EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 - - PowerPoint PPT Presentation

EC537 Microeconomic Theory for Research Students, Part II: Lecture 2

Leonardo Felli

CLM.G.4

15 November 2011

Moral Hazard:

Consider the contractual relationship between two agents (agent 1 and 2) summarized in the following problem generated by the take-it-or-leave-it

• ffer that agent 1 makes to agent 2:

max

e,wi N

• i=1

pi v(e, yi, wi) s.t.

N

• i=1

pi u(e, wi) ≥ U (1) where: v(·, ·, ·) is agent 1’s utility function; u(·, ·) is agent 2’s utility function; U is the reservation utility of agent 2.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 2 / 86

Moreover: e can be interpreted as agent 2’s effort or investment; e enhances the random variable y interpreted as expected profit or expected outcome; wi is a transfer contingent on yi from agent 1 to agent 2; θ is the state of nature and pi is the probability of state θi. Assume that: e is chosen by agent 2 before the state of nature θ is realized; e is only observed by agent 2. It is his private information.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 3 / 86

Label: agent 2, who exerts effort, as the agent A; agent 1, who benefits from the effort, as the principal P. Assume that: y is verifiable information (observable to all agents involved in the contract court included). Moreover, it is critical for the problem to be interesting that: y is not in a one-to-one relation with the effort e. In other case, the contracting problem will result to a highly simplified version of an optimal risk-sharing problem.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 4 / 86

First Best:

Assume first that the effort chosen by the agent e is verifiable. Then w can be a function of e. Let e∗ be the optimal effort from the principal’s view point. The Principal’s optimal contract then will specify:

a state contingent payment {w1(e), . . . , wN(e)} to the agent that is individually rational if and only if e = e∗:

N

• i=1

pi u(e∗, wi(e∗)) = U a state contingent payment {w1(e), . . . , wN(e)} to the agent that is not individually rational (a punishment) otherwise.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 5 / 86

This can of course also be achieved with a y∗ that corresponds to e∗ in a one-to-one fashion. Therefore the problem is interesting only when output is a noisy signal of effort: y = f (e, θ). The principal is thus restricted to offer the contract w(f (e, θ)) and e is chosen by the agent so as to maximize his expected utility: ˆ e ∈ arg max

e N

• i=1

pi u(e, w(f (e, θi))). The contract w(f (e, θ)) must be such that it is in the agent’s best interest to choose the “right” (desired) level of effort ˆ e.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 6 / 86

The principal’s problem is: max

ˆ e,w(f (e,θ)) N

• i=1

pi v(ˆ e, f (ˆ e, θi), w(f (ˆ e, θi))) s.t.

N

• i=1

pi u(ˆ e, w(f (ˆ e, θi))) ≥ U ˆ e ∈ arg max

e N

• i=1

pi u(e, w(f (e, θi))) The latter constraint is known as the agent’s incentive compatibility constraint. The former constraint is the agent’s individual rationality constraint.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 7 / 86

The solution to the principal’s problem with both constraints is in general not a trivial matter. Key tradeoff: the one between insurance and incentives. Recall that without moral hazard the choice of e achieves ex-post allocative efficiency, while the choice of w achieves optimal risk sharing. This cannot be done in the presence of moral hazard.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 8 / 86

Second Best: risk neutral principal

Assume that the principal is risk neutral: v(e, f (e, θ), w) = f (e, θ) − w Optimal risk-sharing is achieved by giving the agent full insurance: w(f (e, θi)) = τ, ∀i ∈ {1, . . . , N}. The optimal choice of e will then be: ˆ e ∈ arg maxe u(e, τ). Clearly the agent will choose the same effort level independently of the contract (τ is independent of e). This means that any conflict of interest between the principal and the agent will not be ameliorated by the contract.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 9 / 86

Special case: effort as pure cost:

Assume that e is a pure cost for the agent: u(e, w) = U(w) − c(e), u′ > 0, c′ > 0, c′′ > 0 Then the agent will minimize effort. If e ≥ 0 then: ˆ e = 0. This differs from the effort level ¯ e that the principal desires: max

e N

• i=1

pi f (e, θi) − c(e)

• r

N

• i=1

pi ∂f (¯ e, θi) ∂e = c′(¯ e) Optimal risk sharing implies no incentives.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 10 / 86

Risk neutral agent:

The special case with no tradeoff between incentives and insurance is when the agent is risk neutral. The solution is to sell the firm/activity to the agent. Let w(f (e, θi)) = f (e, θi) − κ Then ˆ e = ¯ e since: ˆ e ∈ arg max

e N

• i=1

pi f (e, θi) − κ − c(e) While κ is independent of e and such that

N

• i=1

pi f (ˆ e, θi) − κ − c(ˆ e) = U

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 11 / 86

General Characterization:

