Inequality and Social Discounting

We explore steady‐state inequality in an intergenerational model with altruistically linked individuals who experience privately observed taste shocks. When the welfare function depends only on the initial generation, efficiency requires immiseration: inequality grows without bound and everyone’s consumption converges to zero. We study other efficient allocations in which the welfare function values future generations directly, placing a positive but vanishing weight on their welfare. The social discount factor is then higher than the private one, and for any such difference we find that consumption exhibits mean reversion and that a steady‐state, cross‐sectional distribution for consumption and welfare exists, with no one trapped at misery.


I. Introduction
Societies inevitably choose the inheritability of welfare.Some balance between equality of opportunity for newborns and incentives for altru-For useful comments and suggestions we thank the editor and two anonymous referees, as well as Daron Acemoglu, Laurence Ales, Fernando Alvarez, George-Marios Angeletos, Abhijit Banerjee, Gary Becker, Bruno Biais, Olivier Blanchard, Ricardo Caballero, Francesca Carapella, Dean Corbae, Mikhail Golosov, Bengt Holmstrom, Chad Jones, Narayana Kocherlakota, Robert Lucas, Pricila Maziero, Casey Mulligan, Roger Myerson, Chris Phelan, Thomas Philippon, Gilles Saint-Paul, Robert Shimer, Nancy Stokey, Jean Tirole, and various seminar and conference participants.This work began sparked by a seminar presentation by Chris Phelan at MIT.We also gained insight from a manuscript by Michael Sadler and the late Scott Freeman that Dean Corbae brought to our attention.
Each generation is composed of a continuum of individuals who live for one period and are altruistic toward a single descendant.There is a constant aggregate endowment of the only consumption good in each period.Individuals are ex ante identical but experience idiosyncratic shocks to preferences that are privately observed.Feasible allocations must be incentive compatible and must satisfy the aggregate resource constraint in all periods.
When only the welfare of the first generation is considered, the planning problem is equivalent to that of an economy with infinite-lived individuals.Intuitively, immiseration then results because rewards and punishments, required for incentives, are best delivered permanently to smooth dynastic consumption over time.As a result, the consumption process inherits a random-walk component that leads cross-sectional inequality to grow without bound.This is consistent with a constant aggregate endowment only if everyone's consumption converges to zero.As a result, no steady-state, cross-sectional distribution with positive consumption exists.
Interpreted in the intergenerational context, this solution requires a lockstep link between the welfare of parent and child.This perfect intergenerational transmission of welfare improves parental incentives, but it exposes future generations to the risk of their dynasty's history.Future descendants value insurance against the uncertainty of their ancestors' past shocks, and our welfare criterion captures this.
When future generations are weighted in the social welfare function, it remains optimal to link the fortunes of parents and children, but no longer in lockstep.Rewards and punishments are distributed over all future descendants, but in a front-loaded manner.This creates a meanreverting tendency in consumption-instead of a random walk-that is strong enough to bound long-run inequality.The result is a steady-state distribution for the cross section of consumption and welfare, with no one at misery.Moreover, mean reversion ensures a form of social mobility, so that families rise and fall through the ranks incessantly.
It is worth emphasizing that our exercise is not predicated on any paternalistic concern that individuals do not discount the future appropriately.Rather, the difference between social and private discounting used in our Pareto-efficient analysis arises because the social welfare function gives direct weight to future generations.However, our formal analysis can be applied whatever the motivation, paternalistic or not, for a difference in social and private discounting.For example, Caplin  and Leahy (2004) make a case for a higher social discount factor within a lifetime.
A methodological contribution of this paper is to reformulate the social planning problem recursively in a way that extends the ideas introduced by Spear and Srivastava (1987) to a general-equilibrium sit-journal of political economy uation in which private and social objectives potentially differ.We are able to reduce the dynamic program to a one-dimensional state variable, and our analysis and results heavily exploit the resulting Bellman equation.
The paper most closely related to ours is Phelan (2006), which considered a social planning problem with no discounting of the future.Phelan shows that if a steady state for the planning problem exists, then it must solve a static maximization problem, and solutions to this problem have strictly positive inequality and social mobility.Our paper establishes the existence of a steady-state distribution for any difference in social and private discounting.In contrast to the case with no discounting, there is no valid static problem, so our methods are necessarily different.Our work is also indirectly related to that of Sleet and Yeltekin  (2004), who study a utilitarian planner that lacks commitment and always cares for the current generation only.The best equilibrium allocation without commitment is equivalent to the optimal one with commitment but with a more patient welfare criterion.Thus our approach and results provide an indirect, but effective, way of characterizing the problem without commitment and establishing the existence of a steadystate distribution.In effect, lack of commitment, or other political economy considerations, can provide one motivation for the positive Pareto weights that future generations command.
The rest of the paper is organized as follows.Section II contains some simple examples to illustrate why weighing future generations leads to a higher social discount factor and why mean-reverting forces emerge from any difference between social and private discounting.Section III introduces the economic environment and sets up the social planning problem.In Section IV, we develop a recursive version of the planning problem and establish its relation to the original formulation.The resulting Bellman equation is then used in Section V to characterize the mean reversion in the solution.Sections VI and VII prove and discuss the main results on the existence of a steady state for our planning problem.Section VIII offers some conclusions from the analysis.Proofs are contained in the Appendix.

