Hyperbolic Discounting and Uniform Savings Floors

Transcription

1 This work is istribute as a Discussion Paper by the STANFORD INSTITUTE FOR ECONOMIC POLICY RESEARCH SIEPR Discussion Paper No Hyperbolic Discounting an Uniform Savings Floors By Benjamin A. Malin Stanfor University August 3, 2005 Stanfor Institute for Economic Policy Research Stanfor University Stanfor, CA (650) The Stanfor Institute for Economic Policy Research at Stanfor University supports research bearing on economic an public policy issues. The SIEPR Discussion Paper Series reports on research an policy analysis conucte by researchers affiliate with the Institute. Working papers in this series reflect the views of the authors an not necessarily those of the Stanfor Institute for Economic Policy Research or Stanfor University.

2 Hyperbolic Discounting an Uniform Savings Floors Benjamin A. Malin August 3, 2005 Abstract I evelop a general euilibrium moel populate by agents with varying egrees of hyperbolic iscounting who vote for a uniform savings oor. Although partial euilibrium intuition suggests that all iniviuals will prefer to have some constraint on their consumption/savings ecision, I n that even the smallest amount of heterogeneity in preferences leas to very large i erences in preferre policies. In fact, policy preferences are extreme: each iniviual either prefers having no oor impose on the population or having a oor so high that it eliminates all borrowing an lening. I emonstrate that both enogenously etermine prices an ynamically inconsistent preferences are necessary for this result. Finally, I consier how the euilibrium savings oor epens on the average amount of self-control in the population. Keywors: Hyperbolic Discounting, General Euilibrium, Commitment, Voting JEL: E2, H0, H4, H55 Department of Economics, Stanfor University, 579 Serra Mall, Stanfor, CA ( bmalin@stanfor.eu). I woul like to thank Manuel Amaor, Doug Bernheim, Bob Hall, Pete Klenow, Soo Lee, Antonio Rangel, Felix Reichling, Mark Wright, an especially Narayana Kocherlakota for helpful iscussions throughout the course of this project. The nancial support of the Kohlhagen Fellowship Fun at Stanfor University is gratefully acknowlege.

3 Introuction One common explanation for the public support of government involvement in proviing retirement income is that people realize they will not save aeuately on their own, an thus, they prefer to have a savings oor impose on them. Moelling such preferences reuires a eviation from the stanar economic moel; for example, assuming that iniviuals are hyperbolic rather than exponential iscounters. George Akerlof (998) articulates this line of thought succinctly: "the hyperbolic moel explains the uniform popularity of social security, which acts as a pre-commitment evice to reistribute consumption from times when people woul be tempte to overspen uring their working lives to times when they woul otherwise be spening too little in retirement.... The hyperbolic iscounting moel explains, as the stanar moel will not, why the young as well as the ol shoul be particularly enthusiastic about social security" (p. 87). This reasoning is base upon a hyperbolic iscounting literature that focuses on an iniviual s ecision problem given exogenously xe prices (Laibson et al. (998), Amaor et al. (2003)) or on a representative agent in a general euilibrium setting (Laibson (997)). In these moels, the agent will value commitment evices that restrict her ability to take certain actions: examples of commitment evices inclue minimum savings reuirements (Amaor et al.) an the restriction of nancial market innovation (Laibson (997)). Importantly, the use of the commitment evice by a particular agent either oes not a ect the welfare of other agents (since prices are xe in the ecision-theoretic analysis) or a ects all agents uniformly (by construction in the representative agent setting). A state-manate retirement savings program, however, is funamentally i erent from the policies in the moels just escribe in that it is a public commitment evice. This evice a ects all members of society, not just those that su er from self-control an realize they woul bene t from constraint. Moreover, it may a ect each member i erently. Thus, analyzing its esirability in a general euilibrium framework with heterogenous agents seems appropriate. The representative agent is a stan-in for a continuum of ientical agents. 2

4 Moern public retirement savings programs have many complex features: some are pay-as-you-go, others fully-fune, an most reistribute wealth across generations. A common feature, however, is that they reistribute resources across time for any particular iniviual. From an iniviual s perspective, a state-manate retirement savings program reuces isposable income when she is young an ensures a minimum amount of consumption in ol age; it is a savings oor. Thus, to focus on the role of a legislate retirement savings program as a commitment evice, I will moel it as a uniform savings oor. "Uniform" means that the savings oor applies to all iniviuals in the population in the same way. iniviuals face the same oor. In my moel, conitional on their income level, In this paper, I stuy the esirability of a uniform savings oor for mitigating self-control problems of iniviuals. I consier an enowment economy populate by iniviuals with varying egrees of hyperbolic iscounting who vote for the savings oor. 2 My main ning is that even if all iniviuals have large biases for present consumption, a small amount of variation across iniviuals in the strength of the bias will cause vast i erences in preferences for public commitment. The policy preferences are extreme: each iniviual either prefers having no savings oor impose or having a oor so high that it eliminates all borrowing an lening. The intuition behin this result is as follows. Consier an economy without a savings oor (i.e., no constraints on borrowing). Ceteris paribus, iniviuals who have relatively less present-bias will be net savers. The introuction of a savings oor will reuce the aggregate eman for loans, the interest rate, an thus, the wealth of savers. This negative wealth e ect will counter any welfare gains from the mitigation of the time-inconsistency for these iniviuals. On the other han, iniviuals who are originally borrowers will bene t from a positive wealth e ect of the constraint an irectly from the constraint itself. 3 I emonstrate that the extreme-preferences result epens crucially on both enoge- 2 The focus on an enowment economy in this paper is only for ease of exposition. It can be shown that all results of the moel hol in a prouction economy as well. 3 There is also a substitution e ect that accompanies the changing interest rate. It will negatively a ect both savers an (unconstraine) borrowers. This will be iscusse in more etail in Section 3. 3

