Optimal Altruism in Public Good Provision

39 Pages·2014·0.34 MB·English

ES Working Paper Series, September 2014
Optimal Altruism in Public Good Provision
Robert W. Hahn, Smith School of Enterprise and the Environment, Oxford University Institute
for New Economic Thinking (INET)
Robert A. Ritz , Faculty of Economics, Cambridge University Energy Policy Research Group
(EPRG)
Abstract
We present a model of altruistically-minded—yet rational—players contributing to a public good. A key
feature is the tension between altruism and crowding-out effects. We present three main results: (1) More
altruistic behaviour often reduces social welfare; (2) It is almost always optimal for a player to act more
selfishly than her true preference; (3) A player’s optimal altruistic commitment is often low or zero—even
with strongly altruistic preferences. Applications to a range of public good problems, including climate
policy, are discussed. Our results highlight that it will generally be difficult to infer social preferences from
observed behaviour.
We would like to thank Toke Aidt, David Anthoff, Elizabeth Baldwin, John Feddersen, Reyer Gerlagh, Thomas
Greve, Cameron Hepburn, Charles Mason, Grischa Perino, Rick van der Ploeg, John Quah, Robert Stavins,
Paul Tetlock, Alexander Teytelboym, Richard Tol, and Alistair Ulph for helpful comments and advice, seminar
participants at EPRG, OxCarre, Cambridge, and Toulouse for discussions. The usual disclaimer applies.
1 Introduction
There is a growing recognition that social preferences may play an important role in
explaining economic outcomes such as those arising in problems of public good provi-
sion.1 We study the welfare impact of unsel(cid:133)sh behaviour by altruistically-minded(cid:151)yet
rational(cid:151)players, and ask to what extent a preference for altruism is optimally re(cid:135)ected
in a player(cid:146)s contribution to a public good. To our knowledge, this is the (cid:133)rst attempt in
the literature to understand a notion of (cid:147)optimal altruism(cid:148).
Ouranalysisismotivatedinpartbyrecentexperiencewithclimatepolicy, whichmany
considertobeoneofthebiggestpublicgoodproblemsoftoday(Stern2008). Recentyears
have witnessed a number of unilateral initiatives to combat climate change at the local,
national, and regional levels. For example, the EU has a program to reduce greenhouse
gas emissions by 20% (relative to 1990 levels) by 2020 while the UK aims to cut emissions
by 80% by 2050.2 Such initiatives have taken place in the absence of a global agreement
by countries to jointly reduce emissions, e.g., with a global cap-and-trade scheme.
Relatedly, there is an increasing use of the (cid:147)social cost of carbon(cid:148)(SCC) in regulatory
decision-making. The SCC re(cid:135)ects the marginal bene(cid:133)t to the world from reducing CO
2
emissions(cid:151)ratherthanonlytoanindividualcountryorregion. SeveralEuropeancountries
have applied the SCC (Watkiss and Hope 2012), and the US has also developed a measure
of the SCC (Greenstone, Kopits and Wolverton 2013) which to date has been applied to
selectedenergyandenvironmentalregulations. Atthesametime, manyothercountriesdo
not incorporate the SCC in policymaking, and do not appear to have engaged in emissions
abatement beyond (cid:147)business-as-usual(cid:148).
There is some evidence that the domestic costs associated with unilateral policies ex-
ceed domestic bene(cid:133)ts. For example, Tol(cid:146)s (2012) cost-bene(cid:133)t analysis of the European
Union(cid:146)s 20/20/20 policy package (cid:133)nds a bene(cid:133)t-cost ratio < 1 across a range of scenar-
ios.3,4 In a similar vein, the UK Department of Energy and Climate Change(cid:146)s impact
assessment of the 2008 Climate Change Act (cid:133)nds (cid:147)the economic case for the UK contin-
uing to act alone where global action cannot be achieved would be weak(cid:148)(DECC 2009).
It is di¢ cult to reconcile these unilateral initiatives with standard economic theory,
including the theory of international environmental agreements (Barrett 1994, 2005). Put
simply, if unilateral action by local, national, or regional actors reduces their own domes-
tic welfare, then why are they doing it? But it seems possible that some of these climate
initiatives may be a re(cid:135)ection of (cid:147)unsel(cid:133)sh(cid:148)or (cid:147)altruistic(cid:148)motives, in the sense of incor-
1See Sobel (2005) for an overview of interdependent preferences in economic analysis.
2Similiarclimate-policyinitiatives,manyonarelativelysmallscale,alsoexist,forexample,inAustralia,
California, China, Japan, New Zealand and Norway, as well as at the city level.
3This policy package targets a 20% cut in greenhouse gas emissions, a 20% share of renewable energy,
and a 20% improvement in energy e¢ ciency by 2020.
