Empathy and the Efficient Provision of Public Goods

NBER WORKING PAPER SERIES EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS Geoffrey Heal Working Paper 29255 http://www.nber.org/papers/w29255 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 September 2021 I acknowledge support from Columbia Business School. The views expressed herein are those of the author and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2021 by Geoffrey Heal. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. Empathy and the Efficient Provision of Public Goods Geoffrey Heal NBER Working Paper No. 29255 September 2021 JEL No. H00,H23,H41 ABSTRACT I consider the effect of empathy towards others on the internalization of interpersonal externalities and on private contributions to the provision of public goods. I show that if preferences are empathetic in the sense of depending on the well-being of others, then in an extreme case external effects are fully internalized, and private contributions to the provision of a public good will be sufficient for it to be provided at an efficient level. Furthermore I show that an increase in the level of empathy shown by any agent will lead to an increase in the level of provision of the public good, and that as empathy levels increase towards their upper bound, the level of provision of the public good converges to the efficient level. Under certain conditions an increase in empathy is Pareto improving. As it is well-documented that people display some degree of empathy, it is arguable that our failure to provide public goods at efficient levels is attributable to lack of empathy as well as to the free rider problem. Geoffrey Heal Graduate School of Business 516 Uris Hall Columbia University New York, NY 10027-6902 and NBER [email protected] EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 2 1. Introduction Empathy, according to psychologists and neurologists, is an important determinant of human behavior (Iacoboni [17], Batson et al. [6], Decety et al. [12]). Discussions of it are nevertheless scarce in the economics literature. This paper takes some preliminary steps in investigating its implications for two classic problems in public and environmental economics: the management of external effects and the provision of public goods. In doing this I draw on two strands of literature: behavioral economics, and work on altruism and the private provision of public goods. Behavioral economists have imported ideas from psychology to improve our understanding of individual behavior (see Camerer [11] and Thaler [26]). The literature on public goods, altruism and warm glow (Bergstrom [9], Andreoni [2]) looks at how individual contributions to the cost of a public good vary with factors such as altruism and impure altruism or warm glow. My interest here is in how empathy affects contributions to the cost of a public good, or behavior that generates external costs and benefits for others. Empathy is the ability to feel what others feel, to share their feelings and mental experiences, their pain and their happiness.1 Mirror neurons are thought to be a mechanism - perhaps the main one - through which this happens. When we observe someone undergoing an intense experience, neurons in our brain behave as if we were ourselves undergoing the same experiences: these are referred to as mirror neurons. According to Kilner and Lemon [18], ’Mirror neurons were originally defined as neurons which “discharged both during monkey’s active movements and when the monkey observed meaningful hand movements made by the experimenter” ’ ([14]). Kilner and Lemon [18] comment that “the key characteristics of mirror neurons are that their activity is modulated both by action execution and action observation, and that this activity shows a degree of action specificity.” Acharya and Shoukla [1] emphasize the connection between mirror neurons and empathy: “Studies have shown that people who are more empathic according to self-report questionnaires have stronger activations both in the mirror system for hand actions and the mirror system for emotions, providing more direct support for the idea that the mirror system is linked to empathy.” A famous non-human example concerns rats, which contrary to their public image show empathy for their fellows. In an experiment, rats were given a choice between releasing another rat from a cage or eating chocolate: the great majority proceeded to release the caged rat and then share the chocolate with it (for a summary see Wein [27] and for full details Bartal et al. [5]). The importance of these ideas from an economic perspective is that empathy creates connections between individual welfares and may lead us to at least partially internalize any external effects we impose on others, provided we can see or imagine the impacts that these have on them. If something I do will have a negative effect 1 The Oxford English Dictionary defines empathy as “the ability to understand and share the feelings of another.” It goes on to say that people often confuse empathy and sympathy, whereas sympathy means “feelings of pity and sorrow for someone else’s misfortune.” The OED also says that empathy was first used in the early 20th century. EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 3 on my neighbor, and I am aware of this and can empathize with her, then I may be reluctant to carry our this act: my mirror neurons lead me to feel her response to my action. Similarly interest in the welfare of others may affect willingness to pay for public goods. It is the consequences of this that I investigate below. In section 2 I define empathetic preferences, which extend regular preferences over consumption vectors (both own consumption and that of others) by including as arguments the welfare levels of others. In this case, each person’s welfare depends on that of all others. As in Hori [16], no one’s welfare can be specified independently of that of others.2 I then show in section 3.1 that in the presence of interpersonal externalities, if people behave according to what I call maximum empathy, then a competitive equilibrium is Pareto efficient. If preferences are empathetic but weakly so, only a fraction of the externalities is internalized and the outcome is inefficient. In section 3.2 I consider a simple model of the provision of a public good: each person has to choose personal consumption levels in the light of the amount of a public bad produced in the making of the consumption good (think of the greenhouse gases emitted by traveling). A similar result holds in this case: with individuals choosing according to maximally empathetic preferences, the outcome