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Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/inequalitysocialOOfarh 3 DEWEY HB31 .M415 Massachusetts Institute of Technology Department of Econonnics Working Paper Series INEQUALITY AND SOCIAL DISCOUNTING Emmanuel Farhi Ivan Warning Working Paper 05-1 April 27, 2005 Room E52-251 50 Memorial Drive Cambridge, MA 02142 This paper can be downloaded without charge from the Network Paper Collection at http://ssrn.com/abstract=712803 Social Science Research MAY S 2005 LIBRARIES Inequality and Social Discounting* Emmanuel MIT Ivan Werning MIT, NBER and UTDT [email protected] [email protected] Farhi First Draft: June 2004 This Version: April 2005 Abstract To what degree should societies allow inequality to be inherited? What taxation play in shaping the intergenerational transmission of welfare? tions by modeling altruistically-linked individuals who role should estate We explore these ques- experience privately observed taste or Our positive economy is identical to models with infinite-lived individwhere efficiency requires immiseration: inequality grows without bound and everyone's consumption converges to zero. However, under an intergenerational interpretation, previous work only characterizes a particular set of Pareto-efficient allocations: those that value only the initial generation's welfare. We study other efficient allocations where the social welfare criterion values future generations directly, placing a positive weight on their welfare so that the effective social discount rate is lower than the private one. For any such difference in social and private discounting we find that consumption exhibits mean-reversion and that a steadystate, cross-sectional distribution for consumption and welfare exists, where no one is trapped at misery. The optimal allocation can then be implemented by a combination of income and productivity shocks. uals estate taxation. We find that the optimal estate tax is progressive: fortunate parents face higher average marginal tax rates on their bequests. 1 Introduction Societies inevitably choose the inheritability of inequality. tunity for newborns and incentives for altruistic parents act plays out to determine long-run inequality Some balance between is struck. We explore equality of oppor- how and draw some novel implications for this balancing optimal estate taxation. we thank Daron Acemoglu, Fernando Alvarez, George-Marios Angeletos, Gary Becker, Olivier Blanchard, Ricardo Caballero, Dean Corbae, Bengt Holmstrom, Narayana Kocherlakota, Robert Luccis, Casey Mulligan, Roger Myerson, Chris Phelan, Gilles Saint-Paul, Nancy Stokey, Jean Tirole and seminar and conference participants at Chicago, Minnesota, MIT and the Texas Monetary Conference held at the University of Austin in honor of the late Scott Freeman. This work begun motivated by a seminar presentation of Chris Phelan at MIT in May 2004. We also have gained significant insight from a manuscript by Freeman and Sadler we thank Dean Corbae for bringing it to our attention. 'For useful discussions and comments Abhijit Banerjee, — Existing normative models of inequality reach an extreme conclusion: inequality should be per- and fectly inheritable without bound, with everyone converging to absolute misery and rise steadily a vanishing lucky fraction to bliss. This immiseration result and obtains invariably tions on preferences (Phelan, 1998); Thomas and in partial equilibrium (Green, 1987, Worrall, 1990), in general equilibrium (Atkeson and Lucas, 1992), and across environ- ments with moral-hazard regarding work or productivity (Aiyagari We weak assump- robust; requires very is and Alvarez, effort or with private information regarding preferences 1995).^ depart minimally from these contributions, adopting the same positive economic models, but a shghtly different normative criterion. In a generational context, previous work with lived agents characterizes the instance where future generations are not considered On only indirectly through the altruism of earlier ones. (2005) proposes a social planner with equal weights on infinite- directly, but the opposite side of the spectrum, Phelan all future generations. Our interest here is in exploring a class of Pareto-efficient allocations that take into account the current population along with unborn future generations. We place a positive and vanishing Pareto weight on the expected utility of future generations, this leads effectively to a social discount rate that is lower than the private one. This relatively small change produces a drastically different bounded, a steady-state, cross-sectional distribution exists lower for and everyone avoids misery. Indeed, welfare bility is possible bound that is strictly better result: long-run inequality remains consumption and welfare, typically remains above an social than misery. This outcome holds however small the difference between social and private discounting, and regardless of whether the source of asymmetric mation We is begin by modeling a positive economy that is is identical to the taste-shock setup developed composed of a continuum of individuals who one period and are altruistic towards a single descendant. There ment of the only consumption good cations. Feasible allocations constraint in all live a constant aggregate endow- but experience — thus ruling out must be incentive compatible and must first-best allo- satisfy the aggregate resource periods. only the welfare of the to that of an is in each period. Individuals are ex-ante identical, idiosyncratic shocks to preferences that are only privately observed When infor- privately observed preferences or productivity shocks. by Atkeson and Lucas (1992). Each generation for mo- endogenous economy with first generation is considered, the planning problem infinite-lived individuals. Intuitively, is equivalent immiseration then results from the desire to smooth the dynastic consumption path: rewards and punishments, required for incentives, are best delivered component that leads of the distribution 'Many is permanently. As a result, the consumption process inherits a random-walk cross-sectional inequahty to grow endlessly without bound. consistent with a constant aggregate find the immiseration result perplexing and some even endowment only find it if Infinite spreading everyone's consump- morally questionable, but venient from a practical standpoint. Long-run steady-states often provide a natural benchmark it is also incon- to study dynamic economies, but such long-run analyses are not possible for private-information economies without a steady-state distribution with positive consumption. This has impaired the study of long-run implications of optimal taxation, so common in the Ramsey taxation literature. tion eventually converges to zero. Note, that as a consequence, no steady-state, cross-sectional distribution with positive consumption exists. Across generations, this arrangement requires a lock-step link between the welfare of parent and child. — but By Of course, the perfect intergenerational transmission of welfare improves parental incentives at the expense of exposing newborns to the risk of their parent's luck. contrast, no longer it remains optimal to link the fortunes of parents and children in our model, but Rewards and punishments are distributed over in lock-step. manner. This creates a mean-reverting tendency in a front-loaded random walk — that is strong enough to cross-sectional distribution for bound long-run future descendants, but all consumption in The inequality. result is — instead of a a steady-state consumption and welfare, with no fraction of the population stuck at misery. We also study a repeated Mirrleesian version of our economy and derive imphcations for optimal estate taxation. In this model, individuals have identical preferences with regard to consumption and work but are heterogenous in the productivity of their work effort, productivity and work effort is private — only the resulting output is effort. Information about publicly observable. We show that the analysis from the taste-shock model carries over to this setup, virtually without change. In particular, a very similar Bellman equation characterizes the solution to the social planning prob- lem: consumption exhibits mean-reversion and has a steady-state outcome highlights the do not require any particular asymmetry of information. fact that our results More importantly, the Mirrleesian model locations can be offers new cross-sectional distribution. This insights into estate taxation. implemented by combining income and estate taxes. Specifically, Feasible we al- find that a progressive estate tax, which imposes a higher average marginal tax rate on the bequests of fortu- nate parents, is optimal. This result reflects the mean-reversion of consumption: more fortunate dynasties, with relatively high levels of current consumption, must have a declining consumption path induced by higher estate tax rates that lower the net rates of return across generations. Finally, an important methodological contribution of planning problem recursively. In doing so, (1987) to situations where private and we extend this paper is to reformulate the social ideas introduced by Spear social preferences differ. Indeed, we and Srivastava are able to reduce the dynamic program to a one-dimensional state variable, and our analysis and results heavily exploit the resulting Bellman equation.'^ Our dynamic program Related Literature. Our paper social planning most it strictly positive must future. efficient and accurate 4. closely related to problem with no discounting of the planning problem exists then problem have is an also offers computational strategy, as illustrated by an example in Section Phelan (2005), who considered a He shows that solve a static maximization problem, if a steady state for the and that solutions to this inequahty and social mobility. Our paper establishes the existence of a steady-state distribution for the planning problem for any difference in social and private discounting. Unlike the case with no discounting, there ^We is no associated static planning problem also present a recursive formulation with a two-dimensional state- variable that has a very simple interpretation and is useful for developing an intuitive understanding of our results. for economic steady-state distributions, and as a result, the methods we develop here In overlapping-generation models without altruistic links, all weight on future generations. Bernheim (1989) was the efficient place positive direct out that in the dynastic extension of these models with altruism, are not attainable by the market. The Our work estate tax is negative first to point efficient allocations (1995) argued that these Pareto efficient allocations — it is a subsidy equilibria with estate — so as to internalize the externality contributes to a large literature on dynamic economies with asymmetric information. In addition to the work mentioned above, this includes recent research on dynamic optimal taxation Golosov, Kocherlakota and Tsyvinski, 2003; Albanesi and Sleet, 2004; and Kocherlakota, 2004). This hterature has been handicapped by the immiseration result steady-state distribution with positive consumption, making for Pareto on future generations. of giving (e.g., Kaplow many and that they can be implemented by market are natural social objectives taxation policy. are very different. market equilibria that are Pareto it and by the non-existence difficult to of a draw long-run conclusions optimal taxation. Our results provide an encouraging way to overcome this problem. Our work is also indirectly related to a paper by Sleet and Yeltekin (2004), who study an Atkeson- Lucas environment with a utilitarian planner, who lacks commitment and cares only generation. In this environment, as in Phelan's, hold, so the interesting question is how it is for the current a foregone conclusion that immiseration will not to solve for the best subgame-perfect equilibria. Sleet and Yeltekin derive first-order conditions from a Lagrangian and use these to numerically simulate the solution. Interestingly, it turns out that the best allocation in their no-commitment environment is asymptotically equivalent to the optimal one with commitment but featuring a more patient welfare criterion. Thus, our analysis and results provide an indirect, but effective way of characterizing the no-commitment problem and of formally establishing that a steady-state distribution with no one at misery exists. The and rest of the sets up the paper is organized as follows. Section 2 introduces the economic environment social planning problem. In Section 3, problem and draw its we develop a connection to our original formulation. recursive version of the planning The resulting Bellman equation is then put to use in Section 4 to characterize the solution to the social planning problem. Here we derive our main consumption. some results We on mean-reversion and on the existence of a steady-state distribution discuss these results in Section 5 related problems and reformulations. In Section setup with productivity shocks and focus on its and develop intuition 6, we turn for for them by studying to the canonical optimal-taxation implications for estate taxes. Section 7 offers some conclusions from the analysis. All proofs omitted in the main text are contained in the Appendix. 2 A Social Insurance The backbone of our Problem model requires a tradeoff between insurance and incentives. This tradeoff can be due to private information regarding either productivity or preferences. For purposes of comparison, we 6, first adopt the Atkeson-Lucas taste-shock specification. In Section we adapt our arguments to a repeated Mirrleesian model with privately observed productivity shocks. Similar arguments could be applied to moral-hazard situations with unobservable Our An positive economy infinite-lived agent is can be interpreted as a dynasty of individuals Under altruistically linked. this interpretation, effort choices. — the differences are only normative. identical to that of Atkeson-Lucas who have finite lives but are Atkeson-Lucas and others focus on a particular set of efficient allocations: those that only directly consider the welfare of the initial generation.'^ In contrast, our interest here future generations. with lies This approach infinitely-lived social planner who more patient than is that directly weigh the welfare of efficient allocations individuals. Demography, Preferences and Technology. At any point in time, our economy by a continuum of individuals who have identical preferences, single descendant in the next. Parents by a and all asymptotically equivalent to postulating preferences for an is born in period t live for populated is one period, and are replaced are altruistic towards their only child their utility Vt satisfies = vt where q > is the parent's descendant's utility Vt+i- and continuously and w'(oo) = 0. own consumption and We assume that the differentiable for positive The extremes v = = /3 + pvt+i] G , (0, 1) is utility function consumption u{0) and v the inverse of the utility function c identically Et-i [9tu{ct) = u(oo) or m~^, or cost function. This specification of altruism = is M+ — R > is continuous, concave, may not be The taste shock ^ and independently across individuals and time. For adopt the normalization that E[^] : with the Inada conditions w'(0) levels, may the altruistic weight placed on the u simplicity, finite. = oo We let c{u) denote G distributed we assume is is finite and 1. consistent with individuals having a preference over the entire future consumption of their dynasty given by Vt J2P'^t-l[0t-,sU{(k+s)]. (1) In each period, a resource constraint limits aggregate consumption to be no greater than constant aggregate endowment e > 0. some These specifications of individual preferences and technology are precisely those adopted by Atkeson-Lucas. Social Welfare. We depart from Atkeson-Lucas by assuming that the social welfare criterion can be represented by preferences given by the utility function oo J2P'^-i[9tu{ct)], (2) t=0 with P> p. Thus, social preferences are identical to the individual preferences given by (1), except The final paragraph in Atkeson-Lucas's paper discusses the possible importance of relaxing this assumption. However, the approach taken in the literature has been to impose an exogenous lower bound on ex-post lifetime utility, as in Atkeson and Lucas (1993) and Albanesi and Sleet (2004). for the discount factor. This setup puts weight on the welfare of future generations directly. already indirectly valued through the altruism of the current generation. also directly included in the welfare function the social discount factor Future generations are in addition, If, they are must be higher than To p. see this, consider the utilitarian welfare criterion oo oo 5]a*E_it;4 where and 5^ = /3* + (5^~^a + social preferences are approaches (3) • (2) • J]<5tE_i[^iw(Q)], + [3q^~^ + a*. Then • more $= with = the discount factor satisfies patient. In the limit St+i/St -^ max(/3,tt), so the welfare criterion max(/3, a).^'^ Atkeson-Lucas' analysis applies to the case with weight is (3) /? = so /3, we focus on the case where enough placed on future generations to ensure that the long-run social discount factor remains strictly higher than the private one, $ > /3. Although we adopt the preference show how our analysis can be the rest of the paper, we welfare criterion (3). These two specifications are also briefly identical for the long-run, which is The any finite horizon but are our primary concern. Information and Incentives. Taste shock their descendants. adapted to work with the easily slightly different for in (2) directly for by individuals and realizations are privately observed revelation principle then allows us to restrict our attention to mechanisms that rely on truthful reports of these shocks. Thus, each dynasty faces a sequence of consumption functions {q}, where {0o,9i, that (7* : ...,6't). maps A Ct{9'^) represents an individual's consumption after reporting the history dynasty's reporting strategy a histories of shocks 0* -» 9*. We 9* into = {at} a current report 9t. is a sequence of functions at Any strategy use a* to denote the truth-teUing strategy with Given an allocation {q}, the utility 0*"''^ = 9* 9t for all obtained from any reporting strategy a = — O > : a induces a history cr*(6'*) 9^ of reports e QK is oo U{{ct},a;P)=J2 Yl P%u{ct{a\e')))Pr{9'). t=Q 6(«ee*+i ^Caplin and Leahy (2005) argue that a similar logic applies to intra-personal discounting within a lifetime. This is greater than the private one not only across generations, but within generations leads to a social discount factor that as well. ^One can also adopt the more general welfare criterion E_i ^Zt^o '^t^t ^°^ some sequence of positive Pareto weights {q;}, where the weight at can be interpreted as the probability of being born into generation veil of ignorance. "In particular, all i = 0,1,.. the sequence qq = (1 — 0)/{l — P) and at = aoP for i > 1 t for "souls behind the delivers St+i/5t =$ for } . An allocation {cj} incentive compatible is if truth-telling is optimal: U{{ct},a*:P) >U{{ct},a;P). (4) Social Planning Problem. Following Atkeson-Lucas, we identify each dynasty with a number V, which we interpret as that all its initial entitlement to expected, discounted dynasties with the same entitlement v receive the same treatment. distribution of utiUties v across the population of dynasties: V^(/l) wiU receive expected discounted An allocation For any given feasible utility in the set yl a sequence of functions is that a dynasty with is initial entitlements incentive compatible for it is not exceed the fixed endowment e in social allocations. to all is, ijj and then v. We let tp assume denote a the fraction of dynasties who M. v, where c\{9^) represents the consumption after reporting the sequence of shocks t and resources dynasties; all is We = (ii) 9*' e, we say it delivers expected utility of at that an allocation {c^ average consumption in the population does (iii) periods. optimum maximizes the average That C each entitlement v gets at date initial distribution of (i) if: {cJ'} for least V to all initial dynasties entitled to v\ A utility, vq social welfare function, the social planning problem given an weighed by ip, initial distribution of over all feasible entitlements -(/; is maximize subject to i; = U {{c1} , > a* 0) ; / Steady States. Our focus and C/({c^}, a;/?) for all v, J2 Ct (^*) Pr{e')dw{v) <e is on distributions t = 0,l, ... of utility entitlements (5) ip such that the solution to the planning problem features, in each period, a cross-sectional distribution of continuation utilities Vt of that is also distributed according to consumption to replicate itself over time. these properties a steady state and denote utility constitutes a state variable We also require the cross-sectional distribution term any them by that follows a We xp. ip* . initial distribution of As we Markov shall process, entitlements with demonstrate below, continuation and steady states are then invariant distributions of this process. Note that consumption in the is Atkeson-Lucas case, with P= P, the non-existence of a steady state with positive a consequence of the immiseration result. Starting from any initial distribution of entitlements the sequence of distributions converges weakly to the distribution having at misery and full mass zero consumption for everyone. This distribution constitutes a trivial steady state, which does not exhaust the endowment and is Pareto dominated by an allocations that improves everyone's welfare and the social welfare criterion. In contrast, in our steady states that exhaust the aggregate endowment in all periods. model we seek non-trivial A 3 Bellman Equation In this section First, we study a relaxed version of the social planning problem. This has two advantages: — one — we avoid the need to keep track of the entire population. because the relaxed problem can be solved by studying a set of subproblems dynasty with entitlement v for each Second, each of these subproblems admits a simple recursive formulation, which can be characterized quite sharply. Consider a relaxed planning problem where the sequence of resource constraints by the (5) is /oo oo 5]Q,J]cn^*)Pr(^*)d^(i;)<e5]g„ for replaced single intertemporal condition some positive sequence {Qt} with Yl't^o Qt < (6) can interpret this problem as representing c«- Oi^e a small open economy facing intertemporal prices {Qt}- The relaxed and original versions of the planning problem are related in that any solution to the former which happens to satisfy the A resource constraints in (5) must also be a solution to the latter. Lagrangian argument establishes the converse: there must exist some positive sequence {Qt} such that the solution to the original planning problem also solves the relaxed problem. Most importantly, any steady-state solution to the relaxed problem Our is a steady-state solution to the original one. focus on steady states leads naturally to Qt are only compatible with q — $, so forward.