Start out by talking about examples of situations in which contracts (ei-ther explicit or implicit) are in place and point to the particular form ofasymmetric information.

1) PhD student and GSIA. The school devotes faculty time and money tothe student. In exchange, the student produces papers and interacts with thefaculty. There are two types of uncertainty here: the school does not knowhow smart and skilled is the student. Moreover, there are random shocksto the productivity of the student himself. In general, the school bases theallocation on the results (grades, papers, participation...)

2) Car Insurance (Collision insurance). The contract works in such a waythat you have the accident and make the claim. Based upon the claim, theinsurance pays. However, the insurance does not really know whether theguy had an accident. Example of Naples.

The objective of contract theory is to model formally this kind of rela-tionships, by assuming:

1) objective functions for the agents involved (payoff function)2) distribution of bargaining power3) distribution of information4) an equilibrium solution

and then figure out the arrangement that satisfy that definition of equi-librium. Everything will be clearer as we go along.

Positive approach: under which condition the equilibrium contract iswhat we observe?

Normative approach: what is the equilibrium contract under some con-ditions?

The models of the theory of contracts can be distinguished along sev-eral dimensions. A crucial distinction is between complete and incompletecontracts.

In the case of complete contracts, all variables that may have an impact onthe conditions of the contractual relationship during its whole durantion (i.e.all variables on which the provisions of the contract may depend) are takeninto account when the contract is negotiated and signed. Thus the contractitself may be contingent on a very large number of variables. This assumptionimplies that no unforeseen contingency may arise as the relationship evolves.

1

Any change in the economic environment just activates the ad hoc provisionsof the contract.

When the above assumption does not hold, we say that the contract isincomplete. There are several reasons why, in the real world, a contract maybe incomplete.

• Negotiating a contract is often a costly business, which mobilizes man-agers and lawyers. It must therefore be that at some point the cost oftaking into account an improbable contingency outweigh the benefitsof wrting a specific cluase in the contract. The contract should thenbe signed without this clause.

• The unability (or unwillingness) of courts or other third parties toverify ex post the values taken by certain variables observed by allcontractants is another reason why contract will be incompleted. It isno use conditioning the contract on a variable if nobody can settle thedisputes that may arise.

• Finally, bounded rationality may force the parties to neglect some vari-ables whose effect on the relationship they find difficult to evaluate.

For all of these reasons, contracts typically only take into account a lim-ited number of variables that may be the most relevant ones, or simply thosethat are most easily verifiable by a court. During the relationship, some un-foreseen contingencies may arise, that have an impact on the conditions ofthe relationship and the contract gives no clue as to how the parties shouldreact. As a result, the parties may want to renegotiate the contract.

In this course we will consider exclusively complete contracts. In partic-ular, we will focus on situations in which there are two economic agents: aninformed party, whose information is relevant for the common welfare, andthe uninformed party. In general, this is a bilateral monopoly1 situation.This implies that we cannot go very far unless we specify how the parties aregoing to bargain over the terms of exchange. In this respect , we will makequite a drastic assumption, by allocating all bargaining power to one of theparties. This party, known as principal, will make a take it or leave it offer

1As we will see a the course progresses, principal-agent models also apply to situations

with a continuum of infinitesimal agents, each of whom interacts with the principal but

not with the other agents.

2

to the other party, known as agent. The agent will either accept or reject. Ifthe agent accepts, the parties execute the contract.

According to how the asymmetric information is modeled, we distinguishtwo kinds of principal-agent contracts. Adverse Selection and Moral Hazard.Adverse Selection is the case in which the uncertainty is with respect tosome inherent characteristic of the agent. i.e. there is asymmetric informa-tion before the contract is signed. Example: health insurance.

In the case of Moral Hazard instead, there is no ex-ante asymmetry. Theasymmetry arises after the contract is signed. We distinguish two cases.

Hidden Action. Here the agent has to take an action that as an effecton the Principal’s payoff function, but the action itself is not observable tothe principal. Example: Firm vs CEO.

***************** Time Line from the book by Macho-Stadler

Hidden information.Here after the contract is signed, some event occurs, that is observable

to the agent only. Example: a contract between the owner of a plot of landand the farmer that actually cultivates the land. The harvest is stochastic,and is not observable to the owner.

Since I have never taught this class before, I know only the lower boundon what we will be doing. Hopefully, there is going to be time to do more.At the beginning I will go slow. Then I will go faster and faster.

3

We will proceed as follows:- Start with Hidden Information model.- Then we will see and Hidden Action model.- Finally (ad the end of the course) the adverse selection case.References:Salanie (1997), chapter 1.Macho-Stadler and Perez-Castrillo (1997), chapter 1.

4

Reference: Townsend (1982)Mechanism Design and Revelation principle: Fudenberg and Tirole (1991),

section 7.3.

Bottom line.With public information about individuals’ endowments, the optimal in-

surance arrangement is to have full-insurance. Moreover, this outcome re-mains the same when individuals are allowed to repeat arrangements overtime.

With private information, there is no insurance in the one-shot game:agents will always claim to have received a low endowment and thus receivea constant payment every period. However, as opposite to the public infor-mation case, the repetition of the arrangement can now yield some insuranceby exploring long-term relationship.

