Multiple equilibria and optimal public policy in charitable giving

Transcription

1 Multiple equilibria and optimal public policy in charitable giving Sanjit Dhami Ali al-nowaihi 24 October 2013 Abstract Charitable giving is an important economic activity that is of critical importance to many areas of human life. The vast majority of charitable donations are made by a large-number of dispersed and small donors whose behavior, we argue, is best understood in a competitive equilibrium approach to giving. Multiple equilibria naturally arise in such settings; we show that strong aggregate complementarity is a necessary condition for multiple equilibria. We clarify the role of public policy in moving the economy from a low equilibrium in giving to a high equilibrium in giving using subsidies and direct grants to charity financed by taxation. Using a welfare analysis, we also address the question of the optimal public policy towards charitable giving. Keywords: Charitable giving; multiple equilibria; strong aggregate complementarity; optimal mix of public and private giving; subsidies and direct grants; redistribution; privately supplied public goods. JEL Classification: D6, H2, H4. Corresponding author. Department of Economics, University of Leicester, University Road, Leicester, LE1 7RH. Tel: sd106@le.ac.uk. Department of Economics, University of Leicester, University Road, Leicester. LE1 7RH, UK. Phone: Fax: aa10@le.ac.uk. 1

2 1 Introduction Charitable donation is a significant economic activity. For instance, in 2003 for the USA, 89% of all households gave to charity with the average annual gift being $1620, which gives an aggregate total of about $100 billion; see, Mayr et al. (2009). A recent World Giving Index published by the Charities Aid foundation used Gallup surveys of 195,000 people in 153 nations. It found that more than 70% of the population gave money to charity in Australia, Ireland, United Kingdom, Switzerland, Netherlands, Malta, Hong Kong, Thailand. 1 The following five stylized facts, S1-S5, are particularly relevant for our analysis. S1 There is substantial heterogeneity in giving between countries. 2 S2 Individual private donors are the largest contributors. 3 S3 Government direct grants are significant in terms of magnitude. 4 S4 Contributions to charity are typically tax deductible. For instance, the rate of charitable deductions is 50% in the US and 17-29% for Canada. S5 Direct grants in the form of seed money or leadership contributions (that precede private giving) made by governments, foundations, the national lottery (as in the UK) or exceptionally rich individuals are effi cacious in eliciting further contributions Behavioral assumptions about small donors In this paper we are interested only in the behavior of small donors to charity (stylized fact S2). The vast bulk of charitable donations are quite modest and made by a very large number of dispersed private donors. A recent report by the Charities Aid Foundation (2012) for UK data is illustrative of the pattern of giving. The median amount given by donors is 10 per month. About 40% of the total donors who donate under 10 donate the median amount of 4 per month. Another 30% of the population who donates between 1 See for the index. 2 As a percentage of GDP, for , non-religious philanthropic activity was in excess of 4% for the Netherlands and Sweden; 3-4% for Norway and Tanzania; 2-3% for France, UK, USA; and less than 0.5% for India, Brazil, and Poland; see Salamon et al. (2004). 3 For US data, for 2002, individuals accounted for 76.3% of the total charitable contributions. Other givers are: Foundations (11.2%), bequests (7.5%), corporations (5.1%); see Andreoni (2006). 4 For non-us data, governments are typically the single most important contributors to charities. On average, in the developed countries, charities receive close to half their total budget directly as grants from the government, while the average for developing countries is about 21.6%. See the Johns Hopkins Comparative Nonprofit Sector Project ( 5 See, for instance, Karlan and List (2007), Potters et al. (2007), and Rondeau and List (2008). 1

3 per month donate the median amount of about 14. Only about 7% of the population makes gifts of 100 or over per month. How do these individuals make choices about their small private donations, conditional on the choices made by the government (stylized fact S3) and other large donors (stylized fact S5)? There are the following two possibilities. 1. Strategic approach: The formal literature in philanthropy assumes that donors choose their own giving strategically so as to maximize own utility conditional on the vector of donations of all other givers. The relevant solution concept is a Nash equilibrium. This strategic approach may apply well to very large donors. 6 However, it does not appear plausible when applied to a typical small donor who donates, say 5, to the Red Cross. Indeed, there is no direct or strong evidence that small donors who provide the bulk of contributions behave in accordance with the strategic approach as presupposed in, say, the epistemic foundations of a Nash equilibrium. 2. Competitive approach: In this paper we advance the competitive approach as a means of understanding the behavior of small contributors. As far as we know, this approach is new to the philanthropy literature, but one that, we believe, gives a more compelling account of the behavior of small donors. In this view, the contribution of the i th donor, g i, is individually too small to alter the aggregate amount of charitable giving, G, which the donor takes as given. 7 In a competitive equilibrium, the original beliefs about market aggregates are fulfilled in the sense that the total of individual contributions to charity produce the same aggregate level of contributions, G, that were originally assumed. This is the approach we take. 1.2 Multiple equilibria in the competitive approach The strategic approach has typically focussed on a unique strategic equilibrium in giving, thereby ruling out multiple equilibria by assumption. 8 In contrast, multiple equilibria arise naturally in a competitive approach to charitable contributions. Suppose that for any individual i, the marginal utility of contributing an extra unit, g i, is increasing in the level of aggregate giving, G. 9 Now, if donors believe that G will be high, then they are 6 Unlike small donors, large donors may behave in a strategic manner (say, to establish green credentials) but their share in the total contributions is small (5.1% for the US). 7 In competitive markets, consumers cannot influence the market price, which they take as given. Yet their own actions, through their market demands, determine a market price that, in equilibrium, confirms their orginally believed market price. Our approach here is similar. 8 There are some exceptions. For instance, Andreoni (1998) considered mulitple equilibria in a strategic approach that arise from non-convexities in production. This approach and the reasons for multiple equilibria are unrelated to our paper; see Dhami and al-nowaihi (2012) for more details. 9 There are many possible reasons why this might be so. For instance, a larger charity may be a signal that it is reputable, viable and well-regarded by others. See our examples in sections 3.4 and

