Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 20 ppsx

From these optimal consumption dy-namics, it will be straightforward to compute the ex ante division of gains from then denote the corresponding value of expression 20.2.1b that is pro

Equilibrium without Commitment

20.1 Two-sided lack of commitment

In section 19.3 of the previous chapter, we studied insurance without ment That was a partial equilibrium analysis since the moneylender couldborrow or lend resources outside of the village at a given interest rate Recallalso the asymmetry in the environment where villagers could not make any com-mitments while the moneylender was assumed to be able to commit We willnow study a closed system without access to an outside credit market Anyhousehold’s consumption in excess of its own endowment must then come fromthe endowments of the other households in the economy We will also adopt thesymmetric assumption that everyone is unable to make commitments That is,any contract prescribing an exchange of goods today in anticipation of futureexchanges of goods represents a sustainable allocation only if both current andfuture exchanges are incentive compatible to all households involved in the con-tractual arrangement Households are free to walk away from the arrangement

commit-at any point in time and to defect into autarky Such a contract design problemwith participation constraints on both sides of an exchange represents a problemwith two-sided lack of commitment, as compared to the problem with one-sidedlack of commitment in section 19.3

This chapter draws upon the work of Thomas and Worrall (1988) andKocherlakota (1996b) At the end of the chapter, we also discuss market arrange-ments for decentralizing the constrained Pareto optimal allocation, as studied

by Kehoe and Levine (1993) and Alvarez and Jermann (2000)

– 692 –

re-spot market for labor where a worker is paid y t at time t The worker is always

free to walk away from the firm and work in that spot market But if he does, hecan never again enter into a long-term relationship with another firm The firmseeks to maximize the discounted stream of expected future profits by designing

a long-term wage contract that is self-enforcing in the sense that it never gives

the worker an incentive to quit In a contract that stipulates a wage c t at time

t , the firm earns time t profits of y t − c t (as compared to hiring a worker inthe spot market for labor) If Thomas and Worrall had assumed a commitmentproblem only on the part of the worker, their model would be formally identical

to our villager-money lender environment However, Thomas and Worrall alsoassume that the firm itself can renege on a wage contract and buy labor atthe random spot market wage Hence, they require that a self-enforcing wagecontract be one in which neither party ever has an incentive to renege

Kocherlakota (1996b) studies a model that has some valuable features incommon with Thomas and Worrall’s Kocherlakota’s counterpart to Thomasand Worrall’s firm is a risk averse second household In Kocherlakota’s model,two households receive stochastic endowments The contract design problem is

to find an insurance/transfer arrangement that reduces consumption risk whilerespecting participation constraints for both households: both households must

be induced each period not to walk away from the arrangement to live in autarky.Kocherlakota uses his model in an interesting way to help interpret empiricallyestimated conditional consumption-income covariances that seem to violate thehypothesis of complete risk-sharing Kocherlakota investigates the extent towhich those failures reflect impediments to enforcement that are captured byhis participation constraints

For the purpose of studying those conditional covariances in a stationarystochastic environment, Kocherlakota’s use of an environment with two-sidedlack of commitment is important In our model of villagers facing a moneylender in section 19.3, imperfect risk sharing is temporary and so would notprevail in a stochastic steady state In Kocherlakota’s model, imperfect risk

sharing can be perpetual There are equal numbers of two types of households

in the village Each of the households has the preferences, endowments, andautarkic utility possibilities described in chapter 19 Here we assume that theendowments of the two types of households are perfectly negatively correlated

Whenever a household of type 1 receives y s, a household of type 2 receives 1−y s

We assume that y s ∈ [0, 1], that the distribution of y t is i.i.d over time, and

that the distribution of y t is identical to that of 1− y t Also, now the plannerhas access to neither borrowing nor lending opportunities, and is confined toreallocating consumption goods between the two types of households This

limitation leads to two participation constraints At time t , the type 1 household receives endowment y t and consumption c t, while the type 2 household receives

1− y t and 1− c t

In this setting, an allocation is said to be sustainable1 if for all t ≥ 0 and

for all histories h t

u(c t)− u(y t )+βE t

∞

j=1

β j −1 [u(c t+j)− u(y t+j)]≥ 0, (20.2.1a)

u(1 − c t)− u(1 − y t )+βE t

The function Q( ) depicts a (constrained) Pareto frontier It portrays the

maximized value of the expected life-time utility of the type 2 household, wherethe maximization is subject to requiring that the type 1 household receive anexpected life-time utility that exceeds its autarkic welfare level by at least 

utils To find this Pareto frontier, we first solve for the consumption dynamics

1 Kocherlakota says subgame perfect rather than sustainable.

Recursive formulation 695

that characterize all efficient contracts From these optimal consumption

dy-namics, it will be straightforward to compute the ex ante division of gains from

then denote the corresponding value of expression ( 20.2.1b ) that is promised to a

type 2 agent.3 When the endowment realization y s is associated with a promise

to a type 1 agent equal to x s = x, we can write the Bellman equation as

ap-3 Q s(·) is a Pareto frontier conditional on the endowment realization y swhile