is Pareto efficient, but a weaker form of empathy leads to an inefficient outcome. In section 3.3 I consider a general model of the production of a public good, in which each person has to choose between allocating their budget between private consumption and the financing of a public good. Again a competitive equilibrium is Pareto efficient and meets the Bowen-Lindahl-Samuelson (BLS) conditions in the presence of maximum empathy. With a lower level of empathy the BLS conditions are only partly satisfied and the public good is under-provided. I then show in subsection 4.1 that with log-linear preferences and a linear production technology for the public good, more empathetic preferences lead to more of the public good at equilibrium. Finally in section 4.2 I study the comparative statics of changes in the degree of empathy in a general model, and establish three results. One is that an increase in the degree of empathy on the part of any agent leads to an increase in the level at which the public good is provided, the second is that the equilibrium allocation is a continuous function of the agents’ levels of empathy, and as these converge to maximum empathy the allocation of resources converges to a Pareto efficient outcome, and finally this convergence is Pareto improving under certain conditions. It appears from these results that empathetic preferences may be a possible explanation for the finding that people make much larger contributions to the provision of pubic goods than conventional theory might suggest. More generally, empathetic preferences seem to give an additional explanation for apparently altruistic behavior, which economists have often explained by the “warm glow” effect (Andreoni [2] and [4]: Andreoni [3] looks at altruism as a possible explanation of behavior in public goods experiments). The connection between empathy and altruism is complex. The two give rise to very similar models from an analytical perspective - the preference formulation that I use is identical to that developed by Hori [16] to study altruism in the context of social choice. Social psychologists see empathy as the basis for altruism: we show concern for others, which is interpreted as altruistic behavior, because of empathy 2 For a similar modeling of interdependent preferences in an intertemporal context see Millner [21] EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 4 for others (see for example Batson et al. [6] and [8]). However, in the model that I develop below empathy implies that my wellbeing depends on that of others, so that by improving their welfare I am actually contributing to my own. My aid to them is strictly self-interested, and so not consistent with the Oxford English dictionary’s definition of altruism, which is “disinterested and selfless concern for the well-being of others.” The model is strictly one of individual welfare maximization in the best neoclassical tradition, albeit moderated by a recognition of the realities of psychological and social interdependence. Humans are, we are told by scientists, social animals, and have evolved to function in communities: empathy is one of the mechanisms via which we function communally. It is important to capture its implications for economics. The idea of empathy in economics is not new: the founder of our discipline, Adam Smith, devotes a whole chapter of his The Theory of Moral Sentiments to sympathy,3 and commented that “How selfish soever man may be supposed, there are evidently some principles in his nature, which interest him in the fortune of others, and render their happiness necessary to him, though he derives nothing from it except the pleasure of seeing it. Of this kind is pity or compassion, the emotion we feel for the misery of others, when we either see it, or are made to conceive it in a very lively manner. That we often derive sorrow from the sorrow of others, is a matter of fact too obvious to require any instances to prove it; for this sentiment, like all the other original passions of human nature, is by no means confined to the virtuous and humane, though they perhaps may feel it with the most exquisite sensibility. The greatest ruffian, the most hardened violator of the laws of society, is not altogether without it.” (Adam Smith, Ii1 [24]) Pinker, in his very influential book The Better Angels of Our Nature 4 [22], distinguishes sharply between empathy and sympathy. Empathy, the ability to feel what others feel, he argues, gives rise to sympathy and it is this that leads to altruistic behavior. Batson and various co-authors argue that it is empathy itself that leads to altruism ([7], [6]). Neither Smith nor any other writers on this topic connected empathy with externalities and their internalization, but to a contemporary economist it is a natural step. Although I emphasize cases in which empathy is uniform across all others, in reality it is likely that someone will display empathy for some but not for others. For small communities, for example the extended families in which our early ancestors lived, there may be empathy by all for all, leading to internalization of small-scale interpersonal externalities and efficient provision of local public goods. Pinker [22] makes the important point that in contemporary societies this can lead to nepotism and corruption. But for large groups, or the entire world as is relevant for climate change or biodiversity loss, such uniform empathy is most unlikely. In fact there are scientific studies that establish relevant results: the following is from Decety et al.([12]) 3 The word “empathy” did not exist in Smith’s time: it is a twentieth century invention. “Sympathy” and “compassion” were used in its stead. 