A Kantian Approach to Charitable Behavior Oliver Groth Pettersen Thesis submitted for the degree of Master in Economics 30 credits

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A Kantian Approach to Charitable Behavior Oliver Groth Pettersen

Thesis submitted for the degree of Master in Economics

30 credits

Department of Economics Faculty of Social Sciences

University of Oslo

May 2021

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Acknowledgements

This thesis concludes my studies at the University of Oslo. First, I would like to thank my supervisor Paolo G. Piacquadio for all his help, encouragement, and many insightful discussions throughout the process. Thanks are also due to Thor Olav Thoresen and Oslo Fiscal Studies for allowing me to work on a similar project during the summer of 2020. I need to thank Vemund Vikjord, Arnstein Vestre, Edvard Garmannslund, and Kjell Ingar Pettersen for their help in proofreading the thesis. Finally, I would like to thank Ingrid for all her support through these stressful times. Any errors are my own.

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Abstract

In this thesis, I examine charitable behavior and the government’s role in incentivizing donations. I build on the economic model of moral motivation by Brekke, Kverndokk, and Nyborg (2003). Individuals trade off personal benefit with a moral cost of not doing the right thing. The main novelty of this thesis is to let the individuals’ “morally ideal donation” emerge endogenously as a Kantian equilibrium (Roemer, 2019). Furthermore, I generalize the model to heterogeneous preferences. I show that the optimal public intervention in the more complex heterogeneous setting is to set an incentive scheme for donations that balances efficient provision of public goods, individuals’ moral costs, and the efficiency cost of behavioral responses.

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List of Figures

1 Figure 1: Kantian vs Individual Donation with Self-Image Cost . . . 19 2 Figure 2: Multiplicative Equilibrium . . . 26

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1 Introduction

Why do we do seemingly unselfish deeds? Why do we choose to contribute to good causes, even though the cost of doing so is larger than our personal benefit? Why do we choose to donate parts of our income to organizations providing help to poor individuals on the other side of the globe? Is it sympathy or altruism? Is it social norms and rules? Or is it something entirely different? These questions are not just theoretically interesting.

Philanthropy and charitable giving are a significant part of our economic reality. In 2016, Norwegians gave 3.2 billion NOK in donations to voluntary organizations (Epland, 2019).¹ Similarly, according to Giving USA, Americans donated as much as 450 billion dollars in 2019.² With charitable giving being such a significant phenomenon, there is a need for economic tools that can be applied to better understand charitable behavior. This is not just important to better understand human behavior, but also for the formulation of economic policy, such as tax expenditure policy. In this thesis I will present an economic model for describing charitable behavior.

To study charitable behavior, I build on and extend the model of moral motivation by Brekke, Kverndokk, and Nyborg (2003). In their model the individual first decides what she thinks is the “morally ideal contribution”, before deciding what she actually wants to contribute. The individual is confronted with a trade-off between doing what she thinks is right, and consuming leisure. This is called a dual preferences framework because the preferences consist of an “ethical” and a “subjective” part (Harsanyi, 1955). My first contribution is to adopt the solution concept called Kantian optimization to describe the “ethical” preferences of the population. Kantian optimization is a solution concept, developed by Roemer (2019), meant for modeling individuals’ behavior in cooperative settings. It requires less restrictive assumptions than the solutions considered by Brekke et al. (2003) and can easily be altered to allow for heterogeneous populations. A critique of Kantian optimization by Itai Sher (2020) is that Kantian optimization, contrary to

1This only includes organizations that made the donors eligible for tax deductions. Many organizations that receive donations are therefore not included. It also excludes voluntary work.

2See: Giving USA url (visited May 14 2021)

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the opinion of Roemer (2019), is hard to motivate through self-interest. It is more easily motivated as a form of moral reasoning. This suggests that Kantian optimization can be a pertinent extension of the framework by Brekke et al. (2003).

Modeling charitable behavior is also relevant from a policy perspective. Optimal tax expenditure models for charitable sectors are today mostly based on models of “warm glow” as proposed by Andreoni (1989). Explaining charitable behavior with a model of warm glow implies including the individual donation in the individual’s utility function weighted by an exogenous parameter, or a “warm glow” (henceforth, I will refer to this framework as “the warm glow model”). There are problems associated with such an approach as pointed out by Diamond (2006). For example, tax policy is hard to formulate when the models are based on exogenous parameters that in reality might depend on the policy. The dual preferences framework can in some sense be seen as a model with an endogenous warm glow. Can we formulate an optimal tax policy when the moral reasoning of the individuals is endogenous? And will that solve any of the problems postulated by Diamond (2006)? In this thesis, I look into these questions by constructing optimal tax policy on the basis of my model. I show that I can formulate optimal tax policy both with a homogeneous and a heterogeneous population.

The thesis is organized as follows. In section 2, I present the literature relevant for studying the nature of charitable contributions. I also present the model of Brekke et al. (2003) and discuss the concept of Kantian optimization. In section 3, I introduce the model of this thesis. First, I apply the model in a setting with a homogeneous population, before applying it in two different heterogeneous settings. The heterogeneity is introduced in two different parts of the individuals’ preferences. I formulate public policy recommendations following each model formulation. In section 4, I compare the model of this thesis to that of warm glow. I also discuss to what degree the solution of Brekke et al. (2003) is more restrictive compared to the model presented here. Section 5 provides conclusions and suggestions for further research.

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2 Background

2.1 Charity as a Public Good

In a theoretical master thesis on the subject of charitable giving, it is natural to first ask how we should model the phenomenon. In the literature, charity is mostly modeled as a public good. The reason why might not be so obvious. The paper credited with first noting how altruism can be modeled as a public good is Hochman and Rodgers (1969).

They remark upon how redistribution (and even stabilizing fiscal policy) is something that might increase everyone’s welfare, therefore implying a positive externality. Another paper that is credited with being important in this regard is Becker (1974). In his paper he formalizes the argument.