4 also induced to contribute a high amounts, g i. Ex post, aggregate giving would be high, confirming the donor initial expectations. Similarly, if donors begin with initial beliefs that G will be low, they contribute a low amount, confirming their initial expectations of a smaller charity. This gives rise to multiple competitive equilibria. In some equilibria, aggregate giving is high (H), while in other equilibria, aggregate giving is low (L). Cooper and John (1988) show that for the case of private goods, strategic complementarity is a necessary condition for multiple equilibria. On the other hand, in the case of competitive charitable contributions, which have the nature of a public good, we show that strong aggregate complementarity, a new concept that is a modification of strategic complementarity, is a necessary condition for multiple equilibria Engineering moves between multiple equilibria Suppose that we have two equilibria, as discussed above: A high equilibrium, H, and a low equilibrium, L. How could public policy engineer a move from L to H, if such a move is considered desirable? There is no guarantee that subsidies provided to individual charitable contributions (stylized fact S4) at the equilibrium L will induce more charitable contributions (normal comparative statics). Indeed, as is often the case in models with multiple equilibria, one may have perverse comparative statics at L, that reduce aggregate contributions in response to subsidies to individual giving. Furthermore, an equilibrium with perverse comparative statics may be stable (see the discussion in section 6 below). We show that when perverse comparative statics obtain at L, direct government grants (stylized fact S3), which we denote by D, may engineer a move to H. 11 Suppose that D exceeds the total charitable contributions at L. This leaves H as the only other candidate equilibrium. Once the economy settles at H, one may consider the optimal public policy at H by appropriate welfare analysis. In particular, the comparative statics at H may be normal so that one may wish to phase out the direct government grant, D, replacing it by subsidies to induce the equivalent private amount of charitable contributions. The reason that such a policy is welfare improving in our model is that private giving to charities provides warm glow to the donors that direct government grants do not. The applicability of these ideas is illustrated by examples where voluntary giving contributes towards public redistribution or towards public goods. 10 A useful analogy is the difference between the effi ciency conditions for the optimal provision of private goods under competitive conditions (marginal willingness to pay equals marginal cost) and the effi ciency condition for optimal provision of public goods (sum of marginal willingness to pay equals marginal cost). 11 These results also provide another explanation for the effectiveness of seed money or leadership donations (stylized fact S5). Seed money, leadership contributions and national lottery money in the UK play a role similar to the direct grant, D, in our framework. 3

5 1.4 Results We make six main contributions in this paper. (1) Our proposed condition strong aggregate complementarity is a necessary condition for multiple equilibria in a competitive equilibrium in charitable contributions. (2) Multiple equilibria provide a possible explanation of heterogeneity in charitable giving. (3) Using temporary direct grants, a policy maker can engineer a move from the low (L) to the high (H) equilibrium. This is particularly desirable when comparative statics at the low equilibrium are perverse and those at the high equilibrium are normal. (4) When comparative statics at the low equilibrium are normal and those at the high equilibrium are perverse, the government can do better than simply encouraging subsidy-induced giving at the low equilibrium. Indeed, once the government successfully engineers a move to the high equilibrium using temporary direct grants, the perverse comparative statics at the high equilibrium ensure that a reduction in subsides will induce even greater private giving. (5) By carrying out a welfare analysis, we give conditions that specify the optimal mix of public contributions and private contributions to charity. (6) We show that our results are equally applicable to redistributive and public goods contexts. Section 2 formulates the theoretical model. Section 3 derives the equilibria of the model and their comparative static results. This section also gives two illustrative examples: Example 1 in section 3.4 is about voluntary private contributions to redistribution and Example 2 in section 3.5 is about public good provision. Section 4 examines multiple equilibria in aggregate giving in more detail. Section 5 performs a welfare analysis and characterizes the normatively optimal public policy; it also solves for the optimal public policy for Examples 1, 2 of sections 3.4, 3.5 in sections 5.1 and 5.2 respectively. Section 6 discusses dynamic issues. Section 7 concludes. Except for the proof of Proposition 5, all proofs are in the appendix. 2 Formal model There are three types of players in the economy, consumer-donors, a fiscal authority or government, and a single charity (or several identical charities in which case we may speak of a representative charity). For pedagogical clarity, we make several simplifying assumptions. (1) Our interest is in the behavior of small donors and the government (stylized facts S2, S3), so we focus on the supply side of charitable donations and keep the demand side very simple. Thus, charities are passive players in our model, while donors and the government are the active players. 12 (2) In our model, charities can provide 12 In actual practice charities are often active players that behave strategically to attract donations; see Andreoni (2006). 4

6 redistribution and/or public good provision so we deliberately abstract from the role of government redistribution and public good provision. 13 We now describe the model in more detail. There is a continuum of consumer-donors (henceforth, consumers) located on the unit interval, [0, 1], each consumer is identified by her location, x [0, 1] and, hence, we shall simply refer to her as consumer x. The utility function of consumer x is u (c, g, G, x), (1) where c is private consumption expenditure of individual x, g 0 is her contribution to charity, it reflects the warm glow or prestige (also known as impure altruism) from own contribution 14 and G 0 is the aggregate (which is also the average) level of giving to charity. Remark 1 (Warm glow, taxes and private giving): There is overwhelming experimental and field evidence and also growing neuroeconomic evidence that justifies warm glow from acts of giving. 15 In our model, direct private giving to charities yields a warm glow while paying taxes does not because the former is voluntary while paying taxes is not. Also, empirically, people get warm glow from giving but, generally, resent paying taxes. Further, in actual practice, when an individual makes a tax payment he does not know what fraction will be used by the government towards support of his chosen charity (stylized fact S3). The consumption levels across consumers, c(x), x [0, 1] are bounded below by a continuous function c (x) 0 that denotes a subsistence level of consumption. Let m (x) 0 be the exogenously given income of consumer x. Aggregate income is M = Since aggregate giving cannot exceed total income, we must have m(x)dx <. (2) G M. (3) 13 This is not necessarily restrictive as we could assume that a part of the government revenues are used for redistribution/public good provision. In actual practice, charities often carve a niche for themselves in areas where government provision of public goods or redistribution is not adequate. 14 The introduction of a warm glow motive was suggested by Cornes and Sandler (1984) and Andreoni (1989, 1990). The presence of a warm glow term reflects the fact that individuals no longer consider their contributions to be perfect substitutes for the contributions of others. Hence, there is extra utility from one s own contribution, which mitigates the free rider problem arising from purely altruistic considerations, i.e., from a utility function of the form u (c, G). It also obviously implies that government grants to charities do not completely crowd out private donations because the two are imperfect substitutes from the point of view of givers. 15 For the evidence on altruism, see Andreoni (2006). For experimental evidence on warm glow preferences, see Andreoni (1993, 2006), Palfrey and Prisbrey (1997), Andreoni and Miller (2002) and Crumpler and Grossman (2008). For a survey of the neuroeconomic evidence that supports the warm glow motive see Mayr et al. (2009). 5