5 nously etermine prices an ynamically inconsistent preferences. That is, the result oes not hol in a setting in which iniviuals think prices are xe; nor oes it hol if iniviuals have perfect self-control. After characterizing iniviual preferences, I escribe how the euilibrium savings oor is chosen an consier some comparative statics with respect to the average amount of self-control of the population. This paper will focus solely on analytical results as they most clearly show the economics at work. 4 Aing heterogeneity in time-inconsistency seems like a natural way to enrich moels of hyperbolic iscounting preferences for a number of reasons. First, the experimental stuies from the psychology literature that suggest iscount functions are approximately hyperbolic also provie evience of heterogeneity (Kirby an Herrnstein (995)). In recent work, Ameriks an coauthors (2004) ocument i erences across iniviuals in their level of self-control an n a link between self-control an wealth accumulation. Further evience of heterogeneity of iscount factors comes from the enormous ispersion in the accumulate wealth of families approaching retirement, even after conitioning on lifetime earnings, lifetime nancial resources, an portfolio choice (Venti an Wise (2000)). 5 Finally, aing heterogeneity is a simple way to allow for general euilibrium price e ects while eparting from the representative agent framework. To my knowlege, the only other paper which analyzes the welfare implications of commitment evices in a general euilibrium environment that oes not amit a representative agent is Lumer (2004). He consiers hyperbolic iscounting agents who are ex-ante ientical an trae a limite supply of illiui assets. He shows that the availability of illiui assets will generally not lea to welfare improvements an also escribes asset prices an the portfolio holings of agents. The welfare results in his moel arise through changes in euilibrium asset prices in a manner similar to the interest-rate channel escribe above. My analysis i ers from his in at least two ways. 4 It can be shown that some assumptions necessary for close-form expressions, namely Assumptions an 2, can be relaxe an the ualitative results of interest still shown to hol by solving the moel numerically. 5 Venti an Wise (2000) o not test whether this heterogeneity is in the long-run iscount factor or the time-inconsistency parameter, but the point to be mae is that heterogeneity in preferences appears signi cant. In Section 6, I consier heterogeneity of both types in the moel. 4

6 First, in my moel all iniviuals can make use of the uniform savings oor, whereas only a fraction of agents in his moel can utilize the illiui assets. Secon, the savings oor is enogenously etermine, while the supply of illiui assets is exogenously set. The rest of the paper is organize as follows. Section 2 lays out the basic moel an erives euilibrium conitions, while iniviual preferences regaring a uniform savings oor are characterize in Section 3. Section 4 illustrates the importance of general euilibrium, an Section 5 isolates the role of ynamically inconsistent preferences in the analysis. In Section 6, I generalize the extreme-preference result of Section 3 by consiering richer forms of agent heterogeneity; speci cally, agents i er in their patience (i.e., long-run iscount factor) an self-control (i.e., present-bias). Section 7 analyzes how the euilibrium savings oor is a ecte by changes in the istribution of types an escribes one counterintuitive ning. Section 8 conclues, an an Appenix collects some proofs. 2 Basic Moel In this section, I set up the moel, escribe the agents ecision problems, an erive conitions that must hol in euilibrium. In Section 3, these euilibrium conitions are use to characterize agents preferences regaring a uniform savings oor. 2. Environment There are three perios in the economy, t = 0; ; 2. Perio 0 is a policy-setting ate only. In perios an 2, agents get their enowments, trae with each other, an consume. There is a unit measure of agents, who i er only by their time-inconsistency parameter, i, an a type- i agent has preferences represente by the following utility functions: P erio 0 : u(c ) + u(c 2 ) P erio : u(c ) + i u(c 2 ) P erio 2 : u(c 2 ) 5

