Dynamic Savings Choices with Disagreements

Transcription

1 Dynamic Savings Choices with Disagreements Dan Cao Georgetown University Iván Werning MIT November 2017 We study a fexibe dynamic savings game in continuous time, where decision makers rotate in and out of power. These agents vaue spending more highy whie in power creating a time-inconsistency probem. We provide a sharp characterization of Markov equiibria. Our anaysis proceeds by construction and isoates the importance of a oca disagreement index, b(c), defined as the ratio of margina utiity for those in and out of power. If disagreement is constant the mode speciaizes to hyperboic discounting. We aso provide nove resuts for this case, offering a compete and simpe characterization of equiibria. For the genera mode we shoe that dissaving occurs when disagreements are sufficienty high, whie saving occurs when disagreements are sufficienty ow. When disagreements vary sufficienty with spending, richer dynamics are possibe. We provide conditions for continuous equiibria and aso show that the mode can be inverted for primitives that support any smooth consumption function. Our framework appies to individuas under a behaviora interpretation or to governments under a poitica-economy interpretation. 1 Introduction Time-inconsistency probems that bias behavior towards the present may hep expain a number of phenomena and have received ampe attention from economists. However, the extent of these probems ikey varies significanty according to the situation. In particuar, there is no reason to expect the aure of the present over the future fet by the poor to be comparabe to that experienced by the rich. Simiary, it has been noted that governments may suffer from a simiar present bias for poitica economy reasons, yet the degree of this bias may be quite different for advanced countries than for deveoping countries. The genera point is that the strength of time-inconsistency probems may depend on the First version: Apri For usefu comments and discussions we thank Fernando Avarez, Manue Amador, Jinhui Bai, Abhijit Banerjee, Marco Battagini, Satyajit Chatterjee, Hugo Hopenhayn, Roger Lagunoff, Benjamin Mo, Patrick Rabier, Debraj Ray, Eric Young as we as seminar and conference participants. This project was inspired by conversations with Abhijit Banerjee on sef-contro probems with many goods. Finay, we thank Nathan Zorzi for vauabe research assistance. 1

2 eve of weath or spending. This possibiity has received reativey itte attention from the iterature. This paper introduces and studies an infinite-horizon continuous-time savings game that accommodates fexibe forms of time inconsistency. Decision makers rotate in and out of power. An agent currenty in power contros consumption and savings, choosing how much to spend subject to a borrowing constraint and a constant fow of income. Agents currenty in power retain power for a stochastic interva of time and ose it at a Poisson rate to a successor. Once removed from power, an agent continues to care about the future spending path chosen by other agents. However, spending is enjoyed more whie in power. This disagreement, represented by differences in the utiity functions for those in and out of power, captures a form of present bias and eads to a time-inconsistency probem in savings choices. As a resut, we approach the probem as a dynamic game and study Markov equiibria, a widey used refinement in this iterature. Our mode admits both a behaviora and poitica-economy interpretation. For the behaviora one, foowing Strotz (1956), Laibson (1997) and many others, the mode may describe the probem of a singe consumer paying an intertempora game against future seves (a cosey reated iterature, initiated by Pheps and Poak, 1968, studies paternaistic intergenerationa growth modes). The disagreement on the utiity function that we aow generates a time inconsistency probem that is simiar, but stricty generaizes, hyperboic discounting. For the poitica economy interpretation, the mode describes a situation where the ruing party contros the budget and obtains private benefits from spending whie in power, due to pork spending or outright transfers to ruing party members. This reates our work to poitica economy modes of government debt, such as Aesina and Tabeini (1990), Amador (2002), Battagini and Coate (2008), Azzimonti (2011) and others. 1 With few exceptions, the existing time-inconsistency iterature has focused on saving games that are effectivey variants of the hyperboic discounting setup. In our mode this amounts to the assumption of a uniform disagreement, with utiity out of power proportiona to utiity in power. Our anaysis aso appies to this specia case and actuay deivers new and sharp resuts. Our first contribution, however, is to provide a framework to expore disagreements that vary with spending. To this end, we consider genera differences in the utiity functions for those in and out of power. This may give rise to a non-uniform time-inconsistency 1 A different strand of the iterature modes the endogenous transition of power by examining Markov equiibria in dynamic poitica economy games (e.g. Besey and Coate, 1998; Acemogu and Robinson, 2001; Bai and Lagunoff, 2011), but abstracts from government debt or savings. 2

3 probem, where the incentives to save vary with weath. We are especiay interested in the ong-run dynamics of weath and how this depends on the form disagreements take. Why woud disagreements vary with spending? One straightforward answer is that there is no rea reason to expect them to be constant and so the possibiity that they are not must be contempated. For exampe, in the behaviora context, it is pausibe that presentbiased impuses and behaviors decrease with spending. A deeper answer is offered by Banerjee and Muainathan (2010), who provide a foundation for disagreements based on the notion that spending takes pace over many goods, with disagreements on how to spend across these goods. 2 The overa disagreement on tota spending then varies with the eve of spending, except in specia cases. This perspective expains a bias towards the present, but shifts the focus from intertempora discounting to disagreements across different goods. For exampe, in a behaviora context, agents may fee drawn to consume certain tempting goods today extreme exampes may incude unheathy foods, acoho or drugs but do not vaue the consumption of these goods by future seves. If the margina propensity to spend on such goods fas with greater spending this impies decreasing disagreements. A simiar argument appies in poitica economy contexts. Indeed, the voting mode in Battagini and Coate (2008) impies increasing disagreements because the margina propensity to spend on pork transfers is increasing in spending. One of the goas of this paper is to provide a framework that can encompass a wide cass of assumptions on the form of disagreements, incuding increasing and decreasing disagreements. Our second contribution is both technica and substantive, providing a sharp characterization of a Markov equiibria. As is we known, dynamic saving games in discrete time may be quite i behaved. 3 For exampe, Harris and Laibson (2001, 2002) point out that discontinuous equiibria are reativey pervasive in these standard settings. Kruse and Smith (2003) proved that the hyperboic mode has a continuum of oca soutions with discontinuous poicy functions. Recenty Chatterjee and Eyigungor (2016) show that in discrete time a Markov equiibrium must be discontinuous (see aso Morris and Postewaite, 1997 and Morris, 2002 for the anaogs in finite horizon settings). Properties 2 In a behaviora context, Banerjee and Muainathan (2010) focus on a finite-horizon mode with many goods and additivey separabe utiities, with disagreements over which goods shoud be vaued. In a poitica economy context, Aesina and Tabeini (1990) consider an infinite-horizon mode with a reativey genera form of disagreement in the composition of spending across different goods (see their equations 1). However, for their anaysis they speciaize to corner cases and a more extreme and uniform disagreement (see their equations 4 and 5). 3 Linear equiibria exist in the absence of a borrowing constraint. Such equiibria were first studied by Pheps and Poak (1968) and ater put to good use by many others. However, inear equiibria do not exist in the presence of a binding borrowing constraint, a case that has been the primary focus of the iterature. 3

