Abstract

Note: Thanks to Ian Jewitt and to many researchers and visitors at GPI and the Open Philanthropy Project for helpful corrections and comments. All remaining errors are my own.

I explore the implications of time preference heterogeneity for public good funding. I find that the assumption of a common discount rate is knifeedge: allowing for time preference heterogeneity produces substantially different funding behavior in equilibrium. In particular I find that, across a variety of circumstances, patient funders invest, rather than spend, the entirety of their resources for substantial lengths of time in equilibrium. I also find that the implications of this departure from the common-discount-rate case are economically significant, in that the patient payoff to spending in equilibrium, relative to that of spending according to an intermediate time preference rate, can grow arbitrarily large as a patient funder’s share of initial funding goes to zero. Finally, I discuss applications of these results to the timing of philanthropic spending, and to patient philanthropists’ willingness to pay to avoid legal disbursement minima.

Introduction

The modern literature on public good contribution games begins with Bergstrom et al. (1986). Bergstrom et al. observed that well-behaved static public good contribution games feature a unique equilibrium allocation of resources to particular public goods, in which each individual is indifferent to marginal reallocations of resources among the goods she herself funds and weakly prefers reallocations to goods she funds from goods she does not. Even individuals with relatively similar preferences thus typically find themselves “polarized”, in the sense that they fund entirely or almost entirely non-overlapping sets of projects. Similar observations have been made, seemingly independently, on other occasions: see e.g. Kalai and Kalai (2001).

There is now an extensive literature on dynamic public good contribution games as well. A central concern of this literature is efficiency: that is, roughly, determining when the threats made possible by repeated interaction can induce players to avoid free-riding on each other’s contributions, substituting public good spending for spending on private consumption. It is well understood that in a simple continuoustime model under perfect monitoring, fully efficient spending schedules can obtain in subgame-perfect equilibrium, since the gains from deviation are infinitesimal relative to the losses from future punishments. As a result, authors have explored a vast array of modifications to the simple model, including imperfect monitoring, shocks, and investment irreversibility.

Despite its size, however, the literature on dynamic public good contribution games near-universally assumes that the actors under consideration are equally patient: i.e. that they act under a common discount rate. This common-discounting assumption is pervasive both in the theoretical literature and in applied work on the dynamic provision of public goods, such as country-level efforts to mitigate climate change; see e.g. Ferrari et al. (2015) or Dutta (2017). Fishman (2019) explores bargaining over public good provision in a dynamic setting where the players use different discount rates, and a small literature building on Sorger (2006) likewise explores the implications of bargaining under time preference heterogeneity in dynamic settings, some of which could apply to public good provision problems. Finally, the only other paper on the theory of public good contribution games even to mention time preference heterogeneity, of which I am aware—and so the only paper to do so outside a bargaining context—is Jacobsen et al. (2017), but it is set in a static environment: lower time preference, i.e. greater concern for the future, is simply listed as one reason why individuals may have different preferences regarding the provision of an environmental good in the present.

A deeper analysis of dynamic public good provision under time preference heterogeneity is valuable, I will argue, for at least two reasons.

First, individual rates of time preference vary widely.[1] Most developed-world governments publish discounting guidelines that make explicit the discount rates they use in cost-benefit analysis for public policy, revealing unambiguously that they too act under heterogeneous rates of time preference.[2] Economists’ recommendations of time preferences to use in social discounting differ substantially.[3] Philanthropists’ time preferences appear to vary as well, both with each other and with those of individuals and policymakers. Of course, individuals, policymakers, and philanthropists all regularly contribute to public goods to which other such parties also contribute, and these parties must all decide how to allocate their contributions over time. In doing so, they participate in dynamic public good contribution games. Real-world dynamic public good contribution games likely, therefore, exhibit substantial time preference heterogeneity. Our attempts to model these games, and improve public good provision processes in light of them, will likely fail if we do not account for it.

Second, in practice, many individuals currently hold philanthropically-purposed assets in tax-exempt vehicles where they are earmarked for future charitable giving, such as DAFs. Assets in DAFs in particular, in the United States, currently total almost $150 billion; contributions to them have historically grown at a substantially higher rate than charitable contributions as a whole; and disbursements have not risen as quickly as contributions (National Philanthropic Trust, 2020). As we will see, this pattern can straightforwardly be explained as rational behavior by patient philanthropists given “over-spending” (from their perspective) by less patient other parties. Nevertheless, it is routinely criticized as an unjustifiable withholding of charitable funds, or even as a form of tax evasion. These criticisms have recently reached new prominence in the United States with the June 2021 introduction of the Accelerating Charitable Efforts (“ACE”) Act by Senators Angus King and Charles Grassley, which would impose disbursement requirements on DAFs, effectively requiring their contributors to act less patiently. Due to a lack of literature on dynamic public good provision under time preference heterogeneity, the implications of such a requirement, and of similar proposals to introduce or raise charitable disbursement minima in the United States and elsewhere, have not undergone thorough economic scrutiny.

