas workers in the managerial and skilled positions in periods t and t + 1 provide low effort.

(iv) The firm commits to staff the managerial position in period t + 1 with an individual who was an entry-level worker in period t, and then staffs the position with the period t entry-level worker who produces the highest output.

(v) The skilled positions in period t + 1 are staffed by the remaining period t entry-level workers whose outputs in period t were greater than or equal to [[theta].sup.*] + [e.sup.h], and with workers who were self-employed in period t.

In each period t + 1, the ex post optimal promotion rule staffs the managerial position and the skilled positions with the old workers who have the highest values for expected ability. Proposition 1 tells us that this is not how the firm behaves in equilibrium. If [[theta].sup.*] - [[theta].sub.L] is sufficiently small, then in each period t the firm commits to staff the managerial position in period t + 1 with one of the firm's period t entry-level workers. This means that, even when the best period t entry-level worker has a value for [[theta].sup.E.sub.jt] below [[theta].sup.*], it is still the case that in period t + 1 the managerial position is filled by an insider.

The logic for this result is as follows. In equilibrium, the firm uses the assignment rule in period t + 1 as a way of motivating entry-level workers in period t to provide effort [e.sup.h] rather than [e.sup.l]. In particular, the firm pays a managerial wage in t + 1 higher than what is necessary to attract a worker into the managerial position, and the possibility of receiving the supranormal wage in t + 1 provides the entry-level workers in t with the incentive to choose the high effort level. Given this, consider the wage required to attract young workers in period t into the entry-level position, [W.sup.E.sub.t]. Holding [W.sup.M.sub.t+1] fixed, this wage is negatively related to the probability that an entry-level worker in t will be promoted into the managerial position in t + 1. Hence, to increase this probability and thus reduce the period t entry-level wage, the firm prohibits outsiders from filling the managerial position in t + 1.

A different way of describing the time-inconsistency problem that underlies this result concerns the cost to the firm of paying the prize associated with the managerial position, that is, paying a manager [W.sup.M.sub.t+1] rather than U. From the standpoint of the present discounted value of the firm's profit stream evaluated at any date t, there is no cost to the firm of paying a high wage in period t + 1 to a manager who is promoted from within. The reason is that the entry-level wage in period t adjusts such that over the course of an entry-level worker's lifetime the worker's expected utility exactly equals what he or she would receive in self-employment. In contrast, from the standpoint of the present discounted value of the firm's profit stream evaluated at any date t, there is a cost of paying a high wage in period t + 1 to a manager hired from the outside. The reason is that the entry-level wage in period t does not adjust down in anticipation of a high wage earned by a t + 1 manager hired from the ou tside. The time-inconsistency problem is simply that in deciding in period t + 1 who to promote the firm evaluates promoting from within over hiring from the outside as having equal costs, whereas from the standpoint of period t promoting from within has a distinctly lower cost.

Notice the proposition states that (i)-(v) hold if [[theta].sup.*] - [[theta].sub.L] is sufficiently small. The rationale for this condition concerns the size of the ex post cost of always staffing the managerial position with an insider. When [[theta].sup.*] - [[theta].sub.L] is small, the potential lost productivity in each period t + 1 from not following the ex post optimal promotion rule is also small. The result is that it is optimal for the firm to prohibit outsiders from filling the managerial position. When [[theta].sup.*] - [[theta].sub.L] is large, the potential lost productivity from not following the ex post optimal promotion rule is potentially large, and the result is that it can be optimal for the firm to allow an outsider to fill the managerial position.

Proposition 1 demonstrates that by restricting its own ability to hire from the outside, the firm solves a time-inconsistency problem concerning who fills the managerial position. An assumption in the analysis, however, is that a manager promoted from within is paid the same wage as a manager hired from the outside. Because, as indicated earlier, for many settings this is probably not a valid assumption, a natural question to ask is how the analysis changes when the firm has the option of paying two different managerial wages--one for managers promoted from within and one for managers hired from the outside. I now consider this case.

The following proposition considers what happens when in each period t, 1 [less than or equal to]t [less than or equal to] T - 1, the contract offered to young workers specifies four wages: [W.sup.E.sub.t], [W.sup.S.sub.t+1], [W.sup.M.sub.t+1], and [W.sup.M'.sub.t+1]. [W.sup.M.sub.t+1] now denotes the managerial wage in period t + 1 for a manager who was an entry-level worker in period t, and [W.sup.M'.sub.t+1] denotes the managerial wage in period t + 1 for a manager who was self-employed in period t.

PROPOSITION 2. Suppose that in each period t, 1 [less than or equal to] t [less than or equal to] T - 1, the contract offered to young workers specifies [W.sup.E.sub.t], [W.sup.S.sub.t+1], [W.sup.M.sub.t+1] and [W.sup.M'.sub.t+1] Holding everything (including [[theta].sub.L] and [[theta].sub.H]) fixed except h(*), if (N - 1)g([e.sup.h]) > [beta][delta]([[theta].sub.H] - [[theta].sub.L]) and [[theta].sup.*] - [[theta].sub.L] is sufficiently small, then (i)-(v) of Proposition 1 characterize equilibrium behavior in periods t and t+1 for all 1[less than or equal to] t [less than or equal to] T - 1. (12)

Proposition 2 tells us that even if the firm can specify two managerial wages, for some parameterizations the firm still restricts the managerial position in each period t + 1 to individuals who were entry-level workers in period t. There are now three steps to the argument. First, if the firm specifies a managerial wage for a manager hired from the outside that is below U, then an outsider would never accept it and the outcome would be just as if the firm committed to never hire an outsider for the managerial position (see note 12). Second, in the earlier analysis the firm sometimes committed to never hire an outsider for the managerial position because of a time-inconsistency problem. There was a time-inconsistency problem present because in the absence of a restriction concerning hiring from the outside for the managerial position, the prize associated with promotion was sometimes rewarded to an outsider. This problem, however, is also present when the firm can specify two managerial wages if the firm spec ifies a managerial wage for a manager hired from the outside that is above U and there is a positive probability of hiring from the outside (see again note 12).