Q(·) in (20.2.2a) is an ex ante Pareto frontier before observing any endowment

realization

χ j ≥ 0, j = 1, , S; (20.3.1c)

Q j (χ j)≥ 0, j = 1, , S; (20.3.1d)

where expression ( 20.3.1b ) is the promise-keeping constraint; expression ( 20.3.1c )

is the participation constraint for the type 1 agent; and expression ( 20.3.1d ) is the participation constraint for the type 2 agent The set of feasible c is given

by expression ( 20.3.1e ).

Thomas and Worrall prove the existence of a compact interval that contains

all permissible continuation values χ j:

χ j ∈ [0, x j] for j = 1, 2, , S (20.3.1f ) Thomas and Worrall also show that the Pareto-frontier Q j(·) is decreasing,

strictly concave and continuously differentiable on [0, x j ] The bounds on χ j

are motivated as follows The contract cannot award the type 1 agent a value

of χ j less than zero because that would correspond to an expected future

life-time utility below the agent’s autarky level There exists an upper bound x j

above which the planner would never find it optimal to award the type 1 agent

a continuation value conditional on next period’s endowment realization being

y j It would simply be impossible to deliver a higher continuation value because

of the participation constraints In particular, the upper bound x j is such that

Here a type 2 agent receives an expected life-time utility equal to his autarky

level if the next period’s endowment realization is y j and a type 1 agent is

promised the upper bound x j Our two- and three-state examples in sections

20.10 and 20.11 illustrate what determines x j

Attaching Lagrange multipliers µ , βΠ j λ j , and βΠ j θ j to expressions ( 20.3.1b ), ( 20.3.1c ), and ( 20.3.1d ), the first-order conditions for c and χ j are4

c : − u (1− c) + µu  (c) = 0, (20.3.3a)

χ j : βΠ j Q  j (χ j ) + µβΠ j + βΠ j λ j + βΠ j θ j Q  j (χ j ) = 0 (20.3.3b)

4 Here we are proceeding under the conjecture that the non-negativity

con-straints on consumption in ( 20.3.1e ), c ≥ 0 and 1 − c ≥ 0, are not binding.

This conjecture is confirmed below when it is shown that optimal consumption

levels satisfy c ∈ [y1, y S]

Equation ( 20.3.5a ) can be expressed as

where g is a continuously and strictly decreasing function By substituting the inverse of that function into equation ( 20.3.5b ), we obtain the expression

g −1 (c) = (1 + θ j ) g −1 (c j ) + λ j , (20.4.2) where c is again the current consumption of a type 1 agent and c j is his next

period’s consumption when next period’s endowment realization is y The

optimal consumption dynamics implied by an efficient contract are evidentlygoverned by whether or not agents’ participation constraints are binding For

any given endowment realization y j next period, only one of the participation

constraints in ( 20.3.1c ) and ( 20.3.1d ) can bind Hence, there are three regions

of interest for any given realization y j:

1 Neither participation constraint binds When λ j = θ j = 0 , the

consump-tion dynamics in ( 20.4.2 ) satisfy

g −1 (c) = g −1 (c j) =⇒ c = c j ,

where c = c j follows from the fact that g −1(·) is a strictly decreasing

function Hence, consumption is independent of the endowment and theagents are offered full insurance against endowment realizations so long asthere are no binding participation constraints The constant consumption

allocation is determined by the “temporary relative Pareto weight” µ in equation ( 20.3.3a ).

2 The participation constraint of a type 1 person binds (λ j > 0 ), but θ j= 0

Thus, condition ( 20.4.2 ) becomes

g −1 (c) = g −1 (c j ) + λ j =⇒ g −1 (c) > g −1 (c j) =⇒ c < c j

The planner raises the consumption of the type 1 agent in order to satisfy his

participation constraint The strictly positive Lagrange multiplier, λ j > 0 ,

implies that ( 20.3.1c ) holds with equality, χ j = 0 That is, the plannerraises the welfare of a type 1 agent just enough to make her indifferentbetween choosing autarky and staying with the optimal insurance contract