4 such as these render these modes reativey intractabe and make it difficut to characterize equiibria. 4 The iterature has responded to these chaenges in a number of ways. Harris and Laibson (2001) introduce income uncertainty to derive a Generaized Euer equation. Harris and Laibson (2013) propose a continuous-time mode, focusing on a imit with instant gratification and sma noise in asset returns to appy the theory of viscosity soutions. Chatterjee and Eyigungor (2016) work in a discrete-time setting but introduce otteries to smooth out the soution. Despite these efforts from the hyperboic iterature, many fundamenta questions remain open or have ony received partia answers. For exampe, are there conditions that ensure that the equiibrium invoves saving or dissaving? Can saving and dissaving equiibria coexist for some parameters? For a given equiibrium, can saving and dissaving coexist at different weath eves? Do continuous equiibria exist, and, if so, under what conditions? Do these modes dispay mutipe equiibria? Our mode is cast in continuous time and this turns out to be crucia to our approach and resuts. We show that our continuous-time framework is reativey we behaved, without introducing uncertainty or otteries. Our formuation buids on Harris and Laibson (2013), but extends it to aow for more genera disagreements. In addition, we do not focus on the instant-gratification imit and our soution strategy is different. 5 Our approach is to attack the differentia equations for a Markov equiibrium head on. Due to the presence of singuarities, no off-the-shef resuts exist for such equations, but we provide a simpe method to construct and characterize equiibria. Since weath evoves continuousy over time, we buid up ocay towards a goba equiibrium. Since our proofs proceed by construction, we characterize equiibria sharpy, deivering answers to the questions isted above, as we as providing a straightforward procedure for computation. In fact, our characterization is exhaustive, providing a compete description of the cass of equiibria that are possibe. Finay, we provide sufficient conditions for the existence and uniqueness of a continuous equiibrium. 4 Despite these difficuties the iterature has obtained various resuts on the existence of Markov equiibria, aowing for potentiay discontinuous poicy and vaue functions. Bernheim et a. (2015) shows the existence of Markov equiibrium in the standard mode without uncertainty, assuming an interest rate that is stricty greater than the discount factor. Harris and Laibson (2001) provide existence resuts by adding i.i.d. uncertainty in income in the standard quasi-hyperboic mode (Bernheim and Ray (1989) provide a reated resut in an atruistic growth mode with bounded transfers), whie Chatterjee and Eyigungor (2016) does the same for a mode with otteries. 5 The instantaneous gratification imit is tractabe and provides a good approximation in some appications. Nevertheess, it is of obvious theoretica interest to obtain a more genera characterization, away from this approximation. Moreover, in some appications, such as poitica economy settings, the approximation may be inadequate, since it woud represent a situation of extremey high poitica turnover. 4

5 Athough the entire anaysis and resuts appy to our genera framework, they are of interest even within the specia hyperboic case. For this case, we provide a nove graphica anaysis that deivers a compete characterization for the entire set of equiibria in a particuary simpe and visua manner. For exampe, we show that depending on parameters there can be either saving or dissaving, but never both; a continuous equiibrium may exist and there is at most one such equiibria; whenever the equiibrium invoves savings then the equiibrium is unique. To the best of our knowedge, such resuts have no counterparts in the existing iterature and sette various open questions mentioned above. Our third contribution is to isoate the conditions for saving or dissaving and to characterize the resuting dynamics for weath. A crucia innovation of our anaysis here is our introduction a oca disagreement index b(c), defined as the ratio of margina utiities from spending for agents in and out of power. The shape of this function summarizes how disagreements depend on spending (denoted by c). The hyperboic case amounts to the specia case where b(c) is constant. Our first set of resuts in this regard invove cases where the disagreement index does not vary too much and is either sufficienty high or ow. Under these conditions, we show that an equiibrium exists that features either saving or dissaving at a weath eves. Specificay, there exists a threshod ˆb which depends on the interest rate and other parameters and show that when the disagreement index b(c) ies above ˆb then there is a unique equiibrium with positive savings and this equiibrium is continuous; when b(c) ies beow ˆb a equiibria must feature dissaving and there is at most one continuous equiibrium. Athough these resuts appy more generay, a specia case of interest is the hyperboic case with constant b(c). As a byproduct of our anaysis of the hyperboic case we touch on the issue of oca indeterminacy. For the discrete-time quasi-hyperboic mode, Kruse and Smith (2003) constructed a continuum of oca soutions to the equiibrium conditions. We provide an anaogous resut for our continuous-time setup: for any proposed steady state weath a oca soution exists, with weath converging to this eve. However, we show that these oca constructs are not part of an equiibrium in our mode. 6 This concusion actuay foows as a coroary of our resut showing that there exists a unique Markov equiibrium 6 Indeed, the oca constructs can ony be interpreted as Markov equiibria for a modified saving game that adds ad hoc constraints, forcing the agent to choose weath beow a certain bound; this bound must be cose enough to the proposed steady state. Such an ad hoc constraint is unnatura, however, and not standard in the iterature. The oca constructs do not characterize a part of any equiibrium of the origina saving game, which imposes no upper bounds on weath. 5