Third, the other possibility is that the firm specifies a managerial wage for outsiders that just equals U. This possibility avoids the problem of having a prize for promotion that is sometimes received by an outsider. Nevertheless, this solution does not work if N is large and [[theta].sup.*] - [[theta].sub.L] is small. The problem is that, if [W.sup.M'.sub.t+1] = U and these two conditions hold, then a value for [W.sup.M.sub.t+1] potentially high enough to get entry-level workers in period t to choose [e.sup.h] would cause the firm to always fill the managerial position with an outsider. In other words, the firm would not be able to induce high effort among its entry-level workers because the workers would anticipate a zero probability of receiving the high managerial wage in the future.

As a final point, one might ask whether what we know about real-world promotion practices is consistent or inconsistent with the theoretical arguments presented so far. I have been able to find relatively little hard evidence concerning promotion from within versus hiring an outsider except in the case of CEOs, and even in that case there is not much evidence. I would say what I have found, however, is basically consistent with the argument. For example, one of the basic points is that the firm is more likely to restrict promotions to insiders when there is little potential extra productivity associated with hiring an outsider. Two reasons this could occur are that the firm is large, in which case the best internal candidate is likely to be of high ability, or that the industry is heterogeneous, so that outsiders are unlikely to have the human capital required to be a successful CEO. Both Parrino (1997) and Agrawal et al. (2001) study CEO succession empirically and find results consistent with each prediction.

In summary, in this section I have shown that a firm might want to commit to staff with insiders the position or positions that serve as a reward for prior performance. The logic is that because of time inconsistency, in the absence of such a commitment the firm ignores how its promotion decision in one period affects the entry-level wage in the prior period with the result that the firm hires from the outside too often. I also showed that this result can even arise when the firm has the option of paying two different managerial wages--one for managers promoted from within and one for managers hired from the outside. The logic here is that one potential way to avoid the time-inconsistency problem is to have no prize associated with hiring a manager from the outside. But this can result in an inability on the part of the firm to induce high effort among its entry-level workers because a managerial wage high enough to potentially induce high effort would result in only outsiders being promoted.

IV. DISCUSSION

In the previous section I demonstrated that a firm sometimes maximizes its own profits by establishing an internal labor market in which the managerial position can only be staffed by a worker who was promoted from within. This section covers two issues. First, I discuss how the analysis changes when the firm can commit to a promotion standard for internal promotion rather than only being able to commit to a total prohibition on hiring from the outside. Second, I discuss methods a firm may employ to bind its own future actions.

Promotion Standards

In section III it was assumed that the firm has limited ability to commit to a promotion rule. In particular, in each period t the firm either made no commitment or committed to fill the managerial position in period t + 1 with a worker who was an entry-level worker in period t. In this subsection I discuss how the analysis changes when the firm's commitment ability is more nuanced. That is, I assume the firm is able to commit to a promotion standard rather than only being able to commit to always promote an internal candidate. By a promotion standard I mean the firm commits to an output level such that an internal candidate is promoted in period t + 1 as long as one of the period t entry-level workers produces an output level at least equal to the standard.

I again begin by considering the case in which the managerial wage is the same whether the manager is promoted from within or is hired from the outside. As in section III, in this case the firm will commit to always staff the managerial position with an internal candidate given [[theta].sup.*] - [[theta].sub.L] sufficiently small. That is, the firm will commit to a promotion standard sufficiently low that the best entry-level worker's output is always at least equal to the standard. The logic for this result is the same as the logic for the equivalent result in section III. When [[theta].sup.*] - [[theta].sub.L] is small, the potential lost productivity associated with never hiring an outsider for the managerial position is also small. In turn, because of the savings from the reduced entry-level wage, it is optimal given such a parameterization to always promote an internal candidate.

Despite the similarity discussed, there is an important difference between the analysis of this case in section III and the analysis here. This difference concerns what happens when it is not optimal to always fill the managerial position with an internal candidate. In section III the firm either committed to such a rule or committed to nothing. As a result, if the firm decided not to commit to always promoting an internal candidate because such a commitment is too costly, the firm's promotion decision would simply follow the ex post optimal rule. That is, the firm would only promote an internal candidate when there existed an internal candidate whose expected ability was greater than or equal to [[theta].sup.*].

In contrast, here the firm can commit to a promotion standard. As a result, if the firm decides not to commit to always promote an internal candidate because such a commitment is too costly, the firm's promotion decision would still not follow the ex post optimal rule. Rather, the firm would set a promotion standard such that an outsider would only be hired for the managerial position when the outsider's expected ability is significantly greater than the best internal candidate. The reason is that the reduction in the entry-level wage associated with internal promotion makes it optimal to bias the promotion decision toward insiders even when the promotion of outsiders is allowed.

The other case is that there are two managerial wages-one for managers promoted from within and one for managers hired from the outside. In this case the analysis is quite different from the corresponding analysis in section III. In particular, when the firm can commit to a promotion standard and there are two managerial wages, then the firm offers a contract to young workers that satisfies the following three conditions. First, the managerial wage for a worker promoted from within is high enough to induce high effort among the entry-level workers. Second, the managerial wage for a worker hired from the outside equals U. Third, the promotion standard is set so high that an outsider is hired for the managerial position only when the expected ability of the best entry-level worker is less than [[theta].sup.*].

One interesting aspect of the contract is that as in the previous case the firm uses the promotion standard to reduce the frequency with which an outsider is hired for the managerial position. Given that the managerial wage is higher for workers promoted from within than for outsiders, in the absence of a promotion standard the firm would hire an outsider for the managerial position as long as the outsider's expected ability is not significantly less than the best internal candidate. But this is not efficient from an ex ante standpoint. The reason is that this behavior ignores the fact that there is no ex ante cost to paying the high managerial wage to an insider due to the corresponding reduction in the entry-level wage. Hence, from an ex ante perspective, the firm maximizes its profits by paying a high managerial wage only for workers promoted from within and achieving efficient assignment through the promotion standard.