II. Social Discounting and Mean Reversion
In this section, we preview the main forces at work in the full model using a simple deterministic example.We first explain why weighing future generations maps into lower social discounting.We then show how this affects the optimal inheritability of welfare across generations.Finally, we relate the latter to the mean reversion force, which guarantees a steady-state distribution with social mobility in the full model.Our discussion also provides a novel intuition for the immiseration result in Atkeson-Lucas.

Social Discounting
Imagine a two-period deterministic economy.The parent is alive in the first period, , and is replaced by a single child in the next, .t p 0 t p 1 The child derives utility from his own consumption, so that v p 1 .The parent cares about her own consumption but is also altruistic A welfare criterion that weighs both agents and serves to trace out the Pareto frontier between and is , for some weight with the social discount factor given by .b { b ϩ a The only difference between the welfare criterion and the objective of the parent is the rate of discounting.Social discounting depends on the weight on future generations a.When no direct weight is placed on children, so that , social and private discounting coincide, a p 0 , which is the case covered by Atkeson and Lucas (1992).When-b p b ever children are counted directly in the welfare criterion, , society a 1 0 discounts less than parents do privately, .The child's consumption b 1 b receives more weight in the welfare criterion because it is a public good that both generations enjoy.

A Planning Problem
In Section III we show that the calculations above generalize to an infinite-horizon economy and lead to an objective with more patient geometric discounting: . We now consider a simple planning problem for such an infinite-horizon version.Now, suppose that there are two dynasties, A and B. In each period, a planner must divide a fixed endowment 2e between the two dynasties, giving to A and to B. Suppose that, for some reason, the heads of the dynasties are promised differential treatment, so that the difference in their welfare must be D. The planner's problem is The first-order conditions for an interior optimum are where l is the Lagrange multipliers on the constraint.

Imperfect Inheritability
Suppose that the founder of dynasty A has been promised higher welfare so that .The first-order condition then reveals that every D 1 0 l 1 0 member of dynasty A enjoys higher consumption, .If , as in Atkeson and Lucas (1992), consumption is constant over time for both groups, and initial differences persist forever.The unequal promises to the first generation have a permanent impact on their descendants.The inheritability of welfare across generations is perfect: the consumption and welfare of the child move one-to-one with the parent's welfare.
In contrast, when , the difference in consumption between the b 1 b two dynasties shrinks over time.Consumption declines across generations for group A and rises for group B. The inheritability of welfare across generations is imperfect: a child's consumption varies less than one-for-one with the parent's.Indeed, initial differences completely vanish asymptotically-initial inequality dies out. Figure 1 illustrates these dynamics for consumption.In this simple deterministic example, initial inequality D was taken as exogenously given.However, in the model with taste shocks, inequality is continuously generated in order to provide incentives.The dynamics after a shock are similar to those illustrated here, so that figure 1   (1992), there is no mean reversion, so that shocks accumulate indefinitely and inequality increases without bound.

III. An Intergenerational Insurance Problem
At any point in time, the economy is populated by a continuum of individuals who have identical preferences, live for one period, and are replaced by a single descendant in the next.Parents born in period t are altruistic toward their only child, and their welfare satisfies where is the parent's own consumption, is the altruistic c ≥ 0 b (0, 1) t weight placed on the descendant's welfare , and is a taste shock that is assumed to be identically and independently distributed across individuals and time.We make the following assumption.Assumption 1.(a) The set of taste shocks V is finite.(b) The utility function is concave and continuously differentiable for with This specification of altruism is consistent with individuals having a preference over the entire future consumption of their dynasty given by In each period, a resource constraint limits aggregate consumption to be no greater than some constant aggregate endowment .e 1 0 The following notation and conventions will be used.We refer to as utility and the discounted, expected utility as welfare.
Taste shock realizations are privately observed, so any mechanism for allocating consumption must be incentive compatible.The revelation principle allows us to restrict attention to mechanisms that rely on truthful reports of these shocks.Thus each dynasty faces a sequence of consumption functions , where represents an individual's consump- tion after reporting the history .It is more convenient to work with the implied allocation for utility with . A dynasty's reporting strategy is a sequence of functions that maps histories of shocks into a current report .Any strategy j induces a history of reports .We use to t t ϩ1 tϩ1 j : V r V j * denote the truth-telling strategy with for all .
We identify each dynasty with its founder's welfare entitlement v.We assume, without loss of generality, that all dynasties with the same entitlement v receive the same treatment.We then let w denote a cumulative distribution function for v across dynasties.An allocation is a sequence of functions for each v. (ii) it is incentive compatible for all v: whenever this sum converges; and (iii) total consumption does not exceed the fixed endowment e in all periods: Define to be the lowest endowment e such that there exists an e*(w) allocation satisfying (2)-(4), which is precisely the efficiency problem studied in Atkeson and Lucas (1992).