According to Andreoni (2006), there are typically three lines of reasoning for why people give to charity. All three lead to the conclusion that the individuals care about the total supply of the good provided by the charity, suggesting that it is in fact a public good. The first argument he presents is the “enlightened selfishness” argument. The reason people donate is that the individuals can benefit themselves from the supply of the good. An example could be if the charity in question supplies a cultural service that can be consumed by the individual. Then individuals can themselves benefit from the service. The second argument is the “insurance argument”. Because the donors themselves one day might belong to groups in need of care, they want the institutions that care for these groups to be in place. Finally, there is the more traditional altruistic argument. Individuals take the well-being of other individuals (present or future) into account when they make their consumption decision. All of the arguments lead to the same conclusion: People care about the total supply of the good provided by the charity. Hence, there is a positive externality from donations implying that charity can be modeled as private provision of a public good (Andreoni, 2006).

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2.2 Altruism and Warm Glow

Now that we have discussed the general framework under which charity is modeled, we can move on to discuss how the concept has been modeled more specifically. Altruism and warm glow are two important tools for modeling charitable giving. James Andreoni showed the weakness of the first (Andreoni, 1988) and suggested the second as a solution to the problem (Andreoni, 1989). In his paper on altruism, Andreoni explains how economic theory up until that point in time had modeled private charity “as a pure public good in the Samuelson sense” (Andreoni, 1988, p. 57). By this he means that people only care about the total supply of charity, or the public good. Andreoni (1988) sets up such an altruistic model to show why this is problematic. The problem is that in a large economy the strategy of “free riding”, or only consuming the supply of the public good provided by others while not contributing yourself, dominates in most cases. As a result average giving converges to zero, and only “an infinitely small proportion of the economy contributes” (Andreoni, 1988, p. 58). Meaning that only the richest individuals donate. The model also falls victim to the neutrality hypothesis. This means that the model based on altruism predicts a full crowding out effect. A “full crowding out effect”

means that a public provision of the public good, or in our case the charity, would crowd out private donations on a one-to-one ratio. The results holds under both homogeneous and heterogeneous preferences.

We see, however, that the predictions of the model described in the model of Andreoni (1988) do not hold in the real world. People donate to charity. Studies have also showed little evidence of full crowding out. Instead one sees partial crowding out or even crowding in (Andreoni & Payne, 2013), meaning that people donate more to organizations that receive public funds. According to Andreoni and Payne (2013), evidence also seems to suggest that much of the crowding out one observes is due to changes in the fund-raising behavior of charitable organizations. So the altruism framework has problems explaining empirical data.

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Andreoni (1989) suggests a model purportedly better equipped to explain the patterns observed in the data. He suggests that people also care about the funding of a public good, not only its total supply. Individuals gain utility from the act of giving in itself.

This utility is called “warm glow”. The concept is also called “impure altruism”, because it does not exclude the altruistic argument from the preferences completely. In the model of warm glow, presented in Andreoni (1989), individuals have preferences over both the total supply of the public good and their own contribution. The model of warm glow predicts an incomplete crowding out effect given that individuals are not indifferent between contributions through voluntary donations and taxes. As Andreoni (2006) points out, the warm glow solution to the problem of charity seems a bit “ad hoc”. But as he states in the same chapter, experimental data shows clear evidence of a warm glow effect. So with a simple framework that seems to be confirmed by experimental data, why should we not settle for this method of modeling charity?

2.3 Moral and Economics

Brekke et al. (2003) argue that the “warm glow” approach is too simplistic. They give two reasons why they think it could be improved when it comes to explaining charitable behavior. First (as I will discuss in detail later), they argue that the model is unable to explain certain observations of how price incentives can affect charitable behavior. Secondly, the model does not require any “complex moral reasoning on issues such as individual obligations or social values” (Brekke et al., 2003, p. 1969). Are such considerations important when we model economic behavior? Economic models should do their best to explain human behavior. It is therefore worth taking a further look into how individuals reason when they are confronted with situations in which they make moral choices. To do this examination, they suggest a model with dual preferences. In such a framework, the individuals first reason morally, then they move on making their actual choice, taking their moral reasoning into account. As Brekke et al. (2003) state, this idea is not new.

But it requires some explanation, and it is important for understanding the motivation of the model I will present later. I will therefore spend some time trying to explain the

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reasoning behind the dual preferences framework.

Brekke et al. (2003) state that the idea of dual preferences was suggested by (among others) Sen (1977). In his paper, Sen points out weaknesses and strengths with the model of rational behavior. He highlights problems with the view, he means some people have, that rationality is equivalent to self-interest. Sen (1977) postulates that certain types of behavior are hard to explain through such a system of rational preferences. He specifically argues that the framework lacks the ability to explain actions that do not increase one’s personal wellbeing. These are actions conducted out of what he calls “commitment”. He draws a line between this and what he calls “sympathy”. When an individual acts from sympathy she takes the wellbeing of others into account. Such behavior can be modeled within the rational framework. However, when individuals act from “commitment”, they give up their own wellbeing, not because they feel better knowing that other individuals are better off, but because they act from some other rule. His critique can be summarized in the following statement:

A person is given one preference ordering, and as and when the need arises this is supposed to reflect his interests, represent his welfare, summarize his idea of what should be done, and describe his actual choices and behavior.

Can one preference ordering do all these things?

His answer is that such an individual, who needs no distinction between these quite different concepts, makes a weak approximation of human behavior. (Sen, 1977, p. 335)

Sen (1977) also provides an idea for a solution. The idea draws on the framework of Harsanyi (1955). Harsanyi makes a distinction between a person’s “ethical” and “subjective” preferences. By subjective preferences, he means preferences that define what one actually does. The ethical preferences state what the individual prefers seen from an objective standpoint. They reflect what the individual prefers “only in those possibly rare moments when he forces a special impartial and impersonal attitude upon himself”

(Harsanyi, 1955, p. 315). This is an informal version of the dual preferences framework.

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When interpreting the dual preferences framework it is not only important to make the distinction between moral and altruistic motivation. One should also make a distinction between a social norm and a moral motivation or norm. The following follows the expo- sition of Nyborg (2018). The concepts of moral and social norms are often overlapping in the literature. Whereas a moral norm is a “rule of ethically appropriate behavior, enforced by the individual herself through inner feelings such as guilt, conscience, and cognitive dissonance” (Nyborg, 2018, p. 412), a social norm is “a predominant behavioral pattern within a group, supported by a shared understanding of acceptable actions and sustained through social interactions within that group” (Nyborg et al. (2016), as cited in Nyborg (2018, p.407)). A social norm might also be important when making economic decisions and can influence decisions on a national scale. However, as I here want to look at the moral reasoning happening within an individual I will limit myself to moral norms.

