Organization, Careers and Incentives

Organization, Careers and Incentives Chapter 4 Robert Gary-Bobo March 2018 1 / 31

Introduction Introduction A firm is a pyramid of opportunities (Alfred P. Sloan). Promotions can be used to create incentives. But promotion opportunities are constrained by the organization s structure. Common problem : shortage of promotion possibilities leads to turnover (... or top-heavy organizations : 1 president, 10 vice-presidents...). Other well-known problem : the Peter principle. We study the interaction of organizational constraints and the provision of incentives. 2 / 31

Introduction Introduction 2 Firms manage the workers careers subject to a constraint on promotion opportunities. Firms may change their organizational structure to relax incentive constraints. It can be shown that promotions arise as part of an optimal arrangement. Theory of the internal labor market (Doeringer and Piore (1971)). There is a port of entry at the bottom of the firm s hierarchy. Workers remain at the bottom of the hierarchy or are promoted ; they are never demoted. A career path emerges as an optimal solution for the firm (minimizing the wage bill). Pay is backloaded. Forced turnover may be used as a an instrument to provide incentives. 3 / 31

Introduction Introduction 3 This chapter is mainly based on a recent paper by Ke, Li and Powell (2016). For a survey of work on internal labor markets see e.g. R. Gibbons and M. Waldman (1999), E. Lazear and P. Oyer (2013), M. Waldmann (2013). Pioneering work on this topic is due to E. Lazear and S. Rosen (1981) : the theory of tournaments. Important assumption here : wages are attached to positions in the organization. 4 / 31

The model : basic assumptions A model : basic assumptions and notation A firm and a large mass of identical risk-neutral workers interact repeatedly. Time is discrete, t = 1, 2,..., horizon is infinite and all agents share the same discount factor δ (0, 1). In each period, the firm chooses its personnel policy : wages, promotion rules. We restrict attention to stationary (time-independent) policies. Production requires two activities indexed i = 1, 2. A worker performing activity i chooses effort e i {0, 1}. Cost of effort is c i e i. e i = 0 means shirking ; e i = 1 worker is productive. 5 / 31

The model : basic assumptions A model : basic assumptions and notation 2 Effort is private information of the worker but shirking is detected with probability q i (0, 1) during a given period. The firm employs masses N 1 and N 2 of workers in activities i = 1, 2. The revenue is given by a production function F (N 1, N 2 ). The wage of activity i is denoted w i. Let p ij be the probability of being assigned (promoted) to activity j next period given activity i today. Let d i denote the voluntary departure rate from activity i (spontaneous turnover) : d i N i workers leave in each period. The firm chooses the number of positions N i for each i in each period. Let H i denote the number of new hires for activity i in a given period. 6 / 31

The model : basic assumptions Timing of game Firm chooses N i, i = 1, 2. Positions are filled with incumbent workers and new hires H i. Firm offers a contract (w i, p ij ) with i, j = 1, 2 in each activity i (a wage and assignment policy). Wages are tied to activities by assumption. The outside option of a worker is set equal to zero. A worker caught shirking is fired (most severe punishment is optimal given that it never happens on the equilibrium path). Assume that d 1 + d 2 1 departure rates are low. 1 p i1 p i2 0 is the forced turnover rate in activity i. Total turnover rate = voluntary + forced = d i + (1 d i )(1 p i1 p i2 ). 7 / 31

The model : basic assumptions Parallel careers benchmark Suppose that there are no promotions (parallel careers) : p 12 = 0 and p 21 = 0. Firm treats activities independently and maximizes profit F (N 1, N 2 ) w 1 N 1 w 2 N 2 subject to IR and IC constraints. The payoff of activity i to a worker is by definition v i = w i c i + (1 d i )δv i. The workers chooses e i = 1 if w i c i + (1 d i )δv i w i + (1 d i )(1 q i )δv i this is equivalent to v i R i, where by definition, R i = c i (1 d i )δq i 8 / 31

