Motivation The Two Person Model Results

Informal Insurance Arrangements with LimitedCommittment

Ligon, Thomas and Worrall (2002)

Gharad Bryan

Devo Talk

December 11, 2006

Motivation The Two Person Model Results

Outline

1 Motivation

2 The Two Person Model

3 Results

Motivation The Two Person Model Results

Outline

1 Motivation

2 The Two Person Model

3 Results

Motivation The Two Person Model Results

Risk: A Recap from Class

CostsAgricultural incomes fluctuate greatly.Direct cost to the risk averse.May lead to inefficient production ⇒ poverty begetspoverty.We would like to know the extent to which people canspread risk and how they do it.

SolutionsIncomes smoothing:

Crop/plot diversification, asset ownership, wage labor . . .Consumption smoothing:

Mutual insurance, loans, transfers, saving . . .

Motivation The Two Person Model Results

Risk: A Recap from Class

CostsAgricultural incomes fluctuate greatly.Direct cost to the risk averse.May lead to inefficient production ⇒ poverty begetspoverty.We would like to know the extent to which people canspread risk and how they do it.

SolutionsIncomes smoothing:

Crop/plot diversification, asset ownership, wage labor . . .Consumption smoothing:

Mutual insurance, loans, transfers, saving . . .

Motivation The Two Person Model Results

Risk: A Recap from Class

Mutual Insurance: Townsend 1994Complete Arrow-Debreu markets ⇒ consumptionuncorrelated with individual income;Rejected in data (but fares better than you might havethought)LTW attempt to provide a model which explains the gap.There model is based on ‘dynamic limited commitment’.

Motivation The Two Person Model Results

What is Limited Commitment

DefinitionWith limited commitment, an insurance scheme is only feasibleif promises of future reciprocity are perceived to be credible andsufficiently attractive.

Relation to Other ModelsA natural generalization of Arrow-Debreu.Reciprocity formalized through subgame perfection in theinfinite game.

Motivation The Two Person Model Results

What is Limited Commitment

DefinitionWith limited commitment, an insurance scheme is only feasibleif promises of future reciprocity are perceived to be credible andsufficiently attractive.

Relation to Other ModelsA natural generalization of Arrow-Debreu.Reciprocity formalized through subgame perfection in theinfinite game.

Motivation The Two Person Model Results

What is Limited Commitment

StaticTransfers must be stationary: whenever we observe thesame state, we observe the same transfers.Looks more like gifts/transfers than loans .

DynamicCurrent transfers can depend on history of past transfers,not just on the state.Looks more like a loan and adds more flexibility toconsumption paths.

Motivation The Two Person Model Results

What is Limited Commitment

StaticTransfers must be stationary: whenever we observe thesame state, we observe the same transfers.Looks more like gifts/transfers than loans .

DynamicCurrent transfers can depend on history of past transfers,not just on the state.Looks more like a loan and adds more flexibility toconsumption paths.

Motivation The Two Person Model Results

Outline

1 Motivation

2 The Two Person Model

3 Results

Motivation The Two Person Model Results

Details Details!

Each period there is a draw s from a set of states S.Consumers receive income yi(s).Income is markov with transition probability πsr > 0between states s and r .Households are infinitely-lived, risk-averse and joint utilitymaximizers.

Motivation The Two Person Model Results

Details Details!

Each period there is a draw s from a set of states S.Consumers receive income yi(s).Income is markov with transition probability πsr > 0between states s and r .Households are infinitely-lived, risk-averse and joint utilitymaximizers.

Motivation The Two Person Model Results

Details Details!

Each period there is a draw s from a set of states S.Consumers receive income yi(s).Income is markov with transition probability πsr > 0between states s and r .Households are infinitely-lived, risk-averse and joint utilitymaximizers.

Motivation The Two Person Model Results

Details Details!

Each period there is a draw s from a set of states S.Consumers receive income yi(s).Income is markov with transition probability πsr > 0between states s and r .Households are infinitely-lived, risk-averse and joint utilitymaximizers.

Motivation The Two Person Model Results

What is a Contract

A contract specifies τ(h) a transfer from household 1 to 2conditional on the history.If i breaches:

i is exluded from the mutual insurance system.i suffers a further loss Pi(s) ≥ 0. ‘shame’?

LTW look for the sustainable contracts:

U ti (h

t) ≥ −Pi(st) (1)

Note relationship to infinitely repeated game.What interpretation should we give to P?

Motivation The Two Person Model Results

What is a Contract

A contract specifies τ(h) a transfer from household 1 to 2conditional on the history.If i breaches:

i is exluded from the mutual insurance system.i suffers a further loss Pi(s) ≥ 0. ‘shame’?

LTW look for the sustainable contracts:

U ti (h

t) ≥ −Pi(st) (1)

Note relationship to infinitely repeated game.What interpretation should we give to P?

Motivation The Two Person Model Results

What is a Contract

A contract specifies τ(h) a transfer from household 1 to 2conditional on the history.If i breaches:

i is exluded from the mutual insurance system.i suffers a further loss Pi(s) ≥ 0. ‘shame’?