4 Which, according to its index, uses the word “empathy” 187 times. EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 5 “Nonhuman animals show preference toward in-group members in detection and reaction to the distress of others. ... In one study, a female mouse moving toward a dyad member in physical pain led to a decrease in the physical symptoms of pain (less writhing) in the dyad member only when the mouse was a cage mate of the mouse in pain, not when they were strangers (Langford et al. 2010). Similarly, female mice exhibit higher fear responses when exposed to the pain of a close relative than when exposed to the pain of a more distant relative (Jeon et al. 2010). Importantly, it is not necessarily genetic affiliation that solely facilitates assistive behaviors. Rats fostered from birth with another strain have been shown to help strangers of the fostering strain but not rats of their own strain (Ben-Ami Bartal et al. 2014). Thus, strain familiarity, even to one’s own strain, seems required for the expression of prosocial behavior in rodents.” The same authors go on to comment that “A priori, implicit, value-based attitudes toward conspecifics also modulate the response. For example, study participants were significantly more sensitive to the pain of individuals who had contracted AIDS as the result of a blood transfusion as compared to individuals who had contracted AIDS as the result of their illicit drug addiction ...... . Another fMRI study found modulation of empathic neural responses by racial group membership (Xu et al. 2009). Notably, the response in the ACC to viewing others in pain decreased remarkably when participants viewed faces of racial outgroup members relative to racial in-group members. This effect was comparable in Caucasian and Chinese subjects and suggests that modulations of empathic neural responses by racial group membership are similar in different ethnic groups.” They also add that “Several aspects of empathy, such as accuracy and concern for others, as well as generosity and other-oriented behavior, are influenced by social status. Social class seems to shape not only people’s values and behavior but also their affective responses that relate to sensitivity to the welfare of others. Research shows that lower class individuals, relative to their upper class counterparts, score higher on a measure of empathic accuracy, and judge the emotions of a stranger more accurately (Kraus, Cote, and Keltner 2010). Another set of studies indicates that relative to upper class people, lower class individuals exhibited more generosity, more support for charity, more trust behavior toward a stranger, and more helping behavior toward a person in distress (Piff et al. 2010). Despite their reduced resources and subordinate rank, lower class individuals are more willing than their upper class counterparts to increase another’s welfare, even when doing so is costly to the self.” The conclusion to emerge from the psychology literature is that empathy is an important motivator of behavior towards others, but is generally not uniform and EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 6 indeed may reinforce prejudices. In the next section I formalize the concept of empathetic preferences and investigate the consequences of uniform and non-uniform empathy. 2. Empathetic Preferences We consider an economy in which individuals are indexed by j, j = 1, 2, .., J and firms by i, i = 1, 2, .., I. Consumption vectors are Cj ∈ RN , and C−j ∈ RN (J−1) is the vector of consumption goods for all people other than person j. Firm i produces the i − th good according to the formula Yi = fi (Ki ) where Ki is the capital stock of the i − th firm and Yi the total output of good i and the function fi is strictly concave. I make a distinction between a person’s utility Uj and their welfare Wj : utility depends only on consumption, hers and possibly that of others via external effects, Uj (Cj , C−j ). In some models utility may depend on consumption and on the level of provision of a public good g, Uj (Cj , g). In either case the utility function is strictly concave. As in Hori [16], welfare depends on utility Uj - on consumption - and also on the welfare levels of others Wk , k 6= j. Welfare is therefore a more general concept than utility, and depends on factors over and above consumption. The dependence a person’s welfare on the welfares of others captures the idea of empathy: the more important the welfares of others are, the more empathetic a person is. Others have thought of this formulation as modeling altruism - see Bergstrom [9] and Hori [16]. In the first model considered, we suppose externalities to be generated by other peoples’ consumption, according to the utility-of-consumption function Uj (Cj , C−j ). The inclusion of C−j as an argument of Uj denotes a conventional externality: the consumption of people other than individual j affects j ′ s wellbeing via noise or pollution or some other mechanism. Empathy is then represented by assuming that j ′ s overall welfare depends not only on consumption vectors Cj and C−j but also on the welfare levels of others. We assume j ′ s welfare Wj to be separable and linear in the welfares of others: X (2.1) Wj = Uj (Cj , C−j ) + Wk αj,k P k6=j where of course Wk = Uk (Ck , C−k ) + l6=k Wl and αj,k ≥ 0 is the weight that each agent places on the welfare of others. Note that in this formulation individual j is accepting individual k ′ s specification of her welfare. In particular j may have no empathy for m whereas k may place a lot of value on m′ s welfare: by accepting k ′ s own definition of her welfare, j is indirectly placing value on m. Hori [16] calls this nonpaternalisitic altruism: Millner [21] in the context of intertemporal preferences uses a similar specification and calls the resulting preferences non-dogmatic. This specification implies that changes in other people’s utility levels do not change j ′ s ordering over the space of (Cj , C−j ), but do change the utility level associated with any point in the consumption space: they make people better or worse off without changing their underlying preferences over consumption. Another way of describing this is that changes in others’ welfares renumber indifference surfaces. With this definition, each agent’s welfare depends on that of all others: they are all simultaneously determined, and we cannot tell one person’s welfare unless we know that of all others. We can cut through the interdependence the definition (2.1) gives rise to by using a system of simultaneous equations: EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 7 W1 = U1 + α1,2 W2 + α1,3 W3 + .... + α1,J WJ W2 = α2,1 W1 + U2 + α2,3 W3 + .... + α2,J WJ W3 = α3,1 W1 + α3,2 W2 + U3 + .... + α3,J WJ .................... WJ = αJ,1 W1 + αJ,2 W2 + .... + αJ,J−1 WJ−1 + Uj This system can be rewritten (2.2) AW = U where A is the JxJ empathy matrix  1 −α1,2 ... −α1,J−2  −α2,1 1 ... −α 2,J−2   −α3,1 −α3,2 ... −α3,J−2   ... ... ... ...   −αJ−1,1 −αJ−1,2 ... −αJ−1,J−2 −αJ,1 −αJ,2 ... −αJ,J−2 −α1,J−1 −α2,J−1 −α3,J−1 ... 