The model.We consider an economy with two agents. One, which we call principal,

is risk-neutral. The other, which we call agent, is risk-averse. The principalreceives a constant endowment of y every period. His preferences are repre-sented by the linear utility in consumption: v(c) = c. The risk averse agentreceives an endowment yi ∈ Y (Y finite set) in each period with probabilityπi (where

∑i πi = 1) and has a utility function u(c) that satisfies u′ > 0 and

u′′ < 0, u′

(0) = ∞.

The one-period problem with symmetric information.Consider first the one-period Pareto problem with public information

about the endowments. The efficient allocation is given by solving the fol-lowing problem:

maxTi

∑

i

πi(y − Ti) (1)

subject to:

∑

i

πiu(yi + Ti) = w (2)

where Ti is the transfer that the risk neutral agent makes to the riskaverse one when the state is i.

Assuming an interior solution (alternatively, considering u(c) : < → <,of which an example is the negative exponential utility function, u(c) =

5

− exp(−Ac) with A > 0), the solution for the problem above can be charac-terized by:

−πi + λπiu′(yi + Ti) = 0 (3)

u′(yi + Ti) =1

λ(4)

and we are able to conclude that:

yi + Ti = c for all i (5)

which is to say that agents get full-insurance from the efficient risk allo-cation.

In other words, here the contract is a schedule Ti = u−1 (w) − yi. Bymoving w we can figure out the set of Pareto-Optimal Allocations. For everyw in fact the payoffs are

∑i πi(y − u−1 (w) + yi) for the principal and w for

the agent.

The one-period problem with asymmetric information.Contract theory formalizes this problem as three-step game of incomplete

information, where the income of the agent is private information.In step 1, the principal designs a ’mechanism’ or contract, or incentive

scheme. A mechanism is a game in which the agent send a costless message,and the principal provides an allocation that depends on the realized mes-sages. The allocation is a decision about the level of some observable variable(in this case the transfer). In step 2, the agent either accepts or rejects thecontract. An agent who rejects the contract gets some exogenously specified”reservation utility”. In step 3, if the agent accepted the mechanism, playsthe game specified by the mechanism itself.

A contract here defines a message space Ω and a game form to announcethe messages. In this case (if we limit to pure strategies) the message is amapping Y → Ω. The set Ω can be anything. For simplicity here we assumethat is bounded. Because income is private information, the allocation T candepend on income only through the agent’s message. Thus the allocation willbe a function T : Ω → <. The Bayesian Equilibrium for this game is goingto be functions T ∗ ($) , $∗ (yi) such that:

6

T ∗ ($) = argmax∑

i

πi(y − T ($∗ (yi)))

Here it seems important to me that the function T ∗ ($) is defined forevery $.

∀i: u(yi + T ∗ ($∗ (yi))) ≥ u(yi + T ∗ ($)) ∀$ 6= $∗ (yi) .

Revelation principle (Myerson (1979)). It states that the principalcan content herself with ’direct’ mechanisms, in which the message space isthe income space and all the agents announce their type truthfully. ******Change notation, in particular $. Do not confuse $ and w.

Proposition. Suppose that a mechanism with message space Ω and alloca-tion function T ∗ ($) has a Bayesian Equilibrium

$∗ (·) = $∗ (yi)yi∈Y

Then there exists a direct-revelation mechanism (T∗

= T ∗ $∗) such thatthe message space is the income space Y and such that there exists aBayesian equilibrium in which all agents announce their types truth-fully.

Consider the new message space Y , so that the agent gives a message yi

which in principle is different from the true yi. A Bayesian equilibrium for

the new game is given by functionsT

∗(yi) , y

∗ (yi)

such that

T

∗(yi)

= argmax

∑

i

πi(y − T i (yi (yi)))

∀i: u(yi + T∗(yi (yi))) ≥ u(yi + T

∗(yj)) ∀yj 6= yi.

Basically the Revelation Principle says that y∗ (yi) = yi is a Bayesianequilibrium of the new game.

7

Proof. Consider the allocation rule T∗(yi) ≡ T ∗ ($∗ (yi)).

Straight by definition:

∀i: u[yi + T

∗(yi)

]= u [yi + T ∗ ($∗ (yi))]

The condition for Bayesian equilibrium in the original message game:

∀i: u(yi + T ∗ ($∗ (yi))) = sup$∈Ω

u [yi + T ∗ ($ (yi))] .

Finally:

∀i: sup$∈Ω

u(yi + T ∗ ($ (yi))) ≥ maxyi∈Y

u(yi + T∗(yi)).

, since $∗ (yi)yi∈Y ⊂ Ω.

This last weak inequality expresses the fact that in the direct-revelationmechanism everything is as if the agent picked an announcement in the subsetof messages $∗ (yi)yi∈Y , which is a subset of Ω. The agent thus has, atmost, as many posssibilities for deviating as in the original game. In otherwords:

maxyi∈Y

u(yi + T∗(yi)) = max

$∈$∗(yi)yi∈Y

u(yi + T ∗ ($))

End of First Class.

8

To analyze the one-period game with private information, we make useof the revelation principle and include the truth-telling constraints in thePareto problem. The game is:

Message space: Y .Allocation: T (yi)yi∈Y = Ti. The allocation is what is commonly

referred to as the contract.

maxTi

∑

i

πi(y − Ti) (6)

subject to:

∑

i

πiu(yi + Ti) = w (7)

u(yi + Ti) ≥ u(yi + Tj) ∀i, j. (8)

No insurance result.