4Similar issues have also emerged in the analysis of the recently proposed regulation to reduce carbon
dioxide emissions from the US power sector (EPA 2014). The economic analysis conducted by the EPA
suggeststhattheglobalbene(cid:133)tsofthisregulationexceeditscostsbut,insomescenarios,domesticbene(cid:133)ts
accruing only to the US fallshort ofdomestic costs (e.g., depending on the extent to which health-related
bene(cid:133)ts are taken into account in addition to direct climate bene(cid:133)ts).
2
porating bene(cid:133)ts that accrue outside the borders of the acting jurisdiction. This paper
seeks to understand the role that altruism can play in such public goods problems.5
We begin with a two-player model of (non-cooperative) public good provision with the
following key features. A player(cid:146)s net bene(cid:133)t or (cid:147)national welfare(cid:148)(cid:5) (k = i;j) equals the
k
bene(cid:133)t she derives from total contributions to the public good (by both players) minus
the cost of her own contribution, while (cid:147)global welfare(cid:148)W = (cid:5) +(cid:5) .6 Preferences may
i j
depart from self-interest: A player(cid:146)s true objective function S = (1 (cid:18) )(cid:5) +(cid:18) W places
k k k k
(cid:0)
weight on both her own net bene(cid:133)ts and global welfare, where (cid:18) [0;1] represents her
k
2
degree of altruism.7 More altruistic behaviour by player i leads to an increase in its own
public good contribution but induces player j to cut back ((cid:147)crowding out(cid:148)). We refer to
the rate at which the other player(cid:146)s e⁄ort contracts as the (cid:147)leakage(cid:148)rate. The tension
between altruism and leakage lies at the core of our analysis. Our modelling approach
is also consistent with a characteristic shared by many (global) public good problems:
The absence of a world government means that solutions enforced by a central mechanism
designer play a limited role.8
Ouranalysishighlightsthreemain(cid:133)ndings. First,weobtaintheseeminglyparadoxical
result that more altruistic behavior by an individual player often reduces social welfare.
For example, consider a small commitment to more altruistic behavior by player i. Such
a commitment raises the equilibrium net bene(cid:133)t enjoyed by player j but reduces i(cid:146)s own
net bene(cid:133)t. We show that welfare is more likely to fall if player i derives an above-average
marginalbene(cid:133)tfromcontributions, andtheleakageratefromhercommitmentishigher.9
Conversely, anecessaryconditionformorealtruisticbehaviourtoraisesuchaplayer(cid:146)strue
objective is that her degree of altruism exceeds her leakage rate. This already shows that
whether altruism is privately optimal and/or welfare-augmenting depends crucially on
5Our analysis focuses on (cid:147)international altruism(cid:148)between countries rather than (cid:147)intergenerational
altruism(cid:148)betweendi⁄erentgenerationsofpeopleinasingleeconomy. Wedi⁄erfrommuchoftheliterature
on altruism in that we often think of our unit of analysis as a country rather than an individual. Also,
we do not wish to claim that social preferences are the only possible way of explaining unilateral climate
action; in some cases, other explanations, e.g., domestic political economy, may be important.
6Our results are robust to di⁄erent welfare de(cid:133)nitions. Section 5 provides details.
7Our formulation of altruism has a continuum of preferences, ranging from entirely sel(cid:133)sh to entirely
altruistic preferences. On a historical note, Edgeworth (1881) uses essentially the same formulation, by
writing S =(cid:5) +(cid:18) (cid:5) and calling (cid:18) the (cid:147)coe¢ cient of e⁄ective sympathy(cid:148). Some other formulations of
i i i j i
altruismhaveconditionalelements. Forexample,themodelsofinequityaversionduetoFehrandSchmidt
(1999) and Bolton and Ockenfels (2000) feature utility functions with reference points which determine
the degree of perceived inequity in payo⁄s (and also a⁄ect players(cid:146)actions, e.g., depending on whether
they are (cid:147)ahead(cid:148)or (cid:147)behind(cid:148)). Lange and Vogt (2003) show that a preference for equity can generate
cooperationininternationalenvironmentalnegotiations,whileKosfeld,OkadaandRiedl(2009)arguethat
fairness can play an important role in the formation of institutions geared towards improving public good
provision. See also Rabin (1993) on fairness in economic analysis.
8Keycontributionsonvoluntarypublic-goodprovisionincludeBergstrom,BlumeandVarian(1986)and
CornesandSandler(1996). Weworkwithasimpli(cid:133)edmodelwithreduced-formbene(cid:133)tandcostfunctions,
as is standard in much of the environmental economics literature (e.g., Hoel 1991; Barrett 1994), which
captures the key feature that players(cid:146)contributions are strategic substitutes.