1 −αJ,J−1 −α1,J −α3,J −α3,J ... −αJ−1,J 1         and W = W1, W2 , W3 , ....WJ and U = U1 , U2 , U3 , ....UJ (This is very similar to the formulation used by Hori [16]). If A is non-singular and possesses an inverse we can write (2.3) W = A−1 U By way of example, consider the case of three agents: then the inverse of the matrix A is   (1 − α2,3 α3,2 )/D (−α1,2 + α1,3 α3,2 )/D (−α1,3 + α1,2 α2,3 )/D  (−α2,1 + α2,3 α3,1 )/D (1 − α1,3 α3,1 )/D (α1,3 α31 − α2,3 )/D  (−α3,1 + α2,1 α3,2 )/D (α1,2 α3,1 − α3,2 )/D (1 − α1,2 α2,1 )/D where D is the determinant of A, and D = {1 − α1,2 α2,1 − α1,3 α3,1 + α1,2 α2,3 α3,1 + α1,3 α2,1 α3,2 − α2,3 α3,2 } Using this solution, we can express the Wj s as functions of the Uj s : X Uk (Ck , C−k ) βj,k (2.4) Wj = k where βj,k is the j, k − th element of B = A−1 . So we can express each person’s welfare as a weighted sum of everyone’s utilities. If this result is to make economic sense, we need the weights βj,k ≥ 0 : in fact we shall see below that we sometimes need them to be strictly positive. So we need the inverse of the matrix A to be non-negative or possibly positive: below I give necessary and sufficient conditions for this to be true. Intuitively, it makes sense that in a situation where everyone has empathy for everyone else (all αi,k > 0), the weights they implicitly place on others’ utilities should be positive. I will consider a simple special case below: when αi,j = α > 0 ∀i, j, so that everyone puts the same weights on the welfares of others, implying that the offdiagonal terms in the matrix A are all the same, and this is a symmetric matrix EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 8 Figure 2.1. On-diagonal and off-diagonal elements of inverse and their ratio as functions of a from 0 to -0.5 4 4 2 2 -0.5 -0.5 -0.4 -0.3 -0.2 -0.4 -0.3 -0.2 -0.1 -0.1 -2 -2 -4 20 15 10 5 -0.5 -0.4 -0.3 -0.2 -0.1 with ones on the diagonal and −αs everywhere else. In this case its inverse can easily be shown to have the same form, with all diagonal elements equal and all off-diagonal elements equal. For example, the 4x4 matrix   1 a a a       a 1 a a a a 1 a       a a a 1 has as its inverse the matrix whose diagonal elements are all equal to (1 − 3a2 + 2a3 )/(1 − 6a2 + 8a3 − 3a4 ) and whose off-diagonal elements are all (−a + 2a2 − a3 )/(1 − 6a2 + 8a3 − 3a4 ). I will be interested below in a special case which I call maximum empathy, in which equal weight is put by each agent on all utilities, their own included, so that βij = βi ∀j, in which case the matrix A−1 would be singular. There is an intuitive explanation for this singularity: if each person gives the same weight to the utilities of all others, then all have the same welfare function up to a positive multiplier. (They have the same welfare function but not of course the same utility functions.) Hori’s [16] theorems 1 and 2 address this issue in the context of social choice procedures. Effectively there is only one person and in maximizing her welfare she maximizes the sum of all utilities, which as we will see below is equivalent to finding a Pareto efficient allocation of resources. If βjk is the same for all j and so B is singular, this means that in the system of equations W = BU all equations but one are redundant. Hence W must be on the diagonal of RJ - all welfare levels must be the same - and there is a set of possible solutions which is the intersection of a J − 1 dimensional subspace with the positive orthant of RJ . This makes intuitive sense: if all preferences are the same then all welfare levels will be the same, and clearly there are many utility vectors U that are consistent with the weighted sum being W . Figure 2.1 shows for the 4x4 matrix above the values of the on- and off-diagonal elements as a varies from 0 to −0.5, and also the ratio of these terms. Note that EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 9 this ratio is one - the terms are equal - at a = −1/3 at which point the matrix is singular. As we approach the singularity of the A matrix, the weights being put on all utility functions by the inverse matrix tend to equality. 2.1. Uneven Empathy. The scientific results cited above show that an individual’s feelings of empathy are directed at groups with which she feels close or for which she feels approval, but not necessarily at other groups for which she has no such feelings. This raises an interesting question: suppose A has empathy towards B but not C, and B has empathy towards C. Because A feels for B and B for C, does A effectively feel for C, because of her concerns for B and those of B for C? In other words, is empathy transitive? To investigate this we need to go back to the analysis of empathetic preferences: This example can be expressed as W1 = U1 + α1,2 W2 + 0W3 W2 = 0W1 + U2 + α2,3 W3 W3 = α3,1 W1 + 0W2 + U3 in which case the inverse of the matrix A is  1/D −α1,2 /D  α2,3 α3,1 /D 1/D −α3,1 /D α1,2 α3,1 /D  α1,2 α2,3 /D −α2,3 /D  1/D where D = 1 − α1,2 α2,3 α3,1 . As αi,j < 0, the terms in this matrix are all positive. In this case each individual welfare level depends on the utilities of all others: W1 = U1 /D − α1,2 U2 /D + α1,2 α2,3 U3 /D W2 = U1 α2,3 α3,1 /D + U2 /D − U3 α2,3 /D W3 = −U1 α3,1 /D + U2 α1,2 α3,1 /D + U3 /D and empathy is effectively transitive. Note that agent 1 places no weight on the welfare of agent 3, but does place a weight on her utility proportional to α12 α23 which is the product of the weight she places on 2’s welfare and the weight 2 places on 3’s. Likewise agent 3 places no weight on agent 2’s welfare but places a weight on her utility proportional to α12 α31 . If however we change the problem so that α3,1 = 0 then the inverse matrix becomes   1 α1,2 α12 α23  0 1 α2,3  0 0 1 so that two agents are insensitive to the utility of at least one other. This observation raises the question of conditions under which the elements of the inverse matrix A−1 are all positive. If all elements of A−1 are strictly positive then every agent places weight on the utility of every other agent: everyone cares about everyone. I call this universal empathy, and as these examples show this can arise because of the transitivity of empathy even if some agents place no weight on the welfares of some others, so that there are zeros in the A matrix. The next proposition formalizes this idea, but first we need some definitions. The digraph (directed graph) of an nxn matrix A is constructed as follows: draw a set of n vertices in the plane, and connect vertices i and j with a directed segment from i to j if and only if aij 6= 0. The digraph is strongly connected if it is possible to move from any vertex to any other following the directed segments of the graph. EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 10 Note that the matrix A in (2.2) can be written as A = I − G where if gij is a typical element of G, then gii = 0 and gij = αij ≥ 0. Proposition 1. If the modulus of the largest eigenvalue of A is less than one, then A−1 shows universal empathy (has strictly positive elements) if and only if the digraph of A is strongly connected. Proof. An nxnmatrix A is  reducible if there exists a permutation matrix P such A11 A22 T where P T is the transpose of P and the Ajj and 0 are that P AP = 