7 where c t is the agent s consumption in perio t. I assume u 0 ; u 00 > 0; lim c!0 u 0 (c) = +; an lim c! u 0 (c) = 0. The istribution of agents is represente by a ensity f( i ) with c..f. F ( i ) over the interval B = [; ]; > 0;, an all agents are enowe with w t > 0 units of the consumption goo in perios t = ; 2. ( i ) is the iscrepancy between perio-0 an perio- preferences for an agent of type i. 2.2 Decision Problems Agents have two ecisions: a perio-0 vote an a perio- consumption-savings ecision. Let enote the price in perio of a claim to consumption in perio 2, an let be a savings oor (i.e., borrowing constraint). agent chooses savings, A, to solve max A s.t. u(w A A A ) + i u + w 2 De ne A ( i; ; ) to be the solution of this problem. In perio, given an ; each In perio 0, agents vote (as-if-pivotal) over the perio- savings oor. each agent votes for the that solves A max u (w A ( i ; ; ())) + u ( i ; ; ()) + w 2 () s.t. 2 [; 0] That is, where of this problem. ()w 2 is the natural borrowing limit. 6 De ne ( i ) to be the solution A few comments are in orer concerning the perio-0 voting problem. First, note that agents consier how the savings oor will impact the euilibrium price, (). The euilibrium interest rate, (), must ajust so that the supply of loans euals the uantity emane by borrowers in perio, an agents take this into account when they vote. Secon, the savings oor cannot be greater than 0, because in an enowment economy with no storage technology, markets woul not clear if it were. In a prouction economy, however, a positive savings oor is compatible with market clearing, an the main results of this paper still hol. 6 Given Assumption (to follow), there will exist a uniue. 6

8 2.3 Euilibrium Conitions Necessary conitions for an euilibrium inclue constraine optimality of iniviual choices an market clearing. Thus, given an, euilibrium allocations must satisfy u 0 (w A ) i A u0 + w 2 A ( i ; ; ) (2) () where for constraine agents, () hols with strict ineuality an (2) with euality, an for unconstraine agents, () hols with euality. At the euilibrium interest rate, markets must also clear. That is, Z Z A ( i ; )f( i ) i = F ( C ) + A ( i ; )f( i ) i = 0 C (3) where C solves u 0 (w ) = C () u0 () + w 2 (4) Euations () (4) jointly etermine A ( i; ) 8 i an () for any. C is the cut-o between constraine an unconstraine iniviuals. Agents of type C optimally choose to save ; i > C are unconstraine; an i < C are constraine. As I am primarily intereste in characterizing iniviual preferences over, it is not yet necessary to specify the proceure by which iniviual policy preferences are aggregate to etermine the savings oor for the economy. In Section 6, the policy selection process will be mae explicit. To rule out multiple euilibria, I assume that the euilibrium interest rate, (), is a function an that the following assumption hols: Assumption The interest rate, (), is a (weakly) ecreasing function of the savings oor. (i.e., 0) The intuition for Assumption is as follows: if the savings oor constrains some agents, an increase in the savings oor will shift the aggregate eman curve for loans to the left, an the interest rate will fall to euilibrate the loan market. Of course, 7

9 if the savings oor is not a bining constraint for any agent, the euilibrium interest rate will not change. Su cient conitions 7 for Assumption to hol inclue. u(c) = c ; with ; or 2. u(c) = c ; with 2 an w = w 2. 3 Preferences for Public Commitment I now characterize the perio-0 preferences of iniviuals for a public commitment evice, namely, the perio- savings oor. The main result is that all iniviuals prefer an extreme policy but not the same extreme. Given any nonegenerate istribution of types, some prefer having no oor impose an others prefer a oor high enough to eliminate all borrowing an lening. This result stans in stark contrast to the conventional wisom that ubiuitous lack of self-control leas to a consensus opinion on the esirability of a government-manate savings oor. Let U( i ; ) enote the perio-0 inirect utility of an agent of type i when the savings oor is : U( i ; ) u[w A A ( i ; )] + u ( i ; ) + w 2 () Changes in can a ect U( i ; ) through three channels. The rst is the irect e ect of constraine borrowing. The secon an thir are the substitution an wealth e ects associate with a change in the euilibrium interest rate. I characterize U( i ; ) in a series of lemmas to pinpoint the channels through which changes in are operating. As changes, the entire euilibrium changes to be consistent with euations () - (4). 3. Regions of the Policy Space For a given level of the savings oor, some iniviuals will be constraine an others unconstraine. The rst lemma is that if an agent is constraine for some level of the 7 I have not been able to establish necessary conitions for Assumption to hol, although uantitative solutions of numerous parameterizations of the moel have not reveale any cases in which the assumption is violate. Given the few restrictions impose on the istribution of types an the functional form of utility, however, it is possible that a counterexample coul be constructe. 8

10 savings oor, she will also be constraine by any higher savings oor. Lemma An increase in the savings oor, ; must increase the cut-o between constraine an unconstraine agents, C. Proof Assume increases. The left-han sie of (4) must strictly increase. On the right-han sie, note that = 2 > 0, so u0 + w 2 must strictly ecrease. By Assumption, is also ecreasing. Therefore, C must increase for (4) to hol. QED Another classi cation of iniviuals in this economy is into groups of savers an borrowers. The secon lemma implies that if an iniviual is a borrower for some savings oor, then in an euilibrium with a higher savings oor, the iniviual will not be a saver. For a constraine agent, this follows immeiately from Lemma an < 0. Lemma 2 If an agent is an unconstraine borrower for some, she will not be a saver for any higher savings oor. Proof I j A 0 0. A () is implicitly e ne by euation (). Using the implicit function theorem yiels A () = u0 (c ) + u00 (c 2 ) A 2 u 00 (c ) (5) u00 (c 2 ): Then, note that u 0 (c ) = u0 (c 2 ) implies A () < 0 () A < u0 (c 2 ) u 00 (c 2 ) (6) Finally, note that u 0 (c 2 ) u 00 (c 2 ) > 0 an j A 0 0. QED The intuition for Lemma 2 is as follows: an unconstraine iniviual will not be a ecte by the constraint irectly but will be a ecte by the ecrease euilibrium interest rate. The substitution e ect causes her to borrow more, an since she is 9

