EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 - - PowerPoint PPT Presentation

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 a future date. Denote q ∈ {0, 1} the probability that trade occurs and p the trading price. Let v denote the buyer’s valuation for the good, and c the seller’s production cost. We assume that v and c are uncertain.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 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)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 3 / 63

The Hold-Up Problem (3)

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 4 / 63

The Hold-Up Problem (4)

Notice that the gains-from-trade are positive only if v = v and c = c therefore ex-post efficiency requires: q∗ = 1 if v = v and c = c

• therwise

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

i,j

i j (v − c) − φ(i) − ψ(j) The first order conditions imply: i∗ (v − c) = ψ′(j∗) j∗ (v − c) = φ′(i∗)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 5 / 63

The Hold-Up Problem (5)

Assume now that (v, c) and (i, j) are observable but not verifiable to the parties to the contract. In other words, the parties ex-ante cannot write a contract contingent

• n (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 price for trading): (p0, p1) The court can only assess whether trade occurred (the court cannot assess who did not perform ex-post). This is the incomplete contract assumption.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 6 / 63

The Hold-Up Problem (6)

Of course, once the state of nature (v, c) is realized the two parties can renegotiate, if they want, the terms of the ex-ante contract (p0, p1). Notice that at the renegotiation stage the contract incompleteness does 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 renegotiation stage, known as contracting at will (double auction). Both parties simultaneously and independently send one another new written offers: (pB

0 , pB 1 ),

(pS

0 , pS 1 )

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 7 / 63

The Hold-Up Problem (7)

Then both parties decide simultaneously and independently whether to approve the trade. Only if both approve then trade occurs: q = 1 (the court cannot force the parties to trade, freedom of contracts). The court observes q and enforces:

if q = 1 the payment p1 or any other payment pk

1, k ∈ {B, S}, (dated

more recently than the ex-ante contract) that either party received from the counterpart and is willing to show the court; if q = 0 the payment p0 or any other payment pk

0, k ∈ {B, S}, (dated

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 8 / 63

The Hold-Up Problem (8)

Let (ˆ p1, ˆ p0) be the enforced contract (ex-ante one or renegotiated). The buyer approves the trade if and only if: v − ˆ p1 ≥ −ˆ p0,

• r

v ≥ ˆ p1 − ˆ p0 The seller approves the trade if and only if: ˆ p1 − c ≥ ˆ p0,

• r

ˆ p1 − ˆ p0 ≥ c In other words in equilibrium q = 1 if and only if v ≥ ˆ p1 − ˆ p0 ≥ c

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 9 / 63

The Hold-Up Problem (9)

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 ≥ ˆ p1 − ˆ p0 ≥ c then trade occurs.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 10 / 63

The Hold-Up Problem (10)

Consider now the state of nature v = v and c = c and let’s compute the renegotiation prices. The ex-ante contract (p1, p0) is such that only three cases are possible:

case 1: v ≥ p1 − p0 ≥ c, case 2: p1 − p0 > v > c, case 3: v > c > p1 − p0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 11 / 63

The Hold-Up Problem (11)

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

1 > p1,

pS

0 > p0,

p1 > pB

1 ,

p0 > pB

0 .

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 12 / 63

The Hold-Up Problem (12)

In case 2: p1 − p0 > v > c, although trade is efficient, the buyer finds it 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 and

• nly if v − pS

1 ≥ −p0

In equilibrium the seller’s offer is such that: pS

1 = p0 + v

Since in this case the seller’s payoff is: pS

1 − c = p0 + v − c > p0

Therefore ˆ p1 = pS

1 and ˆ

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 13 / 63

The Hold-Up Problem (13)

In case 3: v > c > p1 − p0, although trade is efficient, the seller finds trade to be too costly for the price difference. However the seller will show the buyer’s offer and approve trade if and

