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

Chapter 20 Equilibrium without Commitment 20.1. Two-sided lack of commitment In section 19.3 of the previous chapter, we studied insurance without commit- ment. That was a partial equilibrium analysis since the moneylender could borrow or lend resources outside of the village at a given interest rate. Recall also the asymmetry in the environment where villagers could not make any com- mitments while the moneylender was assumed to be able to commit. We will now study a closed system without access to an outside credit market. Any household’s consumption in excess of its own endowment must then come from the endowments of the other households in the economy. We will also adopt the symmetric assumption that everyone is unable to make commitments. That is, any contract prescribing an exchange of goods today in anticipation of future exchanges of goods represents a sustainable allocation only if both current and future exchanges are incentive compatible to all households involved in the con- tractual arrangement. Households are free to walk away from the arrangement at any point in time and to defect into autarky. Such a contract design problem with participation constraints on both sides of an exchange represents a problem with two-sided lack of commitment, as compared to the problem with one-sided lack of commitment in section 19.3. This chapter draws upon the work of Thomas and Worrall (1988) and Kocherlakota (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 – Aclosedsystem 693 20.2. A closed system Thomas and Worrall’s (1988) model of self-enforcing wage contracts is an an- tecedent to our villager-money lender environment. The counterpart to our ‘money lender’ in their model is a risk-neutral firm that forms a long-term re- lationship with a risk-averse worker. In their model, there is also a competitive 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, he can never again enter into a long-term relationship with another firm. The firm seeks 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 in the spot market for labor). If Thomas and Worrall had assumed a commitment problem only on the part of the worker, their model would be formally identical to our villager-money lender environment. However, Thomas and Worrall also assume that the firm itself can renege on a wage contract and buy labor at the random spot market wage. Hence, they require that a self-enforcing wage contract be one in which neither party ever has an incentive to renege. Kocherlakota (1996b) studies a model that has some valuable features in common with Thomas and Worrall’s. Kocherlakota’s counterpart to Thomas and 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 while respecting 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 empirically estimated conditional consumption-income covariances that seem to violate the hypothesis of complete risk-sharing. Kocherlakota investigates the extent to which those failures reflect impediments to enforcement that are captured by his participation constraints. For the purpose of studying those conditional covariances in a stationary stochastic environment, Kocherlakota’s use of an environment with two-sided lack of commitment is important. In our model of villagers facing a money lender in section 19.3, imperfect risk sharing is temporary and so would not prevail in a stochastic steady state. In Kocherlakota’s model, imperfect risk 694 Equilibrium without Commitment sharing can be perpetual. There are equal numbers of two types of households in the village. Each of the households has the preferences, endowments, and autarkic utility possibilities described in chapter 19. Here we assume that the endowments 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 planner has access to neither borrowing nor lending opportunities, and is confined to reallocating 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 sustainable 1 if for all t ≥ 0and 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 ∞  j=1 β j−1 [u(1 −c t+j ) −u(1 −y t+j )] ≥ 0. (20.2.1b) Let Γ denote the set of sustainable allocations. We seek the following function: Q()=max {c t } E −1 ∞  t=0 β t [u(1 −c t ) −u(1 −y t )] (20.2.2a) subject to {c t }∈Γ, (20.2.2b) E −1 ∞  t=0 β t [u(c t ) −u(y t )] ≥. (20.2.2c) The function Q() depicts a (constrained) Pareto frontier. It portrays the maximized value of the expected life-time utility of the type 2 household, where the maximization is subject to requiring that the type 1 household receive an expected 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 an efficient contract. 