Social Discounting
We have adopted the same preferences, technology, and informational assumptions as in Atkeson and Lucas (1992).Our only departure is to introduce the planning objective (5) for each dynasty, which is equivalent to the preferences in (1), except for the discount factor .Our motivation for this objective is that b 1 b it can be derived from a welfare criterion that places direct weight on the welfare of future generations.To see this, consider the sum of expected welfare, , using strictly positive weights : where .Then the discount factor so that social preferences are more patient.Future generations are already indirectly valued through the altruism of the current generation.
If, in addition, they are also directly included in the welfare function, the social discount factor must be higher than b. 3 In particular, weighing future generations with geometric t p 1, 2, … Pareto weights gives The first term is identical to the expression in (5); the second b 1 b is a constant when initial welfare promises for the founding gener-v 0 ation are given, as they are in the social planning problem defined below.

Planning Problem
Define the social optimum as a feasible allocation that maximizes the integral of (5) with respect to distribution w. .This problem e 1 e*(w) is convex: the objective is linear, constraints (2) and (3) are linear, and the resource constraints (4) are strictly convex.
The way we have defined the social planning problem imposes that initial welfare entitlements v be delivered exactly, in the sense that the promise-keeping constraints (2) are equalities instead of inequalities.Alternatively, suppose that the founder of each dynasty is indexed by some minimum welfare entitlement , with distribution .The Pareto ṽ w problem maximizes the integral of the welfare criterion (7) subject to delivering or more to the founders and incentive compatibility.The ṽ two problems are related: the solution to the Pareto problem solves the social planning problem for some distribution w that first-order stochastically dominates (so that for all v).In particular, ˜w w (v) ≤ w(v) comparing the terms in ( 7) with (5) implies that the Pareto problem chooses w to maximize

͵
subject to for all v.In general, depending on the given w(v) ≤ w(v) , the constraints of delivering the initial welfare entitlements or ˜w v more may be slack, so that may be optimal.However, we shall w ( w show that setting is optimal for initial distributions of entitle-w p w ments that are steady states, as defined below.Our strategy is to w solve the social planning problem and then show that it coincides with the Pareto problem's solution in Section VII.

Steady States
The social planning problem takes the initial distribution of welfare entitlements w as given.In later periods the current cross-sectional distribution of continuation welfare is a sufficient statistic for the re-w t maining social planning problem: the problem is recursive with state variable .It follows that the solution to the social planning problem w t from any period t onward, , is a time-independent function of the current distribution , which evolves according to a stationary re-w t cursion , for some fixed mapping W. w p Ww tϩ1 t We focus on distributions of welfare entitlements such that the w* solution to the social planning problem features, in each period, a crosssectional distribution of continuation utilities that is also distributed v t according to .In this case, the cross-sectional distribution of con-w* sumption also replicates itself over time.We term any distribution of entitlements with these properties a steady state.A steady state cor-w* responds to a fixed point of this mapping, .w* p Ww* In the Atkeson-Lucas case, with , the nonexistence of a steady b p b state with positive consumption is a consequence of the immiseration result: starting from any nontrivial initial distribution w and resources , the sequence of distributions converges weakly to the distri-e p e*(w) bution having full mass at misery , with zero con-v p U(0)/(1 Ϫ b) sumption for everyone.We seek nontrivial steady states that exhaust w* a strictly positive aggregate endowment e in all periods.
Using the entire distribution as a state variable is one way to ap-w t proach the social planning problem.Indeed, this is the method adopted by Atkeson and Lucas (1992).They were able to keep the analysis manageable, despite the large dimensionality of the state variable, by exploiting the homogeneity of the problem with constant relative risk aversion (CRRA) preferences.In contrast, even in the CRRA case, our model lacks this homogeneity, making such a direct approach intractable. 4Consequently, in the next section, we attack the problem differently, using a dynamic program with a one-dimensional state variable. 5he idea is that the continuation welfare of each dynasty follows a v t Markov process and that steady states are invariant distributions of this process.