7 The government levies a tax on income at the rate t [0, 1], hence, income tax revenue equal t mdx = tm. The tax revenues are used to subsidize private donations to charity at the rate s [0, 1] and to finance direct grants to charities, D 0. Therefore, the (balanced) government budget constraint is tm = D + s gdx; s, t [0, 1], D 0. (4) The social welfare function of the government is introduced in Section 5. The charity collects all donations from private consumers, gdx, and the direct grant, D, from the government. Hence, the aggregate level of giving to the charity is G = D + gdx. (5) The charity uses G to perform two traditional functions of charities: Finance transfers to individuals and finance provision of public goods. Consumer x receives an amount τ (x) G from the charity. The balance (if any), G [ 1 τgdx = G 1 ] 1 τdx is used by the charity to finance provision of public goods. We assume that τ (x) 0 and τdx 1. (6) For the reasons mentioned above, the charity is a passive player in the game. Thus we take τ to be exogenous. Remark 2 (A special case): Suppose that τ (x) = 0 for all x. Then in our model charities engage only in the provision of privately provided public goods, financed by voluntary contributions. Indeed, over human history many important public goods have been provided in this manner at some point in time or the other. 16 In Example 1 of section 3.4, below, the charity uses its entire budget for income redistribution, so τdx = 1. Example 2 in section 3.5, below, considers the case τ (x) = 0 for all x so that the entire budget, G, finances the provision of public goods. The budget constraint of consumer x is given by c(x) + (1 s) g(x) (1 t) m (x) + τ (x) G. (7) The LHS of (7) is total expenditure which is made up of private consumption, c(x), whose price is normalized to unity, plus the (net of subsidy) private charitable giving. The RHS of (7) is total income which consists of the after-tax income, (1 t) m (x), plus the individual-specific transfer, τ (x) G, received from the charity. 16 Examples of such public goods include, defence, lighthouses, information on marine navigation, shipping intelligence, mail services, education, public works, support for the poor and care for the sick. 6

8 2.1 Restrictions on consumer preferences Denote partial derivatives by subscripts (u 1 = u, u c 2 = u, u g 3 = u, u G 12 = 2 u... c g etc.). We introduce the following technical assumptions that guarantee the existence of an interior solution. 17 u (c, g, G, x) is a C 2 function of c, g, G for g > 0 and c > c (x). The condition u 1 > 0, u 11 0, u 2 0, u 22 0, u 3 0 holds for all consumers and u 2 > 0 for a set of consumers of positive measure. For all s [0, 1), (1 s) 2 u 11 2 (1 s) u 12 + u 22 < 0. (8) The marginal utility of consumption tends to infinity as consumption tends to its lower bound from above: u 1 as c c (x). (9) Either u is extended to the boundary, g = 0, as a C 1 function, or (10) u 2 as g 0. (11) (11) allows us to rule out the uninteresting case where no consumer wishes to donate. Aggregate income is suffi cient to cover subsistence consumption, For each consumer, x [0, 1], c (x) < m (x) M y=0 c (x) dx < M. (12) [ ] c (y) dy + τ (x) M c (y) dy. (13) y=0 The above assumptions are satisfied by the Example 1 in section 3.4 and Example 2 in section 3.5. In particular, (10) holds for Example 1 and (11) holds for Example 2. By (2), aggregate income, M, is finite but, by (12), more than adequate to cover aggregate subsistence consumption. (13) then guarantees that a tax rate exists that redistributes income so that each consumer can afford a consumption level greater than her subsistence level, c (x), see Lemma (1) below. Remark 3 : If s = 1, then the budget constraint (7) of consumer x reduces to c (1 t) m (x) + τ (x) G. If u 2 > 0 (and we have assumed this to be the case for a set of consumers of positive measure) then g would become a free good so it follows that G =. This is, clearly, not feasible. Hence, s < 1. Thus warm glow is never a free good. 17 For the measure theoretic assumptions that are needed to make our analysis completely rigorous, see the working paper version, Dhami and al-nowaihi (2012). These measure theoretic assumptions are standard in the literature. 7