11 alreay a borrower, the wealth e ect is positive an reinforces the esire to borrow more. Figure illustrates the implications of Lemmas an 2 for a given agent. Start with the highest savings oor, = 0. As the oor is lowere, the iniviual moves from being constraine to unconstraine, an as the interest rate rises, she possibly moves from being a net borrower to a net saver. Regions of Policy Space for a given agent. Unconstraine Saver Borrower Constraine 0 Figure A few remarks about Figure nee to be mae. First, some agents may never be borrowing constraine, even for the highest possible savings oor. For example, the iniviual with the most self-control in the economy will always have non-negative savings as she will have less esire to consume in perio than other iniviuals. Other iniviuals (consier ) will never be savers, even for the lowest savings oor. 3.2 The Inirect Utility Function I now procee to characterize the shape of a given agent s inirect utility function over each region of the policy space. I move from the left to the right of Figure : rst consiering the unconstraine region an then the constraine region. Figure 2 will show the shape of the inirect utility function for any given agent. For su ciently low values of, no agents will be constraine (i.e., F ( C ) = 0). Thus, a small increase in the savings oor will have no impact on euilibrium uantities nor on the utility of the agents. This is shown by the at portion of the inirect utility function in Figure 2. Figure 2 also graphically epicts Lemmas 3 an 4, which 0

12 characterize the preferences of unconstraine an constraine agents, over the savings oor when some agents are constraine (i.e., F ( C ) 6= 0). Before stating an proving Lemma 3, one more assumption is neee. Assumption 2 The price elasticity of savings is ecreasing over a particular region of savings. Letting A A, I assume 0 for u0 (c ) u 00 (c ) < A < 0. Remarks: This assumption is use only once in the proof of Lemma 3, an below, I will iscuss its role in etail. I woul prefer to make an assumption irectly on agents preferences but have not been able to map Assumption 2 into a more primitive form. A su cient conition for the assumption to hol, however, is that u(c) = ln(c). space. Lemma 3 U(; ) is uasiconvex in over the unconstraine region of the policy Proof See the Appenix. The intuition for Lemma 3 comes from the expression for the slope of the inirect utility function 8 : U(; ) = A u 0 (c 2) 2 + u 0 (c ) A (7) First, recall Assumption : an increase in the savings oor causes the interest rate to ecrease, i.e., 0. Next, the secon aitive term within the brackets is negative because an unconstraine agent will save less as the interest rate ecreases, i.e., A < 0. This is exactly the time-inconsistency problem. perio-0 an perio- preferences that is, the larger is ( cost. The larger the iscrepancy between ) the greater the utility Note that in the usual case of time-consistent preferences, =, the envelope conition applies, an the substitution e ect associate with A is negligible. The rst aitive term in euation (7) can be positive or negative epening on the sign of A. an utility. If the agent is a lener, the ecrease in the interest rate ecreases wealth 8 This intuition is for u0 (c ) u 00 (c ) A u0 (c 2 ) u 00 (c 2 ). If A lies below this interval, utility increases with a ecreasing interest rate, an if A lies above this interval, utility ecreases with a ecreasing interest rate.

13 e ect. If the agent is a borrower, however, utility can increase ue to the positive wealth Thus, a borrower views the ecrease interest rate as a mixe blessing. On the one han, the agent becomes wealthier because she can borrow the same amount at a lower price, but on the other han, this reuce price exacerbates the agent s commitment problem. The magnitue of the price elasticity of savings etermines which force is more important for the borrower. If the price elasticity of savings is large, >, the borrower s time-inconsistency will ominate the wealth e ect, an she will su er from the ecrease in the euilibrium interest rate. Assumption 2 ensures that the price elasticity of savings ecreases with a ecrease in the interest rate, so that for tighter borrowing constraints, inirect utility may be increasing. This elivers the uasiconvexity of the inirect utility function over the unconstraine region of the policy space. Having characterize the policy preferences of an unconstraine iniviual, Lemma 4 escribes how the inirect utility of a constraine agent changes with an increase in the savings oor. space. Lemma 4 Perio-0 utility is increasing over the constraine region of the policy Proof Recall con < C unc. U( con ; ) = u 0 (w )( ) + u 0 + w 2 For an unconstraine agent, u 0 (w A ) = unc u 0 A u 0 (w ) < unc u 0 + w 2 u0 + w 2 U( con ; ) = u0 + w 2 u 0 (w ) QED 9 + w 2 2 (8) an A >. Thus,. Rewriting (8), it is easy to see that 2 u0 + w 2 > 0 (9) 9 Note that the proof relies on at least one type of agent being unconstraine, but this is kosher. If all agents are constraine, = 0 (by market clearing). In other wors, in all cases where < 0, some agent is unconstraine an the proof goes through. 2

