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Contracts, Risk-Sharing, and Incentives

Bruno Van der Linden

Université catholique de Louvain

January 25, 2011

(ESL-UCL) 1 / 102

Introduction

The labour relationship:Generally a long-term one

The labour contract: Oftena relationship of subordination specifying rights and duties;does not list the consequences of every possible eventuality.

Remuneration:In some cases linked to performance, in others not.

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Introduction

Aims

Why firms and workers engage in long term relationships?How the tradeoff between insurance and incentives acts upon theremuneration rule for labor ?Why firms make use of hierarchical promotions and internalmarkets ?The links between seniority, experience and wages.The efficiency wage theory.

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Introduction

Standard approach in economics

Standard stylized (narrow) view about human motivation:Employees seek to earn as much money as possible with minimaleffort on the job (or minimal hours worked).

The work done has no value per se.No possibility of identification with the firm (hence, no effort of thefirm to affect employee’s identification).No norms of behaviour (Which effort is “normal”? Fairness?)Worker’s preferences are the same whether or not there is anincentive scheme or a supervisor.

Part 1: Standard approach in economics. Covered by Chap. 6 ofCahuc and Zylberberg (2004).

Note: For this part, slides adapted from those of Bart Cockx.Part 2: Tries to enlarge (a bit) the standard view.

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Introduction

Road-map

1 1. Stylized narrow view about motivation1.1. The Labour Contract1.2. Risk-sharing1.3. Incentives if results verifiable1.4. Incentives in the Absence of Verifiable Results

2 2. Extension2.1. Intrinsic motivation2.2. Envy

3 References

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1. Stylized narrow view about motivation 1.1. The Labour Contract

1. The Labour Contract

Contract theory explains how such contracts can be understood as arational response to:

1 uncertainty of the environment2 private information of the employer or the employee

Uncertainty⇒ To what extent do labour contracts ensure workers(employers?) against risks?

Private information⇒ How do labour contracts provide incentives toreveal private information?

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Typology of contracts

Key questions:Can the employee’s (and the employer’s) activity be observed and if sois it verifiable by a third party (a court)?

Two types of contract can be distinguished:1 Complete or explicit contracts⇔

All clauses of the contract can be verified by a third party, a court;At the moment of signing, all circumstances can be foreseen.

2 Incomplete or implicit contracts⇔All the clauses of the contract cannot be verified by a court orcircumstances are too numerous;The contract must then be self-enforcing: both parties have amutual interest in continuing the relationship;Only feasible in long-term relationship.

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Agency model or principal-agent model

We study contracts within a principal-agent framework:The principal (=employer) proposes a contract;The agent (=employee) can either accept or refuse.

In this chapter, by assumption,no frictions,workers have no bargaining power,it make sense to talk about the “output of the agent”.

Two textbook cases:1 employee’s effort is observed and verifiable but employee’s output

is random2 employee’s effort is not verifiable: employer faces a moral hazard

problem(ESL-UCL) 8 / 102

1. Stylized narrow view about motivation 1.2. Risk-sharing

2. Risk-sharingWhy could this be important?

Assume a one-worker one-firm setting.A1 The agent’s preferences are represented by a

quasi-concave utility function U(C,L) = U(C,1− h),where C denotes the agent’s consumption, h the numberof hours worked (or “effort”) and, therefore, L = 1− h, theagent’s leisure time. Hours observed and verifiable.

A2 The production of the agent = y = f (h, ε), wherefh > 0, fε > 0, fhh ≤ 0 and fhε > 0. A very simple situationwith two possible states: E ={ε1, ε2}, ε1 < ε2.

A3 The profit of the principal is defined by the equalityΠ = f (h, ε)−W , W being real compensation (orearnings) for h hours of work (wage rate is then W/h).

A4 The random variable ε is verifiable.

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Benchmark casePure competition and “spot markets”

Definition: “spot markets” = markets open when the realisation of ε isknown (i.e. “ex-post”).

Interpretation: ε captures the business cycle.

Under perfect competition, free entry implies Π = 0 = f (h, ε)−W .In equilibrium , ∀ε, the pair (h,W ) verifies:

f (h, ε) = W and fh(h, ε) =

[dWdh

]U

=UL(W ,1− h)

UW (W ,1− h)

(see next figure)

Note: unchanged if the principal was risk-averse and maximizing v(Π),with v ′ > 0 and v” ≤ 0.

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Figure: 6.1 Earnings and hours worked in a perfectly competitive spot market

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Two major implications:1 The allocation of labour is efficient ex-post but, from an “ex-ante”

perspective,1 risks are not shared efficiently (a notion madeprecise below): In particular, the wage fluctuates (a lot) with therealisation of ε.

2 Variations in real earnings are greater than variations in hoursworked if, as empirical analyses suggest, the elasticity of the laborsupply is weak w.r.to wages. Although some care is needed(earnings 6= wage rates and hours 6= employment), empiricalobservations suggest that real wage rigidity and large fluctuationsin employment are standard.

1I.e. before the realisation of the “state” ε is observed(ESL-UCL) 12 / 102


Motivation of this literature(started in the 1970s)

1 Can real-wage rigidity be explained by the desire of workers to beinsured against the risk of layoff (when they are not (fully) insuredby a public unemployment insurance scheme)?

2 Can this real-wage rigidity explain

an inefficient allocation of labour

and/or employment fluctuations?

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2.1 Symmetric or verifiable informationAn individual insurance contract model

Assumptions (slightly different from above)A1 Preferences: Unchanged. Note: “effort” (understood as

hours worked) h is observable and verifiable.A2 Same technology but ε is a continuous random variable

defined over the interval E = [ε−, ε+] the density of whichis denoted by g(ε).

A3 The profit of the principal: Unchanged.A4 Verifiability: Unchanged.A5 The principal may be risk-averse: v(Π), with v ′ > 0 and

v” ≤ 0.

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An individual insurance contract modelA contingent insurance contract

A contingent insurance contractspecifies ex-ante....... a level of earnings W and a number of hours worked...... for each possible realisation of ε (hence, the expression“contingent contract”)

Formally, contract A = {W (ε),h(ε)} solves:

max Ev [Π(ε)] s. to EU (W (ε),1− h(ε)) ≥ Uor

max EU (W (ε),1− h(ε)) s. to Ev [Π(ε)] ≥ Π

where U (resp. Π) measures expected external opportunities

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An individual insurance contract modelOptimality conditions

The optimality conditions lead to two strong properties.1. Whatever the state ε, labour is allocated efficiently (!):

UL [W (ε),1− h(ε)]

UC [W (ε),1− h(ε)]= fh [h(ε), ε] , ∀ε ∈ E (3)

2. The so-called “Arrow-Borch” condition for optimal risk-sharing:

UC [W (ε),1− h(ε)]

UC [W (θ),1− h(θ)]=

v ′ [Π(ε)]

v ′ [Π(θ)], ∀(ε, θ) ∈ E2

Risk-sharing efficiency requires that the marginal rate of substitutionbetween any pair of state contingent payoffs be equal for all economicagents

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An individual insurance contract modelReal wage rigidity?

Employers typically own assets while for most workers theirhuman capital is their only (or major) asset.Hence, employers can diversify risks in financial markets.So, if ε only captures diversifiable risks (hence, notmacroeconomic nor “systemic” risks), taking employers as riskneutral is a sensible assumption.

Then,1 the Arrow-Borch condition implies that the marginal utility of

consumption has to be equal in all states ε;2 this does not in general imply full real wage rigidity.

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An individual insurance contract modelProperties of the contract if employers are risk neutral

Differentiating the FOCs wrt to ε, we obtain(U2

CL − UCCULL

UCUCC− fhh

)dhdε

= fhε anddWdε

=UCL

UCC

dhdε

(4)

1 If we assume that UCC < 0 and (UCL)2 − UCCULL < 0 (sufficientconditions for quasi-concavity2), hours worked are an increasingfunction of ε, since fh increases in ε. No ambiguity “substitution vsincome effects”.