6 with strict savings. 7 Our second set of resuts invove cases where the disagreement index b(c) does not ie on one side of ˆb but instead varies enough that it crosses this threshod. We find that rich dynamics then emerge, with saving and dissaving coexisting at different weath eves. We focus on two poar opposite cases, when b(c) is decreasing and when b(c) is increasing. In particuar, if disagreements fa with spending then we show that a poverty trap may emerge, with dissavings beow a threshod and savings above this same threshod. Intuitivey, at ow weath eves the time-inconsistency probem is reativey severe because spending takes pace in the range where disagreements are high. The incentive to consume is high because agents in power do not want to eave resources that may be spent when they are out of power. There is a feedback oop: the incentive to dissave is reinforced by the anticipation that successors overspend from the point of view of those in power. At high weath eves the time-inconsistency probem is mitigated by the fact that spending takes pace in regions with ower disagreement, so saving emerges. Again, a feedback oop reinforces these incentives: the incentive to save is enhanced if successors or future seves do not overspend too much. Poverty traps cannot arise without these feedback oops. Banerjee and Muainathan (2010) derive a reated resut. As mentioned earier, their paper focuses on various impications of intraperiod disagreements on how to spend across goods. In the context of a two-period mode with a singe savings choice in the first period, they show there may be a downward jump in consumption as a function of weath. This discontinuity may be interpreted as a poverty trap of sorts in their twoperiod setup. There are important differences, however. Obviousy, a two-period framework does not permit the study of ong-run weath. In contrast, in our framework a poverty trap invoves weath getting trapped beow a threshod forever. Moreover, a discontinuity ike theirs cannot arise in a two-period adaptation of our mode. Instead, in our setting poverty traps emerge from the strategic interactions across savings decisions over onger horizons. In this sense, the feedback oop described in the previous paragraph is crucia to our resut, but is absent in their setup. Our poverty trap resut reies on non-uniform disagreements. Indeed, our resuts show that this is not ony sufficient but aso necessary: no Markov equiibrium exists featuring a poverty trap in the hyperboic case. Poverty traps may emerge, however, in the hyperboic case if one drops the Markov equiibrium requirement. Working in a 7 The resuts presented here formay appy ony to our continuous-time mode and shoud be taken ony as suggestive for the discrete-time mode studied by Kruse and Smith (2003). 6

7 discrete time setting, Bernheim et a. (2015) show that subgame perfect equiibria may feature poverty traps. 8 As they argue, trigger strategies may be interpreted as sef punishments that provide an interna and endogenous form of sef contro. Poverty traps may emerge in this context because this kind of sef contro is more effective for the rich, who are far from the asset imit. Our focus on Markov equiibria abstracts from these forms of sef contro and reies instead on the rich suffering ess disagreements with future seves. Thus, Bernheim et a. (2015) focus on non-uniform sef contro with uniform disagreements; whie we focus on non-uniform disagreements, without sef contro. Both are obviousy compatibe with one another and coud act as compements. Interestingy, these two mechanisms have a few different impications: ower abor income or greater access to credit make poverty traps ess ikey in Bernheim et a. (2015), but more ikey in our mode. Such contrasting comparative statics serve to highight the underying differences in the mechanisms at work. Turning to the opposite case, if disagreements rise with spending then the probem of time inconsistency becomes heightened at higher weath eves. We show that an equiibrium exists where the weathy dissave, whie the poor save. Starting from any initia vaue, weath converges to an interior steady state. Stabiity forces of this kind ie at the heart of the mean-reverting resut in the poitica economy mode of Battagini and Coate (2008). Indeed, as we discuss in more detai in Section 5.2, their framework provides a foundation for disagreements that increase with spending. They set up and study a mode of egisative voting, with this body determining the composition of spending. As we discuss, they prove a representation that fits our framework. In addition to soving for an equiibrium, given primitives, using our differentia approach, we show how our mode can be fruitfuy inverted to sove for primitives that support any given postuated equiibrium. We know of no parae of this idea in the existing iterature. In particuar, we back out a disagreement function b(c) for any given smooth consumption function. One advantage of this inverse perspective is that it is extremey tractabe and insightfu. Another is that it may aso be more appropriate, at east conceptuay, if the outside economist or econometrician observes behavior (the consumption function) but has no direct evidence on disagreements and how they vary. Finay, we show that in our continuous-time framework there is at most one continuous equiibrium and provide sufficient conditions for its existence. The potentia for a continuous equiibrium underscores, once again, that equiibria in our continuoustime framework can be extremey we-behaved compared to discrete-time counterparts, where no continuous equiibrium exists. We aso provide conditions for the existence 8 Their definition of a poverty trap is sighty different from ours, as we discuss in Section