With this thesis’ goal of making the moral reasoning of individuals endogenous, one could argue that one should go even further in making distinctions between different types of moral motivations, as done by Welsch (2020). But I leave such interesting extensions to further research.

2.4 The Dual Framework

Let us now turn to the model which is the basis for what I do in this thesis, namely the model of Brekke et al. (2003) (henceforth, called the “dual framework”). The authors model a situation in which individuals voluntarily provide time, called effort, to the production of a public good. What sets this model apart from a regular model of such a situation (like a model with warm glow), is that the individuals have a dual preference ordering. The “ethical” part of their preferences is represented by what the authors call a self-image cost. This is a cost that depends on how far the individuals deviate from what they consider to be “morally ideal”. With this cost taken into account, the individuals make a trade-off between providing effort to the production of the public good or consuming leisure. More formally, the individual has preferences over consumption, leisure,

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the total supply of the public good, and their “moral self-image”. This is represented by a utility function:

Ui =u(xi, li, G, Ii),

wherex_iis individuali’s consumption,l_iher leisure,Gthe total supply of the public good, and I_i the moral self-image. The utility functions of the individuals are increasing and strictly quasi-concave. It is assumed that income and labor supply are exogenous. As the paper focuses on the trade-off of time, the individual’s resource constraint is given by her total available time. This is either consumed as leisure or donated as effort: e_i +l_i =T, where e_i is the effort of the individual, and T is the total time available. The total level of public good G is decided as the sum of private and public provision:

G=G_p+X

i

g_i,

whereG_p is the public provision and g_i is the individuals provision, given byg_i =γ(e_i, θ).

The production function of the individual is increasing in effort e_i and an exogenous productivity parameter θ. Moreover, g_i has a negative second derivative with respect to the individual effort, has a positive cross-derivative, and is equal to zero when the effort is equal to zero. Finally, the self-image is a quadratic cost function:

I_i =f(e_i, e^∗_i) =−a(e_i−e^∗_i)²,

where e^∗_i is the effort that the individual finds to be morally ideal. This implies that the individual experiences a self-image cost for deviating from what she considers to be morally ideal, no matter the direction of the deviation.

An important question is of course how the individual decides what the morally ideal effort is. This is done before choosing their actual behavior. Brekke et al. (2003) assume that every individual shares a utilitarian moral philosophy, meaning that they all think that “the interests of every individual should count equally in evaluations of social wel-

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fare” (Brekke et al., 2003, p. 1971). This gives a welfare functionW =u₁+...+u_N, where N is the total number of individuals. Each individual asks herself the question: “which action would maximize social welfare, given that everyone acted like me?”. This leads the individual to search for the level of effort that maximizesW, given that everyone else chooses the same level of effort as themselves. An important observation is that in this optimum the individuals donate what they find to be morally ideal. Therefore, e_i = e^∗_i holds in the optimum, meaning that the self-image cost will not play into the process of finding e^∗_i. With the morally ideal effort level, the individuals move on to optimize their individual preferences using a Nash solution. The solution suggests that there typically will be under-provision of the public good.

There are two points that the authors point out that I would like to highlight and discuss. First, the utilitarian welfare criterion is chosen for simplicity, they postulate that replacing it with some different welfare function is entirely possible, and will not change the “main logic of the argument” (Brekke et al., 2003, p. 1971). I would argue that using a utilitarian welfare criterion is overly restrictive and might limit the scope of the analysis. Using a simpler criterion such as Pareto efficiency yields the same result without saying anything about the individuals’ moral philosophy. As I later show, with the use of Kantian optimization, we do not need to assume anything concerning the individuals’

moral philosophy.

Second, Brekke et al. (2003) state that a model with heterogeneous individuals could be achieved by adjusting the question the individuals ask themselves. Instead of asking which level would maximize social welfare given that everyone else spent the same level of effort, the new question would be “Which general rule of action would maximize social welfare, as I perceive it, given that everyone acted according to the same general rule as I?” (Brekke et al., 2003, p. 1972) This might be overly complicated. As I show later, with the use of the multiplicative Kantian equilibrium, a model with heterogeneous individuals can be constructed with a simpler rule of behavior.

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Finally, one of the important findings of the Brekke et al. (2003) is that economic incentives can affect the outcome in the opposite direction of what usually is the case in economic models. This result is also backed up by survey data. To show this they introduce a fee that the individual must pay if they choose to not participate in the provision of the public good. The fee is introduced in a binary setting, meaning that the individual either chooses to participate or not. There is no choice concerning the degree to which the individual wants to participate. Under this condition, the fee will have a weakly negative effect on participation if the fee is perceived as big enough to buy equivalent services on the market. If the fee is perceived as only symbolic, it will have a weakly positive effect on participation. As a precursor to the results that follow later in this thesis, this result will be hard to replicate in my model, and I will refrain from trying to do so. There are two reasons why: First, in contrast to Brekke et al. (2003), I model a setting of mone- tary donations to a public good. In that setting, a binary contribution makes less sense intuitively. Second, my model is set in a setting where contributions to the public good are traded off against consumption. Introducing a fee as is done in the model of Brekke et al. (2003) would not set up a similar participation constraint. It would only have the effect of making participation mandatory up to a certain degree.

2.5 Kantian Optimization

The term Kantian optimization is a term first coined in Roemer (2010). He later elab- orates the framework in Roemer (2015) and Roemer (2019). Roemer suggests Kantian optimization as a framework for modeling cooperation in strategic situations. He postulates cooperation as a puzzle that economic theory has yet to solve (Roemer, 2019). Why do people bother to vote in big elections? Why do people take their trash to trash cans, even when no one is watching? And why do people donate to charitable organizations?

Roemer means the answer is cooperation, not through solidarity or altruism, but rather through self-interest: “You and I build a house together so that we may each live in it.

