The Firm-Growth Imperative: A Theory of Production and Personnel Management

The Firm-Growth Imperative: A Theory of Production and Personnel Management Rongzhu Ke Hong Kong Baptist University Jin Li London School of Economics Michael Powell Kellogg School of Management

Management Requires Planning Ahead During WWI, DuPont increased workforce from 5,000 in 1914 to 85,000 in 1918 DuPont s post-war diversification into non-chemical industries in part to have a place to locate some managerial personnel

Production Plans and Personnel Policies Production plans affect personnel policies Future production plans determine opportunities for current employees Abundant promotion opportunities in fast-growing firms Scarce promotion opportunities in slow-growing firms Slow-growing firms need to rely on ways other than promotions to motivate their employees

Production Plans and Personnel Policies Personnel policies also affect production plans Using promotions to motivate employees creates a strong organizational bias toward growth to supply the new positions that such promotion-based systems require. (Jensen, 1986) Such growth is typically derided as wasteful but may serve a purpose

Our Contribution Study how past production decisions affect future production decisions when workers are motivated via long-term, career-based incentives Existing works focus on either firm growth or long-term incentives Firm-growth without long-term incentives: Lucas, 1978; Jovanovic, 1982; Hopenhayn, 1992; Ericson & Pakes, 1995 Long-term incentives without implications for firm growth: Rogerson, 1985; Spear & Srivastava, 1987; Biais, Mariotti, Rochet, 2013

Key Findings Optimal promotion policies favor workers with more seniority Cross-subsidization of incentive constraints across worker cohorts Promotion policy is a modified first-in-first-out rule Production plans are time-inconsistent Firm may grow faster than its business opportunities precisely to create promotion opportunities Distortions evolve throughout the firm s lifecycle

Agenda The Model Optimal Personnel Policies Optimal Production Paths Extensions

Model Ingredients One principal interacts with pool of agents at each t = 1,, T Agents are risk-neutral, infinitesimal, identical, and share discount factor δ Agents choose binary effort (work/shirk); shirking can be caught Principal assigns each agent to one of two activities in each period: activity 1 easier to do/monitor Principal chooses production plan {N 1,t, N 2,t }, where N i,t is mass of agents assigned to activity i that exert effort in period t

Each Period 1 2 3 4 5

Each Period Incumbents from previous period 1 2 3 4 5 Principal assigns Agents Agents choose to exit or not 1: Principal assigns each incumbent worker to activity i {1,2}, and each worker decides whether to stay or leave the firm and get 0

Each Period Incumbents from previous period 1 2 3 4 5 Principal assigns Agents Effort e t chosen Agents choose to exit or not 2: Each agent chooses whether to work or shirk. Agent assigned to activity i chooses effort e t {0,1} at cost c i e t

Each Period Incumbents from previous period 1 2 3 4 5 Principal assigns Agents Agents choose to exit or not Effort e t chosen Shirking detected with prob q i 3: A signal y t {0,1} is realized for each agent. If e t = 1, then Pr[y t = 1] = 1 and if e t = 0, then Pr y t = 1 = 1 q i.

Each Period Incumbents from previous period 1 2 3 4 5 Principal assigns Agents Agents choose to exit or not Effort e t chosen Shirking detected with prob q i Principal pays wages 4: The principal pays wages W t 0 to each agent

Each Period Incumbents from previous period 1 2 3 4 5 Principal assigns Agents Agents choose to exit or not Effort e t chosen Shirking detected with prob q i Principal pays wages Exogenous turnover 5: Each agent leaves the relationship with probability d and receives 0 in all future periods

Contracts Principal can commit to a long-term contract specifying sequence of wage policies and assignment policies that can depend on history: h t = (A 1,, A t ), where A τ {0,1,2}, where 0 denotes inactive agent Wage policies are mappings from histories to nonnegative payments w(h t ) W t (h t ). Assignment policies specify probability p i (h t ) that agent with history h t will be assigned to activity i

Assumptions Assumption 1: All agents with same history are treated the same Without loss of generality Assumption 2: All agents are asked to choose e t = 1, and if an agent leaves the firm, he leaves forever. No distinction between shirking and taking the outside option for one period Simplifying assumption to be relaxed later Assumption 3: If signal is ever y t = 0, agent is fired forever Without loss of generality given assumption 2