In effect, the planner minimizes the change in last period’s relative welfaredistribution that is needed to induce the type 1 agent not to abandonthe contract The welfare of the type 1 agent is raised both through the

mentioned higher consumption c j > c and through the expected higher

future consumption Recall our earlier finding that implies that the newhigher consumption level will remain unchanged so long as there are nobinding participation constraints It follows that the contract for agent 1

displays amnesia when agent 1’s participation constraint is binding, because the previously promised value x becomes irrelevant for the consumption

allocated to agent 1 from now on

Equilibrium consumption 699

3 The participation constraint of a type 2 person binds (θ j > 0 ), but λ j= 0

Thus, condition ( 20.4.2 ) becomes

g −1 (c) = (1 + θ j ) g −1 (c j) =⇒ g −1 (c) < g −1 (c j) =⇒ c > c j ,

where we have used the fact that g −1(·) is a negative number This situation

is the mirror image of the previous case When the participation constraint

of the type 2 agent binds, the planner induces the agent to remain withthe optimal contract by increasing her consumption (1− c j ) > (1 − c) but

only by enough that she remains indifferent to the alternative of choosing

autarky, Q j (χ j) = 0 And once again, the change in the welfare distributionpersists in the sense that the new consumption level will remain unchanged

so long as there are no binding participation constraints The amnesiaproperty prevails again

We can summarize the consumption dynamics of an efficient contract as

follows Given the current consumption c of the type 1 agent, next period’s consumption conditional on the endowment realization y j satisfies

20.4.2 Consumption intervals cannot contain each other

We will show that

y k > y q =⇒ c k > c q and c k > c q (20.4.4)

Hence, no consumption interval can contain another Depending on parametervalues, the consumption intervals can be either overlapping or disjoint

As an intermediate step, it is useful to first verify that the following assertion

is correct for any k, q = 1, 2, , S , and for any x ∈ [0, x q] :

Q k x + u(y q)− u(y k)

= Q q (x) + u(1 − y q)− u(1 − y k ) (20.4.5) After invoking functional equation ( 20.3.1 ), the left side of ( 20.4.5 ) is equal to

Q k x + u(y q)− u(y k)

c, {χj} S j=1

and ( 20.3.1c ) – ( 20.3.1e ) We can then verify ( 20.4.5 ).5 And after

differenti-ating that expression with respect to x,

that Q k(·) is decreasing, it follows from Q k x q + u(y q)− u(y k)

5 The two optimization problems on the left side and the right side,

respec-tively, of expression ( 20.4.5 ) share the common objective of maximizing the

expected utility of the type 2 agent, minus an identical constant The

optimiza-tion is subject to the same constraints, u(c) − u(y q ) + βS

restrictions on future continuation values in ( 20.3.1c ) and ( 20.3.1d ).

We leave it to the reader as an exercise to construct a symmetric argument

to show that y k > y q implies c k > c q

20.4.3 Endowments are contained in the consumption intervals

We will show that

y s ∈ [c s , c s ], ∀s; and y1= c1 and y S = c S (20.4.9) First, we show that y s ≤ c s for all s ; and y S = c S Let x = x s in the functional

equation ( 20.3.1 ), then c = c s and

x j and Q j(·) is strictly concave, and the second weak inequality is given by

( 20.4.8 ) In fact, we showed above that the second inequality holds strictly for

j < S and therefore, by the condition for optimality in ( 20.3.5b ),

Q  S (x S ) = (1 + θ j )Q  j (χ j) with θ j > 0, for j < S; and θ S = 0, which imply χ j = x j for all j After also invoking the corresponding expression ( 20.4.10 ) for s = S , we can complete the argument:

We leave it as an exercise for the reader to construct a symmetric argument

showing that y s ≥ c s for all s ; and y1= c1

20.4.4 All consumption intervals are nondegenerate (unless

autarky is the only sustainable allocation)

Suppose that the consumption interval associated with endowment realization

y k is degenerate, i.e., c k = c k = y k (The last inequality follows from section20.4.3 where we established that the endowment is contained in the consumptioninterval.) Since the consumption interval is degenerate, it follows that the range

of permissible continuation values associated with endowment realization y k is

also degenerate, i.e., χ k ∈ [0, x k] ={0} Recall that χ k is the amount of utilsawarded to the type 1 household over and above its autarkic welfare level, given

where we have invoked the degenerate consumption interval, c = y k, and where

χ j are optimally chosen subject to the constraints χ j ≥ 0 for all j ∈ S It

follows immediately that χ j = 0 for all j ∈ S, given the current endowment

realization y k

Due to the degenerate range of continuation values associated with

endow-ment realization y k , i.e., χ k ∈ {0}, it must be the case that the type 2 household

also receives its autarkic welfare level, given endowment realization y k;6

where we have invoked the degenerate consumption interval, and where

contin-uation values Q j (χ j ) are subject to the constraints Q j (χ j)≥ 0 for all j ∈ S It