IV. A Bellman Equation
In this section we approach the social planning problem by studying a relaxed version of it, whose solution coincides with that of the original problem at steady states.The relaxed problem has two important advantages.First, it can be solved by studying a set of subproblems, one for each dynasty, thereby avoiding the need to keep track of the entire distribution in the population.Second, each subproblem admits a w t journal of political economy one-dimensional recursive formulation, which we are able to characterize quite sharply.We believe that the general route we develop here may be useful in other contexts.
Define a relaxed planning problem by replacing the sequence of resource constraints (4) in the social planning problem (8) with a single intertemporal constraint for some positive sequence with .One can interpret this problem as representing a small open economy facing intertemporal prices . 6The original and relaxed versions of the social planning {Q } t problem are related in that any solution to the latter that happens to satisfy the resource constraints ( 4) is also a solution to the former. 7It follows that any steady-state solution to the relaxed problem is a steadystate solution to the original one, since at a steady state the intertemporal constraint (10) implies the resource constraints (4).A steady state requires .
Consider then the intertemporal resource constraint (10) with : ( 1 1 ) Letting h denote the multiplier on this constraint, we form the Lagrangian (omitting the constant term due to e ͵ where 6 This is related to the decentralization result in Atkeson and Lucas (1992, sec.7, theorem 1), although they do not use it, as we do here, to characterize the solution.
7 Since the problem is convex, a Lagrangian argument establishes the converse: there must exist some positive sequence such that the solution to the original social planning {Q } t problem also solves the relaxed problem.This is analogous to the second theorem of welfare economics for our environment.However, we will not require this converse result to construct a steady-state solution.
We study the maximization of subject to (2) and (3).Maximizing L L is equivalent to the pointwise optimization of for each v: and This recursive formulation imposes a promise-keeping constraint (15) and an incentive constraint (16).The latter rules out one-shot deviations from truth-telling, guaranteeing that telling the truth today is optimal if the truth is told in future periods, which is necessary for full incentive compatibility (3).Full incentive compatibility (3) is taken care of in ( 14) by evaluating the value function defined from the sequence problem at the continuation welfare: in the next period, envision the planner as solving the remaining sequence problem, selecting an allocation for each that is incentive compatible for .Then any pair w(v) t p 1, 2, … that satisfies (15) and ( 16), pasted with the corresponding (u(v), w(v)) continuation allocations for each , describes an allocation that sat-w(v) isfies ( 2) and (3).
Among other things, theorem 1 shows that the maximization on the right-hand side of the Bellman equation is uniquely attained by some continuous policy functions and for u and w, respectively.
We emphasize that these policy functions solve the maximization in the Bellman equation ( 14) using the value function defined from the k(v) sequence problem (13). 8For any initial welfare entitlement v, an allocation can then be generated from the policy functions by , with and .Our next then it attains the maximum in the component planning problem ( 13).
The first part of theorem 2 implies either that the solution to the relaxed planning problem is generated by the policy functions or that there is no solution at all.From the second part of theorem 2, a solution is guaranteed if the limiting condition ( 17) can be verified.The proof proceeds by showing that the allocation generated by the policy functions is optimal if it satisfies the incentive compatibility constraint (3); the role of the limiting condition ( 17) is to ensure the latter. 9Conditions (15) and ( 16) ensure that finite deviations from truth-telling are not optimal; condition (17) then rules out infinite deviations.Condition (17) is trivially satisfied for utility functions that are bounded below; proposition 5 below verifies this condition when utility is unbounded below.

V. Mean Reversion
In this section we use the Bellman equation to characterize the solution to the planning problem and to show that it displays mean reversion.Our first result establishes that is differentiable and strictly concave k(v) with an interior peak. 8If one assumes bounded utility, then when the contraction mapping theorem is applied, the Bellman equation is guaranteed to have a unique solution, which must then coincide with defined from the sequence problem.However, we do not assume bounded utility k(v) and, for our purposes, find it unnecessary to solve fixed points of the Bellman equation or prove that it has a unique solution.Instead, we work directly with defined from k(v) the sequence problem (13) and simply exploit the fact that this function satisfies the Bellman equation. 9The proof of theorem 2 applies versions of the principle of optimality to verify incentive compatibility.In particular, for any ( , ) and an initial , a dynasty faces a recursive u w g g v 0 dynamic programming problem with state variable and with the report as the control.
Conditions ( 15) and ( 16) then amount to guessing and verifying a solution to the Bellman equation of the agent's problem, i.e., that the identity function satisfies the Bellman equation with truth-telling.The limiting condition (17) then verifies that this represents the dynasty's value from the sequential problem.
Proposition 1.The value function is strictly concave; it is k(v) differentiable on the interior of its domain, with .
The shape of the value function is important because mean reversion occurs toward the interior peak, as we show next.
Let l be the multiplier on the promise-keeping constraint (15) and let be the multipliers on the incentive constraints ( 16).The with equality if is interior.Given the limits for in proposition u g (v, v) k (v) 1, the solution for must be interior and satisfy the first-order Incentive compatibility implies that is nondecreasing as a func- vV b In sequential notation, this condition is v To provide incentives, the planner rewards the descendants of an individual reporting a low taste shock.Rewards can take two forms, and it is optimal to make use of both.The first is standard and involves spending more on a dynasty in present-value terms.The second is more subtle and exploits differences in preferences: it is to allow an adjustment in the pattern of consumption, for a given present value, in the direction preferred by individuals relative to the planner. 10Since indi-viduals are more impatient than the planner, this form of reward is delivered by tilting the consumption profile toward the present.Earlier consumption dates are used more intensively to provide incentives: rewards and punishments are front-loaded.
Economically, this mean reversion implies an interesting form of social mobility.Divide the population into two, those above and those below .Then mobility is ensured between these groups: descendants v* of individuals with current welfare above will eventually fall below it, v* and vice versa.This rise and fall of families illustrates one form of intergenerational mobility.
It is convenient to reexpress equation ( 19) as so that the stochastic process reverts toward one.Our next {1 Ϫ k (v )} t result derives upper and lower bounds for the evolution of this process.
Proposition 2. For , The bounds in (21) are instrumental in proving that a steady-state distribution exists, but they also illustrate a powerful force away from misery.Proposition 2 can be seen as providing a corridor around the expected value for the realization . This corridor becomes narrower as is decreased and shrinks to zero as .This implies that welfare must rise, 1 Ϫ k (v) r 0 for all realized shocks, if current welfare is low enough.Indeed, if utility is unbounded below, then next period's welfare remains w g (v, v) bounded even as .No matter how badly a parent is to be pun-v r Ϫϱ ished, the child is always somewhat spared.
When the solution for is interior, For example, in the logarithmic utility case, , implying that 1/U (c) p c the one-step-ahead forecast for consumption mean-reverts Moreover, if the amplitude of taste shocks is not too wide, we can guarantee that and .If , then the ergodic set for is , then the ergodic set for is bounded away from g 1 0 1 Ϫ k (v) zero and consumption is bounded away from zero.In this way, one can guarantee that inequality of welfare and consumption remains bounded.

