efficient risk sharing with limited commitment and storage

Efficient Risk Sharing with LimitedCommitment and Storage

Árpád Ábrahám Sarolta Laczó

EUI IAE and Barcelona GSE

Barcelona GSE Trobada

October 26, 2012

Ábrahám (EUI) & Laczó (IAE & Barcelona GSE) Limited Commitment and Storage

Risk sharing

Consider two agents who each receive a risky endowment.

I For example, suppose that each may receive 2 or 8 euros eachperiod (and the aggregate endowment is always 10).

Agents may share the (idiosyncratic) risk they face by makingtransfers.

I First-best/Unconstrained-efficient outcome: lucky agent makesa transfer of 3 and both consume 5 in each period, for example.

Risk sharing

Limited commitment

However, the agent who is lucky today might be unwilling to make alarge transfer...

I He might be better off in autarky, i.e., consuming 8 today andthen his endowment in all future periods. He will not make atransfer of 3 without commitment.

But he might voluntarily make a smaller transfer in order to benefitfrom future risk sharing.

I More precisely, he is willing to make a transfer that still leaveshim at least as well off as being in autarky (2 euros, say).

Limited commitment

Limited commitmentThe limited commitment friction generates partial risk sharing.

I Income fluctuations partially translate into consumptionfluctuation. When an agent earns 8 he consumes 6, and when heearns 2 he consumes 4.

This outcome is constrained efficient.

Applications:

I Households in villages (Ligon, Thomas, and Worrall, 2002)

I Households in the United States (Krueger and Perri, 2006)

I Members of a household (Mazzocco, 2007)

I Countries (Kehoe and Perri, 2002)

I Partnerships such as law firms

I A farmer and a thief (Schechter, 2007)

Limited commitmentThe limited commitment friction generates partial risk sharing.

I Income fluctuations partially translate into consumptionfluctuation. When an agent earns 8 he consumes 6, and when heearns 2 he consumes 4.

This outcome is constrained efficient.

Applications:

I Households in villages (Ligon, Thomas, and Worrall, 2002)

I Households in the United States (Krueger and Perri, 2006)

I Members of a household (Mazzocco, 2007)

I Countries (Kehoe and Perri, 2002)

I Partnerships such as law firms

I A farmer and a thief (Schechter, 2007)

Storage

In all these applications, agents are likely to have access to somestorage technology, both public and private/hidden.

I Households in villages may keep grain or cash around the houseor use community grain storage facilities.

I Households in the United States may keep savings in cash or‘hide’ their assets abroad.

I Spouses in a household may accumulate both joint assets andsavings for personal use.

I Partners in a law firms may have both common and privateassets.

I The European Stability Mechanism has public funds for euroarea countries.

If so, agents can also use storage to improve insurance.

Storage

In all these applications, agents are likely to have access to somestorage technology, both public and private/hidden.

I Households in villages may keep grain or cash around the houseor use community grain storage facilities.

I Households in the United States may keep savings in cash or‘hide’ their assets abroad.

I Spouses in a household may accumulate both joint assets andsavings for personal use.

I Partners in a law firms may have both common and privateassets.

I The European Stability Mechanism has public funds for euroarea countries.

If so, agents can also use storage to improve insurance.

What we do in this paper

I We investigate what are the consequences if both a public and aprivate storage technology are available, and risk sharing isimperfect because of limited commitment.

I We provide a thorough analytical characterization of thisenvironment.

I In particular, we study the implications of the availability ofstorage on aggregate asset accumulation, individual consumptiondynamics, and welfare.

I We also show that any constrained-efficient allocation can bedecentralized as a competitive equilibrium with endogenousborrowing constraints similar to Alvarez and Jermann (2000).

Literature

I In the existing models of risk sharing with limited commitment,only public and/or observable and contractible individualintertemporal technologies have been considered.

Marcet and Marimon (1992), Ligon, Thomas, and Worrall (2000), Kehoe andPerri (2002), Ábrahám and Carceles-Poveda (2006), Krueger and Perri(2006)

I Both public and private storage technologies have beenconsidered when the deep friction that limits risk sharing isimperfect information (hidden income or effort).

Cole and Kocherlakota (2001), Werning (2001), Ábrahám and Pavoni (2008),Ábrahám, Koehne, and Pavoni (2010)

Our starting point is the two-sided lack of commitment framework ofKocherlakota (1996).

