The Principal-Agent Problem

Leandro Gorno

October 29, 2013

Principal-agent problems:

I Uncertainty might develop after the establishment of a contractual relation:

- when one party has to make unobservable choices.

- when one party obtains private information.

I Principal-agent framework:

- one party (principal) designs the contract.

- the other party (agent) decides whether to accept it or not.

I Two extreme cases:

- Hidden actions (moral hazard).

- Hidden information (or monopolistic screening).

Specification:

I The owner of a firm (principal) wants to hire a manager (agent).

I Manager can exert two levels of effort: e_L= 0 and e_H = 1.

I The revenue of the firm X is stochastic and depends on exerted effort:

- Uncertainty is described by PDFs f₀(x) andf₁(x) -f₁(x) first order stochastically dominates f₀(x).

I Effort is not observable, but realized revenue is:

- Manager compensation can depend on firm performance:

w0 and w1.

Ex-post payoffs:

I Principal: X −W

I Agent: u(W,e) =v(W)−c(e) -v increasing and concave.

-c increasing.

- If she does not accept any contract, she gets u.

Ex-ante payoffs:

I Principal: E{X −W|e}=R

(x−w(x))f_e(x)dx

I Agent: E{u(W,e)|e}=R

v(w(x))f_e(x)dx−c(e)

Observable actions benchmark:

I The problem of the principal is:

maxe,w

Z

(x −w(x))f_e(x)dx Z

v(w(x))f_e(x)dx −c(e)≥u

I We can divide it in two subproblems:

- How to optimally implement any fixed action?

minw

Z

we(x)fe(x)dx Z

v(we(x))fe(x)dx−c(e)≥u

- Which action is optimal?

Observable actions benchmark:

I Lagrangian:

L= Z

w_e(x)f_e(x)dx+γ_e

u+c(e)− Z

v(w_e(x))f_e(x)dx

I FOC:

1

v⁰(w_e(x)) =γ_e a.s.

Observable actions benchmark:

I With agent strictly risk averse, optimal wage is constant:

we(x) = (v⁰)⁻¹(1/γe) =we.

I The wage level is pinned down by reservation utility:

v(w_e) = Z

v(w_e)f_e(x)dx =u+c(e).

I If agent is risk neutral, any compensation scheme which gives her exactly her reservation utility is optimal.

Observable actions benchmark:

I Then, the principal maximizes:

Z

(x −w(x))f_e(x)dx = Z

xf_e(x)dx −v⁻¹(u+c(e)).

Example 1:

I Supposev(w) = ln(x) and c(e) =e.

I Then high effort is optimal for the principal if and only if E{X|e = 1} ≥E{X|e = 0}+ exp (u+ 1)−exp (u).

Non-observable actions:

I Incentives require inducing risk: efficient action vs.

insurance trade-off.

I If the agent is risk-neutral, this is not a trade-off!

I Hence, the outcome of the optimal contract under full information is still implementable when actions are not observable.

Risk-neutral agent:

I The principal sells the firm: w(x) =x −p.

I The agent doing effort e will now get a payoff of:

Z

w(x)f_e(x)dx−c(e) = Z

xf_e(x)dx−p−c(e) which she will willing to take as long as

p ≤pˆ_e = Z

xf_e(x)dx −c(e)−u

I Note that ˆp_e is the expected payoff of the principal under full information.

Risk-neutral agent:

I Then,p^∗ = max ˆpe^∗ is the optimal effort under full information.

I Hence, if the principal choosesp =p^∗, then the agent will accept the contract and do an optimal effort.

I The intuition is simple: fully incentivizing the agent is optimal because she does not demand a premium for bearing the risk.

Risk-averse agent:

I Fully incentivizing the agent is now costly.

I Implementing low effort is still easy:

w(x) =w₀ =v⁻¹(u+c(0))

I Implementing high effort is more tricky.

Implementing high effort:

I To optimally implemente = 1, the principal must solve:

minw

Z

w₁(x)f₁(x)dx subject to the participation constraint:

Z

v(w₁(x))f₁(x)dx−c(1) ≥u and the incentive compatibilityconstraint:

Z

v(w₁(x))f₁(x)dx −c(1) ≥ Z

v(w₁(x))f₀(x)dx −c(0).

Implementing high effort:

I Lagrangian:

L= Z

w₁(x)f₁(x)dx+γ₁

u+c(e)− Z

v(w₁(x))f₁(x)dx

+µ_1,0

c(1)−c(0)− Z

v(w₁(x))(f₁(x)−f₀(x))dx

, whereγ₁ ≥0 and µ_1,0 ≥0 are the multipliers.