6 Obviously, there would be a contradiction if the type 2 household were

to receive Q k (χ k ) > 0 Because then it would be possible to transfer current

consumption from the type 2 household to the type 1 household without ing the type 2 household’s participation constraint and hence, the consumption

violat-interval associated with endowment realization y could not be degenerate

Pareto frontier – ex ante division of the gains 703

follows immediately that Q j (χ j ) = 0 for all j ∈ S, given the current endowment

realization y k Hence, given endowment realization y k, we have continuation

values of the type 2 household satisfying Q j (χ j) = 0 so it must be the case that

the optimally chosen χ j are set at their maximum permissible values, χ j = x j

Moreover, we know from above that the optimal values also satisfy χ j= 0 and

therefore, we can conclude that x j = 0 for all j ∈ S.

We have shown that if one consumption interval is degenerate, then all

consumption intervals must be degenerate, i.e., c s = c s = y s for all s ∈ S This

finding seems rather intuitive A degenerate consumption interval associated

with any endowment realization y k implies that, given the realization of y k,none of the households have anything to gain from the optimal contract – neitherfrom current transfers nor from future risk sharing That can only happen ifautarky is the only sustainable allocation

20.5 Pareto frontier – ex ante division of the gains

We have characterized the optimal consumption dynamics of any efficient tract The consumption intervals{[c j , c j]} S

con-j=1 and the updating rules in ( 20.4.3 ) are identical for all efficient contracts The ex ante division of gains from an effi-

cient contract can be viewed as being determined by an implicit past

consump-tion level, c ∈ [c1, c S ] (by ( 20.4.9 ), this can also be written as c ∈ [y1, y S] )

A contract with an implicit past consumption level c = c1 gives all of the plus to the type 2 agent and none to the type 1 agent This follows immediately

sur-from the updating rules in ( 20.4.3 ) that prescribe a first period consumption level equal to c j if the endowment realization is y j The corresponding promised

value to the type 1 agent, conditional on endowment realization y j , is χ j = 0

Thus, the ex ante gain to the type 1 agent in expression ( 20.2.2c ) becomes

We can similarly show that a contract with an implicit consumption level c =

c S gives all of the surplus to the type 1 agent and none to the type 2 agent The

updating rules in ( 20.4.3 ) will then prescribe a first period consumption level equal to c j if the endowment realization is y with a corresponding promised

value of χ j = x j We can compute the ex ante gain to the type 1 agent as

environ-20.6 Consumption distribution

20.6.1 Asymptotic distribution

The asymptotic consumption distribution depends sensitively on whether thereexists any first-best sustainable allocation We say that a sustainable alloca-

tion is first best if the participation constraint of neither agent ever binds As

we have seen, non-binding participation constraints imply that consumptionremains constant over time Thus, a first-best sustainable allocation can existonly if the intersection of all the consumption intervals {[c j , c j]} S

j=1 is nonempty.Define the following two critical numbers

( 20.4.3 ) that consumption remains unchanged forever and therefore, the

asymp-totic consumption distribution is degenerate

But what happens if the ex ante division of gains is associated with an

implicit initial consumption level outside of this range, or if there does not exist

any first-best sustainable allocation (cmin < cmax)? To understand the vergence of consumption to an asymptotic distribution in general, we make the

con-following observations According to the updating rules in ( 20.4.3 ), any increase

in consumption between two consecutive periods has consumption attaining thelower bound of some consumption interval It follows that in periods of increas-

ing consumption, the consumption level is bounded above by cmax ( = c S) and

hence increases can occur only if the initial consumption level is less than cmax.Similarly, any decrease in consumption between two consecutive periods hasconsumption attain the upper bound of some consumption interval It followsthat in periods of decreasing consumption, consumption is bounded below by

cmin ( = c1) and hence decreases can only occur if initial consumption is higher

than cmin Given a current consumption level c , we can then summarize the permissible range for next-period consumption c  as follows:

if c ≤ cmax then c  ∈ [min{c, cmin}, cmax] , (20.6.2a)

if c ≥ cmin then c  ∈ [cmin , max{c, cmax}] (20.6.2b)