We cooperate, not because of an interest in the other’s welfare, but because cooperative

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production is the only way of providing any domicile” (Roemer, 2019, p. 4).

Roemer (2019) discusses how cooperation tends to be explained in economic theory. He postulates that it is explained either through multi-stage and dynamic games, or through including the welfare of others in the preferences. Both things are done while still using the Nash equilibrium. He points to several problems with these approaches. First, the dynamic approach where one punishes the player who does not cooperate only works when everyone thinks there are several stages left. In the end, there can be an incentive to deviate. If not for the person not cooperating, then for the agent supposed to exert punishment. Second, when tweaking the preferences one is only able to obtain a cooperative solution in relatively simple games. When analyzing more complex games we end up unable to explain the cooperation one sees in the real world. What Roemer (2019) suggests is that instead of tweaking the games and preferences to fit the solution concept, a better idea is to change the solution concept. He suggests Kantian optimization as this new solution concept. It is important to note that Roemer draws a clear line between Kantian optimization and altruism, or solidarity. One chooses to cooperate because it is in one’s own best interest to do so, “it is important to note that the Kantian optimizers ask what common strategy (played by all) would be best for him: he is not altruistic, in thinking about the payoffs of others. To calculate the strategy he would like everyone else to play he need only know his own preferences” (Roemer, 2019, p. 13).

Kantian optimization is a concept where you, instead of taking everyone else’s strategy as given and optimizing your payoff (as one does in the Nash equilibrium), maximize your payoff given that everyone else behaves in a similar fashion as yourself. In some sense, it is a simple version of the Kant’s categorical imperative. But as Roemer specifically says, the link is weak due to the unconditional nature of Kant’s imperative (Roemer, 2019). There are different versions of Kantian optimization, which he calls different sorts of Kantian equilibria. Later I will make use of two of them, and I will explain those here.

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The simple Kantian equilibrium is the simplest and most illustrative version of the Kan- tian equilibrium. In the simple Kantian equilibrium, a player optimizes simply by asking

“what is optimal to do, given that everyone else acted like me?”. To put it in more formal terms we can follow the definition of Sher (2020):

Consider a game with n players, a common strategy space S, from which each player choose a strategy, and a set of utility function Vⁱ : Sⁿ → R for each player i = 1, ..., n. Let [n] = {1, ..., n} be the set of agents. A Strategy S^∗ ∈S is a simple Kantian equilibriumif:

∀i∈[n], ∀s∈S : Vⁱ(s^∗, ..., s^∗)≥Vⁱ(s, ..., s)

(Sher, 2020, p. 47)

So a simple Kantian equilibrium is your preferred outcome when every agent chooses the same strategy. According to Roemer (2019), such an equilibrium will always exist as long as the game has a common diagonal. By common diagonal, he means that the payoff functions of every agent coincide on the diagonal (in an ordinal setting this would imply that the ordering of the outcomes coincides on the diagonal). Such a condition is weaker than saying that the game has to be symmetric.

There is a clear weakness to the simple Kantian equilibrium. With heterogeneous preferences, what each agent would want the other agents to do might not coincide. To make Kantian optimization possible under such circumstances Roemer (2015) suggests a class of equilibria he calls Kantian variations. The two most prominent of these are the multiplicative and additive Kantian equilibria. I will look at the multiplicative Kantian equilibrium here. In short terms, a multiplicative Kantian equilibrium is an allocation from which all individuals would defer from deviating, given that everyone else would deviate with the same non-negative multiplicative factor. To define the multiplicative equilibrium more formally I will again use the definition from Sher (2020). Using the same economic environment as in the simple Kantian equilibrium, a strategy profile (s^∗₁, ..., s^∗_n)

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is a multiplicative Kantian equilibriumif:

∀i∈[n], ∀r ∈R+: Vⁱ(s^∗₁, ..., s^∗_n)≥Vⁱ(rs^∗₁, ..., rs^∗_n)

As we see this type of equilibrium allows for heterogeneous strategies. It is worth men- tioning that the simple and multiplicative equilibria are equivalent when agents are homogeneous (Roemer, 2015).

Roemer (2019) shows several attractive features of Kantian optimization. One is efficiency. In Roemer (2019) he shows that both simple and multiplicative equilibria are Pareto efficient in what he calls “strictly monotone games” (with the caveat that the multiplicative equilibrium also has to be strictly positive). Roemer (2019) defines a game V as (strictly) monotone increasing if “for each i,Vⁱ is (strictly) increasing in the strategies of the other playersj 6=i.”(Roemer, 2019, p. 23) The definition of a monotone decreasing game is analogous.

The Kantian equilibrium framework has some limitations. Sher (2020) reviews some weaknesses of the Kantian equilibrium that needs to be properly addressed before the solution concept can serve as a good modeling tool for cooperation. One of these is the motivation for playing the Kantian equilibrium. Roemer (2019) claims that the Kantian equilibrium can be founded on self-interest. Sher (2020) disagrees with this claim. His argument goes beyond the technical concept of the equilibrium. He argues that you can not optimize over things you do not control yourself. Instead, you can form beliefs concerning the strategies of others and choose a strategy based on these beliefs. Imagine that a player expects everyone else to play a strategy consistent with a Kantian equilibrium. If the player has a beneficial deviation, given that everyone else plays the Kantian equilibrium, then from a purely self-interested perspective, why would the player not make this deviation? Sher (2020) argues that this line of reasoning shows that the motivation for playing the Kantian equilibrium has to be founded in something else than self-interest.

Instead, the motivation for playing the Kantian equilibrium has to be founded in some

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sort of moral consideration, as it would be in a setting of dual preferences.

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3 The Model

In this section, I present the theoretical framework of the thesis. First, I present a homogeneous model. In this part, I make use of the simple Kantian equilibrium to define the ethical preferences of the individual. Then I investigate how two different kinds of heterogeneity affect the problem. I first present a model with a heterogeneous self-image cost. Surprisingly, I am still able to use the simple Kantian equilibrium. Then I look at how a heterogeneous valuation of the public good affects the outcome. In this part, I use a multiplicative Kantian equilibrium to derive the morally ideal donation. After each part, I discuss the form of a possible public intervention.

