EC537 Microeconomic Theory for Research Students, Part II ...econ.lse.ac.uk/staff/lfelli/teach/EC537 Slides Lecture 3.pdf · EC537 Microeconomic Theory for Research Students, Part

EC537 Microeconomic Theory for Research Students,Part II: Lecture 3

Leonardo Felli

CLM.G.4

22 November 2011

The Hold-Up Problem (Hart and Moore 1988)

A buyer and seller want to trade one indivisible unit of a good at afuture date.

Denote q ∈ {0, 1} the probability that trade occurs and p the tradingprice.

Let v denote the buyer’s valuation for the good, and c the seller’sproduction cost.

We assume that v and c are uncertain.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 322 November 2011 2 / 63

The Hold-Up Problem (2)

In particular, v is such that:

v ∈ {v , v}, v < v , Pr{v = v} = j

The buyer can increase j by undertaking an ex-ante investment at the(strictly convex) costs ψ(j). The buyer’s ex-ante payoff is then:

v q − p − ψ(j)

Moreover, c is such that:

c ∈ {c , c}, c < c , Pr{c = c} = i

The seller can increase i by undertaking an ex-ante investment at the(strictly convex) costs φ(i). The seller’s ex-ante payoff is then:

p − c q − φ(i)



Assume that:c > v > c > v

Timing:

The parties write an ex-ante contract.

The parties choose simultaneously their investments (i , j).

Both parties learn the state of nature (v , c).

The parties renegotiate the ex-ante contract if they want to.

Trade may or may not occur.



Notice that the gains-from-trade are positive only if v = v and c = ctherefore ex-post efficiency requires:

q∗ =

{1 if v = v and c = c0 otherwise

Given this trading rule ex-ante efficiency requires that the ex-anteinvestments are such that:

maxi ,j

i j (v − c)− φ(i)− ψ(j)

The first order conditions imply:

i∗ (v − c) = ψ′(j∗)

j∗ (v − c) = φ′(i∗)



Assume now that (v , c) and (i , j) are observable but not verifiable tothe parties to the contract.

In other words, the parties ex-ante cannot write a contract contingenton (v , c) and (i , j).

That is the ex-ante contract can only specify a price (transfer)contingent on whether q ∈ {0, 1} (a price for not trading and a pricefor trading):

(p0, p1)

The court can only assess whether trade occurred (the court cannotassess who did not perform ex-post).

This is the incomplete contract assumption.



Of course, once the state of nature (v , c) is realized the two partiescan renegotiate, if they want, the terms of the ex-ante contract(p0, p1).

Notice that at the renegotiation stage the contract incompletenessdoes not play any role: no need to contract on (v , c) and (i , j).

Therefore Coase Theorem applies and this negotiation is efficient.

Assume that the following extensive form applies to the renegotiationstage, known as contracting at will (double auction).

Both parties simultaneously and independently send one another newwritten offers:

(pB0 , p

B1 ), (pS

0 , pS1 )



Then both parties decide simultaneously and independently whetherto approve the trade.

Only if both approve then trade occurs: q = 1 (the court cannot forcethe parties to trade, freedom of contracts).

The court observes q and enforces:

if q = 1 the payment p1 or any other payment pk1 , k ∈ {B,S}, (dated

more recently than the ex-ante contract) that either party receivedfrom the counterpart and is willing to show the court;

if q = 0 the payment p0 or any other payment pk0 , k ∈ {B,S}, (dated

more recently than the ex-ante contract) that either party receivedfrom the counterpart and is willing to show the court.



Let (p1, p0) be the enforced contract (ex-ante one or renegotiated).

The buyer approves the trade if and only if:

v − p1 ≥ −p0, or v ≥ p1 − p0

The seller approves the trade if and only if:

p1 − c ≥ p0, or p1 − p0 ≥ c

In other words in equilibrium q = 1 if and only if

v ≥ p1 − p0 ≥ c



Result (Hart and Moore 1988)

The outcome of the renegotiation process is ex-post efficiency:

q = 1 iff v ≥ c

In other words:

if v = v or c = c then q = 0,

if instead v = v and c = c then q = 1.

Proof: As we have seen above if

v ≥ p1 − p0 ≥ c

then trade occurs.



Consider now the state of nature v = v and c = c and let’s computethe renegotiation prices.

The ex-ante contract (p1, p0) is such that only three cases arepossible:

case 1: v ≥ p1 − p0 ≥ c ,

case 2: p1 − p0 > v > c ,

case 3: v > c > p1 − p0.