^ Attaching a multiplier A we adopt > = q^ for some q > this 'value for the relaxed 0. Indeed, steady states problem from to the intertemporal resource constraint (6), this point we can form L = f L^dip{v) where the Lagrangian oo and study the optimization of L subject to v = U {{c^} , a* P) ; > U{{c^},a;P) equivalent to the pointwise optimization, for each v, of the subproblem k{v) V = U {{c^} a* P) , ; superscript v Our first > f/({C(}, when denoting cr;/3). Where there is no risk of confusion, = for all v. This is supL*' subject to we henceforth drop the allocations {c^}. result characterizes the value function k{v) = m&xE\eu{e) - and shows that Xc(u{e)) it satisfies the Bellman equation + pk(w{e))] (7) subject to V ''A simple variational argument can be used = to distribution with a finite average value for c'{ut). E[9u{9) + pw{9)] show that Qt+i/Qt (8) =P for any relaxed problem at an invariant + pw{0) > eu{e) for all ^,^' e0." Theorem 1 The value function k{v) is 9u{e') + pw{e') (9) continuous, concave, and satisfies the Bellman equation (V-(9). The the latter rules out one-shot deviations from truth-telling, guaranteeing that telling the truth today optimal is if the truth the value function at the continuation told in future periods. is incentive-compatibility condition (4). full and an incentive constraint recursive formulation imposes a promise-keeping constraint (8) (9). Intuitively, The rest for utility: This is necessary to satisfy the implicitly taken care of in (7) is by evaluating any given continuation value w[9), envision the planner in the next period solving the remaining sequence problem by selecting an entire allocation that is Then k{w{9)) incentive compatible from then on. Taken together, a this continuation allocation. pair u{9) represents the value to the planner of and w{9) that satisfies (8)-(9) pasted with the corresponding continuation allocations for each w{9), yields an allocation that satisfies the incentive-compatibility (4). The full objective function in (7) then represents the value to the planner of this allocation. Among other things, We tained. Theorem 1 shows that the respectively. For any initial utility entitlement = policy functions (^",y^) by setting Uo{9o) for t > 1 using ut{9') Our next in this is = and g^{9t,v^{9'-')) result elucidates the connection way and Theorem maximum 2 (a) Vq, (7) is at- an allocation {ut} can then be generated from the g^{9o,vo) v^+,{e') = and defining Mt(^*) and Wt+i(^*) recursively g^{9t,Vt{9'-')). between allocations generated from the policy functions An allocation {ut} is optimal for the relaxed problem, given Vq, if generated by the policy functions {g^,g^) starting at has limt_»oo Bellman equation solutions to the planning problem. lifetime utility of Vq; (h) vq, in the the policy functions g^{9,v) and g^{9,v) denote the unique solutions for u and w, let vo, is incentive compatible, and only and if it delivers a an allocation {ut} generated by the policy functions (g^ig^), starting /?*IE-if((^*~'^) — policy functions {g^,g^), starting and delivers utility vq; (c) from Vq, is incentive an allocation compatible limsupE_i/3*Ui((j*-^(^*-^)) at {ut.,Vt} generated by the if >0 t—»oo for all reporting strategies a. Part (a) of Theorem 2 implies that either the solution to the relaxed planning generated by the policy functions of the Bellman equation, or there and (c) of the theorem show that the ^We study a Bellman equation first case is guaranteed if is we can that also characterizes the problem for g 7^ /? in no solution at problem all. is Parts (b) verify the limit condition in Section 5. The part (b). we latter will see that We automatically satisfied for is and state the Bellman close this section with a detour associated with maximizing the welfare criterion to f = functions that are all utility bounded below and can be verified in other cases of interest.^ it U{{dl},(T*;/3) > (3). U{{c^},a]P). Then very Define k{v) for the relaxed = sup Ylu^o planning problem '^*^-i['^t~-^cJ'] subject similar arguments imply that the value function k{v) satisfies the Bellman equation = maxETu - Xc(u{e)) + Because this Bellman equation is ~k{v) ak(w{e))] u,w subject to (8) and and analysis (9). results that follow almost identical to the one in the (7), can easily be adapted to this case. Optimal Inequality 4 In this section We we exploit the connection between the Bellman equation and the planning problem. characterize the solution and derive a key equation that illustrates mean-reverting forces in the dynamics of consumption. The main enough to imply the existence of result of the section is to establish that these forces are strong an invariant distribution with no misery. sufficient conditions that verify the requirements of Theorem Finally, we provide and ensure that a solution to the 2, planning problem exists. Mean Reversion We now are in a position to study the Bellman equation's optimization problem. justify the use of first-order conditions with the following Lemma 1 The value function k{v) main, with \iiny^yk'{v) \imy^yk'{v) Let A let = = = — oo. is strictly If utility is unbounded we begin, concave and differentiable on the interior of below, then liiny^yk'{v) = 1. do- its Otherwise oo. k'{v) be the multiplier on the left-hand side of the promise- keeping constraint n{9,9') be the multipliers on the incentive constraints (9). 1 ^Theorem To lemma: - Xc'{u{e))]p{9) - e\p{e) The + Y^ Oi^ie, e') - (8) and first-order condition for u{6) ^ e'^{e, e') 2 involves various applications of versions of the Principle of Optimality. < is o, For example, given policy functions {g'\g^) and an initial value Vo, the individual dynasty faces a recursive dynamic programming problem then amount to guessing and verifying a solution to the Bellman equation of the agent's problem. In particular, that the value function that solves the Bellman equation, with truth telling, is the identity function. However, one needs to verify that this value function indeed represents the true value from the msjcimization of the dynasty's sequential problem. This verification is accomplished by part (c) of with state variable Theorem v. Conditions (8) and (9) 2. 10 with equality u if (9) is interior. for k{v) derived in Lemma 1, The and must pk'{w{e))p{9) - must be solution for w{9) interior, given the f3\p{9) +PY1 i^{9, 9')-PY1 ^(^' ^') = 0e' Using the envelope condition k'{v) = Inada conditions satisfy the first-order condition e' A and adding up across Y,k'{w{9))p{9) = this 9, becomes ^k'{v). This key equation can be represented in sequential notation as = E,.^[k'{vt+,{9'))] where {vt} generated by the policy function g^. is CLAR) Markov Regressive (hereafter: process. ^k'{vt{9'-')). Thus, {k'{vt)} (10) is a Conditional Linear Auto Note that we can translate anything about the process {k'{vt)} into implications for the process {vt}, since the derivative k'{v) strictly decreasing. Likewise, using the policy function g^{9,v), conclusions is continuous and about the process {vt} provide information about the process for consumption. The conditional expectation in (10) illustrates that for the process {k'{vt)} maximum, is toward zero. Lemma /3//3 < 1 so reversion occurs towards an interior utility level this case To derive is = /3 a CLAR reversion By interior This feature contrast, in the Atkeson-Lucas equation similar to (10) also holds, but the relevant value function in monotone. we see this CLAR its /?, mean — away from misery. key to our results on the existence of invariant distributions. model with creates a force for imphes that the value function k{v) has an 1 write the Bellman equation for the relaxed problem in the Atkeson-Lucas case and Euler equation. Even though no steady state exists in this model, if we assume a constant relative risk aversion utility function, then the relaxed problem does characterize the social planning equation problem for an appropriately chosen value of the price qal- The associated Bellman is KAL{v)=mmE[c{u{9))+qALKAL{w{0))] U,W subject to (8) and K'^i [v) > 0. (9). L / \ Crucially, here the value function The envelope and \ KAiiv) first-order conditions yield the is CLAR E ^"alH0))p{9) = ^K,{v), 11 / J strictly increasing, so that Euler equation similar to condition (10) when qal > P-^ However, the critical difference that here {-^^^^(^4)} is non-negative process, which implies by the Martingale Convergence Theorem that almost surely utilities g^(9,v) follows, Vt ^ to (a.s.) some V and q — anything other than K'^j^{vt) non-monotonicity, which in turn hinges on the lower social discounting — Immiseration then a.s. > [3 > fi. Economically, the mean-reversion equation implies a form of social mobility. That of individuals with current welfare above v will eventually and rise is, descendants below v* and vice versa. The resulting , result pushes the characterization of reversion past the average behavior of the {k[} process by deriving bounds for its evolution. These an invariant distribution with no mass and fall of famiUes illustrates a strong intergenerational mobility in the model. fall Our next of a This highlights the technical importance of our value function's a.s. ;> is must converge Since incentives must be provided using continuation finite value. g'^{0,v), this rules out ^ it bounds are We at misery. critical for guaranteeing the existence summarize the previous CLAR equation state the next result in the following proposition: Proposition 1 The policy function satisfies J2k'{g'"{e,v))p{9) dee = ^k'{v). P Define the constants Then (a) if utility is 7(1 and - (b) if utility is 7 = iP/P) 7 = iP/P) 2min^(l max{{l + er.