2 Earnings W is insensitive to the cycle if UCL = 0. Reflectsrisk-sharing.

2i.e. UCC (UL/UC)2 − 2UCL (UL/UC) + ULL < 0 ∀(C, L)

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About dWdε = UCL

UCC

dhdε :

3 Earnings W are pro-cyclical only if UCL < 0.Usually, UCL > 0 is assumed. For leisure to be a normal good, itcan be shown that the following condition should be fulfilled :

UCC

UL⊕︸︷︷︸

− UCL UC⊕< 0

A sufficient condition is therefore UCL ≥ 0.1+3⇒ when leisure is a normal good, in a “bad state”, there ismore leisure and hence the marginal utility of consumptionincreases. Then, to equate this marginal utility across states,higher earnings are needed in states with more leisure.

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4 If leisure is a normal good, the real wage rate w evolvescounter-cyclicly:Deriving W (ε) ≡ w(ε) · h(ε) wrt ε and using (4), we find that:

dwdε

=(UCL − wUCC)

hUCC

dhdε

< 0

5 Other implications if leisure is a normal good:1 the agent’s utility is counter-cyclical:

dU/dε = UCdW/dε− ULdh/dε = [UC(UCL/UCC)− UL]dh/dε < 0.2 the firm has an interest to announce to the worker that the “state” ε

is good, since then the worker accepts a contract in which he worksmore and receives a lower compensation W : profits then increase(see below)

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An individual insurance contract model

ExerciseConsider the utility function U[W + φ(1− h)] withU ′ > 0, U” ≤ 0, φ′ > 0 and φ” ≤ 0.

What are the signs of UCL and UCCUL − UCLUC?Knowing this revisit Properties 1 to 5 above.

Properties 3, 4, 5-1 are odd. Extension: “Insurance and ex-postlabor mobility”.

Prop. 5-2⇒ what if ε non verifiable? Extension: “Asymmetric orunverifiable information”.

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Insurance and ex-post labor mobilityp. 314-317

What if ex-post, there is a spot market where the worker can get thecompetitive spot wage W (ε)? Assume a stationary framework:

A1’ The number of hours worked is fixed: utility of anemployee is a function of wage only U(W ); infinite life anddiscount factor δ ∈ ]0,1[ .

A2’ f (h, ε) = ε

A3’ Expected lifetime profits = E [ε−W (ε)]/(1− δ),

A4 Verifiability: Unchanged.A5’ The principal is risk neutralA6 The worker can quit at each moment and earn the outside

wage W (ε), where W′(ε) > 0

Without mobility, insurance contract⇒ constant W (and w).

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Insurance and ex-post labor mobility

Expected present value outside the contract:

V (ε) = U[W (ε)] + δEV (θ)

Expected present value in a firm offering a contract A = {W (ε)}:

V (ε) = U [W (ε)] + δEMax[V (θ), V (θ)

]State-specific participation constraints:

V (ε) > V (ε), ∀ε

If U denotes EU[W (ε)], the participation constraints can be written as:

U [W (ε)] +δ

1− δEU [W (θ)] ≥ U[W (ε)] +

δ

1− δU, ∀ε (6)

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Maximizing expected profits s. to the participation constraints (6):

maxA={W (ε)}

∫ε−W (ε)

1− δ

+λ(ε)

[U [W (ε)] +

δ

1− δEU [W (θ)]− U[W (ε)]− δ

1− δU]

g(ε)dε

where λ(ε) ≥ 0 is the Lagrangian multiplier associated to (6).

For any ε, the F.O.C. w.r. to W (ε) can be written as:

[(1− δ)λ(ε) + δEλ(θ)] U ′ [W (ε)] = 1 (7)

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Exploiting (7), i.e. [(1− δ)λ(ε) + δEλ(θ)] U ′ [W (ε)] = 1:

Let Λ+ = {ε |λ(ε) > 0} (participation constraints are binding).From (7), W (ε) is no more fully rigid for ε ∈ Λ+.

Let ε1 > ε2, both in Λ+. Subtracting equalities (6), i.e.

U [W (ε)] +δ

1− δEU [W (θ)] = U[W (ε)] +

δ

1− δU

leads to

U [W (ε1)]− U [W (ε2)] = U[W (ε1)

]− U

[W (ε2)

]> 0 sinceW

′(ε) > 0

⇒W ′(ε) > 0 over the set Λ+.

Then , from (7), in Λ+, as U ′′ < 0, one has λ′(ε) > 0.⇒ Λ+ = {ε | ε ≥ ελ}, i.e. the set Λ+ is made of all ε above a certainthreshold value (ελ).

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Figure: 6.2 The wage contract with labor mobility

If principal could break the contract if profits become too low, thenwages should not be completely rigid downwards.

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Asymmetric or Unverifiable InformationStatic model

Only the (risk-neutral) principal observes the true values of theproductivity shock ε. The principal may have an incentive to hide thisinformation to the agent.How can one design a contract such that the principle has an incentiveto reveal the correct information?

The Revelation PrincipleConsider any contingent contract A = {W (ε),h(ε)}.For any true state ε, the principal announces the state m(ε) thatmaximizes its profits:

m(ε) = ArgMaxθ{f [h(θ), ε]−W (θ)} (8)

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The Revelation Principle

Consider now a contingent contract

A = {W (ε), h(ε)} = {W (m(ε)),h(m(ε))}

♦ Contract A is incentive-compatible, i.e. the principal can never makemore profit by “lying” then by revealing the truth.Assume that A is in force, the principal announces that the state is θwhile it is actually ε. Then the principal makes a loss:

f[h(θ), ε

]− W (θ) ≡ f {h[m(θ)], ε} −W [m(θ)]

≤ Maxs{f [h(s), ε]−W (s)}

≡ f [h(m(ε)), ε]−W (m(ε)) = f [h(ε), ε]− W (ε)

♦ Contracts A and A lead to the same allocations and compensations.

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Asymmetric or Unverifiable InformationThe Incentive-Compatible contract

Since it is possible to associate any contract with anincentive-compatible contract that leads to the same allocation ofresources, the search for the optimal contract can be confined to theset of incentive-compatible contracts.This “second-best” contract solves: max

AE Π(ε) s. to

EU (W (ε),1− h(ε)) ≥ Uf [h(ε), ε]−W (ε) ≥ f [h(θ), ε]−W (θ), ∀(ε, θ) (I-C)

In general, complex problem!

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Asymmetric or Unverifiable Information

♦ The first-best contract (i.e. the one when (I-C) contraints areignored) is also optimal under asymmetric information if leisure is notaffected by the income level. For then, from the optimality conditions,announcing that the state is better than the true one increases realoutput and the wage bill to the same extent (Formula (12) in CZ2004).

♦If UCL ≥ 0 (hence, leisure normal good), we saw (see Prop 5.2above) that the first-best contract is characterized bydh/dε > 0 > dW/dε. So, the firm has an interest to announce to theworker that the state is good (actually, ε+) since then the workeraccepts a contract in which he works more and receives a lowercompensation W .

♦ If leisure is an inferior good, the firm announces ε−.

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Asymmetric or Unverifiable InformationAn example with two equiprobable states of nature ε+ > ε−

A3” f (h, ε) = εh (⇒ fh(h, ε) = ε).