14 The interpretation of Lemma 4 is as follows: an increase in the savings oor irectly bene ts agents who are borrowing constraine through giving them more commitment. This is re ecte by the term in suare brackets in (9). The term outsie of the brackets shows that a ecrease in the interest rate also makes these agents better o because they are borrowers, i.e., < 0. One might at rst be surprise by Lemma 4. As increases, the euilibrium interest rate is ecreasing. It may seem that for low enough values of the interest rate, a tighter constraint cannot possibly be welfare improving. In euilibrium, however, the interest rate will never get too low. It is boune below by the marginal rate of substitution of agent in an autarkic euilibrium. Shape of Perio-0 Inirect Utility Function U(; ) Unconstraine Constraine Saver Borrower 0 Figure Extreme Preferences The lemmas of the previous two sections escribe the policy preferences of any particular agent. For the entire istribution of agents, a two-part "Extreme Preferences" result emerges. All iniviuals prefer an extreme policy but not the same extreme: some iniviuals prefer having no savings oor impose while others prefer a oor so high that it eliminates all borrowing an lening. First, the "all" part is shown by Proposition an Figure 3. Proposition Each iniviual s preferre policy is either (e ectively) no savings oor,, or the maximum savings oor, = 0. 3

15 QED Proof It follows irectly from Lemmas -4 that U( i ; ) is uasiconvex in, 8 i. Savings Floor Preferences of Various Iniviuals U(; ) meian 0 Figure 3 The "not the same" part of the result is shown by Proposition 2 an Figure 4. Recall that () enotes the preferre savings oor of a type agent. Proposition 2 There exists an iniviual e 2 (; ) who is ini erent between an. For <, e =. For >, e =. Proof See the Appenix. Figure 4 shows that iniviuals with more of a time-inconsistency problem (lower ) prefer = 0, while those whose preferences change relatively less over time prefer =. In this particular setting, iniviuals who are net savers in the no-savings- oor euilibrium are the ones who woul have a ecrease in utility by moving to an euilibrium with a high savings oor. Preferre Savings Floor () 0 ~ Figure 4 4

16 4 Euilibrium Prices vs. Fixe Prices Allowing for the interest rate to change with movements in the savings oor is crucial for the extreme-preferences result of the previous section. To emonstrate this, I consier a xe price version of the moel in which the interest rate is no longer pinne own by market clearing. Instea, it is a constant that oes not epen on the euilibrium savings oor. Euations () an (2) still characterize the consumption/savings ecision of perio- iniviuals. Let U P E ( i ; ) enote the perio-0 inirect utility of an agent of type i when the savings oor is : U P E ( i ; ) u[w A A ( i ; )] + u ( i ; ) + w 2 : (0) As changes, it only impacts an agent s welfare through her savings, A ( i; ); an has no e ect on the interest rate,. Furthermore, it is apparent from euations () an (2) that savings only change if the savings oor is a bining constraint. In this setting, a two-part result emerges that is the opposite of the previous section s two-part extreme-preferences result: all iniviuals prefer the same savings oor, an the preferre policy may not be an "extreme". The use of uotes re ects the fact that the extremes of the policy space are no longer e ne by euilibrium conitions, an thus, they can be arbitrarily chosen. will assume 2 [ w 2 ; 0]: For comparison to the previous section, I Proposition 3 All iniviuals prefer the same savings oor. Proof See the Appenix. Because iniviuals are ientical except for their time-inconsistency parameter an because the interest rate oes not vary with the policy, it is easy to see from (0) that the optimal savings from the perio-0 perspective is inepenent of i : Denote the optimal savings as A 0 : All iniviuals will prefer the savings oor in the policy space that is closest to A 0. If this savings oor is a bining constraint on their perio- savings ecision, the iniviual strictly prefers it to other oors; if this savings oor oes not bin, the iniviual is ini erent between this oor an others. 5

17 Proposition 4 The preferre policy may not be an extreme policy. Proof See the Appenix. Proposition 4 is most easily emonstrate by consiering the case in which w = w 2 = w. The optimal (from a perio-0 perspective) savings satis es u 0 (w A 0 ) = A 0 u0 + w : If >, then A0 > 0 an the agents preferre savings oor is an extreme, = 0. If = A 0. <, then A0 < 0 an the agents preferre savings oor is an interior policy, Figures 5 an 6 epict iniviuals policy preferences for these two cases. U P E (; ) Savings Floor Preferences: A 0 > 0 meian 0 Figure 5 U P E (; ) = Savings Floor Preferences: A 0 < 0 meian A 0 0 Figure 6 6