Payoffs In period t, agent assigned to activity i = 1,2 receives w h t agent assigned to activity 0 receives w(h t ) c i e t and Principal s payoff in period t is F t N 1,t, N 2,t h t H t w h t l h t, where l(h t ) is mass of agents with history h t

Outline Recall that a production path is sequence N = N 1,t, N 2,t t=1 T For a given production path N, find optimal personnel policies Constraints and program Solution and features Optimal production path Two-period example General features

Agenda The Model Optimal Personnel Policies Optimal Production Paths Extensions

The Program: Notation Define v(h t ) to be agent s continuation payoff with history h t Define c(h t ) to be agent s cost of effort with history h t c h t = c i if h t = i 1,2 and c h t = 0 if h t = 0 For h t = i 1,2, define q h t q i

The Program: Objective & Constraints subject to min w, p i T t=1 h t H t δ t 1 w h t l h t v h t = w h t c h t + δ 1 d i 0,1,2 p i h t v h t i v h t (1 q h t )(w h t + δ 1 d i {0,1,2} p i h t v(h t i)) for h t 1,2 v h t 0 h t h t =i l ht = N i,t for i {1,2} l h t i = 1 d p i h t l h t for i {1,2} (PK) (IC) (IR) (Flow)

The Program: Objective & Constraints subject to min w, p i T t=1 h t H t δ t 1 w h t l h t v h t = w h t c h t + δ 1 d i 1,2 p i h t v h t i v h t (1 q h t )(w h t + δ 1 d i {0,1,2} p i h t v(h t i)) for h t 1,2 v h t 0 h t h t =i l ht = N i,t for i {1,2} l h t i = 1 d p i h t l h t for i {1,2} (PK) (IC) (IR) (Flow)

The Program: Objective & Constraints subject to min w, p i T t=1 h t H t δ t 1 w h t l h t v h t = w h t c h t + δ 1 d i 1,2 p i h t v h t i v h t (1 q h t )(w h t + δ 1 d i {1,2} p i h t v(h t i)) for h t 1,2 v h t 0 h t h t =i l ht = N i,t for i {1,2} l h t i = 1 d p i h t l h t for i {1,2} (PK) (IC) (IR) (Flow)

The Program: Objective & Constraints subject to min w, p i T t=1 h t H t δ t 1 w h t l h t v h t = w h t c h t + δ 1 d i 1,2 p i h t v h t i 1 q ht v h t (1 qc h t )(w h t + δ 1 d i {0,1,2} p i h t v(h t i)) for h t 1,2 q ht R ht for h t {1,2} ht v h t 0 h t h t =i l ht = N i,t for i {1,2} l h t i = 1 d p i h t l h t for i {1,2} (PK) (IC) (IR) (Flow)

Minimizing Rents to New Hires Denote agent s initial-hire history by n t = (0,, 0, n t ) where n t {1,2} Cost-minimizing personnel policies minimize rents paid to new hires: min subject to (PK), (IC), and (Flow) T w, p i t=1 h t H t δ t 1 l n t v(n t ) Focus on steady production paths, which for all t, satisfy N 2,t+1 < 1 d N 1,t + N 2,t N i,t+1 1 d N i,t for i = 1,2

Internal Labor Markets Under a steady production path, there is an optimal personnel policy with the following properties: 1. All new workers are assigned to activity 1 2. Workers in activity 1 have v h t [R 1, R 2 ] and are not fired if y t = 1 3. Workers in activity 2 have v h t = R 2 and stay in activity 2 unless exiting exogenously Comments: 1. No unique optimal personnel policy 2. Lots of payment possibilities: we pick payment paths that partially frontload, ruling out paths with v h t > R 2 and w h t > 0 3. Lots of assignment possibilities 4. The hiring policy is unique

ILMs: Wage and Promotion Policies Under a steady production path, there is an optimal personnel policy with the following properties τ 1. Wages are non-decreasing: w 1,t+1 2. Seniority: if τ < τ, then w τ 1,t τ w 1,t τ w 1,t, p τ τ t p τ t, and v 1,t+1 3. Modified FIFO: if p τ t, p τ τ t (0,1), then v 1,t+1 τ = v 1,t+1 τ v 1,t+1