-E{e<9n])/en), l<n<N all 9 > ^ i ^„]/^„_i). unbounded below, k\v)) - + (^1 < 1 bounded below with u{0) - E Q. For values of v such that k'{g-{e, v)) = < = w{e) = p-^v>v k'{v) = < - 7(1 k'{v)) + (1 for values of v such that k'{v) 0, u{e) k'{w{e)) for + ^n-i - E[^ > (11) 1, we have {p/p)k\v) 1, the lower bound in (11) holds; the upper bound in (11) holds for sufficiently high v. °With logarithmic Thus, it is with qal = P features constant average consumption. problem that imposes an aggregate resource constraint in every period. With the utility the partial cqiiilibrinm sohition also a solution to the constant relative risk aversion utility function u{c) has the same effect. Indeed, for ct < 1 = c^'" /{I — a) there the appropriate qal satisfies qal 12 is > a value of qal, for each value of a, that P- Proposition illustrates 1 a powerful tendency away from misery. unbounded below, continuation how much a Main it utility g'^{9, v) remains bounded even as supposed to be punished, is his child state the main result of this section: if The proof admits an invariant distribution with no misery. mean- reversion properties discussed Proposition 2 The is ^ — oo. Thus, no matter No Misery a solution to the relaxed planning problem exists, conditional-expectation equation (10) and the bounds in Proposition for the t" always somewhat spared. is Result: Existence of an Invariant Distribution with We now then parent For example, with utility existence of an invariant distribution Markov process bounded above, or j Proposition in the previous subsection. g'^ is {vt} implied by < ip* Thus, it ^° with no mass at misery, Tp*{{v}) if either: utility is unbounded (a) of Theorem 2, leaves open only two exists This situation contrasts strongly with the Atkeson-Lucas case, with but does not admit a steady section we show We state, /? = /?, illustrate this 0, no solution (ii) and everyone ends up at misery. Towards the end of an also provides efficient method for explicitly solving (i) where a solution that a solution to the planning problem can be guaranteed so that case Our Bellman equation lem. = below, utility possibihties: the rela:xed problem admits a steady-state invariant distribution with no misery; or exists. on the makes use of both 1. when combined with part 2, guaranteed of this result relies 1. (i) this holds. the planning prob- with two examples, one analytical and another numerical. Example 1. Suppose utility is CRRA with a — 1/2, so that u{c) = 4c^/^ for c > and c{u) = u^/2 for IX > 0. Atkeson-Lucas cover this case for P = $ and show that the optimum involves consumption inequality growing without bound, leading to immiseration. Consider the relaxed problem where we ignore the non-negativity constraints on u and w, k{v) = ma^E\eu{e) - XuiOf 12 + Mw{e))] \ U,W subject to (8) and (9). This is / J , a linear quadratic dynamic programming problem, so the solution a quadratic value function with linear policy functions in g^{9,v) = j-{e)v is v. + ro{0) For taste shocks with sufficiently small amplitude we can guarantee, by continuity with the deterministic case = [vl,vh] with vl 9, > that 0. -y'^{9) < 1 and 7"'(^) Moreover, g'^{9,v) > > 0, for u implying a unique bounded ergodic set for utility G [vl-,vh\- Hence, since the planning problem is — which is ^"When utility is bounded below, we either require that utility be bounded above, or that 7 ensured for a small dispersion of the shocks as a simple way of ensuring that the ergodic set from misery. It seems very plausible, however, that these conditions could be dispensed with. — 13 < is 1 bounded away Figure convex and original 1: utility To 2. iterated c{v{l — is = P)) against c{g"'{9,v){l reveals a unique, low values of u. p= 0.9, = 0.975, e (A)-i = 0.6, Oh = we compute the 1.2, Oi = 0.75 solution and p = 0.5. for k{v) until convergence.^^ plots the policy function for continuation utility in consumption-equivalent units, consumption, c{v{l 1 with P on the Bellman equation 1 Policy function g'^{6,w) illustrate the numerical value of our recursive formulation, for the logarithmic case Figure 2: turns out to be strictly positive at the steady state, this solution does solves the problem with non-negativity constraints on Example We Figure Policy functions g'^{6,v) v. — P)) — /?)), while Figure 2 does the same for the pohcy function for against g'^{9,v). bounded ergodic Both policy functions are monotonic and smooth. Figure set for v. This illustrates the Note that both policy functions become nearly result, discussed immediately after Proposition 1, flat for that utility kept above some endogenous bound. Figure 3 displays the steady-state, cross-sectional distribution of dynastic utility measured in consumption-equivalent units, c{v{l — P)) implied by the solution to the planning problem. ^^ The '^The details of this numerical exercise where as follows: we solved for u{9) as a function of w {0) using the and promise-keeping constraints. We then maximized over w{9). We employed a grid for v defined in terms of equally spaced consumption-equivalent units c((l — (3)v) = {0.01, ..., 2}. Results with a grid size of 100 and 300 were similar; we report the latter. We used Matlab's splines package to interpolate the value function and used fmincon.m as our optimization routine over w{9). Our iterations were initialized with the value function corresponding to the feasible plan that features constant consumption: incentive v{l - ,6) - \c{v{l - ,6)) ko{v) 1-13 We stopped the iterations when ||fcn(u) — A':„_i(w)|| < 10~''-° and verified that the policy functions had also converged. Note that g"'(0,t') is well within the interior of [.01, 2], so that the arbitrary upper and lower bounds from our grid choice were not found to be binding. '^^The invariant distribution was approximated by generated a Monte Carlo simulation for the dynamics of the Vt process generated by g^, with an arbitrary initial value of vq. Since this process converges to a unique invariant distribution ip, starting from any initial value of Vq, the frequencies in a long time-series sample approach the frequencies of 0. To create the figure we used Matlab's Wavelet Toolbox to approximate the density from the simulated Monte-Carlo sample. 14 0.61 utility in Figure 0.62 consumption equivalent units Steady-State Distributions of DjTiastic Utility 3: long-run distribution has a smooth bell-curve shape mean-reverting dynamics of the model, since The distribution of taste shocks. /?. The degree cannot be a direct consequence of our two-point shows the invariant distributions of inequality appears to decrease with higher values of by the intuitively figure also it — a feature that must be due to the smooth, on the coefficient CLAR equation (10) /?. for various values of This outcome and the discussion is in Section 5 suggested on features of the impulse response to shocks. These simulations also support the natural conjecture that as we approach the Atkeson-Lucas case, 3 -^ (3, the resulting sequence of invariant distributions blows up. since no steady state with positive consumption exists when $ = (3. We now tmn briefly to issues of uniqueness and stability for the invariant distribution guaranteed by Proposition of social 2. This question is of economic interest because it represents an even stronger notion mobihty than that implied by the mean-reversion condition subsection. That utiUty level vq, is, if convergence toward the distribution tp" (10) discussed in the previous occurs starting from any initial then the fortunes of distant descendants — the distribution of their welfare — is independent of the individual's present condition. At the optimum, the past always exerts some influence on the present, but its influence is bounded and or disadvantages of distant ancestors are eventually Indeed, under if is wiped out. some conditions we can guarantee that the display this strong notion of social mobility. process dies out over time, so that the advantages compact. This is optimum is monotone from any in v, the invariant distribution ip* initial distribution ipQ, 15 in our model does see this, suppose the ergodic set for the {kf} guaranteed, for example, by applying Proposition the policy function g^{9,v) in the sense that, starting To social 1 when 7 < is 1. Then, unique and stable the sequence of distributions {^t}, generated by (10) ensures converges weakly to g'^, ip* w policy functions for continuation utility 1 This follows since the conditional-expectation equation . enough mixing to apply Hopenhayn-Prescott's Theorem. ^^ The monotonicity of the seems intuitive and plausible, as illustrated by Examples 2.14 and Another approach suggests uniqueness and convergence without relying on monotonicity. Grunwald, Hyndman, Tedesco and Tweedie (1999) show that one-dimensional, with the Feller property that are bounded below and satisfy a a unique and stable invariant distribution. initial distribution is fast, in ments of their which of irreducibility, distribution. We is likely to it occurs at the geometric rate P/$. All the require- verified already for our hold if such as (10), have Moreover, convergence to this distribution from any the sense that theorem have been Markov processes irreducible CLAR condition, we were model, except for the technical condition assume that the taste shock has a continuous to do not pursue this formally other than to note that the forces for reversion in (10) could be further exploited to establish uniqueness and convergence. Our focus on steady states, where the distribution of utility entitlements replicates itself over time, has exploited the fact the relaxed rithmic utility case we can do more: we and original planning problems then coincide. For the loga- characterize transitional dynamics for any initial distribution of entitlements. Proposition 3 (a) If utility is logarithmic, then for any there exists a value of e such that the solution to the relaxed ijj endowments first-order stochastic dominance, then the associated Proof, its and original (h) Suppose two distributions of utility entitlements satisfy coincide, (a) of utility entitlements initial distribution ip"' satisfy ea Consider the relaxed social planning problem for some A = social planning "problem > > e~^. et- We solution also solves the original social planning problem for any distribution / k'{v; — e)dip{v) Since utility is 0. We then show that there exists a value for A unbounded below, we have k'{vt) = Ei_i[l = in the sense of ip^ first show that that satisfies ipQ e~^ for any initial distribution. — Xc'{ut)]. Applying the law of iterated expectations to (10) then yields E_i[l-Ac'K(^*))] With logarithmic t = 0, 1, ... Theorem Now J The 2, it utihty c'{u) allocation follows that is it = c{u), so that = 0, and the k'{v)d'ip{v) = imphes jE^iCtdip incentive compatible by Proposition 2 below, must = -1 A = e for all and applying part (c) of solve the original planning problem. consider any initial distribution k'{v; e)dip{v) J = (^0^'^ V ip. We argue that we can result then follows immediately. We find a value of A = e"^ such that denote the value function for the Lucas and Prescott (1989). can be shown that g^ (v, 9) is strictly increasing in v. However, although we know of no counterexample, we have not found conditions that ensure the monotonicity of g^{6, (O.v)) for 9^6. ^'^See pg. 382-38:i in Stokey, ^'^Indeed, it 16 = relaxed problem with A e by ^ k{v; e). The homogeneity problem implies that of the sequential the value function must satisfy ^^^' ^^ Note that e ^^ — oo, k'{v — log(e)/(l — /?); k'{v] e)dij{v) defines a unique value of e for (b) 1) is strictly log(e); 1 increasing in e and limits to 1 and — oo as any initial distribution, identically at the outset = I k' (v - -^ log(e); 1 j d^p{v) * oo and = (12) initial distribution ^q- the situation where the planning problem Then logarithmic coincides with the solution to the relaxed problem. Then our utility level v* that solves k'{v*) '0* can be reinterpreted in terms of the stability of the cross-sectional distribution in the population. stable then the cross-sectional distribution of welfare will eventually settle The economic problem is down interest behind part attains a distribution of entitlements creates a Pareto is If Markov process the and consumption in the population to the steady state. (b) is that it implies that the solution to the planning ex-post Pareto efficient in the following sense. then implies that there modified to select is dynasties are treated all initial =0. With and started with the optimum then the social is instead of taking one as given. previous discussion of convergence to a unique invariant distribution is — a strictly decreasing function. Perhaps of particular interest the best {vt} e This part follows immediately from equation (12) which determines e by using the fact that 1) is utility ^— respectively. It follows that / A;'(-; " VI^ ^^^^^^ + k(v- ifj Consider the optimal allocation that with some constant endowment level no alternative allocation that improves the improvement among individuals e. The proposition social welfare criterion in the current generation. That is, the optimal allocation also solves the original social planning problem modified to require [/({cJ'},cr*;/3) place of U{{dl}, a*;P) = v. Since this result holds for any distribution of entitlements for the distribution of continuation utilities implied in this sense, the optimal allocation is by the optimal allocation ex-post Pareto in and it >v\n. also holds any period. Thus, efficient. Sufficient Conditions for Verification We conclude this section by describing sufficient conditions for a solution to the planning problem to exist at the steady state steps. First, we ip* identified by the policy functions establish that allocations generated compatible by verifying the condition in part consumption is finite (c) of under the invariant distribution 17 in Proposition 2. This involves two by the policy functions are indeed incentive Theorem ip*. 