The Principal’s Problem

Max(hi ,W i)i=+,−

[12(ε+h+ −W +

)+

12(ε−h− −W−)]

Subject to constraints:

12

U(W +,1− h+) +12

U(W−,1− h−) ≥ U (14)

ε+h+ −W + ≥ ε+h− −W− (15)

ε−h− −W− ≥ ε−h+ −W + (16)

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Asymmetric or Unverifiable InformationAn example with two equiprobable states of nature ε+ > ε−

The optimal contract when leisure is a normal goodThe principal only has an incentive to lie if the bad state realises;The incentive compatible contract avoids this, since it requires theprincipal to pay a high wages if it announces a good state. ⇒wages (and hours) and economic conditions unambiguously movein the same direction: h+ > h−,W + > W−.The FOC also results in over-employment in good states, but notin underemployment in bad states:

MRSCL(ε+) =UL(W +,1− h+)

UC(W +,1− h+)> ε+ = MPL(ε+) (22)

MRSCL(ε−) =UL(W−,1− h−)

UC(W−,1− h−)= ε− = MPL(ε−) (23)

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Risk-sharingConclusion

It is standard to conclude that taking risk-sharing between employersand employees into account

can help understanding the low variability of real wages,can help understanding the procyclicity of hours andcompensation,does not yield insight into the reasons for underemployment inbad states.

The last conclusion is however questioned by e.g. Drèze (1997) whostresses that coordination failures sustained by wage rigidities causepervasive underutilisation of resources. For him, risk-sharing is anatural reason for these wage rigidities.

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1. Stylized narrow view about motivation 1.3. Incentives if results verifiable

3. Incentives if results are verifiable3.1 The principal-agent model with hidden action

The problem1 Actions followed by the realisation of some random process lead

to results of an agent’s activity.2 Actions of the agent (effort) are not verifiable by a third party,3 but

results of actions are.3 Consequence: a trade-off between providing incentives (to induce

effort) and insurance.

The solution:An explicit performance contract:

1 Define the action of the agent as a reaction to a wage contract,before a random shock, affecting the result of this action, realises.

2 Determine the choice of the principal, anticipating the action of theagent

3In the words of some authors: “effort is not contractible”.(ESL-UCL) 34 / 102


The principal-agent model with hidden actionAssumptions and Notations: Timing of decisions

Decisions unfold in the following sequence:1 the principal offers a contract;2 the agent accepts the contract, or turns it down;3 if the agent turns it down, the protagonists go their separate ways,

but if the agent accepts it, he or she then supplies an effort; let e∗

denote the optimally chosen effort level.4 a random event ε which affects the result of the agent’s effort e

occurs;5 the principal and the agent observe the result;6 the principal remunerates the agent according to the terms of the

contractNote: This requires that the notion of “the result of an agent” makessense. OK for a salesman. Also in the case of teams of workers???

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The principal-agent model with hidden actionAssumptions and Notations

A1 Preferences of the agent of the CARA type:

U [W − C(e)] = −exp {−a [W − C(e)]} (24)

W is wage income, e the agent’s unverifiable effort, C(e)is the cost of effort, where C′ > 04 and C

′′> 0, and a > 0

is the index of absolute risk aversion, equal to −U ′′/U ′.A2 Effort results in a certain level of verifiable but random

production: y = e + ε, where ε is a normal randomvariable with zero mean and standard error σ.

A3 The explicit performance wage contract is linear in y :W = w + by , where w is a fixed wage and b is apiece-rate on production.

4With a more general specification U(W , e) one could let ∂U/∂e > 0 over somerange. Yet, the optimum cannot be in this range since the worker would unilaterallyincrease e and hence profits and his own utility.

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The principal-agent model with hidden actionThe agent’s effort level

If the agent accepts the contract, he takes the wage contract as givenand chooses his effort as to maximise expected utility:

EU = −exp {−a [w + be − C(e)]}E [exp(−abε)]

= −exp{−a[w + be − C(e)− ab2σ2

2

]}since the exponential of a Normal random variable X ∼ N (µ, σ2) islog-normally distributed with mean exp

[µ+ σ2

2

].

The chosen effort e∗ trades-off the benefits and costs of marginallyincreasing effort:

C′(e∗) = b defines e∗ = e∗(b) withde∗

db=

1C′′(e∗)

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The principal-agent model with hidden actionThe Principal’s Behaviour

The expected profit of the principal isE(y −W ) = E(1− b)(e∗ + ε)− w = (1− b)e∗ − w .

Max{w , b}

[(1− b)e∗ − w ] s. to C′(e∗) = b, EU ≥ U ≡ −exp{−a · x} (25)

The binding participation constraint can be rewritten as:

w = x − be∗ + C(e∗) +ab2σ2

2(26)

The optimisation problem can therefore be rewritten as:

Max{b}

[e∗(b)− C[e∗(b)]− a {C′[e∗(b)]}2 σ2

2− x

]

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The principal-agent model with hidden actionThe Optimal Remuneration Rule

The piece rate (or variable part of total remuneration) is

b∗ =1

1 + aC ′′(e∗)σ2 < 1⇒ ∂b∗

∂a< 0,

∂b∗

∂σ2 < 0, (27)

and w is given by (26) evaluated at b∗. (27) captures the trade-offbetween providing incentives and insurance:The magnitude of the piece rate decreases with

absolute risk-aversionthe variance of the “noise” εthe concavity of the cost of effort (i.e. how the marginal cost ofeffort varies)

Note: risk-aversion is taken into account by the principal because ofthe participation constraint EU ≥ U.

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The principal-agent model with hidden actionThe Optimal Remuneration Rule: An example

Let C(e) = c · e2/2, c > 0.

Then as C′′(e) = c i.e. is constant:b∗ = 1/(1 + aσ2 c), where aσ2 is called the “risk factor”, ande∗ = b∗/c.

Contrary to Formula (29) in the book, I find that

w∗ = x − 1− aσ2 c2c (1 + aσ2 c)2 = x − (b∗)2 1− aσ2 c

2c

A rise in σ or in a induces an increase in the fixed wage component w∗

if aσ2 c < 3.

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The principal-agent model with hidden actionTwo limit cases for comparison

♦ Comparison with the first-best outcome: e verifiable⇒ is part of thecontract chosen by the firm. The latter solves:

Max{w , b,e}

[e − (w + be)] subject to w + be − C(e)− ab2σ2

2≥ x

Indexing the first-best with superscript o:

bo = 0,C′(eo) = 1,wo = x + C(eo)

i.e. full insurance; eo > e∗ since C′(e∗) = b < 1.

♦ If risk-neutral workers (a = 0), b∗ = 1.

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In practice, how does the employer know theemployee’s best response?

The employer can arrive at an estimate of the best effort response

C′(e∗) = b

by varying b and observing the effects on output since on average

Ey = e

It is here assumed that the estimate e∗(b) is accurate.

(ESL-UCL) 42 / 102


Empirical findingsp. 327-8

1 Empirical findings confirm theory: move to incentive contractraises productivity;increases variance of performance;reduces (increases) quit rate of most (least) productive workers.attracts more productive applicants.

2 However, financial incentives can have counterproductive effects:“experimental and field evidence indicates that extrinsic motivation(contingent rewards) can sometimes conflict with intrinsicmotivation (the individual’s desire to perform the task for its ownsake)” (Bénabou and Tirole, 2003, p. 490).

3 1+ 2 : In what cases should financial incentives be used withcaution? A growing literature that requires a less narrow viewabout human motivation... → see Part 2.

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3.2 Should Remuneration Always Be Individualised?

Initial idea: Why a contrat influenced exclusively by individualproduction?If there are other verifiable variables, their utilization could lead to moreefficient contracts.The Agency Model with Two Signals

A5 The principal observes a verifiable signal ε ∼ N(0, σ2)that is possibly correlated with the component ofindividual production y that is unrelated to effort, i.e. withε ∼ N(0, σ2): cov(ε, ε) = ρσ2.