9Hoel(1991)obtainsarelatedresultinanimportantearlymodelofunilateralcommitmentinenviron-
mentalpolicythatdoesnotfeaturesocialpreferences. Heshowsthatasmall(exogenous)commitmenttoa
higherpublicgoodcontributionbyaplayer,startingfromaworldinwhichallplayersmakeentirelysel(cid:133)sh
contributions, can reduce global welfare. Our work goes further by examining a world in which players
can behave altruistically to di⁄erent degrees, and deriving a notion of optimal (endogenous) altruism.
3
the details of the environment; a player may thus wish to (cid:133)nd ways of making public
contributions that departs from her true objective.
Second, we show that a player who genuinely wants to maximize global welfare almost
always does best by being at least somewhat sel(cid:133)sh. Toseethis, supposethatplayeri(cid:146)strue
preference is entirely altruistic, (cid:18) = 1, while player j is altruistic only to some degree,
i
(cid:18) < 1. Should i make the contribution that maximizes its underlying global-welfare
j
objective? No. Intuitively, a small decrease in its own contribution only leads to a second-
order loss in global welfare (by the envelope theorem). But the resulting induced increase
in the other player(cid:146)s e⁄ort leads to a (cid:133)rst-order gain (whenever the other player is not
already choosing the (cid:133)rst-best e⁄ort).10 This is what we call (cid:147)reverse leakage(cid:148)(cid:151)a weaker
commitment reduces free-riding by other players, and this can raise social welfare.
Third, we (cid:133)nd that a player(cid:146)s optimal altruistic commitment is often (cid:147)low(cid:148)or zero(cid:151)
even with strongly altruistic preferences. In some cases, it is optimal for a player who cares
aboutglobalwelfaretoactentirelysel(cid:133)shly,maximisingonly herownnetbene(cid:133)t. Wethus
highlight that caution is required in inferring whether or not players are (cid:147)being sel(cid:133)sh(cid:148)
from their observed behaviour; sel(cid:133)sh behaviour may be a welfare-maximising response to
crowding-out e⁄ects, especially with heterogeneous players.
We characterize optimal altruistic commitments using the following modelling device:
Player k has a strategic objective function (cid:10) = (1 (cid:21) )(cid:5) + (cid:21) S , where (cid:21) 0 is
k k k k k k
(cid:0) (cid:21)
her strategic preference. A player chooses a public good contribution according to her
true preference if (cid:21) = 1, but whenever (cid:21) < 1 ((cid:21) > 1) acts more (less) sel(cid:133)shly than
k k k
would be her true preference. We determine a player(cid:146)s optimal commitment (cid:21) ((cid:18) ;(cid:18) )
(cid:3)k i j
to incorporate its altruism into public good contribution. In particular, we always have
(cid:21) 1, almost always (cid:133)nd (cid:21) < 1 (for k = i;j), and, in a range of cases, (cid:21) 0 and/or
(cid:3)k (cid:3)k (cid:3)i
(cid:20) (cid:25)
(cid:21) 0.11 Only where all players have entirely altruistic preferences (cid:18) = (cid:18) = 1, is a full
(cid:3)j i j
(cid:25)
commitment (cid:21) = 1 (for k = i;j) optimal, in which case the (cid:133)rst-best outcome obtains.
(cid:3)k
Weshowthattheseresultsareveryrobusttoavarietyofdi⁄erentmodelspeci(cid:133)cations.
This includes the generalization to n 3 players(cid:151)where we exploit the fact that players(cid:146)
(cid:21)
contributions are made in an (cid:147)aggregative game(cid:148)(in the sense of Corch(cid:243)n 1994; see also
CornesandSandler2007);moderatedegreesofcross-countrycostspillovers(e.g.,inrenew-
ableenergytechnologiessuchassolarorwind); anddi⁄erentrepresentationsofaltruismin
players(cid:146)objective functions, including the (cid:147)warm glow(cid:148)of Andreoni (1989, 1990). Thus,
our results apply with both (cid:147)pure(cid:148)and (cid:147)impure(cid:148)forms of altruistic preferences.
One way of thinking about how a player can commit to actions that depart from her
true preference is in terms of the theory of strategic delegation. For example, citizens may
delegate decision-making on abatement targets to politicians, and may wish to appoint
politicians whose climate-policy preference di⁄ers from their own (e.g., from those of the
10Thesebasicinsightsrelyoncrowding-oute⁄ectsbutnotonwhetherleakageratesare(cid:147)high(cid:148)or(cid:147)low(cid:148).
11To illustrate, two countries(cid:146)true preferences may be to apply the global SCC to 100% and 46% of
projectsrespectively,thatis,((cid:18)i;(cid:18)j)=(1;163). Butif((cid:21)(cid:3)i;(cid:21)(cid:3)j)=(12;0),say,thenoptimalaltruisminvolves
using the SCC only in 50% (= 1 1) of projects for country i and not at all for country j (= 0 6 ).