2. Second, we verify that average Lemma 2 The allocation generated from the policy functions {g^,g^), starting from any vq, guaranteed bounded below; is compatible in the following cases: (a) utility to be incentive < or (d) ^ (c) utility is logarithmic; 1 bounded below, then the Proof, (a) If utility is also utility unbounded below, but bounded above. Then is implies that \imy^_cok'{g^(9,v)) a vl > —00 such that g'^(6,v) > = It 1. - g'"{e,v) and 9 for all pairs of histories 9 part (b) of Theorem 2 < (d) If 7 > we have Then then the bound If for all v by Vh- We 7 < 1, - — = A. Yimt^oo (3^'^-i[vt{9^~^)] > < It (c) of Theorem 2. one can show that: log {0/9). The incentive constraint follows that Vt{0^~^) > - tA Vt{e^-^) - pHA. = 0. Since Y\m.t^^l3hA < = 0, it follows 0. in (11) implies that k' {g^ (^i f^)) have g^{9, v) now apply 1, e' g^{0,v) {0/ [3)\og{e/e) then we can define k < vh ^e can 1 attained, so there exists Then . that limsupt^^/3*E_i[t;(((j*-i(^*-^)] immediately. is immediately from part result follows P'¥._,[vt{a'-\9'-^))] >/3*E_i[i;,(^*-i)] From So suppose (b). continuous and Proposition is , utility this implies that g^{6,v) then implies that g'^{e,v) from part result follows k'{g^{6, )) follows that m.ax^ k' {g'^" (9 v)) d{u{e_)) With logarithmic is (b) utility 0. Using the first-order conditions from the proof of Proposition (c) bounded above; vl- bounded below, the (b) If utility is > or 7 is v. = It 1 — (1 — (3/P)/{l — ^ 1 7), ~ P/P ^^^ follows that the unique ergodic set the argument in (a) so there exists a vl the result follows and define vh by k'{vH) > —00 is = k. bounded above such that g^ (9, v) > vl- U We now find sufficient conditions that guarantee average consumption is finite under the invariant distribution ip* . the ergodic set for utility v If bounded and average consumption is is is bounded away from the extremes, then consumption trivially finite. Even when a bounded ergodic set for utility V cannot be ensured, finite average consumption can be guaranteed for a large class of utility functions. Lemma 3 Average consumption is finite under the invariant distribution ip* 9 if either (a) the ergodic set for v of c. Proof. Part For part (a) is is bounded; or (b) utility is such that c'{u{c)) a convex function is immediate since by continuity of the policy functions, consumption (b), recall that J k'{v)dip*{v) = under the invariant distribution 18 ijj* . If is bounded. utihty is un- bounded below then all solutions for consumption are interior. corner solutions with g^{9, v) g'^iO, v) is bounded, = l-k'{v) some for for all 6 in this = compact If utility is bounded below, then low enough 9 can only occur for levels of v, so that Recall that for interior solutions set. XE[c'{g^{e,v))]=XE[c'{;u{c{g''ie,v))))] Applying Jensen's inequality we obtain c' The (u (jE[c{g''{9,v))]dr{v)]) < result then follows since c!{u{c)) Note that a bounded ergodic is set is an increasing function of guaranteed by 7 with sufficiently small amplitude; and condition aversion utility functions with The a > < 1, (b) holds, for which and thus has full range. is ensured for taste shocks example, for all constant relative risk 1. utility, For instance, in the case of average steady state consumption is a power function of A, In fact, in this case the entire solution for consumption of degree one in the value of the 1- c. value of average consumption depends on the value of A. constant relative risk aversion endowment planning problem for any endowment 5 j E[d {u{c{g'^{9,v))))]dr{v) = is homogenous This ensures a steady state solution to the social e. level. Discussion: Mean-Reversion This section develops an intuitive understanding of the key mean-reversion property discussed pre- We viously. first revisit the full derive the impulse response of consumption to a one-time taste shock. problem with an alternative Bellman equation that We then useful as a source of intuition. is Impulse Response Consider a version of our model where only the first generation faces uncertainty. period, there are two possible values for the taste shock ^0 is deterministic: = I for t > We 1. compare {9l-, this to the case In the we adopt logarithmic first 9h}, but thereafter the economy with no uncertainty in the This allows us to trace out the consumption response to the taste shock over time. period. simplify, We 9t ^ first To utility. begin by studying a subproblem of the deterministic planning problem from the second generation onward, that planning problem is for t = 1,2,.... For a given, promised continuation utility is kdet{vi) = max V"^ 19 (logQ - Xct] , Vi, the subject to <=1 The associated Bellman equation is kdetivt) = max ct,vt+l subject to Vt = logQ + /3ft+i. The ( ct {k'^eti'^t}} - Aq + ^A;det(^^i+i) J and envelope conditions imply that first-order k',Jvt+i) Condition (13) shows that log(Q) \ = ^X^tivt), (13) = X~\l-k',Jvt)). (14) reverts geometrically towards zero at the rate a deterministic version of the conditional-expectation equation (10). translates directly into back to a common consumption by the discounting coincide, so that new level c{vi/{l Turning to the is In the logarithmic case, it Thus, consumption reverts steady state at the same rate; deviations from the steady-state level of consump- tion have a half-life of (log2 (/?//?) )~^. q = first-order condition (14). This /?//?. /? = Note that in the Atkeson-Lucas case when social and private consumption remains perfectly constant /?, after the shock at its — /?)). first generation at i = 0, the planning problem solves maxE[^olog(co(^o)) - AoCo(^o) + ^fcdet(yi(^o))l Uo,Vl subject to + Pv,{eL) > OLiog{co{eL)) where we omit the other incentive constraint since is convex. At the optimum, Cq{6h) > eLiog{co{dH)) it is not binding at the optimum, and the problem < Cq(6l), vi{6h) + PvjiOH). viidi), and EKet(^l(^))]=0, implying that Vi{9h) and equal to A < v* < Vi{9l) where k'{v*) = 0. Note that average consumption Figure 4 shows the consumption response to a taste shock in the periods. That is constant in all periods. is, we use (13) and (14) for t > 1 first starting at Vi{9l), v* period, for subsequent and v-[{9h)- on consumption from the shock dies out over time and consumption returns to a The common effect steady- state level. Again, this illustrates that the influence of past fortunes eventually vanishes for distant descendants. We also plot the Atkeson-Lucas case with has a permanent impact on the consumption of To provide incentives for the first all (3 = $, where the luck of the first generation descendants. generation, society rewards the descendants of an individual 20 Figure Consumption path 4: reporting a low taste shock. first is i > 1 in response to taste shock at ^ = Rewards can take two forms and society makes use of both. consumption, for exploits differences in preferences: it allows an adjustment in the pattern of a given present value, in the direction preferred by individuals.^'^ Since individuals are more impatient than the planner, this latter form of reward by The standard and involves increased consumption spending, in present-value terms. The second more subtle and is for tilting the consumption profile toward the present. Similarly, punishments involve consumption path toward the future. intensively to provide incentives returns to a affecting the common first delivered tilting the In both cases, earlier consumption dates are used — rewards and punishments are front-loaded. more Indeed, consumption steady-state level in the long-run regardless of the initial shock because consumption of very distant descendants incentives to the is is not an efficient way for society to provide generation. Another Bellman Equation Here we develop another Bellman equation that holds This alternative formulation is useful, for any value of q not necessarily equal to $. both as a source of intuition and to motivate our focus on Consider the following cost minimization problem oo ^^Some readers may recognize parents when conceding this last method some goods more than the parent wishes, system of rewards and punishments used by TV-time. In these instances, the child values as the time- honored their child's favorite snack or reducing their and the parent uses them to provide 21 incentives. subject to the incentive compatibility constraint U {{ct} a* P) , ; > U{{ct},a; P) and oo V = J2p'E.,[9Mct)] t=o _- oo while delivering utility v and v for the planner and individual, respectively. must satisfy the Then the value function Bellman equation K{v,v) = maxE[c(u(^)) u,w,w + qK(w (6) ,w [9))], subject to V = E[eu{9) + Pw{0)] V = E[9u{9) + Pw{9)] and 9u{9) + Pw{9) > This formulation could be used to derive equation (7) is slightly formulation, however, The more convenient is that it 9u{9') all + I3w{9') for all 9, 9' Bellman of our results, although the lower-dimensional The advantage for that purpose. of this cost-minimization lends itself naturally to economic interpretations. following story provides a useful reinterpretation infinite-lived e 6. and source for intuition. Consider an household with two members, husband and wife, and assume that consumption public good — there disagree on how is is a no intra-period resource allocation problem. However, husband and wife more patient, but only the husband problem characterizes the constrained Pareto problem for this to discount the future. Suppose the wife is can observe and report taste-shock realizations. Then this cost-minimization household, in the sense that the isocost curve K{v,v) Pareto frontier between husband and wife. = Kq The Pareto represents, given resources Kq, the frontier is non-standard in that not it is everywhere decreasing and does not represent the usual transfer of private goods between two agents. Instead, it arises from differences in preferences that generate a disagreement about the optimal consumption path for the only public good. Since disagreement on preferences