VI. Existence of a Steady State
In this section we show that a steady-state invariant distribution exists.The proof relies on the mean reversion in equation ( 19) and the bounds in proposition 2.
Proposition 3. The Markov process implied by has an in- , where a solution b p b exists but does not admit a steady state.Later in this section we verify the second part of theorem 2 to confirm that a solution to the social planning problem can be guaranteed and a steady state exists.
Our Bellman equation also provides an efficient way of solving the planning problem.We illustrate this with two examples, one analytical and another numerical.
Example 1. Suppose that utility is CRRA with , so that j p 1/2 for and for .Atkeson-Lucas show the optimum involves consumption inequality growing b p b without bound and leading to immiseration.
The Bellman equation for is subject to (15) and ( 16).If we ignore the nonnegativity constraints on u and w, this is a linear-quadratic dynamic programming problem, so For taste shocks that are not too wide, we can guarantee strictly positive consumption and a bounded ergodic set for welfare.The nonnegativity constraints are then satisfied, so that this solves the problem that imposes them.
Example 2. To illustrate the numerical value of our recursive formulation, we now compute the solution for the logarithmic utility case with , , , , tions have a smooth bell curve shape.This must be due to the smooth, mean-reverting dynamics of the model, since it cannot be a direct consequence of our two-point distribution of taste shocks.Dispersion appears to increase for lower values of , supporting the natural conjecture b that as we approach , the Atkeson-Lucas case, the invariant dis-b r b tributions diverge.
We now briefly discuss issues of uniqueness and stability of the invariant distribution guaranteed by proposition 3.This question is of economic interest because it represents an even stronger notion of social mobility than that implied by the mean reversion condition (19) discussed in the previous section.Suppose that the economy finds itself at a steady state .Then convergence from any initial toward the w* v 0 distribution means that the distribution of welfare for distant de-w* scendants is independent of an individual's present condition.The past exerts some influence on the present, but its influence dies out over time.The inheritability of welfare is imperfect, and the advantages or disadvantages of distant ancestors are eventually wiped out.
Indeed, the solution may display this strong notion of social mobility.To see this, suppose that the ergodic set for is compact, which is {k (v )} t guaranteed if . Then if the policy function is monotone is unique and stable: starting from w* any initial distribution , the sequence of distributions , generated w {w} 0 t by , converges weakly to . 11The required monotonicity of the policy w g w* functions was satisfied by examples 1 and 2 and seems plausible more generally. 12Another approach suggests uniqueness and convergence without relying on monotonicity of the policy functions.Grunwald et  al. (2000) show that one-dimensional, irreducible Markov processes with the Feller property that are bounded below and are conditional linear autoregressive, as implied by ( 19), have a unique and stable invariant distribution.All their hypotheses have been verified here except for the technical condition of irreducibility. 13Although we do not pursue this further, our discussion illustrates how the forces for reversion in ( 19) might be exploited to establish uniqueness and convergence.
We have focused on steady states in which the distribution of welfare replicates itself over time.However, for the logarithmic utility case we can also characterize transitional dynamics.
Proposition 4. If utility is logarithmic, , then for any U(c) p log (c) initial distribution of entitlements w there exists an endowment level such that the solution to the social planning problem is gen-ê p e(w) erated by the policy functions starting from w.The function ˆê(w ) ! e(w ) One can apply this result to the case with no initial inequality, where dynasties are all started at solving .The cross-sectional v* k (v*) p 0 distribution of welfare and consumption fans out over time starting from this initial egalitarian position.The issues of convergence and uniqueness discussed above now acquire an additional economic interpretation.It implies that the transition is stable, with the cross-sectional distributions of welfare and consumption converging over time to the steady state.
12 Indeed, it can be shown that must be strictly increasing in v.However, although w g (v, v) we know of no counterexample, we have not found conditions that ensure the monotonicity of in v for .