Methods of Committment

Section III and the previous subsection discussed the idea that a firm may be able to increase its profitability by committing to a promotion rule that limits its own ability to hire from the outside. An interesting issue is how a firm can bind its own future actions. One possibility was suggested by Milgrom and Roberts (1988). They argued that the establishment of a rule-oriented personnel department may be a way that firms bind themselves to what I have called the ex ante optimal promotion rule. In their argument the firm takes the decision of who is to be promoted "out of the hands of decision makers who have an interest in making the best possible assignment and turns it over to someone whose interests are in following the stated policy" (Milgrom and Roberts, 1988, S176).

If the argument captures the role of the personnel department, then we would expect it to behave in a bureaucratic fashion and, in particular, be unresponsive to the views and complaints of line managers. (13) In fact, this seems to be the view within many firms concerning the behavior of the personnel department. Notice that in this interpretation the bureaucratic and unresponsive nature of the personnel department is part of the efficient design of the firm. Without such a department the firm follows the ex post rather than the ex ante optimal promotion rule, with the subsequent result being a decrease in the firm's long-run profitability.

The discussion provides a useful contrast between the explanation put forth here for why firms frequently favor internal candidates and alternate explanations for the phenomenon. As briefly discussed in the introduction, two alternate explanations are that specific human capital is important in performing the tasks associated with promotions and that lack of information concerning outside candidates makes it difficult to hire outsiders for managerial positions. (14) In both explanations the firm promotes the best candidate, but because of lack of skills or lack of information the best candidate is typically an internal one. In other words, one difference between the explanation put forth here and these alternates concerns the efficient design of the firm. The explanation put forth here is consistent with a personnel department that frequently stops the firm from choosing the best candidate, whereas the alternate explanations provide no reason to have such a department.

The other possibility concerning how a firm limits its own ability to hire outsiders is that the firm does not actually bind its future actions but rather achieves the ex ante optimal promotion rule through reputation considerations. It has been assumed that the firm is sufficiently short-lived that it is unable to establish a reputation for following the ex ante optimal rule. Potentially a longer-lived firm, however, could avoid the time-inconsistency problem through the establishment of the appropriate reputation. For example, suppose the firm was in the market for an infinite number of periods rather than only through period T, and that from an ex ante standpoint the firm's optimal behavior was to never hire an outsider for the managerial position. As long as the firm discounted the future sufficiently slowly, standard arguments yield that never hiring an outsider would in fact be an equilibrium even if the firm had no actual ability to commit. (15)

One qualification to the argument is that even if the firm itself is infinitely lived, it is typically the case that many of the decision makers inside the firm have relatively short time horizons. This suggests that due to standard agency problems between the stockholders and the managers making the promotion decisions, even the infinitely lived firm may find it difficult to follow the ex ante optimal promotion rule. (16) In turn, if this is the case, then it is possible that even an infinitely lived firm may find it advantageous to bind its own future actions through something like a rule-oriented personnel department.

One interesting point is that in terms of thinking about these two possible methods of commitment, the importance of the method probably depends on what type of promotion is being considered. At low levels of the job ladder the personnel department probably has significant ability to block promotions, that is, to block the firm from hiring someone from the outside if not hiring an outsider is the optimal ex ante rule. Thus, for promotions at low levels of the job ladder a bureaucratic and unresponsive personnel department is probably a very important commitment device. However, for promotions to high levels of the job ladder the situation is likely quite different. The personnel department at most firms does not have the ability to block the promotion and hiring preferences of high-level senior managers. Thus, for high-level promotions a bureaucratic and unresponsive personnel department is probably not very important as a commitment device. Rather, for such positions, it is the ability for a firm to establi sh a reputation that potentially allows it to achieve the ex ante as opposed to the ex post optimal promotion rule.

V. CONCLUSION

A firm deciding on its promotion practices will be concerned both with the efficient assignment of workers to tasks and with rewarding prior performance. This can lead to a time-inconsistency problem because from an ex ante standpoint firms would want to take into account both aspects of the promotion decision, but from an ex post standpoint it is only assignment that matters. This article explores this time-inconsistency problem and in particular argues that the common practice of favoring internal candidates for promotion can be understood as a response by firms to the problem of time inconsistency.

There are a number of directions in which the analysis in this article could be extended. First, one could investigate the extent to which other promotion practices are the result of time inconsistency. One potential example is the now outlawed practice of mandatory retirement. Lazear (1979) argued that mandatory retirement is a consequence of imperfect monitoring leading to upward-sloping age-earnings profiles. In his argument there is the potential for workers to shirk, and firms employ upward-sloping age-earnings profiles to avoid the shirking problem. Thus mandatory retirement is needed because old workers would not voluntarily retire at the optimal retirement age because at that age the wage exceeds the reservation wage.

In Lazear's argument there is a reason for an age at which the firm has the option of terminating the employment relationship, but there is no reason why the firm would want to constrain its own ability to retain workers beyond a certain age. This is of interest because, as found by Kittner (1977), for many of the workers who were subject to mandatory retirement the practice was binding on both the worker and the firm. That is, there was both an age at which the firm had the option of forcing the worker to retire and an age (sometimes the same age) at which the firm was prohibited from retaining the worker. This suggests that Lazear's argument is not a complete explanation for the use of mandatory retirement, and that at least part of the motivation for its use involved time inconsistency.

Second, in this article I have explored a relatively simple model so that the logic of the argument is clear. It might be worthwhile investigating the basic idea in a more realistic setting to see whether the conclusions of such an analysis match well with what we know concerning real-world promotion practices. There are a number of ways in which the analysis could be made more realistic. For example, I have focused on a single firm where the firm's external candidates for the managerial position are the self-employed from the previous period. A more realistic approach would be to assume competition in the labor market between firms, where a firm's external candidates are the employees of the firm's rivals. Such a model would be more realistic concerning who fills managerial positions when an internal candidate is not chosen and potentially would also make more realistic predictions concerning the relevant pay of those promoted from within and those hired from the outside.