I Two (types of) infinitely-lived, risk-averse, ex-ante identicalagents.

I Income has discrete support (S states), is i.i.d. over time, and isperfectly negatively correlated across the two agents, i.e. there isno aggregate risk in the sense that the aggregate endowment isconstant.

Our starting point is the two-sided lack of commitment framework ofKocherlakota (1996).

I Two (types of) infinitely-lived, risk-averse, ex-ante identicalagents.

I Income has discrete support (S states), is i.i.d. over time, and isperfectly negatively correlated across the two agents, i.e. there isno aggregate risk in the sense that the aggregate endowment isconstant.

Setup continued

We consider a storage/saving technology with exogenous return−1 ≤ r ≤ 1/β − 1. Borrowing is not allowed.

I The planner can use this storage technology.

– public storage

I Agents have access to the same technology in a hidden(non-observable and/or non-contractible) way.

– private storage (later)

Model with public storage

The planner’s problem is

maxci(st),B(st)

∞∑t=1

βt Pr(st)u(ci(st))

s.t.∑i

ci(st)≤ Y + (1 + r)B(st−1)−B(st), B(st) ≥ 0, ∀st,

∞∑r=t

βr−t Pr(sr | st

)u (ci (s

r)) ≥ Uaui (st) , ∀st,∀i,

where Uaui (st) is the value function of autarky.

(Alternative outside options are possible as long as the value isincreasing in current income.)

Model with public storage

maxci(st),B(st)

∞∑t=1

βt Pr(st)u(ci(st))

s.t.∑i

∞∑r=t

βr−t Pr(sr | st

)u (ci (s

where Uaui (st) is the value function of autarky.

(Alternative outside options are possible as long as the value isincreasing in current income.)

CharacterizationI Define Mi (s

t) ≡ λi +∑tτ=1 µi (s

τ ), where sτ is the subhistory of

st up to time τ , υi(st) ≡µi(st)Mi(st)

x(st)≡ M1 (s

M2 (st)=u′ (c2 (s

u′ (c1 (st)),

which is the ‘temporary’ relative Pareto weight of agent 1.

I The law of motion of x is given by

x(st)= x

(st−1

) 1− υ2 (st)1− υ1 (st)

I The planner’s Euler (optimality condition for Bt+1) is

u′(ci(st))≥ β(1 + r)

∑st+1

Pr(st+1|st

) u′ (ci (st+1))

1− υi (st+1),

where 0 ≤ υi(st+1

)≤ 1.

CharacterizationI Define Mi (s

t) ≡ λi +∑tτ=1 µi (s

τ ), where sτ is the subhistory of

st up to time τ , υi(st) ≡µi(st)Mi(st)

x(st)≡ M1 (s

M2 (st)=u′ (c2 (s

u′ (c1 (st)),

which is the ‘temporary’ relative Pareto weight of agent 1.

I The law of motion of x is given by

x(st)= x

(st−1

) 1− υ2 (st)1− υ1 (st)

I The planner’s Euler (optimality condition for Bt+1) is

u′(ci(st))≥ β(1 + r)

∑st+1

Pr(st+1|st

) u′ (ci (st+1))

1− υi (st+1),

where 0 ≤ υi(st+1

)≤ 1.

Characterization

I The key variable for the characterization of the model’s dynamicsis the time-varying relative Pareto weight of agent 1, x.

I Given B, the constrained-efficient allocation is described byoptimal state-dependent intervals,

[xj(B), xj(B)

], j = 1, ..., S.

I The lower (upper) limit is determined by the bindingparticipation constraint of agent 1 (agent 2).

I The dynamics of the relative Pareto weight is given by

xj(B) if xt−1 > xj(B)xt−1 if xt−1 ∈

[xj(B), xj(B)

]xj(B) if xt−1 < xj(B)

I Similar intervals and dynamics on consumption:[cj(B), cj(B)

Characterization

I The key variable for the characterization of the model’s dynamicsis the time-varying relative Pareto weight of agent 1, x.

I Given B, the constrained-efficient allocation is described byoptimal state-dependent intervals,

[xj(B), xj(B)

], j = 1, ..., S.

I The lower (upper) limit is determined by the bindingparticipation constraint of agent 1 (agent 2).