Note that, since u (·) is strictly monotone, we obtain from (8) that Ti ≥ Tj

, ∀j. By interchanging i and j in (8), we are also able to conclude that Tj ≥ Ti

, ∀i.Hence, and without further considerations, we obtain that Ti = Tj = T

for all i and for all j, which is to say that individuals do not get any insurancein the one-period game with private information.

The two-period problem with symmetric information.

Consider now the two-period version of the above environments.Here the contract is simply given by functions T :Y → < and T ′:Y ×Y →

< or schedules Ti, Tij.The Pareto problem under public information is the solution to the fol-

lowing problem:

maxTi,Tij

∑

i

πi

[(y − Ti) + β

∑

j

πj(y − Tij)

](9)

subject to the consistency requirement:

∑

i

πi

[u(yi + Ti) + β

∑

j

πju(yj + Tij)

]= w (10)

9

where Tij is the transfer from the risk neutral agent to the risk averseagent when the first period state was i and the second period state is j.

We thus allow agents to engage in long-term arrangements. That is: weallow for transfer to depend on current and past outcomes. Notice thatTij = Tj for all i is a particular case of the transfer system that the economyis allowed to choose. (That is: it is the case of no long-term arrangements).

Also, this formulation assumes that both agents have the same discountfactor 0 < β < 1.

Ignoring nonnegativity constraints (or with a negative exponential utilityfor the risk averse agent), the solution to the problem can be characterizedby:

−βπiπj + λβπiπju′(yj + Tij) = 0 ⇔ u′(yj + Tij) =

1

λ(11)

−πi + λπiu′(yi + Ti) = 0 ⇔ u′(yi + Ti) =

1

λ(12)

and hence we can conclude that:

yi + Ti = yj + Tij = c ∀i, j (13)

where the fact that we get exactly ci = cj has to do with the assumptionof equal discount factors for both agents.

We thus obtain the full-insurance result, despite the fact that the game isrepeated one more time. Observe that (13) implies that allowing for secondperiod transfers to depend on the first period state is irrelevant: the left-hand-side of this equation says that the same level of (constant) consumption canbe obtained with repeated static arrangements.

Bottom line: repetition does not add anything. Exactly the same alloca-

tion can be achieved by repeating the static contract twice.

The two-period problem with asymmetric information.Message space: Y .Allocations: T : Y → < and T ′ : Y × Y → <. Realization are iid. We

will use the following notation:T (yi)yi∈Y = Ti and T ′ (yi, yj)(yi,yj)∈Y ×Y

= Tij

You have to tell that a modified version of the Revelation Principle holds.You have to make sure it is optimal to reveal truthfully at any information

10

node. The first IC below is standard. The second imposes that the agentwon’t lie in the first period, given that he won’t lie in the last.

The allocation is what is commonly referred to as the contract.On theother hand, the Pareto problem under private information is the solution to:

maxTi,Tij

∑

i

πi

[(y − Ti) + β

∑

j

πj(y − Tij)

](14)

subject to:

∑

i

πi

[u(yi + Ti) + β

∑

j

πju(yj + Tij)

]= w (15)

u(yj + Tij) ≥ u(yj + Tik) ∀i, j, k (16)

u(yi + Ti) + β∑

j

πju(yj + Tij) ≥ (17)

≥ u(yi + Tk) + β∑

j

πju(yj + Tkj) ∀i, k

where (17) is the first period truth-telling constraint and (16) the secondperiod one. This last constraint says that it is always better to tell the truthin the second period, no matter the first period state (i), for any actualsecond period endowment (j) and for any potential first period report (k).

Following the same reasoning as before, we can interchange j and k in (16)and use the strict monotonicity of u (·) to conclude that Tij = Tik = Ti foreach i and for all j and k. That is, agents receive no insurance in the secondperiod. This is really a result we should expect, since the second periodis essentially like a static one-shot arrangement: there is no next period tomake its outcome contingent on this period’s reports.

Using this result, we can simplify the notation by defining T ′i ≡ Tij , the

second period transfer as a function of the first period state only. Addi-tionally, since second period transfers are independent of the second periodreports, the constraint (16) actually becomes irrelevant under this notation.

The above problem can thus be rewritten as:

11

maxTi,T

′

i

∑

i

πi

[(y − Ti) + β

∑

j

πj(y − T ′i )

](18)

subject to:

∑

i

πi

[u(yi + Ti) + β

∑

j

πju(yj + T ′i )

]= w (19)

u(yi + Ti) + β∑

j

πju(yj + T ′i ) ≥ (20)

≥ u(yi + Tk) + β∑

j

πju(yj + T ′k) ∀i, k.

If we prevent long-term arrangements in this problem, by constrainingT ′

i = T ′k = T ′ ∀i, k , then the summations in (20) cancel. We can again

interchange i and k in this constraint and, by using u′ > 0, we conclude thatTi = Tk and so agents get no insurance.

Therefore, we have to allow for long-term arrangements for some insur-ance to be possible here.

Consider a particular case of this problem. We assume there are twostates only, with y1 > y2.

The incentive constraint (20) can be now written as:

u(y1 + T1) + β∑

j

πju(yj + T ′1) ≥ u(y1 + T2) + β

∑

j

πju(yj + T ′2) (21)

u(y2 + T2) + β∑

j

πju(yj + T ′2) ≥ u(y2 + T1) + β

∑

j

πju(yj + T ′1).(22)

It is also useful to rewrite the above problem as one of choosing directlyutility levels (current and future) instead of transfers. Hence, define thefollowing variables (for i = 1, 2):

ui ≡ u(yi + Ti)

ci ≡ yi + Ti

w′i ≡

∑

j

πju(yj + T ′i )

12

and define also c ≡ u−1 (·), the level of consumption required to attain agiven level of utility. From the properties of u we can readily conclude thatc′ > 0 and c′′ > 0.

