Contracting With Synergies Alex Edmans1

Itay Goldstein2

John Zhu3

LSE Finance Seminar

October 2012

1 The

Wharton School, University of Pennsylvania, NBER, and ECGI Wharton School, University of Pennsylvania 3 The Wharton School, University of Pennsylvania 2 The

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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Introduction

Introduction Most compensation models consider a single agent (e.g. CEO). But most work is conducted in teams Team members’actions are synergistic (e¤ort by one agent reduces colleague’s cost of e¤ort). Structure of synergies is complex – Asymmetric: CEO has greater impact on VP than vice-versa – Number of synergistic relationships varies across agents

Broad framework of contracting under synergies where – E¤ort is continuous (bi-level contracting problem) – Synergies are asymmetric – Number of synergistic relationships di¤ers

Solve for optimal e¤ort and wages (absolute and relative); total wage bill

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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Introduction

Introduction (cont’d) Purpose of model: – Address questions that cannot be explored in a single-agent framework, e.g. XS pay and e¤ort di¤erences, team composition – Explain real-life practices that contradict single-agent models, e.g. high incentives given to employees with little direct e¤ect on output

Existing literature on multi-agent principal-agent problems: – Free-rider problem, e¤orts are perfect substitutes: Holmstrom (1982) – Itoh (1991), Ramakrishnan and Thakor (1991): multi-tasking problem – Symmetric production complementarities: Kremer (1993), Che and Yoo (2001), Winter (2004, 2006, 2010) – Contracting with externalities: * Kandel and Lazear (1992): peer pressure; e¤ort a¤ects others’utility * Segal (1999): agents impact others’utilities rather than costs. Action is participation not e¤ort: no output * Bernstein and Winter (2012) extend to heterogeneous externalities Edmans, Goldstein, Zhu

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Introduction

Roadmap

General model illustrating synergy concept and e¤ect of changing i’s in‡uence on his e¤ort and pay Speci…c model allows additional study on j’s e¤ort and pay – Two-agent model – Three-agent model

Extensions – Negative synergies – First-best analysis – Production complementarities

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Contracting With Synergies

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The General Model

The General Model RN principal (“…rm”) and N agents (“workers”). LL, w = 0 Output is r 2 f0, 1g, contractible. Production function: Pr(r = 1) = P (p1 , . . . , pN ) = P (p ) with Pi > 0, Pii < 0 E¤ort is pi 2 [0, 1]

i’s overall cost function is C i (p ) = ki gi (pi ) Πj 6=i hji (pj )

– gi (pi ) is i’s individual cost function with gi0 > 0, gi00 > 0 – hji pj , with hji0 ( ) < 0, represents j’s in‡uence on i * A¤ects marginal cost, not just total cost * Cost reduction or private bene…t enhancement

– ki = 1 initially

P and C i are common knowledge before contracting. Agents choose e¤orts pi simultaneously in a Nash equilibrium Edmans, Goldstein, Zhu

Contracting With Synergies

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The General Model

Analysis

Agent i is paid 0 upon failure and wi upon success; principal must choose optimal wi – wi captures both level and sensitivity of pay

Agent i solves maxpi wi P (p )

C i (p ). Assuming FOA, wi =

Principal solves maxfpi g,fw i g P (p1 , . . . , pN ) (1

Edmans, Goldstein, Zhu

Contracting With Synergies

C ii Pi

∑i wi ) = PM.

October 2012

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The General Model

Analysis (cont’d) FOC wrt pi is " 0 =Pi 1 P

∑ wi

i 00 gi (pi

#

)Πj 6=i hji (pj )Pi

P ∑ gj0 pj

Pi2

gi0 (pi )Πj 6=i hji (pj )Pii

hij0 (pi )Πm 6=i ,j hmj (pm )Pj Pj2

j 6 =i

Πj 6=i hij (pi )Pij

First term: increased production multiplied by principal’s share Second term: increased wage required to induce a higher pi Third term results from complementarities – First part: e¤ect of pi on colleagues’cost (in‡uence) – Second part: e¤ect of pi on colleagues’productivity Edmans, Goldstein, Zhu

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The General Model

E¤ect of Increasing 1’s In‡uence 0 0 8 j 6 = 1, with h1j (pj ) shifts to h˜ 1j (pj ), where h˜ 1j h1j and h˜ 1j h1j 0 < h 0 for at least one j. Two e¤ects: h˜ 1j < h1j and h˜ 1j 1j 1

Reduces j’s cost C j and MC Cjj to C j (p ) = gj (pj ) e h1j (p1 )

Cjj (p ) = gj0 (pj ) e h1j (p1 )

2

∏

hmj (pm )