We provide a more detailed characterization of the moral hazard problem by considering its simplest general form. The principal hires the agent to perform a task. The agent chooses his effort intensity, e, which affects the outcome of the task, q. The principal only cares about the outcome, but effort is costly for the agent, hence the principal has to compensate the agent for incurring the cost of effort. Effort is observable only to the agent, hence the agent’s compensation has to be contingent on the outcome q: a noisy signal of effort.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 12 / 86

Assume that the outcome of the task can take only two values: q ∈ {0, 1}. We assume that when q = 1 the task is successful and when q = 0 the task is a failure. The probability of success is: P{q = 1|e} = e. The principal’s preferences are represented by: V (q − w), V ′(·) > 0, V ′′(·) ≤ 0 where w is the transfer to the agent.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 13 / 86

The agent’s preferences are represented by the utility function separable in income and effort: U(w) − c(e), U′(·) > 0, U′′(·) ≤ 0 where c′(·) > 0, c′′(·) ≥ 0. For convenience we take c(e) = e2/2 and we normalize the agent’s outside option: U = 0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 14 / 86

First Best Contract:

The first best contract can be contingent on e. It is obtained as the solution to the problem: max

e,wi

e V (1 − w1) + (1 − e) V (−w0) s.t. e U(w1) + (1 − e) U(w0) ≥ e2 2 The optimal pair of transfers w∗

1 and w∗ 0 are such that the following

FOC (Borch optimal risk-sharing rule) are satisfied: V ′(1 − w∗

1 )

U′(w∗

1 )

= V ′(−w∗

0 )

U′(w∗

0 )

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 15 / 86

These transfers are paid only if the effort level coincides with e∗ satisfying: e∗ = [U(w∗

1 ) − U(w∗ 0 )] +

U′(w∗

1 )

V ′(1 − w∗

1 ) [V (1 − w∗ 1 ) − V (−w∗ 0 )]

Finally the agent’s expected utility coincides with the outside option: e∗U(w∗

1 ) + (1 − e∗) U(w∗ 0 ) = (e∗)2/2

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 16 / 86

If the principal is risk neutral: V (x) = x Then the conditions above become: w∗

1 = w∗ 0 = w∗

and U(w∗) = (e∗)2/2, e∗ = U′(w∗) If the agent is risk neutral: U(x) = x Then the optimum entails: w∗

1 − w∗ 0 = 1,

e∗ = 1.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 17 / 86

Second Best Contract:

If e is not verifiable then for every w1 and w0 it is determined so that: max

e

e U(w1) + (1 − e) U(w0) − e2/2 (2) The second best contract can be contingent only on q. It is obtained as the solution to the problem: max

ˆ e,wi

ˆ e V (1 − w1) + (1 − ˆ e) V (−w0) s.t. ˆ e U(w1) + (1 − ˆ e) U(w0) ≥ (ˆ e)2/2 ˆ e ∈ arg max

e

e U(w1) + (1 − e) U(w0) − e2/2

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 18 / 86

The FOC of the incentive compatibility constraint are: ˆ e = [U(w1) − U(w0)] (3) A first observation: from this condition it is clear that full insurance leads to no incentives: ˆ e = 0 Notice also that our assumptions imply that the solution to the agent’s incentive problem is unique for any pair (w0, w1). This allow us to replace the agent’s (IC) by (3): the solution to the FOC of the agent’s incentive problem. In general, replacing the (IC) constraint with the FOC of the agent’s effort choice problem is not a valid approach as we will see later on.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 19 / 86

Consider now the case in which the agent is risk neutral: U(x) = x we have seen that first best optimality requires e∗ = 1 In this case the FOC of the (IC) constraint becomes: ˆ e = (w1 − w0) Therefore setting w1 − w0 = 1 leads to the first best allocation: optimal risk sharing and optimal incentives.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 20 / 86

The reason is that:

• ptimal risk sharing requires that the agent bears all the risk in the

environment,

• ptimal incentives requires that the agent is residual claimant.