18 The partial euilibrium intuition allue to in the introuction is easily seen in Figures 5 an 6. If prices are xe, all iniviuals will reach a consensus regaring the optimal savings oor. In general euilibrium, however, a small bit of heterogeneity in iniviuals nee for commitment leas to big i erences in policy preferences. Those agents with less self-control problems (high ) prefer to get their commitment from the high interest rate associate with a low savings oor rather than having a high oor. 5 Time-Inconsistency vs. Impatience The extreme preferences result above i ers from results in the time-inconsistency (i.e., hyperbolic iscounting) literature partly because the interest rate changes with the euilibrium savings oor. The general euilibrium e ects of a policy change, however, will occur even with stanar, time-consistent preferences. To isolate the role of time-inconsistency for the result, I consier an environment ientical in all ways to the one in Section 2 except that iniviuals now have time-consistent preferences. Speci cally, all iniviuals have perfect self-control, but some are more impatient than others. Preferences are represente by P erio 0 : u(c ) + i u(c 2 ) P erio : u(c ) + i u(c 2 ) P erio 2 : u(c 2 ) where the istribution of types is represente by a ensity f() over the interval D = [; ]; > 0;. 5. Qualitative Features of () When agents ha time-inconsistent preferences, I showe that an agent s most preferre uniform savings oor was an extreme. The analysis that follows shows that timeinconsistency of preferences is necessary for this extreme result. h i Let U(; ) u[w A (; )] + u A (;) () + w 2 be the time-0 inirect utility of an agent of type when the savings oor is. Thus, 7

19 U(; ) A = u0 (c 2) u 0 (c ) u 0 (c 2) A 2 () Lemmas analogous to Lemmas an 2 will hol in this environment as well, so that Figure will continue to escribe the regions of the policy space. Therefore, I characterize preferences rst for an unconstraine iniviual (Lemma 5) an then for one who is constraine (Lemmas 6 an 7). space. Lemma 5 U(; ) is uasiconvex in over the unconstraine region of the policy Proof By the envelope theorem, the rst term in () is eual to 0. Thus, U(;) has the opposite sign of A (; ). Finally, by Lemma 2, as the interest rate ecreases, a borrower will never become a saver. QED For constraine iniviuals, Lemma 6 characterizes the slope of the inirect utility function at the point in the policy space when the iniviual becomes constraine, an Lemma 7 consiers the slope as the constraint approaches its upper boun. Lemma 6 For a just"-constraine agent, U(;) is positive. Proof This follows very closely from the proof of Lemma 5. Note that for a justconstraine" iniviual, the rst term is to a rst-orer approximation 0, while the secon term is strictly positive. QED Lemma 7 For a constraine agent, as! 0, U(;) is negative. Proof As! 0, the secon term in () goes to 0, while the rst term is strictly negative. QED 8

20 In contrast to the extreme preferences result obtaine with time-inconsistent preferences, Lemmas 5-7 establish that an iniviual with time-consistent preferences can have a moerate policy as his preferre policy. When iniviuals ha time-inconsistent preferences, it was not possible for them to be overly constraine, but with stanar preferences, the irect e ect of a constraint is to ecrease utility. There is a wealth e ect that allows for a small amount of constraint to be welfare improving, but this e ect is overwhelme by the istortion of optimal choice that becomes more severe as the constraint increases. The extreme-preferences result, therefore, crucially epens on the presence of time-inconsistency in preferences. 6 Heterogeneity in Present-Bias an Patience In Section 2, consumers i er only by the strength of their present-bias,, whereas in Section 5, they i er only in their level of patience,. This section characterizes conitions uner which the extreme preferences result hols for the more general case in which iniviuals are heterogenous in both imensions. Unless otherwise note, the moel is ientical to the one evelope in earlier sections. 6. Environment There is a unit measure of people, inexe by i, who are heterogeneous in their presentbias factor, i, an long-run iscount factor, i. They have preferences represente by the following utility functions P erio 0 : u(c ) + i u(c 2 ) P erio : u(c ) + i i u(c 2 ) P erio 2 : u(c 2 ) where ( i ; i ) 2 [; ] [; ]; ; > 0; ; : Assume that i an i are stochastically inepenent. 9

21 6.2 Policy Preferences If there is variation in the present-bias factor across iniviuals but no heterogeneity in the long-run iscount factor, Proposition states that iniviuals most preferre policy will be an extreme. On the other han, with heterogeneity only in the long-run iscount factor, Lemmas 6 an 7 establish that an interior policy may be most preferre. The following proposition gives su cient conitions for an extreme policy to be the most preferre policy of an iniviual. Proposition 5 Iniviual i will prefer an extreme policy if i. Proof See the Appenix. Iniviuals make a trae-o between gains from trae an gains from commitment when choosing the level of the uniform savings oor. Smaller gains from trae because iniviuals have similar preferences (i.e., less variation in ) an a greater nee for commitment (i.e., lower levels of ) are associate with iniviuals preferring an extreme constraint. For the special case of heterogeneity only in, all iniviuals prefer an extreme policy. If, on the other han, is the only from of variation, an extreme policy is guarantee to be the euilibrium policy only if is low enough an there is not much variation in. 7 Political Economic Comparative Statics Having characterize iniviual preferences for a uniform savings oor, I now consier some comparative statics of the moel. I return to the baseline moel from Section 2 an consier how a shift in the istribution of types a ects the savings oor selecte in euilibrium. To o so, it becomes necessary to specify how iniviual preferences are aggregate to select the euilibrium policy. 20