∏

hmj (pm )

m 6=1,j m 6=1,j

Reduction would occur even if p1 held constant at p1 . Could be achieved by reducing kj , so present in a model w/o synergies 0 (p ) h1j Reduces cross-partial C1jj = gj0 (pj ) e 1 ∏m 6=1,j hmj (pm ) < 0. Speci…c to a model of synergy Edmans, Goldstein, Zhu

Contracting With Synergies

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The General Model

E¤ect of Increasing 1’s In‡uence (cont’d)

We thus shift 1’s in‡uence to reduce C1jj , but keep C j and Cjj unchanged at p1 . Set kj =

h 1j (p 1 ) , h˜ 1j (p 1 )

so j’s new cost function is:

e j (p ) = h1j (p1 ) gj (pj )h˜ 1j (p1 ) C h˜ 1j (p ) 1

∏

hmj (pm ) .

m 6=1,j

e j (p , p2 , ..., pN ) Note that C j (p1 , p2 , ..., pN ) = C 1 e Principal’s new objective function is P M

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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The General Model

E¤ect of Increasing 1’s In‡uence: Results ∂ ∂p j

e = 0, so jp P M

e > 0 is su¢ cient for p jp P M e1 > p1 " # ej 1 C ∂ j e = P1 1 C1 jp P M ∂p1 P1 j∑ P 6 =1 j " # e j Pj C e j Pj 1 1 P C C11 C11 P11 j1 j i +P ∑ P12 Pj2 j 6 =1

=

∂ ∂p 1

∂ P jp PM + ∑ ∂p P | 1 {z } j 6 =1 j

Cjj1

=0

P =∑ P j 6 =1 j

∏ hij (pi )gj0 (pj )

ej C j1

0 h1j (p1 )

0 kj h˜ 1j (p1 )

i 6 =1

No hij terms (taken care of by rescaling), only hij0 terms Pj 1 drops out: already taken care of in original maximization Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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The General Model

E¤ect of Increasing 1’s In‡uence: Results (cont’d)

0 Since h˜ 1j

0 < 0 and k h1j j

1,

∂ ∂p 1

e > 0 so p jp P M e1 > p1

0 p – Robust to any P, g , hij . All we need is h˜ 1j i

0 p < h1j i

If P1j = 0 8 j and hj 1 ( ) = 1 8 j, w e i > wi

– Not automatic that " p !" w . In single-agent models with RN and LL, w = 12 , independent of p (and thus productivity and cost)

Here, wage unambiguously increases in in‡uence – In‡uence has no direct e¤ect on output; only a¤ects colleague’s cost – 1 does not consider this e¤ect, as he takes j’s e¤ort as given – Sharper contract causes him to internalize the externality. 1 is “over-incentivized” compared to a model without synergy

Edmans, Goldstein, Zhu

Contracting With Synergies

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The Two-Agent Model

The Model

Pr(r = 1) =

p 1 +p 2 2

gi (pi ) = 14 pi2 hji (pj ) = 1

hji pj where hij

0

is an in‡uence parameter. A¤ects marginal cost, not just total cost Thus, Ci (p ) = 41 pi2 (1

hji pj )

Synergy is the sum of the in‡uence parameters: s = h12 + h21

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Contracting With Synergies

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The Two-Agent Model

Results (i) For nonzero synergy, e¤orts are equal: p1 (s ) = p2 (s ) a critical synergy level s > 0 s.t.: ( p 4 3s 2 s 2 (0, s ) 3s p (s ) = 1 s s .

p (s ). 9

Seems that more in‡uential agent should exert higher e¤ort. But this decreases colleague’s cost, inducing greater e¤ort – Equal e¤ort is model-speci…c. But, idea that pi depends on i’s in‡uence and j’s in‡uence (not just i’s individual production and cost functions) is general – While in‡uence parameters are individual and asymmetric, the synergy is common to both agents; an “echo”. A su¢ cient statistic for e¤ort

E¤ort p (s ) is increasing in synergy s – Direct e¤ect on output unchanged, but reduces colleague’s cost Edmans, Goldstein, Zhu

Contracting With Synergies

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The Two-Agent Model

Results (cont’d) (ii) Total wages, and expected total wages, are both increasing in s on (0, s ] – Again, not automatic: even though p is higher, cost of e¤ort is lower

Suggests high equity incentives should be given to low-level employees, if large indirect e¤ect (e.g. e¢ cient analyst reduces cost of director going to a meeting) – Synergies will be particularly high in small and young …rms (blurred job descriptions, frequent interactions, ‡at hierarchy) – Hochberg and Lindsey (2010): broad-based option plans – Oyer (2004): used for retention; Oyer and Schaefer (2005): evidence for retention and screening; Bergman and Jenter (2007): exploit employees’irrational overvaluation