Modified First-in-First-Out Promotions For each period t, there are two thresholds: τ 1 t τ 2 (t) Workers entering the firm after τ 2 t will not be promoted this period Workers entering between τ 1 (t) and τ 2 t probabilities, where p τ t decreases in τ have positive promotion Workers entering before τ 1 (t) have the same (highest) promotion probability If τ 1 t = t, then promotion policy is seniority-blind

Three-Period Example Suppose T = 3. Let d = 0, δ = 1, N 1,t N 2,t = 1, and q 1 = q 2 = 1 2, so R i = c i Assume: 1. g 2 > c 1 c 2 c 1 2. c 1 > 1 g 1 1 g 1 1+g 2 +1 c 2 Then the optimal internal labor market strictly favors cohort-1 workers in promotion decisions in period 2: p 2 1 > p 2 2

Cross-Subsidizing Incentive Constraints Assumption 1 implies that if p 2 1 = p 2 2, the IC constraint for cohort-2 workers is slack (even if they are paid a wage of 0) By reducing the probability of promotion for cohort-2 workers, the IC constraints for cohort-1 workers can be relaxed The firm can reduce cohort-1 workers wages Assumption 2 ensures cohort-1 workers wages must be positive if p 2 1 = p 2 2

Agenda The Model Optimal Personnel Policies Optimal Production Paths Extensions

General Production Problem Formulation Firm s problem: max N i,t,h t,w s 1,t,z s t,v 1,t s,p t s T t=1 T t=1 δ t 1 F t N 1,t, N 2,t w 2,t N 2,t t s=1 H s w s s 1,t z t subject to the flow constraint for N 1,t, the flow constraint for N 2,t, H t + t 1 s=1 N 2,t 1 1 d + H s z t s = N 1,t, s z t+1 = z s s t 1 p t 1 d. and IC and promise-keeping constraints and the additional constraints t 1 s=1 s s H s z t 1 p t 1 = N 2,t, H t : number of new hires in period t who are assigned to activity 1 z t s : fraction of cohort-s workers who remain in activity 1 in period t

General Results Suppose T <. Under a steady production path, then for all t T, the following hold: 1. F t t N 1,t w 1,t 0 and: 2. F T N 2,T w T 2,T T s=t F t t w N 1,t 1,t 0 and: δ s t 1 d s t F t+1 t+1 w N 1,t+1 1,t+1 F s s w N 2,s 0 2,s

Distortions in the Bottom Position The distortion in the bottom position is measured by: D t F t t w N 1,t 1,t In the static case, these distortions would be zero in each period. D t > 0: for the same wage, fewer workers are hired than in static case Add l bottom position today puts pressure on future opportunities D t > 0 reflects the shadow cost of incentive provision Distortion is bigger for workers hired at t than at t + 1 (D t D t+1 ) 1. Workers hired in t puts pressure on opportunities at t + 1 2. From t + 2 on, workers hired in t have higher wages and promotion probabilities than those hired in t + 1

Distortions in the Top Position Unlike bottom position, number of top positions can be greater or smaller than corresponding static case because two conflicting forces: 1. An extra top position today crowds out future promotion opportunities 2. An extra top position today provides more promotion opportunities for previously hired workers At T, this first force is nonexistent: F T N 2,T w T 2,T 0

Distortions in the Top Position More generally, consider perturbation which creates 1 position at t, 1 d positions at t + 1, 1 d 2 positions at t + 2, and so on This perturbation increases promotion opportunities in period t and does not affect promotion opportunities in periods after t, so T s=t δ s t 1 d s t F s s w N 2,s 0 2,s This is the shadow benefit of creating a line

Optimal Production: Two-Period Example Suppose T = 2, and firm has fixed-factor production: Let N t min{n 1,t /s,n 2,t } and θ t be a demand parameter Revenues in period t are θ t ln N t The firm s maximization problem is max θ 1 ln N 1 N 1 sw 1,1 + w 2,1 + δ(ln N 2 N 2 sw 1,2 + w 2,2 ) N 1, N 2, {w i,1, w i,2 } subject to (IC), (IR), and (Flow) Question: How do demand conditions affect production decisions?