2 (cid:2) (cid:2) 13
(The details underlying this numerical example are at the end of Section 6.)
4
median voter). Commitment can also be achieved by political or regulatory institutions(cid:151)
perhaps independent of government(cid:151)which adopt particular rules and practices.
The classic reference on such delegation is Schelling (1960), and the idea has been
applied widely to di⁄erent contexts such as bargaining (Segendor⁄1998), monetary policy
(Persson and Tabellini 1993), and the theory of the (cid:133)rm (Vickers 1985). It is fairly well-
known that an incentive to misrepresent preferences exists in virtually any game (Heifetz,
Shannon and Spiegel 2007)(cid:151)although this, in itself, says little about how preferences will
be distorted in a particular game.
Wedi⁄erfromthisliteratureinseveralrespects. Tobeginwith, weconsideradi⁄erent
class of game, and examine a setting in which agents are not driven by pure self-interest;
many of our themes thus have no analog in previous models.12 Moreover, our main
application to climate policy has at least two advantages compared to other delegation
applications. First, thereissigni(cid:133)cantempiricalevidencethatplayers(cid:146)e⁄ortsarestrategic
substitutes: A very large majority of work on unilateral climate policy (cid:133)nds that carbon
leakage rates are positive, as in our model.13 Second, climate policy is characterized by
something close to an (cid:147)informational level playing (cid:133)eld(cid:148)between countries: The climate-
change debate is highly public and global (based, in part, on scienti(cid:133)c evidence) and
countries(cid:146)abatement policies are commonly known (perhaps with a few exceptions), as is
whether or not they have adopted the SCC.14
Our analysis also shows that altruism can, at least in principle, neutralize the strategic
incentive to distort preferences which is emphasized by this literature. Our model always
features strategic substitutes, so contributing less induces a favourable response from the
other player; but if both players are fully altruistic, they recognize that such preference
distortion no longer yields any gain (as it induces a move away from (cid:133)rst-best). So an
incentive to distort play may exist in a standard game with sel(cid:133)sh players(cid:151)but not in an
otherwise identical game featuring social preferences.
Other applications. Questionsofaltruisticbehaviourariseinotherenvironmentalprob-
lems. For example, there is an ongoing debate about the motivations behind the Montreal
Protocol to reduce chloro(cid:135)uorocarbons (CFCs) which deplete the ozone layer. While
Barrett (1994) argues that the protocol was broadly consistent with the outcome of a
12Perhapsclosestto us,though in a ratherdi⁄erentsetup withoutaltruism,Roelfsema (2007)considers
amodelofimperfectcompetitionwithstrategictradepolicies,inwhichdelegationtoapoliticianwhocares
more about the environment than the median voter can be optimal because this induces other countries
to do the same. (This result relies on a particular form of competition in product markets.)
13Many empirical estimates are derived from numerical simulations of multi-sector, general equilibrium
models which focus on climate initiatives by OECD countries that result in carbon leakage to non-OECD
countries. These typically (cid:133)nd leakage in the range of 5(cid:150)40%, with many estimates below 20%. Leakage
estimates for individual sectors (such as the cement and steel industries in the EU Emissions Trading
Scheme)arefrequentlyhigher,e.g.,above50%,butalsorarelyexceed100%. SeeBabiker(2005),Copeland
and Taylor (2005), Ritz (2009), and the references cited therein.
By contrast, in delegation models on the theory of the (cid:133)rm, it is often di¢ cult to tell with con(cid:133)-
dencewhethercompetitionbetween(cid:133)rmsisinstrategicsubstitutes(Cournot)orinstrategiccomplements
(Bertrand)(cid:151)and many results are known to depend critically on this unobservable feature of the model.
14Inpractice,thereissigni(cid:133)cantuncertaintyoverthecostsandbene(cid:133)tsofCO abatement;thekeypoint
2
for us is that actions are easily observable and informational asymmetries between countries are small.
5
non-cooperative game, Sunstein (2007) notes that the US used a relatively low discount
rate in evaluating its commitment(cid:151)which might be interpreted as a form of altruism.