Main application: If the results of the agent’s effort arecorrelated to the results of colleagues (because, say, ofcommon shocks).

A3’ The linear wage contract takes the following form:W = w + by − bε

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The Agency Model with Two Signals

Now, the expected utility is:

EU = −exp {−a [w + be − C(e)]}E[exp(−a

[bε− bε

])]

= −exp{−a[w + be − C(e)− aσ2

2

(b2 + b2 − 2ρbb

)]}The chosen effort still verifies:

C′(e∗) = b which defines e∗ = e∗(b)

As E ε = 0, the problem of the principal is:

Max{w ,b,b}

[(1− b)e∗ − w ] s.to C′(e∗) = b, EU ≥ −exp{−a · x}

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The Optimal Compensation Rule

The binding participation constraint can be rewritten as:

w = x − be∗ + C(e∗) +aσ2

2

(b2 + b2 − 2ρbb

)(30)

The optimisation problem can therefore be rewritten as:

Max{b,b}

[e∗(b)− C[e∗(b)]− aσ2

2

(b2 + b2 − 2ρbb

)− x

]

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The Optimal Compensation Rule

From the f.o.c.’s:

b∗ =1

1 + a[1− ρ2]C ′′(e∗)σ2 < 1⇒ ∂b∗

∂|ρ|> 0;

∂b∗

∂a,∂b∗

∂σ2 < 0,

b∗ = ρb∗ ⇒ db∗

dρ= b∗ + ρ · ∂b∗

∂ρ> 0 if ρ > 0,

W ∗ = w∗ + b∗y − b∗ε, w∗ from (30)

1 ρ = 0⇒ b = 0:The second signal just adds in “noise”⇒ ignore it

2 1 ≥ ρ > 0⇒ b∗ ≥ b∗ > 0.An increase in the signal ε reduces the compensation W ∗ sincethe positive correlation induces that (statistically) y is higher“because” ε is bigger, too.

3 −1 ≤ ρ < 0 ≤⇒ b∗ < 0 < b∗.(ESL-UCL) 47 / 102


3.3 Some reasons why performance pay may beinefficient3.3.1 Multitasking

A1’ The preferences of the agent are as in A1: CARA utilityfunction but with two efforts; effort (time) devoted to twodistinct tasks is perfectly substitutable:C(e1,e2) = C(e1 + e2).

A2’ y1 = e1 + ε1, y2 = e2 + ε2 where εi ∼ N(0, σ2i ), i = 1,2.

cov(ε1, ε2) = σ12A3” W = w + b1y1 + b2y2. Hence, W = W (ε1, ε2).

The agent’s behaviour: Max{e1,e2}

EU

= Max{e1,e2}

− exp{−a[w + b1e1 + b2e2 − C(e1 + e2)

− a2

Var [W (ε1, ε2)]

]}(ESL-UCL) 48 / 102


Multitasking

The FOC’s are:bi = C′(e∗1 + e∗2), for i = 1,2.

This implies that b∗1 = b∗2. The principal must remunerate the effortdevoted to each activity equally. Otherwise the agent will devote alleffort to the activity that is best remunerated.It can be checked that:

b∗1 = b∗2 =1

1 + aC′′(e∗1 + e∗2)(σ21 + 2σ12 + σ2

2)

Consequence: Suppose that the performance of one of the agent’stasks cannot be measured: e.g. σ1 →∞. Then b∗1 = b∗2 = 0. So,whenever it’s difficult to measure the performance of one task,performance pay becomes inefficient if e1,e2 perfect substitutes.

(ESL-UCL) 49 / 102


3.3.2 Supervision and rent-seeking

Often the principal does not observe agent’s outputbut well supervisors who are themselves agents:

1 To avoid friction with collaborators, supervisors tend to writefavourable performance reports⇒ problem of measurement ofperformance.

2 Or, agents try to influence performance reports by undertakingactions that attempt to "impress" supervisors: rent-seeking orlobbying

The book focuses on case 2.

(ESL-UCL) 50 / 102


A Model with Rent-Seeking

A1” Preferences and cost of effort as in A1. In addition, thereis a cost for rent-seeking activity α: K (α), where K ′ > 0and K ′′ > 0.

A3”’ Output is y but the supervisor reports to the principal thatit is y + α⇒W = w + b(y + α)

Observe that α has no productive value: Assumption A2 is maintained:y = e + ε

The behaviour of the agent

C′(e∗) = K ′(α∗) = b

The Optimal Pay Scheme

Maxb

[e∗(b)− C[e∗(b)]− K [α∗(b)]− a {C′[e∗(b)]}2 σ2

2− x

](ESL-UCL) 51 / 102


A Model with Rent-SeekingThe Optimal Pay Scheme

b∗ =1

1 + C′′(e∗)K ′′(α∗) + aC′′(e∗)σ2

The less costly rent-seeking, the smaller K ′′(α∗), the smaller is theperformance pay. With risk-neutral agents (a = 0) the first-bestsolution, b∗ = 1, can no longer be attained.

Conclusion: If performance is not verifiable due to complex tasks,components of which are not measurable or due to rent seekingactivities, then other contractual should be designed to provideincentives:

Promotions on the basis of relative performance;Seniority rules in long-term contracts.

(ESL-UCL) 52 / 102

1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results

4. Incentives when results are not verifiable

What if both, effort and individual performance are unverifiable?Double moral hazard problem. Answers:

4.1 An internal market + a system of promotions1 if relative performance (= ordinal) is easier to measure than

absolute performance (= cardinal);2 if relative performance is verifiable:

One can publicly announce in advance the wage increase to whichthe promotion entitles and the number of promoted workers =verifiable clauses;One should limit rent-seeking behaviour:(i) if the firm systematically favours candidates without merit, this willharm the firm’s reputation and one will no longer be willing to work inthis firm.(ii) One may design procedures such that rent-seeking behaviour ishampered: e.g. assign outside members to the promotion committee.

4.2 Compensation rules based on seniority

(ESL-UCL) 53 / 102


4.1 Promotions and Tournaments4.1.1 A (single period) Tournament Model

A1P (Homogeneous) agents are risk-neutral: U = W − C(e),where C(0) = 0, C′ > 0 and C′′ > 0.

A2P Effort (e ≥ 0) results in a certain level of unverifiable andrandom production: y = e + ε, where ε is a normalrandom variable with zero mean and standard error σ.CDF Φ(·), φ(·) ≡ Φ′(·). All the workers’ specific ε are ⊥⊥.

A3P A large firm with N employees, among which L will bepromoted and get a bonus b in addition to the base wagew0. ⇒Wage bill = w0N + bL. Announcing {w0,b,L} isformally equivalent to a contract {w0,b, y}, wherepromotion takes place if y > y > 0. Reason:if N sufficiently large, L

N = [1− Φ(y − e∗)].

A4P The expected profit per capita of the firm is:Eπ = E(y −W ) = e − w0 − b [1− Φ(y − e∗)]

(ESL-UCL) 54 / 102


A Tournament Model

The Behaviour of the Agent

Maxe

EU = Maxe{w0 + b [1− Φ(y − e)]− C(e)} (33)

The FOC leads to the following incentive constraint:

b · φ(y − e∗) = C′(e∗)⇒ e∗ = e∗(b, y) (34)

The 2nd order condition requires: bφ′ + C′′ > 0.One can check that de∗/db 6= 0 and de∗/dy 6= 0.