3.1 The Homogeneous Individuals Model

There is a unit mass of individuals. Each individual i has preferences over consumption x_i ≥ 0, a public good G≥ 0, and a moral self-image I_i. The preferences of individual i are represented by a utility function:

Ui =u(xi, G, Ii) (1)

The utility function is assumed to be increasing in each argument, concave, twice differentiable, and have weakly positive cross derivatives between x_i and G. The disposable income of the individual can either be used for consumption or a donation D_i ≥0 to the provision of the public good. Individuals have a fixed income m, and pays a tax τ(D_i).

This means that the budget constraint is:

x_i+D_i ≤m−τ(D_i) (2)

The sum of individual donations and contributions by the government (denoted Gp) is transformed into the public good by the increasing, concave, and twice differentiable

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function f:

G=f

G_p+ Z 1

0

D_idi

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The moral self-image of the individual is determined by the difference between the donation level the individual considers “morally ideal” (denoted D^e_i) and the donation level she actually makes. The weight of the moral self-image is given by the parameter a >0.

I_i =−a(D_i−D_i^e)² (4)

The model is solved by first deciding the morally ideal donation. This represents the ethical part of the preferences and corresponds to the thought experiment of finding out what the individual would want from an objective standpoint as outlined by Harsanyi (1955). After finding the morally ideal donation the individuals move on to optimize over their subjective preferences while taking their ethical parameters into account.

3.1.1 Finding the Morally Ideal Donation

For illustrative purposes, I will first discuss the setting where Gp = τ(Di) = 0, meaning that there is no government intervention. To find the morally ideal donation I will use Kantian optimization, as discussed earlier. Because we here look at a homogeneous model, we can use the simple Kantian equilibrium. In the simple Kantian equilibrium, each individual asks herself the question: “what is optimal to do, given that everyone else acted like me?”. This leads the individual to look for the donation level that, if donated by everyone, will maximize her utility.

One possible source of confusion for the reader, which might make the Kantian thought experiment hard to imagine is how the self-image cost plays into the problem. We are here looking for the equilibrium that characterizes the morally ideal donation D^e_i. The donation level we derive from the Kantian equilibrium will be the morally ideal one, meaning that in the equilibrium D_i =D^e_i. This means that the self-image cost will not affect the

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solution because I_i = 0 in the equilibrium. To remove any confusion I will here insert Di =D_i^e and maximize over D^e_i.

maxD_i^e u(x_i, G, I_i)

s.t. x_i+D_i^e≤m (5)

G=f Z 1

0

D^e_idi

(6) I =−a(D_i^e−D^e_i)² = 0 (7)

After substituting for (5), (6), and (7) in the utility function, the first-order condition becomes:

FOC_D^e : u_x uG

=f⁰ Z 1

0

D^e_idi

which implicitly determines the morally ideal donation D^e_i. It is worth noting that this solution coincides with a decentralized Samuelson condition. To see why imagine that we take the integral over all the individuals’ Kantian solutions. We would get that the aggregated marginal rate of substitutions equal the marginal rate of transformation, which corresponds to the Samuelson condition. This suggests that the outcome of the Kantian thought experiment is Pareto efficient. The result is consistent with Brekke et al. (2003) who also finds that the moral thought experiment yields an efficient outcome. A more direct interpretation of the condition above is that how much of the public good the individual is willing to give up for one more unit of consumption must correspond to how much more public good the individual would gain if the average donation changed by one unit. This interpretation follows the reasoning of the Kantian thought experiment, in the sense that it links individual and aggregate behavior.

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3.1.2 Individual Optimization

Now that we have found the morally ideal donation, how much will the consumer actually decide to donate? To answer this question we can use a Nash solution, meaning that each agent takes the contribution of the other agents as given when optimizing. This gives the following optimization problem:

max

Di

u(x_i, G, I_i) s. t. (2) - (4)

For illustrative purposes I setR1

0 D_idi≡G_D. This, because there is no public provision of the public good, will constitute the entire input into the production of the public good.

Substituting from the conditions and taking the derivative with respect to Di gives the following first-order condition:

FOC_D_i :−u_x−2a(D_i−D^e_i)u_I+ ∂G_D

∂D_i f⁰(G_D)u_G= 0

⇔ D_i =D_i^e− u_x 2au_I

The last step follows from the observation that the derivative of GD with respect to D_i must be approximately zero. This is because there are enough individuals so that one individual’s contribution has a minimal effect on the average contribution. Because u_I > 0, u_x > 0, and a ≥0, 0 ≤ D_i ≤ D^e must hold. The result is intuitive. The higher the marginal value of consumption an individual has, the lower is the donation. If the individuals are more “moral”, in the sense that they experience a high cost by deviating from what they think is morally right, they will choose to actually donate more. Finally, a higher morally ideal donation yields a higher actual donation. An interpretation of the condition is that we will never have over-provision of the public good G. By over- provision I mean a situation in which the level of the public good is higher than the efficient outcome. To illustrate how the individual optimization differs from the Kantian thought experiment we can compare the two conditions in figure 1:

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D_i

D^e_i Kantian

Individual

ux

2aui

Figure 1: Kantian vs Individual Donation with Self- Image Cost

As illustrated in figure 1, the individuals will donate the level they find morally ideal in the Kantian thought experiment. This leads to the 45-degree line in the figure. When individuals apply the Nash solution with a self-image cost they will donate weakly less than what they consider to be morally ideal. This difference is the cost of giving up consumption weighted by the marginal self-image cost.