20.6.2 Temporary imperfect risk sharing

We now return to the case that there exist first-best sustainable allocations,

cmin ≥ cmax, but we let the ex ante division of gains be given by an implicit initial consumption level c ∈ [cmax, cmin] The permissible range for next-

period consumption, as given in ( 20.6.2 ), and the support of the asymptotic

We have monotone convergence in ( 20.6.3a ) for two reasons First, consumption

is bounded from above by cmax Second, consumption cannot decrease when

c ≤ cmin and by assumption cmin ≥ cmax, so consumption cannot decrease

when c ≤ cmax It follows immediately that cmax is an absorbing point that

is attained as soon as the endowment y S is realized with its consumption level

c S = cmax Similarly, the explanation for monotone convergence in ( 20.6.3b ) goes as follows First, consumption is bounded from below by cmin Second,

consumption cannot increase when c ≥ cmax and by assumption cmin≥ cmax, so

consumption cannot increase when c ≥ cmin It follows immediately that cmin

is an absorbing point that is attained as soon as the endowment y1 is realized

with its consumption level c1= cmin

These convergence results assert that imperfect risk sharing is at most porary if the set of first-best sustainable allocations is nonempty Notice thanwhen an economy begins with an implicit initial consumption outside of theinterval of sustainable constant consumption levels, the subsequent monotoneconvergence to the closest end point of that interval is reminiscent to our earlieranalysis in section 19.3 of the money lender and the villagers with one-sidedlack of commitment In the current setting, the agent who is relatively disad-vantaged under the initial welfare assignment will see her consumption weaklyincrease over time until she has experienced the endowment realization that ismost favorable to her From thereon, the consumption level remains constantforever and the participation constraints will never bind again

tem-Alternative recursive formulation 707

20.6.3 Permanent imperfect risk sharing

If the set of first-best sustainable allocations is empty (cmin< cmax), it breaksthe monotone convergence to a constant consumption level The updating rules

in ( 20.4.3 ) imply that the permissible range for next-period consumption in ( 20.6.2 ) will ultimately shrink to [cmin, cmax] , regardless of the initial welfareassignment If the implicit initial consumption lies outside of that set, consump-tion is bound to converge to it, again because of the monotonicity of consumption

when c ≤ cmin or c ≥ cmax And as soon as there is a binding participationconstraint with an associated consumption level that falls inside of the interval

[cmin, cmax] , the updating rules in ( 20.4.3 ) will never take us outside of this

interval again Thereafter, the only observed consumption levels belong to the

[cmin, cmax]2

{c j , c j } S j=1

(

with a unique asymptotic distribution Within this invariant set, the pation constraints of both agents occasionally bind, reflecting imperfections inrisk sharing

partici-If autarky is the only sustainable allocation, then each consumption interval

is degenerate with c j = c j = y j for all j ∈ S, as discussed in section 20.4.4.

Hence, the ergodic consumption set in ( 20.6.4 ) is then trivially equal to the set

of endowment levels, {y j } S

j=1

20.7 Alternative recursive formulation

Kocherlakota (1996b) used an alternative recursive formulation of the contractdesign problem, one that more closely resembles our treatment of the money-lender villager economy of section 19.3 After replacing the argument in the

function of ( 20.2.2a ) by the expected utility of the type 1 agent, Kocherlakota

writes the Bellman equation as

P (v) = max

{cs ,w s} S s=1

u(c s ) + βw s ≥ u(y s ) + βvaut, s = 1, , S; (20.7.1c)

u(1 − c s ) + βP (w s)≥ u(1 − y s ) + βvaut, s = 1, , S; (20.7.1d)

Here the planner comes into a period with a state variable v that is a promised

expected utility to the type 1 agent Before observing the current endowment

realization, the planner chooses a consumption level c sand a continuation value

w s for each possible realization of the current endowment This state-contingentportfolio {c s , w s } S

s=1 must deliver at least the promised value v to the type

1 agent, as stated in ( 20.7.1b ), and must also be consistent with the agents’ participation constraints in ( 20.7.1c ) and ( 20.7.1d ).

Notice the difference in timing with our presentation, which we have based

on Thomas and Worrall’s (1988) analysis Kocherlakota’s planner leaves the

current period with only one continuation value w s and postpones the question

of how to deliver that promised value across future states until the beginning

of next period but before observing next period’s endowment In contrast, in

our setting, in the current period the planner chooses a state-contingent set ofcontinuation values for the next period, {χ j } S

j=1 , where χ j is the number ofutils that the type 1 agent’s expected utility should exceed her autarky level

in the next period if that period’s endowment is y j We can evidently expressKocherlakota’s one state variable in terms of our state vector,

where vaut is the ex ante welfare level in autarky as given by ( 19.3.2 ) Similarly,

Kocherlakota’s upper bound on permissible values of next period’s continuation

value in ( 20.7.1f ) is related to our upper bounds {x j } S

Pareto frontier revisited 709

20.8 Pareto frontier revisited

Given our earlier characterization of the optimal solution, we can map

Kocher-lakota’s promised value v into an implicit promised consumption level c ∈

[c1, c S ] = [y1, y S ] Let that mapping be encoded in the function v(c ) Hence,

given a promised a value v(c ) , the optimal consumption dynamics in section20.4.1 instructs us to set Kocherlakota’s choice variables,{c s , w s } S

s=1, as follows:

c s = c + max{0, c s − c } − max{0, c − c s }, (20.8.1a)