the Pareto frontier is non-monotone and the highest possible interior utility level for the the left The of this point must husband, where Ki{v*,v*) — 0. is bounded, utility for the wife is attained for Reductions in the husband's an utility to also decrease utility for the wife, for a given level of resources. first-order conditions can be rearranged to deliver PK2{v,v)=qK2{w{9),w{9)) 22 (15) K\{w{e),w{9)) K2{v,v) (16) K2{w{9),w{9)) J3 Condition (15) can then be used to argue that a steady-state requires q {K2t} would increase without bound; likewise, if g > = p. Indeed, ^, then {K2t} decreases q ii toward < 0, then zero. Both situations clearly do not lend themselves to the existence of an invariant distribution for {v,v). On the other hand, distribution When -f^2(f^0) "f^o)- g = /? then K2{vt,Vt) constant along the optimal path and an invariant is possible. is = q if P, the state {vt,Vt) moves along a one-dimensional locus given by K2{v,v) Intuitively, since no incentives are required for the wife, she is perfectly insured in the sense that the marginal cost of delivering welfare to her Figure Figure 5 shows that the curve K2{v,v) = curves from below, and cuts Ki{v,v) perfect insurance for the — is held constant across time. K{v,v) Isocost curves of 5: K2{vo,Vo) for continuation from above. = utilities cuts the isocost Intuitively, incentives require foregoing husband and accepting fluctuations Starting from {v*,v*), rewards can be dehvered in two ways. in v as rewards The optimum makes and punishments. use of both forms of rewards, explaining the shape of the schedule for continuation utilities. The form of rewarding involves increasing resources K, and can be seen as an upward first movement along the diagonal Ki{v,v) an allocation of these resources that movements along the Pareto frontier, both forms of rewards and, as a K\{y,v) = and above the = is 0. more which result, However, the husband to his liking, at {v*,v*) is 23 also rewarded by allowing which can be represented as horizontally {v,v) travels along K2{v,v) initial isocost curve. is flat. — The lateral solution combines /C2(fo,t'o) to the right of Condition (16) is the analog of the conditional-expectation equation (10) obtained from the one-dimensional Bellman equation. frontier), —K1/K2 Here, it implies that the slope of the isocost curve (Pareto reverts geometrically toward 0. Thus, {vt,Vt) and eventually reverts toward {y* v*). Intuitively, husband with it is incentives, but , efficient to revert the wife: Patience ensures that the wife has her moves along K2{v,v) = K2{vq,Vq) the solution deviates from [v* ,v*) to provide the way back to this point of ma:x:imum efficiency for in the long-run. Estate Taxation 6 We now turn to a repeated Mirrleesian economy and study optimal taxation. In this version of our model, individuals have identical preferences over consumption and work regarding their labor productivity, which is but are heterogenous effort privately observed by the individual We distributed across generations and dynasties. and independently continue to focus on the case where the social welfare criterion discounts the future at a lower rate than individuals. Unlike the taste-shock model, here even between parent and child, there would still if we were to restrict allocations to feature no link be a non-trivial planning problem. Indeed, in each period the situation would then be identical to the static, nonlinear income tax problem originally studied by Mirrlees (1971). Moreover, in the absence of altruism, so that With actually coincides with this static solution. altruism, however, (5 we — Q, the social shall see optimum below that it is always optimal to link welfare across generations within a dynasty to enhance incentives for parents. Despite differences between the Mirrleesian economy and our taste-shock model, our previous analysis can be adapted virtually without change. In particular, a recursive representation can be derived, and the Bellman equation can be used to characterize the solution and to establish that a steady-state, invariant distribution exists. This highlights the fact that our model requires asymmetric information, but not any particular form of We some it. focus on an implementation of the allocation that uses income and estate taxes, and derive interesting results for the latter. We find that estate taxation should be progressive: more fortunate parents should face a higher average marginal tax rate on their bequests. reflects the mean reversion in consumption explained in the previous section. A This result higher estate tax ensures that the fortunate face a lower net rate of return across generations, and that consequently their consumption path decreases over time toward the mean. Repeated Mirrlees: Productivity Shocks Each period Utility of this economy depends on the generation t is level of identical to the canonical optimal taxation setup in Mirrlees (1971). consumption c and work effort n. have identical preferences that satisfy Vt = Et.Mct)-h[nt)+l3Vt+i], 24 We assume that individuals in , but diflFer An regarding their productivity in translating work effort into output. productivity w, exerting work effort n, produces output y = wn. We individual with assume that productivity w is independently and identically distributed across dynasties and generations. Thus, the productivity talents of parent assumption, if the and child are unrelated optimum — innate skills are assumed nonheritable. features intergenerational transmission of welfare, then social decision to provide altruistic parents it Given this represents a with incentives in this way, and not a mechanical result originating from the eissumed physical environment.-'^ For convenience, we adopt the power disutility function h{n) we can = rC /^ so that, defining w~'^ OO CXD 5]/3*E_i[«(q) -^,/i(y,)] = X^/?^E_i[r'MQ_i) t=0 The right-hand problem we — write total utility over consumption and output as being subject to taste shocks -^,/i(y,)] - E_i [^o%o)] i=l side of this equation leads to a convenient recursive representation of the planning in the continuation utility defined by Vt — ^^Q/5*Et_i[u(ct+s-i) — Ot+sh{yt+s)\ — where are abusing notation shghtly by folding the /3~^ into the definition of the utility function u{c). The resource constraint requires total consumption not to exceed total output plus some fixed constant endowment Y,c'i{e')Y>v{e')d^{v) where individuals are indexed by < fY,yt{0')Pr{9')di^{v) + e, their initial utility entitlement v, with distribution in the ip population. We continue to assume social discounting is lower than private discounting: $ > problem is to choose an allocation {c^{6^),y^{6^)} to incentive-compatibility constraints Using the last expression for and the resource maximize average (3. The planning social welfare subject to the constraints. the utility function and applying similar reasoning as in the taste- shock model yields the Bellman equation for the associated relaxed problem k{v) = max E\u^ - Ac(«_) - eh{9) u-,h,w v + Xyih{e)) + ]3kiw{e))] = E[u_-eh{e) + i3w{e)] -eh{9) + pw{e) > -eh{6') + (3w{e'), where the function y{h) represents the inverse of the disutihty function, y = h~^ . The arguments that justify the study of this Bellman equation, are similar to those that underlie Theorems in the context of the taste-shock previously, and imply that an invariant distribution ^^As in the taste shock model, here the case with Albanesi and Sleet (2004), 1 and 2 model. The results regarding steady states parallel those obtained who impose an exogenous /3 = /? lower 25 exists with no immiseration, as in Proposition leads to immiseration. bound on dynastic This case has been studied by welfare to circumvent immiseration. Implementation with Income and Estate Taxation Any allocation that incentive compatible is taxes on labor income and estates. Here and we feasible can be implemented by a combination of describe this implementation, and explore first some features of the optimal estate tax in the next subsection. For any incentive-compatible and feasible allocation {cj(^*), In each period, conditional on the history of their tation along the lines of Kocherlakota (2004). and any inherited wealth, individuals report dynasty's reports 9 consume, pay taxes and bequeath wealth subject to the following ct where Rt-i^t + bt< yt{e') - Tt{9') + we propose an implemen- y]^{6^)} - (1 their current Tt{9'))Rt-,,tbt-i = t 0,l, the before-tax interest rate across generations, and initially 6_i is shock Of, produce, budget constraints set of (17) ... = 0. Individuals are subject to two forms of taxation: a labor income tax Tt{9^), and a proportional tax on inherited wealth Rt-i^th-i at rate at rate Tt{9 Given a tax policy {Tj"(^'), )}'^ t'^{9*), an equihbrium consists of a sequence of interest yf{9'')}, income and bequests rates {Rt^t+i}', an allocation for consumption, labor reporting strategy {crj(^*)} such that: (i) {q, jbt, taking the sequence of interest rates {Rt^t+i} and the asset market clears so that J E^j[b'^{9^)]d(j){v) equilibrium For any is incentive compatible if, — tax; policy {Tj, for all in addition, it interest rates {Rt-i^t}- r^, 0,1,... induces truth 6J'(^')}; yt} as given; We and a utility subject to (17), and (ii) the say that a competitive telling. we construct an incentive-compatible = u'{c^{9 pRt-i,t any sequence of = t feasible, incentive-compatible allocation {dl,y^}, competitive equilibrium with no bequests by setting T^{9^) for {cj'(^*), at} maximize dynastic yt{9^) — Ct[9^) and )) These choices work because the estate tax ensures that for any reporting strategy a, the consumption allocation {Cf{a\9*))} satisfies the consumption Euler equation «'(c^(a*(^*))) = ,5i?,,+i J]^x'(c^^l(cT'+l(^^^,+l)))(l and the labor income tax allocation is and no bequests, - rr+i(o-*+^(^',^m))) Pr(^m), such that the budget constraints are satisfied with this consumption 6J'(^*) = 0. Thus, this no-bequest choice is optimal for the individual and labor income y# is mandated given M* being implemented are invertible, then by the taxation principle we can rewrite T and r as functions of this history of labor income and avoid having to mandate labor income. Under this arrangement, individuals do not make reports on their shocks, but instead simplj^ choose a budget-feasible allocation of consumption and labor income, taking as given prices and the tax system. ^^In this formulation, taxes are a function of the entire history of reports, this history. However, if the labor income histories j/' : 6' 26 — > regardless of the reporting strategy followed. Since the resulting allocation by hypothesis, it follows that truth teUing is budget constraints then ensure that the asset market As noted above, in our economy without The optimal. is clears. ^^ capital only the after-tax interest rate matters so the implementation allows any equilibrium before-tax interest rate {Rt^it}. we In the next subsection, set the interest rate to the reciprocal of the social discount factor, Rt-i,t natural because it incentive compatible, resource constraints together with the =P This choice is represents the interest rate that would prevail at the steady state in a version of our economy with capital. Optimal Progressive Estate Taxation In this subsection we derive an important intertemporal condition that must be satisfied by the optimal allocation. This condition has interesting implications for the optimal estate tax, computed using (18) at the optimal allocation. Let A be the multiplier on the promise-keeping constraint and on the incentive constraints. Then the first-order conditions for let /x(^, 6') represent the multipliers w_ and w{9) are = c'(t/_)-A-A M'W^))pW-/3Ap(^)-J]M^,^') + E^(^''^) = e' and the envelope condition is k' {v) = e' Together these imply X. Y,k'{wi9))p{e) Using c'(«_) = l/u'{c-) = A -I- k'{v) ^ The = left-hand side together with the equation. negative we = L'{v), P e arrive at the Modified Inverse Euler equation ii:;4^M«)-A(i-i). first term on the right-hand side The second term on the right-hand side is novel, since it is is (19) the standard inverse Euler zero when (3 = (5 and is strictly when $ > p}^ In our environment, the relevant past history is encoded tax r(0*~\^i) can actually be reexpressed as a function of in the continuation utility so the estate Vt{6*'~^) and Ot- Abusing notation we then denote the estate tax by Tt{v,9t)- Since we focus on the steady-state, invariant distribution, we also drop the time subscripts and write t{v, 9). ^^Since the consumption Euler equation holds with equality, the same estate tax can be used to implement alloca- tions with any other bequest plan with income taxes that are consistent with the budget constraints. ^^This equation can also be derived from an elementary variation argument. This is done in Farhi, Kocherlakota and Werning (2005), who also show that this equation, and its implications for estate taxation, generalize to an economy with capital and an arbitrary process for skills. 