22 Fortunately, a meian voter theorem applies in this setting because, by Proposition 2, an iniviual s preferre savings oor is a monotonic function of her type. Since () is monotonic, I assume that = ( meian ) 0 an evaluate how changes in the istribution of types, F (); a ect. Restricting the class of utility functions to those that satisfy the rst su cient conition for Assumption, the following proposition shows that a ecrease in the average amount of self-control in the economy can actually lea to less commitment in this environment. Proposition 6 Any ownwar (in FOSD sense) shift in F () that leaves meian unchange leas to a (weak) ecrease in. Proof A ownwar shift in the istribution leas to a ecrease in. To see this, note that if stays constant, the left-han sie of euation (3) ecreases because A > 0. An increase in woul ecrease A (); 8. Thus, must ecrease in orer for (3) to hol. A ecrease in increases U(; ); 8. Lemma 3) that U(;) has the same sign as To see this note (following the proof of A + ( ) A. The secon term is always negative, an the sign of the rst term is negative if the agent is a saver. If A the agent is a borrower, we nee A > ( ), an this is true in the class of utility functions uner consieration. Therefore, ~ ecreases. If ~ new < meian ~ ol, 2 ecreases. Otherwise, remains the same. QED Thus, a shift in the istribution of agents that lowers the average amount of personal commitment ability in the economy may also lea to policies that provie less commitment ability, not more as is suggeste in many moels of a representative agent with time-inconsistent preferences. If the average self-control of agents relative to the meian voter is smaller, the meian voter will bene t more from others lack of constraint an may vote against public commitment. 0 My results will still hol for any super-majority voting rule. Either of the two su cient conitions for Assumption woul have been su cient for this step. 2 I assume that agent of type ~ votes for = 0. 2

23 On the other han, if the shift also changes the ientity of the meian agent to a lower type, may increase. The result will epen on the sign of ~ new new meian. 8 Conclusion I exten the analysis of the economics of commitment to a general euilibrium setting in which iniviuals vote over a public commitment mechanism, namely a uniform savings oor. The moel emonstrates that even if iniviuals have very small i erences in their levels of self-control, they will have large i erences in their preferre policies. Each iniviual either prefers no savings reuirement or a savings reuirement that forces everyone to save the same amount. This extreme result is riven by the enogenously etermine interest rate an woul not hol in moels of private commitment evices or moels that amit a representative agent. It shoul be emphasize that the simplicity of the moel (enowment economy with heterogeneity in preferences) was primarily for expositional purposes, but the extreme result will hol for a wier class of moels. Speci cally, it continues to hol in a prouction economy where iniviuals can transfer resources across perios in the form of prouctive capital. The main complications inclue ree ning the policy space to allow for a positive minimum savings oor an consiering the e ect that changes in the capital stock have on wages. Another extension is to allow for alternative forms of heterogeneity, such as i erences in the slope of the agents enowment pro les. This is formally euivalent to the case of heterogeneity in the long-run iscount factors iscusse in sections 5 an 6. Throughout the paper, I moelle the public commitment evice as a uniform savings oor an suggeste that government-manate retirement savings programs essentially have this feature. This is, of course, a rough approximation. Although all U.S. workers have 2.4% of their earnings automatically save for them through Social Security, it is well known that the payroll tax is regressive (because of the cap on taxable earnings) an the bene t scheule is progressive. This paper shows that even if we abstracte from the reistribution aspects of social security an focuse 22

24 solely on its role as a commitment evice, we still woul not expect to see a consensus opinion on its esirability. 9 Appenix 9. Proof of Lemma 3 First, I nee to establish some properties of A. Euations (5) an (6) imply that a ecrease in the interest rate leas to an increase in perio- consumption if an only if perio- savings are not too high. Secon, note by the implicit function theorem that A () A A = 2 (2) = u0 (c ) A u00 (c ) 2 u 00 (c ) + u00 (c 2 ) (3) Thus, A () < 0 () A > u0 (c ) u 00 (c ) (4) In wors, a ecrease in the interest rate leas to a ecrease in perio-2 consumption if an only if perio- savings are not too low. Now, I prove the lemma. U(; ) = = u 0 (w A )( ) A A + u0 + w 2 u 0 (c A 2)( 2 ) + u 0 (c ) A A A (5) : (6) The secon euality comes by substituting the rst-orer conition of the agent s perio- ecision problem into the expression an collecting like terms. Since we are intereste in the sign of this expression, ivie through by u 0 (c 2 ) > 0 to get U(; ) = = A 2 + u0 (c ) u 0 (c 2 ) A 2 + A (7) A : (8) 23