The Behaviour of the PrincipalThe principal maximises expected per capita profits taking theincentive and participation constraints (EU > U) into account. Thelatter binds ex ante (not ex post for those promoted) and verifies thefollowing expression:

w0 + b [1− Φ(y − e∗)] = U + C(e∗) (36)

(ESL-UCL) 55 / 102


A Tournament Model

Inserting these in the expression of expected profits (per capita)results in the following optimisation problem:

Max{b,y}

{e∗(b, y)− C[e∗(b, y)]− U

}The FOC determines the optimal level of effort (if de∗/db 6= 0 andde∗/dy 6= 0):

C′(e∗) = 1 (37)

Note that:Given the assumed properties of C(·), e∗ is unique.Condition (37) entails the first-best level of effort: a consequenceof risk-neutrality of the agent.

(ESL-UCL) 56 / 102


The contract is characterized by {w0,b, y}.

Given the first best value e∗, this vector solves the incentive constraint:

b · φ(y − e∗) = C′(e∗) = 1 (34)

andw0 + b [1− Φ(y − e∗)] = U + C(e∗) (36)

where as before U is exogenously given.So there remains a degree of freedom (say, w0). In the original paper(Malcomson, 1984), the model has 2 periods and the promotion takesplace between the 1st and the 2nd one (in between the worker canleave and take a job in a period-2 “spot market”). This complicates themodel and imposes more constraints on the optimal contract.

The number of promoted is L = N [1− Φ(y − e∗)] (assuming that thelaw of large numbers can be applied!).

(ESL-UCL) 57 / 102


A Tournament ModelIncreasing risk i.e. σ

The density of a normal distribution satisfies the following conditions:

φ(0) = 1/σ√

2Π and φ′(0) = 0

Therefore, if e∗ is sufficiently close to y , the incentive and participationconstraints (34 and 36) jointly with the FOC (37) can be approximatedas follows:

b ' σ√

2Π and [1− Φ(y − e∗)] ' C(e∗) + U − w0

σ√

2Π

Increasing risk (σ) increases the wage gap, b, and reduces theproportion of promotions, (1− Φ).

(ESL-UCL) 58 / 102


A Tournament ModelIncreasing risk i.e. σ

If the risk increases with the hierarchical grade (e.g. because jobsbecome increasingly complex), we should observe the wage gainto increase with this number;(Unmodelled) workers’ risk-aversion, on the other hand, shoulddecrease the wage gain.

The following figure illustrates the first prediction. It concerns a largeUS firm of the service sector. Eight levels are distinguished. Theaverage wage corresponding to each grade increases at an increasingrate as we move up the hierarchy.This figure also shows a dispersion of wages within each grade (⇒ inaddition to the tournament, another incentive scheme is in place).

(ESL-UCL) 59 / 102


A Tournament ModelEmpirical evidence

Figure: Salary ranges by level of hierarchy

(ESL-UCL) 60 / 102


Tournaments and Rent-seekingAre seniority rules so bad after all?

A1P’ In addition to A1, the worker can engage in a rent-seekingactivity, α to impress his supervisor. This entails cost,K (α), where K ′ > 0 and K ′′ > 0.

A3P’ If y + α = e + ε+ α > r , then W = w0 + b, b > 0;Otherwise, W = w0.

For any r and b, the agent maximises expected utility wrt e and α:

Max{e,α}

{w0 + b [1− Φ(r − e − α)]− C(e)− K (α)}

This results in the following FOC:

bφ(r − e∗ − α∗) = C′(e∗) = K ′(α∗) (40)

(ESL-UCL) 61 / 102


Tournaments and Rent-seekingAre seniority rules so bad after all?

Inserting this together with the participation constraint

w0 + b [1− Φ(r − e∗ − α∗)] = U + C(e∗) + K (α∗) (41)

in the maximisation problem of the principal, this yields:

Max{b,r}{e∗(b, r)− C[e∗(b, r)]− K [α∗(b, r)]}

This results in the following FOC:

K ′(α∗) = C′(e∗) =K′′

(α∗)

C ′′(e∗) + K ′′(α∗)< 1 (42)

The interest in staging promotions is lessened since effort falls.

Other problems: sabotage by competing colleagues; issues of fairness.(ESL-UCL) 62 / 102


Exercise1 What is an “explicit performance” contract? To answer that question, you do not

need to model such a contract. Explain in words the economic assumptionsneeded for such a contract and the main features of it.

2 Why do employers offer such a contract rather than a fixed wage contract?3 Linear performance contracts, maximizing profits, usually index wages only

partially and not completely to performance (single task, no rent-seeking). Whatare the key factors that determine the degree in which wages should be linked toperformance? How do they affect the optimal performance contract and therebyprofits? What’s the intuition behind?

4 Why may it be sensible – rather than directly relating wages to measuredperformance - to link wages to a system of promotions ?

(ESL-UCL) 63 / 102


4.2 The role of seniority4.2.1 The “shirking” model

Dynamic model where the agent’s production is not verifiable. Wesearch for a self-enforcing contract (the partners stay together if andonly if the two parties have a mutual interest to do so).

A1S The worker is risk neutral and exerts two levels of effort: 0or e > 0 to which are associated costs of respectively 0and C > 0.

A2S Production at date t , yt > 0, is only realised if e > 0, butdue to costs of supervision the firm inspects theproduction level (and therefore effort) only with anexogenous probability p ∈ (0,1).

A3S Since effort is unverifiable the wage cannot be linked tothe level of effort. Instead the employer offers a contract{wt ; t = 0,1, ..,+∞} that is paid until the worker is caughtshirking; if caught the worker is paid the wage until theend of the period and is fired.

(ESL-UCL) 64 / 102


The modelThe Behaviour of the Principal

In a discrete time framework, δ ∈ (0,1] being the discount factor andq ∈ (0,1) an exogenous probability of separation, the expecteddiscounted profit solves:

Πt = yt − wt + δ[(1− q)Max(Πt+1,Π

st+1) + qΠt+1

](43)

where Πt+1 designates the profit if the contractual relationship endsand Πs

t+1 the expected profit is the firm “cheats” (i.e. breaks thecontract, wrongly claiming that the worker “shirks”). The latter is givenby

Πst = yt − wt + δΠt+1 (44)

(ESL-UCL) 65 / 102


The “shirking” modelThe Behaviour of the Principal

The employer’s incentive constraint is Πt ≥ Πst :

Πt − Πst = δ(1− q)

[Max(Πt+1,Π

st+1)− Πt+1

]≥ 0, ∀t ≥ 0

⇔Πt+1 ≥ Πt+1, ∀t ≥ 0

Only future rents matter! If one also considers the participationconstraint, Π0 ≥ Π0, we obtain

Πt ≥ Πt , ∀t ≥ 0

(ESL-UCL) 66 / 102


The “shirking” modelThe Behaviour of the Agent

Vt = wt − C + δ[(1− q)Max(Vt+1,V s

t+1) + qVt+1]

(45)

where V st+1 is the expected lifetime utility of the worker who shirks and

Vt+1 is the outside utility. The former is equal to

V st = wt + (1− p)δ

[(1− q)Max(Vt+1,V s

t+1) + qVt+1]

+ pδVt+1 (46)

The worker’s incentive constraint is then

Vt − V st = −C + pδ(1− q)

[Max(Vt+1,V s

t+1)− Vt+1]≥ 0, ∀t ≥ 0

m

Vt+1 − Vt+1 ≥C

pδ(1− q)> 0 , ∀t ≥ 0 (47)

(ESL-UCL) 67 / 102


The “shirking” modelRent and Set of Feasible Contracts

The set of feasible contracts, P, is defined by:{(Πt ,Vt )

∣∣∣∣Πt ≥ Πt ,Vt+1 − Vt+1 ≥C

pδ(1− q),V0 ≥ V0,∀t ≥ 0

}(48)

Surplus and the Existence of Self-Enforcing Contracts

The global surplus is the sum of rents procured by the contract:

St ≡ Vt − Vt + Πt − Πt , ∀t ≥ 0

By adding up (43) and (45), the surplus is defined by:

St − δ(1− q)St+1 = yt −C + δ(Πt+1 + Vt+1)− (Vt + Πt ), ∀t ≥ 0 (49)

⇒ the surplus is exogenous since the RHS is taken as exogenous bythe partners.