3.1.3 The Effect of Public Intervention

I have now investigated how the dual framework with Kantian optimization works in absence of public intervention. Is there a case for public intervention? And if so, of what form? I now assume G_p ≥ 0. The government has to finance the contribution through a tax τ(D_i). The model is otherwise the same. First, we need to decide the criterion from which the government in our model decides their behavior. One natural candidate is, since all individuals are alike, social efficiency. If such a government has all tools available to them, a natural behavior would be to follow the result of Samuelson (1954). Since the morally ideal donation is efficient, as described above, this would mean that the government would provide the aggregate morally ideal level of the public good

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everyone else donates the same as you, will then be zero. To see why we can consider the Kantian thought experiment when the government already provides the efficient level of the public good. Any donation above this level would yield an over-provision. And since all individuals are alike, they would all reach this conclusion. Looking at the morally ideal donation above I can conclude that the individual donation will then be zero since negative donations are not allowed. The self-image cost will then also be zero. The presence of a self-image cost would not affect this outcome, rather it would facilitate it. The self- image cost will provide a further incentive for the government to not deviate from the efficient level. This because a deviation will not only decrease the social welfare through a lower level of public good but also inflict a further cost on all individuals through their moral self-image. It is worth noting that this result requires a certain sophistication from the individuals. They have to take the government actions into account when they do their Kantian thought experiment. The result also depends on the functional form of the production function which here makes no distinction between public and private funds.

3.2 The Heterogeneous Self-Image Model

As stated earlier, one of the goals of this thesis is to extend the model by allowing for heterogeneity. The question is where to introduce it. A natural place to start is in how important their moral self-image is to them. It seems intuitive that individuals value

“doing the right thing” to a varying degree. It will also, as I explain later, allow for a simple solution. I introduce this heterogeneity by defining the weight on the self-image a_i >0. The model otherwise looks the same:

U_i =u_i(x_i, G, I_i) (8)

x_i+D_i =m−τ(D_i) (9)

G=f

G_p+ Z 1

0

D_idi

(10)

I =−a_i(D_i −D^e)² (11)

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The same assumptions apply as earlier. The only difference is that a_i now varies across the individuals. Again the individuals first find their morally ideal donation before the individual optimization.

I first look at the case where G_p = τ(D_i) = 0. The reader might think that since we are in a heterogeneous setting, the simple Kantian equilibrium will not hold and that we, therefore, have to use the multiplicative Kantian equilibrium. Surprisingly, as I will show, this is not the case. To explain we can repeat the discussion I also had in the homogeneous model. How will the self-image cost affect the solution? The same line of reasoning holds as earlier even though a_i varies. In the Kantian equilibrium, the individual finds the morally ideal donation. Therefore, D_i = D^e_i still holds in the equilibrium and I_i = 0 no matter the value of a_i. As we know that the self-image cost is equal to zero in the equilibrium, the heterogeneity will not affect the equilibrium outcome. The individuals can in this heterogeneous setting still do the simple Kantian thought experiment and expect other individuals to end up with the same conclusion as them. This leads to the conclusion that the simple Kantian equilibrium can be applied. If the heterogeneity was included somewhere else in the model, for example in the income or another part of the preferences, the same reasoning would not hold. For the simple Kantian equilibrium to break down, the heterogeneity has to be included so that the individuals prefer different levels of the public good when doing the Kantian optimization. Since the calculation and result will be analogous to those in the homogeneous model, I will not include them here.

To find the actual donation level I use the Nash solution. The calculation is analogous to the one in the homogeneous model. The solution also looks very similar to the one we had in the homogeneous model, only that we now have heterogeneity in the weight of the

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self-image cost:

D_i =D_i^e− u_x_i 2aiuIi

(12)

whereD_i^eis the same as in the homogeneous model. The interpretation will be the same as earlier, but now the size of the positive effect from the self-image cost will vary positively with the variableai. The condition implies that the individuals with a higheraiwill donate more. We see that with this type of heterogeneity we also never get over-provision.

Let us again look at the case where a government wants to intervene in the economy.

Another implication of the morally ideal donation being unaffected is that the Kantian equilibrium stays efficient. If the government now can collect a lump-sum tax to finance the efficient outcome, why would they not want to do so? To show that they would I can argue analogously to how I did in the homogeneous setting. If the government already provides the aggregate morally ideal level, then any further donation will lead to over- provision. Therefore, the morally ideal donation will be zero. This follows from the fact that everyone will have the same level of morally ideal donation. Since the self-image cost is zero in the Kantian equilibrium this holds no matter the self-image weight of each individual. In the individual optimization, they will all have a morally ideal donation of zero and therefore end up donating zero and consuming their entire disposable income.

3.3 The Heterogeneous Preferences Model

As shown above, allowing for a heterogeneous self-image will not have a great impact on the outcome of the model. In this part, I will look at how the results change if we introduce heterogeneity in another part of the preferences. As stated above the model will not be solved as easily if people prefer different levels of the public good. Such a situation might be more realistic, especially in a setting of charity. Let us, therefore, look at how the solution changes when we allow for heterogeneity in the valuation of the public

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good. The model will be similar to the one we had earlier. The difference now is that I introduce heterogeneity in the valuation of the public good by introducing the parameter bi >0. Also, a is again a fixed parameter. The model looks as follows:

U_i =u(x_i, b_iG, I_i) x_i+D_i =m−τ(D_i) G=f

Gp+

Z 1 0

Didi

I =−a(Di−D^e_i)²

Otherwise, the same assumptions apply as earlier. For notational purposes, I will in the following section use G_D as total donations andG^e_D as the total morally ideal donations.

To illustrate I will first solve the model without any public intervention, meaning that I start by looking at the case where G_P = τ(D_i) = 0. As discussed in the background section, heterogeneity that affects the preferred level of public good makes the simple Kantian equilibrium break down. An individual valuation of the public good will make the individuals prefer different levels of the public good. To illustrate why this becomes a problem, we can think of an individual who uses the simple Kantian equilibrium while knowing that everyone values the public good differently. She will ask the question “what is optimal to do, given that everyone else acted like me?”. The individual might find a strategy, but can not expect it to hold. The reason is that when everyone asks this question they will not come up with the same result. One can of course argue that the simple equilibrium still is applicable in the dual preferences framework. Here the Kantian equilibrium is just meant as a thought experiment, and the individuals are not expected to actually play their Kantian optimization. However, not taking the strategy of others into account when making the thought experiment seems less realistic. Therefore, we can instead apply the solution concept of multiplicative Kantian equilibrium discussed in the background section. The individuals will now search for a level of donations from which

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nobody would prefer to deviate if that meant that everyone else would deviate by the same non-negative factor. In more formal terms each individual will find a level D^e_i such that:

∀i, ∀r ≥0, U_i(D^e)≥U_i(rD^e)

Where D^e is the vector of every individual’s morally ideal donation D_i^e and Ui the utility function of individual i. How do we characterize such an equilibrium? To identify the equilibrium I will use a method outlined in Roemer (2019). The solution will have to be a point where the cost of deviating, given that everyone else deviates with the same non-negative factor is greater (for every non-negative factor) than the benefit from doing so. So it will have to be a point where there is no benefit from changing r when r= 1. It seems logical that one should be able to characterize such a point by optimizing over r, in the point where r= 1. Since the preferences are concave this is a possible solution.