For a given value of c , we define the following three sets that partition

the set S of endowment realizations:

So (c )≡'j ∈ S : c ∈ (c j , c j)

(

S− (c )≡'j ∈ S : c ≥ c j

According to our characterization of consumption intervals in ( 20.4.4 ), these

three sets are mutually exclusive and their union is equal to S So (c ) is theset of states, i.e., endowment realizations, for which the optimal consumption

level is c s = c But if the endowment realization falls outside of So (c ) ,

the optimal consumption c s is determined by either the upper or lower bound

of the consumption interval associated with that endowment realization In

particular, for s ∈ S − (c ) , consumption should drop to the upper bound of

the consumption interval, c s = c s ; and for s ∈ S+(c ) , consumption should

increase to the lower bound of the consumption interval, c s = c s

The continuation value v(c ) can then be expressed as

20.8.1 Continuous in implicit consumption

Both v(c ) and P (v(c )) are continuous in the implicit consumption level c

From ( 20.8.4 ) and ( 20.8.5 ) this is trivially true when variations in c do not

change the partition of states given by the sets So(·), S −(·) and S+(·) It

can also be shown to be true when variations in c do involve changes in thepartition of states As an illustration, let us compute the limiting values of

v(c ) when c approaches c k from below and from above, respectively, where

we recall that c k is the upper bound of the consumption interval associated with

endowment y k

We can choose a sufficiently small  > 0 such that

{c s , c s } S s=1

2

[c k − , c k + ] = c k

In particular, the findings in ( 20.4.4 ) ensure that we can choose a sufficiently small  so that this intersection contains no upper bounds on consumption intervals other than c k Similarly,  can be chosen sufficiently small so that the intersection does not contain any lower bound on consumption intervals unless

there exists a consumption interval with a lower bound that is exactly equal to

c k , i.e., if for some j ≥ 1, c k+j = c k We will have to keep this possibility in

mind as we proceed in our characterization of the sets So(·), S −(·) and S+(·).

Pareto frontier revisited 711

All the three sets are constant for an implicit consumption c ∈ [c k −, c k)with max{S − (c

)} = k−1 Concerning an implicit consumption c ∈ [c k , c k+

] , the set S − (c ) is constant with max{S − (c

)} = k while the configuration

of the other two sets depends on which one of the following two possible casesapplies

Case a): c k = c s for all s ∈ S Here it follows that the set S+(c ) is constant for

any implicit consumption c ∈ [c k −, c k +] Using ( 20.8.3 ) the limiting values

of v(c ) when c approaches c k from below and from above, respectively, arethen equal to

Case b): c k = c k+j for some j ≥ 1 Here it follows that the set S+(c )

is constant with min{S+(c )} = k + j for any implicit consumption c ∈

[c k − , c k] ; and S+(c ) is constant with min{S+(c )} = k + j + 1 for any

implicit consumption c ∈ (c k , c k + ] Using ( 20.8.3 ) the limiting values of

v(c ) when c approaches c k from below and from above, respectively, arethen equal to

where we have invoked the fact the c k+j = c k.

We have shown that v(c ) is continuous at the upper bound of any sumption interval even though the partition of states changes at such a point

con-Similarly, we can show that v(c ) is continuous at the lower bound of any

con-sumption interval And in the same manner, we can also establish that P (v(c ))

is continuous in the implicit consumption c

20.8.2 Differentiability of the Pareto frontier

Consider an implicit consumption level c ∈ [y1, y S] that falls strictly inside

at least one consumption interval We can then use expressions ( 20.8.4 ) and ( 20.8.5 ) to compute the derivative of the Pareto frontier at v(c ) by differen-

tiating with respect to c :

P  (v(c )) =

dP (v(c ))

dc dv(c )

dc

=− u (1− c )

u  (c ) . (20.8.8)

It can be verified that ( 20.8.8 ) is the derivative of the Pareto frontier so long as

the set So (c ) remains nonempty That is, changes in the set So (c ) induced

by varying c do not affect the expression for the derivative in ( 20.8.8 ) This

follows from the fact that the derivatives are the same to the left and to the

right of an implicit consumption level where the set So (c ) changes; and the

fact that v(c ) and P (v(c )) are continuous in the implicit consumption level,

as shown in section 20.8.1 It can also be verified that the derivative in ( 20.8.8 )

exists in the knife-edged case that happens when So (c ) becomes empty at asingle point because two adjacent consumption intervals share only one point,

i.e., when c k = c k+1 which implies that So (c k) = So (c k+1) =∅.