(ESL-UCL) 68 / 102


The “shirking” modelRent and Set of Feasible Contracts

Using that Πt − Πt = St − (Vt − Vt ) for all t ≥ 0, the set P can becharacterised in relation to the exogenous surplus. So, P can berewritten as the set of {Vt}t≥0 such that:

St+1 ≥ Vt+1 − Vt+1 ≥C

pδ(1− q), S0 ≥ V0 − V0 ≥ 0, ∀t ≥ 0 (50)

A self-enforcing contract will therefore only exist if

P 6= ∅⇐⇒ S0 ≥ 0 and St+1 ≥C

pδ(1− q), ∀t ≥ 0 (51)

The non-verifiability of effort and performance leads to Paretoinefficiency: if the surplus is positive, but not large enough, mutuallyadvantageous production is not realised: explains why workers with alow productivity are excluded from long-term contractual relationships.

(ESL-UCL) 69 / 102


The “shirking” modelRent and the Optimal Contract

The optimal contract is found by maximising the expected profitsΠt = −Vt + (Vt + Πt + St ) over P,∀t . As Vt + Πt + St is exogenous,this results in choosing the lowest value of Vt over the set P:

(i) if P 6= ∅, then V0 = V0, Vt+1 = Vt+1 + C/pδ(1− q),∀t ≥ 0(ii) if P = ∅, no self-enforcing contract exists.

(52)

The worker does not obtain any rent at time t = 0: V0 = V0;Incentives to provide effort result from a strictly positive rent in allsubsequent periods (t > 0);The principal captures the entire surplus at t = 0:Π0 − Π0 = S0 ≥ 0.

(ESL-UCL) 70 / 102


4.2.2 Seniority, Experience and WageThe Reasons Why Wages Rise with Seniority

1 The accumulation of general human capital leads to an increasingrelationship between experience (i.e.

∑employment durations)

and wages;2 The accumulation of specific human capital leads to an increasing

relationship between seniority (i.e. duration of employment in thesame firm) and wages if workers have some bargaining power;

3 On-the-job search leads to an increasing relationship betweenexperience and wages;

4 Revelation of information on the worker’s abilities allows to assignto worker to tasks that match their abilities and may therefore leadto an increasing relationship between seniority and wages;

5 A “deferred payment” scheme provides incentives to work hardand explain why wages rise with seniority.

(ESL-UCL) 71 / 102


Seniority, Experience and WageThe Optimal Wage Profile in the Shirking Model

Combining the expression for the expected utility of the worker (45)and the relation (52) defining an optimal contract and assuming thatthe outside opportunities procure an instantaneous utility wt − C, theoptimal wages are:

w0 = w0 −Cp< w0 (55)

wt = wt +Cp

[1

δ(1− q)− 1]> wt , ∀t ≥ 1 (56)

Proof of (56): By (45) in the set P,

Vt = wt − C + δ(1− q)[Vt+1 − V t+1

]+ δV t+1

wt = V t − δV t+1︸︷︷︸w t−C

+ C +Cp

[1

δ(1− q)− 1]

(53)

(ESL-UCL) 72 / 102



Figure: The profile of optimal wages with general human capital and deferredpayments.

(ESL-UCL) 73 / 102


Seniority, Experience and WageA bonding mechanism

The optimal wage profile described by relations (55) and (56) isinterpretable as an incentive mechanism in which there is a “bond” theagent is obliged to post at the time of hiring which will be graduallypaid back to him or her.

Is such a bonding mechanism plausible? Arguably, not (see Section4.4.).

(ESL-UCL) 74 / 102



The deferred payment takes an extreme form:Only the hiring wage w0 is lower than the outside wage w0 (e.g.the competitive wage).As of t = 1, dwt/dw t = 1.General human capital affects the outside wage and hence wt ...... but specific human capital does not affect w t , hence nor wt(recall that workers bargaining power = 0 !)

More generally, a less extreme form of deferred payment ifheterogeneity in the ability of workers;and time required for these abilities to be revealed to the employer.

(ESL-UCL) 75 / 102


4.3 Empirical Elements on Experience, Seniority, andWages4.3.1. Estimating the return to seniority

Let wijt be the wage of individual i in firm j in year t . With "ed" foreducation, "ex" for experience and "te" for seniority and ignoring nonlinearities, the basic equation writes:

ln wijt = β + βt t + βededijt + βexexit + βteteijt + εijt (57)

Problem:

1 Unobserved heterogeneity bias: More efficient workers also havelarger seniority and experience.

2 Endogeneity bias: participation and therefore experience ispositively related to wages; quits are negatively related to wagesand therefore seniority positively.

(ESL-UCL) 76 / 102


Estimating the return to seniority

To see this more clearly we decompose the error term as follows:

εijt = ηi + µij + ζijt (58)

where ηi is a fixed unobserved individual component, µij is anunobserved match specific component of worker i employed in firm jand ζijt is a random term.In general, ηi and µij are correlated with experience and seniority andthe OLS estimator of equation (57) is therefore inconsistent.

(ESL-UCL) 77 / 102


Estimating the return to seniority

Solutions

1. Abraham and Farber (1987)

ln wijt = β + βt t + βededijt + βexexijt + βteteijt + βduduij + εijt

where duij is the total effective job duration of worker i in firm j .Problems:

right censoring of these durations;does not account for correlation between unobserved individualeffects and experience;It does not account for variation of the match specific effect overthe job spell: µijt instead of µij .

(ESL-UCL) 78 / 102


Estimating the return to senioritySolutions

2. Topel (1991)

ln wijt = b + βt t + bededijt + bexex0ij + bteteijt + εijt

where the term ex0ij designates the experience of individual i at the

time he or she is hired for job j .

1 Estimate this equation in differences, retaining only workers whodo not change jobs between 2 dates. This identifies a consistentestimator of bte;

2 In a second stage Topel fits the following equation:

ln wijt − bteteijt − βt t = b + bededijt + bexex0ij + εijt (59)

where bte − bex is the return to seniority.

(ESL-UCL) 79 / 102


Estimating the return to seniorityTopel (1991)

Problems:1 First step is consistent only if the wages grow at same rate

independently of whether workers stay or not within the firm.Justifies this by the finding that all wages vary according to arandom walk;

2 In the second step, bed may be biased upwards, since experiencecan be positively correlated with the unobserved productivitycomponents of workers⇒ bte − bex is a lower bound to the returnof seniority.

ConclusionResearchers tackling these problems find results close to Topel’s: 10years of seniority increases wages by 25% and 10 years of experienceby 34%. One also finds that returns to seniority in an industry are moreimportant than within a firm.

(ESL-UCL) 80 / 102


4.3.2. On the Mechanism of Deferred Payment

1 See figure 6.5 in CZ: The time profile of the wage income of asalesperson follows closely the productivity profile (output of thesalesperson is arguably verifiable); For managers the wages startoff below productivity and ends above (output arguably notverifiable).

2 Formula (56), namely wt = wt + Cp

[1

δ(1−q) − 1], predicts

∂wt/∂p < 0. Firms (hospitals) pay higher wages the lower thesupervisor/employee ratio (proxy for the probability p that effort isverified).

(ESL-UCL) 81 / 102


4.4. Efficiency wages and involuntary unemployment

Since credit markets are imperfect, Shapiro and Stiglitz (1984)argue that the bonding mechanism is infeasible;Assume instead that employer must pay the same wage in everyperiodThis results in involuntary unemployment as an incentive device.

Shapiro and Stiglitz (1984) consider an economy populated withhomogeneous workers and firms.