8 of discontinuous equiibria in our mode. As a coroary, since both conditions turn out to be compatibe in some cases, these resuts prove the possibiity of mutipe Markov equiibria. 2 A Dynamic Savings Game This section introduces a continuous-time savings framework with a singe consumption good that can accommodate reativey genera forms of disagreement. We then offer one interpretation or motivation for our primitives, in a setting with many goods. 2.1 Preferences Time is continuous with an infinite horizon, denoted by t 2 [0, ). The fow utiity obtained from consumption by an agent in power is whie utiity for an agent out of power is U 1 (c t ), U 0 (c t ). The utiity functions U 1 : R +! R and U 0 : R +! R are concave, increasing, continuous and differentiabe. We aso assume U1 0 (0) = and im c! U1 0 (c) =0.9 Agents in power are removed at a constant Poisson arriva rate 0. To simpify the exposition we assume power cannot be regained, but show ater that this is without oss of generaity (see Section 3.3). The continuation ifetime utiity at time t for an agent in power is then appe Z t V t E t e rs U 1 (c t+s )ds + e rt W t+t = 0 Z 0 e (r+)s (U 1 (c t+s )+W t+s )ds (1) where r > 0 is the discount rate and t the random time at which the agent currenty in power is removed. 10 Here W t is the continuation ifetime utiity for an agent out of power W t Z 0 e rs U 0 (c t+s )ds. (2) 9 Concavity and differentiabiity of U 0 are not crucia for the anaysis but simpify the exposition for most of the resuts. An earier version of the paper focused on a case that had a convex kink in U 0 (c). Theorem 8 beow aso reaxes the concavity assumption and assumes a concave kink in U The ony uncertainty present in the mode is the timing for the ateration of power, t. However, consumption is deterministic and does not depend on the reaization of this uncertainty because of the symmetry of preferences (i.e. agents stepping up to power have identica preferences to those stepping down) and our focus on Markov strategies. 8

9 The difference between U 1 and U 0 is a form of disagreement that creates a timeinconsistency probem. The framework can be interpreted iteray in a poitica economy setting as describing the ateration of power of different ruers or egisative majorities. Aternativey, a behaviora interpretation is that the different agents represent different seves or states of mind within an individua. Crucia to our anaysis is the introduction of a oca disagreement index, which summarizes these differences, defined as the ratio of margina utiities b(c) U0 0 (c) U 0 1 (c). When b(c) =1 for a c there is no disagreement and the utiity functions coincide, up to a constant. As we sha show, the function b(c) summarizes the reevant difference between the utiity functions U 1 and U 0. Throughout the paper we assume that the margina utiity from consumption is higher whie in power. Assumption 1 (Present Bias). The utiity functions U 1 and U 0 are such that, for some b > 0 b(c) 2 [b,1] for a c > 0. When b(c) < 1 agents prefer to consume reativey more whie in power, eading to a present-bias time-inconsistency probem. Those out of power want those in power to exercise restraint, to consume ess and save more. Likewise, those currenty in power woud ike to commit their successors somehow, but have no means to do so. This simpe and fexibe framework aows us to capture different patterns of disagreement. In particuar, for some appications it is natura to assume that disagreements are stronger at ower consumption eves, so that b(c) is increasing. Yet in other cases it may be reasonabe to suppose that disagreements grow with spending, so that b(c) is decreasing. Hyperboic Discounting. An important specia case occurs when disagreements are constant: b(c) = b < 1 so that U 0 (c) = bu 1 (c). The mode is then equivaent to the continuous-time hyperboic discounting mode introduced by Harris and Laibson (2013), which in turn buids on discrete-time quasi-hyperboic counterparts in Harris and Laibson (2001), Laibson (1997) and Pheps and Poak (1968). It is aso common to adopt power utiity functions: U 1 (c) =c 1 s /(1 s) for s > 0. Harris and Laibson (2013) focus on the imit as!, the so-caed Instantaneous Gratification imit. They show that tractabiity is gained from the fact that then V t = W t in the imit, so that a singe continuation vaue function suffices. As part of our anaysis, we revisit this specia hyperboic case and provide some nove and sharp resuts. Indeed, we show that the equiibrium is unique in some cases, or 9

10 beongs to a simpe cass in others, and offer a tight characterization (see Section 4.3). As we expain beow, we aso everage these resuts for more genera cases where b(c) is not constant. Throughout, we do not focus on the instant gratification imit, but instead aow for any finite. Tai Assumptions. One of the goas of our framework is to aow for reativey genera differences in utiities, extending disagreements past the hyperboic discounting assumption. It is convenient to restrict these differences to an arbitrary bounded interva and assume that disagreements are constant outside this interva. This amounts to assuming hyperboic discounting in the tai. 11 Assumption 2. At high consumption eves disagreements are constant and utiities are power functions: b(c) = b appe 1 and U 1 (c) = 1 1 s c1 s with s > 0 for a c c for some c > 0. Furthermore, (1 s)r < r. For a few resuts, it is convenient to adopt the same hyperboic assumption at the ower tai. Assumption 3. At ow consumption eves disagreements are constant and utiities are power functions: b(c) = b appe 1 and U 1 (c) = 1 1 s c1 s with s > 0 for a c appe c for some c > ra. Assumption 2 constrains preferences ony above some arbitrariy high eve of consumption and paces no restrictions on the disagreement index or utiity functions beow this threshod. Assumption 3 constrains preferences ony over an arbitrariy sma interva. In this sense, both can be viewed as reativey weak constraints. These mid assumptions are not required for a our resuts, but greaty simpify the anaysis for others. We empoy them in two reated ways. First, when the equiibrium features positive savings, our constructive approach is aided by pinning things down above some asset eve, and focusing on the remaining probem over a bounded interva of assets (when the equiibrium features dissaving, the boundary is automaticay provided at the asset imit). As we sha see, Assumption 2 effectivey provides us with a boundary condition at some high enough eve of weath ā. Secondy, for the hyperboic mode we prove uniqueness or show that the equiibrium beongs to a restricted cass, offering a tight characterization. Assumptions 2 and 3 aow us to everage these resuts outside the hyperboic case. 11 The condition that (1 s)r < r is a standard growth condition to ensure finite ifetime discounted utiity. 10