The model : basic assumptions Parallel careers benchmark 2 LEMMA 1. If a firm maximizes profits subject to parallel careers (i.e., p 12 = 0 and p 21 = 0), then, (i) the firm chooses wages ŵ i = c i + (1 (1 d i )δ)r i yielding rents v i = R i to each worker, i = 1, 2 ; (ii) hires H i = d i N i in every period, i = 1, 2 ; (iii) ŵ i = F (N 1, N 2 ) N i > c i. This result is consistent with efficiency-wage theory à la Shapiro-Stiglitz (1984) : wages are higher than outside options (ŵ i c i > 0) and R i > 0. Employment levels are lower than optimal levels (first-best optimality requires F / N i = c i ). 9 / 31

The model : basic assumptions Proof of Lemma 1 Easy. Incentive compatibility should hold with equality (to save on wage bill), thus, v i = w i c i + (1 d i )δv i = w i + (1 d i )(1 q i )δv i. from this we derive v i = c i /(q i δ(1 d i )) = R i. Finally, Q.E.D. w i = c i + [1 (1 d i )δ]r i > c i. 10 / 31

Managing careers Managing careers 1 : PK and IR constraints More generally. Given (N 1, N 2 ), the firm wants to minimize the wage bill w 1 N 1 + w 2 N 2. Promise-keeping constraints PK i must hold, i = 1, 2 v 1 = w 1 c 1 + (1 d 1 )δ(p 11 v 1 + p 12 v 2 ) v 2 = w 2 c 2 + (1 d 2 )δ(p 21 v 1 + p 22 v 2 ) Individual rationality constraints IR i must hold, i = 1, 2 v 1 0, v 2 0. 11 / 31

Managing careers Managing careers 2 : IC constraints Workers IC constraints denoted IC i, i = 1, 2, can be written, v 1 w 1 + (1 d 1 )(1 q 1 )δ(p 11 v 1 + p 12 v 2 ), v 2 w 2 + (1 d 2 )(1 q 2 )δ(p 21 v 1 + p 22 v 2 ). It is then easy to see that IC i constraints are equivalent to, p 11 v 1 + p 12 v 2 R 1, p 21 v 1 + p 22 v 2 R 2. 12 / 31

Managing careers Managing careers 3 : Flow (FL) constraints The recruitment policy must be feasible. The FL i constraints are as follows, p 11 (1 d 1 )N 1 + p 21 (1 d 2 )N 2 + H 1 = N 1, p 12 (1 d 1 )N 1 + p 22 (1 d 2 )N 2 + H 2 = N 2. Probabilities p ij must be nonnegative and we must have p i1 + p i2 1 for all i. 13 / 31

Managing careers Solution of the problem in two steps Fix (N 1, N 2 ). First step, minimize wage bill w 1 N 1 + w 2 N 2 subject to IR i, IC i, PK i and FL i. We find (H i, p ij, w i ) as a function of (N 1, N 2 ). Second step, choose (N 1, N 2 ) to maximize profits F (N 1, N 2 ) w 1 N 1 w 2 N 2. (Hi, p ij, w i ) is a personnel policy. Define W (N 1, N 2 ) = Min{w 1 N 1 + w 2 N 2 PK, IC, IR, IC} 14 / 31

Optimal personnel policy Optimal personnel policy We assume R 2 R 1, activity 2 is called the high-rent activity (without loss of generality). We now study the case in which N 1 N 2. Activity 1 workers will be called the bottom workers...activity 2 is called the top job. We will soon understand why. Since d 1 + d 2 1 and N 1 N 2 we have N 2 d 2 N 1 (1 d 1 ). There are less top workers who quit than there are bottom workers who stay. In other words, there are enough incumbent bottom workers to fill all the top-job vacancies generated by departures. 15 / 31

Optimal personnel policy Existence of a port of entry LEMMA 2. All new workers are hired in the bottom job, i.e., H 2 = 0. Intuition : Hiring directly in the top job requires the payment of a rent v2 to the new worker. Hiring in the bottom job and promoting requires the payment of a rent v1 < v 2. Hiring in the bottom job makes IC and IR constraints easier to satisfy, because of a positive promotion probability, p 12 > 0. Promotions increases motivation of bottom workers, using the rents of top workers. This allows the firm to reduce the bottom wage w 1. 16 / 31