25 Again, the secon euality comes from substituting in the rst-orer conition. (8) has the same sign as If A (; ) > u0 (c 2 ) u 00 (c 2 ), U(;) with an increase in. A + ( ) A : (9) is negative as consumption in both perios ecreases If 0 A (; ) u0 (c 2 ) u 00 (c 2 ), U(;) is negative as both terms in (9) are negative. If u0 (c ) u 00 (c ) < A (; ) < 0, the rst term in (9) is positive. Thus, (9) is negative if an only if Since cuto level, A A = A A > : (20) 2 A u 0 (c )+u00 (c 2 ) 2 u 00 (c )+u00 (c 2 ) is ecreasing in for u0 (c ) u 00 (c ) < A () < 0, there is a A ~, below which U(;) is positive. Since A U(;) is positive for some, it will be positive for any greater. < 0 on this region, once Finally, if A (; ) u0 (c ) u 00 (c ), U(;) increases with an increase in. QED is positive as consumption in both perios 9.2 Proof of Proposition 2 By Proposition, () 2 f; 0g; 8: If = 0, all agents have the same utility since they have ientical enowments an perio-0 preferences. Simply put, U(; 0) = u(w ) + u(w 2 ) U(0); 8 If = ()w 2, all agents are unconstraine, an perio-0 utility is enote by U(; ). Furthermore, U(; ) < U(0) since an agent of type is the rst to become 24

26 constraine, an hence, his utility is nonecreasing over the policy space. At the other extreme, U( ; ) > U(0) since a type agent will always be an unconstraine saver, an thus, her utility is nonincreasing over the policy space. Finally, U(; ) is strictly increasing in. U(; ) A = ( )u 0 (w A () ()) = ( ) u0 (c 2 ) A () > 0 + u 0 A () A () + w 2 Thus, there exists some ~ 2 (; ) such that U(; ) < U(0); if < ~ U(; ) = U(0); if = ~ U(; ) > U(0); if > ~ QED 9.3 Proof of Proposition 3 Let e ( i ) enote the perio- savings choice of an unconstraine agent of type i. That is, e ( i ) is implicitly e ne by u 0 [w e ] = i u0 " e + w 2 Let ( i ) = enote the optimal (from a perio-0 perspective) savings choice of a # : type- i agent. is implicitly e ne by u 0 [w ] = u0 + w 2 : Note that ( e i ) for i, with strict ineuality unless i =. More generally, note that () e has the same sign as ( ) for > 0: I procee by characterizing the shape of U P E ( i ; ) over the policy space. For < ( e i ), the agent is unconstraine an U P E ( i ; ) is at. For ( e i ), the agent 25

27 is constraine by the savings oor. For a constraine agent, U P E () = u0 + w 2 u 0 (w ) h u 0 e i + w 2 = ( ) ; where is chosen so that () e = : Thus, U P E ( i ; ) is increasing over the region e ( i ) an ecreasing over the region. If is in the policy space, it is the preferre policy of all agents. agents prefer the point in the policy space closest to. QED Otherwise, all 9.4 Proof of Proposition 4 Let w = w 2 = w an <. An iniviual s optimal (from a perio-0 perspective) savings A 0 satis es u 0 (w A 0 ) = A 0 u0 + w : Thus, A 0 < 0 an the agents preferre savings oor is an interior policy, = A 0. QED 9.5 Proof of Proposition 5 Let U( i ; i ; ) u[w A ( i; i ; )]+ i u utility of agent i when the savings oor is. h A ( i ; i ;) () + w 2 i enote the time-0 inirect It is straight-forwar to exten Lemmas, 2, an 3 to this more general case. Thus, once an agent becomes constraine, she will also be constraine in an euilibrium with a higher savings oor. Furthermore, U( i ; i ; ) is uasiconvex in for any unconstraine agent. For a constraine agent, U( i ; i ; ) i = u0 + w 2 u 0 (w ) {z } substitution e ect 26 + i u 0 + w 2 2 {z } wealth e ect

28 The wealth e ect is always positive but approaches 0 as! 0. The substitution e ect is non-negative for a just-constraine iniviual an becomes smaller as increases. If the substitution e ect is always non-negative, then the iniviual will have an extreme policy as her most preferre policy. Thus, we nee to check the sign of i u w u 0 (w ) as! 0. The euilibrium price when = 0 must satisfy This implies Thus, i (0) u0 (w 2 ) u 0 i (w ) = u 0 (w ) i i (0) u0 (w 2 ) 8i: (0) = u0 (w 2 ) u 0 (w ) : u 0 (w ) 0 () i : QED References [] Akerlof, George A. Comments on Self-Control an Saving for Retirement," Brookings Papers on Economic Activity,Vol. 998, No.. (998), pp [2] Amaor, Manuel, George-Marios Angeletos, an Ivan Werning. Commitment vs. Flexibility," NBER Working Paper No. 05, [3] Ameriks, John, Anrew Caplin, John Leahy, an Tom Tyler. "Measuring Self- Control," NBER Working Paper No. 054, [4] Kirby, Kris N. an R.J. Herrnstein. Preference Reversals Due to Myopic Discounting of Delaye Rewar," Psychological Science, Vol. 6, No. 2, March 995, pp [5] Laibson, Davi. Golen Eggs an Hyperbolic Discounting," Quarterly Journal of Economics, 997, [6] Laibson, Davi, Anrea Repetto an Jeremy Tobacman. Self-Control an Savings for Retirement," Brookings Papers on Economic Activity, Vol. 998, No.. (998), pp

29 [7] Lumer, Sebastian. "Illiui Assets an Self-Control," Working Paper, Princeton University,