• nly if pB

1 − c ≥ p0

In equilibrium the buyer’s offer is such that: pB

1 = p0 + c

Since in this case the buyer’s payoff is: v − pB

1 = v − c − p0 > −p0

Therefore ˆ p1 = pB

1 and ˆ

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 14 / 63

The Hold-Up Problem (14)

Whenever the state of nature is such that v = v and c = c then (ˆ p1, ˆ p0) are such that: v ≥ ˆ p1 − ˆ p0 ≥ c and trade occurs q = 1. We can now consider the ex-ante efficiency of the parties’ investment decision. Notice that the buyer’s payoff in equilibrium is: i j [v − (ˆ p1 − ˆ p0)] − p0 − ψ(j) While the seller’s payoff in equilibrium is: i j [(ˆ p1 − ˆ p0) − c] + p0 − φ(i)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 15 / 63

Underinvestment

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

j

i j [v − (ˆ p1 − ˆ p0)] − p0 − ψ(j) max

i

i j [(ˆ p1 − ˆ p0) − c] + p0 − φ(i) The first order conditions of these problems are then: i∗∗ [v − (ˆ p1 − ˆ p0)] = ψ′(j∗∗) j∗∗ [(ˆ p1 − ˆ p0) − c] = φ′(i∗∗) We can evaluate ex-ante efficiency by comparing the latter FOC and the ex-ante efficiency conditions i∗ (v − c) = ψ′(j∗), j∗ (v − c) = φ′(i∗)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 16 / 63

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 ≥ ˆ p1 − ˆ p0 ≥ c we have that only one of the following three cases may arise: v > ˆ p1 − ˆ p0 > c, v = ˆ p1 − ˆ p0 > c, v > ˆ p1 − ˆ p0 = c

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 17 / 63

Underinvestment (3)

In other words we have that: v − (ˆ p1 − ˆ p0) ≤ v − c, (ˆ p1 − ˆ 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∗).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 18 / 63

Underinvestment (4)

Consider now the first case, from the FOC j∗∗ [(ˆ p1 − ˆ p0) − c] = φ′(i∗∗) we obtain using implicit function theorem: 0 < d i∗∗ d j = (ˆ p1 − ˆ p0) − c φ′′(i∗∗) ≤ v − c φ′′(i∗∗) While from the FOC condition: i∗∗ [v − (ˆ p1 − ˆ p0)] = ψ′(j∗∗) we obtain: d i∗∗ d j = ψ′′(j∗∗) v − (ˆ p1 − ˆ p0) ≥ ψ′′(j∗∗) v − c > 0 With at least one of the inequalities strict.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 19 / 63

Underinvestment (5)

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

✲ ✻

i j

q q

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . j∗∗ j∗ i∗∗ i∗ i∗

ψ(j)

i∗

φ(j)

i∗∗

φ (j)

i∗∗

ψ (j)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 20 / 63

Free-Rider Problem

The basic intuition behind this result is the same as the one behind the standard free-rider-problem. In the absence of a complete ex-ante contract each party making the investment will be expropriated ex-post (at the renegotiation stage) of the returns from his/her investment. Since by subgame perfection this party can forecast this expropriation then at the investment stage the party will reduce at the margin his investment. The expropriation is what is usually known as the hold-up problem. Therefore the incompleteness of contracts leads to an ex-ante inefficiency that takes the form of under-investment.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 21 / 63

Foundations of Incompleteness:

What is an Incomplete Contract? A contract is incomplete if it is not as fully contingent on the state of the world (the resolution of uncertainty about the future) as the parties to the contract might like it to be. The Literature offers three main reasons for contractual incompleteness:

Some aspects of the state of the world may not be common knowledge

• r commonly observable: in particular to the enforcer (observable but

not verifiable). Some aspects of the state may be unforeseen or indescribable by the parties in advance. Even if certain aspects are foreseen, writing them into a contract may be too costly.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 22 / 63