9 2.2 Sequence of moves The charity moves first to announce the transfer that it will make to each individual in the economy, τ (x) 0, x [0, 1]. These transfers are exogenously determined by the charity but satisfy (6). The government moves next to announce the policy parameters s, t and D. 18 Finally, the consumers move simultaneously. Consumer x chooses c(x) and g(x) so as to maximize her utility, u (c, g, G, x), subject to her budget constraint (7). Since each consumer is of measure zero, she takes G, s, t, m (x) and τ (x) as given. An equilibrium, G, is a value of G that equates supply and demand for charitable giving. The solution is given by backward induction. 2.3 Some preliminary and intermediate results This section collects some intermediate results that are used later in the paper. Lemma 1, below, shows that by a feasible choice of s and t, the government can ensure that each consumer has a level of after-tax income that allows for the consumption of the subsistence level of consumption, c (x). Lemma 1 : Consider s = 0, t = c (y) dy, D = tm then 0 < t 1 and for M y=0 each consumer, x [0, 1], c (x) < (1 t) m (x) + τ (x) G. Since u 1 > 0, the budget constraint (7) holds with equality. Hence, we can use it to eliminate c from (1). Letting U (g, G, s, t, x) be the result, we have U (g, G, s, t, x) = u ((1 t) m (x) + τ (x) G (1 s) g, g, G, x). (14) From (8) and (14) it follows that U is concave in own giving, U 11 < 0. (15) Consumer x s objective is given by 19 g arg max U (g, G, s, t, x), subject to 0 g 1 [(1 t) m (x) + τ (x) G c (x)], 1 s given s, t, G, m (x), τ (x), c (x). The constraint follows from (7) and from the assumption that g, c are bounded below by zero and c (x), respectively. If c (x) = (1 t) m (x) + τ (x) G then the maximization problem (16) has the unique solution, g = 0. But if c (x) > (1 t) m (x) + τ (x) G then it has no solution. In this case we set g = 0 and U =. The more interesting case, c (x) < (1 t) m (x) + τ (x) G, is considered in Proposition We shall see below that announcing D can help coordinate private expectations of the level of G. 19 Recall from Remark 3 that s < 1. (16) 8

10 Proposition 2 : Suppose c (x) < (1 t) m (x) + τ (x) G. (a) Consumer x s maximization problem (16) has a unique solution, g (G, s, t, x), (b) g is a continuous function of G, s, t, x, (c) 0 g (G, s, t, x) < 1 [(1 t) m (x) + τ (x) G c (x)], 1 s (d) 0 tm + (1 s) g dx < 2M, (e) g is an integrable function of x, (f) If, in addition, (11) holds, then g (G, s, t, x) > 0, In Lemma 3 we now consider the comparative static effects of s, t, G on g. Lemma 3 : Suppose that g (G 0, s 0, t 0, x 0 ) > 0. Let c (G, s, t, x) = (1 t) m (x) + τ (x) G (1 s) g (G, s, t, x). Then, at G 0, s 0, t 0, x 0, g (G 0, s 0, t 0, x 0 ), c (G 0, s 0, t 0, x 0 ): (a) U 1 = 0, (b) (1 s) u 1 = u 2, (c) g = U 12 G U 11 = u 23 (1 s)u 13 +τ[u 12 (1 s)u 11 ], 2(1 s)u 12 (1 s) 2 u 11 u 22 (d) g = U 13 s U 11 = u 1+[u 12 (1 s)u 11 ]g (e) g t = U 14 U 11 =, 2(1 s)u 12 (1 s) 2 u 11 u 22 m[(1 s)u 11 u 12 ]. 2(1 s)u 12 u 22 (1 s) 2 u Complements and Substitutes The traditional concepts of strategic complements and strategic substitutes do not presuppose strategic interaction among players, rather these are conditions on the cross partial derivatives of the utility function. Definition 1 (Bulow et al., 1985): g and G are strategic complements (substitutes) at s 0, t 0 if 2 U g G = U 12 > 0 (< 0) at s 0, t 0 for all g, G, x. Let g and G be strategic complements at s 0, t 0. From Definition 1 and Lemma 3c we then get that for all x, g > 0. Definition 2, below, introduces a new concept that is G fundamental to our paper. Definition 2 (Strong aggregate complementarity): g and G are strong aggregate complements at s 0, t 0, G 0 if g dx 1 G 1 s 0. Strategic complementarity (Definition 1) for all individuals gives g dx > 0. G Clearly, this is not suffi cient for strong aggregate complementarity (Definition 2). Furthermore, strategic complementarity ( g G > 0 for all x) is not necessary for strong aggregate complementarity either. This is because, in Definition 2, we require that g 1 G 1 s 0 on average, not at each x. 20 holds 20 Strategic complementarity, strictly increasing differences, single crossing and supermodularity are increasingly more general concepts, but they all coincide for our model. 9

11 3 Equilibrium giving and public policy 3.1 The aggregate desire to give to charity 21 When consumer x makes her charity decision, she has some original beliefs about the aggregate donations to charity, G, that determines her optimal charitable contributions, g (G, s, t, x). The aggregate of all desired public contributions, in the form of direct government grants, D, and private donations, D + g (G, s, t, x) dx, need not equal the originally believed aggregate contributions, G. Therefore, we introduce a new function, F, which represents the aggregate of all desires (public and private) to give to charity, F = D + g (G, s, t, x) dx. (17) Remark 4 : The situation described above is analogous to competitive markets. Conditional on some originally believed price vector, p, consumers form individual demands that lead to an aggregate demand. However, the aggregate demand need not equal actual aggregate supply except at a competitive equilibrium price vector, p. The government budget constraint, (4), can be written as D (s, t, G) = tm s g (G, s, t, x) dx; s, t [0, 1], D 0, (18) i.e., any tax revenues that are not spent by the government to subsidize private charitable giving can be used to make direct grants to charities. From (17) and (18) we can substitute out D to get F (s, t, G) = tm + (1 s) g (G, s, t, x) dx. (19) Remark 5 : In (19) we have eliminated D and we will focus on the remaining two instruments of government policy, s, t. Therefore, the government budget constraint will always hold in the analysis below, and s, t [0, 1] and D 0 in (18). From (19) and Proposition 2c,0 a simple calculation shows that 22 0 F (s, t, G) < 2M. (20) For given s and t, G F (s, t, G) is a mapping from [0, 2M] to [0, 2M]. The above discussion suggests the following definition. 21 Our solution method has some similarities with the techniques developed in Cornes and Hartley (2007). 22 We use here the fact that the lower bound on c is zero, the upper bound on G is M (from (3)) and the upper bound on the sum of transfers made by charities is one (from (6)). 10