(ESL-UCL) 82 / 102


Efficiency wagesA particular stationary version of the shirking model

Shapiro and Stiglitz (1984) assume a stationary version whereV t = V ,∀t ≥ 0. Then, (53) simplifies:

wt = w = V t − δV t+1︸︷︷︸(1−δ)V

+ C +Cp

[1

δ(1− q)− 1]

(60)

Moreover in the set P, ∀t , one has V − V = C/pδ(1− q) > 0, whereV is here interpreted as the expected utility of an unemployed person:

V = z + δ[sV + (1− s)V

]⇒ (1− δ)V = z + s

Cp(1− q)

in which z is the gains in unemployment and s is the probability ofbeing hired. Note: shirking = a misconduct⇒ z 6= unemploymentbenefits as these are normally only paid to the “involuntaryunemployed”.

(ESL-UCL) 83 / 102



Inserting (1− δ)V in (60) yields:

w = z + C +Cp

[1

1− q

(s +

1δ

)− 1]

(61)

In a stationary state, the flow qL of entries into unemployment equalsthe flow of exits s(N − L) out of it. Consequently we haves = qL/(N − L). Carrying this over in equation (61) results in theincentive curve (IC):

w = z + C +Cp

[1

1− q

(qL

N − L+

1δ

)− 1]

(62)

It is increasing and possesses a vertical asymptote at point L = N.This property signifies that there is never full employment atequilibrium: the fear for unemployment can only provide incentives ifone does not immediately find a new job.

(ESL-UCL) 84 / 102



Involuntary unemployment. Since V − V = C/pδ(1− q) > 0

Unemployment is thus involuntary in nature, since anyonelooking for work prefers to accept a job at the current wage w(which offers him or her an expected utility V0), rather thanremain unemployed (which offers an expected utility equal toV ). (CZ2004, p. 355-6)

If the principal was able to extract all the rent (V0 = V ), one hardlycould use the expression “involuntary unemployed”.

(ESL-UCL) 85 / 102



To close the model, we must specify the behaviour of the firms. Thegain Π expected from a filled job is given by the equality:

Π = y − w + δ[(1− q)Π + qΠ

](63)

where Π designates the expected profit of a vacant job. Assume that:Information on the labour market is perfect;The number of unemployed workers exceeds the number ofvacancies: s < 1. Consequently, all jobs get immediately filled⇒vacant jobs offer the same prospect as filled jobs⇒ Π = Π.

Free entry on the labour market: Π = Π = CK , where CK is thefixed equipment cost of opening a vacancy.

Inserting these conditions in (63) defines the efficiency wage w∗:

w∗ = y − (1− δ)CK

(ESL-UCL) 86 / 102



Figure: The equilibrium of the model of Shapiro and Stiglitz

(ESL-UCL) 87 / 102


Final Remarks on Efficiency Wage Theory

If the principal has a sufficiently wide range of strategies, then(s)he can always extract all the surplus from the match.⇒ unemployment, if any, cannot be “involuntary”.⇒ A rent for the agent in a moral hazard setting is based on(justified?) restrictions on the set of strategies accessible to theprincipal.Back to the bonding critique. A minimum wage wm could bebinding generating a rent wm − w0 > 0 and unemploymentbecomes involuntary again. However, not inherent to the incentivescheme. Similar properties in a competitive (hence,perfect-information) setting.Asymmetric information with respect to the types of employers:good or bad, i.e. high or low y ,⇒ provide a rent to worker to signalquality. However, there are cheaper ways to signal this quality.

(ESL-UCL) 88 / 102


Final Remarks on Efficiency Wage Theory

As V0 > V , the Shapiro-Stiglitz model⇒ the unemployed couldunderbid prevailing wages but employers would not lower wages.Empirical evidence on that?

Survey of employers: Bewley (1998), Agell and Bennmarker (2007).Firms are reluctant to lower wages even if unemployment is large.Experiments in labs.: Fehr and Falk (1999) conclude: “(i) Workers’underbidding is very frequent. (ii) Firms do not take advantage ofworkers’ low wage offers. Instead, they offer and pay generouswages. (iii) The opportunity to underbid has no wage decreasingeffect, giving rise to wage rigidity.(iv) Workers respond to low (high)wages with low (high) effort. These results indicate that - underconditions of incomplete contracts - the wage formation processmay well be driven by the anticipation of a positive wage-effortrelationship and that competitive pressures need not affect wages.”

Moen and Rosen (2006): Output is verifiable but firms neitherobserve the workers’ effort nor their match-specific productivity.The employed enjoy a rent and unemployment is “involuntary”.

(ESL-UCL) 89 / 102


Exercice about efficiency wage theoryInspired by Solow (1979) and Layard, Nickell and Jackman (1991)

ExerciseAssume that the effort of a worker employed in firm i, Ei , is notverifiable. Assume moreover a black-box relationship Ei = (wi − B)λ ,where

wi is the real wage rate in firm i,B = we(1− u) + bu,we = the wage rate in the rest of the economy,u = the unemployment rate (0 ≤ u ≤ 1),b = real unemployment benefits,

(ESL-UCL) 90 / 102


Exercice

Exercise1. Assuming that the employer sets the wage and labour demand,what are the first-order conditions of the following problem :maxLi ,wi siFi

((wi − B)λ Li

)− wiLi , where

Li = labour demand in firm i,si = a firm-specific technological parameter (si > 0).F [...] = the revenue function (F ′ > 0,F ′′ < 0 ).

2. Show that the optimal value of wi is independent of si and is amark-up over B. Interpret this result.3. In general equilibrium, if all firms are identical, one has wi = we, ∀i .Assume an exogenous constant replacement ratio β:b = βwe(0 < β < 1). Compute the unemployment rate in equilibrium.Interpret the role of β and of λ.

(ESL-UCL) 91 / 102

2. Extension 2.1. Intrinsic motivation

Motivation

Related to earlier quotation of Bénabou and Tirole (2003) (p. 490):

“Psychologists have provided compelling evidence that rewards cancrowd-out intrinsic motivation (...). Economists became recentlyinterested in this issue (...) and empirical (mostly experimental)evidence has been collected on this ambiguous effect of rewards(...).When studying explicit punishment incentives, experimentalresearch also indicates a potentially detrimental effect of sanctions(...). The negative effect of sanctions is due to a breach of thesocial norm of trust or to the reduction of the psychological costarising from the violation of a norm. (...)” (Dickinson and Villeval,2008, p. 58)“Our results [from controlled laboratory experiments] indicate thatthe disciplining effect of monitoring predicted by agency theorydominates in both our distant and interpersonal relationshiptreatments. However, we also find evidence in the interpersonaltreatment that increased monitoring crowd-out effort.” (Dickinsonand Villeval, 2008, p. 57)

(ESL-UCL) 92 / 102


Akerlof and Kranton (2008) enlarge preferences in the following way.

Social Psychology: intrinsic motivation depends on “how workerssee themselves in relation to the firm” (p. 212)Ethnography:

Workers resent supervision;In the absence of supervision, workers develop their own output(effort) norm.

On this basis, they assume:Workers have an identity that determines their intrinsic motivation:

No Supervision+Monitoring⇒ a work group identity (“G”) to whichis associated a specific ideal (“norm”) of effort (eΓ).Supervision+Monitoring⇒ an outsider identity (“O”) to which isassociated another, low, ideal of effort (eB < eΓ).

(ESL-UCL) 93 / 102


Preferences: Utility is the sum of three termsan increasing and concave function of consumption (say, ln W ),a decreasing function of effort (say, −e),a loss if effort deviates from the ideal e∗(c), c ∈ {G,O} (say,−tc · | e∗(c)− e |), where tc ≥ 0.