Thekeyfeaturesofourmodelaresharedbyotherproblemsofthecommons. In(cid:133)sheries
policy, forexample, thereisastrongtendencytowardsoverexploitation; individualplayers
have a suboptimal incentive to limit their catch (Stavins 2011) and catch reductions are
typically strategic substitutes (Levrahi and Mirman 1980), leading to a leakage problem
analogous to ours. Similarly, in a classic paper, Olson and Zeckhauser (1966) suggest
that small countries tend to free-ride on the defense investments of large countries, and
observe that countries(cid:146)military expenditures are often strategic substitutes. It is more
di¢ cult to pinpoint altruism empirically in these applications, partly because there is no
clear equivalent to the adoption of the SCC. However, it seems conceivable that individual
European countries, say, also care about the welfare of the EU as a whole when it comes
to policies a⁄ecting the environment or defense.15
Our results can also apply to problems from other domains that share public good
characteristics. For example, suppose family member j pursues some useful activity;
family member i derives indirect bene(cid:133)ts from the activity, and can help out at some cost.
If altruistic, i also cares about the bene(cid:133)ts accruing to j in choosing how much to help.
But the more i helps, the less j does himself(cid:151)the leakage problem. While i(cid:146)s help always
raises j(cid:146)s private payo⁄, it need not raise overall welfare or i(cid:146)s own altruistic objective.
Optimal altruism typically involves (cid:21) < 1, so i(cid:146)s help falls short of her true preference.16
(cid:3)i
To be concrete, a parent may want to help a child with its homework on 4 out of 5 days a
week ((cid:18) = 4, say), but realizes that, because of incentive e⁄ects, (cid:21) = 1, say, is optimal(cid:151)
i 5 (cid:3)i 2
and thus only helps twice a week. In practice, such a well-meaning but stern commitment
may be achieved by putting certain rules into place, or the parent may engage a tutor (or
sibling) twice a week and abstain from helping directly.17
Plan for the paper. Section 2 sets up our benchmark model. Section 3 examines the
impact of (cid:147)small(cid:148)altruistic commitments. Section 4 analyzes in detail players(cid:146)optimal
commitments, and Section 5 shows that our main results are robust in a variety of di-
rections. Section 6 points out some further properties of our model, with a focus on its
empiricalimplications. Finally, Section7discussesrecentclimatepolicyinitiativesinlight
ofourresults,ando⁄erssomesuggestionsforfutureresearch. (TheproofsareinAppendix
A, and the details of the robustness analysis are in Appendix B.)
15ArelatedapplicationistheproblemfacedbylargecharitiesliketheBill&MelindaGatesorRockefeller
foundations. It seems clear that the broad objective of such organizations is to enhance some measure of
global welfare. At the same time, there are well-known concerns that their contributions can (cid:147)crowd out(cid:148)
others,suchaslocalgovernments,theprivatesector,andsmallercharities. Thiscorrespondsquitedirectly
to the tension between altruism and leakage in our analysis.
16The (cid:147)rotten kid theorem(cid:148)(Becker 1974) does not apply in our model. It states that, under certain
conditions(Bergstrom 1989),an altruistichead whomakestransferstoself-interestedhouseholdmembers
induces the e¢ cient outcome(cid:151)despite limited altruism in the family overall. By contrast, our setup does
not feature a design with transfer payments (see also our concluding discussion in Section 7). (Recall also
that the rotten kid theorem itself can fail in public-good settings, despite transfers.)
17Ourmodelassumesthe tutorisoptimally chosen and incentivized by the parent(strictly speaking,at
zero cost) and has no special skills(cid:151)although this is clearly not essential for the results.
6
2 A model of altruism in public good provision
Setup of the model. Two players, i and j, contribute to the provision of a public good.
Player k (k = i;j) makes a contribution (e.g., shared investment, emissions reduction, or
(cid:147)e⁄ort(cid:148)) denoted by X , and derives bene(cid:133)ts B (X +X ) which depend on the aggregate
k k i j
e⁄ortbythetwoplayers. Themarginalbene(cid:133)tsatis(cid:133)esB ( ) > 0andB ( ) < 0. Thecost
k0 (cid:1) k00 (cid:1)
function C (X ) is player-speci(cid:133)c, with marginal cost satisfying C ( ) > 0 and C ( ) > 0.
k k k0 (cid:1) k00 (cid:1)
To guarantee an interior solution, assume C (0) = C (0) = 0 and B (X ) C (X ) < 0
k k0 k0 k (cid:0) k0 k
for some X < . De(cid:133)ne a player k(cid:146)s (cid:147)net bene(cid:133)t(cid:148)or (cid:147)national welfare(cid:148)as (cid:5) =
k k
1
B (X +X ) C (X ), and (cid:147)social surplus(cid:148)or (cid:147)global welfare(cid:148)as W = (cid:5) +(cid:5) .
k i j k k i j
(cid:0)
In our model, each player(cid:146)s preferences may be at least partly altruistic. In particular,
player k(cid:146)s true objective function is given by
S = (1 (cid:18) )(cid:5) +(cid:18) W, (1)
k k k k
(cid:0)
where the parameter (cid:18) [0;1] represents her true preference for altruism. Player k is
k
2
purely self-interested if (cid:18) = 0 (so S = (cid:5) ), and entirely altruistic if (cid:18) = 1 (so S = W),
k k k k k
in which case her preference re(cid:135)ects the full global bene(cid:133)t of contributions, B +B . More
i j
generally, a higher value of (cid:18) represents a (cid:147)more altruistic(cid:148)preference that gives more
k
weight to the other player(cid:146)s net bene(cid:133)t. For our application to climate policy, we can
interpret (cid:18) = 0 as an underlying preference for the (cid:147)business-as-usual(cid:148)(BAU) level of
k
emissions, while (cid:18) = 1 corresponds to a desire to incorporate the global (cid:147)social cost of
k
carbon(cid:148)(SCC) into decision-making.