Optimal personnel policy Proof of Lemma 2 Denote M i = (1 d i )N i. Substitute first PK i and then FL i in the wage bill W. We have, W = w 1 N 1 + w 2 N 2 = i [v i + c i δ(1 d i )(p i1 v 1 + p i2 v 2 )]N i = i = i c i N i + i c i N i + i v i [N i δ((1 d 1 )p 1i N 1 + (1 d 2 )p 2i N 2 )] v i [N i δ(n i H i )]. We must therefore minimize, v 1 [(1 δ)n 1 + δh 1 ] + v 2 [(1 δ)n 2 + δh 2 ] 17 / 31

Optimal personnel policy Proof of Lemma 2, ctd We assume (and check later that indeed) v1 v2. By way of contradiction, suppose now that H2 > 0. Then, since d 2N 2 < (1 d 1)N 1 = M 1, we have N 2 = M 2 + d 2N 2 < M 1 + M 2. Now, if p 12 = 1 and p 22 = 1, it follows from the FL constraints that we must have M 1 + M 2 + H 2 = N 2. But this is impossible if H 2 0. Too many workers arrive in the top job. Hence either p12 < 1 or p22 < 1. Assume first that p 12 < 1. Consider a perturbation of the personnel policy (H i, p ij ). Fewer workers are hired into activity 2, more are hired in activity 1. H 1 = H 1 + M 1ɛ, p 11 = p 11 ɛ, H 2 = H 2 M 1ɛ, p 12 = p 12 + ɛ. We compute the wage bill with this change, and ɛ > 0 small enough, we find W = W δɛm 1(v 2 v 1 ) W. If v 2 > v 1, the original personel policy is not optimal. If v 2 = v 1, the perturbation has no effect, we can choose H 2 = 0. Reasoning is the same if p 22 < 1 (we then keep slightly more workers in top job). Q.E.D. 18 / 31

Optimal personnel policy Sufficient separation rents d 2 N 2 top positions are freed in any period. This frees up an amount d 2 N 2 R 2 of rents that can be reallocated. If d 2 N 2 R 2 (1 d 1 )N 1 R 1 we are in the case of sufficient separation rents. LEMMA 3. If v1 v 2, and we have sufficient separation rents, in an optimal personnel policy, bottom workers receive zero rents, v1 = 0. Top workers receive v2 = R 2. There are no demotions, p21 = 0. Workers receive full job security, pi1 + p i2 = 1, for all i. Top workers are never promoted, it follows that they must receive at least R 2 to satisfy the IC constraint. With sufficient separation rents, promotion prospects alone provide enough motivation for bottom workers, so IC 1 is slack. Bottom worker s per-period payoffs are lower than 0 but they hope to be promoted. If top workers were demoted, they should receive more. 19 / 31

Optimal personnel policy Proof of Lemma 3 We want to minimize W = v 1[(1 δ)n 1 + δh 1] + v 2(1 δ)n 2 since we know that H 2 = 0. IC constraints impose p 21v 1 + p 22v 2 R 2 (IC 2) and p 11v 1 + p 12v 2 R 1 (IC 1). It follows that if the solution satisfies v 2 v 1, IC 2 implies (p 21 + p 22)v 2 R 2 and finally v 2 R 2. On the other hand, IR constraints impose v 1 0. So, if the solution (v 1, v 2 ) = (0, R 2) is feasible (satisfies IR, IC, FL and PK) constraints, it is the optimal solution (given that we assume v 1 v 2 ). Remark that N 1 > N 2 implies (1 δ)n 1 + H 1 > (1 δ)n 2. It follows that a solution of the form v 1 = R 2 and v 2 = 0 is necessarily inefficient. It is sufficient to find transition probabilities (p ij ) such that (v 1, v 2 ) = (0, R 2) satisfies all the constraints. Take p 22 = 1 and p 12 = d 2N 2/((1 d 1)N 1) 1. Constraint IC 2 is trivially satisfied and IC 1 becomes p 12R 2 R 1 which is equivalent to d 2N 2R 2 (1 d 1)N 1R 1 (true by assumption here). Clearly, p 21 = 1 p 22 = 0 implies no demotions. 20 / 31