20.3. Recursive formulation We choose to study Kocherlakota’s model using the approach proposed by Thomas and Worrall. 2 Thomas and Worrall (1988) formulate the contract de- sign problem as a dynamic program, where the state of the system prior to the current period’s endowment realization is given by a vector [x 1 x 2 x s x S ]. Here x s is the value of expression (20.2.1a)thatispromisedtoatype1agent conditional on the current period’s endowment realization being y s .LetQ s (x s ) then denote the corresponding value of expression (20.2.1b)thatispromisedtoa 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 Q s (x)= max c, {χ j } S j=1  u(1 −c) −u(1 − y s )+β S  j=1 Π j Q j (χ j )  (20.3.1a) subject to u(c) −u( y s )+β S  j=1 Π j χ j ≥ x, (20.3.1b) 2 Kocherlakota uses an approach that can be regarded as extending the ap- proach that we used in the villager-money lender model of section 19.3 to an environment with two-sided lack of commitment. We followed Kocherlakota in using this approach in chapter 15 of the first edition of this book. However, when applied to problems with two-sided lack of commitment, this approach encounters a technical difficulty associated with possible kinks in the Pareto frontier. (We first encountered this difficulty when we assigned a version of exercise 20.3 to our students.) Thomas and Worrall’s approach avoids this non-differentiability problem by using conditional Pareto frontiers, one for each realization of the endowment. 3 Q s (·) is a Pareto frontier conditional on the endowment realization y s while Q(·)in(20.2.2a)isanex ante Pareto frontier before observing any endowment realization. 696 Equilibrium without Commitment χ j ≥ 0,j=1, ,S;(20.3.1c) Q j (χ j ) ≥ 0,j=1, ,S;(20.3.1d) c ∈ [0, 1], (20.3.1e) 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 ]forj =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 Q j (x j )=0. (20.3.2) 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 ,andβΠ j θ j to expressions (20.3.1b), (20.3.1c), and (20.3.1d), the first-order conditions for c and χ j are 4 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 ≥ 0and1− c ≥ 0, are not binding. This conjecture is confirmed below when it is shown that optimal consumption levels satisfy c ∈ [ y 1 , y S ]. Equilibrium consumption 697 By the envelope theorem, Q  s (x)=−µ. (20.3.4) After substituting (20.3.4) into (20.3.3a)and(20.3.3b), respectively, the opti- mal choices of c and χ j satisfy Q  s (x)=− u  (1 −c) u  (c) , (20.3.5a) Q  s (x)=(1+θ j )Q  j (χ j )+λ j . (20.3.5b) 20.4. Equilibrium consumption 20.4.1. Consumption dynamics From equation (20.3.5a), the consumption c of a type 1 agent is an increasing function of the promised value x. The properties of the Pareto frontier Q s (x) imply that c is a differentiable function of x on [0, x s ]. Since x ∈ [0, x s ], c is contained in the non-empty compact interval [c s , c s ], where Q  s (0) = − u  (1 −c s ) u  (c s ) and Q  s (x s )=− u  (1 −c s ) u  (c s ) . Thus, if c = c s , x = 0 so that a type 1 agent gets no gain from the contract from then on. If c = c s , Q s (x)=Q s (¯x s ) = 0 so that a type 2 agent gets no gain. Equation (20.3.5a) can be expressed as c = g(Q  s (x)) , (20.4.1) 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 j .The 698 Equilibrium without Commitment optimal consumption dynamics implied by an efficient contract are evidently governed 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 the agents are offered full insurance against endowment