Optimal Production Optimal production plans satisfy the following 1: dn 1 dθ 2 0, 2: 0 d ln N t d ln θ t 1, 3: dn 2 dθ 1 0. These inequalities are strict whenever θ 2 θ 1 (l, l) for some l < l which are independent of θ 1 and θ 2. Compare with the static benchmark dn 1 static static = 0; dn 2 = 0 dθ 2 dθ 1 static d ln N t = 1 d ln θ t

Agenda The Model Optimal Personnel Policies Optimal Production Paths Extensions Unsteady Production Path Temporary Layoff

Unsteady Production Paths So far focused on steady production paths satisfying: 1. N 2,t+1 < 1 d N 1,t + N 2,t 2. N i,t+1 N i,t (1 d) for i = 1,2 There is breakneck growth at t + 1 if N 2,t+1 > 1 d N 1,t + N 2,t There is deep downsizing in i at t + 1 if N i,t+1 < N i,t (1 d)

Permanent Deep Downsizing Suppose N satisfies N 1,t+1 < 1 d N 1,t, N 2,t+1 > 1 d N 2,t+1, and N 1,t+1 + N 2,t+1 < (1 d)(n 1,t + N 2,t ) for all t. Then the following hold: 1. Laid-off workers receive severance pay 2. τ τ If τ < τ, then p 0,t p 0,t 3. Conditional on being laid off, workers hired earlier get more severance pay τ Intuition similar to seniority result: for τ < τ, v 1,t+1 τ v 1,t+1 Higher continuation payoff: higher promotion and lower layoff prob (modified) FIFO for promotions (modified) LIFO for layoffs Conditions in proposition can be weakened

Permanent Deep Downsizing Suppose N satisfies N 1,t+1 < 1 d N 1,t, N 2,t+1 > 1 d N 2,t+1, and N 1,t+1 + N 2,t+1 < (1 d)(n 1,t + N 2,t ) for all t. Then the following hold: 1. Laid-off workers receive severance pay 2. τ τ If τ < τ, then p 0,t p 0,t 3. Conditional on being laid off, workers hired earlier get more severance pay The conditions state that the firm is permanently downsizing. τ Intuition similar to the seniority result: for τ < τ, v 1,t+1 τ v 1,t+1 Higher continuation payoff: higher promotion and lower layoff prob (modified) FIFO for promotions (modified) LIFO for layoffs

Agenda The Model Optimal Personnel Policies Optimal Production Paths Extensions Unsteady Production Path Temporary Layoff

Temporary Layoffs So far, have focused on permanent layoffs: if assigned to activity 0 at t + 1, assigned to activity 0 in all future periods We now allow for temporary layoffs: a worker assigned to activity 0 at t + 1 may be assigned to activity 1 or 2 in the future Allowing for temporary layoffs can reduce firm s wage bill in two ways With (temporary) deep downsizing Without deep downsizing

Temporary Deep Downsizing Suppose there is a t 1 at which N 1,t1 +1 < 1 d N 1,t1 and N 1,t1 +1 + N 2,t1 +1 < (1 d)(n 1,t1 + N 2,t1 ) and there is a t 2 > t 1 at which N 1,t2 +1 + N 2,t2 +1 > (1 d)(n 1,t2 + N 2,t2 ). Then the following hold: 1. No workers are permanently laid off in period t 1. τ 2. v 1,t2 +k t v 2 +k 1,t2 +k for all τ < t 2 and for all k 1. Condition states that the firm will recover after some period of downsizing. Part 1: rehiring existing workers reduces rents given to workers Part 2: same intuition as seniority result

No Deep Downsizing: Example Suppose T = 3. Let d = 0, δ = 1, N 1,t N 2,t = 1, and q 1 = q 2 = 1/2 so R i = c i Suppose the following two assumptions hold 1. g 2 > max 2. c 1 > c 2 /2 c 1 c 2 c 1, 2 Under these assumptions, optimal personnel policies lay off cohort-1 workers in period 2 and rehire them in period 3

No Deep Downsizing: Intuition Cross-subsidizing incentive constraints: Previously: subsidize through increasing promotion probability Now: subsidize through reducing the effort cost of the cohort-1 worker Implemented through hiring new workers one period earlier and ask them to put in effort Marginal cost of rewarding the agent is nonlinear Equal to 0 up to R 2 Equal to 1 afterwards

Conclusion Model of personnel policies and production Optimal personnel policy features seniority-based internal labor markets Promotion policy is a modified FIFO rule Optimal production is time-consistent Further implications/work Externally: greenfield investment versus acquisitions Internally: spin-off, encouraging workers to create start-ups