journal of political economy
As mentioned in Section IV, for any utility function one can characterize the solution for any as the solution to a relaxed problem (w, e) with some sequence that is not necessarily exponential, that is, {Q } t imposing the general intertemporal constraint (10) instead of ( 11 ), the long-run dynamics are dominated by the policy functions u 1 ( g , from the problem with that we have characterized. We have shown the existence of a steady state generated by the w* policy function .We now provide sufficient conditions to ensure that w g is also a steady state for the social planning problem.This involves w* two things.We first establish that allocations generated by the policy functions are indeed incentive compatible by verifying the limiting condition (17) in theorem 2; this guarantees that, given and h, the w* allocation maximizes the Lagrangian ( 12).Second, we verify that average consumption is finite under , so that there exists some endow-w* ment e for which the resource constraints ( 4) and ( 10 (g , g ) v 0 any of the following cases: (a) utility is bounded above, (b) utility is bounded below, (c) utility is logarithmic, or (d) or .g ! 1 g 1 0 Next, we give sufficient conditions to guarantee that total consumption is finite at the steady state .If the ergodic set for welfare v is w* bounded away from the extremes, then consumption is bounded and total consumption is finite.Even when a bounded ergodic set for welfare v cannot be ensured, we can guarantee that total consumption is finite for a large class of utility functions.
Proposition 6.Total consumption is finite under the invariant distribution , It is worth remarking that the hypotheses of all these propositions are met for a wide range of primitives.In particular, they hold for any utility function as long as the amplitude of taste shocks is not too wide, so that we can ensure that or .They also hold for any arbitrary g 1 0 g ! 1 amplitude of taste shocks when utility is logarithmic or when utility is bounded above and is asymptotically convex.For example, At-1/U (c) keson and Lucas (1992) focused on the CRRA specification U(c) p with .All our results apply in this specification: for 1Ϫj c /(1 Ϫ j) j 1 0 with any shock distribution and for with shocks j [1, ϱ) j (0, 1) that are not too wide.
The steady-state distribution w and the implied value of total consumption will generally vary with h.Thus different values h translate into different required endowments e.For the CRRA case, we can say that steady-state consumption is a power function of h and thus has full range.In fact, in this case the entire solution for consumption is homogeneous of degree one in the value of the endowment e.This ensures a steady-state solution to the social planning problem for any endowment level.

VII. Pareto Problem
We now return to the Pareto problem ( 9) and its relation to the social planning problem.Recall that the former is exactly as the latter except that the promise-keeping constraints are inequalities instead of equalities.The next result establishes that these inequality constraints bind for steady-state distributions with strictly positive consumption.Thus w* the solutions to the Pareto and social planning problems coincide.The proof relies on the fact that a marginal increase in v contributes to the welfare criterion (9) (see also [7]) and that , k (v) Ϫ 1 k (v) ! 1 unless consumption is zero for some agents.Recall that was S(w; e) defined in (8).
Proposition 7. Let denote a steady state for the social planning w* problem.Suppose that consumption is strictly positive for all agents, so that for all v in the support of .Then solves Thus steady states for the social planning problem coincide with steady states for the Pareto problem as long as consumption is positive.That is, if the Pareto problem is started with the distribution , then it is replicated over time and for all w p w* w p w* t p 0, 1, t .By implication, the Pareto optimum is then time-consistent: the … journal of political economy initial solution at also solves the Pareto problem at any future t p 0 period t.In other words, this Pareto-efficient allocation is ex post Pareto efficient.Note that the condition that consumption be strictly positive is guaranteed if utility is unbounded below or if the amplitude of the taste shocks is not too wide so that (see proposition g 1 0 2).
In the Pareto problem ( 9), the welfare of future generations was aggregated using geometric Pareto weights.We now map these weights into their welfare implications and discuss a planning problem that is cast directly in terms of welfare, without Pareto weights. Starting

VIII. Conclusions
How should privately felt parental altruism affect the social contract?What are the long-run implications for inequality?To address these questions, we modeled the trade-off between equality of opportunity for newborns and incentives for altruistic parents.In our model, society should exploit altruism to motivate parents, linking the welfare of children to that of their parents.If future generations are included in the welfare function, this inheritability should be tempered and the existence of a steady state is ensured, where welfare and consumption are mean-reverting, long-run inequality is bounded, social mobility is possible, and misery is avoided by everyone.
The backbone of our model requires a trade-off between insurance and incentives.The source for this trade-off is inessential.In this paper, we adopted the Atkeson-Lucas taste-shock specification for purposes of comparison.In Farhi and Werning (2006), we study a dynamic Mirrleesian model-with productivity shocks instead of taste shocks-and find that a progressive estate tax implements efficient allocations by providing the necessary mean reversion across generations.

Proof of Theorem 1
The value function defined by ( 13) is weakly concave since the objective k(v) function is concave and the constraint set convex.Weak concavity implies continuity over the interior of its domain: . If utility is bounded below, then (v, v) continuity at is established as follows.Define the first-best value function v Then is continuous and with equality at .Since is weakly a contradiction since .Thus must be continuous at .This We first show that the constraint (11) with implies that utility and q p b continuation welfare are well defined.Toward a contradiction, suppose that is not defined, for some .This implies that s ≥ Ϫ1 T t lim b max ‫ޅ{‬ vu , 0} p ϱ.