In case 1: v ≥ p1 − p0 ≥ c trade occurs at price p1.

In this case necessarily:

pS1 > p1, pS

0 > p0,

p1 > pB1 , p0 > pB

0 .

But then neither the buyer nor the seller has an incentive to revealthe offers that they received.



In case 2: p1 − p0 > v > c , although trade is efficient, the buyer findsit too expensive and hence at the price p1 will not approve trade.

However the buyer will show the seller’s offer and approve trade if andonly if v − pS

1 ≥ −p0

In equilibrium the seller’s offer is such that:

pS1 = p0 + v

Since in this case the seller’s payoff is: pS1 − c = p0 + v − c > p0

Therefore p1 = pS1 and p0 = p0 and trade occurs: v = p1 − p0 > c .



In case 3: v > c > p1 − p0, although trade is efficient, the seller findstrade to be too costly for the price difference.

However the seller will show the buyer’s offer and approve trade if andonly if pB

1 − c ≥ p0

In equilibrium the buyer’s offer is such that:

pB1 = p0 + c

Since in this case the buyer’s payoff is: v − pB1 = v − c − p0 > −p0

Therefore p1 = pB1 and p0 = p0 and trade occurs: v > p1 − p0 = c .



Whenever the state of nature is such that v = v and c = c then(p1, p0) are such that:

v ≥ p1 − p0 ≥ c

and trade occurs q = 1.

We can now consider the ex-ante efficiency of the parties’ investmentdecision.

Notice that the buyer’s payoff in equilibrium is:

i j [v − (p1 − p0)]− p0 − ψ(j)

While the seller’s payoff in equilibrium is:

i j [(p1 − p0)− c] + p0 − φ(i)


Underinvestment

The equilibrium investments i∗∗ and j∗∗ are then characterized by thesolution to the following problems:

maxj

i j [v − (p1 − p0)]− p0 − ψ(j)

maxi

i j [(p1 − p0)− c] + p0 − φ(i)

The first order conditions of these problems are then:

i∗∗ [v − (p1 − p0)] = ψ′(j∗∗)

j∗∗ [(p1 − p0)− c] = φ′(i∗∗)

We can evaluate ex-ante efficiency by comparing the latter FOC andthe ex-ante efficiency conditions

i∗ (v − c) = ψ′(j∗), j∗ (v − c) = φ′(i∗)


Underinvestment (2)

Result (Hart and Moore 1988)

When contracts are incomplete ex-ante inefficiency may arise.

The parties’ investments choices are such that

j∗∗ ≤ j∗, i∗∗ ≤ i∗

with at least one of the inequality being strict.

Proof: Notice first that from the condition that guarantees trade:

v ≥ p1 − p0 ≥ c

we have that only one of the following three cases may arise:

v > p1 − p0 > c , v = p1 − p0 > c , v > p1 − p0 = c


Underinvestment (3)

In other words we have that:

v − (p1 − p0) ≤ v − c , (p1 − p0)− c ≤ v − c

with at least one of the two inequalities strict.

Assume now that i∗ > 0 and j∗ > 0 exists.

For this to be the case we need to impose conditions on φ′(i0), ψ′(j0),φ′′(i∗) and ψ′′(i∗).


Underinvestment (4)

Consider now the first case, from the FOC

j∗∗ [(p1 − p0)− c] = φ′(i∗∗)

we obtain using implicit function theorem:

0 <d i∗∗

d j=

(p1 − p0)− c

φ′′(i∗∗)≤ v − c

φ′′(i∗∗)

While from the FOC condition:

i∗∗ [v − (p1 − p0)] = ψ′(j∗∗)

we obtain:d i∗∗

d j=

ψ′′(j∗∗)

v − (p1 − p0)≥ ψ′′(j∗∗)

v − c> 0

With at least one of the inequalities strict.


Underinvestment (5)

In other words the situation can be represented in the following graph:

-

6i

j

qq................................................................

...........................................................

.............................................

...............................................

j∗∗ j∗

i∗∗

i∗

i∗ψ(j)

i∗φ(j)

i∗∗φ (j)

i∗∗ψ (j)


Free-Rider Problem

The basic intuition behind this result is the same as the one behindthe standard free-rider-problem.

In the absence of a complete ex-ante contract each party making theinvestment will be expropriated ex-post (at the renegotiation stage) ofthe returns from his/her investment.

Since by subgame perfection this party can forecast this expropriationthen at the investment stage the party will reduce at the margin hisinvestment.