11 2.2 Budget Constraints and Borrowing Limits The agent in power chooses consumption and assets a t subject to the budget constraint The interest rate r > 0 and income y. a t = ra t + y c t. (3) 0 are constant and given. Weath is subject to an asset or borrowing imit given by The so-caed natura borrowing constraint sets a = a appe a t. (4) y r appe 0 and aows borrowing against the fu present vaue of income. We mainy focus on tighter constraints, with a > y r. Whenever a t = a we require c t appe y + ra to ensure that ȧ 0. Asset imits can aso be interpreted as commitment devices. In the individua agent context, this may capture forced savings such as socia security or iiquid assets. In the poitica-economy context, it may capture weath funds with imits on discretionary spending from natura resources. Without oss of generaity we set income to zero, y = 0, and consider a positive asset imit, a > 0. This is a normaization since, by a change of variabes, one can transform a probem with positive income, y > 0, to one with zero income as foows. Defining ã t = a t + y r, then ã. t = rã t c t and ã ã a + y r. As a resut of this transformation, the asset imit becomes a positive ower bound on assets, ã > 0, except in the natura borrowing imit case where a = Many Goods, Enge Curves and Disagreements One interesting motivation for time-inconsistency probems is to consider spending across various goods, with disagreement on how to spend across these goods. This notion is popuar in poitica economy modes on government spending and debt, appearing in Persson and Svensson (1989), Aesina and Tabeini (1990), Amador (2002) and Azzimonti (2011) among others. However, this iterature typicay adopts simpe specifications that impy uniform disagreements. An important exception is Battagini and Coate (2008) which instead derives specific non-uniform disagreements. Banerjee and Muainathan (2010) work in a behaviora context and consider reativey rich and fexibe disagreements across goods, providing a reationship between the shape of Enge curves and disagreements. We appy these ideas to our formuation. Suppose consumption takes pace over two goods, c A and c B, and normaize prices to unity. Utiities are additivey separabe, with h(c A ) perceived by both those in and out of power, whie g(c B ) is perceived ony by agents in power (ess extreme assumptions work simiary). The static subprobem of spending 11

12 across c A and c B given tota spending defines indirect utiity functions U 1 (c) = max {h(c A)+g(c B )} and U 0 (c) =h(ĉ A (c)), (5) c A +c B =c where (ĉ A (c), ĉ B (c)) denotes the soution to the maximization. The next resut shows that we can generate any desired U 1 and U 0 in this way, by appropriate choices of h and g. Proposition 1. Given U 1 and U 0 satisfying Assumption 1, there exists stricty concave functions h and g so that (5) hods. Proof. Appendix A.1. Note that U1 0 (c) =h0 (ĉ A (c)) and U0 0 (c) =h0 (ĉ A (c))ĉ 0 A (c), impying b(c) =ĉ 0 A (c) =1 ĉ0 B (c), so that a high margina propensity to spend on c B increases disagreement, since those out of power do not vaue this good. When ĉ A (c) is concave, so that the margina propensity to spend on c A is decreasing, b(c) is decreasing. In this way, the shape of the Enge curve dictates the shape of our disagreement index b(c) Markov Equiibria We focus on Markov equiibria with weath as the state variabe. The poicy function for consumption maximizes utiity for the agent in power (1), taking as given the vaue function for those out of power, W(a), satisfying (2). We provide a more technica and detaied recursive definition of our equiibrium concept beow. We aso incorporate a standard refinement from the iterature. 3.1 Reguar Equiibria Our Markov equiibrium concept is defined in terms of the functions (ĉ(a), V(a), W(a)) and imposes standard conditions. The foowing differentia equations pay a crucia roe: rv (a) = max{u 1 (c) + V 0 (a)(ra c) + (W (a) V (a))}, (6a) c rw (a) = U 0 (ĉ (a)) + W 0 (a)(ra ĉ (a)), (6b) 12 For exampe, under hyperboic discounting, U 0 (c) = bu 1 (c), the functions h, g in Proposition 1 become cā cb h(c A )= bu 1 and g(c B )=(1 b)u 1, b 1 b which impies that ĉ A (c) = bc and ĉ B (c) =(1 goods. b)c, i.e. constant margina propensities to consume on both 12

13 where ĉ (a) denotes the soution to the maximization in (6a), which is equivaent to the first-order condition U 0 1 (ĉ (a)) = V 0 (a), for a > a. Equation (6a) is the Hamiton-Jacobi-Beman equation providing a recursive representation of the probem of maximizing (1) taking the vaue function W(a) as given. The ast term takes into account the probabiity of transitioning out of power with probabiity, at which point the continuation vaue jumps from V(a) to W(a). Equation (6b) is a recursive representation of condition (2), which defines W(a) given the poicy function ĉ(a). Finay, the impied weath must satisfy (6c). a t = ra t ĉ(a t ). (7) For any initia conditions, we require a soution path to this differentia equation to exist and impose appropriate transversaity conditions aong this path (see beow). The eements of a Markov equiibrium speed out above are famiiar enough and a that is required when deaing with continuousy differentiabe V and W and continuous ĉ. However, we aow jumps in ĉ and W, as we as points of non-differentiabiity in V or W, since there is a priori reason to excude such non smooth behavior. Indeed, in some cases jumps naturay arise or are even essentia, as in our poverty trap equiibria. Accordingy, we adapt our definition of a Markov equiibrium as a tripet of functions (ĉ(a), V(a), W(a)) with the foowing properties: (a) V is piecewise continuousy differentiabe; (b) at a points of differentiabiity of V, equations (6a) and (6c) hod and equation (6b) hods if in addition ĉ(a) 6= ra; and (c) for any initia vaue a 0 a the differentia equation ȧ t = ra t ĉ(a t ) admits a path {a t } t2[0, ) that satisfies the asset imit a t a and impies that: W(a t ) is continuous for a t 0, the transversaity conditions im t! e (r+)t V(a t )=0 and im t! e rt W(a t )=0 hod, and whenever ĉ(a 0 )=ra 0 we aso require that W(a 0 )= R 0 e rt U 0 (ĉ(a t ))ds. These conditions are reativey straightforward. The ony subte issue worth highighting is the smoothness requirements for V and W. The function V must be everywhere continuous because it represents the vaue from a continuous-time optima contro probem with a controabe state (assets a) and payoffs that are continuous in the contro (consumption c). Discontinuities in W do not induce discontinuities in V because c (hence ȧ) is unrestricted for a a > a. Athough V is continuous, by (6a) it inherits kinks at points where W is discontinuous; at these same points ĉ must be discontinuous. In contrast, the function W may be discontinuous, because it is not the vaue from a smooth optimization. Jumps are severey imited, however. Condition (c) impies that W(a t )= R t e r(s t) U 0 (ĉ(a s ))ds so that the vaue function W must be continuous and 13