Optimal personnel policy Proof of Lemma 3, ctd. We now prove full job security at the bottom (no forced turnover). Adding up FL 1 and FL 2 with H 2 = 0 yields and it is easy to see that H 1 > 0. H 1 = N 1 M 1(p 11 + p 12) + N 2 M 2, Assume that p 11 + p 12 < 1. Set H 1 = H 1 M 1ɛ, and p 11 = p 11 + ɛ with ɛ > 0 small. The Fl contraints are still satisfied ; IC 1 is not affected ; all other constraints remain satisfied. The perturbation reduces the wage bill W weakly, since H 1 is reduced : it follows that p 11 + p 12 = 1 is part of an optimum. Q.E.D. Remark : IC 1 is slack because of sufficient separation rents : p 11v 1 + p 12v 2 = p 12v 2 = d 2N 2 (1 d 1)N 1 R 2 > R 1. Remark 2 : Full job security is optimal but it is not the only optimum. 21 / 31

Optimal personnel policy Insufficient separation rents In the second case, we can prove LEMMA 4. Assume that v1 v 2. If there are insufficient separation rents, i.e., d 2 N 2 R 2 < (1 d 1 )N 1 R 1, in a optimal policy, bottom workers receive zero rents, i.e., v1 = 0, top workers receive more than R 2, i.e., v2 > R 2 ; there are no demotions, i.e., p21 = 0 ; there is full job security at the bottom,i.e., p11 + p 12 = 1, and forced turnover at the top, p 22 < 1. Intuition : To increase incentives for bottom workers, some top workers must be fired. It s never optimal to increase wages at the bottom. The firm can recapture increased wages at the top by lowering wages at the bottom. Everything depends on the ratio N 1 /N 2, called the span. 22 / 31

Optimal personnel policy Proof of Lemma 4 Define the excess rent i = p i1 v 1 + p i2 v 2 R i. Using FL i ; we have M 1 1 + M 2 2 = i ((M 1p 1i + M 2p 2i )v i M i R i ) = (N 1 H 1)v 1 + (N 2 H 2)v 2 M 1R 1 M 2R 2. The objective function W can be rewritten as follows, W = i [N i c i + [(1 δ)(n i H i ) + H i ]v i ] = i [N i c i + H i v i + (1 δ)(n i H i )v i ] = i [N i c i + H i v i + (1 δ)(m i i + M i R i )]. We then find a lower bound for W, W N 1c 1 + N 2c 2 + (1 δ)(m 1R 1 + M 2R 2). If v 1 = 0 and 1 = 2 = 0 are feasible with H 2 = 0, we reach the lower bound. 23 / 31

Optimal personnel policy Proof of Lemma 4, ctd We first show that i = 0 for all i is feasible. With v1 = 0, IC becomes p 12v2 = R 1, p 22v2 = R 2. Using FL i with i = 0 yields, M 1R 1 + M 2R 2 = Σ i (N i H i )vi = N 2v2. Therefore, v2 M1R1 + M2R2 (d2n2 + M2)R2 = > = N2R2 = R 2. N 2 N 2 N 2 From IC i, with v1 = 0 we derive the probabilities, p 12 = R1 v 2 = N 2R 1 M 1R 1 + M 2R 2, p 22 = R2 v 2 Since R 2 > R 1 and under insufficient separation rents, we have p 12 < p 22 < = N 2R 2 d 2N 2R 2 + M 2R 2 = 1. N 2R 2 M 1R 1 + M 2R 2. 24 / 31

Optimal personnel policy Proof of Lemma 4, ctd 2 We finally need to show no demotion at the top and full job security at the bottom. If p 21 > 0 choose ɛ > 0 small and define the perturbation, p 21 = p 21 ɛ, H 1 = H 1 + M 2ɛ. Constraint FL 1 is still satisfied and decreasing p 21 doesn t affect IC 2 since v1 = 0. It follows that p 21 = 0 is optimal, but we have forced turnover at the top since p22 < 1. The same logic as in Lemma 3 shows that p 11 + p 12 = 1 is part of an optimum. Q.E.D. 25 / 31