Another possible change concerns why high-level positions are associated with high wages. I have followed most of the tournament literature and assumed that the only reason that higher-level jobs are associated with high wages is that the firm is attempting to induce high effort among its entry-level workers. But from a real-world standpoint there are other reasons why high-level jobs are associated with high wages. For example, the signaling argument originally developed in Waldman (1984b) and Ricart i Costa (1988) makes this prediction. In that argument promoting a worker serves as a positive signal of the worker's ability with the result that promoted workers receive large wage increases to stop them from being bid away by prospective employers. (17) One interesting direction for future research might be to investigate how alternative reasons for why high-level jobs are associated with high wages influence a firm's incentive to favor internal candidates for promotion.

APPENDIX

Proof of Proposition 1. In the proof that follows I assume that [W.sup.S.sub.t+1] is paid both to skilled workers in period t + 1 in who were entry-level workers in period t and to skilled workers in period t + 1 who were self-employed in period t. However, because in equilibrium both types of workers would be paid the same wage even without this assumption, this assumption is not necessary to prove the proposition, but it does significantly simplify the proof. In the proof t also assume that given a contract for entry-level workers in period t such that there are multiple outcomes for the effort choices in period consistent with equilibrium behavior, the outcome that occurs is the one that maximizes the expected number of entry-level workers in period t who choose high effort. This assumption is also not necessary to prove the proposition but significantly simplifies the proof.

Consider first how the firm forms inferences concerning workers' abilities. There are three possibilities. First, if in period the equilibrium is such that each entry-level worker chooses [e.sup.h] with probability one, then [[theta].sup.E.sub.jt] = [y.sub.jt] - [e.sup.h] for all j. Second, if in period t the equilibrium is such that each entry-level worker chooses [e.sup.l] with probability one, then [[theta].sup.E.sub.jt] = [y.sub.jt] - [e.sup.l]. Third, if in period t the equilibrium is such that entry-level worker j chooses [e.sup.h] with probability p and [e.sup.l] with probability (1 - p), then the firm draws its inferences using Bayes's rule.

The first result is that [W.sup.M.sub.t+1] [greater than or equal to] U because the firm will be unable to staff the managerial position in period t+1 if [W.sup.M.sub.t+1] < U. Similarly, we know [W.sup.S.sub.t+1] [greater than or equal to] U. The second result is that because in period t + 1 the wages for the managerial and skilled positions are fixed, in period t + 1 the workers in these positions provide low effort.

Suppose for now that there exists a contract that satisfies (i)-(v). Let X denote the firm's aggregate wage bill for this contract for the young in period t and the old in period t + 1, and let Y denote the firm's expected aggregate output for the young in period t and the old in period t + 1. Given this, we first focus on the family of contracts for which the managerial position in period t + 1 is restricted to period t entry-level workers. We know that in period t + 1 the firm will want to maximize the aggregate output of its managerial and skilled positions. This in combination with the assumption [delta] > [lambda] > 1 yields that the firm will staff the managerial position in period t + 1 with the period t entry-level worker with the highest value for [[theta].sup.E.sub.jt], and the skilled positions will be staffed by a combination of the remaining period t entry level workers with values for [[theta].sup.E.sub.jt] greater than or equal to [[theta].sup.*] (see note 6) and workers who were self-employed i n period t. In turn, given this contract for which all period entry-level workers provide high effort, the condition translates into (iv) and (v).

Continuing to restrict attention to contracts for which the managerial position in period t + 1 is restricted to period t entry-level workers, if the contract is such that all period t entry-level workers provide high effort, it must be the case that either [W.sup.M.sub.t+1] > U or [W.sup.S.sub.t+1] > U for otherwise all entry-level workers in period t would have an incentive to provide low effort. Further, if [W.sup.S.sub.t+1] = U, then the individual rationality constraint for entry-level workers in period t yields (ii). Given this, consider a contract for which [W.sup.S.sub.t+1] > U and each period t entry-level worker chooses [e.sup.h] with probability one. The firm's expected aggregate output for the young in period t and the old in period t + 1 equals Y. Let [DELTA] = [beta]([W.sup.S.sub.t+1] - U) and let c denote the expected number of skilled positions in period t + 1 that will be filled with outsiders. The individual rationality constraint for entry-level workers in period t now yields that the firm' s aggregate wage bill for the young in period t and the old in period t + 1 must be greater than or equal to X + c[DELTA] The reason is that c[DELTA] of the premium associated with being a skilled worker is not going to period t entry-level workers, and thus the aggregate wage bill must rise by at least c[DELTA] to compensate for this. In turn, because productivity is the same and the wage bill is higher, any such contract is dominated by a contract that satisfies (i)-(v).

We now have that among contracts for which the managerial position in period t + 1 is restricted to period t entry-level workers, if we further restrict attention to contracts for which all entry-level workers in period t provide high effort, then the best contract satisfies (i)-(v). Given this, consider a contract in this family in which each period t entry-level worker does not choose [e.sup.h] with probability one. Let c now denote the expected number of entry-level workers in period t who provide low rather than high effort. For this contract the firm's expected aggregate output for the young in period t and the old in period t + 1 is less than or equal to Y - c([e.sup.h] - [e.sup.l]). Furthermore, the individual rationality constraint for entry-level workers in period t now yields that the firm's aggregate wage bill for the young in period t and the old in period t + 1 is greater than or equal to X - cg([e.sup.h]). The reason is that there is a decrease in the expected aggregate disutility of working at the entry-level positions equal to cg([e.sup.h]), and thus the aggregate wage bill potentially decreases by as much as cg([e.sup.h]). In turn, because [e.sup.h] - [e.sup.l] > g([e.sup.h]), any such contract is dominated by a contract that satisfies (i)-(v).