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 future payoffs, then the fact that they cannot describe the possible physical states in advance is irrelevant to welfare. That is, the parties can devise a contract that leaves them no worse off than were they able to describe the physical states ex ante. It should be noted that this result does not mean that the parties can do as well as though they could write a fully contingent contract.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 23 / 63

An Example:

Two agents 1 and 2. They contemplate trading a single indivisible good that agent 1 will produce 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 2 and agent 2 chooses investment e2 determining the cost c for 1: v ′(e1) > 0, c′(e2) < 0 Date 2: uncertainty realized, production and trade.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 24 / 63

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 von Neumann-Morgenstern utility functions. Investments (e1, e2) and (v, c) are common knowledge to 1 and 2 but not verifiable. A state of the world corresponds to the characteristics of the good together with the properties of the intermediate input: the state is verifiable at date 2.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 25 / 63

Example (3)

Efficiency: choice of e∗

1 and e∗ 2 such that:

max

e1 v(e1) − e1,

min

e2 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 contract specifies:

e1 and the penalty that 1 pays 2 if he fails to deliver on these properties, e2 and the penalty that 2 pays 1 if he fails to deliver on these properties.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 26 / 63

An Optimal Mechanism (Moore and Repullo 1988):

The thrust of the Irrelevance Result stated above is that even if no description is available ex-ante this should not affect the parties’ welfare. Consider therefore the following mechanism/contract signed at date 0 and executed at date 2:

Stage 1: Agent 1 announces ˆ v while agent 2 announces ˆ c, clearly in general it is possible that ˆ v = v(e1) and ˆ c = 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∗ˆ v − p∗ > q∗∗ˆ v − p∗∗

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 27 / 63

Optimal Mechanism (2)

Notice first if ˆ v = v(e1) then agent 2 will choose (q∗, p∗) since q∗ˆ v − p∗ > q∗∗ˆ v − p∗∗ The challenge succeeds if agent 2 chooses (q∗∗, p∗∗) revealing that agent 2 has lied. In this case (q∗∗, p∗∗) is implemented: agent 1 produces and delivers q∗∗ units of the good for price p∗∗. The mechanism then concludes.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 28 / 63

Optimal Mechanism (3)

The challenge fails if agent 2 chooses (q∗, p∗) revealing that agent 2 has 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 challenged unsuccessfully. The mechanism then concludes. If agent 1 does not challenge the mechanism moves to Stage 3.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 29 / 63

Optimal Mechanism (4)

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∗ − ˆ cq∗ > p∗∗∗ − q∗∗∗ˆ c Once again if ˆ c = c(e2) then agent 2 will choose (q∗, p∗). The challenge succeeds if agent 1 chooses (q∗∗∗, p∗∗∗) revealing that agent 1 has lied. In this case (q∗∗∗, p∗∗∗) is implemented: agent 1 produces and delivers q∗∗∗ units of the good for price p∗∗∗. The mechanism then concludes.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 30 / 63

Optimal Mechanism (5)

The challenge fails if agent 1 chooses (q∗, p∗) revealing that agent 1 has 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 challenged unsuccessfully. The mechanism then concludes. If agent 2 does not challenge the mechanism moves to Stage 4.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 31 / 63

Optimal Mechanism (6)

Stage 4: Agent 2 delivers the input with properties corresponding to the realized state. Agent 1 produces and delivers a unit of the good with characteristics corresponding to the realized state. Agent 2 pays the price p(ˆ v, ˆ c), where given a constant κ p(ˆ v, ˆ c) = ˆ v + ˆ c + κ The mechanism then concludes.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 32 / 63

Incentive Compatibility

Notice first that if ˆ v = v(e1) there exists a pair (p∗, q∗) and (p∗∗, q∗∗), such that q∗ˆ v − p∗ > q∗∗ˆ v − p∗∗ q∗v(e1) − p∗ < q∗∗v(e1) − p∗∗ In other words, agent 1 successfully challenges agent 2 if and only if agent 2 has lied. Moreover, if F is big enough agent 1 has an incentive to challenge successfully.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 33 / 63