14 differentiabe aong the equiibrium outcome path {a t } t2[0, ). Condition (b) then impies that (c, W) can ony jump together to induce ȧ = ra may jump from dissaving ȧ < 0 to saving ȧ ĉ(a) to cross zero. For exampe, we 0 (a jump that is important for our poverty trap equiibrium); or we may jump from dissaving ȧ < 0 to a steady state ȧ = 0 (this kind of jump occurs in discontinuous dissaving equiibria). 13 In this atter type of jump we must have V(a) = V(a), so jump cannot occur at any point. Finay, we impose a standard refinement form the iterature. When utiity is unbounded beow it is difficut to rue out the possibiity that future seves consume an infinitesima amount which eads to ever ower continuation vaues. This in turn may induce the current sef to save more and consume ess. This feedback effect coud potentiay ead to an equiibrium with vanishing consumption and utiity that is unbounded beow. We have not been abe to construct such an equiibrium, but it is usefu to focus away from this possibiity using the foowing refinement. Definition 1 (Reguar equiibria). A Markov equiibrium is reguar if there exists n > 0 such that ĉ(a) na for a a a. Harris and Laibson (2013) impose precisey this refinement and ca equiibria not satisfying it pathoogica. In the same spirit, Bernheim et a. (2015) add the constraint c t na t to the decision maker probem. 14 We are abe to show that a Markov equiibria are reguar for an important case of interest. Lemma 1. If U 1 is bounded beow and satisfies Assumption 2 with s < 1 then a Markov equiibria are reguar. From now on we imit ourseves to reguar Markov equiibria and refer to such an equiibrium simpy as equiibrium, for short. 3.2 Soution Approach The system (6) can be thought of as a differentia system in (V, W). Equation (6a) can be soved for V 0 (a) at any asset a given any pair of vaues (V(a), W(a)) satisfying (r + 13 A symmetric case occurs with savings, jumping from a steady state ȧ = 0 to saving ȧ > 0. We can rue out, however, a jump from saving ȧ > 0 to dissaving ȧ < 0, since such a stabe steady state requires continuity of W and hence ĉ. 14 Adding a constraint in the decision maker probem is not exacty the same as focusing on equiibria satisfying these constraints. However, both approaches prevent situations where utiity goes to. 14

15 )V(a) W(a) U 1 (ra) 0. Indeed, there are two soutions or roots, one root associated with saving ĉ(a) appe ra and one root associated with dissaving ĉ(a) coincide if and ony if (r + )V(a) W(a) U 1 (ra) = 0 impying ĉ(a) =ra. Define the vaues of constant weath by V(a) = r U 1 (ra) + U 0 (ra) r + r r + r and ra; the roots W(a) = U 0 (ra). (8) r The vaue for those not in power, W, is the present vaue of utiity from consuming ra forever using U 0. For those in power the vaue V is a weighted average using U 1 and U 0. These functions pay an important roe in our anaysis, since the equiibrium vaue functions (V, W) must coincide with ( V, W) at steady states. Indeed, note that (r + ) V(a) W(a) U 1 (ra) =0 whenever (V, W) =( V, W), so that ĉ(a) =ra is the unique root. Our method for characterizing equiibria is to construct soutions to the differentia system (6) by attacking these equations directy. Thus, we do not appea to genera existence or uniqueness resuts for the system (6). Indeed, we are unaware of any off-the-shef resuts of this form for such equations for finite. 15,16 One technica chaenge is that the differentia system (6) features a singuarity at steady states. As a resut, we cannot appy standard existence theorems for reguar ODEs. 17 Thus, we must provide our own existence resut. The foowing emma proves the existence around singuar points when b < ˆb, which wi turn out to be the reevant case. Lemma 2. Suppose b(ra 0 ) < ˆb. Then the differentia system (6) with initia condition (V(a 0 ), W(a 0 )) = ( V(a 0 ), W(a 0 )) admits a soution over the interva [a 0, a 0 + w] for some w > 0, with (i) V (a) > V (a) for a > a 0 ; 15 This may seem surprising at first. After a, (6a) is a Hamiton-Jacobi-Beman for V given W, and for which various existence resuts may appy (at east for a reguar enough cass of W functions). However, the main difficuty is not with soving (6a) for V given W. The probem ies in soving the system (6) jointy for both V and W (a fixed point). In particuar, (6b) is reminiscent of a Hamiton-Jacobi-Beman equation, but it is not since ĉ(a) does not maximize the right hand side (6b), it instead maximizes (6a). 16 Harris and Laibson (2013) appy genera existence resuts, using viscosity theory, in the hyperboic instantaneous-gratification imit as!. They show that, under some conditions, (6) is then equivaent to the condition for the vaue function of a time-consistent consumer with modified utiity, impying existence and uniqueness. 17 First, the differentia system (6) seen as an ODE is not Lipschitz continuous around steady states. Second and more seriousy, W 0 (a) is not even determined at steady state points. When we rewrite system (6) as a differentia agebraic equation (DAE), the steady states correspond to critica singuar points. However, the DAE at this point does not satisfy the sufficient conditions provided in the iterature, for exampe in Rabier and Rheinbodt (2002), for the existence and uniqueness of soutions around singuar points of DAEs, except for the case = 0. 15