The original choice variables in terms of the transformed problem can berecovered by (for i = 1, 2):

Ti = c(ui) − yi

T ′i ≡ V (w′

i)

where we can conclude from the properties of c that V ′ > 0 and V ′′ > 0.The Pareto problem written in terms of choosing (u1, u2, w

′1, w

′2) instead

of (T1, T2, T′1, T

′2) is then:

maxu1,u2,w′

1,w′

2

∑

i

πi [y − c(ui) + yi + β (y − V (w′i))] (23)

subject to:

∑

i

πi (ui + βw′i) = w (24)

u1 + βw′1 ≥ u (y1 + c(u2) − y2) + βw′

2 (25)

u2 + βw′2 ≥ u (y2 + c(u1) − y1) + βw′

1 (26)

or still, by defining ∆ ≡ y1 − y2 > 0 (y and yi are constants):

minu1,u2,w′

1,w′

2

∑

i

πi [c(ui) + βV (w′i)] (27)

subject to:

∑

i

πi (ui + βw′i) = w (28)

u1 + βw′1 ≥ u (c(u2) + ∆) + βw′

2 (29)

u2 + βw′2 ≥ u (c(u1) − ∆) + βw′

1 (30)

where we will attach a multiplier λ to (28) and multipliers µ1 and µ2 tothe IC constraints, respectively (29) and (30). It follows from the propertiesof c and V that the objective function in problem P is strictly convex.

The two benchmark solutions in terms of this transformed problem are:

13

• Full-insurance:

When the endowments are publicly observable, the two IC constraintsbecome irrelevant and the (interior) solution is u1 = u2 (implying c1 = c2,since u′′ < 0) and w′

1 = w′2.

• No insurance:

When we constrain transfers to be noncontingent, i.e. T1 = T2 = T , thesolution is trivially given by c1 = y1 + T and c2 = y2 + T , which means thatc1 = c2 + ∆.

Therefore, in the intermediate case where endowments are private infor-mation but we allow for long-term arrangements, we also expect an interme-diate solution c1 such that cFI

1 = c2 < c1 < cNI1 = c2 + ∆.

This is exactly the content of the next proposition.In the solution to problem P, c1 > c2, w

′1 > w′

2, (29) binds and (30) isslack.

w′1 > w′

2 and (29) binding imply that c1 < c2 + ∆, as we want to show.The proof follows in 5 steps.

Step 1. Both (29) and (30) cannot be slack.

Suppose they are both slack. Then the solution to problem P can beobtained without these two constraints. However, as we saw above, thisyields the full-insurance solution, u1 = u2 and w′

1 = w′2. By replacing this

solution in (29), we find that u(c1) > u(c1 + ∆), a contradiction since ∆ > 0and u′ > 0.

Step 2. (29) must bind.

By contradiction, assume (29) is slack. Then (30) must bind by Step 1.Hence, (29) and (30) together imply that w′

2 ≥ w′1. Otherwise, if w′

1 > w′2,

we could slightly increase w′2 and decrease w′

1 such that∑2

i=1 πiw′i is constant,

so (28) and both (29) and (30) still hold. Strict convexity of V would in turnimply that we were not at a minimizer in the first place, a contradiction.

Also, if (29) is slack we can conclude that u2 ≥ u1. Otherwise, if u1 > u2,we could slightly increase u2 and decrease u1 such that

∑2i=1 πiui is constant,

14

so (28) and both (29) and (30) still hold. Strict convexity of c would in turnimply that we were not at a minimizer in the first place, a contradiction.

Therefore, w′2 ≥ w′

1 and u2 ≥ u1 if (29) is slack.

However, if (29) is slack, then:

u1 + βw′1 > u (c(u2) + ∆) + βw′

2

≥ u (c(u1) + ∆) + βw′1

> u (c(u1)) + βw′1

where the first step follows from c′ > 0 and since w′2 ≥ w′

1 and u2 ≥ u1

and the second step follows from ∆ > 0 and u′ > 0. This is obviously acontradiction, since u (c(u1)) = u1.

Hence, (29) must bind, whereas (30) may or may not bind.

Suppose for now that (30) actually binds.

Step 3. u1 > u2.

The Lagrangean:

∑

i

πi [c(ui) + βV (w′i)] + λ

[∑

i

πi (ui + βw′i) − w

]+

µ1 [u1 + βw′1 − u (c(u2) + ∆) − βw′

2] + µ2 [u2 + βw′2 − u (c(u1) − ∆) − βw′

1]

The first order conditions for problem P are:

(u1) : −πic′(u1) + λπ1 + µ1 − µ2u

′(c1 − ∆)c′(u1) = 0 (31)

(u2) : −πic′(u2) + λπ2 + µ2 − µ1u

′(c2 + ∆)c′(u2) = 0 (32)

(w′1) : −π1V

′(w′1) + λπ1 + µ1 − µ2 = 0 (33)

(w′2) : −π2V

′(w′2) + λπ2 + µ2 − µ1 = 0. (34)

Notice that c′(u1) = 1u′(c1)

.