This is achieved by selling the activity to the agent at a fix price −w0 > 0 so that the risk averse principal receives full insurance. Notice that in this case we need the agent to have deep enough pockets: when the outcome is q = 0 the agent’s payoff is w0 < 0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 21 / 86

In other words, the agent must be willing to incur a loss with a strictly positive probability. It is often natural to assume that the agent has no resources to put in the activity. This implies a resource or limited liability constraint: wi ≥ 0. In this case the problem becomes: max

ˆ e,wi

ˆ e V (1 − w1) + (1 − ˆ e) V (−w0) s.t. ˆ e w1 + (1 − ˆ e) w0 ≥ (ˆ e)2/2 ˆ e = (w1 − w0) wi ≥ 0 ∀i ∈ {0, 1}

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 22 / 86

In the situation in which the agent is resource constrained not all the risk can be transferred to the agent: the constraint wi ≥ 0 will be binding for the transfer w0: w0 = 0 It is still possible to create first best incentives but for this purpose the agent’s needs to be rewarded. If e∗ = w1 − w0 = w1 = 1 then the agent’s payoff is: e∗w1 − (e∗)2/2 = 1/2 > 0 In other words the (IR) constraint is not binding. This is not necessarily optimal for the principal.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 23 / 86

In particular, if we assume that the principal is risk neutral as well: V (x) = x then the principal’s problem is: max

ˆ e,wi

ˆ e (1 − w1) − (1 − ˆ e) w0 s.t. ˆ e w1 + (1 − ˆ e) w0 ≥ (ˆ e)2/2 ˆ e = (w1 − w0) wi ≥ 0 ∀i ∈ {0, 1} The solution implies that w0 = 0, w1 = ˆ e

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 24 / 86

Moreover, ˆ e solves the constrained problem: max

ˆ e

ˆ e (1 − ˆ e) s.t. (ˆ e)2/2 ≥ 0

• r

ˆ e = 1/2 We then conclude: ˆ e = 1/2 < e∗ = 1. The resource constraint implies a second best level of effort.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 25 / 86

The principal trades off the lower effort choice by the agent against the higher compensation that the agents needs to provide first best level of effort. However, the agent still gets a strictly positive payoff:

• ΠA = (ˆ

e)2/2 = 1/8 > 0 Indeed, the principal also received a strictly positive profit:

• ΠP = ˆ

e(1 − ˆ 2) = 1/4

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 26 / 86

Recall that social surplus associated with the first best problem i.e. the solution to the social planner problem is such that: ΠP + ΠA = e∗ − (e∗)2/2 = 1/2 Compare it to the social surplus in equilibrium:

• ΠP +

ΠA = 3/8 The gap is the result of the agency problem.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 27 / 86

Continuous Outcomes:

Consider now the more general environment in which the state-space representation of the effort’s outcome is the random variable q(e, θ), where θ ∈ [θ, θ]. First, for simplicity assume that φ(e) = e. Second, let us consider the parameterized distribution characterization

• f the same problem: q ∈ Q = [q, q] is the support of the density

f (q, e) > 0 and cdf F(q, e). We assume that: Fe(q, e) < 0, ∀q ∈ [q, q] In other words, the effort e produces a first-order stochastic dominant shift on Q. If e0 < e1:

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 28 / 86

✲ ✻

F(q, e) F(q, e1) F(q, e0) q q q 1

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 29 / 86

First Best:

In this setup the first best contract solves: max

e,w(·)

q

q

V (q − w(q)) f (q, e)dq s.t. q

q

U(w(q)) f (q, e)dq ≥ e That is: V ′(q − w∗(q)) U′(w∗(q)) = λ, ∀q ∈ Q q

q

U(w∗(q)) f (q, e∗)dq = e∗ and q

q

[V (q − w∗(q)) + λU(w∗(q))] fe(q, e∗)dq = λ

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 30 / 86

Second Best:

The second best contract solves: max

ˆ e,w(·)

q

q

V (q − w(q)) f (q, ˆ e)dq s.t. q

q

U(w(q)) f (q, ˆ e)dq ≥ ˆ e ˆ e ∈ arg max

e

q

q

U(w(q)) f (q, e)dq − e The (IC) constraint implies, assuming an interior optimum: q

q

U(w(q)) fe(q, ˆ e)dq = 1 and q

q

U(w(q)) fee(q, ˆ e)dq ≤ 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 31 / 86

We proceed using the so called first order approach: substitute the (IC) constraint with the FOC of the agent’s incentive problem. In this case the lagrangian is: L = q

q

V (q − w(q)) f (q, ˆ e)dq + λ q

q

U(w(q)) f (q, ˆ e)dq − ˆ e

• + µ

q

q

U(w(q)) fe(q, ˆ e)dq − 1

• Leonardo Felli (LSE)

EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 32 / 86

There exists no constraint on the first and second derivative of w(·). Therefore it is possible to solve the problem by pointwise maximization: V ′(q − w(q)) U′(w(q)) = λ + µ fe(q, ˆ e) f (q, ˆ e) , ∀q ∈ Q This is, once again, the modified Borch rule.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 33 / 86

Assume that: the agent is strictly risk averse: U′′(·) < 0, the distribution F(q, e) has fixed support Q: Fe(q, e) = Fe(q, e) = 0, the distribution F(q, e) satisfies first order stochastic dominance: Fe(q, e) < 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 34 / 86

Result (Holmstr¨

• m, 1979)

Assume that the first order approach is valid then at the optimum: µ > 0. Proof: Assume not: µ ≤ 0. From the necessary condition ∂L ∂ˆ e = 0 and the modified (IC) constraint we obtain: q

q

V (q − w(q)) fe(q, ˆ e)dq + µ q

q

U(w(q)) fee(q, ˆ e)dq

• = 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 35 / 86

Using the second order condition of the agent’s incentive problem q

q

U(w(q)) fee(q, ˆ e)dq ≤ 0 when µ ≤ 0 it becomes: q

q

V (q − w(q)) fe(q, ˆ e)dq ≤ 0 Define w0(q) as the function that solves, for µ = 0: V ′(q − w0(q)) U′(w0(q)) = λ, ∀q ∈ Q From U′′(·) < 0, w′

0(·) exists and is such that:

0 ≤ w′

0(q) =

V ′′ U′ V ′′ U′ + V ′ U′′ < 1

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 36 / 86

Moreover, when µ ≤ 0 we have from V ′(q − w(q)) U′(w(q)) = λ + µ fe(q, ˆ e) f (q, ˆ e) , ∀q ∈ Q that w(q) ≤ w0(q) ⇔ fe(q, ˆ e) ≥ 0 w(q) > w0(q) ⇔ fe(q, ˆ e) < 0 Since ∂(V ′(q − w(q)/U′(w(q))) ∂w(q) = −V ′′ U′ + V ′ U′′ (U′)2 > 0 We therefore conclude that for all q ∈ Q: V (q − w(q)) fe(q, ˆ e) ≥ V (q − w0(q)) fe(q, ˆ e)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 37 / 86

Integrating over Q we then obtain: q

q

V (q − w(q)) fe(q, ˆ e) dq ≥ q

q

V (q − w0(q)) fe(q, ˆ e) dq Integrating by parts and using the fixed support and the first order stochastic dominance we conclude: q

q

V (q − w0(q)) fe(q, ˆ e) dq = =

• V (q − w0(q)) Fe(q, ˆ

e) q

q−

− q

q

V ′(q − w0(q)) (1 − w′

0(q)) Fe(q, ˆ

e) dq > 0 A contradiction of the necessary conditions above.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 38 / 86

Observations:

We have assumed that e is in the interior of E, it is a much simpler problem if the agent’s incentive problem leads to a corner solution. The assumption that Q does not depend on e is crucial, moving support may lead to first best outcome. Commitment to the contract is critical: risk will be renegotiated away from the agent between choice of e and realization of q. The fe/f is the derivative of the ln f and the gradient for a MLE of e. We proved 0 ≤ w′

0(·) < 1 but we have not proved that w(·) is

monotonic.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 39 / 86

MLRP:

The monotone likelihood ratio property (MLRP) is satisfied for F(·) and f (·) iff d dq fe(q, e) f (q, e)

• > 0

Notice that MLRP implies FOSD, Fe(q, e) < 0 for every q ∈ Q. In fact: Fe(q, e) = q

q

fe(s, e) f (s, e) f (s, e)ds (4) Since q

q

fe(s, e) f (s, e) f (s, e)ds = Fe(q, e) − Fe(q, e) = 0 Then for every q < q the fact that the likelihood ratio is increasing in s implies that (4) is strictly negative.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 40 / 86

Result (Holmstr¨

• m, 1979, Shavell 1979)

Under the first order approach, if f (·) satisfies the MLRP, then the wage contract w(·) is increasing in output. The proof is a direct consequence of the definition of MLRP. Notice that if the agent can freely dispose of output then monotonicity of w(·) is a constraint that the solution must satisfy.

Result (Holmstr¨

• m, 1979, Shavell 1979)

Under the first order approach, if w(q) is the solution then there exists a new contract w(q, s) that strictly Pareto dominates w(q) if and only if [fe(q, s, e)/f (q, s, e)] varies with s.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 41 / 86

First Order Approach:

The first order approach of first finding w(·) using the relaxed problem and then checking that the principal’s choice of ˆ e actually solves the agent’s incentive problem is logically invalid (additional restrictions).