This paper therefore begins more seriously to explore the implications of time preference heterogeneity for the dynamic provision of public goods. To do so, I use a simple model in which there is a single public good, and the utility produced by spending on it is isoelastic in total flow spending on it. Players with constant but different rates of time preference make decisions about how much to contribute to the good, and how much to invest for future contribution, over an infinite horizon in continuous time.[4]

To highlight the implications of time preference heterogeneity per se, I assume that the size of each player’s total contribution budget is fixed. Contributors decide only the schedule on which to deploy their spending, not the extent to which they will spend on public as opposed to private goods each period. As a result, the model does not resemble the existing literature on dynamic public good provision so much as Bergstrom et al.’s original paper. The relevant change is that I explore what happens when individuals choose their contribution levels for an infinite stream of public goods over which their respective preferences differ—i.e. funding at t, for all t ≥ 0—in sequence, rather than simultaneously. (I sometimes refer to the public good contributors as “philanthropists”, and focus on applications to philanthropy, but the mathematical results are applicable to decentralized public good contributors more generally.)

The model, despite its simplicity, allows us to draw some important and broad conclusions about the implications of heterogeneous discounting for public good provision. First, the common discounting assumption is a knife-edge condition: even slight differences in patience, even (indeed especially) by small (i.e. poorly-funded) players, give rise to very different equilibria. In particular, they can give rise to relatively simple and natural equilibria in which spending is “polarized” in the sense given above, with impatient parties exclusively responsible for public good funding before some future date and patient parties exclusively responsible after. Second, this knife-edge condition is payoff-relevant: the equilibria that obtain under heterogeneous discounting can offer payoffs, at least for unusually patient parties, which differ dramatically from the payoffs they achieve under common discounting. Third, and relatedly, disbursement requirements can reduce the payoff to philanthropy dramatically, from a patient perspective.

Besides the literature on public good provision in static settings, and in dynamic settings under homogeneous time preferences, I will now briefly discuss connections to adjacent strands of literature which do not directly concern public good provision but in which the implications of time preference heterogeneity have been explored more extensively.

One such strand concerns the collective allocation of private consumption over time—i.e., under certain preference aggregability assumptions, the discounting behavior of a representative agent—in a population of individuals or lineages with heterogeneous time preferences. One classic observation from this literature is that consumption (Rader, 1981) and/or wealth (Becker, 1980; Ryder, 1985) can, in the limit, become entirely concentrated in the hands of society’s most patient members, simply because they consume less and invest more. Another, closely related, as shown in an exchange economy by Gollier and Zeckhauser (2005) (and with variations elsewhere), is that a representative agent (if one exists) will, under complete markets, exhibit a discount rate that declines with time to that of society’s most patient members. In other words, interest rates fall as patient parties lend to their less patient counterparts and command an ever-growing share of the financial market. As we will see, similar dynamics play out among agents spending on public goods, and the effects are often even more extreme.

A second body of relevant research concerns optimal taxation by policymakers more patient than their constituents. Farhi and Werning (2007, 2010) analyze optimal taxation in an intergenerational model where individuals save insufficiently, from the patient social planner’s perspective, for their descendants. Household consumption, in these models, is effectively a public good: a good whose provision satisfies the preferences of multiple parties (the policymaker and the household itself) nonexcludably and nonrivally. Similarly, von Below (2012), Belfiori (2017), and Barrage (2018) study optimal carbon taxation and/or investment subsidization in contexts where present production confers both future costs and future benefits (from climate damage and capital accumulation respectively). An important lesson from this literature is that patient policymakers might like to invest resources for future spending, but that to avoid crowding out private investment, it is often optimal for them instead to subsidize private investment and tax private consumption.

Time preference heterogeneity has different implications in the context of optimal taxation than in the context of private spending on public goods, however. The former setting involves an asymmetry in the players’ strategy sets: households cannot tax or subsidize policymakers, but policymakers can tax and subsidize households. At least in the absence of political or informational constraints, policymakers endorsing a given time preference rate can often use these tools to implement population-wide behavior that is optimal or near-optimal from the policymaker’s perspective. Finally, therefore, a literature has emerged on the implications of time preference heterogeneity among more symmetrically empowered agents, who must collectively set discount rate policy for use in public good provision. Gollier and Zeckhauser (2005), as noted above, find conditions under which a group of agents with heterogeneous time preferences give rise to a representative agent whose time preferences are determined by the intertemporal allocation of private consumption, and the corresponding interest rate schedule, that clears the agents’ financial contracts. As Heal and Millner (2014) argue, there is a natural sense in which policymakers aiming to set discounting policy for the provision of a public good, while deferring to the time preferences of their constituents, do best to defer to this aggregated discounting schedule.