12 Definition 3 : The aggregate desire to give is defined by the mapping, F (s, t, G), From Definition 3, we get that F (s, t, G) = tm + (1 s) g (G, s, t, x) dx. F G (s, t, G) = F (s, t, G) G = (1 s) g (G, s, t, x) dx. (21) G Using Definition 2 and (21) we immediately get a useful alternative characterization of strong aggregate complements in Lemma 4. Lemma 4 : g and G are strong aggregate complements at s 0, t 0, G 0 if, and only if, F G (s 0, t 0, G 0 ) 1. (22) From Lemma 4, the strong aggregate complements property simply requires that for every unit increase in G that the private sector believes will take place, the aggregate desire to give (public and private) should rise more than one for one. 3.2 Equilibria We now define equilibrium in charitable giving that can be motivated by Remark 4. Definition 4 (Equilibrium in giving): The economy is in an equilibrium if the aggregate of all desires to donate to charity, F, equals the aggregate of all donations, G, i.e., G [0, 2M] is an equilibrium if it is a fixed point of the equation G = F (s, t, G ), (23) since F incorporates (18), at any such G, the government budget constraint holds. Definition 5 (Isolated equilibrium): An equilibrium, G, is isolated if there is a neighborhood of G in which it is the only equilibrium. Proposition 5 : (a) An equilibrium, G [0, 2M], exists and satisfies 0 G < 2M. (b) If F G < 1 for all G [0, 2M] or if F G > 1 for all G [0, 2M], then an equilibrium, G, is unique. (c) If [F G ] G 1, then G is an isolated equilibrium. 11

13 Figure 1: Unique and multiple equilibria in charitable giving. Proof of Proposition 5: Let H (s, t, G) = G F (s, t, G). (a) From (23), at an equilibrium, G, G = F (s, t, G ). (i) F (s, t, 0) < 0 is not feasible because D 0 and g 0 for all x. (ii) If F (s, t, 0) = 0 then, clearly, 0 is an equilibrium. (iii) Now, suppose that F (s, t, 0) > 0. Since F (s, t, 0) > 0, it follows that H (s, t, 0) = F (s, t, 0) < 0. From (20), F (s, t, G) < 2M, and G [0, 2M], so, in particular, F (s, t, 2M) < 2M. Hence, H (s, t, 2M) = 2M F (s, t, 2M) > 0. Since H (s, t, G) is continuous, it follows that H (s, t, G ) = 0, for some G [0, 2M). Hence G is an equilibrium and 0 G < 2M. (b) Since H is continuous, the set of equilibria {G : H (s, t, G) = 0} is a closed subset of [0, 2M] and, hence, is compact. By (a), it is not empty. Hence, it must contain minimum and maximum elements, G min and G max, respectively, G min G max, H (s, t, G min) = 0, H (s, t, G max) = 0. If G min < G max then, since H G is continuous, it follows that H G = 0 for some G (G min, G max). Thus, F G = 1 for some G (G min, G max), which cannot be the case because of the restriction F G < 1 or F G > 1 in Proposition 5(b). Hence, G min = G max, which implies that the equilibrium is unique. (c) Suppose [F G ] G 1. Then [H G ] G 0. Hence, either [H G ] G < 0 or [H G ] G > 0. Since H G is continuous, it follows that H G < 0 (or H G > 0) in some neighborhood of G. Thus, using an argument similar to that deployed in (b), G can be shown to be an isolated equilibrium.. Figure 1 illustrates the results in Proposition 5. Three possible shapes of the function F (s, t, G) are shown. Along the curve AED, the suffi cient condition for uniqueness, F G < 1 for all G [0, 2M], holds, and we have a unique equilibrium at E. Along the two curves, ABCD and AHD (which appears as a dotted line), we observe values of G for which F G < 1 and other values of G for which F G > 1. In this case, since Proposition 5b gives suffi cient, 12

14 but note necessary, conditions, we could have a unique equilibrium (for AHD) or multiple equilibria (for ABCD). All equilibria shown in Figure 1 satisfy [F G ] G 1, hence, they are all isolated equilibria. 3.3 Equilibrium analysis: Normal, neutral and perverse comparative statics We now investigate how equilibrium aggregate giving, G, responds to the policy instruments, s, t at an isolated equilibrium where F G 1 (see Proposition 5c). Using the implicit function theorem, we can then regard G as a C 1 function, G (s, t), of s and t in that neighborhood. Let G s, G t be the partial derivatives of G with respect to s and t respectively. We begin with a useful technical result. Lemma 6 : ( g + ) g s G G s dx = 1 1 s ( G s + g dx Proposition 7 : Let G be an equilibrium at which F G (s, t, G ) 1. Then G is isolated and (a) G s (s, t) = Fs 1 F G, (b) G t (s, t) = ). Ft 1 F G, (c) G tt (s, t) = (Ftt+F tgg t )(1 F G)+F t(f tg +F GG G t ). (1 F G ) 2 We now define the critical concepts of normal, neutral and perverse comparative statics, which have to do with the response of G to changes in the subsidy, s. Definition 6 (Normal, neutral and perverse incentives): Comparative statics are normal if G s > 0, neutral if G s = 0 and perverse if G s < 0. From Definition 6, comparative statics are perverse when an increase in subsidy reduces total giving in equilibrium. Proposition 7 and Definition 6 immediately imply Corollary 8, below. Corollary 8 : Let G be an equilibrium at which F G (s, t, G ) 1. (a) Comparative statics are normal if (i) F s > 0 and F G < 1 at G or if (ii) F s < 0 and F G > 1 at G. (b) Comparative statics are neutral if F s = 0 at G. (c) Comparative statics are perverse if (i) F s > 0 and F G > 1 at G or if (ii) F s < 0 and F G < 1 at G. Table 1 provides a quick reference to the possible comparative static results that are obvious from Proposition 7a,b, and Definition 6. We assume that G is an equilibrium at which F G (s, t, G ) 1 and all partial derivatives are evaluated at G. Figure 2 illustrates the two cases of normal and perverse comparative statics in Table 1 when F s > 0 for all values of G and s 1 < s 2. 13