The expropriation is what is usually known as the hold-up problem.

Therefore the incompleteness of contracts leads to an ex-anteinefficiency that takes the form of under-investment.


Foundations of Incompleteness:

What is an Incomplete Contract? A contract is incomplete if it is notas fully contingent on the state of the world (the resolution ofuncertainty about the future) as the parties to the contract might likeit to be.

The Literature offers three main reasons for contractualincompleteness:

Some aspects of the state of the world may not be common knowledgeor commonly observable: in particular to the enforcer (observable butnot verifiable).

Some aspects of the state may be unforeseen or indescribable by theparties in advance.

Even if certain aspects are foreseen, writing them into a contract maybe too costly.


Indescribability (Maskin and Tirole 1999):

Consider unforeseen or indescribable states of nature.

Result (Maskin and Tirole 1999)

If parties can assign a probability distribution to their possible futurepayoffs, then the fact that they cannot describe the possible physicalstates in advance is irrelevant to welfare. That is, the parties can devise acontract that leaves them no worse off than were they able to describe thephysical states ex ante.

It should be noted that this result does not mean that the parties can doas well as though they could write a fully contingent contract.


An Example:

Two agents 1 and 2.

They contemplate trading a single indivisible good that agent 1 willproduce and agent 2 will consume.

There are three dates:

Date 0: the two agents negotiate the contract.

Date 1: agent 1 chooses investment e1 determining the value v for 2and agent 2 chooses investment e2 determining the cost c for 1:

v ′(e1) > 0, c ′(e2) < 0

Date 2: uncertainty realized, production and trade.


Example (2)

Agent 1’s payoff:u1(p − c(e2)− e1)

Agent 2’s payoff:u2(v(e1)− p − e2)

Where p is the price of the good and u1 and u2 are vonNeumann-Morgenstern utility functions.

Investments (e1, e2) and (v , c) are common knowledge to 1 and 2 butnot verifiable.

A state of the world corresponds to the characteristics of the goodtogether with the properties of the intermediate input: the state isverifiable at date 2.


Example (3)

Efficiency: choice of e∗1 and e∗2 such that:

maxe1

v(e1)− e1, mine2

c(e2) + e2

Assume that production and trade are desirable:

v(e∗1)− e∗1 > c(e∗2) + e∗2

If parties can foresee the properties of e1 and e2 then the contractspecifies:

e1 and the penalty that 1 pays 2 if he fails to deliver on theseproperties,

e2 and the penalty that 2 pays 1 if he fails to deliver on theseproperties.


An Optimal Mechanism (Moore and Repullo 1988):

The thrust of the Irrelevance Result stated above is that even if nodescription is available ex-ante this should not affect the parties’welfare.

Consider therefore the following mechanism/contract signed at date 0and executed at date 2:

Stage 1: Agent 1 announces v while agent 2 announces c , clearly ingeneral it is possible that v 6= v(e1) and c 6= c(e2).

Stage 2: Agent 1 can challenge agent 2’s announcement.

If the challenge occurs agent 2 first pays the fine F to agent 1.

Then agent 1 offers agent 2 the choice between (p∗, q∗) and(p∗∗, q∗∗), where q∗, q∗∗ ∈ {0, 1} and q∗v − p∗ > q∗∗v − p∗∗


Optimal Mechanism (2)

Notice first if v = v(e1) then agent 2 will choose (q∗, p∗) since

q∗v − p∗ > q∗∗v − p∗∗

The challenge succeeds if agent 2 chooses (q∗∗, p∗∗) revealing thatagent 2 has lied.

In this case (q∗∗, p∗∗) is implemented: agent 1 produces and deliversq∗∗ units of the good for price p∗∗.

The mechanism then concludes.



The challenge fails if agent 2 chooses (q∗, p∗) revealing that agent 2has told the truth.

In this case (q∗, p∗) is implemented: agent 1 produces and delivers q∗

units of the good for price p∗.

In this case agent 1 must pay the fine 2F for having challengedunsuccessfully.


If agent 1 does not challenge the mechanism moves to Stage 3.



Stage 3: Agent 2 can challenge agent 1’s announcement.

If the challenge occurs agent 1 first pays the fine F to agent 2.

Then agent 2 offers agent 1 the choice between (p∗, q∗) and(p∗∗∗, q∗∗∗), where q∗, q∗∗∗ ∈ {0, 1} and p∗ − cq∗ > p∗∗∗ − q∗∗∗c

Once again if c = c(e2) then agent 2 will choose (q∗, p∗).