We next introduce a modelling device in form of a strategic objective function:
(cid:10) = (1 (cid:21) )(cid:5) +(cid:21) S . (2)
k k k k k
(cid:0)
A strategic objective is a convex combination of a player(cid:146)s net bene(cid:133)t (cid:5) and her true
k
objective S , with a relative weight given by the strategic preference (cid:21) [0;(cid:18) 1]. If
k k 2 (cid:0)k
(cid:21) = 0, the strategic objective is entirely sel(cid:133)sh, so (cid:10) = (cid:5) (regardless of the underlying
k k k
true objective S ). If (cid:21) = 1, the player(cid:146)s strategic objective is identical to her true
k k
objective, so (cid:10) = S . We restrict attention to (cid:21) (cid:18) 1 (cid:21) (cid:18) 1 to focus on
k k k (cid:20) (cid:0)k , k k (cid:20)
the typical situation where each player contributes too little to the public good from a
social-welfare perspective(cid:151)rather than too much. (Whenever (cid:18) < 1, we do allow for
k
the possibility that (cid:21) > 1 so the strategic objective could place more weight on altruism
k
than the true objective(cid:151)although we will see that, in equilibrium, this does not occur.)
This ((cid:21) ;(cid:21) )-modelling device allows us to analyze the welfare impact of players following
i j
through on their altruistic preferences, and, building on this, to understand the extent to
which players optimally engage in altruistic behaviour.18
The timing of the model is as follows. At Date 0, each player is endowed with a
18Intheliteratureoninternationalenvironmentalagreements,countriestypicallymakeabinarydecision
onjoininganagreement((cid:147)inorout(cid:148)). Bycontrast,countriesherechoosetheintensityoftheircommitment.
7
bene(cid:133)t function and a cost function, B ( ) and C ( ), as well as with a true objective
k k
(cid:1) (cid:1)
S ( ) that re(cid:135)ects her degree of altruism, (cid:18) [0;1]. Then, at Date 1, each player chooses
k k
(cid:1) 2
her strategic preference (cid:21) [0;(cid:18) 1] to maximize her true objective S . Finally, at Date
k 2 (cid:0)k k
2, each player(cid:151)or her agent(cid:151)chooses e⁄ort according to the strategic objective function
(cid:10) . (For environmental applications, a country(cid:146)s choice of X is equivalent to choosing a
k k
domestic price on emissions.19)
We focus on the subgame-perfect Nash equilibrium of the game, and follow the dele-
gation literature in assuming that players(cid:146)strategic objective functions, (cid:10) and (cid:10) , form
i j
credible commitments.20 The plausibility of this assumption will, of course, vary de-
pending on the application in question. As explained in the introduction, we think that
commitment value is reasonably likely to obtain in the climate-policy context, given some-
thing close to an informational level playing (cid:133)eld between countries, as well as in other
public good problems.
Key properties of the model. Webeginbyestablishingthekeypropertiesofthemodel
at Date 2. For player i, say, the (cid:133)rst-order condition for its contribution is
@(cid:10) =@X = (B C )+(cid:21) (cid:18) B = 0. (3)
i i i0 i0 i i j0
(cid:0)
The (cid:147)(cid:133)rst-best(cid:148)benchmark is nested where (i) both players have entirely unsel(cid:133)sh true
preferences, (cid:18) = (cid:18) = 1, and (ii) both players choose their respective e⁄ort levels accord-
i j
ingly, (cid:21) = (cid:21) = 1. Inthiscase, playersatDate2makecontributiondecisionstomax W
i j Xk
(where k = i;j), thus each incorporating the full global bene(cid:133)t of their actions.