LTW look for the sustainable contracts:

U ti (h

t) ≥ −Pi(st) (1)

Note relationship to infinitely repeated game.What interpretation should we give to P?

Motivation The Two Person Model Results

What is a Contract

A contract specifies τ(h) a transfer from household 1 to 2conditional on the history.If i breaches:

i is exluded from the mutual insurance system.i suffers a further loss Pi(s) ≥ 0. ‘shame’?

LTW look for the sustainable contracts:

U ti (h

t) ≥ −Pi(st) (1)

Note relationship to infinitely repeated game.What interpretation should we give to P?

Motivation The Two Person Model Results

What is a Contract

A contract specifies τ(h) a transfer from household 1 to 2conditional on the history.If i breaches:

i is exluded from the mutual insurance system.i suffers a further loss Pi(s) ≥ 0. ‘shame’?

LTW look for the sustainable contracts:

U ti (h

t) ≥ −Pi(st) (1)

Note relationship to infinitely repeated game.What interpretation should we give to P?

Motivation The Two Person Model Results

Thoughts About P

Acts as a fit parameter.Means that the Townsend model is nested.Measure of quality of ‘instititions’?

Motivation The Two Person Model Results

Solving the Model

Equation (1) implies an upper and lower bound on surplus,Us

i ∈ [Usi , Us

i ].LTW look for the efficient contracts, therefore work with thepareto frontier.Define Us

2(Us1) to be the pareto frontier for the problem.

Motivation The Two Person Model Results

The Pareto Problem

Us2(Us

1) = maxτs,(U r

1)r∈S

u2(y2(s) + τs)− v(y2(s)) + δ∑r∈S

πrsU r2(U

r1)

subject to:

u(y1(s)− τs)− u(y1(s)) + δ∑r∈S

U r1 ≥ Us

1

U ri ≥ U r

i ∀r ∈ S

Motivation The Two Person Model Results

First Order Conditions

τ :u′2(y2(s)+τs)

u′1(y1(s)−τs)= λ (2)

U1 : − U r2′(U r

1) = λ+φr1+µr

, ∀r ∈ S (3)

ET : − Us2′(Us

1) = λ (4)

Where φ and µ are the lagrange multipliers on the sustainabilityconstraints.

Motivation The Two Person Model Results

Solution

Theorem

There exist λr and λr ∀r ∈ S such that:

λ(ht+1) =

λr if λ(ht) < λr

λ(ht) if λ(ht) ∈ [λr , λr ]

λr if λ(ht) > λr

Noteλr is the slope of the pareto frontier (and ratio of MRS)when 1 receives U1

λr is the slope of the pareto frontier when 1 receives U1

Motivation The Two Person Model Results

A Four State Example

States: LL, HL, LH, HH

Starting point is relevant.

GIves all contracts.

Surpluses

Motivation The Two Person Model Results

A Four State Example

State LL

1 is constrained.

Transfer to 2.

Surpluses

Motivation The Two Person Model Results

A Four State Example

State HL

Transfer to 2.

Less than full insurance.

Surpluses

Motivation The Two Person Model Results

A Four State Example

State HH

Same income - transfer to 1.

Resembles a loan payment.

Surpluses

Motivation The Two Person Model Results

A Four State Example

State HH

Transfer to 1.

‘Repayments’ stop.

Surpluses

Motivation The Two Person Model Results

Comparative Statics

If the penalties are high enough, you can support any fullinsurance outcome.If people are sufficiently patient, there is some sustainablefull insurance contract.If there are no penalties and people are sufficiently myopic,then you get back the autarkic outcome.So the model nests the Townsend and Autarkic models.

Motivation The Two Person Model Results

Outline

1 Motivation

2 The Two Person Model

3 Results

Motivation The Two Person Model Results

Results

A Test of FitAssume CRRA utility with parameter γ.Find λ intervals.Simulate consumption path given observed income.Estimate the triple θ = {γ, δ, P} which is the OLS estimatorof:

log c it = log c i

t (λit , ct | θ) + µi

t (5)

logσi

t+1

σit

= logσi

t+1(θ)

σit(θ)

+ ν it+1 − ν i

t (6)

Quasi-likelihood gives a measure of fit.

Motivation The Two Person Model Results

Results

Motivation The Two Person Model Results

Actual

Motivation The Two Person Model Results

Levels

Motivation The Two Person Model Results

Changes

Motivation The Two Person Model Results

Lorenz Curves

Motivation The Two Person Model Results

Comments

CriticismsA three parameter model must fit better than a oneparameter model.The difference between the two estimators is a bignegative.Best evidence is in consistency of γ.

Alternatives?Strong predictions on movement of consumptions - whynot test this?

Motivation The Two Person Model Results

Comments

CriticismsA three parameter model must fit better than a oneparameter model.The difference between the two estimators is a bignegative.Best evidence is in consistency of γ.

Alternatives?Strong predictions on movement of consumptions - whynot test this?

Motivation The Two Person Model Results

Final Thoughts

Provides a model which allows loans.Intermediate between Autarky and Townsend.Do we believe the mechanism.Can it act as a benchmark.Is P a relevant statistic.