15 F s > 0 F s < 0 F t > 0 F t < 0 F G < 1 F G > 1 G s > 0 (normal) G s < 0 (perverse) G s < 0 (perverse) G s > 0 (normal) G t > 0 G t < 0 G t < 0 G t > 0 Table 1: Summary of the various comparative static results Figure 2: Normal and perverse comparative static results. From Remark 5 recall that D has been substituted out but the government budget constraint holds. Thus, the higher subsidies could be financed by extra taxation, say, with the tax rate increasing from t 1 to t 2 or simply by reducing direct government grants, D, or a combination of both. In Figure 2 we have assumed that taxes increase from t 1 to t 2 but the net impact of an increase in s, t on F is positive. In Figure 2, the 45 o line, F = G, is shown as the dark line. The case 0 < F G < 1 is illustrated by the two, thin, continuous, straight lines, while the other case, F G > 1 is shown by the two dashed lines. Thus, in the latter case, and from Lemma 4, g and G are strong aggregate complements. A. Normal comparative statics (G s > 0; cell 1,1 in Table 1): The upward shift of the continuous, light, curve F (s 1, t 1, G) to F (s 2, t 2, G) in Figure 2 illustrates the case F s > 0, 0 < F G < 1. The equilibrium moves from A to B. Hence, G s > 0. B. Perverse comparative statics (G s < 0; cell 2,1 in Table 1). This case is shown in Figure 2 by an upward shift of the dashed curve F (s 1, t 1, G) to F (s 2, t 2, G), which assumes F s > 0, F G > 1. The equilibrium moves from B to A. Equilibrium aggregate contributions, G, decrease as the price of giving reduces (larger s). Because F G > 1, consumers over-react to an increase in G (see Lemma 4). Thus, paradoxically, G needs to fall in order to restore equilibrium in the market for charity. The tax reforms of the 1980 s in the US increased the price of charitable giving by reducing subsidies, yet charitable contributions continued to rise in the following years; see, Clotfelter (1990). The following lemma will be useful in welfare analysis, in section 5 below. 14

16 s Lemma 9 : = D Figure 3: A representation of Example 1. 1 s. sg s + g dx 3.4 Example 1: Charitable contributions as public redistribution We consider an economy where some consumers have no income. Their consumption expenditure is financed entirely by either charitable donations, g, made by other caring consumers with positive income and/or by tax-financed direct government grants, D (which now have the interpretation of social welfare payments). In this example we will show that we have two equilibria in aggregate giving, G < G +. The structure of the example is illustrated in Figure 3. Formally, the assumptions are as follows. 1. m (x) > 0 for 0 x p < 1, m (x) = 0 for p < x The aggregate of all donations to charity (private and public), G, is divided among the consumers with no income. Hence, τ (x) = 0 for x [0, p] and τdx = G. x=p 3. If x [0, k], where 0 < k p, then consumer x cares about the plight of those with no income. Each of these caring consumers has the utility function u (c, g, G, x) = ln c + a (x) gg, a (x) > 0, x [0, k], (24) so there is strategic complementarity between g and G (see Definition 1). following technical condition holds: The 1 a (x) G < 1 t m (x), x [0, k]. (25) 1 s 4. Some individuals with positive incomes do not care about individuals with zero incomes; these are individuals x (k, p]. 23 The utility function of all individuals x (k, 1] is given by u = ln c, x (k, 1]. (26) 23 In the special case of k = p all consumers with postive incomes are caring. We allow for both cases, k = p and k < p. 15

17 Let m be the aggregate income of the caring consumers. Then m = k m (x) dx p m (x) dx = Recalling that a (x) > 0 is a parameter in (24), define A = k m (x) dx = M. (27) 1 dx > 0. (28) a (x) Proposition 10 : (a) Multiple equilibria: The only economically interesting cases occur when [m + t (M m)] 2 > 4(1 s)a. In this case, we have two distinct, real, positive, equilibria 0 < G (s, t) < G + (s, t). These are given by G ± (s, t) = 1 [ ] m + t (M m) ± [m + t (M m)] 2 4(1 s)a, 2 and, correspondingly, g ± ( s, t, G ± (s, t), x ) = { 1 t 1 1 s a(x)g ± (s,t) if x [0, k] 0 if x (k, 1]. (b) Increasing and concave desire to contribute: For all G, the aggregate desire to give, F, (i) responds positively to subsides, i.e., F s > 0 and, (ii) it is increasing and concave, i.e., F G > 0, F GG < 0. So we have the case depicted in Figures 5, 6A, and 6B. Furthermore, { G G = + F G < 1 G F G > 1 (c) Perverse and normal comparative statics: The comparative statics with respect to the subsidy are perverse at the low equilibrium and normal at the high equilibrium, i.e., G s < 0, and G + s > 0. For m < M (and so k < p) the same holds for the comparative static result with respect to the tax rate, i.e., G t < 0, G + t > 0. For k = p, G ± t = 0. We consider optimal public policy for this example in Section 5.1 below. 3.5 Example 2: Voluntary contributions to a public good Individuals often voluntarily contribute to, and directly use, several kinds of public goods such as health services and education 24. Let the utility function of consumer x be [ u (c, g, G, x) = [1 a (x)] ln c b (x) ] + a (x) ln g, (29) G 24 For the US, education, health and human services account for the greatest proportion of private giving after religion; see Table 3 in Andreoni (2006). A similar picture holds true of the UK. For the year 2011/12, the proportion of donors donating to medical research, hospitals, children and animal charities were, respectively, 33%, 30%, 23% and 16%; see the 2012 report by Charities Aid Foundation ( 16