I FOC:

1

v⁰(w₁(x)) =γ₁+µ_1,0

1− f₀(x) f₁(x)

Implementing high effort:

I γ₁ >0:

- Since effort affects profits,f0(x)>f1(x) on some open set.

- Then, ifγ₁ = 0, the FOC would imply v⁰(w(x))<0.

- This is impossible, so γ₁ >0.

I µ_1,0 >0:

- Ifµ= 0, then the FOC would imply a constant wage.

- But facing that schedule, the agent would choosee = 0.

I Hence, both PC and IC bind at the optimum.

The likelihood-ratio:

I Given an optimal contract, define ˆw = (v⁰)⁻¹(1/γ₁).

I Define thelikelihood ratio as r(x) =f₀(x)/f₁(x).

I Then, the FOC can be written:

1

v⁰(w₁(x)) = 1

v⁰( ˆw) +µ₁(1−r(x))

I Thus,w1(x)>wˆ whenever r(x)<1 and viceversa.

I Intuitive way of providing incentives: pay relatively more in states more likely to occur under high effort.

The likelihood-ratio:

I But also surprising implication: optimal wages may decrease with performance!

I What do we need to obtain motonicity?

- MLRP:r(x) must be decreasing.

Example 2:

I Supposev(w) = 1−exp (w), c(e) = e/2, u = 0 and f_e(x) = exp (−x/(1 + 5e)).

I Then, low effort is implemented by w0 = 0.

I To implement high effort, note that the FOC yields:

w₁(x) = ln (γ₁+µ_1,0(1−r(x))),

wherer(x) = 6 exp (−5x/6). Note that the MLRP holds.

I Finally, it is a matter of finding multipliers such that both constraints hold with equality.

Example 2 (cont’d):

I The following picture shows those values of γ₁ and µ_1,0 for which the constraints bind:

Optimal effort level:

I Low effort is implemented through full-information wages.

I In order to implement high effort, higher wages (in expected value) are required to compensate the agent for the risk.

I Therefore, non-observability distorts effort downwards (this result does not generalize to multiple effort levels).

I Whenever high effort is optimal under full information, non-observability generates a welfare loss.

Multiple effort levels:

I Suppose nowe ∈[0,∞).

I Then, a typical IC constraint would look like this:

Z

v(we(x))fe(x)dx −c(e)≥ Z

v(we(x))fe⁰(x)dx−c(e⁰).

I But there are [0,∞)×[0,∞) of them!

The first-order approach:

I If e >0 and fe(x) is smooth in e, the following FOC is necessary:

Z

v(w_e(x))

∂f_e(x)

∂e

dx =c⁰(e).

I Then, we can use it instead of the double continuum of ICs in the cost-minimization problem of the principal and hope for the best!

I Obvious problem: it may not be sufficient.

The first-order approach:

I Lagrangian:

L= Z

w_e(x)f_e(x)dx+γ_e

u+c(e)− Z

v(w_e(x))f_e(x)dx

+µ_e

c⁰(e)− Z

v(w_e(x))

∂f_e(x)

∂e

dx

,

I FOC:

1

v⁰(w_e(x)) =γ_e+µ_e

∂fe(x)

∂e

f_e(x)

!

Example 3:

I If v(w) = ln(w) and f_e(x) = exp (−λ(e)x)/λ(e), where λ(e) = 1/(1 +ke), the FOC derived using the first-order approach yields:

w(x) = γ_e+µ_e

(1−λ(e)x)k λ(e)(1 +ke)²

I Hence, an optimal contract implementing positive effort should be linear!

A dynamic model:

I Based on “Optimal unemployment insurance” by Hopenhayn and Nicolini (JPE 1997).

I Unemployed worker seaches for jobs by exerting efforte ≥0.

I Probability of finding a job this month isp(e)∈(0,1), with p⁰(e)>0 and p⁰⁰(e)<0.

I Worker’s monthly payoff isu(c(h^t))−e(h^t) after historyh^t while searching and u(w) after becoming employed.

I Government wants to ensure the unemployed worker (expected) utility V in the cheapeast possible way.

I Common discount rateβ ∈(0,1).

Worker’s problem:

I Value after finding a job:

V_t^E =V^E = u(w) 1−β

I Optimal effort satisfies V_t^U = max

e

u(c_t)−e+β p(e)V^E + (1−p(e))V_t+1^U )

I FOC for e >0:

βp⁰(e)(V^E −V_t+1^U ) = 1

Government’s problem:

I Problem doesn’t look recursive, but can be made so by using promised utility as a forward-looking state variable (“APS trick”).