The(cid:133)rst-orderconditionalsode(cid:133)nesplayeri(cid:146)sbestresponsetoplayerj(cid:146)scontribution,
R (X ). The slope of this function is given by
i j
B +(cid:21) (cid:18) B
i00 i i j00
R (X ) = ( 1;0). (4)
i0 j ( B(cid:16) +C (cid:21) (cid:18)(cid:17)B ) 2 (cid:0)
(cid:0) i00 i00(cid:0) i i j00
A key property of the model is that players(cid:146)e⁄orts are strategic substitutes. This captures
a(cid:147)crowdingout(cid:148)e⁄ect: Ifoneplayerincreaseshere⁄ort, thisreducesthemarginalbene(cid:133)t
of e⁄ort for the other player, who therefore responds by cutting back. In the context of
climate policy, L [ R (X )] (0;1) is the marginal rate of (cid:147)carbon leakage(cid:148)(IPCC,
i (cid:17) (cid:0) j0 i 2
2007) resulting from country i(cid:146)s e⁄ort. Borrowing this terminology:
Lemma 1 The leakage rate due to player k(cid:146)s e⁄ort is given by L (0;1).
k
2
19To see this, imagine splitting country k(cid:146)s abatement decision at Date 2 into two parts. At Date 2b, a
representative, price-taking (cid:133)rm chooses emissions abatement X to maximize its pro(cid:133)ts p X C (X ),
k k k k k
(cid:0)
where pk is the domestic emissions price, such that pk =Ck0(Xk), in equilibrium. This de(cid:133)nes an upward-
sloping abatement supply curve with dXk=dpk = 1=Ck00(Xk) > 0. At Date 2a, policymakers choose the
domesticpricep tomaximizethestrategicobjective(cid:10) . Thissetupisexactlyequivalenttothebenchmark
k k
model since choosing the domestic emissions price is equivalent to choosing an abatement e⁄ort.
20This is essentially equivalent to assuming that players(cid:146)contributions are publicly observable, which,
inturn,correspondstoiknowingj(cid:146)s(cid:21) (cid:18) whenchoosinghercontributionpolicyatDate2(asbene(cid:133)tand
j j
cost functions are commonly known). (We do not require that i then knows j(cid:146)s true preference (cid:18) .)
j
8
Leakage rates quantify the severity of the crowding-out problem; they are positive but
less than 100%. This is a common feature of public good models across di⁄erent domains,
including environmental problems, military protection, (cid:133)sheries, and charitable giving.
We next con(cid:133)rm the intuition that more altruistic behaviour by a player leads to an
increase in her e⁄ort. (The result from Lemma 1 ensures that the equilibrium is unique,
stable, and exhibits well-behaved comparative statics.)
Lemma 2 If player k(cid:146)s true preference (cid:18) > 0, her e⁄ort satis(cid:133)es dX =d(cid:21) > 0.
k k(cid:3) k
A higher value of (cid:21) in(cid:135)ates the marginal return to public good contribution, which,
k
by stability, also increases its equilibrium level. So an increase (cid:21) , say, raises X (Lemma
i i(cid:3)
2) and also raises X +X , but not by as much (Lemma 1).
i(cid:3) j(cid:3)
To complete our preliminary discussion, we show that a player with an entirely sel(cid:133)sh
true preference, (cid:18) = 0, does not want to engage in a strategic commitment.
k
Lemma 3 If player k(cid:146)s true preference (cid:18) = 0, her optimal e⁄ort solves max (cid:5) .
k Xk k
As a notational convention, we refer to such an optimal commitment as (cid:21) = 0.21
(cid:3)k
3 The welfare impact of small altruistic commitments
To build intuition, we begin our analysis by considering (cid:147)small(cid:148)commitments. Suppose
that player i(cid:146)s true preference (cid:18) > 0 is altruistic at least to some extent (while (cid:18) 0),
i j
(cid:21)
and that initially both players act purely in their self-interest, i.e., (cid:21) = 0 for k = i;j.
k
What is the impact of a small commitment d(cid:21) > 0 by player i towards incorporating her
i
true altruistic preference in her public good contribution?
Proposition 1 The impact of a small unilateral commitment d(cid:21) > 0 by player i on her
i
equilibrium true objective
dS dX
i(cid:3) = (cid:18) B B L i(cid:3)
d(cid:21) i j0 (cid:0) i0 i d(cid:21)
i(cid:12)(cid:21)i=(cid:21)j=0 (cid:20) i (cid:21)(cid:21)i=(cid:21)j=0
(cid:12) (cid:0) (cid:1)
(cid:12)
is (a) positive if the ratio o(cid:12)f marginal bene(cid:133)ts satis(cid:133)es B B and her true preference
i0 (cid:20) j0
exceeds the leakage rate (cid:18) > L , and (b) negative for a ratio of marginal bene(cid:133)ts B =B
i i i0 j0
su¢ ciently large or for a true preference (cid:18) su¢ ciently small.
i
Whether a small altruistic commitment is bene(cid:133)cial for a player depends crucially on
the details of the environment. If she either derives a relatively large marginal bene(cid:133)t, or
her true preference contains only a small degree of altruism, it is never a good idea for
someone to make such a commitment. However, two simple conditions which are jointly
21This convention makes an altruistic player(cid:146)s strategic commitment directly comparable with a sel(cid:133)sh
player; note that, if (cid:18) >0, then the contribution solves max (cid:5) if and only if (cid:21) =0. (Recall that we
k Xk k k
are restricting attention to cases where (cid:21) 0.)
k
(cid:21)
9
su¢ cient for dS > 0 are that the player has a relatively low marginal bene(cid:133)t as well as
i(cid:3)
a true preference that exceeds the rate of leakage.