0 A33 sub-matrices. Otherwise it is irreducible. The digraph of a matrix M is strongly connected if and only if M is irreducible.5 A Z-matrix is a matrix for which aij ≤ 0 for i 6= j. An M -matrix is a matrix that can be expressed as M = τ I − G, where G is non- negative and τ ≥ ρ (G) where ρ (G) is the spectral radius of G, that is the modulus of the largest eigenvalue. Note that the matrix A has ones on the diagonal and is non-positive off the diagonal. It is thus A Z-matrix. As noted above it can be written (2.5) A=I −G where I is the identity matrix and G is a non-negative matrix with zeros on the diagonal: gij = −aij ∀i, j. As we have assumed the modulus of the largest eigenvalue of A to be less than one, it follows that A is an M -matrix as well as Z-matrix. We can now use the following theorem (Meyer and Stadelmaier ([20]): (1) If A is a non-singular M -matrix then A is inverse positive. (2) If A is a Z-matrix then the converse of (1) is also true. (3) If A is an irreducible non-singular M -matrix then A is strictly inverse positive. (4) If A is a Z-matrix then the converse of (3) is also true. Proposition 4 now follows: the digraph of A being strongly connected means that it is irreducible, and it is a Z-matrix and an M -matrix so we can invoke the Theorem cited by Meyer and Stadelmaier to asset that A is strictly inverse positive and so displays universal empathy. 6  There is an intuitive basis for this result: the directed graph being strongly connected means that everyone directly or indirectly cares about everyone else, and this is reflected in the positivity of the inverse matrix. The condition that the modulus of the largest eigenvalue of G be less than one essentially means that G is a contraction mapping that shrinks vectors on which it operates. We have some information about the largest eigenvalue of G : we know that the sum of the eigenvalues of a matrix equals the trace, which in this case is zero, and we know that the largest eigenvalue is bounded below by the lowest row sum of P and above by the greatest row sum.7 A further intuitive understanding of this result comes P∞from −1 the standard result from input-output analysis that A−1 = [I − G] = t=1 Gt so that the inverse of A is the sum of the powers of G, provided that this series 5This is a standard result - see for example https://www.stat.berkeley.edu/~mmahoney/s15stat260-cs294/Lectures/lecture03-29jan15.pdf 6There is an alternative proof of Proposition 4 which is simpler but perhaps less intuitive. Berman and Plemmons [10] prove the following: Let an nxn matrix A be irreducible. Then each of the following conditions is equivalent to the statement “A is a non-singular matrix.” (1) A−1 >> 0 and (2) Ax > 0 for some x >> 0. We assume A to be irreducible and know from the definition of A (2.2) that AW = U where W, U are non-negative vectors in RJ . As a strictly positive vector U must exist we know that A satisfies the second condition here and so A−1 >> 0. 7This is part of the Frobenius-Perron theorem - see Heal Hughes and Tarling page 120 [15]. EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 11 converges, which it does if and only if the maximum root of G is less than one. The i, j elements of the t − th power of G gives the weight placed on j ′ s welfare by i via paths of length t in the digraph of G. So the i, jth element of G is just the weight i places directly on j ′ s welfare, and the same element of G2 is the weight i places on the welfares of people who place weight directly on j, and so on. Next we consider the implications of empathetic preferences for three models, one of a system with consumption externalities, one with a public good or bad produced as a side effect of the production of consumption goods, and finally a classical case of public goods. 3. Externalities and Public Goods with Maximum Empathy In this section I show that interpersonal externalities are internalized, and pubic goods efficiently provided, if everyone has maximum empathy. This is a very special case, but is interesting for two reasons. One is that as levels of empathy increase, the system converges to this special case, which therefore provides a benchmark for evaluating other states. Another is that, as I argue in section 6, this is a case that is privileged by many important ethical systems. In the next on comparative statics I show that as empathy increases from levels below the maximum, allocations move towards efficiency, with the level of public good provision increasing monotonically with the levels of empathy. Under certain conditions this process is Pareto improving: an increase in anyone’s empathy makes everyone better off. 3.1. Consumption Externalities. As mentioned above, I assume that firm i produces the i − th good according to the formula Yi = fi (Ki ) where Ki is the capital stock of the i − th firm and Yi the total output of good i, and there are externalities in consumption, so that utility is Uj (Cj , C−j ) . To characterize Pareto efficiency we need to solve the following optimization problem which maximizes the weighted sum of welfares subject to a resource constraint: (3.1) X X X X X M ax w j Wj = wj βjk Uk (Ck , C−k ) , Cij = fi (Ki ) , Ki = K j j j k i ′ where wj > 0 is the weight assigned to agent j s welfare: variations in these weights trace out theP utility possibility or Pareto frontier. Rearranging the double sum, and letting γj = k [wk βkj ] , this problem gives the Lagrangian   ) (  X X X X Cij − fi (Ki ) − θ Ki − K λi γj Uj (Cj , C−j ) − (3.2) L =   j j i i and the corresponding first order conditions (FOCs) are X ∂Uk ∂Uj (3.3) γj + = λi γk ∂Ci,j ∂Ci,j k6=j (3.4) ′ λi f i = θ or equivalently (3.5) X ∂Uk ′ ∂Uj = θ/fi γj − ∂Ci,j ∂Ci,j k6=j  γk γj  EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 12 Clearly we can think of the second term on the RHS here as a Pigouvian tax on j’s consumption of good i. It reflects the impact of this consumption on the welfare levels of others. Next we consider the individual’s choice with empathetic preferences. If the individual faces prices p ∈ RN and has income Yj , then the individual optimization problem is then X (3.6) M ax βj,j Uj (Cj , C−j ) + βjk Uk (Ck , C−k ) s.t. p.Cj = Yj k6=j giving as FOCs (3.7) βjj X ∂Uk ∂Uj + βj,k = µj pi ∂Ci,j ∂Ci,j k6=j where µj is the Lagrange multiplier on individual j ′ s budget constraint. Equivalently X ∂Uk  βjk  ∂Uj = pi µ j − (3.8) ∂Ci,j ∂Ci,j βjj k6=j Again we can think of the second term on the RHS as a Pigouvian tax on consumption. Compare (3.5) with (3.8): these have the same general form, with the marginal utility of the i − th good equal to its social marginal cost or price, minus a function of the