The Pareto frontier becomes nondifferentiable when two adjacent tion intervals are disjoint Consider such a situation where an implicit consump-

consump-tion level c ∈ [y1, y S] does not fall inside any consumption interval, which

implies that the set So (c ) is empty Let y k and y k+1 be the endowment alizations associated with the consumption interval to the left and to the right

re-Continuation values ` a la Kocherlakota 713

of c , respectively That is,

c k < c < c k+1

According to ( 20.8.3 ), the continuation value for any implicit consumption level

c ∈ [c k , c k+1] is then constant and equal to

By using expression ( 20.8.8 ), we can compute the derivative of the Pareto

fron-tier on the left side and the right side of ˆv ,

and hence, the Pareto frontier is not differentiable at ˆv 7

7 Kocherlakota (1996b) prematurely assumed that Thomas and Worrall’s

(1988) demonstration of the differentiability of the Pareto frontier Q s(·) would

imply that his conceptually different frontier P ( ·) would be differentiable Koeppl

(2003) uses the approach of Benveniste and Scheinkman (1979) to establish a

sufficient condition for differentiability of the Pareto frontier P (v) For a given value of v , the sufficient condition is that there exist at least one realization

of the endowment such that the participation constraints are not binding for

any household in that state, i.e., our set So (c ) should be nonempty for the

implicit consumption level c associated with that particular value of v That condition is sufficient but not necessary, since we have seen above that P (v) is also differentiable at a knife-edged case with c = c k = c k+1, even though the

set So (c ) would then be empty

20.9 Continuation values ` a la Kocherlakota

20.9.1 Asymptotic distribution is nondegenerate for imperfect risk

Here we assume that there exist sustainable allocations other than autarky butthat first-best outcomes are not attainable, i.e., there exist sustainable alloca-tions with imperfect risk sharing Kocherlakota (1996b, Proposition 4.2) statesthat the continuation values will then converge to a unique nondegenerate dis-tribution Here we will verify that the claim of a nondegenerate asymptotic

distribution is correct except for when there are only two states ( S = 2 ) The assumption that the distribution of y t is identical to that of 1− y t

means that

y j= 1− y S+1−j , (20.9.1b) for all j ∈ S The symmetric environment bestows symmetry on the consump-

tion intervals of section 20.4

c j = 1− c S+1−j , (20.9.1c) for all j ∈ S; and symmetry on the continuation values of the type 1 and type

2 household

v(c ) = P (v(1 − c )) (20.9.1d)

As discussed in section 20.6.1, the condition for the nonexistence of

first-best sustainable allocations is that cmin< cmax, which by ( 20.6.1 ) is the same

as

c1< c S =⇒ c S > 0.5 (20.9.2) where the implication follows from using c1= 1− c S as given by ( 20.9.1c ) It is quite intuitive that the consumption interval [c S , c S] associated with the highest

endowment realization y S cannot contain the average value of the stochasticendowment, S

i=1Πi y i = 0.5 Otherwise, there would certainly exist first-best

sustainable allocations, i.e., a contradiction

To prove the existence of a nondegenerate asymptotic distribution of tinuation values, it is sufficient to show that the continuation value of an agent

con-Continuation values ` a la Kocherlakota 715

experiencing the highest endowment, say, the type 1 household, exceeds thecontinuation value of the other agent who is then experiencing the lowest en-dowment, say, the type 2 household Given her current realization of the highest

endowment y S, the type 1 household is awarded the highest consumption level

c S ( = cmax) in the ergodic consumption set of ( 20.6.4 ) Conditional upon next period’s endowment realization y i, the type 1 household’s consumption ˆc i in

the next period is determined by ( 20.8.1a ), where c = c S From ( 20.4.4 ) we know that c S ≥ c i for all i ∈ S, so next period’s consumption of the type 1

household as determined by ( 20.8.1a ) can be written as

ˆ

c i= min{c i , c S } (20.9.3)

Given the vector {ˆc i } S

i=1 for next period’s consumption, we can use ( 20.8.3 )

to compute the type 1 household’s outgoing continuation value in the currentperiod,

for all i ∈ S, and at least one of them holds with strict inequality The proof

proceeds by considering four possible cases for each i ∈ S.