Problem

if the SOC of the agent’s incentive problem are not globally satisfied, then the solution to the principal’s problem satisfies the agent’s FOC but not necessarily the principal’s ones. This is because the global maximum to the principal’s problem might involve a corner solution and so the principal’s selected ˆ e may not satisfy the necessary Kuhn-Tucker conditions of the relaxed problem.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 42 / 86

Definition

A distribution F(·) satisfies the Convexity of Distribution Function Condition (CDFC) if and only if for every γ ∈ [0, 1]: F(x, γe + (1 − γ)e′) ≤ γF(x, e) + (1 − γ)F(x, e′)

• r

Fee(x, e) ≥ 0. Special case of CDFC is the linear distribution condition: let f (q) FOSD f (q) then f (q, e) = ef (q) + (1 − e)f (q)

Result (Mirrlees 1976, Rogerson 1985)

The first order approach is valid if F(q, e) satisfies MLRP and CDFC.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 43 / 86

“Proof:” Integrating by parts the agent’s payoff becomes, if w(·) is differentiable: q

q

U(w(q)) f (q, e)dq − e =

• U(w(q)) F(q, e)

q

q −

− q

q

U′(w(q)) w′(q) F(q, e) dq − e Differentiating it with respect to e twice we get from CDFC for every e ∈ E: − q

q

U′(w(q)) w′(q) Fee(q, e) dq ≤ 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 44 / 86

We conclude the proof with the observation that under MLRP and the first order approach µ > 0 and w′(·) > 0. This reasoning is however wrong. We know that µ > 0 only if the first order approach is valid. Therefore in proving that the first order approach is valid we cannot take for granted that µ > 0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 45 / 86

Proof: Consider an alternative principal’s relaxed problem where the (IC) constraint is substituted by d de q

q

U(w(q)) f (q, e)dq − e

• ≥ 0

Inequality constraint implies that the constraint multiplier µ′ ≥ 0 therefore under MLRP the solution to this new problem is such that w′(q) ≥ 0 We still need to show that:

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 46 / 86

Lemma (Rogerson 1985)

The solution to the new principal’s problem is such that d de q

q

U(w(q)) f (q, e)dq − e

• = 0

Proof: If µ′ > 0 then we are done. Assume then that µ′ = 0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 47 / 86

The lagrangian is then: L′ = q

q

V (q − w(q)) f (q, e)dq + + λ q

q

U(w(q)) f (q, e)dq − e

• +

+ µ′ d de q

q

U(w(q)) f (q, e)dq − e

• Therefore from the necessary first order condition with respect to e we get:

∂L′ ∂e = q

q

V (q − w(q)) fe(q, e)dq + + λ d de q

q

U(w(q)) f (q, e)dq − e

• = 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 48 / 86

Moreover if µ′ = 0 then w(q) ≡ w0(q) and we already proved that 0 ≤ w′

0(q) < 1.

Recall further that integrating by parts the principal’s expected utility we get: q

q

V (q − w0(q)) fe(q, e) dq = = − q

q

V ′(q − w0(q)) (1 − w′

0(q)) Fe(q, e) dq > 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 49 / 86

Therefore the necessary first order conditions above imply that λ > 0 and: d de q

q

U(w(q)) f (q, e)dq − e

• < 0

Clearly a contradiction. Very few distribution satisfy MLRP and CDFC. An example is the generalization of the uniform distribution for e ∈ [0, 1): F(q, e) = q − q q − q

• 1

1−e

Jewitt (1988) provides a collection of alternative sufficient conditions on f (q, e) and U(w(q)) which are weaker than CDFC.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 50 / 86

First Order Approach (Grossman and Hart 1983)

Consider now the original principal’s problem and assume that there are

• nly a finite number of outputs

q1 < q2 < . . . < qN. Clearly in this case f (qi, e) = Pr{q = qi|e} Assume that the principal is risk neutral, V (x) = x and that U(x) satisfies Inada conditions for x → 0 and E is compact. Assume that the agent’s utility function is (K(e)U(w) − φ(e)): U(w) − e

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 51 / 86

The second best contract is then the solution to the following problem: max

ˆ e,w(·) N

• i=1

f (qi, ˆ e)(qi − w(qi)) s.t.

N

• i=1

U(w(qi)) f (qi, ˆ e) ≥ ˆ e

N

• i=1

U(w(qi)) f (qi, ˆ e) − ˆ e ≥ ≥

N

• i=1

U(w(qi)) f (qi, e) − e ∀e ∈ E

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 52 / 86

We solve the problem in two steps: find the least cost way to implement a given action e, find the optimal e from the principal’s view point.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 53 / 86

Step 1: min

wi N

• i=1

f (qi, e) wi s.t.