Again we have to discuss how the self-image variable affects the outcome. Now that we have heterogeneity that affects the preferred level of the public good, so that people will have individual morally ideal donation parameters, will the self-image cost play into the calculation of the morally ideal donation? Again the answer is no. Even though every individual now gets a unique ideal donation, the donation they find through the multiplicative Kantian process will still be their morally ideal donation. In equilibrium Di =D_i^e, meaning that Ii = 0. To simplify I replaceDi with D_i^e below. I also include the factor r in the model, to get:

U_i =u(x_i, b_iG, I_i) (13)

x_i+rD^e_i =m (14)

G=f(rG^e_D) (15)

I =−a(rD^e_i −rD^e_i)² = 0 (16)

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I substitute for the arguments of U_i by using (14) - (16). This allows me to characterize the point by taking the first-order condition of the utility function with respect to r in the point where r= 1:

d dr

r=1

u(m−rD^e_i, b_if(rG^e_D), I_i) = 0

⇔f⁰b_iu_GG^e_D =D^e_iu_x (17)

⇔D_i^e=f⁰ b_iu_G

u_x G^e_D (18)

Now we have found the characterization of the morally ideal donation level across all individuals. Taking the derivative with respect to r again yields the following second- order condition:

f⁰⁰u_G(rGê_D)²+ [f⁰(rGê_D)rGê_D]² ∂²u

∂G² +D_i^e2∂²u

∂x² < f⁰D^e_irG^e_D( ∂²u

∂x∂G + ∂²u

∂G∂x) Examining this second-order condition, we see that the LHS always will be negative due to the concavity assumptions. This means that as long as the RHS is weakly positive, the condition will hold. The implication is that the cross derivatives will have to be positive or zero for the point to be a maximum point, which they are under our assumptions. The interpretation is that the public and private good has to be weak complements.

Now that we have found a solution it is worth taking a moment to discuss it. Let us first look at how the condition is formulated in (18). The individual weight on the public good b_i has a positive relation with the morally ideal donation of the individual. This makes sense on an intuitive level. The higher you value the public good, the higher is the desired donation level. One assumption has to be made for this to be true. A higher b_i also implies less consumption, and therefore a higheru_x. To say that a higherb_ileads to a higher morally ideal donation, we also have to say that when we compare two individuals with different b’s, the difference inbis larger than the difference inu_x. The total donation

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level has two opposite effects on the individual i’s morally ideal donation. First, a higher total donation level will give a lower individual ideal donation because f is concave. At the same time, a higher total donation also has a positive effect on the individual’s ideal donation. If everyone donates more, the effect from a deviation by everyone will be larger.

Another way of interpreting the condition is found by looking at (17). When the condition is formulated as it is there, we can see that the marginal benefit from deviating, given that everyone deviates in a similar fashion, has to equal the marginal cost of doing so.

The RHS is the cost of deviating when everyone deviates in a similar fashion, and the LHS is the benefit. The RHS is increasing in r, whereas the LHS is decreasing. In the equilibrium, these two will be equal. To illustrate this interpretation we can illustrate (17) in figure 2:

D

_i^e

u

_x

f

⁰

b

_i

u

_G

G

^e_D

r = 1

Equili- brium

Figure 2: Multiplicative equilibrium r Marginal

cost/benefit

Figure 2 illustrates the multiplicative Kantian equilibrium found above. I should point out that the condition is illustrated in the graph as if the utility function satisfies the Inada conditions. The assumptions of the model presented here are weaker.

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As pointed out by pointed out by Sher (2020), there also exists another possible solution to the multiplicative equilibrium, namely the point where D^e = 0 (here 0 denotes the null vector). To understand we can think of a situation in which everyone donates zero. Deviating from zero with a nonnegative multiplicative factor will only yield zero, meaning that people will be indifferent between any r. This is a special case of the devi- ations we are checking in our equilibrium candidate above, namely r= 0. As we checked earlier this will not be a beneficial deviation. Another argument against this equilibrium is that there is no case for the RHS of (18) to become zero unless b_i = 0.

3.3.2 Efficiency

An interesting question is whether our multiplicative Kantian equilibrium is efficient. The answer is not intuitive. To check the efficiency of the solution we can solve the planner’s problem. If the equilibrium corresponds to a decentralized version of the planner’s solution the equilibrium solution is efficient.³ By decentralized I mean that all the individual solutions aggregate to the planner’s solution. For the planner to solve the problem under the same circumstances as the individuals, they solve it when the self-image cost is zero.

The planner’s problem will look as follows:

maxD_i^e,xi

Z 1 0

u(x_i, b_iG, I_i)di s. t. G=f

Z 1 0

D_i^edi

Z 1 0

(xi+D^e_i)di= Z 1

0

m di

3Proposition 3.1 in Roemer (2019) states that a strictly positive multiplicative equilibrium of a monotone game is Pareto efficient in the game. With a “strictly monotone” game Roemer means a game in which every individuali’s utility is strictly increasing or strictly decreasing in the strategy of every other player (simple externalities). But since this setting is quite different from the one Roemer is looking at I have taken the time to check myself. Another way of proving efficiency is by solving the planner’s problem using Negishi weights. If we can find weights so that the solution above and the solution of the planner’s problem are equivalent the solution is efficient. However, such a method will become quite complex with an infinite amount of individuals.

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The first-order condition becomes:

Z 1 0

(b_i u_Gf⁰)di= Z 1

0

(u_x)di

Taking the integral of the equilibrium condition over all individuals yields the same result.

Therefore, we can conclude that the multiplicative Kantian equilibrium found above is efficient.