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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The Two-Agent Model

Results (cont’d)

(iii) " in either in‡uence parameter raises e¤ort and pay

– A rise in in‡uence raises s, and thus e¤ort (i) and wages (ii). But, no independent e¤ect other than through s – Common synergy, not individual in‡uence parameters, determines p – That s is a su¢ cient statistic is model-speci…c, but idea that e¤ort and pay depends on your in‡uence and colleague’s in‡uence is general

(iv) The more in‡uential agent receives higher pay – Holds even though both agents exert same e¤ort (so " w is not a compensating di¤erential) and have same direct productivity – A greater externality to internalize – More in‡uential agents should receive higher pay, even if perform same tasks (e.g. senior vs. junior faculty)

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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The Two-Agent Model

Results (cont’d)

(v) " i’s relative in‡uence (i.e. " hij and # hji so that s is unchanged) !"relative and absolute w (vi) " hij !" wj i¤ hji >

1 6p (s )

– Optimal to reinforce e¤ect of hij by incentivizing j more i¤ j is su¢ ciently in‡uential

(vii) The more in‡uential agent receives higher utility

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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The Three-Agent Model

The Model

Pr(r = 1) =

p 1 +p 2 +p 3 3

Ci (p ) = 16 pi2 1 De…ne:

∑j 6=i hji pj

A = h12 + h21 B = h13 + h31 C = h23 + h32 . Synergy pro…le s is the vector (A, B, C ). – In two-agent model, this was a scalar

A, B and C are the synergy components of the synergy pro…le Size of s is s = jj(A, B, C )jj.

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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The Three-Agent Model

Results

Edmans, Goldstein, Zhu

Contracting With Synergies

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The Three-Agent Model

Results: Interior E¤ort If s small and synergy component > sum of the other two, e¤ort pro…le p (s) is interior – Rise in s augments e¤ort pro…le

Simplex …xes A + B + C at K and changes relative components Middle triangle studies interior e¤ort pro…le – All three synergy components matter for the relative size of the individual e¤ort levels: i¤ B > C , p1 > p2 . – p1 vs p2 depends on total synergy with agent 3 (B vs C), not just h13 vs h23 . h31 doesn’t a¤ect 1’s productivity, but does a¤ect his cost. As in 2-agent case, p depends on common synergy, not individual in‡uence

Agents no longer exert same e¤ort. More synergistic agents work harder – In 2-agent case, there is only one synergy component – Here, 3 SCs, allowing for asymmetry in e¤ort levels Edmans, Goldstein, Zhu

Contracting With Synergies

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The Three-Agent Model

Results: Boundary E¤ort

If one synergy component > sum of other two, model collapses to 2-agent model – Third agent is ignored, despite having same direct productivity. His participation depends on extrinsic parameters. Implications for composition of team / boundaries of …rm – Only the largest synergy component matters for optimal e¤ort pro…le * For interior solution, p1 > p2 i¤ B > C * Here, p1 = p2 regardless of B vs. C . Synergy between 1 and 2 is so important that individual synergies with 3 are irrelevant

– As in 2-agent model, wages depend on relative in‡uence of each agent

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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The Three-Agent Model

Results: Boundary E¤ort (cont’d) If a single synergy component is close to zero, the two non-synergistic agents can be aggregated into a single agent and model again collapses to 2-agent model – Third agent exerts same e¤ort as others combined; their e¤orts depend on relative size of synergies with third agent – Applies to CEO and divisional managers * Bebchuk, Cremers, and Peyer (2011): high pay to CEO is ine¢ cient rent extraction * Rising pay over time consistent with improving communication technologies (cf. Garicano and Rossi-Hansberg (2006)) * Relevant measure of …rm size (e.g. in Gabaix and Landier (2008), Terviö (2008)) is scope and depth of synergies, rather than assets or pro…ts. CEO of holding company should be paid less than CEO of small focused …rm

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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The Three-Agent Model

Results: Wages

Total wages depend on total synergy Relative wages depend on in‡uence parameters: an increase in one agent’s in‡uence parameter increases his wage in both absolute and relative terms If in‡uence parameters are symmetric across a pair of agents, relative wages within this pair are the same. Entire wage pro…le can be fully solved; ratios of optimal wages coincide with ratios of optimal e¤ort

Edmans, Goldstein, Zhu

Contracting With Synergies

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Extensions

Extensions

Negative in‡uence parameters: if s > 0, results hold; if s < 0, collapses to a single-agent model Complementary e¤ort: Pr(r = 1) = min (p1 , p2 , ..., pN ). Results remain robust: e¤ort and wages increase in synergy First-best analysis: more in‡uential agent exerts less e¤ort – Hence, moral hazard problem !" relative e¤ort of the more in‡uential agent – First-best: wages are compensation for disutility (quadratic total costs) * Cost reduction due to 1’s in‡uence is h12 p1 p22 , so " h12 causes principal to " p2 more than p1