realizations so long as there 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 planner raises the welfare of a type 1 agent just enough to make her indifferent between choosing autarky and staying with the optimal insurance contract. In effect, the planner minimizes the change in last period’s relative welfare distribution that is needed to induce the type 1 agent not to abandon the contract. The welfare of the type 1 agent is raised both through the mentioned higher consumption c j >cand through the expected higher future consumption. Recall our earlier finding that implies that the new higher consumption level will remain unchanged so long as there are no binding 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 , wherewehaveusedthefactthatg −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 with the 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 distribution persists in the sense that the new consumption level will remain unchanged so long as there are no binding participation constraints. The amnesia property 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 c j =    c j if c<c j (p.c. of type 1 binds), c if c ∈ [c j , c j ] (p.c. of neither type binds), c j if c>c j (p.c. of type 2 binds). (20.4.3) 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 parameter values, 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,andforanyx ∈ [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 )  =max c, {χ j } S j=1  u(1 −c) −u(1 − y k )+β S  j=1 Π j Q j (χ j )  700 Equilibrium without Commitment subject to u(c) −u( y k )+β S  j=1 Π j χ j ≥ x + u(y q ) −u(y k ) and (20.3.1c)–(20.3.1e); and the right side of (20.4.5) is equal to Q q (x)+u(1 −y q ) −u(1 −y k ) =max c, {χ j } S j=1  u(1 −c) −u(1 − y q )+β S  j=1 Π j Q j (χ j )  + u(1 −y q ) −u(1 −y k ) subject to u(c) −u( y q )+β S  j=1 Π j χ j ≥ x and (20.3.1c)–(20.3.1e). Wecanthenverify(20.4.5). 5 And after differenti- ating that expression with respect to x, Q  k  x + u( y q ) −u(y k )  = Q  q (x). (20.4.6) To show that y k > y q implies c k > c q ,setx = x q in expression (20.4.5), Q k  x q + u(y q ) −u(y k )  = u(1 −y q ) −u(1 −y k ) > 0, (20.4.7) wherewehaveusedQ q (x q ) = 0. After also invoking Q k (x k )=0 andthefact that Q k (·) is decreasing, it follows from Q k  x q + u(y q ) −u(y k )  > 0that x k > x q + u(y q ) −u(y k ). So by the strict concavity of Q k (·), we have Q  k (x k ) <Q  k  x q + u(y q ) −u(y k )  = Q  q (x q ), (20.4.8) 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 )+β  S j=1 Π j χ j ≥ x and (20.3.1c)–(20.3.1e). Hence, they are identical well-defined optimization prob- lems. The observant reader should not be concerned with the fact that Q k (·) ontheleftsideof(20.4.5) might be evaluated at a promised value outside of the range [0, ¯x k ]. This constitutes no problem because the optimization prob- lem imposes no participation constraint in the current period, in contrast to the restrictions on future continuation values in (20.3.1c)and(20.3.1d). Equilibrium consumption 701 where the equality is given by (20.4.6). Finally, by using function (20.4.1) and the present finding that Q  k (x k ) <Q  q (x q ), we can verify our assertion that c k = g(Q  k (x k )) >g  Q  q (x q )  = c q . 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;andy 1 = c 1 and y S = c S . (20.4.9) First, we show that y s ≤ c s for all s;andy S = c S .Letx = x s in the functional equation (20.3.1), then c = c s and u(1 − c s ) −u(1 −y s )+β S  j=1 Π j Q j (χ j )=0 (20.4.10) with {χ j } S j=1 being optimally chosen. Since Q j (χ j ) ≥ 0, it follows immediately that u(1 − c s ) −u(1 −y s ) ≤ 0=⇒ y s ≤ c s . To establish strictly equality for s = S,wenotethat Q  j (χ j ) ≤ Q  j (x j ) ≤ Q  S (x S ), where the first weak inequality follows from the fact that all permissible χ j ≤ 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 )withθ j > 0, for j<S;andθ 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: β S  j=1 Π j Q j (x j )=0 =⇒ u(1 −c S ) −u(1 −y S )=0 =⇒ y S = c S . [...]... (20. 10.2a) (20. 10.2b) where v is given by (20. 