external costs that it imposes on others (which will be negative if these are costs rather than benefits). But in the two equations the functions of external costs are different so that we cannot in general say that the private and social FOCs are aligned. However in some special cases we can say more than this. Assume that in the matrix B all elements are the same, i.e. βjk = βik = β ∀i, j. So all agents give the same weight to all other agents’ utilities - maximum empathy. Then using the definition of γj above we see that γj = γk ∀j, k Hence in the social FOC (3.5) we see that γk /γj = βjk /βjj so in this case the two sets of FOCs are the same and hence the privately optimal outcome is Pareto efficient. There is a clear intuition behind this case: every individual agrees on the weights to put on each utility function and these determine the social weights in the maximand used to characterize Pareto efficiency. Note that in this case the matrix B is singular: as noted in the discussion in the previous section, if everyone agrees on the weights to place on each utility function, then all have the same welfare function and we have a community of identical individuals. Proposition 2. With interpersonal externalities, if individuals choose according to preferences showing maximum empathy, then the competitive equilibrium is Pareto efficient. 3.2. Public Goods I. Individuals are again indexed by j, j = 1, 2, .., J and firms by i, i = 1, 2, .., I. As before, firm i produces the i−th good according to the formula Yi = fi (Ki ) where Ki is the capital stock of the i − th firm and Yi the total output of good i. In this case firm i also produces a public bad Xi in an amount directly proportional to its output of consumption good: Xi = αi Y : think of greenhouse gases produced as a by-product of transportation or of the production of electric power. Individual j ′ s utility of consumption depends on her P private consumption P Cj ∈ RN and on the public good or bad X = i Xi = i αi Yi and is given by EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 13 Uj (Cj , X) where Cj = Cij, i = 1, .., I. As in the previous section, her overall welfare Wj depends also on the welfares of others: X Wj = Uk (Ck , X) βj,k k We consider a case when the individual has empathetic preferences and so seeks to maximize Wj subject to p.Cj = P Y where P p is a price vector P and Wj the individual’s income. Recall that X = i Xi = i αi Ci and j Cij = Ci . We can therefore write the Lagrangian for the individual’s optimization problem as ! X X X Cik + µj {Wj − p.Cj } αi βjk Uj Cj , (3.9) Lj = i k k which gives as FOCs (3.10) βjj X ∂Uk ∂Uj + αi βjk = pi µj , i = 1, 2, ..I ∂Cij ∂X k We next characterize Pareto efficiency, which is the solution to: (3.11)X X X X X αi fi (Ki ) Ki = K, X = wj Cij = fi (Ki ) , M ax βjk Uk (Ck , X) , j k i i j where K is the total capital available, which using the γj s defined in the previous section gives the Lagrangian. From this the relevant FOCs are X ∂Uk ∂Uj (3.12) γj +α = λi γk i ∂Cij ∂X k (3.13) ′ λi f i = θ We can write (3.10) and (3.12) as follows: X ∂Uk  βjk  ∂Uj = pi µ j − α i (3.14) ∂Cij ∂X βjj k and (3.15) X ∂Uk ∂Uj = λi /γj − α i ∂Cij ∂X k  γk γj  Clearly these equations are the same as the FOCs of the previous section: under the same assumptions as there we have the following results: Proposition 3. If public goods (or bads) are associated with the production of private consumption goods and consumers choose their consumption levels according to maximally empathetic preferences , then the outcome is Pareto efficient. 3.3. Public Goods II. Next we consider a model in which consumers have the choice of purchasing private goods for consumption or contributing to a fund that provides public goods. Utility depends on consumption of the private good and the provision of the public good: Uj (Cj , g). As before, we consider an economy in which individuals are indexed by j, j = 1, 2, .., J and firms by i, i = 1, 2, .., I. Consumption vectors are Cj in R so there is only one consumption good, and the amount of the public good provided is g. We suppose that each person has a budget that can be divided between the regular consumption goods and contributing to the provision EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 14 of the public good. The total amount of the public good provided is a function P  of the total amount contributed by all individuals: g = f where gj is j ′ s j gj contribution to the public good and f is concave. Preferences are P again empathetic, represented according the the arguments of section 2 by Wj = k Uk (Ck , g) βjk . Hence the individual optimization problem is (setting the price of the private good equal to one and letting Yk be the value of k ′ s endowment) ! X X M axCj ,gj Uk (Ck , g) βjk , g = f g k , Ck = Yk − g k k k In this problem the agent is optimizing over her choice of consumption of the private good and contribution to the public good, taking as given the contributions that she thinks others are making (gk , k 6= j) as in a Nash non-cooperative equilibrium, giving rise to the first order conditions X ∂Uk ∂Uj βjj = λj , βjk f ′ = λj ∀j ∂Cj ∂g k from which we get X ∂Uk /∂g βjk 1 = ′ ∂Uk /∂Ck βkk f (3.16) k Next we characterize a Pareto efficient allocation. This is a solution to the problem ! X X X M axCj ,gj wj Uk (Ck , g) βjk , g = f g k , Ck = Y k − g k j k k and as before consolidating the double sum gives us the first order conditions for efficiency X ∂Uk ∂Uj (3.17) γj = µ, f′ = µ γk ∂Cj ∂g k which give the standard Bowen-Lindahl-Samuelson conditions that the sum of the marginal rates of substitution should equal the marginal rate of transformation: X ∂Uk /∂g γk 1 = ′ (3.18) ∂Uk /∂Ck γj f k Comparing these conditions with (3.16) we see as in the previous two sections that if we assume that in the matrix B all elements are the same, then γk /γj = βjk /βjj = βk /βj and hence Proposition 4. If each person has maximally empathetic preferences over private and public goods, then privately optimal choices lead to an efficient allocation of resources, in particular to the efficient provision of the public good. Next I present an example that illustrates this proposition: it is a simple general equilibrium model in which we can compute the equilibria analytically and compare the efficient and privately optimal levels of provision of the public good. As to be expected, the two levels of provision are the same when the levels of empathy are maximal: less obvious but also true is that as the empathy levels increase, the level of provision of the public good increases monotonically towards the efficient level. EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 