The challenge succeeds if agent 1 chooses (q∗∗∗, p∗∗∗) revealing thatagent 1 has lied.

In this case (q∗∗∗, p∗∗∗) is implemented: agent 1 produces anddelivers q∗∗∗ units of the good for price p∗∗∗.




The challenge fails if agent 1 chooses (q∗, p∗) revealing that agent 1has told the truth.

In this case (q∗, p∗) is implemented: agent 1 produces and delivers q∗

units of the good for price p∗.

In this case agent 2 must pay the fine 2F for having challengedunsuccessfully.


If agent 2 does not challenge the mechanism moves to Stage 4.



Stage 4: Agent 2 delivers the input with properties corresponding tothe realized state.

Agent 1 produces and delivers a unit of the good with characteristicscorresponding to the realized state.

Agent 2 pays the price p(v , c), where given a constant κ

p(v , c) = v + c + κ



Incentive Compatibility

Notice first that if v 6= v(e1) there exists a pair (p∗, q∗) and(p∗∗, q∗∗), such that

q∗v − p∗ > q∗∗v − p∗∗

q∗v(e1)− p∗ < q∗∗v(e1)− p∗∗

In other words, agent 1 successfully challenges agent 2 if and only ifagent 2 has lied.

Moreover, if F is big enough agent 1 has an incentive to challengesuccessfully.


Incentive Compatibility (2)

Conversely, agent 1 will never challenge if agent 2 has been truthful.

So agent 1 would expect any challenge to fail: q∗v(e1)− p∗ >q∗∗v(e1)− p∗∗, recall that 1 still collects F from 2 but pays 2F .

Agent 2 expects to be challenged and fined if and only if heannounces untruthfully.

Agent 2 therefore has the incentive to set v = v(e1).

Similarly, agent 1 has the incentive to set c = c(e2).


Individual Rationality

We can show that for both agents: ei = e∗i , i ∈ {1, 2} and theirpayoff is non-negative.

Given IC agent 1’s date 1 payoff is:

p(v(e1), c(e2))− c(e2)− e1

Substituting p(v , c) we get that agent 1’s problem is:

maxe1

v(e1) + κ− e1


Individual Rationality (2)

In other words e1 = e∗1 .

Similarly e2 = e∗2 .

Recall that:v(e∗1)− e∗1 − c(e∗2)− e∗2 > 0

We can then choose κ so that both agents obtain a strictly positivepayoff.


Renegotiation (Segal 1999, Hart and Moore 1999):

Consider the payment of the fine 2F by agent 1.

Clearly, this fine cannot be paid to agent 2, for a large enough Fagent 2 would then have an incentive to choose (p∗, q∗) even if 2 hasnot been truthful.

Therefore, 2F has to be paid to a third party.

Now consider the renegotiation proposal by agent 1 right before hehas to pay 2F : we share the fine 2F between each other.

The problem is that if agent 2 gets a large enough share than he stillhas an interest to let the challenge fail.


Risk Aversion (Maskin 2002):

The problem is to conceive a fine that punishes agent 1 withoutrewarding agent 2.

This is possible, in principle, when agent 1 is risk averse. Assume thatagent 2 is risk neutral.

Replace the fine 2F with the fine paid to agent 2:{G with probability 1/2

−G with probability 1/2


Risk Aversion (2)

Clearly a large enough G can be a very effective punishment for 1.

However, it is not a reward for agent 2.

Moreover, if uncertainty is realized as soon as the challenge failsex-post renegotiation has no reason to take place.

A straightforward argument shows that also ex-ante renegotation hasno reason to take place.


Undescribable Events (Al-Najjar, Anderlini and Felli 2006):

Consider now the third cause of contractual incompleteness: theimpossibility to describe events in a contract.

Consider a situation where the parties to a contract are:

able to understand the consequences and probabilities associated withthe environment in which they operate,

but are unable to describe adequately certain complex contingenciesthat may arise.


Example Tenure

The ex-ante description of the event an academic gets tenure.

This includes the full set of papers that deserves tenure: in principle afinite set.

However, for all intents and purposes the features that fully identifythe academic’s output that deserves tenure might be taken to belongto an infinite set.

Any finite description of the research output that deserves tenure willnot capture exactly the set of states of nature in which tenure is given.

The probability that the academic gets tenure with m publishedpapers may even be 1. However, surely the probability of gettingtenure with k < m published papers is in fact neither 0 nor 1.


What Can be Described

States are described by means of a language with a countable infinityof elementary statements.