16 and (ii) ĉ (a) > ra for a > a 0, im a#a0 ĉ(a) =ra 0 and im a#a0 ĉ 0 (a) =. Proof. Appendix C. Fortunatey, with this emma in hand, our constructive method is reativey straightforward and aso provides an immediate characterization. We construct equiibria by soving the ODEs starting at the bottom and working up, when dissaving; or by starting at the top and working down, with saving; or by combining these procedures. In more detai, the construction invoves the foowing: (i) imposing boundary conditions that serve as initia conditions; (ii) soving the ODEs with a saving or dissaving root over an interva of weath; (iii) decide whether to engineer a jump in W. The great advantage of this approach is that (ii) is oca in nature. Aso, (iii) is aided by the fact that V must be continuous. Finay, the boundary conditions required for (i) are naturay suppied at the asset imit or at high enough eve of weath, appeaing to Assumption Recovery of Power We now justify our focus on a situation where agents do not return to power. Consider a situation where power can be recovered at Poisson rate r > 0. The differentia system (6) becomes rv (a) = max{u 1 (c) + V 0 (a)(ra c) + (W (a) V (a))}, c rw (a) = U 0 (ĉ (a)) + W 0 (a)(ra ĉ (a)) + r (V (a) W (a)). The ast term in the second equation captures the vaue from returning to power. Athough this creates a difference with system 6, our next resut estabishes that the two settings are observationay equivaent. Proposition 2. Consider an economy with utiities and Poisson rates (U 1, U 0,, r ) with positive recovery probabiity r > 0. Equiibria for this economy coincide with equiibria for an economy with utiities and Poisson rates (U 1, Ũ 0,,0) with no possibe recovery of power, where + r and Ũ 0 (c) + r U 0 (c) + r + r U 1 (c). Proof. The pair (V, W) satisfies the differentia system with power recovery above for (U 1, U 0,, r ) if and ony if the pair (V, W) with W + r W + r + r V satisfies the differentia system without power recovery (6a) (6b) for (U 1, Ũ 0,,0) with + r and Ũ 0 (c) + r U 0 (c) + r + r U 1 (c). Intuitivey, the possibiity of recovering power makes an agent more invested in consumption after being ousted from power, which is simiar to pacing a higher vaue on 16

17 consumption whie out of power. In a poitica economy setting, Amador (2002) and Azzimonti (2011) assume that there are no benefits from consuming out of power, U 0 = 0. Then with r = 0 there is no time-inconsistency probem, ony greater discounting, at rate r + > r. Thus, they assume that the agent returns to power with positive probabiity, r > 0. Proposition 2 shows that this is equivaent to a mode without recovery of power but with a positive utiity for those out of power: Ũ 0 = bu 1 where b = /( + r ) 2 (0, 1), i.e. a hyperboic mode. 3.4 Generaized Euer Equation To economize on space we omit the cacuations, but we have shown that d dt U0 1 (c t) =(r r)u 0 1 (c t)+ Z T 0 e R s 0 (r+ r+ĉ0 (a t+z ))dz U 0 1 (c t+s)(1 b(c t+s ))ĉ 0 (a t+s )ds + e R T 0 (r+ r+ĉ 0 (a t+z ))dz U 0 1 (ra T ) (1 b(ra T )), where T denotes the moment weath reaches a steady state, i.e. ĉ(a T )=ra T ; if T = the ast term is zero. This is an adaptation of the Generaized Euer equation, derived by Harris and Laibson (2001) in the context of the discrete-time quasi-hyperboic mode, to our continuous-time mode. When = 0 or b(c) =1 this reduces to the standard Euer equation. However, when > 0, b(c) < 1 and ĉ 0 (a) > 0 the second and third terms on the right side are positive, acting in the same direction as higher r. 4 Dissaving and Saving In this section we construct and characterize equiibria focusing on situations where the disagreement index b(c) is reativey stabe, impying goba savings or dissavings. When r < r a time-consistent agent ( = 0 or b(c) =1) dissaves and time inconsistency (i.e. > 0 and b(c) < 1) ony reinforces this concusion. When r > r a time consistent agent saves, but time inconsistency may overturn this concusion. What turns out to be crucia is the vaue of our oca disagreement index b(c) reative to a cutoff defined by ˆb r r 1 r r. (9) Note that ˆb < 1 if and ony if r > r. Moreover, whenever ˆb > 0 then ˆb is decreasing in r and increasing in r, and increasing (decreasing) in if r > r (if r < r). 17