Optimal personnel policy Full characterization of optimal personnel policy A case remains to be studied : when equilibrium payoffs satisfy v 1 > v 2. It is possible to show that this can happen only if N 2 > N 1. PROPOSITION 1. If N 1 > N 2 the optimal policy satisfies v2 v 1. So Lemmata 2-4 characterize the solution, with the following features, (i) Hiring occurs only in the bottom job (port of entry). (ii) Bottom workers stay at the bottom or are promoted (well-defined career path). Top-workers are never demoted but may be fired. (iii) Bottom-job wages correspond to rents that are lower than R 1. Top-job wages correspond to rents that are higher than or equal to R 2, and v 2 > R 2 when there are insufficient separation rents. (iv) wages and probabilities of transition depend only on the span N 1 /N 2. 26 / 31

Optimal personnel policy Full characterization of optimal personnel policy 2 COROLLARY 1A. If N 1 > N 2 the optimal policy satisfies the following, (i) If there are sufficient separation rents, the wages are w 1 = c 1 δ d 2R 2 N 2 N 1 w 2 = c 2 + (1 δ)(1 d 2 )R 2 + d 2 R 2. (ii) Under insufficient separation rents, the wages are, w 1 = c 1 δ(1 d 1 )R 1 w 2 = c 2 + (1 δ)(1 d 2 )R 2 + N 1R 1 (1 d 1 ) N 2. 27 / 31

Optimal personnel policy Full characterization of optimal personnel policy 3 COROLLARY 1B. If N 1 > N 2 the optimal policy satisfies the following, (i) If there are sufficient separation rents, the labor cost function is W (N 1, N 2 ) = c 1 N 1 + c 2 N 2 + (1 δ)n 2 R 2. (ii) Under insufficient separation rents, the labor cost function becomes, W (N 1, N 2 ) = c 1 N 1 + c 2 N 2 + (1 δ)[(1 d 1 )N 1 R 1 + (1 d 2 )N 2 R 2 ] Remark : The labor-cost function is linear in (N 1, N 2 ). The coefficients on N i are the marginal labor costs. These coefficients are typically larger than c i (inefficiency). 28 / 31

Optimal production Optimal production We can now determine the organizational structure (N1, N 2 ). The firm solves the following problem, Maximize F (N 1, N 2 ) W (N 1, N 2 ) with respect to (N 1, N 2 ). A standard problem in microeconomics. First, given the production level y, choose the structure (N 1 (y ), N 2 (y )) that minimizes W subject to F (N 1, N 2 ) y. A technical assumption on the MRTS F 1 /F 2 would make sure that N 1 > N 2. Drawing : isoquants, iso-cost lines. 29 / 31

Optimal production Important consequences Let MRPi have be the marginal revenue product of the firm for activity i, we At the optimum, we have MRP i = F (N 1 (y ), N 2 (y )) N i w 1 < MRP 1 and w 2 > MRP 2. This departure from neo-classical marginal productivity theory is due to moral hazard (asymmetric information ; imperfectly observable effort). Backloading of compensation implies that creating additional positions at the top relaxes the firm s incentive constraints, so the firm will go beyond w 2 = MRP 2. Creating additional positions at the bottom tightens the incentive constraints so the firm will stop hiring at the bottom before reaching the point at which w 1 = MRP 1. 30 / 31

Empirical relevance Empirical relevance Internal labor markets : Workers tend to be hired into lower-level positions ; there are internal promotions ; demotions are exceptional. Empirical facts : There are wage jumps at promotion. Higher wages at the top serve a dual role of motivating both top and bottom workers. Firms may exhibit a preference for internal candidates (insider bias in hiring at the top). External hires at the top are typically more difficult (they have a cost in terms of incentives at the bottom). Firms may adopt forced turnover policies to create promotion opportunities (e.g., Mandatory-retirement policies binding on the firm s side). 31 / 31