I Bellman equation:

C(V) = min

c,e,V^U

c +β(1−p(e))C(V^U) subject to apromise keeping constraint

u(c)−e+β(p(e)V^E + (1−p(e))V^U ≥V

I and the (first-order) incentive compatibility constraint βp⁰(e)(V^E −V^U) = 1.

Analysis:

I Lagrangian:

L=c+β(1−p(e))C(V^U) +µ 1−βp⁰(e)(V^E −V^U) +γ V −u(c) +e−β(p(e)V^E + (1−p(e))V^U

I FOCs forc,e and V^U:

1 = γu⁰(c) C(V^U) =−µp⁰⁰(e)

p⁰(e)(V^E−V^U) +γ 1

βp⁰(e) −(V^E −V^U)

C⁰(V^U) = γ−µ

p⁰(e) 1−p(e)

Analysis:

I Envelope theorem:

C⁰(V) = γ

I More on theV^U FOC:

C⁰(V^U) =C⁰(V)−µ

p⁰(e) 1−p(e)

<C⁰(V)

I So, if we assume conditions for C convex:

V^U <V

I Sinceu⁰(c)C⁰(V) = 1, the replacement rate must be decreasing over time!

Analysis:

I More on thee FOC:

C(V^U) = −µp⁰⁰(e)

p⁰(e)(V^E −V^U)

Sincep⁰⁰(e)<0, the optimal scheme requires V^U <V^E whenever the program is running a deficit.

I CombiningV^U <V and βp⁰(e)(V^E −V^U) = 1, we can see that search effort increases over time!

Motivation:

I Suppose actions are observable but the agent only learns the cost of implementing different actions after signing the contract.

I In principle, the agent could communicate her private information to the principal, but the principal may not be able to verify the agent’s claim.

Assorted examples:

I Delegating construction.

I Debt restructuring.

I Public policy and constitutional design.

Model:

I Letx(e) be the (deterministic) gross profit of the principal when the agent exerts efforte ∈[0,∞).

I We assume x(0) = 0, x⁰(e)>0 and x⁰⁰(e)<0 for alle.

I Suppose the uncertainty over agent’s costs is summarized by a state variableθ ∈ {θ_L, θ_H}.

I The probability ofθ =θ_H is λ∈(0,1).

Model:

I The utility of an agent receiving wage w and exerting effort e in stateθ is

v(w −g(e, θ)),

wherev is increasing and strictly concave and g satisfies:

-g(0, θ) = 0.

-ge(0, θ) = 0 and ge(e, θ)>0 for alle >0.

-g_ee(e, θ)>0.

-g_θ(e, θ)<0.

-geθ(0, θ) = 0 andgeθ(e, θ)<0 for all e >0.

Full information benchmark:

I Suppose information is verifiable.

I A contract consists of two wage-effort pairs: (wL,eL) and (w_H,e_H).

I The principal solves:

maxw,e λ(x(e_H)−w_H) + (1−λ)(x(e_L)−w_L) subject to the participation constraint:

λv(w_H −g(e_H, θ_H)) + (1−λ)v(w_L−g(e_L, θ_L))≥u.

Full information benchmark:

I Lagrangian:

L=λ(x(e_H)−w_H) + (1−λ)(x(e_L)−w_L)

+γ(λv(w_H −g(e_H, θ_H)) + (1−λ)v(w_L−g(e_L, θ_L))−u), whereγ ≥0 is the multiplier.

I Obtain FOCs.

Full information benchmark:

I From the FOCs:

v⁰(w_H^∗ −g(e_H^∗, θ^∗_H)) = 1

γ^∗ =v⁰(w_L^∗−g(e_L^∗, θ^∗_L)).

I Strict risk aversion then implies

v(w_H^∗ −g(e_H^∗, θ_H^∗)) =v(w_L^∗−g(e_L^∗, θ_L^∗)), so utility is fully equalized across states.

Full information benchmark:

I Show that optimal effort levels must be positive.

I Show that optimal effort levels satisfy:

x⁰(e_H^∗) =g_e⁰(e_H^∗, θ_H) x⁰(e_L^∗) =g_e⁰(e_L^∗, θ_L).

I Given optimal effort levels, wages are:

w_H^∗ =g(e_H^∗, θ_H) +v⁻¹(u) w_L^∗ =g(e_H^∗, θ_L) +v⁻¹(u).

Full information benchmark:

I Separation undelying optimal contract:

Unverifiable states:

I Insurance vs. incentives trade-off.

I The space of possible contracts is huge:

- The owner sets a performance-based wagew(x) together with some constraint on effort choices.