– Second-best: wages are incentives for e¤ort (linear marginal costs)

Edmans, Goldstein, Zhu

Contracting With Synergies

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Extensions

Production Synergies p Pr (r = 1) = a p1 +2 p2 + b p1 p2 ci (p ) = 14 pi2 p1 = p2 =

a +b 4

To implement (p1 , p2 ), must o¤er w1 =

pq 1 p a +b p 2

and w2 =

1

w1 = w2 =

1 4

and is independent of a or b

pq 2 p a +b p 1 2

a and b enter symmetrically: both increase agent i’s impact on output Production function and cost function are not isomorphic – Contracts are contingent upon output but cannot be contingent upon cost of e¤ort. Thus, agent internalizes e¤ect on production but not on colleague’s costs – Cost synergies are true externalities, unlike production synergies

Edmans, Goldstein, Zhu

Contracting With Synergies

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Extensions

Production Synergies: General Model

With production synergies, i’s FOC changes from wi Pi (pi , pj )

gi0 (pi ) Πj 6=i hji (pj )

ei (pi , pj ) wi P

gi0 (pi ) Πj 6=i e hji (pj ) ,

to and so the synergy is internalized. With cost synergies, Pi is unchanged. hij changes to hij , but this does not show up in i’s FOC so is not internalized

Edmans, Goldstein, Zhu

Contracting With Synergies

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Extensions

Conclusion Paper studies e¤ect of synergies on optimal e¤ort and wages General model: – Additional term in e¤ort determination equation – " i’s in‡uence !" pi

2-agent framework – Wages di¤er, even if same e¤ort and direct productivity – Total wages increase with synergy, consistent with high equity incentives in start-ups – E¤ort depends not only on individual production and cost functions, but your in‡uence and colleague’s in‡uence

3-agent framework: – E¤orts di¤er and depend on synergies, not in‡uence – If synergies between two agents are strong, third agent is excluded – Agents that exert synergies over more colleagues are paid more Edmans, Goldstein, Zhu

Contracting With Synergies

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Complementary E¤ort

Complementary E¤ort Production function: Pr(r = 1) = min (p1 , p2 , ..., pN ) . Cost function: hi (pi ) =

κi 2 p . 2 i

FOCs: p1 = p2 = . . . = pN wi ( p ) = κ i p

1

∑ hji p

j 6 =i

p !

All agents exert same e¤ort (as perfect complementarities) Agents with more di¢ cult tasks (higher κ i ) receive higher wages. Synergy is sum of each agent’s total in‡uence: s = ∑i ∑j 6=i hij κ j Di¢ culty is sum of the cost parameters, κ ∑i κ i . Edmans, Goldstein, Zhu

Contracting With Synergies

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Complementary E¤ort

Complementary E¤ort: Results (i) 9 a critical synergy level s (κ ) > 0 s.t.: ( p κ κ 3s s 2 [0, s (κ )) 3s p (s ) = 1 s s (κ ) . Optimal e¤ort p (s ) is strictly increasing on [0, s (κ )]. (ii) Total wages, and expected total wages, are both strictly increasing on [0, s (κ )]. (iii) An increase in any in‡uence parameter of any agent raises e¤ort and wages (iv) Suppose agent i’s relative in‡uence increases (total in‡uence increases but total synergy is constant), then his relative and absolute wealth increases Edmans, Goldstein, Zhu

Contracting With Synergies

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Complementary E¤ort

General Model: Wages g 0 (p ) Πj 6=i hji (pj ) C11 (p1 ) = 1 1 P1 (p ) P1 (p ) 0 1 0 g (p e ) Πj 6=i hji (p ej ) C (p˜ ) w˜ 1 = 1 1 = 1 1 . P1 (p˜ ) P1 (p e ) w1 =

g10 (p e1 ) > g10 (p1 ), since p e1 > p1 and costs are convex P1 (p e1 , ) < P1 (p1 , ), since p e1 > p1 and prod. fn. is concave Potentially confounding e¤ects: – Even though P1 (p e1 , ) < P1 (p1 , ), P1 may not have decreased because it depends on p2 , ..., pN . If " pj and P1j > 0, or # pj and P1j < 0, this tends to !" P1 and # w1 – If " pj , hj 1 p ej < hj 1 pj – These two e¤ects disappear if P1j = 0 8 j (so P1 (p e ) < P1 (p )) and hj 1 ( ) = 1 8 j

Edmans, Goldstein, Zhu

Contracting With Synergies

October 2012

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