Now that we have found the morally ideal donation we can move on to the individual optimization. With all preference parameters of the dual preferences framework in place, how much will the individual actually donate? Using the Nash solution, by taking everyone else’s donation as given, we end up with a first-order condition similar to the one in the homogeneous model:

D_i =D_i^e− u_x

2au_I (19)

The difference now is that D_i^e is a function of the weight on the public good bi and the total level of donations from the Kantian equilibrium. As discussed earlier, the morally ideal donation of the individual i will increase in the weight on the public good. This means that an individual with a higher valuation of the public good typically will donate more. This effect is further strengthened by the effect from the self-image. As in the other models, the condition shows that there will not be over-provision of the public good.

Let us now open for a government intervention by removing the assumption thatτ(D_i) = G_P = 0. First, a short discussion is in order. Is this case any different than the ones we had earlier? Why should the government in this case not provide the efficient level of the public good and finance it through a lump-sum tax? To answer we can imagine a government providing the outcome of the multiplicative Kantian equilibrium above.

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If the government provides this level of the public good, some individuals would prefer a lower level if that meant that they could get a higher income. Were they to set the level lower than this, some individuals would prefer a higher level of the public good and therefore experience a self-image cost. Finding an optimal level of the public good will not be as simple with this sort of heterogeneity, as long as the government is unable to tax based on the individual valuation of the public good. Maybe a better outcome could be achieved if the government instead facilitated further donations by the individuals themselves? In this way, one could get a higher provision and lower the self-image cost.

I will now introduce a linear tax schedule. The tax is decreasing in the donation of the individual. More specifically, the tax will take the form τ(D_i) = T − tD_i. All the tax revenue of the government will go to providing the public good, meaning that R1

0 τ(D_i)di = G_P. I will here focus on the optimal tax deduction t, and keep the tax T exogenous. To understand the behavior of the government we need to consider how the tax will affect the individuals’ behavior. Therefore, we need to solve the model while we are including the tax schedule. To find the multiplicative Kantian equilibrium we can use the same method as earlier. We derive the point from which no deviation is beneficial when r = 1. One clarification I need to make before doing the calculation is whether the individuals are sophisticated enough to consider the effect their donations will have on public spending. In the case where we only use a Nash equilibrium, the individuals will take the government behavior as exogenous. The conclusion follows from the fact that there are so many individuals that each donation will have a negligible effect on the tax revenue of the government. With Kantian optimization, on the other hand, people will take the strategy of everyone else as part of their own strategy. Assuming that they consider the effect of a change in their strategy on public spending as negligible therefore seems less plausible. The circumstances under which the individuals now find the multiplicative equilibrium looks much the same as earlier. The difference is in the budget constraint and the input into the production function of the public good. I will also assume that the utility function is additively separable in its arguments. The budget

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constraint now looks as follows:

x_i+ (1−t)D_i ≤m−T

The condition tells us that the sum of donations and consumption has to be less or equal to the disposable income of the individual. The input of the function for the provision of the public good has been changed to include the tax revenue of the government:

G=f Z 1

0

(T −rtD_i^e)di+ Z 1

0

(rD^e_idi)

=f Z 1

0

(T +r(1−t)D_i^e)di

I substitute for all the arguments to get:

d dr

r=1

u

m−T −r(1−t)D^e_i, b_if Z 1

0

(T +r(1−t)D^e_i)di

, I_i

= 0

⇔ −u_x(1−t)D^e_i +b_iu_Gf⁰ Z 1

0

(T + (1−t)D_i^e)di Z 1

0

((1−t)D_i^e)di = 0

⇔D_i^e= biuG

u_x f⁰ Z 1

0

(T + (1−t)D_i^e)di Z 1

0

D^e_idi

The final step follows from (1−t) canceling out on each side. Let us first consider the effect of the tax deductiont. At first glance, the only effect that we clearly see is that the morally ideal donation will increase with the tax deduction because it reduces the public provision of the good. This effect is also strengthened by u_G, which will increase as G decreases from the change in public provision. An interpretation is that the individuals will feel a greater responsibility to donate, when the price of doing so decreases, and when the public provision goes down as a result. Looking closer, there are several less obvious effects from the tax deduction. The effect of donations having become cheaper relative to consumption will come in through u_x. With the donations becoming cheaper, consumption will typically change as a result. Which direction depends on whether the substitution effect or the income effect dominates. Some effects will be harder to identify due to the fixed point character of the solution. How will the individual’s morally ideal donation be affected by the change in the total donation level? This level is of course

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again affected by each individual donation. It is hard to conclude decisively as to which direction the tax deduction affects the morally ideal donation. Still, it is reasonable to assume that the deduction leads to a higher morally ideal donation.

Even though T is exogenous it is interesting to consider its effects. The effects of the tax T will be similar to that of the tax deduction. Because of T’s direct relation to the public provision of the good the tax will have a negative effect on the morally ideal donation. This can be seen as the crowding out effect of the public good. Furthermore, T will lead to a decrease in the morally ideal donation through the derivative of u with respect toGandx. An increase in G as a result of an increase in T will lead touG decreasing. At the same time, u_x will weakly increase as a result of less disposable income. Even though all the effects above seem to suggest that a higher T leads to a decreased morally ideal donation the fixed point character of the solution makes it hard to conclude decisively.

Having discussed the effects the tax scheme has on the morally ideal donation, we can move on to look at the actual donation. The optimization is the same as earlier, only that we get t and T in the budget constraint and the provision of the public good. It is worth noting that with the Nash solution individuals will go back to not considering the effect their donation has on public provision of the public good. This leads to the following characterization of the individual donation:

D_i =D^e_i − ux(1−t) 2au_I

How can we interpret the effect of the tax scheme on the individual donation? First, the donation is increasing in the morally ideal donation. And since the ideal donation is included as it is, we can say that many of the effects we discussed when interpreting D^e_i works in a similar fashion here. To consider the other effects, let us start with the effects of the tax deduction. First there is the direct effect of t in the expression. This can be seen as a direct effect of increased disposable income. The effect onux will be of a similar character as the one in the morally ideal donation. There are effects going in both

A Kantian Approach to Charitable Behavior Oliver Groth Pettersen Thesis submitted for the degree of Master in Economics 30 credits