Each elementary statement represents a particular feature that can beeither present (1) or not (0) in the description of a given state ofnature (the sky can be either blue or not blue).

The complete description of each state is a complex object itself.

Given a language to describe the states, it is natural and compellingto restrict attention to finite sentences in the language.


State Space and Probability

Consider a finite S ... unfortunately if does not work for our purposes:finitely many states can always be “separated” by looking at finitelymany of their constituent “characteristics”.

Consider a nice continuous, compact etc. S ... unfortunately it doesnot work for our purposes.

Assume Utility continuous in actions (money transfers). And thatExpected Utility is well defined

Then we can integrate U(a(s), s) w.r.t. s. But this means a(s) canbe approximated by a step function.

Hence a sequence of finite descriptions will approximate “first best”utilities. (Anderlini and Felli 1994).


State Space and Probability (2)

Consider a countably infinite S ... it might work for our purposesdepending on the probability measure.

A countably additive measure over S ... unfortunately does not work.

Then there exists a finite set of states that captures almost all theprobability mass .... we are back in the finite case.

A countable infinity of states, with an atomless measure does work.


The Risk-Sharing Problem

State space S, with element s.

Let S = Z ∪ Z and pZ = Pr(Z).

Two agents i = 1, 2. Ui (·, s) is state-dependent utility function:

U1(t, s) =

{V (1 + t) if s ∈ ZV (t) if s ∈ Z

U2(t, s) =

{V (−t) if s ∈ ZV (1− t) if s ∈ Z

V is increasing, strictly concave, twice continuously differentiable andInada.

Party 1 makes a take-it-or-leave-it offer of t(·) to 2


The Risk-Sharing Problem (2)

First best contract prescribes transfers t∗(s)

Up to a “measure zero” set of states t∗(·) satisfies.

t∗(s) =

{tZ if s ∈ ZtZ if s ∈ Z

With 1 + tZ = tZ , so that both are fully insured.

U1(t(s), s) = V (1 + tZ) = V (tZ) ∀ s ∈ SU2(t(s), s) = V (−tZ) = V (1− tZ) ∀ s ∈ S

And 2 is sitting on his participation constraint so that

pZV (−tZ) + (1− pZ)V (1− tZ) = pZV (0) + (1− pZ)V (1)


The State Space

The language to describe states consists of a countable infinity ofelementary statements (characteristics of a state).

Definition (State Space)

The state space S is a countably infinite set:

S = {s1, s2, . . . sn, . . .}

were sn is an infinite sequence of the type {s1n , . . . , s in, . . .} with s in ∈ {0, 1}for every i and n.

A complete description of a typical state sn is then an object of the type

010011011101011110 . . .


Probabilities

We define an atomless measures on S.

Definition (Density)

Given any Z ⊆ S define the density of Z as

µ(Z) = limN→∞

1

N

N∑n=1

IZ(sn) (1)

when the limit in (1) exists. Otherwise µ(Z) is left undefined (IZ = thecharacteristic function of Z).


Finite Additivity

The density of a set µ(Z) is its frequency within S:

any finite set of states will be assigned zero probability,

an infinite set consisting of say “every third state” will receive aprobability of 1/3.

Result (Finitely Additive Prob. Measure)

There exists a finitely additive probability measure µ over (S,Σ) such that

µ(B) = µ(B) ∀B ∈ D.


Finitely Definable Sets and Contracts

Define A(i , j) the set of states that have/do not have the i-th feature:

A(i , j) = {sn ∈ S such that s in = j}, j ∈ {0, 1}

Definition (Finitely Definable Sets)

Let A be the algebra of subsets of S of the type A(i , j).

A ∈ A can be described using finitely many elementary statements of thelanguage.


Finite Contracts

We limit attention to contracts that take finitely many possible values.

Definition (Finite Contracts)

A contract t(·) is finite if and only if it is measurable with respect to A.The set of finite contracts is denoted by F .


Computing Expected Utilities

Parties need to be able to evaluate their expected utilities and thereforethe (lim) probability measure:

µ(A ∩ Z), ∀A ∈ A

First result guarantees that we can construct S so that parties canevaluate µ(A), A ∈ A.

Result

There exists a state space S as defined above such that every A ∈ A has awell defined density µ(A) (A ⊆ D).


Intuition of the Proof:

Consider a (countably additive) uniform µ over {0, 1}N.

Let S = {s1, . . . , sn, . . .} be a typical realization of countably manyi.i.d. draws from µ on {0, 1}N.