I The dynamics of the relative Pareto weight is given by

xj(B) if xt−1 > xj(B)xt−1 if xt−1 ∈

[xj(B), xj(B)

]xj(B) if xt−1 < xj(B)

I Similar intervals and dynamics on consumption:[cj(B), cj(B)

Characterization

I β determines the width of the intervals.

I High β: wide intervals and more risk sharing.

I Low β: narrow intervals and less risk sharing.

We say that risk sharing is high if the consumption distribution istime invariant in the long run.

I This happens when β is above some threshold.

We say that risk sharing is low if the consumption distribution variesover time.

Characterization

I β determines the width of the intervals.

I High β: wide intervals and more risk sharing.

I Low β: narrow intervals and less risk sharing.

We say that risk sharing is high if the consumption distribution istime invariant in the long run.

I This happens when β is above some threshold.

We say that risk sharing is low if the consumption distribution variesover time.

Short-run asset dynamics

Proposition. Given inherited assets, aggregate storage is greater whenconsumption inequality is higher.

Intuition: the planner stores for the agents and to improve risksharing in the future.

I When inequality is higher today, the high-income agent has moreincentive to store. The planner inherits this incentive.

I When inequality is higher today, the gains from improving risksharing in the future are higher.

Short-run asset dynamics

Proposition. Given inherited assets, aggregate storage is greater whenconsumption inequality is higher.

Intuition: the planner stores for the agents and to improve risksharing in the future.

I When inequality is higher today, the high-income agent has moreincentive to store. The planner inherits this incentive.

I When inequality is higher today, the gains from improving risksharing in the future are higher.

Long-run behavior of assets

Proposition. Assume that β is such that agents obtain low risksharing in the sense that the consumption distribution is time varyingwhen B′ = B = 0 is imposed.

(i) There exists r1 such that for all r ∈ [−1, r1], B′ = 0 for all incomelevels, that is, public storage is never used in the long run.

(ii) There exists a strictly positive r2 > r1 such that for allr ∈ (r1, r2), B remains stochastic but bounded in the long run.

(iii) For all r ∈ [r2, 1/β − 1), B converges almost surely to a strictlypositive constant where the consumption distribution is timeinvariant but perfect risk sharing is not achieved.

(iv) Whenever r = 1/β − 1, B converges to a strictly positiveconstant and perfect risk sharing is self-enforcing.

If β is such that the consumption distribution is time invariant whenB′ = B = 0 is imposed, then only (i), (iii), and (iv) occur.

Intuition:

There is a trade-off between improving future risk sharing, and usingan inefficient storage technology.

I As r increases, the first becomes relatively more important.

I No trade-off when β(1 + r) = 1.

Decentralization/Competitive equilibrium

Public storage can be interpreted as a special case of capital⇒ decentralization is a special case of Ábrahám and Carceles-Poveda(2006).

I Agents trade Arrow securities subject to endogenous borrowingconstraints (as in Alvarez and Jermann, 2000).

I Intermediaries issue Arrow securities to finance storage (i.e.,agents do not own it directly) and make zero profits.

I Asset prices take into account future binding borrowingconstraints.

Our model provides a theory of endogenously growing (and shrinking)Lucas trees in equilibrium.

Decentralization/Competitive equilibrium

Public storage can be interpreted as a special case of capital⇒ decentralization is a special case of Ábrahám and Carceles-Poveda(2006).

I Agents trade Arrow securities subject to endogenous borrowingconstraints (as in Alvarez and Jermann, 2000).

I Intermediaries issue Arrow securities to finance storage (i.e.,agents do not own it directly) and make zero profits.

I Asset prices take into account future binding borrowingconstraints.

Our model provides a theory of endogenously growing (and shrinking)Lucas trees in equilibrium.

Model with both public and private storage

maxci(st),B(st)

∞∑t=1

βt Pr(st)u(ci(st))

s.t.∑i

∞∑r=t

βr−t Pr(sr | st

)u (ci (s

u′(ci(st))≥ β(1 + r)

∑st+1

Pr(st+1

)u′(ci(st+1

)), ∀st,∀i.

(We show that the first-order condition approach is valid.)

maxci(st),B(st)

∞∑t=1

βt Pr(st)u(ci(st))

s.t.∑i

∞∑r=t

βr−t Pr(sr | st

)u (ci (s

u′(ci(st))≥ β(1 + r)

∑st+1

Pr(st+1

)u′(ci(st+1

)), ∀st,∀i.

maxci(st),B(st)

∞∑t=1

βt Pr(st)u(ci(st))

s.t.∑i

∞∑r=t

βr−t Pr(sr | st

)u (ci (s

u′(ci(st))≥ β(1 + r)

∑st+1

Pr(st+1

)u′(ci(st+1

)), ∀st,∀i.