N

• i=1

U(wi) f (qi, e) ≥ e

N

• i=1

U(wi) f (qi, e) − e ≥ ≥

N

• i=1

U(wi) f (qi, e) − e ∀e ∈ E

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 54 / 86

This is not a convex programming problem, to be able to use Kuhn-Tucker theorem we change variable. Define h(·) = U−1(·) we then use as control variable: ui = U(wi) The now convex programming problem is: min

ui N

• i=1

f (qi, e) h(ui) s.t.

N

• i=1

ui f (qi, e) ≥ e

N

• i=1

ui f (qi, e) − e ≥

N

• i=1

ui f (qi, e) − e ∀e ∈ E

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 55 / 86

Result (Grossman and Hart 1983)

If either K ′(e) = 0 or φ′(e) = 0 then the (IR) constraint of this problem is binding. Notice that in this case the agent’s preferences over action lotteries is independent of income. Define: C(e) =      inf

• i

f (qi, e)h(ui)

• ,

if e obtained; +∞,

• therwise.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 56 / 86

Step 2: max

e N

• i=1

f (qi, e) qi − C(e)

Result (Grossman and Hart 1983)

Step 2’s solution exists and the inf in C(e) is a min. Assume that E = {eL, eH} and ˆ e = eH then h′(ui) = 1 U′(wi) = λ + µL f (qi, eH) − f (qi, eL) f (qi, eH)

• µL > 0 and wi increases with (f (qi, eL)/f (qi, eH)).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 57 / 86

CARA Utility, Normal and Linear Contracts:

Let the principal be risk neutral V (x) = x and the agent’s preferences be represented (a denotes effort, r is the index of absolute risk aversion) by: U(x, a) = −e−r(x−φ(a)) Let the outcome q be such that: q = a + ε where ε ∼ N(0, σ2) The cost of effort is quadratic: φ(a) = a2 2 and we normalize U = −1.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 58 / 86

Restrict now the principal to offer only linear contracts: w(q) = β q + γ Recall also that if x ∼ N(µ, σ2) then: Ex

• et x

= eµ t− 1

2 σ2 t2 Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 59 / 86

First best: max

w(·),a

Eq [q − β q − γ] s.t. Eq

• −e−r(β q+γ−φ(a))

≥ −1 effort choice a∗ = 1 transfer to the agent w∗(q) = 1/2 for every q. expected profit of the principal Π∗ = 1/2

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 60 / 86

Second best:

The principal’s problem is: max

w(·),ˆ a

Eq [q − β q − γ] s.t. Eq

• −e−r(β q+γ−φ(ˆ

a))

≥ −1 ˆ a ∈ arg max

a

Eq

• −e−r(β q+γ−φ(a))

Notice that: Eq

• −e−r(β a+β ε+γ−φ(a))

= − e−r(β a+γ−φ(a))+ 1

2 β2 r2 σ2 Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 61 / 86

Therefore the agent’s incentive problem becomes: ˆ a ∈ arg max

a

β a + γ − a2 2 − r 2 β2 σ2 The unique solution to this problem is then ˆ a = β Then the second best problem becomes: max

γ,ˆ a

ˆ a − ˆ a2 − γ s.t. ˆ a2 + γ − ˆ a2 2 − r 2 ˆ a2 σ2 ≥ 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 62 / 86

The solution is then: ˆ a = β = 1 1 + r σ2 , γ = r σ2 − 1 2 (1 + r σ2)2 Moreover: ˆ a < a∗, Π = 1 2 (1 + r σ2) < Π∗ Comparative static: if either r or σ2 decreases the power of the

• ptimal incentive scheme increases (less distortion).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 63 / 86

Unrestricted Contracts (Mirrlees 1999)

Question: what if the contract space is unrestricted?