Theseresultscanbeunderstoodasfollows. Withaslightabuseofnotation,letdX > 0
i(cid:3)
denote the increase in i(cid:146)s e⁄ort due to its small unilateral commitment d(cid:21) > 0. (More
i
formally,dX = (dX =d(cid:21) ) d(cid:21) > 0byLemma2.) Duetothecrowding-oute⁄ect
i(cid:3) i(cid:3) i (cid:21)i=(cid:21)j=0 i
(Lemma 1), j adhjusts its e⁄ort by idXj(cid:3) = ((cid:0)Li)dXi(cid:3) < 0 in response. By the envelope
theorem, the direct e⁄ect of a small change in each player(cid:146)s e⁄ort on its own net bene(cid:133)t
is zero. The reason is that both players were initially choosing their respective e⁄orts
sel(cid:133)shly to maximize their own net bene(cid:133)t, so any (small) change their own contribution
only has a second-order e⁄ect. However, the unilateral commitment by i also has two
strategic e⁄ects, one positive and one negative. First, the increase in i(cid:146)s e⁄ort yields an
increase in the bene(cid:133)ts enjoyed by the other player j of B dX > 0. Second, the induced
j0 i(cid:3)
reductioninj(cid:146)se⁄ortmeansthati(cid:146)sbene(cid:133)tchangesbyB dX = ( B L )dX < 0. Player
i0 j(cid:3) (cid:0) i0 i i(cid:3)
i(cid:146)s true objective S = (cid:5) +(cid:18) (cid:5) places weight (cid:18) [0;1] on the (cid:133)rst (positive) strategic
i i i j i
2
e⁄ect and full weight on the second (negative) strategic e⁄ect. The weighted sum of these
e⁄ects, ((cid:18) B B L )dX , thus determines the impact of a small unilateral commitment
i j0 (cid:0) i0 i i(cid:3)
on its own true objective function and behaves according to Proposition 1.
Intuitively, a unilateral commitment by i increases the net bene(cid:133)t (cid:5) enjoyed by j
(cid:3)j
but acting unsel(cid:133)shly hurts its own net bene(cid:133)t (cid:5) . The commitment thus enhances its
(cid:3)i
own true objective if (and only if) the former e⁄ect outweighs the latter. The positive
e⁄ect will be large if j(cid:146)s marginal bene(cid:133)t is large, and receives large weight according to
i(cid:146)s degree of altruism, (cid:18) . The negative e⁄ect will be small if there is little leakage, and if
i
i(cid:146)s own marginal bene(cid:133)t is small.
We can also address when a small altruistic commitment improves global welfare:
Proposition 2 The impact of a small unilateral commitment d(cid:21) > 0 by player i on
i
equilibrium global welfare
dW dX
(cid:3) = B B L i(cid:3)
d(cid:21) j0 (cid:0) i0 i d(cid:21)
i (cid:12)(cid:21)i=(cid:21)j=0 (cid:20) i (cid:21)(cid:21)i=(cid:21)j=0
(cid:12) (cid:0) (cid:1)
(cid:12)
is (a) positive if the ratio of(cid:12) marginal bene(cid:133)ts satis(cid:133)es B B , and (b) negative for a
i0 (cid:20) j0
ratio of marginal bene(cid:133)ts B =B su¢ ciently large.
i0 j0
The logic underlying Proposition 2 follows that of Proposition 1. Again, the direct
e⁄ects on each player(cid:146)s net bene(cid:133)t are both zero by the envelope theorem. The only
di⁄erence arises because, from a global-welfare perspective, the combined e⁄ect of the
two strategic e⁄ects depends on their unweighted sum. So the increase in the bene(cid:133)ts
enjoyed by the other player j of (B )dX > 0 plus the induced reduction in i(cid:146)s bene(cid:133)t
j0 i(cid:3)
of [B ( L )]dX < 0 yield an overall welfare impact dW = (B B L )dX . The sign
i0 (cid:0) i i(cid:3) (cid:3) j0 (cid:0) i0 i i(cid:3)
of this expression, too, is ambiguous. However, note that Proposition 2(a) implies that a
small commitment must be global welfare-enhancing for at least one of the two players.
10