Characterization

Proposition. When the planner’s Euler is satisfied, the agents’ Eulersare satisfied as well. Therefore, the solution of the model with bothpublic and hidden storage corresponds to the solution of the problemwith only public storage, as long as the outside option is the same.

Planner: u′(ci(st))≥ β(1 + r)

∑st+1

Pr(st+1|st

) u′ (ci (st+1))

1− υi (st+1).

Agent: u′(ci(st))≥ β(1 + r)

∑st+1

Pr(st+1|st

)u′(ci(st+1

I Intuition: public storage relaxes future participation constraints,thus improves risk sharing – only the planner internalizes thisexternality.

Characterization

∑st+1

Pr(st+1|st

) u′ (ci (st+1))

1− υi (st+1).

∑st+1

Pr(st+1|st

)u′(ci(st+1

Characterization

∑st+1

Pr(st+1|st

) u′ (ci (st+1))

1− υi (st+1).

∑st+1

Pr(st+1|st

)u′(ci(st+1

Does access to hidden storage matter?

We have just seen that hidden storage does not matter in our modelwith public storage. In other words, agents’ Euler inequalities aresatisfied.

Does hidden storage matter in the basic model without storage?

Proposition. Suppose that partial insurance occurs. There existsr < 1

β − 1 such that for all r > r, agents’ Euler inequalities areviolated at the constrained-efficient allocation of the basic model.

I r can even be negative

Intuition: under bounded support of income and imperfect risksharing, high income agents face weakly decreasing consumption nextperiod.

Does access to hidden storage matter?

We have just seen that hidden storage does not matter in our modelwith public storage. In other words, agents’ Euler inequalities aresatisfied.

Does hidden storage matter in the basic model without storage?

Proposition. Suppose that partial insurance occurs. There existsr < 1

β − 1 such that for all r > r, agents’ Euler inequalities areviolated at the constrained-efficient allocation of the basic model.

I r can even be negative

Intuition: under bounded support of income and imperfect risksharing, high income agents face weakly decreasing consumption nextperiod.

Individual consumption dynamics

We overturn 3 counterfactual predictions of the basic model withrespect to consumption dynamics (Attanasio, 1999; Broer, 2011).

In our model,

I the amnesia property does not hold in general.

I the persistence property does not hold in general.

I agents’ Euler inequalities hold.

Long-run welfare

Proposition.

(i) There exists r1 such that for all r ∈ [−1, r1] storage is not usedeven in autarky, therefore access to storage is welfare neutral.

(ii) There exists a strictly positive r2 > r1 such that for allr ∈ (r1, r2] storage is used in autarky but not in equilibrium,therefore consumption dispersion increases and welfaredeteriorates as a result of access to storage.

(iii) There exists a strictly positive r3 > r2 such that for allr ∈ (r2, r3) public storage is (at least sometimes) positive, butaccess to storage is still welfare reducing. Access to storage iswelfare neutral in the long run at the threshold r = r3.

(iv) For all r ∈ (r3, 1/β − 1] access to storage is welfare improving inthe long run. Consumption dispersion may or may not be lowerthan in the basic model without storage.

Long-run welfare

Proposition.

Long-run welfare

Proposition.

Long-run welfare

Proposition.

Long-run welfare

Big picture

Hidden storage has been considered in models with hidden income oreffort (Cole and Kocherlakota, 2001; Ábrahám, Koehne, and Pavoni,2010). The problem is a similar joint deviation: storage andshirking/misreporting vs. storage and default.

I Limited commitment: storage by the planner both improvesinsurance and relaxes the incentive problem, hence access tostorage may be welfare improving.

I Hidden income/effort: public asset accumulation would makeincentive provision more expensive, so it does not take place inthe optimal allocation, and access to storage is detrimental towelfare.

Applications

I Quantitative implications of storage on the dynamics ofconsumption.

I Risk sharing in villages – rationale for public storage facilities.

I Partnerships – rationale for ‘common’ capital.

I Couples – economic reasons for not providing rights toaccumulated capital for initiator of divorce.

I Economic unions – rationale for building up stability funds.

efficient risk sharing with limited commitment and storage

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