We now have that among contracts for which the managerial position in period t+1 is restricted to period t entry-level workers, the best contract satisfies (i)-(v). Given this, we now consider contracts for which the managerial position in period t + 1 is not restricted to individuals who were entry-level workers in period t. Consider such a contract for which there is a positive probability that at least one entry-level worker in period chooses [e.sup.h]. For a period t entry-level worker to choose [e.sup.h] with positive probability, it must be that either [W.sup.M.sub.t+1] [greater than or equal to] U + (g([e.sup.h])/[beta]) or [W.sup.S.sub.t+1] [greater than or equal to] U + (g([e.sup.h])/[beta]). Also, for such a contract the firm will staff the managerial position with a worker who was self-employed in period t if [[theta].sup.e.sub.jt] < [[theta].sup.*] for all j. Let v be the probability this occurs and let c again be the expected number of entry-level workers in period t who provide low rather than high effort.

Given a contract of the type described, the firm's expected aggregate output for the young in period t and the old in period t + 1 must be strictly less than Y - c([e.sup.h] - [e.sup.l]) + [beta]v[delta]([[theta].sup.*] - [[theta].sub.L]). The reason is that the firm loses in expected productivity c([e.sup.h] - [e.sup.l]) because on average c entry-level workers provide low rather than high effort and gain something less than [beta]v[delta]([[theta].sup.*] - [[theta].sub.L]) because the managerial position can be staffed with a worker who was self-employed in period t. Suppose [W.sup.M.sub.t+1] [greater than or equal to] U + (g([e.sup.h])/[beta]). Given this and the individual rationality constraint for an entry-level worker in period t, the firm's expected aggregate wage bill for the young in period t and the old in period t + 1 must be greater than or equal to X + vg([e.sup.h]) - cg([e.sup.h]). The reason is that on average, at least vg([e.sup.h]) of the premium associated with being the manager in period t + 1 is not going to period t entry-level workers, and thus the aggregate wage bill must rise by vg([e.sup.h]) to compensate for this, whereas there is a decrease in the expected aggregate disutility of working at the entry-level positions equal to cg([e.sup.h]) and the aggregate wage bill decreases by as much as cg([e.sup.h]) to compensate for this. We now have that expected aggregate output minus the expected aggregate wage bill for the young in period t and the old in period t + 1 must be strictly less than Y - X - c([e.sup.h] - [e.sup.l]) + [beta]v[delta]([[theta].sup.*] - [[theta].sub.L]) - vg([e.sup.h]) + cg([e.sup.h]). Given this and [e.sup.h] - [e.sup.l] > g([e.sup.h]), expected aggregate output minus the expected aggregate wage bill for the young in period t and the old in period t + 1 must be strictly less than Y - X + v([beta][delta]([[theta].sup.*] -[[theta].sub.L]) - g([e.sup.h])). But this means that if [[theta].sup.*] - [[theta].sub.L] is sufficiently small, then expected aggregate output minus the expected aggregate wage bill for the young in period t and the old in period t + 1 must be strictly less than Y - X. Suppose [W.sup.S.sub.t+1] [greater than or equal to] U + (g([e.sup.h])/[beta]). The expected number of skilled positions in period t + 1 that are filled by workers who were self-employed in period t is greater than v. Thus we again have that the firm's expected aggregate wage bill for the young in period t and the old in period t + 1 is greater than or equal to X + vg([e.sup.h]) - cg([e.sup.h]). Hence, if [[theta].sup.*] - [[theta].sub.L] is sufficiently small, then a contract that satisfies (i)-(v) dominates the best contract for which the managerial position in period t + 1 is not restricted to individuals who were entry-level workers in period t and there is a positive probability at least one entry-level worker in period t chooses [e.sup.h].

Within the family of contracts for which the managerial position in period t + 1 is not restricted to individuals who were entry-level workers in period t, consider a contract for which there is a probability one that every entry-level worker in period t chooses [e.sup.l]. Let v again be the probability that the firm staffs the managerial position with a worker who was self-employed in period t. The expected aggregate output for the young in period t and the old in period t + 1 must be less than Y - N([e.sup.h] - [e.sup.l]) + [beta]v[delta]([[theta].sup.*] - [[theta].sub.L]). The reason is that the firm loses in expected productivity N([e.sup.h] - [e.sup.l]) because entry-level workers provide low rather than high effort and gain something less than [beta]v[delta]([[theta].sup.*] - [[theta].sub.L]) because the managerial position can be staffed with a worker who was self-employed in period t. Further, from the individual rationality constraint for the entry-level workers in period t we know that the aggregate wage bill for the young in period t and the old in period t + 1 is greater than or equal to X - Ng([e.sup.h]). The reason is that there is a decrease in the expected aggregate disutility of working at the entry-level positions equal to Ng([e.sup.h]), and thus the aggregate wage bill decreases by as much as Ng([e.sup.h]). Given [e.sup.h] - [e.sup.l] > g([e.sup.h]), if [[theta].sup.*] - [[theta].sub.L] is sufficiently small, then -N([e.sup.h] - [e.sup.l]) + [beta]v[delta]([[theta].sup.*] - [[theta].sub.L]) < -Ng([e.sup.h]). Hence, if [[theta].sup.*] - [[theta].sub.L] is sufficiently small, then a contract that satisfies (i)-(v) dominates the best contract for which the managerial position in period t + 1 is not restricted to individuals who were entry-level workers in period t.

Combining the results, we now have that a contract that satisfies (i)-(v) dominates the best contract that does not satisfy (i)-(v). The final step of the proof, therefore, is to demonstrate that there exists a contract that satisfies (i)-(v). Consider a contract that satisfies (i), results in all entry-level workers in period t providing high effort, and contains a commitment by the firm to staff the managerial position in period t+1 with an individual who was an entry-level worker in period t. From above we know that such a contract will also satisfy the remaining parts of (iii), (iv), and (v). Further, given each entry-level worker has a 1/N probability of becoming the manager in the following period and an (N - 1)/N probability of receiving U in the same period, the individual rationality constraint for entry-level workers in period t yields (ii).