By adding (29) and (30):

15

u1 + u2 = u(c2 + ∆) + u(c1 − ∆). (35)

Define f(x) ≡ u(c2 + x) + u(c1 − x). Some useful properties of f arethat it is strictly concave since f ′(x) = u′(c2 + x) − u′(c1 − x) and f ′′(x) =u′′(c2 + x) + u′′(c1 − x) < 0 and also that f(0) = f(∆), since f(0) = u1 + u2.Hence, f(x) must look like:

which implies that we must have f ′(0) > 0 (otherwise we violate strictconcavity).

But then f ′(0) = −u′(c1) + u′(c2) > 0 implies that

u′(c2) > u′(c1).

Since u′′ < 0, u′ is decresasing and thus c1 > c2. In turn, by monotonicityof u, we have that u1 > u2, which is what we wanted to show.

Step 4. w′1 > w′

2.

From (??) and (??) we get:

c′(u1) = λ+1

π1

[µ1 − µ2

u′(c1 − ∆)

u′(c1)

](36)

c′(u2) = λ+1

π2

[µ2 − µ1

u′(c2 + ∆)

u′(c2)

]. (37)

Since u1 > u2 from Step 3, c′′ > 0 implies that c′(u1) > c′(u2) and this isequivalent to:

1

π1

[µ1 − µ2

u′(c1 − ∆)

u′(c1)

]>

1

π2

[µ2 − µ1

u′(c2 + ∆)

u′(c2)

](38)

16

where u′(c1−∆)u′(c1)

> 1 and u′(c2+∆)u′(c2)

< 1 by virtue of strict concavity of u.

Since µ1 ≤ µ2 makes the LHS of (??) negative and the RHS positive, theonly way this inequality may hold is when µ2 < µ1.

Using this fact on (??) and (??) we conclude that V ′(w′1) > λ > V ′(w′

2),and from strict convexity of V we finally get that w′

1 > w′2.

Now, recall we concluded that u1 > u2 and w′1 > w′

2 by assuming that(30) actually binds.

If it slacks instead we have µ2 = 0. However, we can check by replacingµ2 = 0 in the proofs of steps 3 and 4 that they will still follow, and hence wehave u1 > u2 and w′

1 > w′2 irrespective of (30) being binding or slack.

Step 5. (30) is slack.

Suppose instead that (30) binds.Then we saw that, by adding (29) and (30) we get:

u1 + u2 = u(c2 + ∆) + u(c1 − ∆) (39)

which is equivalent to:

u(c2)− u(c2 − (c2 + ∆− c1)) = u(c2 + ∆)− u(c2 + ∆ − (c2 + ∆ − c1)) (40)

Define ∆′ ≡ c2 + ∆ − c1. Then this equation can rewritten as:

u(c2) − u(c2 − ∆′) = u(c2 + ∆) − u(c2 + ∆ − ∆′) (41)

Recall from our previous remark that w′1 > w′

2 and (29) binding implythat c1 < c2 + ∆, and so ∆′ > 0.

However, (??) contradicts strict concavity of u, as can be seen from thenext figure:

and hence (30) must be slack.

17

This completes the proof of the Proposition.Note that the above Proposition says that truthful revelation in the

present entails giving a smaller transfer in the future if an agent revealslow endowment today (truthfully or not). That is, w′

1 > w′2 means T ′

1 > T ′2.

End of Second Class

18

Borrowing and Lending Scheme.

A relevant question is whether the apparent similarity between the effi-cient insurance contract and the pure borrowing and lending scheme actuallymeans that the two are equivalent. It turns out that they are not, whichmeans the pure borrowing and lending scheme is not optimal.

The pure borrowing and lending scheme would equalize marginal utilityof consumption across the two periods, that is:

u′(ci) = β(1 + r)∑

j

πju′(cij) (42)

where i and j are the states in the first and second period.By evaluating (??) at first period states i = 1, 2

u′(c1)

u′(c2)=

∑j πju

′(c1j)∑j πju′(c2j)

. (43)

Recall that w′i ≡

∑j πju(yj + V (w′

i)). By totally differentiating:

dw′i = dV

∑

j

πju′

(yj + V (w′i))

we get V ′(w′i) =

[∑j πju

′(cij)]−1

, and so (??) can be rewritten as:

u′(c1)

u′(c2)=V ′(w′

2)

V ′(w′1)

⇔V ′(w′

1)

c′ (u1)=V ′(w′

2)

c′ (u2).

In the efficient contract, we have (by replacing µ2 = 0 in the first orderconditions for problem P, (??)-(??)):

c′ (u1) = λ+µ1

π1

c′ (u2) = λ−µ1

π2

u′ (c2 + ∆)

u′ (c2)

V ′(w′1) = λ+

µ1

π1

V ′(w′2) = λ−

µ1

π2

.

19

Hence, c′ (u1) = V ′(w′1) and c′ (u2) > V ′(w′

2), implying that the efficientcontract is characterized by:

V ′(w′1)

c′ (u1)>V ′(w′

2)

c′ (u2)

where we had an equality in the pure borrowing and lending scheme.The departure from efficiency in the pure borrowing and lending scheme

occurs because agents would like to transfer income from the low endow-ment state to the high endowment one. Instead, the efficient contract makesconsumption in the bad state relatively less desirable so that agents havethe correct incentives not to lie when y2 occurs in order to obtain the hightransfer.