s t t tp0 Trϱ
Since utility is concave, (i.e., ) for some Taking the limit yields .Since there are finitely many measure of such agents, this implies a contradiction of the intertemporal constraint (11) and thus of at least one resource constraint in (4).Thus, for both journal of political economy the relaxed and the original social planning problems, utility and continuation welfare are well defined, which is important for the recursive formulation.
We now prove two lemmas that imply the rest of the theorem.Consider the optimization problem on the right-hand side of the Bellman equation: subject to ( 15) and ( 16).Define and m subject to ( 15) and ( 16).The objective function in (A2) is nonpositive, which simplifies the arguments below.Lemma A1.The supremum in (A1), or equivalently (A2), is attained.
Proof.Suppose first that utility is unbounded below.We show that and then use this result to restrict, without loss, the optimization within a compact set, ensuring that a maximum is attained.To establish these limits, define the function . Since this corresponds to the same problem but without the incentive constraints, it follows that .Since This is a standard allocation problem, with solution for some , increasing in v and such that and (A3), using the fact that .
Then and satisfies constraints ( 15) and ( 16), . Then, since the objective is nonpositive, we can restrict the ˆb)) Ϫ m ϩ bk(v) maximization over so that .Since is concave with k(w(v)) the limits (A3), this defines a closed, bounded interval for for each w(v) v .Similarly, we can restrict the maximization over so that Proof.Suppose that for some v where the maximization is subject to (15) and ( 16).Then there exists such for all (u, w) that satisfy (15) and ( 16).But then by definition for all allocations that yield and are incentive compatible.Substituting, ũ w (v) we find that for all incentive-compatible allocations that deliver v, a contradiction with the definition of ; namely, there should be a plan with value arbitrarily close to k(v) .We conclude that k(v u,w subject to (15) and ( 16).

journal of political economy
By definition, for every v and , there exists a plan that is Let Consider the plan and for Since was arbitrary, it follows that subject to ( 15) and ( 16).QED bk(w(v))]

Proof of Theorem 2
We establish the following results from which the theorem follows: (a) An allocation is optimal for the component planning problem ( 13 for all reporting strategies j. Part a: Suppose that the allocation is generated by the policy functions {u } t starting from , is incentive compatible, and delivers welfare .After repeated v v 0 0 substitutions of the Bellman equation ( 14), we arrive at Conversely, suppose that an allocation is optimal given .Then by defi-{u } v t 0 nition it must be incentive compatible and deliver welfare .Define the con-v 0 tinuation welfare implicit in the allocation and suppose that either or for some . Since the original plan is incentive compatible, and satisfy (15) and ( 16).The Bellman equation then implies that The first inequality follows since does not maximize ( 14); the second in-u 0 equality follows the definition of .Thus the allocation cannot be k(w (v)) {u } 0 t optimal, a contradiction.A similar argument applies if the plan is not generated by the policy functions after some history and .We conclude that an t v t ≥ 1 optimal allocation must be generated from the policy functions.
Part b: First, suppose that an allocation generated by the policy functions {u , v } t t starting at satisfies .Then, after repeated sub- stitutions of (15), we obtain Taking the limit, we get so that the allocation delivers welfare .Next, we show that for any allocation generated by starting

Ϫ1 t t rϱ
Suppose that utility is unbounded above and .Then implies that .Since the value function is nonconstant, is concave, and reaches an interior maximum, we can bound the value function so that , with .Thus and then (A5) implies that , a contradiction since there are feasible k(v ) p Ϫϱ 0 plans that yield finite values.We conclude that .
Similarly, suppose that utility is unbounded below and that .Since , this implies that The two established inequalities imply that .
Part c: Suppose that for every reporting strategy Then after repeated substitutions of ( 16), As shown in the proof of theorem 2, and deliver welfare and , respectively.This implies that delivers welfare for some history .Consider iterating T times on the Bellman equations and averaging we obtain for large enough T where the strict inequality follows from the strict concavity of the cost function , the fact that we have the inequality (A7), and the weak concavity of the C(u) value function k.The last weak inequality follows from iterating on the Bellman equation for since the average plan ( , ) satisfies the constraints of the a a a v uv Bellman equation at every step.This proves that the value function is strictly k(v) concave.
Differentiability.-Since is concave, it is subdifferentiable: there is at least k(v) one subgradient at every v.We establish differentiability by proving that there is a unique subgradient by variational envelope arguments.
Suppose first that utility is unbounded below.Fix an interior value .In a v 0 neighborhood of define the test function Since is the value of a feasible allocation in the neighborhood of , it W(v) v 0 follows that , with equality at .Since exists, it follows, by application of the Benveniste-Scheinkman theorem (see Stokey and Lucas 1989,  theorem 4.10), that also exists and k Finally, since , this shows that .The limit is Otherwise it is inherited by the upper bound in- Next, suppose that utility is bounded below but unbounded above.Without loss of generality, we normalize so that .Then U(0) p 0 for all reporting strategies j so that, when we Benveniste-Scheinkman theorem that exists and equals .
The proof of is the same as in the case with utility unbounded lim k (v) p Ϫϱ vrv ¯

Proof of Proposition 3
Consider first the case with utility unbounded below.Since the derivative k (v) is continuous and strictly decreasing, we can define the transition function for all if utility is unbounded below.For any probability measure m, let x ! 1 be the probability measure defined by x (Ϫϱ, 1) T Q probability measures over (Ϫϱ, 1] to probability measures over (Ϫϱ, 1), and for all .T (d ) p T (d ) x (Ϫϱ, 1) We next show that the sequence is tight in that for any there {T (d )} Tightness implies that there exists a subsequence that converges T (d ) weakly, that is, in distribution, to some probability measure p.Since is Q(x, v) journal of political economy continuous in x, converges weakly to .But the linearity of T (T (d )) . In both cases, since , we must have that .Finally,   w* can be loosely interpreted as an impulse response function.If , shocks b p b have a permanent effect on inequality and consumption inherits a random-walk component.If , the impact of shocks decays over time b 1 b and consumption is mean-reverting.As long as , inequality vanishes in the long run in this deter-b 1 b ministic example.With ongoing taste shocks, inequality remains positive in the long run and the mean-reverting force ensures that inequality remains bounded.By contrast, when as in Atkeson and Lucas b p b