The only result left to show is that, given [W.sup.S.sub.t+1] = U and the firm commits to staff the managerial position in period t + 1 with an individual who was an entry-level worker in period t, there exists a value [W.sup.M.sub.t+1] such that it is equilibrium behavior for every entry-level worker in period t to provide high effort. Suppose all entry-level workers in period t provide high effort. We can show that this is consistent with equilibrium if no worker has an incentive to deviate. Consider worker i. If the worker chooses high effort, then the worker's discounted expected utility over the two periods, denoted U([e.sup.h]), is given by (A1).

(A1) U([e.sup.h]) = [W.sup.E.sub.t] - g([e.sup.h])

+[beta][Pr([e.sup.h])[W.sup.M.sub.t+1] + (1 -Pr([e.sup.h]))U],

where Pr([e.sup.h]) is the probability the firm believes that worker i is the highest-ability worker given the worker chooses [e.sup.h]. If the worker chooses low effort, then the worker's discounted expected utility over the two periods, denoted U([e.sup.l]), is given by (A2).

(A2) U([e.sup.l]) = [W.sup.E.sub.t] + [beta][Pr([e.sup.l])[W.sup.M.sub.t+1] + (1 - Pr([e.sup.l]))U],

where Pr([e.sup.l]) is the probability the firm believes that worker i is the highest-ability worker given the worker chooses [e.sup.l]. From earlier we know that given an equilibrium in which each entry-level worker chooses high effort, the firm promotes the worker with the highest output. This yields Pr([e.sup.h]) = 1/N > Pr([e.sup.l]). In turn, (Al) and (A2) now yield U([e.sup.h]) > U([e.sup.l]) if [W.sup.M.sub.t+1] is sufficiently large. (Note: there is no upper bound on [W.sup.M.sub.t+1] because [W.sup.E.sub.t] can always be lowered sufficiently such that (ii) holds.) Hence, there must exist a contract such that (i) through (v) are satisfied.

Proof of Proposition 2. From the proof of Proposition 1, we know that if [[theta].sup.*] - [[theta].sub.L] is sufficiently small and the firm commits to staff the managerial position in period t + 1 with an individual who was an entry-level worker in period t, then (i)-(v) of Proposition 1 must be satisfied. This in turn implies that if an equilibrium contract does not satisfy (i)-(v) of Proposition 1 and [[theta].sup.*] - [[theta].sub.L] is sufficiently small, then the firm must not restrict the managerial position in period t + 1 to an individual who was an entry-level worker in period t.

Given the above, suppose [W.sup.M'.sub.t+1] < U and the firm does not restrict the managerial position in period t + 1 to an individual who was an entry-level worker in period t. Because the managerial wage in period t + 1 for a manager who was self-employed in period t is less than the utility such a worker would receive in self-employment, the firm would not be able to attract into the managerial position in period t + 1 a worker who was self-employed in period t. Thus, it is as if the firm restricted the managerial position in period t + 1 to an individual who was an entry-level worker in period t, and from the proof of Proposition 1 we therefore know that if [[theta].sup.*] - [[theta].sub.L] is sufficiently small then (i)-(v) of Proposition 1 must be satisfied (see note 12).

Suppose the firm does not restrict the managerial position in period t + 1 to an individual who was an entry-level worker in period t but [W.sup.M'.sub.t+1] is set sufficiently high that the firm never hires into the managerial position in period t + 1 a worker who was self-employed in period t. It is again as if the firm restricted the managerial position in period t + 1 to an individual who was an entry-level worker in period t. Thus, the proof of Proposition 1 again implies that if [[theta].sup.*] - [[theta].sub.L] is sufficiently small then (i)-(v) of Proposition 1 must be satisfied (see again note 12).

In the rest of the proof we assume [W.sup.S.sub.t+1] = U. This follows given arguments similar to those presented in the proof of Proposition 1. Suppose [W.sup.M'.sub.t+1] = U and the firm does not restrict the managerial position in period t + 1 to an individual who was an entry-level worker in period t. There are two possibilities. First, the contract could be such that the managerial position in period t + 1 is always staffed with a worker who was self-employed in period t. For such a contract the entry-level workers in period t will provide low rather than high effort, and the expected output of the manager in period t + 1 will be strictly less than if the managerial position in period t + 1 was restricted to an individual who was an entry-level worker in period t. Thus, the expected aggregate output for the young in period t and the old in period t + 1 must be strictly less than Y - N([e.sup.h] - [e.sup.L]). Furthermore, the individual rationality constraint for the entry-level workers in period t yields that the aggregate wage bill for the young in period t and the old in period t + 1 must be greater than or equal to X - Ng([e.sup.h]). Because [e.sup.h] - [e.sup.l] g([e.sup.h]) we have that this contract is dominated by a contract that satisfies (i)-(v) of Proposition 1.

Second, the contract could be such that there is a strictly positive probability that the managerial position in period t + 1 is filled with an individual who was an entry-level worker in period t and a strictly positive probability that the managerial position in period t + 1 is filled with an individual who was self-employed in period t. This means U [less than or equal to] [W.sup.M.sub.t+1] [less than or equal to] U + [delta]([[theta].sub.H] - [[theta].sup.*]). Given (N - 1)g([e.sup.h]) > [beta][delta]([[theta].sub.H] - [[theta].sub.L]), the incentive compatibility constraint for entry-level workers in period t now yields that not all entry-level workers in period t provide high effort. Rather, the expected number of entry-level workers in period t who provide high effort cannot exceed [beta][delta]([[theta].sub.H] - [[theta].sup.*])/g([e.sup.h]) < N - 1. Let c again be the expected number of entry-level workers in period t who provide low rather than high effort. We know c > 1. We now have that the expected aggregate output for the young in period t and the old in period t + 1 must be strictly less than Y - c([e.sup.h] - [e.sup.l]) + [beta]([[theta].sup.*] - [[theta].sup.L]). Further, the individual rationality constraint for the entry-level workers in period t yields that the expected aggregate wage bill for the young in period t and the old in period t + 1 must be greater than or equal to X - cg([e.sup.h]). We now have that expected aggregate output minus the expected aggregate wage bill for the young in period t and the old in period t + 1 must be strictly less than Y - c([e.sup.h] - [e.sup.l]) + [beta]([[theta].sup.*] - [[theta].sup.L]) - X + cg([e.sup.h]). Given this, c > 1, and [e.sup.h] - [e.sup.l] > g([e.sup.h]), we have that expected aggregate output minus the expected aggregate wage bill for the young in period t and the old in period t + 1 must be strictly less than Y - X - ([e.sup.h] - [e.sup.l]) + g([e.sup.h]) + [beta]([[theta].sup.*] - [[theta].sub.L]). But this means that for [[theta].sup.*] - [[theta].sub.L] sufficiently small this contract is dominated by a contract that satisfies (i) - (v) of Proposition 1.