27 The average estate tax rate t{v) is then defined by l-f(i;) = 5^(l-r(^,^))p(^) (20) e Using the modified inverse Euler equation (19) we obtain f{v) = -A«'(c_(^^))(^-l) In particular, this impfies that the average estate tax rate are subsidized. is negative, f{v) < so that bequests 0, However, recaU that before-tax interest rates are not uniquely determined in our (18). With our particular choice for the before-tax; interest rate, however, the tax rates are pinned down and As a consequence, neither implementation. acquires a corrective, Pigouvian role. externalities are the estate taxes computed by Differences in discounting can be interpreted as a form of from future consumption, and the negative average tax can then be seen as a way of countering these externalities as prescribed depends on the choice of the before-tax be a robust steady-state outcome In our model it is more in by Pigou. In our setup without interest rate. a version of our economy with capital. interesting to understand how the average tax varies with the history of past shocks encoded in the promised continuation utility v. function of consumption, which, in turn, progressive: the average tax Proposition 4 In capital, this result However, the negative tax on estates would on transfers is for The average tax v. higher. the repeated Mirrlees economy, the optimal allocation can be The is face lower net rates of return so that their 7 Conclusions Should privately- felt parental altruism implications for inequality? the optimal The fortunate consumption path decreases towards the mean.^° affect the social contract? To address ip* , increasing in promised continuation utility v. progressivity of the estate tax reflects the mean-reversion in consumption. must is implemented by a combination of income and estate taxes. At a steady-state, invariant distribution average estate tax f{v) defined by (18) and (20) an increasing Thus, estate taxation an increasing function of more fortunate parents is is these questions, If so, we model a what are the long-run central tension in society: the tradeoff between ensuring equality of opportunity for newborns and providing incentives for altruistic parents. Our model's answer is that society should indeed exploit altruism to motivate parents, linking the welfare of children to that of their parents. However, of future generations directly, the inheritability of ^*^Farhi, we also find that if we value the welfare good or bad fortune should be tempered. This Kocherlakota and Werning (2005) explore more general versions of this result and discuss several related intuitions. 28 produces a steady-state outcome in which welfare and consumption are mean-reverting, long-run inequality What of our is bounded, social mobility is and misery possible is avoided by everyone. instruments should society use to implement such allocations? For a Mirrleesian version model we progressive: find an important role for the estate tax. more fortunate parents should The optimal tax on face a higher average marginal tax rate on inheritances is their bequests. This result illustrates an interesting way in which the conflict between corrective and redistributive taxation is optimally resolved. Further examination of other situations with similar conflicts remains an interesting direction for future work.^^ Appendix Proof of Theorem Weak 1 concavity of the value function k{v) follows because the relaxed sequence problem has a concave objective and a convex constraint its continuity over the interior of also be established as follows. Define the k*{v) subject to at any 1) finite = bounded, continuity at the extremes can first-best value function Then Then v. concavity of the value function k{v) implies If utility is = max^^* ^^Q/3*E_i[^4Ui(^*)]. extremes v and The weak set. domain. its E.^[etut{e') k*{v) is - Xc{ut{9'))] continuous and k{v) < k*{v), with equality continuity of k{v) at finite extremes follows. Thus, k{v) is continuous. The constraint (6) with q — J3 implies that utihty and continuation utility are well-defined. Toward a contradiction, suppose T lim Y(3'Esetu{ct) ' ^—>oo T—>oo ^ t=0 is not defined, for some s utility is concave 9u{c) < > — 1. Ac-[- B This implies that limy^oo St=o/3*'^^^(-^s^t'"(^)7 0) for some A, T ^/5*max(£;,^tu(Q),0) < Taking the limit yields lim^^oo X^f=o /^ ^^-iCt 5> 0, so it ^^- Since follows that T A^/?%q + S < = = T /I J]] ^*E,q oo. Since there are finitely -f many 5 histories 9^ G 0*+^ St=o/5 E_iQ = oo. If there is a non-zero measure of such agents this implies (6). Thus, for both the relaxed and unrelaxed problems utility and continuation this implies limj-^oo a contradiction of utility are well defined given the other constraints on the problem. This is recursive formulation below. 'Some progress along these lines can be found in Amador, Angeletos and Werning (2005). 29 important for our We next prove two lemmas that imply the rest of the theorem. Consider the optimization problem on the right hand side of the Bellman equation: - supE[^^i(^) v Ou Define m= + pw{e) > (e) maXc>o,6iee(^^(c) = — Xc{u{9)) (21) E[eu{9)+Pw{e)] Ou {9') and k{v) Ac) + pk{w{e))] + pw{9') = (22) — m/{\ — k{v) eQ for all 9,9' $) < (23) The problem 0. in (21) is equivalent to the following optimization with non-positive objective: supE[^u(^) - Xc{u{9)) -m + pk{w{9))] (24) subject to (22) and (23). Lemma Proof. The supremum in 1 If utility is (21), or equivalently (24), is attained. bounded the result follows and compactness of the constraint arguments apply when utility is immediately by continuity of the objective function So suppose set. utility is unbounded above and below only unbounded below or only unbounded above. We — similar first show that lim k{v) = and then use maximum To is without this result to restrict, lim k{v) = — oo (25) i;—»— oo v—*oo the optimization within a compact loss, set, ensuring a attained. establish these limits, define oo h{v] p) subject to f = = sup Y^ ^*E_i [9tu{9') E_i Yl^ol^^^tu{9^)- Since incentive constraints it then the desired limits (25) follow. Since 9u h{v, P) < h{v, \c{u{9')) h{v,P). — Xc{u) If — - m] same problem but without the this corresponds to the < follows that k{v) - \miy^ooh{v, $) m< P)=v- XC{v, and (3) - (3 = <$ limy^^oo h{v,/3) it = — oo, follows that 777 ^—^, (26) where CO C{v,l3) = miY^(3'E_^c{u{9')) t=o subject to f = ^^g/3*E_i[^iu(^')]. problem, with solution u{9^) = Note that C{v,P) {c')~^{9t'y{v)) for some 30 is a standard convex first-best allocation positive multiplier 7(f), increasing in v and = such that hmij^-ooTlf) ^^^ hm^_oo7 C{v,P) so that Um^j^-oo h{v, hm„_^_oo h{v, Fix a (3) /?) = — oo = — oo and and hm^,^oo Take any allocation that V. bounded Then oo- h{v, = — oo. /3) = — oo, verifies the constraints (22) and (23) and > Since {9u — we can Xc{u)) is this defines a closed, that we can with {9u — \u{9)\ < continuous over this restricted Lemma \c{u)) — for (25), < oo — m > k/p{9). oo or m ^ — oo, Thus, there exists an My^ < oo such > when Xc{u{9)) either u — My^u- maximum must be set. Since the objective function attained. ). some v k{v) where the maximization My^^ * — — oo 2 The value function k{v) satisfies the Bellman equation (7)-(9 Proof. Suppose that is > maxE[^M(^) - Xc{u{9)) subject to (22) and (23). k{v) for all the set, oo be M^„;. interval for u{9), for each 9. maximization to < A; restrict the concave with the limits maximization over u{9) so that 9u{9) strictly concave, bounded < Hence, we can restrict the ma:ximization in (24) to a compact is is let we can non-positive, interval for w{9), for each 9. It follows that there exists restrict the restrict the is k/{$p{9)). Since k{w{9)) such that we can restrict the maximization to |w(^)| Similarly, Using the inequahty (26) this estabhshes which, in turn, imply the hmits (25). Then, since the objective maximization to w{9) such that k{w{9)) this defines a closed, = = ^E[c[{cr\9^{v)))], hm,;_^oo h{v,$) the corresponding value of (24). i'^) Then + pk[w{9))] A> there exists such that >¥.[9u{9)-\c{u{9))^[5k{w{9))]^/l {u,w) that satisfy (22) and (23). But then by definition oo > k{w{9)) for all allocations u that yield w [9] J];3*E_i[^,M,(^*) - Xc{ut{9'))] and are incentive compatible. Substituting, we find that oo k{v) > Y^p'e_,[9M0') - HMO'))] + A t=o for all incentive-compatible allocations that deliver v, a contradiction with the definition of k{v). Namely, that there should be a plan with value arbitrarily close to k{vo). max„,^ E[^u(^) - Xc{u{9)) + Pk{w{9))] subject to (22) and (23). 31 We conclude that k{v) < By and definition, for every v and e > there exists a plan {U((^*; v, e)} that is incentive compatible delivers v with value oo Y,J5'¥._^[etUt(e';v,e)-Xc{ut{e'-v,e))] Let {u* {9) w* {6)) e argmax„,^E[6'ti(6') , ondutie') =Ut-i{{9i,...,et);w*{eo),€) - Xc{u{6)) fori> + Pk{w{6))]. >k{v)-£. Consider the plan Uo{9o) = u*{6q) Then 1. oo > k{v) 5];3'E_i[0,u,(^*)-Ac(«i(^*))] f=0 oo = E_i [eoU*{9o) - Ac(«*(^o)) +^^^*Eo[^i+iMt+i(^*+^) - Xc{ut+i{e'+'))]j t=o > Since £ (22) > and was arbitrary it Xu{e) + (3k(w{e))] - Pe. follows that k{v) > max^^u; E[^ii(^) — Xc{u{9)) + (3k{w{6))] subject to (23). both inequalities imply k{v) Finally, together to (22) - maxE[6'u(^) and = — maXu^.u,E[^'u(0) Xc{u{6)) -\- pk{w{9))\ subject (23). Proof of Theorem 2 Suppose the allocation {ut} Pcirt (a). incentive compatible equation (7), we and is generated by the policy functions starting from delivers lifetime utility Vq. Vq, is After repeated substitutions of the Bellman arrive at T k{vo) = J^P'^-ii^Me') - Xc{u,{0'))]+p^E_,k{vT{9^)). (27) t=0 Since k{vo) is bounded above this implies that oo k{vo) so {tit} is < ^p'E_,[eMo') - xc{ut{e'))], optimal, by definition of k{vo). Conversely, suppose an allocation {uf} incentive compatible and is optimal given vq. Then, by definition it must be deliver utility vq. Define the continuation utility implicit in the allocation oo t=l and suppose that either uq (9) ^ (7" {9; vq) or wq g"^ {9; Vq), for (9) 7^ 32 some 9 E Q. Since the original plan {ut} The Bellman equation incentive compatible, Uo{9) and wo{9) satisfy (22) and (23). is then implies that k{vo) = E[g''{e;vo)-Xc{g%9;vo))+Pk{g^{9;vo))] > E[uo{9) > E_,[uo{9o) - + pk.