Result (Mirrlees 1999)

If f (q, a) is a normal distribution with mean a and variance σ2 and unlimited punishments are possible, the first best outcome can be approximated arbitrarily closely. Let the principal be risk neutral V (x) = x and the agent’s preferences be U(w, a) = −e−r(w−φ(a)).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 64 / 86

Proof: By assumption: f (q, a) = 1 √ 2 π σ e

−(q−a)2 2σ2

Therefore: fa(q, a) f (q, a) = d da ln f (q, a) = (q − a) σ2 In other words: detection is efficient for very small q.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 65 / 86

Recall that the first-best contract is the solution to the following problem: max

w(q),a

Eq [q − w(q)] s.t. Eq

• −e−r(w(q)−φ(a))

≥ −1 The first order conditions of this problem imply: e−r(w(q)−φ(a)) = 1 λ r , ∀q ∈ R

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 66 / 86

This condition clearly implies full insurance: w(q) = w, ∀q ∈ R We can now re-write the principal’s problem as: max

w,a

[a − w] s.t. −e−r(w−φ(a)) ≥ −1

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 67 / 86

• r equivalently:

max

w,a

[a − w] s.t. w − φ(a) ≥ 0 A linear programming problem where the objective function is monotonic decreasing in w, while the constraint is monotonic increasing in w. Therefore in equilibrium the constraint is necessarily binding: w∗ = φ(a∗), φ′(a∗) = 1

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 68 / 86

Consider the following approximate first best contract: for every q ≥ q0 (q0 very small) the agent is paid the constant transfer w∗, for every q < q0 the agent is paid the very small constant amount k. The amount k is chosen (low enough) so that it is optimal for the agent to choose ˆ a = a∗: q0

−∞

U(k) fa(q, a∗) dq + + +∞

q0

U(w∗) fa(q, a∗) dq = φ′(a∗)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 69 / 86

This can always be done choosing a low enough k since from the necessary first order conditions of the agent’s (IC) constraint: q0

−∞

U(k) fa(q, ˆ a) dq + + +∞

q0

U(w∗) fa(q, ˆ a) dq = φ′(ˆ a) we obtain: sign d ˆ a d k

• = sign

q0

−∞

U′(k) fa(q, ˆ a) dq

• =

= sign

• U′(k) Fa(q, ˆ

a)

• Therefore d ˆ

a d k < 0

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 70 / 86

We still need to show that this contract is such that the agent’s (IR) constraint can be satisfied arbitrarily closely. The loss with respect to the first best is: ∆ = q0

−∞

(U(w∗) − U(k)) f (q, a∗)dq Define M(q0) = fa(q0, a∗) f (q0, a∗) Since the normal distribution satisfies MLRP, for all q < q0, we have that for a low enough q0 such that fa(q0, a∗) < 0: f (q, a∗) < fa(q, a∗) M(q0)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 71 / 86

Therefore ∆ ≤ 1 M(q0) q0

−∞

(U(w∗) − U(k)) fa(q, a∗)dq Using the necessary first order conditions of the (IC) constraint we obtain that: q0

−∞

U(k) fa(q, a∗) dq = − +∞

q0

U(w∗) fa(q, a∗) dq + φ′(a∗)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 72 / 86

We can, therefore, rewrite this bound as: ∆ ≤ 1 M(q0) +∞

−∞

U(w∗)fa(q, a∗)dq − φ′(a∗)

• The bound on ∆ is then a constant (with respect to k and q0) divided by

M(q0). Therefore as q0 decreases unboundedly by MLRP also M(q0) decreases

• unboundedly. Therefore ∆ → 0

In other words, the (IR) is approximately satisfied for a low enough q0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 73 / 86

Intuition: In the lower tail of the normal distribution q is very informative about ˆ a. Therefore, a harsh punishment can achieve first best incentives at a cost to the principal. The unboundedness of the support of the distribution allows the principal to render these costs arbitrarily small. The result generalizes to other distributions with unbounded support and to general utility functions.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 74 / 86

Multitasking (Homstrom and Milgrom 1991, 1994)

Assume that the agent can carry out multiple tasks that affect output. Consider our previous model: both the principal and the agent are risk neutral but the agent has limited liability: wi ≥ 0. The agent now controls two tasks: one is “standard” (S) and one is “noisy” (N). The agent chooses two effort levels: eS and eN, eS ∈ [0, 1], eN ∈ [0, 1]. Disutility of effort: (e2

S + e2 N)/4.

Assume that the two tasks are perfect complements in the stochastic technology: P{q = 1|eS, eN} = min{eS, eN}

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 75 / 86

ΠP is expected output, minus expected wage. ΠA equals the agent’s expected wage, minus the disutility of effort. Assume that q is not contractible. Task S generates the contractible binary signal σS ∈ {0, 1} that is equal to 1 with probability eS. Task N generate the contractible binary signal σN ∈ {0, 1} that is equal to 1 with probability [eNp + (1 − eN)(1 − p)] with p ∈ [1/2, 1]. If p = 1/2 then σN contains no information about eS, while if p = 1, the signals σS and σN are equally informative.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 76 / 86

A contract is now a quadruple of wages (wS0, wS1, wN0, wN1), one for each task, and for each possible value of the corresponding signal. Limited liability implies wS0 = wN0 = 0. The (IC)s constraints pin down eS and eN as eS = 2wS1, eN = 2wN1(2p − 1) The principal’s problem of maximizing ΠP subject to the restrictions above gives: eS = eN = max{0, 1/2 − (1 − p)/(8p − 4)} Notice that when p = 1 we get the same outcome as in the single task case, if instead p ≤ 3/5 then eS = eN = 0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 77 / 86

Notice that as p decreases (task N becomes more noisy) two changes

• ccur.