Suppose [W.sup.M'.sub.t+1] > U and the firm does not restrict the managerial position in period t + 1 to an individual who was an entry-level worker in period t. There are again two possibilities. First, the contract could be such that the managerial position in period t + 1 is always staffed with a worker who was self-employed in period t. For such a contract the entry-level workers in period t will provide low rather than high effort and the expected output of the manager in period t + 1 will be strictly less than if the managerial position in period t + 1 was restricted to an individual who was an entry-level worker in period t. Thus, the expected aggregate output for the young in period t and the old in period t + 1 must be strictly less than Y - N([e.sup.h] - [e.sup.l]). Furthermore, the individual rationality constraint for the entry-level workers in period t combined with [W.sup.M'.sub.t+1] > U yields that the aggregate wage bill for the young in period t and the old in period t + 1 must be greater than X - Ng([e.sup.h]). Because [e.sup.h] - [e.sup.l] > g([e.sup.h]) we have that this contract is dominated by a contract that satisfies (i) - (v) of Proposition 1.

Second, the contract could be such that there is a strictly positive probability that the managerial position in period t + 1 is filled with an individual who was an entry-level worker in period t and a strictly positive probability that the managerial position in period t + 1 is filled with an individual who was self-employed in period t. This means [W.sup.M.sub.t+1] [less than or equal to] [W.sup.M'.sub.t+1] + [delta]([[theta].sub.H] - [[theta].sup.*]). Let c again be the expected number of entry-level workers in period t who provide low rather than high effort, and let v again be the probability that in period t + 1 the managerial position is filled by a worker who was self-employed in period t. Suppose [W.sup.M'.sub.t+1] [greater than or equal to] U + [delta]([[theta].sup.*] - [[theta].sub.L]). We know that the expected aggregate output for the young in period t and the old in period t + 1 must be less than Y - c([e.sup.h] - [e.sup.l]) + v[beta][delta]([[theta].sup.*] - [[theta].sub.L]). Furthermore, the individual rationality constraint for the entry-level workers in period t combined with [W.sup.M'.sub.t+1] [greater than or equal to] U + [delta]([[theta].sup.*] - [[theta].sub.L]) yields that the expected aggregate wage bill for the young in period t and the old in period t + 1 must be greater than or equal to X - cg([e.sup.h]) + v[beta][delta]([[theta].sup.*] - [[theta].sub.L]). Because [e.sup.h] - [e.sup.l] > g([e.sup.h]) we have that this contract is dominated by a contract that satisfies (i) - (v) of Proposition 1. Hence, U < [W.sup.M'.sub.t+1] < U + [delta]([[theta].sup.*] - [[theta].sub.L]).

Suppose U < [W.sup.M'.sub.t+1] < U + [delta]([[theta].sup.*] - [[theta].sub.L]). Given [W.sup.M.sub.t+1] [less than or equal to] [W.sup.M'.sub.t+1] + [delta]([[theta].sub.H] - [[theta].sup.*]) and (N - 1)g([e.sup.h]) > [beta][delta]([[theta].sub.H] - [[theta].sub.L]), the incentive compatibility constraint for entry-level workers in period t now yields c > 1. We know that the expected aggregate output for the young in period t and the old in period t + 1 must be less than Y - c([e.sup.h] - [e.sup.l]) + v[beta][delta]([[theta].sup.*] - [[theta].sub.L]). Furthermore, the individual rationality constraint for the entry-level workers in period t yields that the expected aggregate wage bill for the young in period t and the old in period t + 1 must be greater than X - cg([e.sup.h]). We now have that expected aggregate output minus the expected aggregate wage bill for the young in period t and the old in period t + 1 must be strictly less than Y - c([e.sup.h] - [e.sup.l]) + v[beta][delta]([[theta].sup.*] - [[theta].sup.L]) - X + cg([e.sup.h]). Given this, c > 1, and [e.sup.h] - [e.sup.l] > g([e.sup.h]), we have that expected aggregate output minus the expected aggregate wage bill for the young in period t and the old in period t + 1 must be strictly less than Y - X - ([e.sup.h] - [e.sup.l]) + g([e.sup.h]) + v[beta][delta]([[theta].sup.*] - [[theta].sub.L]). But this means that for [[theta].sup.*] - [[theta].sub.L] sufficiently small this contract is dominated by a contract that satisfies (i) - (v) of Proposition 1.

(1.) Papers concerning thc assignment aspects of the promotion decision include Sattinger (1975), Rosen (1978, 1982), Waldman (1984a), and Gibbons and Waldman (1999a). See Sattinger (1993) for a survey. Papers that investigate promotions as a reward for prior performance include Lazear and Rosen (1981), Malcomson (1984), Rosen (1986), Lazear (1989), and Prendergast (1993). Gibbons and Waldman (1999b) survey both topics.

(2.) One piece of evidence that firms typically favor internal candidates for promotion is that between 1984 and 1994 over 85% of the individuals appointed to the CEO position at a Fortune 500 firm were promoted from within. See Chan (1996) for a discussion.

(3.) In a recent paper Tsoulouhas et al. (2001) extend Chan's analysis to try to explain a number of facts concerning CEO succession. Their model is closer than Chan (1996) to the one considered here in that they do allow for a production stage after the promotion decision takes place.