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 he announces untruthfully. Agent 2 therefore has the incentive to set ˆ v = v(e1). Similarly, agent 1 has the incentive to set ˆ c = c(e2).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 34 / 63

Individual Rationality

We can show that for both agents: ei = e∗

i , i ∈ {1, 2} and their

payoff is non-negative. Given IC agent 1’s date 1 payoff is: p(v(e1), c(e2)) − c(e2) − e1 Substituting p(ˆ v, ˆ c) we get that agent 1’s problem is: max

e1 v(e1) + κ − e1

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 35 / 63

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 positive payoff.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 36 / 63

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 F agent 2 would then have an incentive to choose (p∗, q∗) even if 2 has not been truthful. Therefore, 2F has to be paid to a third party. Now consider the renegotiation proposal by agent 1 right before he has 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 still has an interest to let the challenge fail.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 37 / 63

Risk Aversion (Maskin 2002):

The problem is to conceive a fine that punishes agent 1 without rewarding agent 2. This is possible, in principle, when agent 1 is risk averse. Assume that agent 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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 38 / 63

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 fails ex-post renegotiation has no reason to take place. A straightforward argument shows that also ex-ante renegotation has no reason to take place.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 39 / 63

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

Consider now the third cause of contractual incompleteness: the impossibility to describe events in a contract. Consider a situation where the parties to a contract are:

able to understand the consequences and probabilities associated with the environment in which they operate, but are unable to describe adequately certain complex contingencies that may arise.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 40 / 63

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 a finite set. However, for all intents and purposes the features that fully identify the academic’s output that deserves tenure might be taken to belong to an infinite set. Any finite description of the research output that deserves tenure will not capture exactly the set of states of nature in which tenure is given. The probability that the academic gets tenure with m published papers may even be 1. However, surely the probability of getting tenure with k < m published papers is in fact neither 0 nor 1.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 41 / 63

What Can be Described

States are described by means of a language with a countable infinity

• f elementary statements.

Each elementary statement represents a particular feature that can be either present (1) or not (0) in the description of a given state of nature (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 compelling to restrict attention to finite sentences in the language.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 42 / 63

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 finitely many of their constituent “characteristics”. Consider a nice continuous, compact etc. S ... unfortunately it does not work for our purposes. Assume Utility continuous in actions (money transfers). And that Expected Utility is well defined Then we can integrate U(a(s), s) w.r.t. s. But this means a(s) can be approximated by a step function. Hence a sequence of finite descriptions will approximate “first best”

• utilities. (Anderlini and Felli 1994).

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 43 / 63

State Space and Probability (2)

Consider a countably infinite S ... it might work for our purposes depending 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 the probability mass .... we are back in the finite case. A countable infinity of states, with an atomless measure does work.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 44 / 63

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 ∈ Z V (t) if s ∈ Z U2(t, s) = V (−t) if s ∈ Z V (1 − t) if s ∈ Z V is increasing, strictly concave, twice continuously differentiable and Inada. Party 1 makes a take-it-or-leave-it offer of t(·) to 2

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 45 / 63

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 ∈ Z tZ if s ∈ Z With 1 + tZ = tZ, so that both are fully insured. U1(t(s), s) = V (1 + tZ) = V (tZ) ∀ s ∈ S U2(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)

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 46 / 63

The State Space

The language to describe states consists of a countable infinity of elementary 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 {s1

n, . . . , si n, . . .} with si n ∈ {0, 1}

for every i and n. A complete description of a typical state sn is then an object of the type 010011011101011110 . . .

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 47 / 63

Probabilities

We define an atomless measures on S.