Fig. 1 .
Fig. 1.-Consumption paths for groups A and B. Solid lines represent the case with ; the dotted line at is the steady state.The horizontal dashed lines represent b 1 b c p e p 1 the Atkeson-Lucas case with .b p b *) p 0 strictly higher than misery .Reversion occurs toward this interior point. u the following conditions holds: utility is un-∫ k (v)dw*(v) p 0 bounded below, utility is bounded above, , or .g ! 1 g 1 0 When any of the conditions of proposition 3 are satisfied, theorem 2 leaves open only two possibilities.Either the social planning problem admits a steady-state invariant distribution or no solution exists.This contrasts with the Atkeson-Lucas case, with

Ϫ1U
(c) p log (c) b p 0.9 e p h p 0.6 v p 1.2 v p 0.75 p p h l , and several values of .Figure 2 displays steady-state distributions 0.5 b of welfare in consumption-equivalent units .The distribu-C(v(1 Ϫ b)) ) hold.It follows that the allocations generated by solve the social planning prob-u w (g , g ) lem, given e and .w* Proposition 5.The allocation generated by the policy functions starting at any solves the component planning problem in u w Ϫϱ lim h(v, b) p Ϫϱ vrϱ and , which, in turn, imply the limits ˆlim h(v, b) p Ϫϱ lim h(v, b) p Ϫϱ vrϪϱ vrϱ an invariant distribution under Q on (Ϫϱ, 1).Proof.The bounds (21) derived in proposition 2 imply that, for all, v V b w lim Q(x, v) p lim k (g (v, v)) p ! 1. xF1 vrϪϱ bWe first extend the continuous transition function Q(x, v) : (Ϫϱ, 1) # V r to a continuous transition function with (Ϫϱ, 1)Q(x, v) : (Ϫϱ, 1) # V r (Ϫϱ, 1) and , for all .It follows that mapsˆQ(1, v) p b/b Q(x, v) p Q(x, v) k (v )} ≤ ‫ޅ‬ [k (v(v ))] ≤ T (d )[(Ϫϱ, ϪA)](ϪA)

∫
is continuous with .The rest of theQ(x, v) ≤ k (v ) ! ϱ Largument is then a straightforward modification of the one spelled out for the case with utility unbounded below, except that plays the role of 1. Indeed, k (v ) L things are slightly simpler here since we do not need to construct an extension of Q.If , then the bound in (21) implies that , and thew ĝ 1 0 k (g (v, v)) ≤ 1 Ϫ (b/b) result follows immediately.Proof of Proposition 4Consider indexing the relaxed planning problem by e and setting forϪ1h p e the associated component planning problem, with associated value function .We first show that if an initial distribution w satisfies the condition k(v; e), then it is a solution to the relaxed and original social planning k (v; e)dw(v) p 0 ∫ problems.We then show that for any initial distribution there exists a value for e that satisfies .k (v; e)dw(v) p 0 Since utility is unbounded below, we have .Ap-v t k (v ; e) p ‫ޅ‬ [1 Ϫ hC (u (v ))] hC (u (v ))] p k (v; e).
u )]dw(v) p h p e for all t p 0, 1, … .͵Ϫ1 tThat the limiting condition (17) is satisfied is shown in proposition 5.It then follows that allocations generated this way solve the relaxed and original social planning problems with .Ϫ1 e p h Now consider any initial distribution w.We argue that we can find a value of 1 Ϫ b with equality whenever w, h, and e are such that at the allocation that attains the intertemporal resource constraint (11) holds, which is true at the k(v, h) constructed steady state.Integrating the right-hand side by parts gives e R S (w; e) Ϫ vdw(v) ≤ [1Ϫ w(v)][k (v; h) Ϫ 1]dv ϩ h .e) Ϫ vdw(v) Ϫ S(w*; e) Ϫ vdw*(v) w; e) Ϫ vdw(v) Ϫ S (w*; e) Ϫ vdw*(v) (v) Ϫ w(v)][k (v; h) Ϫ 1]dv.
For any given initial distribution to (2) and (3).We call this subproblem, for a given v and h, the component planning problem.Its connection with the relaxed problem is that for any e there exists a positive multiplier h such that an allocation solves the relaxed planning problem with endowment e if and only v {u } t given v and h(Luenberger 1969, chap.8).
from the steady state , the optimum for the Pareto problem e)} that p is an invariant measure under Q on (Ϫϱ, 1).QEDThe argument for the case with utility bounded below is very similar.Define

͵
Ifon the support of , then the last term is strictly negative for allk (v; h) ! 1 Ϫ ∫ vdw(v) for all v. Proposition 1 implies that if utility is unbounded w(v) ≤ w*(v) k (v; h) ! 1 below,in which case consumption is strictly positive.By proposition 2, when utility is bounded, for a positive measure of agents under if and k (v; h) ≥ 1 w* only if consumption is zero for a positive measure of agents under .