(4.) Another related analysis is Kahn and Huberman (1988). Kahn and Huberman consider a time-inconsistency problem that arises in environments characterized by a single task. In their analysis, the possibility of future wage increases provides an incentive for workers to invest in specific human capital, but a time-inconsistency problem concerning the probability of receiving a future wage increase causes workers to underinvest. The conclusion is that the employment of up-or-out contracts is one way that the firm can avoid this time-inconsistency problem. See Prendergast (1993) and Zabojnik (1998) for alternative approaches to the problem of workers underinvesting in specific human capital.

(5.) See Novos (1992, 1995) for related analyses.

(6.) To simplify the exposition, it is also assumed that if a worker is indifferent between working at the firm and self-employment, the worker chooses the firm, and that if the firm is indifferent between internal and external candidates in staffing either the managerial position or a skilled position, the firm chooses the internal candidate.

(7.) It can be shown that, as long as [delta] - [lambda] is not too large, the firm never has an incentive to commit to fill the skilled positions in period t + 1 only with workers who were in entry-level positions in period t.

(8.) One drawback of the model described concerns the implicit assumption that promotions are used to reward prior performance, rather than bonuses. Although this is consistent with many if not most employment situations, as argued by Baker et al. (1988), there are theoretical reasons why bonuses should be used to reward prior performance and promotions should be used to assign workers to tasks rather than having promotions used for both purposes. Their reasoning is that using promotions to achieve both goals leads the firm to trade off incentives and assignment, and this trade-off can be avoided by using bonuses rather than promotions to reward prior performance. Fairburn and Malcomson (2001) address the Baker et al. (1988) argument. Fairburn and Malcomson allow for the possibility of bribery between a worker and the manager who evaluates the worker's performance. They show that bribery eliminates the ability of the firm to offer incentives through bonus payments, with the result that the firm uses the possi bility of promotion to a higher-paying job as an incentive for effort. Although I do not formally model the bribery process as in Fairburn and Malcomson, the analysis is consistent with their conclusion that incentives are provided through the possibility of future promotions. I could demonstrate the main results of the paper in a Fairburn and Malcomson--type setting, but such an analysis would be quite involved, and the intuition underlying the time-inconsistency problem (which is the focus here) would be more difficult to follow.

(9.) It is not necessary to specify whether in any period t + 1 the firm is restricted to pay the same wage to a skilled worker who was an entry-level worker in period and to a skilled worker who was self-employed in period t. The reason is that in equilibrium the two types of workers are paid the same wage whether or not there exists such a restriction.

(10.) Notice that even in the ex post case the firm will staff the managerial position with an outsider relatively rarely (only when all the entry-level workers in the previous period have a value for [[theta].sup.E.sub.jt] below [[theta].sup.*]). This is consistent with the discussion in the introduction that one reason a firm may staff higher levels of a job ladder entirely or almost entirely with insiders is that the firm has better information about the abilities of its own workers.

(11.) Note that [W.sup.E.sub.t] and [W.sup.M.sub.t+1] are not uniquely defined.

(12.) As was true for Proposition 1, [W.sup.E.sub.t] and [W.sup.M.sub.t+1] are not uniquely defined. Also, rather than directly specifying that the managerial position in period t + 1 is restricted to individuals who were entry-level workers in period t, the contract could simply specify a value for [W.sup.M'.sub.t+1] sufficiently high or low that the firm never fills the managerial position in period t + 1 with an outsider.

(13.) For this practice to solve the firm's time-inconsistency problem it must be that at the date of the actual promotion decision the firm cannot reverse its earlier action and take authority over the decision away from the personnel department. One reason the firm might be unable to reverse itself and take authority away from the personnel department is if the personnel department has specialized knowledge required for the evaluation of candidates. For example, it might be that only the personnel department has the experience necessary to properly interpret a worker's performance evaluations and references.

(14.) See Becker (1962) for an early discussion of the importance of specific human capital, and Novos (1992, 1995) for analyses that show that lack of information concerning outside candidates reduces the number of outsiders that firms hire. The models considered by Novos build on the earlier work of Greenwald (1986).

(15.) As discussed in the previous subsection, the ex ante optimal promotion rule can be that the firm sometimes hires an outsider, but that it does this less often than would maximize current profits. Such a rule would be difficult to achieve through reputation formation because it would be unclear to a future potential employee whether the hiring of an outsider today was consistent or inconsistent with the firm's ex ante optimal rule. My conjecture is that given an infinitely lived firm facing this situation, there exists an equilibrium in which the firm does better than the ex post optimal promotion rule, but there is no equilibrium in which the firm achieves the best ex ante rule. Formal analysis of this case, however, is beyond the scope of this article.

(16.) One might argue that as long as the managers making the promotion decisions own some stock in the firm, then there will be no agency problems between the stockholders and the managers. Stein (1989) shows that this argument is incorrect in a world in which outsiders are imperfectly informed about the actions of the managers and the current stock price serves as a signal of long-run profitability. He finds that in such a world managers behave myopically, that is, they stress current profits more than is optimal from the standpoint of the long-run profitability of the firm.

(17.) Other articles that consider this signaling argument include Waldman (1990), Bernhardt (1995), and Zabojnik and Bernhardt (2001). Zabojnik and Bernhardt's analysis is particularly relevant in that they incorporate the job assignment signaling story into a tournament-style model. However, their analysis does not consider a firm's incentive to favor internal candidates for promotion over external ones due to time inconsistency.

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MICHAEL WALDMAN *

* I would like to thank Dave Butz, Harold Demsetz, Bruce Fallick, Robert Gibbons, Bengt Holmstrom, Robert Hutchens, Paul Milgrom, Olivia Mitchell, Ian Novos, Mark Rebiek, Jean Rosenthal, Ken Sokoloff, two anonymous referees, the editor William S. Neilson, and participants at workshops at Cornell University, UCLA, and the Stanford Institute for Theoretical Economics for helpful comments.

Waldman: Charles H. Dyson Professor in Management and Professor of Economics, Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853. Phone 1-607-255-8631, Fax 1-607-254-4590, E-mail mw46@cornell.edu