18 We wi show that when b(c) < ˆb the agent dissaves and when b(c) > ˆb the agent saves. To anticipate this key roe payed by b reative to ˆb it is usefu to cover some specia cases. We first state a very simpe resut showing that in the borderine case b = ˆb an equiibrium exists with zero savings. Theorem 1 (Zero Savings). Assume that b(c) = ˆb for a c ĉ(a) =ra is an equiibrium. ra. Then, (V, W) = ( V, W) and Proof. Appendix D. Next we present a resut on inear equiibria for the hyperboic case that extends the characterization offered aready in Harris and Laibson (2013, Appendix F). The resut requires no asset imit and power utiity functions. Theorem 2 (Linear Equiibria). Suppose b(c) = b appe 1, U 1 (c) = 1 1 s c1 s and a = 0. Then if (1 s)r < r there exists a unique inear equiibrium ĉ(a) =ya with saving y < r when b > ˆb and dissaving y > r when b < ˆb. When b > ˆb the resut hods even if a > 0. Proof. Appendix D. These inear equiibria are simiar to the we-known inear equiibria empoyed in discrete-time quasi-hyperboic settings, such as Pheps and Poak (1968), Laibson (1996) and others. An important difference is that Theorem 2 states that in continuous time inear equiibria are aways unique, echoing Harris and Laibson (2013, Appendix F). Cruciay, we show that the sign of the saving rate y r depends on b versus ˆb. When the interest rate is ow enough or disagreements are high enough, b < ˆb, the agent dissaves. When the interest rate is high enough or disagreements are ow enough, b > ˆb, the agent saves. The inear equiibrium breaks down if b(c) is not constant, if utiity functions are not power functions, or in the presence of a non-trivia borrowing constraint. Athough inear equiibria are specia, the condition for savings and dissavings turn out to hinge on b versus ˆb more generay. 4.1 Dissaving Our first resut constructs an equiibrium with dissaving, when b(c) < ˆb. Indeed, we show that a equiibria must have this property. 18

19 Theorem 3 (Dissaving). Suppose Assumptions 1 and 2 hod and b(c) < ˆb for a c ra. Then there exists an equiibrium with dissavings, ĉ(a) ra for a a, and V(a) > V(a) in a neighborhood of a steady state. Moreover, a equiibria satisfy these properties. If in addition Assumption 3 hods then there is at most one continuous equiibrium and this equiibrium features strict dissavings, ĉ(a) > ra for a a > a. Proof. Appendix E. When r < r dissaving is guaranteed, even in a time consistent situation. When r r dissaving occurs if the time-inconsistency probem created by disagreements is sufficienty strong, so that b < ˆb. Whenever ˆb > 0 then ˆb is decreasing in r and increasing in r and (given r > r). Thus, ower r, higher r, higher and ower b a promote dissaving. Our constructive proof soves the differentia system (6) starting at a with initia conditions V(a) = V(a) and W(a) = W(a), empoying Lemma 2. We move up using the root for V 0 (a) associated with dissaving. There are two possibiities. First, in some cases the soution may be continued indefinitey as a!, providing a continuous equiibrium; Assumption 3 then ensures that there is ony one such soution. Aternativey, the soution to the differentia system may reach a point where ĉ(a) =ra and V(a) < V(a). Past this point there is no root associated with dissaving. An equiibrium must then invove a jump in W and ĉ at the asset point where V crosses V from above. The system is then restarted at this point, by setting (V, W) =( V, W) again and continuing as before. Note that in this construction jumps ony occur at steady states. As this discussion makes cear, both continuous and discontinuous equiibria are possibe and they may even coexist in some cases. We postpone a more detaied discussion regarding these possibiities unti Section 7, where we provide sufficient conditions for the existence of continuous equiibria. The existence portion of Theorem 3 ony requires Assumption 1 and b(c) < ˆb. Assumption 2 is invoked to prove that a equiibria have the same property; this everages our tight characterization of the hyperboic case contained in Section 4.3. Assumption 3 is used to provide a unique oca soution immediatey above a, which then impies that there is at most one continuous equiibrium. Commitment Devices. Time inconsistency probems generay create a demand for commitment devices. In our setting, a simpe form of commitment can be captured by raising the asset imit. This amounts to removing iquidity from the hands of those in power. Amador et a. (2006) argue that such minimum savings poicies are optima within a re- 19

20 ated cass of modes. Here we simpy expore whether they woud be adopted by those in power. The most extreme form of such a commitment device sets the new asset imit at the present asset eve, effectivey imposing a budget baance rue to hod assets constant. Such a commitment is desirabe to those in power if V(a) < V(a). Is such a commitment device desirabe for the agents in power in our mode? Theorem 3 shows that V(a) V(a) near steady states. Moreover, the proof shows that there aways exists an equiibrium (possiby discontinuous) with the property that V(a) V(a) for a a a. Thus, when this inequaity hods, the agent in power never wiingy ties himsef to the mast, so to speak, to adopt a budget baance commitment. This is not obvious, since by adopting a commitment device today the agent trades off ower consumption today with greater commitment in the future. Athough our mode has this somewhat surprising feature, that an equiibrium with this property aways exists, in some cases a continuous equiibrium exists and features V(a) < V(a) for arge enough assets. In such cases, the agent in power woud ike to commit to a baanced budget rue immediatey to raise utiity to V(a). We concude that immediate commitment to a baanced budget may arise in some cases, but ony when weath is sufficienty high. Even if V(a) V(a), so that the current decision maker woud not impose upon itsef a budget baance rue, whenever r r one can show that W(a) < W(a) away from steady states. This impies that agents out of power woud ike to bind those in power to a budget baance rue to hod assets constant. 18 Equivaenty, the agent in power woud ike to commit its successors to such a rue. There are two ways the agent in power can achieve something simiar. First, it may commit to a budget-baance rue that ony comes into effect in the future, after a grace period. Second, it may commit immediatey to an asset imit that is beow the current asset eve, so that this constraint binds in the future but not immediatey. 4.2 Saving We now consider the opposite case, where disagreement is ow enough and b > ˆb. We show that this ensures positive savings. Indeed, we estabish that there is a unique equiibrium. Theorem 4 (Saving). Suppose that b( c) > ˆb and that Assumption 1 and 2 hod. Then there exists a unique equiibrium for a a â for some â 2 [0, c r ). Indeed, there exists ā > r c and a unique consumption function ĉ(a) for a â with the property that ĉ is the unique equiibrium for any a â and ĉ(a) =ya for a ā, with y given by Theorem 18 When r < r then W > W is possibe since some dissaving is desirabe even for a time-consistent agent. 20