{wo{9))] Xc{uo{9)) CXD - Xc{uoi9o))] +J2f^'^-^[Md') ~ HMO'))]. t=i The first inequality follows since uq does not definition of k{wo{9)). argument applies t > 1. We if maximize while the second inequahty follows the (7), Thus, the allocation {ut} cannot be optimal, a contradiction. the plan is A similar not generated by the policy functions after some history ^* and conclude that an optimal allocation must be generated from the policy functions. Pcirt (b). First, suppose an allocation {ut, Vt} generated by the policy functions (^", g^) starting = at vo satisfies limt^oo/3*E_ift(^*) 0. Then, after repeated substitutions of (8), we obtain T V = Y, /5'E_i [9Md')] + /3^E_i [vt{9^)] (28) . t=o Taking the limit we get Vq Vq. Next, we show Suppose utility is Yl't^o /5*E-i[^<^f (^*)] so = oo. Since the value function k{v) maximum, we can bound < Similarly, = — cx), liminfi^oo/3 E_ift(^*) utility is = — oo. (3 > implies that (5 < av + b, with a < 0. Thus, + b= -oo a contradiction since there are feasible plans that yield conclude that limsupj^^/3*E_it;j(^*) suppose Then we have t—>oo (27) implies that k{vo) We 0- Vq, non-constant, concave and reaches an alimsup^*E_iz;f(^*) i—»oo finite values. is the value function so that k{v) limsup/9'E_iA;(z;t(^*)) and then that the allocation {ut} delivers lifetime utility unbounded above and limsupj_Qo/^*E-i^<(^*) > limsupj^Q^/? E_ifi(^') interior = that for any ahocation generated by (p",p"'), starting from finite < 0. unbounded below and that liminfj^oo Using k{v) < av + b, with a liminf /?*E_iA;(z;i(^*)) > 0, /3*E_ii;i(^*) < 0. Then Then after we conclude that = -oo t—«x implying A;(i;o) The two Part = — oo, a contradiction. Thus, we must have liminft^oo/5*JE_ii't(^*) established inequalities imply limt^oo P^E_iVt{9^) (c). Suppose limsup^^^ P^E-iVt{a^{9^)) 33 > = > 0. 0. for every reporting strategy a. repeated substitutions of (9), implying T ^"" Therefore, {ut} Proof of Part (a) is incentive compatible, since v Lemma (Strict Concavity) Let Theorem and is attainable with truth telling from part (b). 1 functions starting at vq (note: utility values Va t=o Vb, with {ut{6'^ vq) , Va^ Vb. is = aut{d\va) + {I v^{e') = avt{e';va) + {l-a)vt{9';vb) and required). Take two initial utilities <(^*) 2 part (b) implies that {ut{9^,Va)} T Vt{9\ Vq)} be the plans generated from the policy Define the average immediately implies that {uf{9^)} delivers there exists a finite time , no claim of incentive compatibility - a)ut{e\vb) {ut(^*,Vf,)} deliver Va initial utility v°' = aVa + (1 and — Vb, respectively. ce)vb- It This also implies that such that T T so that ut{9'va)i^u^{9'vb), for some history and Vb 9^ € 0'+^. Consider iterating T (29) times on the Bellman equations starting from Va : T k{v,) = J2p'E.,[9M0';Va)-c{ut{9';Va))]+fE_,k{vT{9^;Va)) Kvb) = T 5];5*E_l[^,^i,(^^^;6)-c(«i(^^i;,))]+^Vl^^(t;T(^^;^;6)), 34 . and averaging we obtain T ak{va) + J2p'E_i[etunO') - = {l-a)k{v,) [ac{ut{9';Va)) + (1 - a)c{ut{9';v,))]] t=0 +fE^^[ak{vT{e'^;Va)) T + - (1 a)k{vT{9^;Vk))] 4=0 where the that inequahty follows from the strict we have the inequality (29), concavity of the cost function c(u), the fact strict and the weak concavity of the value function k. inequality follows from iterating on the Bellman equation for satisfies is The last weak since the average plan {u°',v°') the Bellman equations constraints at every step. This proves that the value function k{v) strictly concave. (b) (Differentiability) Since the value function k{v) is, v°' there is concave, it is — that can then be established by at least one sub-gradient at every point v. Differentiability is sub-differentiable the following variational envelope arguments. Suppose first that utility neighborhood around W{v) Since W{v) is unbounded below. Fix an E[9{g^{9, vo) + {v vq. - Vo)) Since W'{vo) exists Theorem (see 4.10, in Stokey, k'ivo) Finally, since d{u) >0 = shows that this Next, suppose utility - Xc[g\9, = + {v - Vo)) is it follows, + Pk{g^{9, vo))] it W'ivo) k'{v) bounded below, < = W{v) < k{v), by application of the Benveniste-Scheinkman -A 1 follows that k'{vo) also exists and E[c'{u*{9))]. (30) 1. say, by > zero. Then the argument above 0, for all 9 establishes E Q. However, corner solutions with are possible here even with Inada assumption on the utility function, so a different envelope argument If utility is that, applying any vo) Lucas and Prescott, 1989), that differentiability at a point Vq as long as g'^{9, vq) g'^{9,Vo) interior value vq for initial utility. For a the value of a feasible allocation in the neighborhood of vq with equality at Theorem = is Vq define the test function is required. We provide one that exploits the homogeneity of the constraint bounded below, then limsupj_^^E_i/3*f((a-(^*)) > Theorem 2, for all reporting strategies a solution {uj} to the planner's sequence problem interior vq, the plan {{v/vQ)uf} is incentive compatible = V^*E_i \9t-ut{9') 35 - Xc(-ut{9')] a so ensured. Then, for and attains value v addition the test function W{v) is set. for the agent. In W{v) < satisfies The Hmit wise it is = W{vo) k{v), Scheinkman Theorem, that lim.y^yk'{v) = — oo inherited by the upper limy^ooh{v,p) = -oo is differ enti able. It follows immediately from lim^^c A;(w) bound h{v,P) introduced = \imy^y^k'{v) from the Benveniste- follows and equals W'{vo). and lim^^oo §^h{v,P) we show that FinaUy, and k{vo) k'{vo) exists = = — oo, in the proof of if i; Theorem < oo. Other- since both 1, -oo. Consider the deterministic planning problem oo. oo k{v) = uiaxy^P {ut — \c{ut)) t=Q = ^^0 /5*'"i- Note that k{v) is plans are trivially incentive compatible, it subject to v must have liiny^yk'{v) = CLAR < follows that k{v) = oo. Since deterministic k{v), with equality at v. Then we oo to avoid a contradiction. Proof of Proposition The differentiable with liiny^yk^{v) 1 equation was shown in the main text, so we focus here on the bounds. Consider the program maxy^ Pn{9nUn " n V 9nUn This problem and constraint on u. its But of neighboring agents + PlUn > which is = y^^PnjOnUn+PWn) 6'„M„+i + /3Wn+l ioT is U = We notation require some discussion. this notation allows us to consider 1,2, ..., K- 1, do not incorporate the monotonicity bunching in the following way. If bunched, then we group these agents under a single index and the total probability of this group. this group, + Pk{Wn)} c(m„) ' what is Likewise let any let set p„ be On represent the conditional average of 9 within relevant for the promise-keeping constraint Let 9n be the taste shock of the highest agent in the group. The and the objective function. incentive constraint must rule the highest agent in each group from deviating and taking the allocation of the group above him. Of course, every combination of bunched agents leads to a different program. We study all of them. The optimal allocation of our problem must solve one of these programs, although not necessarily the one that yields the highest value, since this one condition violated. The is may not be feasible if the monotonicity first-order conditions are Pn{9n - Ac' (tin) PniPk'iWr,) where, by the Envelope theorem, A = - Xdn} + On/J^n " - PX} + /?(/X„ - k'{v). Summing 36 ^n-lA^n-l Ai„_i) = = the first-order conditions for consumption, we get = \E[c'{u{9))] The first-order conditions for n — (l-X)+ I l-k'{v) imply ^^ ^^^("i) = ^^^^("^)1 < 1^ ^ This imphes Pi :i-A)^ ~ ^1 Using we get 3 > A;'K) l P + 1 ^1 (^ n Similarly, writing the first-order conditions for '1 _ A) - ^J£zlt^I<^ = = K, we A(/(uk) > ^i" ,// X k'{v) :r- 13 '0^ +^ P Ui (^iJ — 1 - ^1 get XE[c'{ue)] ^ 1-A This implies f-i>i^-(l-A)/^ Using ,// P^ X , P we P l^K-1 P Pk get k'{v) -f fA'-l For any n, 1 + lU/c < i^n 1-^ ^1 ^1 < u^i, A:'(i;) c/A'-1 we have +i P for every h_}_ h /5L e K-l ^K-\ 'K-l (^K-l n < k'{Wn) < P 0, 1 37 + k'{v) 'K-l ^K-1 After rearranging, we obtain ^1 1 + :i-k'(v)) l^i > l-k'{g''{e,v)) 1 > e (1 - k'{v)) + 1-1 ^A'-l To arrive at the expression in the text that is Turning to the bounded utihty satisfied w we take the worst most unfavorable to each bound, noting that when A > implies k'{w{9)) 1 with ^„ = = k'{f3~^v) — case scenario: > k'{v) we choose the subproblem 0. case, note that all the first-order conditions = and u{9) = 1 ((5 / (3)k' {v) and w {9) = P~^v > Since the problem . is v. The and constraints are first-order condition for strictly convex, this represents the unique solution. Since establishing the lower for all V. bound involved no assumption on The upper bound, on the high enough v, i.e. for low enough > other hand, required u{9) interior solutions for for all 9, u this holds which must be true for k'{v). Proof of Proposition 2 Since the derivative k'{v) is continuous and strictly decreasing, we can define the transition function = Q{x,9) for all X < 1. k'{g-{{krHx),0)) For any probabihty distribution //, let Tq{ii) be the probability distribution defined by Tq{ij){A) for any Borel For example, TQ^ni^x) = Lemma is _ Tq + TI + = ... + T^ n the empirical average of k'{vt) over all histories of length n starting with X. 4 For each x < there exists a subsequence TQ^^(n){5x) that converges weakly 1 distribution) to an invariant distribution Proof. The bounds (11) derived on (— oo, in Proposition 1 limg(x,^) xj,l We first / ^{Q(x,e)eA}d^i{x)dp{9) set A. Define ^^'" k'{vo) = = lim t;—-0O 1) : [— oo, 1) x 6 — » k'ig'\9,v)) (— oo, 38 in imply that extend the continuous transition function Q{x, transition function Q{x,9) (i.e. under Q. 1), 9) : = P/p < (— oo, 1) with Q{1,9) 1. x —^ (— oo, = P/P 1) to a continuous and Q{x,9) = Q{x,9), for all X G (—00, 1). It follows that Tq maps and = distributions over (—00, We next set is 1), Tq{Sj:) probability distributions over [—00, 1) to probability Tq{5x), for show that the sequence {Tq^{5x)} A^ such that Tq^{6x){Ai;) > 1 simply E^i[k'{vtie'-^))] with x = — e, is for all k'{vo) < all x G (—00, tight, in that for ri. The expected 1). any £ > there exists a compact value of the distribution T^{Sx) = Recall that E_i[k'{vt{9^-^))] 1. k'{vo){P/PY -^ 0. This imphes that < mm{0,k'{vo)} E^,[k'{vt{e'-'))] < T^{Sx){-cx,,-A)i-A) + {l-T^{5x){-cx,,-A))l for all A> 0. Rearranging, T^fe)(-cx..-^)< '-7;\°-^> which implies that {Tq{6x)}, and therefore {Tq^niSx)}, is tight. Tightness implies that there exists a subsequence Tqj^,.{Sx) that converges weakly some distribution) to Since Q{x,9) tt. is (i.e. in continuous in x, then Tq{Tq ^,J5x)) converges weakly to TQ{n). But the linearity of Tq implies that _T^^''^^\5x)-Tq{5x) and since 0(n) Recall that (—00, for 1). ^ 00 we must have Tq{7v) = Tq maps This implies that such distribution it tt. probability distributions over [—00, 1) to probability distributions over tt = ^^(Tr) has follows that n = no probability mass Tq{'k)^ so that tt is at {!}. Since Tq and Tq coincide an invariant distribution on (—00, 1) under Q. 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