Finally, an alternative approach to collective social discounting is to consider, in a social-choice-theoretic framework, the decision-making of committees of social planners with different rates of time preference. Such an approach, however, faces varieties of the preference aggregation impossibilities faced in other social choice contexts, as explored in detail by Chambers and Echenique (2018). Millner (2020) proposes a method by which the discounting planners might reach a kind of consensus, but such proposals are themselves inevitably vulnerable to disagreement. The social choice approach to discounting must also confront the issue of time inconsistency. In particular, Jackson and Yariv (2015) show that any social welfare function used in this setting must be either dictatorial or time-inconsistent, in that future committee meetings will, if they use the same forward-looking social welfare function, decide to revise the plans made by previous meetings—at least if these were made naively, without taking the possibility of future revisions into account. Millner and Heal (2018) therefore examine the collective decision-making of discounting committees aware that they are playing a dynamic game with their future selves. One finding is that attempts to implement “weighted utilitarian” social discounting in such a dynamic game will generally be inefficient. (By contrast, I find that decentralized private actors strategically allocating public good contributions over time can implement efficient, weighted utilitarian social discounting.)

The structure of this paper is as follows.

I begin in §2 with the benchmark scenario in which the good has only a single funder. Barring certain technicalities, the spending schedule obtaining in this case is the same as that which obtains given multiple funders with a shared rate of time preference.

In §3, I explore the interaction between a patient and an impatient philanthropist. I open by discussing, as a second benchmark, what follows when one party is “warmglow”, in the sense of Andreoni (1990) that he is concerned only with his own schedule of contributions to the public good, whereas the other party is “altruistic”, in that she cares about the schedule of total contributions. Modeling both parties as altruistic is most standard in the literature on public good contribution games, but I open with a discussion of warm-glow behavior both because it is a widely documented pattern of philanthropic behavior in practice and because it serves as a simple introduction to the nature of the parties’ interactions.

I then discuss the interaction between altruistic funders across a sequence of increasingly complex game specifications: one in which the parties simultaneously commit to a spending schedule; one in which the impatient party commits to a spending schedule and the patient party then chooses his best response; and finally one, ultimately of most interest, in which the parties play a dynamic game, choosing strategies that govern their spending rates at each point in time as a function of the joint contribution history up to that time. I find that every game specification produces a unique and Pareto-inefficient equilibrium, except the last, which produces many equilibria of which some are efficient and some are not. I also find that there is a sense in which the impatient party has an advantage in the dynamic game, relative to her status in the game with simultaneous commitment, analogous to the first-mover advantage she enjoys in the second game specification with altruistic funders.

In §4, I highlight the economic significance of time preference heterogeneity by estimating a funder’s willingness to pay to move from spending his budget according to a schedule that is optimal given an alternative time preference rate to spending his budget according to the schedule that is optimal given his own time preference rate, even though doing so may induce an inefficient heterogeneous-discounting equilibrium. I find that, for an atypically patient party (but not for an atypically impatient party), this willingness to pay approaches the entirety of his budget as his budget share—his fraction of the sum of the parties’ budgets—goes to zero. That is, when most of the funding for his chosen cause is governed impatiently, the payoff that the patient party attains by acting patiently grows arbitrarily higher than the payoff he attains by spending as he would under common, less patient discounting.

Finally, §5 illustrates conclusions from the earlier sections with what I believe to be plausible parameter values, across examples in which the patient funder’s endowment is initially much smaller than, the same size as, and much bigger than the impatient funder’s. I focus primarily on the first of these cases. In particular, I consider the extreme possibility that, when taken to its logical conclusion, the game theory of the earlier sections recommends that globally impartial patient philanthropists should, at least under some circumstances, invest their wealth for centuries—until they have amassed a substantial share of global wealth—before disbursing.

§6 concludes.

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  1. A helpful review of the econometric and experimental literature on time preference heterogeneity among individuals and households can be found in Alan and Browning (2010), pp. 1252–3. ↩︎

  2. Compare US Office of Management and Budget (2019) and HM Treasury (2020), for instance. ↩︎

  3. As surveyed by Drupp et al. (2018). ↩︎

  4. I use roughly the notation and framework for dynamic optimization problems and games in continuous time presented by Simon and Stinchcombe (1989) and Stinchcombe (2013), and as summarized in Appendix A. ↩︎

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