18 where 0 < a (x) < 1, b (x) > 0, b (x) < (1 t)m (x). (30) G Condition (30) ensures that consumer x has suffi cient disposable income, (1 t)m (x), to sustain a level of consumption, c, greater than b(x) and also a positive level of donation to G charity, g (x). It is straightforward to check that u 1 > 0, u 2 > 0, u 3 > 0. This example can be given the following interpretation. Private (voluntary) contributions to public goods, gdx, plus public contribution, D, financed from income taxation, provide the necessary complementary goods for private consumption, c. An increase in aggregate expenditure on, say, health care, G = D + gdx, leads to a higher level of utility for a given level of consumption. Define the constants B, C as: B = a (x) m (x) dx + t [1 a (x)] m (x) dx, C = a (x) b (x) dx. (31) The main results for this example are listed in Proposition 11, below. The proof of this Proposition is along the same lines as Proposition 10, hence, it is omitted. 25 Proposition 11 : (a) Multiple equilibria: The only economically interesting cases occur when B 2 > 4C. In this case, we have two distinct real positive equilibria 0 < G (s, t) < G + (s, t). These are given by G ± (s, t) = 1 2 and, correspondingly, the levels of private giving are g ± (s, t, G ± (s, t), x) = a (x) 1 s ( B ± ) B 2 4C, (32) [ (1 t) m (x) b (x) G ± (s, t) ], x [0, 1]. (b) Increasing and concave desire to contribute: The aggregate desire to give, F, (i) responds positively to taxes, i.e., F t > 0, (ii) is unresponsive to subsidies, i.e., F s = 0, and, (iii) it is increasing and concave, i.e., F G > 0, F GG < 0 (see Figures 5, 6A, 6B). Furthermore, G = { G + F G < 1 G F G > 1 (c) Neutral comparative statics: Comparative statics are neutral, i.e., G ± s = 0. Thus, subsidies are ineffective in influencing aggregate giving. Furthermore, G t < 0, G + t > 0. From Proposition 11, we know that the economy has two equilibria. 25 Readers interested in the formal proof can consult Dhami and al-nowaihi (2012). 17

19 1. The low equilibrium is characterized by low voluntary contributions to the public good, causing low aggregate spending on the public good, G. From (29), to achieve any specific utility level, high private consumption expenditure is needed. From the budget constraint, (7), we see that, as a consequence, less income can be contributed to the public good, which is a strategic complement, thus, perpetuating the low expenditure on the public good. 2. The high equilibrium is characterized by high contributions to the public good, causing high aggregate expenditure on the public good, G +. In turn, this implies that relatively less private consumption expenditure is needed to reach any specific utility level. Hence, relatively more income is left over to donate to charity, perpetuating the high expenditure on infrastructure. We consider optimal public policy for this example in Section 5.2 below. 4 Equilibrium analysis with multiple equilibria Suppose that we have two equilibria in aggregate giving, G < G +. Suppose that the economy is at G and we wish to move it to G +. The following argument is crucial: If it is possible for the government to give a direct grant, D, such that G < D < G +, then, from (5), we see that G + becomes the only feasible candidate for an equilibrium. But we wish to go further. Does such a D exist? Once G + is established, can we phase out D? Would this cause G + to decline or increase further? Could the economy revert back to G? It is questions like these that we address below. There is no reason to suppose that an equilibrium with perverse comparative statics is unstable and, therefore, can be ignored. Similarly, it cannot be assumed that an equilibrium with normal comparative statics is stable; see section 6 below for a consideration of the relevant arguments. Table 1 suggests several possible cases. We investigate one case in detail: F s > 0, F t > 0, F G > 0, F GG < 0 (subsection 4.2) The case F s = G s = 0 (neutral comparative statics) is considered in detail via Example 2 in subsection 3.5 and 5.2. All other cases can be dealt with in a similar fashion. We begin with a reasonable assumption which we call stability of beliefs that typically underlies any static analysis of multiple equilibria in economics. 4.1 Stability of beliefs Suppose that we have two equilibria, G < G +. Suppose that G + is more desirable than G. Ideally, in considering a move from G to G + one would like to adopt an explicitly dynamic model in which beliefs endogenously evolve over time as individuals engage in 18

20 Figure 4: Stability of beliefs. learning. 26 In the absence of a satisfactory resolution of this problem, one needs to fall back on some ad-hoc (but hopefully plausible) assumption about the stability of beliefs. This is an issue in all static economic models. 27 As an illustrative example, consider a coordination problem where people decide whether to drive on the left or the right of the road. People in the UK (respectively USA) drive on the left (respectively right), safe in their beliefs that all others will also drive on the same side. This example illustrates powerfully the idea that when a good equilibrium gets established, people come to expect that it will prevail. In other words once established, beliefs may exhibit inertia to change, i.e., they may be stable. In a charity context, once the Red Cross is established as a large charity then typical beliefs are that it will be large next period. We call this the stability of beliefs assumption and illustrate it in Figure 4. Suppose that we have two loci of equilibria, G (s, t) and G + (s, t) with t fixed and s variable, such that G (s, t) < G + (s, t). Suppose that the comparative statics along the G locus are perverse, i.e., G s < 0, while those along G + are normal, i.e., G + s > This is reflected in the shapes of the two loci. Suppose that we begin at point a on the G locus, where G s < 0. As we vary s we will trace out various points on the G locus. Now suppose that we can somehow engineer a jump to the G + locus, say, to point b where G + s > 0 (point b need not be directly above point a). Once the economy is on the G + locus (at point, b, say) for a suffi cient length of time then the stability of beliefs requires that as we adjust s the economy moves along the G + locus. 29 Without this assumption it would be diffi cult to perform even comparative 26 Most traditional learning models do not perform too well when taken to the evidence although some behavioral models of learning do better; see Camerer (2003). 27 See Section 6, below for further discussion of these issues. 28 This follows from Table In order to make this argument completely rigorous one would have to specify what a suffi cient length 19