An illustration can be given for β (1 + r) = 1.In this case, the borrowing and lending scheme is characterized by

u′ (c1) = (V ′(w′1))

−1

u′ (c2) = (V ′(w′2))

−1

and the efficient contract by

u′ (c1) = (V ′ (w′1))

−1

u′ (c2) < (V ′(w′2))

−1.

If agents were allowed to borrow or lend on top of the efficient contract,they would rather lend part of the transfer they get in the bad state (whilethe borrowing and lending possibility would not be used in the good state).The borrowing and lending scheme does not provide enough insurance sinceindividuals get an higher c2 in the efficient contract.

An alternative way to show the nonequivalence between the optimal andthe borrowing and lending contracts is to note that the solution to the latteris obtained by replacing the incentive compatibility constraint in the Paretoproblem by a “zero expected transfers” constraint: T1 +βT ′

1 = T2 +βT ′2 = 0.

It can be shown that the incentive constraints for both states are satisfiedon inequality when agents can freely borrow or lend at rate r, which impliesthat we can improve on this allocation by, namely, increasing c2 until theincentive constraint for the good state is satisfied with equality.

20

1 The infinite horizon case - The study of

wealth distribution.

A final note concerns the study of wealth distribution. Since it is appropriateto think of expected utility as being approximately equal to wealth, thissetup can be used to study the properties of the distribution of wealth acrossagents. Notice, for instance, that in the efficient contract incentives providean intertemporal dependence of expected utilities: if an agent reports a lowendowment today, expected utility must be lower, and conversely if an agentreports an high endowment.

A question of interest is to consider the infinite horizon version of theprevious setup and to investigate whether there exists a limiting distributionof expected utilities. We know that this distribution, if it exists, will not bedegenerate since different agents have different histories of shocks.

The full insurance contract when the time horizon is infinite yieldsw = u (c) + βw or w = u(c)

1−β.

A parametric case that is widely used since it provides a closed-formsolution is the negative exponential one. That is, u(c) = −e−Ac with A > 0and c ∈ <, implying that consumption has a logarithmic form: c (u) =− 1

Aln (−u).

Sequence Problem.The risk-averse agent is offered some utility ω.

We use yt (yt) to denote the report an agent plans (at date 0) to giveabout his date t endowment if he has actually experienced the history yt =y0, y1, ..., yt.

A reporting strategy is thus y = yt (yt)∞t=0, where for all t, yt :

Y t+1 → Y .

The truthful reporting strategy is is denoted by y∗ = y∗t (yt)∞t=0 where

y∗t (yt) = yt for all t and yt ∈ Y t+1.

Let T (yt) denote the transfer that the individual w receives at date t onthe basis of the reporting history yt.

21

Let −e−A(yt+T(yt)) = −e−A[yt−yt]e−A(yt+T(yt)) ≡ e−A[yt−yt]u [yt] be theutility he receives from his consumption.

Thus the principal will assign a sequence u = u (yt)∞t=0 to the risk-averse

agent.

If the principal chooses the sequence u and the agent chooses the reportingstrategy y , then the agent receives a total discounted expected utility givenby

U (u, y) =∞∑

t=0

∫

Y t+1

βtut

[yt

(yt

)]e−A[yt−yt]dµt+1.

Define an allocation as a sequence u such that

w = U (u, y∗)

(i.e. the allocation awards utility w if truthful reporting is chosen)and

U (u, y∗) ≥ U (u, y) ∀y.

(truthful reporting is optimal for the agent).

The principal will choose the allocation u that minimizes

∞∑

t=0

βt

∫

Y t+1

c[u

(w, yt

)]dµt+1.

Recursive Formulation.Think now of the principal as choosing a sequence of functions (ft, gt) ,

functions of (w, y).

ft (w, y) is the current utility an agent receives if his expected utilityentitlement from t on is w and he announces the income y.

wt (w, y) is the expected utlity entitlement an agent receives if his ex-pected utility entitlement from t on is w and he announces the income y.

That is, we can think of the planner as summarizing a con-sumer’s entire report history and his initial entitlement in a single

22

number w that represents his expected utility entitlement from thecurrent period on.

Letσ = ft, gt

∞t=0

denote a sequence of such functions.

A sequence σ defines a sequence u as follows. Let the sequence wt∞t=0

solve the difference equation wt+1 = gt (wt, yt), with initial value w0. Then

ut

(w0, y

t)

= ft

[wt

(w0, y

t−1), yt

]

In other words, a sequence σ generates a plan u. Do the sequencething.

We call the sequence σ an allocation rule if it satisfies the two followingconditions:∀t and ∀w

w =2∑

i=1

πi [ft (w, yi) + βgt (w, yi)] ,

ft (w, y1) + βgt (w, y1) ≥ ft (w, y2) e−A[y1−y2] + βgt (w, y2)

ft (w, y2) + βgt (w, y2) ≥ ft (w, y1) e−A[y2−y1] + βgt (w, y1)

Obviously an allocation rule will also generate a cost for the principal,given by

V (w0) =2∑

i=1

πic [f0 (w0, yi)] +∞∑

t=1

βt

2∑

i=1

πic [ft (gt−1 (wt−1, yt−1) , yi)]

V (w0) =2∑

i=1

πi c [f0 (w0, yi)] + βV (g0 (w0, yi))

What we can prove in this environment:a) If the allocation u attains w with total cost C, then there is an alloca-

tion rule σ that attains w with total cost C.