15 4. Comparative Statics 4.1. Log-Linear Preferences. In this section we consider a special case of public good provision in which it is possible to derive analytical solutions to the FOCs. We can then show that in the case of extreme empathy, the Nash equilibrium outcome is Pareto efficient, and also that an increase in empathy leads to an increase in the provision of the public good. Let utilities be uj (cj , g) = θlog (g) + log (cj ) . Welfare is X Wj = uj (cj , g) + αjk Wk k6=j P so that Wj = k uk (ck , g) βjk . Let J be the total number of people, W the total endowment and W/J the share of each person. Assume the public good is produced from the private according to the linear technology g (z) = z where z is the amount of the private good allocated in total to the production of the public good. A Pareto efficient allocation is the solution to X XX cj = W {θlog (g) + log (ck )} βjk , z + M ax j j k The Lagrangean is L= XX j or βjk {θlog (g) + log (ck )} + λ k L = βθlog (g) + XX j    W −z− ( log (ck ) βjk + λ W − k X cj j X k ck − g    ) P P where β j = k βjk is the j − th column sum and β = j β j . The FOCs with respect to g and ck are θ βj β = λ, =λ g ck from which it follows that the efficient allocation satisfies ck = βj W , β (1 + θ) ge = Wθ 1+θ The typical individual solves the problem X X M axWj = u (ck , g) βjk = βjk {θlog (g) + log (ck )} , k k W − zj J Substituting into the utility function we find the FOCs are X θ βjj βjk − W =0 g J − zj k cj = or cj θ X k βjk = βjj g g = zj + X k6=j zk , EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS and summing over j gives X cj = W − g = j or g= θ+ 16 g X βjj P θ j k βjk Wθ P j Pβjj k βjk P β Wθ . This is equal to the Pareto For simplicity let σ = j P jjβjk so that g = θ+σ k efficient level of provision if σ = 1, and less if σ > 1. Then the following are true: (1) (2) (3) (4) (5) Pβjj k βjk is the fraction of the total weight assigned by individual j to her own utility, and if she places equal weight on all - maximum empathy - it will be 1/J. In this case σ = 1 and the private and social levels of provision are identical. If people are self-oriented in the sense of placing more weight on their utility β than on that of others, then P jjβjk > J1 so that σ > 1 and the provision of k the public good will fall short of the efficient level. If we think of the special case in which βjj ≥ βjk , then it is clear that as βjk → βjj , σ → 1 and the private level of provision tends monotonically from below to the efficient level. In a situation such as that in point 2 above, an increase in any off-diagonal element βjk of the matrix B reduces σ and increases the amount of the public good: ∂g/∂βjk > 0 ∀j 6= k. An increase in an off-diagonal element corresponds to an increase in empathy. Similarly an increase in a diagonal element - the weight agent j places on her own utility - reduces the level of the public good. If each agent places weight only on her own utility - βjj > 0, βjk = 0 ∀j 6= k - then σ = J the total number of agents. In this case we have an efficient outcome only if J = 1. 4.2. General Comparative Statics. Next we generalize the results of the example of the previous subsection and establish some important results about changes in the level of empathy. We show that an increase in any off-diagonal element αij increases the welfare levels of all agents connected to agent i, meaning that if the graph is strongly connected then it increases the welfares of all agents. We also show that an increase in the weight that any agent places on the utility of another (an increase in any βij , i 6= j) increases the level of provision of the public good in the model of subsection 3.3, and that in that model the equilibrium allocation is a continuous function of the elements βij of the matrix B, so that if the off-diagonal elements of B converge from below to unity then the associated allocations converge to the efficient allocation. Finally we show that an increase in βij , i 6= j, is Pareto improving: all welfares increase or at least remain constant. 4.2.1. Increases in αij . An increase in any αij , i 6= j, increases the welfare of agent i and also that of any agent k for whom αki > 0, because her welfare depends on that of i. Let Ki denote the set of agents for whom αki > 0. It also increases the welfare levels of agents for whom αlm > 0 for any m ∈ Ki . In short it increases the welfare of any agent connected directly or indirectly to agent i in the digraph of A. EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 17 If the matrix is irreducible and the graph strongly connected then it increases the welfares of all agents. 4.2.2. Increase in empathy increase public good provision. We can use the FOC 3.16 for the individual optimization in section 3.3 as an implicit function X ∂Uk (Ck , g) /∂g βjk 1 (4.1) − ′ =0 ∂Uk (Ck , g) /∂Ck βkk f (g) k and from this ∂Uk /∂g ′′ f ∂g ∂U /∂C = − n ∂U ∂ 2 Uk ∂Uk ∂ 2 U o − ′ 2 k ∂βjk βjk ∂Ck ∂g2k − ∂gk ∂C ∂g (f ) k k (4.2)  2 ∂Uk ∂Ck 2 2 ∂Uk ∂ Uk ∂Uk ∂ Uk which is positive if ∂C 2 − ∂g ∂C ∂g < 0: the first term here is negative by k ∂g k concavity of the utility function and the second will be non-positive if the cross partial is positive or zero. Its being positive implies that an increase in the public good raises the marginal utility of the private good, and it is zero if the utility function is separable in the public and private goods. As the last term on the RHS ′′ of (4.2) is positive ( f being negative), this is a sufficient but not a necessary condition. Hence we have: Proposition 5. If the public good enhances (or does not decrease) the marginal utility of the private good, an increase in the weight placed by any agent on any other agent’s utility leads to an increase in the provision of the public good. This is consistent with the analytical example in 4.1. So under these conditions more empathy leads to more of the public good. 4.2.3. Convergence to Pareto efficiency. Assume that βjj = 1∀j and βij < 1∀i 6= j. So all agents place less weight on the utilities of others than on their own, and we set the own weights at one. We can now show that if βij → 1 ∀i, j, i 6= j, then the equilibrium allocations of the public and private goods converge to the Pareto efficient allocation that arises when βij = 1 ∀i, j. To show this we prove that the equilibrium allocations are continuous functions of the elements of the matrix B ∈ RJxJ . Consider agent j’s optimization problem !! X X βjk + λj {Yj − gj − Cj } L= Uk Ck , f gk k Cj∗ k gj∗ (B, Yj ) and (B, Yj ) be the solutions. Let Ĉj , gˆj be the values of Cj and and let gj when all agents place a weight of one on every utility function: we know these to be Pareto efficient from proposition 4. We are assuming that the utility function is strictly quasi-concave and we note that the budget set is compact provided that gj , Cj ≥ 0. In this case we can apply Theorem 3.1 of Terazano and Matani [25] to assert that Cj∗ and gj∗ are continuous functions of their arguments. Hence as βjk → 1 ∀j, k, Cj∗ (B, Yj ) , gj∗ (B, Yj ) → Ĉj , gˆj , leading to Proposition 6. As every agent becomes more empathetic and the system moves toward maximum empathy, the allocation of public and private goods converges to the efficient allocation. EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 18 Let g (B) be the Nash equilibrium with the matrix B of weights on utilities, and let B 1 be the matrix of fweights on utilities in which all elements are funity f (bij = 1 ∀i, j). Let B − B 1 be the distance between B and B 1 , given by B −