21 Figure 5: The case s 1 < s 2 with t 1 fixed. statics in any static model in economics even when an equilibrium is unique (think of G, G + in this case to be different equilibria that arise from some policy intervention). 4.2 Engineering moves between equilibria: The case F s > 0, F t > 0, F G > 0, F GG < 0 Suppose that the objective of the government is to move the economy from a low equilibrium with perverse comparative statics, G, to a high equilibrium with normal comparative statics, G +. We now discuss this central question in our paper. As pointed out above, we shall consider in detail the case, F s > 0, F t > 0, F G > 0, F GG < 0. Figure 5 sketches the case { F (s, t, G) = 0 for G [0, G (s, t)] F (s, t, G) > 0, F G > 0, F GG < 0 for G > G (33) (s, t) From (33), F is strictly increasing and strictly concave in G, for G > G (s, t). We assume that s 1 < s 2. Since F s > 0, it follows that the graph of F (s 2, t 1, G) is strictly above that of F (s 1, t 1, G). There are four equilibria: a and d corresponding to the parameter values (s 1, t 1 ); and b and c corresponding to the parameter values (s 2, t 1 ). Furthermore, we see that F G > 1 at points a and b but F G < 1 at points c and d. Suppose now that the economy is at point a, with G = G (s 1, t 1 ). Also suppose that (for whatever reason) the government wants to shift the economy from a to d, where the latter point corresponds to G = G + (s 1, t 1 ). Since F s > 0 and F G > 1 at point a it follows, from Corollary 8ci (or see Table 1), that comparative statics at point a are perverse (G s < 0). of time is. But this would require a dynamic model of learning and updating of beliefs that is subject to the caveats mentioned above. This observation applies to the very large literature on multiple equilibria in static models in economics. 20

22 Suppose that the equilibrium G = G (s 1, t 1 ) has had time to get established, so using the argument about the stability of beliefs (see section 4.1) and Figure 4, any change in the parameter s would move the economy along the G locus. Hence, an increase is s, from s 1 to s 2, would reduce aggregate giving, from G (s 1, t 1 ) to G (s 2, t 1 ). A decrease in s would move the economy to another point on the G locus but not enable the economy to move to the G + locus. Thus subsidies are ineffective in moving the economy from a to d. We will now show that, by contrast, the other instrument, t, can be effective. The argument follows the following steps that we will expand on below. 1. Starting from the policy parameters, s 1, t 1, at point a, alter the policy parameters to 0, t 2, which gives rise to two equilibria, G (0, t 2 ) and G + (0, t 2 ). 2. We now give a public grant equal to D > G (0, t 2 ) which rules out G (0, t 2 ) as an equilibrium. Using (5), this leaves G + (0, t 2 ) as the only equilibrium. We know that this equilibrium will exist because of the existence result in Proposition Once the new equilibrium, G + (0, t 2 ), which is a point on the G + locus in Figure 4, gets established, the stability of beliefs argument (see subsection 4.1) can be used to argue that the economy is now on the G + locus. One can then choose the policy parameters, s, t, D, optimally. In particular, one can choose the parameters s 1, t 1 in which case the economy reaches the equilibrium G + (s 1, t 1 ) corresponding to point d. We now elaborate on these steps in greater detail. Figure 6A plots the locus F (s 1, t 1, G), which is exactly as in Figure 5. We also sketch the graph of F (0, t 1, G) in Figure 6A, which lies strictly below the graph of F (s 1, t 1, G), if s 1 > 0, because F s > 0. Figure 6A shows four equilibria a, d, e and f. Equilibria a and d, in Figure 6A, are exactly the same as in Figure 5 and correspond to the parameter values (s 1, t 1 ). However, e and f correspond to the parameter values (0, t 1 ). Figure 6B plots the locus F (0, t 1, G), which is exactly as in Figure 6A. We also sketch the graph of F (0, t 2, G), t 1 < t 2, in Figure 6B, which lies strictly above the graph of F (0, t 1, G), because F t > 0. Figure 6B shows four equilibria e, f, g and h. Equilibria e and f, in Figure 6B, are exactly the same as in Figure 6A and correspond to the parameter values (0, t 1 ). However, g and h correspond to the parameter values (0, t 2 ). We now consider a policy that moves the economy from a to d (see Figure 5) Moving the economy from a to d. Suppose that the economy is at point a of Figure 5, the original policy instruments are (s 1, t 1 ) and the equilibrium level of contributions are G (s 1, t 1 ). Once, G is established, changes in s,t will move the economy along the G locus (see Figure 4). 21

23 Figure 6: Engineering moves between multiple equilibria in two steps. In the first step (see Figure 6A), the government changes the values of the instruments from (s 1, t 1 ) to (0, t 1 ). At the new policy (0, t 1 ), the government budget constraint (4) is given by t 1 M = D. This will give rise to a new equilibrium G (0, t 1 ) and the corresponding level of direct grants by the government are D (0, t 1, G (0, t 1 )). Substituting s = 0 in (4) the government budget constraint is given by t 1 = D(0,t 1,G (0,t 1 )) ; this gives the tax rate M that is necessary to finance direct grants to the charity in the absence of any subsidies to private charitable donations. In the second step (see Figure 6B), the government changes the values of the instruments from (0, t 1 ) to (0, t 2 ). Since s = 0, the government budget constraint (4) is given by t 2 M = D. Let us determine t 2 such that the tax-financed direct public grant to the charity is D = G (0, t 1 ). Thus, t 2 = G (0,t 1 ). If the original level of subsidies s M 1 > 0 then G (0, t 1 ) > G (s 1, t 1 ) (see Figure 6A); this arises on account of the perverse comparative statics, G s < 0, on the locus G (s, t 1 ). By definition (see (5)), when the policy is (0, t 1 ) we get G (0, t 1 ) = D(0, t 1, G (0, t 1 )) + g (0, t 1, G (0, t 1 ), x)dx, so, if at least one individual contributes a positive amount, g > 0, then G (0, t 1 ) > D(0, t 1, G (0, t 1 )). Thus, t 1 = D(0,t 1,G (0,t 1 )) < G (0,t 1 ) = t M M 2. Since G t < 0 (because F t > 0, F G > 1 and from Proposition 7, G t = Ft 1 F G ), it follows that G (0, t 2 ) < G (0, t 1 ) = D as in Figure 6B. Hence, the sole equilibrium, consistent with the stated level of D, is now the high equilibrium, G + (0, t 2 ), point h of Figure 6B. 22