The effort level takes three possible values: eA > eΓ > eB.Verifiable firms revenues are either high (πH ) or low (πL). Let0 ≤ γ ≤ 1,

πH πLeA 1/2 1/2eΓ γ/2 1− γ/2eB 0 1

Table: Probability of revenue for the firm as a function of effort level.

When there is a supervisor, a low effort, eB, is verified with anexogenous probability p while the high effort level, eA, is neververifiable.

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Akerlof and Kranton (2008) compare two settings:

1 With Supervision+Monitoring: the choice is between eA and eBInstruments of the principal: contingent wages wO

H ,wOL and a fine f

if the agent exerts eB;5

These instruments have to maximize expected profits subject to(i) a participation constraint,(ii) an incentive constraint to elicit effort eA, knowing thate∗(c) = eB,(iii) an upper-bound on f .

2 No Supervision+Monitoring:Instruments of the principal: contingent wages wG

H ,wGL

These instruments maximize expected profits subject to(i) a participation constraint and either(ii) incentive constraints to elicit effort eA instead of eΓ or eB,knowing that e∗(c) = eΓ or(ii′) an incentive constraint to elicit eΓ instead of eB.

5Akerlof and Kranton (2008) assume that getting πL for sure cannot be preferable.(ESL-UCL) 95 / 102


No detailed study of the solution.(many configurations are possible)

Some intuition of properties only:

Obviously, the interest of a supervisor heavily depends on thedetection probability p and the upper-bound on f .Introducing a supervisor can be detrimental to the principal sincethere is a need to compensate for tO· | eB − eA |.

In the absence of a supervisor, for high enough γ and tG, theprincipal prefers to elicit eΓ than eA.

Comparing across the principal’s options, there is a thresholdp′ > 0 under which the principal does not want to institute asupervisor and contingent wages.

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2. Extension 2.2. Envy

Motivation

Starting point: Section 3.2 of Part 1, i.e.“Should remuneration always be individualised?”:

If the principal observes a signal ε positively correlated (ρ > 0) with theagent’s productivity, the optimal payment of this agent is negativelyinfluenced by ε.

Application:An industry with, say, 2 firms i and j .It is likely that there are common random shocks (like demand shocks).So, ε above could reveal information on the performance of the otherfirm j . The incentive payment to the Chief Executive Officers (CEO) in ishould be negatively related to the performance of firm j .

Fershtman, Hvide and Weiss (2003) (henceforth, FHW) notice the lackof evidence of such a negative relationship in the empirical literature.

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FHW develops a setting where managers’ wages can be positivelyrelated in an industry. This effect goes against the standard one andhence could explain this lack of empirical evidence.

Key ingredient: interdependent preferences or social preferences,i.e. agents care about the distribution of material goods and not simplytheir own allocation. Such preferences can be of different types. Here:♦ The principal (firm owner) is risk-neutral (maximizes his expectedprofits⇒ no interdependent preferences).♦ The agent is the CEO of firm i with a utility function given by:

U(Wi ,Wj ,ei ;β) = −exp{−α[Wi + β(Wi −Wj)− e2i /2]}, α > 0, (64)

Wi is the wage of the manager in firm i ,

Wj , the wage of the other manager, affects negatively the utility ofmanager i (if β > 0; the higher β, the higher the intensity of “envy”),ei is non verifiable effort of manager i .

And symmetrically for the CEO in firm j .(ESL-UCL) 98 / 102


FHW assume in addition:

A2 Effort ei results in a verifiable but random output: yi = ei + εi withεi ∼ N (0, σ2), and similarly in firm j .They assume that the random shocks εi and εj are independent.Seems odd. Why? To get that with standard preferences like (24)the incentive contract i should not be affected by characteristics infirm j , and conversely.

A3∗ The explicit performance wage contract is linear in outputs:

Wi = wi + aiyi + biyj and Wj = wj + ajyj + bjyi (65)

where wi is a fixed wage and ai and bi are piece-rates.

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The optimal effort level maximizes expected utility. So,

e∗i = max(ai + β(ai − bj),0), with ∂ei/∂bj ≤ 0. (66)

Expected profit of the principal in firm i is:

ei · (1− ai)− ej · bi − wi (67)

The two principals compete for (mobile) managers by offeringcontracts.

Without proof, the reaction function of the two firms are of the form:

a∗i =1

1 + ασ2 + A · bj (68)

b∗j = B · ai (69)

where 1 > A > 0 and B > 0 are positive and functions of parameters{β, α, σ}. Note that β = 0⇒ A = B = 0.

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Strategic interaction: Firm i raises ai as bj increases since a rise in bjincreases Wj and hence make it more costly for i to exert effort (e∗ishrinks, which is detrimental to the principal). The rise in ai intends tooffset this effect.This increase in ai induces in turn firm j to rise bj and so on.

Assume that the two firms are symmetric.

It can be shown that a unique Nash equilibrium with positive effort e∗

exists if the “risk factor” ασ2 is “sufficiently large” relative to theintensity of the envy effect β.This equilibrium is such that:

a∗ =1

1 + ασ2 + A · b∗ (70)

b∗ = B · a∗ (71)e∗ = (1 + β)a∗ − βb∗ (72)

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♦ Ifβ = 0: We are back to the solution of Sec 3.2 (Part 1) in theparticular case of a quadratic Cost of effort function.

a∗ =1

1 + ασ2 , b∗ = 0

♦ Ifβ > 0: The wage of agent i is in increasing in the wage of agent j .

In sum,In the presence of envy, strategic interaction can induce a positivecorrelation between the wages of CEOs who are sufficiently closefor a comparison to take place.In reality, there is presumably a positive correlation between therandom components εi and εj . So, in the absence of envy, anegative correlation between their wages would be optimal.The combination of these two opposite effects can explain the lackof evidence about clear correlations between compensation ofCEOs.

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References

Agell, J. and H. Bennmarker (2007) “Wage incentives and wagerigidity: A representative view from within”.Labour Economics, 14:347–369.

Akerlof, G. and R. Kranton (2008) “Identity, supervision, and workgroups”.American Economic Review, 98(2):212–217.

Bénabou, R. and J. Tirole (2003) “Intrinsic and extrinsicmotivation”.Review of Economic Studies, 70(2):489–520.

Bewley, T. F. (1998) “Why not cut pay?”European Economic Review, 42:459–490.

Cahuc, P. and A. Zylberberg (2004) Labor economics.MIT Press, Cambridge.

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References

Dickinson, D. and M.-C. Villeval (2008) “Does monitoring decreasework effort? The complementarity between agency andcrowding-out theories”.Games and Economic Behavior, 63:56–76.

Drèze, J. (1997) “Walras-keynes equilibria coordination andmacroeconomics”.European Economic Review, 9:1735–1762.

Fehr, E. and A. Falk (1999) “Wage rigidities in a competitiveincomplete contract market. an experimental investigation”.Journal of Political Economy, 107:106–134.

Fershtman, C., H. Hvide and Y. Weiss (2003) “A behavioralexplanation of the relative performance evaluation puzzle”.Annales d’Economie et de Statistique, 71-72:349–361.

Layard, R., S. Nickell and R. Jackman (1991) Unemployment :Macroeconomic performance and the labour market.Oxford University Press, Oxford.

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References

Malcomson, J. (1984) “Work incentives, hierarchy, and internallabor markets”.Journal of Political Economy, 92(3):486–507.

Moen, E. and A. Rosen (2006) “Equilibrium incentive contracts andefficiency wages”.Journal of the European Economic Association, 4(1165–1192).

Shapiro, C. and J. Stiglitz (1984) “Equilibrium unemployment as aworker discipline device”.American Economic Review, 74:433–444.

Solow, R. (1979) “Another possible source of wage stickiness”.Journal of Macroeconomics, 1:79–82.

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