23

b) Suppose the allocation rule σ that attains w with total cost C and u

is the utility plan generated by σ. Then u is an allocation, and u attains wwith total cost C.

c) The function V (w), that gives the minimum cost of achieving a givenvalue w, solves the following Bellman equation.

d) The functions ft, gt are stationary. That is ft = f and gt = g for everyt.

We can characterize the efficient contract, heuristically from the finitehorizon setup, as the solution to the following Bellman equation:

V (w) = minu1,u2,w′

1,w′

2

∑

i

πi [c(ui) + βV (w′i)] (44)

subject to:

∑

i

πi (ui + βw′i) = w (45)

u1 + βw′1 ≥ u (c(u2) + ∆) + βw′

2 (46)

u2 + βw′2 ≥ u (c(u1) − ∆) + βw′

1. (47)

Here you have to say that: f (w, y1) = u1, f (w, y2) = u2, g (w, y1) = w′2,

g (w, y2) = w′2.

In the finite horizon problem, the form of the value function was knownto be given by w′

i ≡∑

j πju(yj +V (w′i)), whereas we do not know the general

form of V (w) in the above problem.Some care is needed to formally justify the legitimacy of using the recur-

sive formulation above.

Two interpretation of the above problem were given in the literature, bothleading to this same formulation. The setup used in each of them differs insome respects:

24

• [Thomas and Worral (JET, 1990)] Two agents, one risk neutral andthe other risk averse, leading to a principal-agent formulation. Therisk-averse future utility becomes arbitrarily negative with probabilityone. The borrower gets deeper and deeper into debt (exponential util-ity allows for negative consumption), and consumption moves down asdebt increases. The contract is therefore not very good at stabilizingconsumption over time; nevertheless what apperas to be happening isthat making future utilities low reduces the cost of inducing incentivecompatibility, which is obtained by variations in future utility. So sta-bility in consumption in the initial periods is obtained at a cost ofvariation in consumption over time. Thomas and Worral show that asβ → 1, the Second-Best Pareto frontier or (Costrained-Pareto optimalutilities) converge pointwise to the first-best Pareto Frontier (first-bestutilitites), but not uniformly. The principal becomes richer andricher. The agent becomes poorer and poorer. Crucial rolefor negative consumption. Ayiagari & Alvarez impose non-negative consumption and obtain a non-degenerate limitingdistribution.

• [Green (1987)] Large number of risk averse agents plus an intermediarywhich can borrow/lend from some outside source at the rate β−1 − 1 .The risk averse agents do not have access to neither a storing technologynor to (say, foreign) borrowing/lending possibilities. The communityof individuals is rich at the beginning, because w > −1, but it becomespoor over time. That implies that the intermediary must be borrowingfrom the outside source at the beginning of time and then paying back.

Differently from Atkeson and Lucas, a period-by-period feasi-bility constraint is not imposed.

An educated guess for V is of the same form as c, that is,

V (w) = κ1 −1

κ0

ln (−w)

.Since V is strictly convex and strictly increasing, we can use the previous

results to conclude that (??) binds and (??) is slack, and can be ignored.After replacing the guess for V in the Bellman equation, the first order con-ditions are: (maximize the negative of the objective function)

25

π1

Au1

+ λπ1 + µ = 0

π2

Au2

+ λπ2 − µe−A∆ = 0

π1

κ0w′1

+ λπ1 + µ = 0

π2

κ0w′2

+ λπ2 − µ = 0.

***********************************************************************************Adding the first two equations (multiply and divide the two conditions

by u1 and u2, respectively):

π1

A+ λπ1u1 + µu1 +

π2

A+ λπ2u2 − u2µe

−A∆ = 0.

and thus:

1

A+ λ (π1u1 + π2u2) + µ

(u1 − u2e

−A∆)

= 0

Adding the last two equations (multiply the two conditions by w′1 and

w′

2, respectively):

βπ1

k0

+ λβπ1w′1 + βµw′

1 +βπ2

k0

+ λβπ2w′

2 − µβw′

2 = 0

and thus:

β

κ0

+ λβ (π1w′1 + π2w

′2) + βµ (w′

1 − w′2) = 0.

Adding the two resulting equations:

1

A+β

k0

+ λ

[∑

i

πi (ui + βw′i)

]+ µ [u1 + βw′

1 − u (c(u2) + ∆) − βw′2] = 0.

Using the incentive compatible and promise keeping constraints:

1

A+β

κ0

+ λw = 0

26

and we can solve for

λ = −1

w

(1

A+β

κ0

)

***********************************************************************************By the Envelope theorem, V ′ (w) = λ.Thus

−1

wκ0

= −1

w

(1

A+β

κ0

)

and thusκ0 = A (1 − β) .

We have obtained the first parameter of the value function.And also

λ = −1

w

1

A (1 − β)

***********************************************************************************

Now we want to figure out the parameter µ.Replacing κ0 in the first order conditions:

u1 = −π1

A (λπ1 + µ)(48)

u2 = −π2

A (λπ2 − µe−A∆)(49)

w′1 = −

π1

κ0 (λπ1 + µ)(50)

w′2 = −

π2

κ0 (λπ2 − µ)(51)

and replacing these in the promise keeping constraint:

−π2

1

A (λπ1 + µ)−

π22

A (λπ2 − µe−A∆)− β

[π2

1

κ0 (λπ1 + µ)+

π22

κ0 (λπ2 − µ)

]= w.