Consider

A(i , j) = {sn ∈ S | s in = j}, j ∈ {0, 1}, i ∈ N

By the law of large numbers there exist a µ-measure 1 set of S suchthat for every A(i , j) ⊆ S:

limN→∞

1

N

N∑n=1

IA(i ,j)(sn) = µ(A(i , j)) = µ(A(i , j))

For example A(1, 0) = {sn ∈ [0, 1/2]}, µ(A(1, 0)) = 12 .


Well-Defined Frequencies

Consider now an event Z that has well-defined frequencies: it is such thatparties can evaluate µ(Z).

Definition (Well-Defined Frequencies)

The characteristic function IZ has well defined frequencies if

Z ∩ A ∈ D ∀ A ∈ A


Expected Utilities

Definition (Expected Utilities)

Consider S that satisfies the result above and Z that has well-definedfrequencies. The parties expected utilities are then:

EU1(t) =M∑i=1

V (1 + ti )µ[t−1(ti ) ∩ Z] +

+M∑i=1

V (ti )µ[t−1(ti ) ∩ Z ]

EU2(t) =M∑i=1

V (−ti )µ[t−1(ti ) ∩ Z] +

+M∑i=1

V (1− ti )µ[t−1(ti ) ∩ Z ]


Finite Invariance

We can now characterize the desired properties of Z.

First property: Z displays finite invariance if the percentage of states in Zis the same in every A ∈ A as in the entire S.

Definition (Finite Invariance)

The characteristic function IZ displays finite invariance if for every subsetA ⊆ S such that A ∈ A and µ(A) > 0,

µ(Z|A) = µ(Z)


Finite Invariance (2)

The characteristic function IZ displays finite invariance if the densitiesof the sets Z are the same, conditional on all finitely definable subsetsof A.

In other words, if IZ displays finite invariance knowing that s belongsto any finitely definable subset A does not help us to predict betterthe values that IZ will take.

Trivial cases of finite invariance:

Z = S, Z = ∅


Fine Variability

The crucial point is that in our state space there exist non-trivial finiteinvariant characteristic functions IZ (second property).

Definition (Fine Variability)

The characteristic function IZ displays fine variability if and only if it isfinite invariant and:

µ(Z) > 0, µ(Z) > 0.


Existence

Result

Let pZ ∈ [0, 1] and A ∈ A be given, with µ(A) > 0. There exists an Sand a set Z ⊂ S with characteristic function IZ that

has well defined frequencies,

displays finite invariance,

displays fine variability,

and is such thatµ(Z) = µ(Z|A) = pZ .

In other words, ... finite invariance does not limit actual variability.

Not true in the continuum.



For simplicity, let pZ = 1/2 and construct S as in the result above.

We set IZ(sn) equal to 0 or 1 with equal probability, and with i.i.ddraws across all the states sn.

The law of large numbers guarantees that there exist a measure 1 setof IZ constructed as above such that:

they have well-defined frequencies: µ(Z) =1

2

they exhibit finite invariance: µ(Z|A) =1

2,∀A ∈ A

they exhibit fine variability: µ(Z) = µ(Z) =1

2> 0.


Undescribable Events:

Consider the parties’ risk-sharing problem constrained to finite contracts.

maxt

EU1(t)

s.t. EU2(t) ≥ µ(Z)V (0) + µ(Z)V (1)t ∈ F

Result

There exist an S, µ and Z with µ(Z) ∈ (0, 1) with the followingproperties.

1 The optimal finite contract t∗∗ exists unique.

2 The characteristic function of Z is well defined in terms offrequencies.

3 The optimal finite contract t∗∗ is such that t∗∗(s) = 0,∀s ∈ S

In other words, no transfer occurs.



Construct S as in the result above and choose Z ∈ D so that IZ exhibitswell-defined frequencies, finite invariance and fine variability withpZ = µ(Z) as above.

By finite invariance, Z is such that any attempt to condition on a finitelydescribable set A leaves the parties with a set of states of which only afraction µ(Z) are in Z.

Risk aversion implies that any finite contract t that is not constant on S isdominated by another one that is constant (average).

Therefore t∗∗ exists unique and constant and party 2’s IR constraintsimply zero transfers.


Ultimate Incomplete Contract

The optimal finite contract yields the no-contract allocation: theultimate incomplete contract.

In the setting above agent 1’s expected utility is bounded away fromthe his full-insurance expected utility: the approximation result fails.

We have provided a formal model of undescribable events as eventsthat are fully understood but it is impossible to do them justice tryingto describe them in advance.


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