- The manager makes an annoucement ˆθ about the observed state and the owner sets a compensation w(ˆθ) without constraints on effort.

- The owner sets a wage w(e) depending on the level of effort the manager chooses.

I The optimal contract seems hard to find. However...

Revelation mechanisms:

I The manager annouces ˆθ after observing the true state θ.

I The contract specifies an outcome (w(ˆθ),e(ˆθ)) for each possible announcement.

I For every stateθ, the manager finds it optimal to report the true state.

The Revelation Principle:

I Every implementable outcome can be implemented through the revelation mechanism.

General program:

I The principal solves:

maxw,e E{x(e(θ))−w(θ)}

subject to the participation constraint

E{v(w(θ)−g(e(θ), θ))} ≥u and the incentive compatibility constraints

w(θ)−g(e(θ), θ)≥w(θ⁰)−g(e(θ⁰), θ).

Infinite risk-aversion:

I Consider the limit case in which the manager is infinitely risk-averse.

I Then, the participation constraint becomes:

minθ v(w(θ)−g(e(θ), θ))≥u.

I Equivalently,

w(θ)−g(e(θ), θ)≥v⁻¹(u) for all θ∈ {θ_L, θ_H}.

Infinite risk-aversion:

I Any contract satisfying the constraints, must also satisfy w(θH)−g(e(θH), θH) ≥ w(θL)−g(e(θL), θH)

> w(θ_L)−g(e(θ_L), θ_L)

≥ v⁻¹(u).

I So, we can ignore all but the lowest participation constraint.

I The lowest participation is binding (otherwise a contract with a slight constant reduction in wages would be feasible.)

Optimal contract:

I ˆe_L ≤e_L^∗. Whenever e_L >e_L^∗, the owner can get more profits from settingeL =e_L^∗.

I ˆeH =e_H^∗. Given eL ≤e_L^∗, tangency between the

θ_H-indifference curve and an isoprofit curve is necessary for optimality. And all such tangency points occur at e_H =e_H^∗.

I ˆe_L <e_L^∗. If not, a slight decrease in e_L would allow a big increase ine_H (the optimal level of e_L is such that the marginal cost of reducing e_L equals the marginal benefit of the highere_H.)

Optimal contract:

I The optimal effort for θ_L satisfies the FOC:

[x⁰(ˆe_L)−g_e⁰(ˆe_L, θ_L)] + λ

1−λ[g_e⁰(ˆe_L, θ_H)−g_e⁰(ˆe_L, θ_L)] = 0

I Differentiating and noting thatg_e⁰(e, θH)−g_e⁰(e, θL) is negative and decreasing ine, we can get:

∂ˆe_L

∂λ <0

I Moreover, when λ→0, we have ˆe_L→e_L^∗ and the solution approximates the first-best.

Price discrimination by a monopolist:

I Suppose consumers have difference preferences for a good.

I The monopolist can offer a menu of quantity-price (qi,pi) bundles (non-linear pricing).

I The utility of (q_i,p_i) for a consumer of type θ is given by U(q_i,p_i|θ) = v(q_i, θ)−p_i.

I The consumer gets 0 if she doesn’t buy.

I We assume that v satisfies properties analogous those of g.

Monopolist’s problem:

I The monopolist choosest(θ) and q(θ) to maximize E{t(θ)−cq(θ)}

subject to the participation constraint v(q(θ), θ)−t(θ)≥0 and the incentive compatibility constraint

v(q(θ), θ)−t(θ)≥v(q(θ⁰), θ)−t(θ⁰).

Two types:

I If θ_L and θ_H > θ_L, the binding constraints are:

v(q_L, θ_L)−t_L = 0 and

v(q_H, θ_H)−t_H =v(q_L, θ_H)−t_L.

I Relax and verify: ignoring non-binding constraints and solving this relaxed version of the original problem provides a solution.

Example:

I Supposev(q, θ) = 2θq^1/2 and assumeλ∈(0, θ_L/θ_H).

I Then, ˆ qL =

θ_L−λθ_H (1−λ)c

2

ˆ qH =

λθ_H c

2

I Note that is possible that ˆq_L>qˆ_H using the formulae above.

But in this case, the optimal contract is pools both types.

More than two types:

I Every IC contract is (weakly) monotonic.

I Only the participation constraint for the lowest type binds.

I Only downward local incentive compatibility constraints may bind in any IC contract.

I Hence, we can relax the problem by looking at monotonic contracts satisfying the relevant subset of constraints.

I Pooling some types might be optimal (since separation is costly).