Divide everywhere by w:

−π2

1

A (λπ1w + µw)−

π22

A (λπ2w − wµe−A∆)−β

[π2

1

κ0 (λπ1w + µw)+

π22

κ0 (λπ2w − µw)

]= 1.

27

To solve for µ, we guess it to be of the same form as λ: µ = − 1wµ∗ with

µ∗ > 0 being a constant. Now use λ = − 1w

1A(1−β)

and µ = − 1wµ∗. Replacing

this guess above,

−π2

1

A(

−π1

A(1−β)− µ∗

)− π22

A(

−π2

A(1−β)+ µ∗e−A∆

)−β

π21

κ0

(−π1

A(1−β)− µ∗

) +π2

2

κ0

(−π2

A(1−β)+ µ∗

)

= 1

and (since k0 = A (1 −B))

−π2

1

A(

−π1−µ∗k0

A(1−β)

)− π22

A(

−π2+k0µ∗e−A∆

A(1−β)

)−β

π21

κ0

(−π1−µ∗k0

A(1−β)

) +π2

2

κ0

(−π2+µ∗k0

A(1−β)

)

= 1

or, equivalently,

π21 (1 − β)

π1 + µ∗κ0

+π2

2 (1 − β)

π2 − k0µ∗e−A∆+ β

[π2

1

π1 + µ∗k0

+π2

2

π2 − µ∗κ0

]

or

π21

π1 + µ∗κ0

+π2

2 (1 − β)

π2 − µ∗κ0e−A∆+ β

π22

π2 − µ∗κ0

= 1. (52)

***********************************************************************************Define

φ (µ∗) ≡π2

1

π1 + µ∗κ0

+π2

2 (1 − β)

π2 − µ∗κ0e−A∆+ β

π22

π2 − µ∗κ0

and note that φ (0) = 1 , φ′

(0) < 0 and that φ′′ > 0 since φ is the sum ofthree convex functions in µ∗. Also φ (µ∗) → +∞ as µ∗ → π2

κ0> 0.

The graph of this function φ (µ∗) thus looks like:

The function φ (µ∗) is continuous. It follows that there exists 0 < µ∗ < π2

κ0

28

that solves (??). Note that µ∗ = 0 (and hence µ = 0 as well) is not asolution to (??) since it implies that the IC constraint is slack and thus thefull-insurance obtains. We know that it cannot be.

***********************************************************************************Using µ = − 1

wµ∗ to solve for current and expected future utilities in

(??)-(??):

u1 =π1w (1 − β)

π1 + µ∗κ0

(53)

u2 =π2w (1 − β)

π2 − µ∗κ0e−A∆(54)

w′1 =

π1w

π1 + µ∗κ0

(55)

w′2 =

π2w

π2 − µ∗κ0

(56)

and it follows thatu1 > u2

and thatw′

2 < w < w′1

.We can replace (??)-(??) in the Bellman equation and solve for κ1. We

do not do it here.***********************************************************************************To characterize the limiting distribution of expected utilities, observe

that:

wt+1 =

π1wt

π1+µ∗κ0if yt = y1

π2wt

π2−µ∗κ0if yt = y2

and so,

ln (−wt+1) =

ln (−wt) + ln π1

π1+µ∗κ0if yt = y1

ln (−wt) + ln π2

π2−µ∗κ0if yt = y2

.

We can write that

ln (−wt+1) = ln (−wt) + ε , E (ε) = π1 lnπ1

π1 + µ∗κ0

+ π2 lnπ2

π2 − µ∗κ0

29

Therefore,

E ln (−w) = π1 ln (−w′1) + π2 ln (−w′

2)

= ln (−w) + π1 lnπ1

π1 + µ∗κ0

+ π2 lnπ2

π2 − µ∗κ0

≡ ln (−w) + ψ (µ∗) . (57)

Since endowment shocks are i.i.d., ln (−wt) is i.i.d. as well and ln (−wt)is a random walk with positive drift (ψ (µ∗) > 0). The drift is positive since:

ψ′ (µ∗) = −π1κ0

π1 + µ∗κ0

+π2κ0

π2 − µ∗κ0

⇒ ψ′ (0) = 0

ψ′′ (µ∗) =π1κ

20

π1 + µ∗κ0

+π2κ

20

π2 − µ∗κ0

> 0.

Since ln (−wt) is a random walk with positive drift (sub-martingale), itfollows that ln (−wt) → +∞ as t → +∞ and hence that wt → −∞ andut → −∞.

We thus conclude that there does not exist a stationary distribution ofexpected utilities, as risk-averse agents get increasingly poorer over time,while the risk neutral agent gets increasingly richer (so that the aggregateresource constraint keeps balanced).

The results will change dramatically if we impose c ≥ 0 (or some otherlower bound for consumption). In particular, a stationary distribution forutilities obtains in this case (see Aiyagari and Alvarez (1995)).

A final point is due about the relationship between the finite horizon andthe infinite horizon setups.

In both cases, β = 0 leads to no insurance. This follows directly from theincentive compatible constraints.

However, as β → 1 we converge to the full-insurance result in the infinitehorizon setup, since we can check from (??)-(??) that w′

1 → w, w′2 → w and

u1

u2→ 1.Radner and Townsend provide an interpretation of this result. They set

β = 1 and let T → ∞. The economy is then able to sustain full-insuranceallocations as the time horizon extends to infinity. An important point is thatagents only care about in how many periods they will receive a compensationwhen signing the contract. If β < 1, the argument of Radner and Townsenddoes not work anymore.

30