f PP 1 2 B1 = i,j βij − βij . Then Proposition 7. If the public good enhances the marginal utility of the private good, f f there exists ǫ > 0 such that whenever B − B 1 < ǫ, an increase in any element βij , i 6= j of B is Pareto improving. Proof. First we prove that an increase in any βij whichPis less than one leads to a new Nash equilibrium at which the sum of all utilities j Uj (Cj , g) is greater. At a Nash equilibrium we know from (3.16) that X ∂Uk /∂g βjk 1 = ′ ∂Uk /∂Ck βkk f k Here βkk = 1 and βjk < 1. So X ∂Uk /∂g 1 < ′ ∂Uk /∂Ck f (4.3) k  Look at the effect of increasing g on sum of all utility functions. Suppose the public good is increased by ∆g > 0 and private good decreased by ∆Ck < 0. The gain in total utility from more of the public good is X ∂Uk >0 ∆g ∂g and the loss from the decrease in the private good is ∂Uk ∆Ck <0 ∂Ck P P Recall that Ck = Yk − gk so that g = f ( gk ) = f ( (Yk − Ck )) and dg/dCk = ′ ′ −f or ∆g = −f ∆Ck We want to know if utilities increase in total as a result of these marginal changes: X ∂Uk X ∂Uk ∂Uk ∂Uk ∆g > 0 or ∆g + ∆Ck > −∆Ck ∂g ∂Ck ∂g ∂Ck k k Hence we need to know if −f ′ ∆Ck X ∂Uk > −∆Ck P < ∂g k or : ∂Uk k ∂g ∂Uk ∂Ck ∂Uk ∂Ck 1 f′ Hence from (4.3) any increase in g leads to an increase in the sum of utilities. We know from the argument for Proposition 6 that the equilibrium is a continuous function of the matrix B. We also know from Proposition 5 that an increase in any bij leads to an increase in g. Hence it leads to an increase in the sum of utilities. If the matrix of utility weights B has all elements equal to unity, then clearly an increase in the sum of utilities leads to an increase in every welfare level EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 19 and is Pareto improving. This is still true one P for a matrix B whosePelements are P 2 or close to one: if j ∂Uj /∂g > 0 then j bij ∂Uj /∂g > 0 ∀i if j (1 − βij ) < δi for some δi > 0. The ǫ of the proposition is now given by ǫ = mini δi . 5. Local Public Goods Suppose that instead of benefitting everyone, a public good g benefits only a subset of people: let L = 1, 2, ..., G, G < J be the strict subset of people who gain from a public good. Suppose also that within this group, empathy levels are strong, but are not necessarily strong towards non-members (as discussed in 2.1). Then if bij = 1 ∀i, j ∈ L, by the arguments of section 3.3, the public good will be provided efficiently to this group. The same would be true of a second public good ge which e a strict subset of the total group of people with benefits only members of the set L, no intersection with L. A society could therefore be divided into non-intersecting groups that provide public goods efficiently to their members, but nevertheless provide a pubic good that benefits everyone inefficiently. In this context closelyknot religious or ethnic groups come to mind as examples. The comparative statics results of propositions 5 and 6 would hold for each of these groups, so that an increase in within-group empathy would lead to greater levels of provision of the group-specific public good. In these cases the digraphs of the A matrix would not e be strongly connected, but the graphs restricted to the groups such as L and L would be strongly connected. 6. Discussion Empathy is a widely-recognized phenomenon in psychology, neuroscience and philosophy. Its implications for economics merit consideration. We can now see that they are quite substantial. In an economy where everyone has maximally empathetic preferences, interpersonal externalities in consumption do not cause inefficiency, and public goods are provided at a Pareto efficient level. This is a special case, but it is one that is held out as an ideal by many religions and systems of moral philosophy. For example, according to Christ, Thou shalt love thy neighbour as thyself. (Matthew 22:36-40, King James Version). Hinduism has a similar maxim: Do not to others what ye do not wish done to yourself (The Mahabarata, Brihaspati Anusasana Parva-Section CXIII Verse 8). In Buddhist texts we find Hurt not others in ways that you yourself would find hurtful (Udanavarga, 5:18, Tibetan Dhammapada). Confucianism echos these principles: Do not do to others what you do not want done to yourself (Confucius, Analects, 15:23, 6:28). And of course in Groundwork of the Metaphysic of Morals (1785) Kant stated that we should Act only according to that maxim whereby you can, at the same time, will that it should become a universal law.8 All of these precepts require that we think of the wellbeing of others exactly as we think of our own wellbeing, and this is what the preference formulation (2.1) does when the weights βjk are all equal to one. So in an ethically ideal society, two of the main problems of public economics - externalities and public goods - are automatically resolved. Religious precepts, if taken seriously, lead to efficiency in the public sphere. In addition an increase in empathy will lead to an increase in the provision of the public good if the public 8For interesting discussions of Kant’s implications for economics see Laffont [19] and Roemer [23]. EMPATHY AND THE EFFICIENT PROVISION OF PUBLIC GOODS 20 and private goods are complements, and may be Pareto improving. This holds for public goods that benefit all, and also for local pubic goods that benefit only a subset of the population. Einstein [13] made a remark this is relevant here, seeing empathy and compassion as important: “A human being is part of a whole, called by us the “Universe,” a part limited in time and space. He experiences himself, his thoughts and feelings, as something separated from the rest - a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty.” (Einstein [13]). 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