In equilibrium, eN decreases (the increased noise increases the cost of N incentives). The effort eS decreases as well: increased noise yields softer incentives on the standard task, as well as the noisy one. Complementarity of tasks dictates that as eN becomes more expensive the principal chooses to induce lower eS as well. When p ≤ 3/5, σN is not informative enough. In this case eS = eN = wS1 = wN1 = 0 (no incentive contract is signed).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 78 / 86

Informed Principal (Maskin and Tirole 1990, 1992)

Effort is e non-contractible and private information of A. Output q ∈ {0, 1} is contractible and P{q = 1|e} = e Assume that also the principal has private information, creating a potential signaling role for the contract offer. Consider a simple “Common values” case as in Maskin and Tirole (1992). There are two types of principal: PH and PL.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 79 / 86

Assume that P = PH with probability φ = 18/29 and P = PL with probability 1 − φ = 11/29. If P is of type H, A’s outside option is U = 9/32, while if P is of type L then A’s outside option is U = 0. If PH and PL separate in their contract offer than in equilibrium two IRs need to be satisfied for A. If pooling obtains A faces a single IR and his expected outside option is φ U = 81/464 A has limited liability as described above.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 80 / 86

Timing: First P learns his type. P offers a contract to A, which may take the form of a menu (wages contingent on output and P’s announcement). After seeing the offer A updates his beliefs about P’s type and then decides whether to accept or reject. As in any signaling game, the issue of off-the-equilibrium-path beliefs arises: assume that A’s beliefs after observing an “unexpected” offer are that P is of type H with probability 1. After a contract is signed P announces to A which part of the menu applies in his case (if the contract is in fact a menu). Finally, A chooses effort, output is realized and payoffs are obtained.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 81 / 86

In a separating equilibrium PH and PL offer two distinct pairs of

• utput-contingent wages: (wH1, wH0) and (wL1, wL0).

A’s ICs dictate that after being offered (wH1, wH0) effort is eH = wH1 − wH0, while after being offered (wL1, wL0) effort is eL = wL1 − wL0. Separation requires that neither PH nor PL has an incentive to offer the other type’s wage pair. Since P’s private information does not enter directly his payoff, this can only be true if the expected profits for the two types of principals, ΠH and ΠL are the same truth telling constraint: ΠH = eH(1 − eH) = eL(1 − eL) − wL0 = ΠL

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 82 / 86

Since U = 9/32 for the H agent then his IR does bind. Using IC this yields eH = wH1 = 3/4. Using the principal’s truth telling constraint we obtain eL = 1/2, wL0 = 1/16 and wL1 = 9/16. In other words, ΠH = ΠL = 3/16.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 83 / 86

Consider now the possibility of pooling equilibria, in which the contract is a menu. Both PH and PL offer the same menu (wM

H1, wM H0, wM L1, wM L0), which

A has to accept or reject based on his expected IR. After a contract is signed, P tells A which pair of output-contingent wages applies. The truth-telling constraint still applies, since both PH and PL have to be willing to indicate to A the appropriate wage pair.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 84 / 86

Using IC and wM

H0 = 0, the truth telling constraint reads:

ΠM

H = eM H (1 − eM H ) = eM L (1 − eM L ) − wM L0 = ΠM L

The single binding expected IR and the ICs yield (18/58)(eM

H )2 + (11/29)[(eM L )2 + wM L0] = 81/464

From the truth-telling constraint we then get: eH = wH1 = 5/8, eL = 1/2, wL0 = 1/64 and wL1 = 33/64. In other words: ΠH = ΠL = 15/64.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 85 / 86

Both types of P enjoy strictly higher profits under pooling than under separation. Pooling relaxes A’s IR which binds in expectation. PH can lower wH1 which increases ΠM

H relative to the separation case.

The increased profit for PH affects PL via the principal’s truth-telling constraint. PL lowers both output-contingent wages to satisfy the truth-telling constraint, which in turn increases ΠM

L to keep it in line with ΠM H .

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 86 / 86