10.1 ) We graphically illustrate how c is chosen in order to maximize (20. 10.2a) subject to (20. 10.2b ) in Figures 20. 10.1 and 20. 10.2 for utility function (1 − γ)−1 c1−γ and parameter values (β, γ, y) = (.85, 1.1, 6) It can be veri ed numerically that c = 536 Figure 20. 10.1 shows (20. 10.2a) as a decreasing function of c in the interval [.5, 6] Figure 20. 10.2... 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 the term “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 promise keeping constraints (20. 10.3a) and (20. 10.3b ) and the participation constraint (20. 10.4b ) of a type 2 agent when it receives 1 − y at equality: v u(1 − c+ ) + βP (ˆ) = u(1 − y) + βvaut (20. 10.5) Equation (20. 10.3b ) and (20. 10.5 ) are two equations in (c+ , P (vmax )) After solving them, we can solve (20. 10.3a) for vmax Substituting (20. 10.5 ) into (20. 10.3b ) gives P (vmax ) = 5[u(1... continuation value in (20. 7.1f ) is related to our upper bounds {xj }S , j=1 S vmax = vaut + Πj xj = vaut + j=1 max Pareto frontier revisited 709 20. 8 Pareto frontier revisited Given our earlier characterization of the optimal solution, we can map Kocherlakota’s promised value v into an implicit promised consumption level c ∈ [c1 , cS ] = [y 1 , y S ] Let that mapping be encoded in the function v(c... According to (20. 9.1c) cS+1−i = 1 − ci , so inequality (20. 9.5 ) can then be written as ci ≥ 1 − cS+1−i = ci which is true since ci > ci for all i ∈ S, as established in section 20. 4.4 Case b): ci ≤ cS and cS+1−i > cS According to (20. 9.1c) ci = 1 − cS+1−i , so inequality (20. 9.5 ) can then be written as ci = 1−cS+1−i ≥ 1−cS which is true since cS+1−i < cS for all i = 1, as established in section 20. 4.2... cS+1−i ≤ cS According to (20. 9.1c) cS+1−i = 1 − ci , so inequality (20. 9.5 ) can then be written as cS ≥ 1 − cS+1−i = ci which is true since cS > ci for all i = S, as established in section 20. 4.2 Case d): ci > cS and cS+1−i > cS The inequality (20. 9.5 ) can then be written as cS ≥ 1 − cS which is true since cS > 0.5 as established in (20. 9.2 ) We can conclude that the inequality (20. 9.5 ) holds with strict... type 1 household’s consumption ci in ˆ the next period is determined by (20. 8.1a), where c = cS From (20. 4.4 ) we know that cS ≥ ci for all i ∈ S, so next period’s consumption of the type 1 household as determined by (20. 8.1a) can be written as ci = min{ci , cS } ˆ (20. 9.3) Given the vector {ˆi }S for next period’s consumption, we can use (20. 8.3 ) c i=1 to compute the type 1 household’s outgoing continuation... of (20. 2.2a) by the expected utility of the type 1 agent, Kocherlakota writes the Bellman equation as S P (v) = Πs u(1 − cs ) + βP (ws ) max {cs ,ws }S s=1 (20. 7.1a) s=1 subject to S Πs [u(cs ) + βws ] ≥ v, s=1 (20. 7.1b) 708 Equilibrium without Commitment u(cs ) + βws ≥ u(ys ) + βvaut , s = 1, , S; (20. 7.1c) u(1 − cs ) + βP (ws ) ≥ u(1 − y s ) + βvaut , s = 1, , S; (20. 7.1d) cs ∈ [0, 1], (20. 7.1e)... 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 ws and postpones the question of how to deliver that promised value across... goes Since ck < ck+1 it follows from (20. 6.4 ) that both ck and ck+1 belong to the ergodic set of consumption Moreover, (20. 4.4 ) implies that So (ck ) = So (ck+1 ) = ∅ , where So (·) is defined in (20. 8.2a) Using expression (20. 8.3 ), we can compute a common continuation value v(ck ) = v(ck+1 ) = v , where v ˆ ˆ is given by (20. 8.9 ) when that expression is evaluated for any c ∈ [ck , ck+1 ] Given this . Chapter 20 Equilibrium without Commitment 20. 1. Two-sided lack of commitment In section 19.3 of the previous chapter, we studied insurance without commit- ment. That was a. Jermann (200 0). – 692 – Aclosedsystem 693 20. 2. A closed system Thomas and Worrall’s (1988) model of self-enforcing wage contracts is an an- tecedent to our villager-money lender environment. The. represents a problem with two-sided lack of commitment, as compared to the problem with one-sided lack of commitment in section 19.3. This chapter draws upon the work of Thomas and Worrall (1988)