Case a): c i ≤ c S and c S+1−i ≤ c S According to ( 20.9.1c ) c S+1−i= 1− c i, so

inequality ( 20.9.5 ) can then be written as

c i ≥ 1 − c S+1−i = c i which is true since c i > c i for all i ∈ S,

as established in section 20.4.4

Case b): c i ≤ c S and c S+1−i > c S According to ( 20.9.1c ) c i= 1− c S+1−i, so

inequality ( 20.9.5 ) can then be written as

c i= 1−c S+1−i ≥ 1−c S which is true since c S+1−i < c S for all i = 1,

as established in section 20.4.2

Case c): c i > c S and c S+1−i ≤ c S According to ( 20.9.1c ) c S+1−i= 1− c i, so

inequality ( 20.9.5 ) can then be written as

c S ≥ 1 − c S+1−i = c i which is true since c S > c i for all i = S,

positive if there are more than two states and hence, the asymptotic distribution

of continuation values is nondegenerate But what about when there are only

two states ( S = 2 )? Since c1 < c S by ( 20.9.2 ) and c S > c S, it follows that

case b) applies when i = 1 and case c) applies when i = 2 = S Therefore, the difference in ( 20.9.4 ) is zero and thus, the continuation value of an agent

Continuation values ` a la Kocherlakota 717

experiencing the highest endowment is equal to that of the other agent who isthen experiencing the lowest endowment Since there are no other continuationvalues in an economy with only two possible endowment realizations, it followsthat the asymptotic distribution of continuation values is degenerate when there

are only two states ( S = 2 ).

A two-state example in section 20.10 illustrates our findings The intuitionfor the degenerate asymptotic distribution of continuation values is straightfor-ward On the one hand, the planner would like to vary continuation valuesand thereby avoid large changes in current consumption that would otherwise

be needed to satisfy binding participation constraints But, on the other hand,different continuation values presuppose that there exist “intermediate” states

in which a higher continuation value can be awarded In our two-state example,the participation constraint of either one or the other type of agent always bindsand the asymptotic distribution is degenerate with only one continuation value

20.9.2 Continuation values do not always respond to binding

participation constraints

Evidently, continuation values will eventually not respond to binding pation constraints in a two-state economy, since we have just shown that theasymptotic distribution is degenerate with only one continuation value But theoutcome that continuation values might not respond to binding participationconstraints occurs even with more states when endowments are i.i.d In fact,

partici-it is present whenever the consumption intervals of two adjacent endowment

realizations, y k and y k+1 , do not overlap, i.e., when c k < c k+1 Here is howthe argument goes

Since c k < c k+1 it follows from ( 20.6.4 ) that both c k and c k+1 belong

to the ergodic set of consumption Moreover, ( 20.4.4 ) implies that S o (c k) =

So (c k+1) = ∅, where S o

(·) is defined in (20.8.2a) Using expression (20.8.3),

we can compute a common continuation value v(c k ) = v(c k+1) = ˆv , where ˆ v

is given by ( 20.8.9 ) when that expression is evaluated for any c ∈ [c k , c k+1] Given this identical continuation value, it follows that there are situations wherehouseholds’ continuation values will not respond to binding participation con-straints

As an example, let the current consumption and continuation value of the

type 1 household be c k and v(c k) = ˆv , and suppose that the household next

period realizes the endowment y k+1 It follows that the participation constraint

of the type 1 household is binding and that the optimal solution in ( 20.8.1 ) is to award the household a consumption level c k+1 and continuation value v(c k+1) That is, the household is induced not to defect into autarky by increasing its con-

sumption, c k+1 > c k , but its continuation value is kept unchanged, v(c k+1) = ˆv

Suppose next that the type 1 household experiences y k in the following period.This time it will be the participation constraint of the type 2 households that

binds and the optimal solution in ( 20.8.1 ) prescribes that the type 1 hold is awarded consumption c k and continuation value v(c k) = ˆv Hence,

house-only consumption levels but not continuation values are adjusted in these tworealizations with alternating binding participation constraints

We use a three-state example in section 20.11 to elaborate on the point thateven though an incoming continuation value lies in the interior of the range of

permissible continuation values in ( 20.7.1f ), a binding participation constraint

still might not trigger a change in the outgoing continuation value because theremay not exist any efficient way to deliver a changed continuation value Con-tinuation values that do not respond to binding participation constraints are

a manifestation of the possibility that the Pareto frontier P ( ·) need not be

differentiable everywhere on the interval [vaut, vmax] , as shown in section 20.8.2

20.10 A two-state example: amnesia overwhelms

memory

In this example and the three-state example of the following section, we use theterm “continuation value” to denote the state variable of Kocherlakota (1996b)

as described in the preceding section.8 That is, at the end of a period, the

continuation value v is the promised expected utility to the type 1 agent that

will be delivered at the start of the next period

Assume that there are only two possible endowment realizations, S = 2 ,

with {y1, y2} = {1 − y, y}, where y ∈ (.5, 1) Each endowment realization is

equally likely to occur, {Π1 , Π2} = {0.5, 0.5} Hence, the two types of agents

8 See Krueger and Perri (2003b) for another analysis of a two state example