Definition (Density)

Given any Z ⊆ S define the density of Z as µ(Z) = lim

N→∞

1 N

N

• n=1

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 48 / 63

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 a probability 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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 49 / 63

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 si

n = 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 the language.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 50 / 63

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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 51 / 63

Computing Expected Utilities

Parties need to be able to evaluate their expected utilities and therefore the (lim) probability measure: µ(A ∩ Z), ∀A ∈ A First result guarantees that we can construct S so that parties can evaluate µ(A), A ∈ A.

Result

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 52 / 63

Intuition of the Proof:

Consider a (countably additive) uniform ˆ µ over {0, 1}N. Let S = {s1, . . . , sn, . . .} be a typical realization of countably many i.i.d. draws from ˆ µ on {0, 1}N. Consider A(i, j) = {sn ∈ S | si

n = j},

j ∈ {0, 1}, i ∈ N By the law of large numbers there exist a ˆ µ-measure 1 set of S such that for every A(i, j) ⊆ S: lim

N→∞

1 N

N

• n=1

IA(i,j)(sn) = ˆ µ(A(i, j)) = µ(A(i, j)) For example A(1, 0) = {sn ∈ [0, 1/2]}, µ(A(1, 0)) = 1

2.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 53 / 63

Well-Defined Frequencies

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

Definition (Well-Defined Frequencies)

The characteristic function IZ has well defined frequencies if Z ∩ A ∈ D ∀ A ∈ A

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 54 / 63

Expected Utilities

Definition (Expected Utilities)

Consider S that satisfies the result above and Z that has well-defined

• frequencies. 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 ]

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 55 / 63

Finite Invariance

We can now characterize the desired properties of Z. First property: Z displays finite invariance if the percentage of states in Z is the same in every A ∈ A as in the entire S.

Definition (Finite Invariance)

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

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 56 / 63

Finite Invariance (2)

The characteristic function IZ displays finite invariance if the densities

• f the sets Z are the same, conditional on all finitely definable subsets
• f A.

In other words, if IZ displays finite invariance knowing that s belongs to any finitely definable subset A does not help us to predict better the values that IZ will take. Trivial cases of finite invariance: Z = S, Z = ∅

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 57 / 63

Fine Variability

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

Definition (Fine Variability)

The characteristic function IZ displays fine variability if and only if it is finite invariant and: µ(Z) > 0, µ(Z) > 0.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 58 / 63

Existence

Result

Let pZ ∈ [0, 1] and A ∈ A be given, with µ(A) > 0. There exists an S and a set Z ⊂ S with characteristic function IZ that has well defined frequencies, displays finite invariance, displays fine variability, and is such that µ(Z) = µ(Z|A) = pZ. In other words, ... finite invariance does not limit actual variability. Not true in the continuum.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 59 / 63

Intuition of the Proof:

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.d draws across all the states sn. The law of large numbers guarantees that there exist a measure 1 set

• f 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.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 60 / 63

Undescribable Events:

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

t

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 following properties.

1 The optimal finite contract t∗∗ exists unique. 2 The characteristic function of Z is well defined in terms of

frequencies.

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

In other words, no transfer occurs.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 61 / 63

Intuition of the Proof:

Construct S as in the result above and choose Z ∈ D so that IZ exhibits well-defined frequencies, finite invariance and fine variability with pZ = µ(Z) as above. By finite invariance, Z is such that any attempt to condition on a finitely describable set A leaves the parties with a set of states of which only a fraction µ(Z) are in Z. Risk aversion implies that any finite contract t that is not constant on S is dominated by another one that is constant (average). Therefore t∗∗ exists unique and constant and party 2’s IR constraints imply zero transfers.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 62 / 63

Ultimate Incomplete Contract

The optimal finite contract yields the no-contract allocation: the ultimate incomplete contract. In the setting above agent 1’s expected utility is bounded away from the his full-insurance expected utility: the approximation result fails. We have provided a formal model of undescribable events as events that are fully understood but it is impossible to do them justice trying to describe them in advance.

Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 63 / 63