Werner's Blog — Opinion, Analysis, Commentary
The economics of altruism

One of the topics we do not much teach in microeconomics courses is the notion that people care about other people. Our standard tool for analyzing economic behaviour is the utility function that only depends on one's own use of goods and services, our own income, and the prices we face in the economy. Our standard "selfish" model of utility does not allows for the fact that most of us in society care about the well-being of others. Behavioural and experimental economics is making much progress studying social behaviour related to the pro-social preferences. Specifically, there are three motives why we care about others:

  • Fairness expresses the notion that we care about the income distribution in society, that is, the avoidance of inequity. This notion is often referred to as inequity aversion.
  • Reciprocity is linked to the expectation that if we are nice to others, they will be nice to us (for example, when we need their help).
  • Altruism is selflessness out of intrinsic personal values or societal norms, or even evolutionary biological imperatives. Sometimes a distinction is made between pure and impure altruism. In the latter case, a donor derives prestige or utility from the act of giving itself, sometimes referred to as warm glow. On the other hand, pure altruism does not reward the donor with any benefit.

Embedding notions of fairness (inequity aversion) in economic models is not altogether difficult. Consider a simple model in which person \(i\) has utility \(U_i\) that is derived both from own consumption as well as consumption by others. Let \(0\le\beta<1\) be a coefficient that describes how much person \(i\) cares about person \(j\). Further assume that each person has an endowment of goods \(y_i\) and has the choice of transferring \(0\le x_{ij}\lt y_i\) to someone else. Than person \(i\)'s utility is given by \[ U_i = v(y_i-x_{ij})+\beta\cdot v(y_j+x_{ij}) \] To simplify things, let us assume that the utility function \(v(\cdot)\) is logarithmic. Then maximizing utility generates a first-order condition \[ \frac{\mathrm{d} U_i}{\mathrm{d} x_{ij}}=0 \quad\Leftrightarrow\quad x_{ij} = y_i\cdot\left(\frac{\beta-y_j/y_i}{1+\beta} \right)\] What this says is that someone with income \(y_i\) will give a fraction of that income to the other person as long as that person's income is sufficiently small: \(y_j\lt\beta y_i\). This constraint is very important as it says that altruistic behaviour only happens when the income gap is sufficiently large. After redistribution, consumption for both persons becomes \[ \tilde{y}_i = \frac{y_i+y_j}{1+\beta}\quad;\quad \tilde{y}_j=\beta\frac{y_i+y_j}{1+\beta}\] From this it is also apparent that \(\tilde{y}_j/\tilde{y}_i=\beta\). The coefficient \(\beta\) of altruism puts a lower bound on the poorer person's income.

The Gini coefficient is a widely-used measure for income dispersion. In the case of two incomes \(a\) and \(b\) with \(a\lt b\), it is simply given by \(G(a,b)=(b-a)/(a+b)\). The Gini coefficient is zero when incomes are distributed equally, and it is one when the income is perfectly concentrated. At the outset, we started with an income inequality of \(G=(y_i-y_j)/(y_i+y_j)\). After the altruistic redistribution, the new level of inequality is defined by the altruism parameter; it is \(\tilde{G}=(1-\beta)/(1+\beta)\). This must be an improvement. Because we know that transfers occur only when \(y_j<\beta y_i\), it is easy to show that \(G\lt(1-\beta)/(1+\beta)\), and therefore \(\tilde{G}<G\). Modern societies all employ some form of income redistribution through the system of taxation. In Canada, we observe an income inequality that has been about 0.32 since the early 2000's. Translated into our two-person model world, a Gini coefficient of 0.32 would imply an altruistic preference of roughly 0.5. The richer person cares about the poorer person about half as much as about himself! The richer person would start transferring consumption to the poorer person if the richer person started out at least twice as rich as the poorer person. Quite an astounding insight. How far does inequity avoidance go? If someone cared equally about another person in the model \(\beta=1\), then the income distribution would be perfectly equal.

The above model has some interesting implications and testable predictions. If we increased the endowment of person \(i\) by an infinitesimal amount, that person would increase the transfer to the other person by \(\beta/(1+\beta)\). If the rich person in our model has a preference parameter of \(\beta=0.5\), then this person would give a third of his additional (marginal) income to others. Interestingly, that is not very far away from typical marginal rates of income taxation in many OECD countries.

You may wonder if there was something special in the assumption of logarithmic utility, and making the utility additively separable. It made the algebraic results particularly easy, but similar results can be obtained under other types of specifications and with other, more complex, utility functions.

What have we learned from this back-of-the envelope calculus above? First, we should build altruistic motives more into our economic models. The empirical evidence clearly supports the notion that societies care about their economically weaker members, although there is great heterogeneity across cultures and countries, and variation over time within societies. Second, embedding non-selfish behaviour in economic models can explain a number of economic paradoxes. Among them is the voting paradox—the fact that people vote in elections even though the benefit of participating (casting the deciding vote) is usually tiny while the cost of participating (the act of casting a ballot) is much larger. Voters bear the cost of casting a ballot because they think that their preferred policy will benefit society.

Advanced Altruism Algebra

If you are interested more in the formal modeling of altruistic behaviour in economics, I have a little more to offer. When we have more than just two individuals in the model, it is often useful to work with a bit of matrix algebra. Assume that person \(i\)'s utility depends on everybody else's utility with weight \(w_{ij}\gt0\): \[ U_i = \alpha v_i + \sum_{i\ne j} w_{ij} U_j \] That is, \(w_{ij}\) is how much person \(i\) cares about person \(j\). The scalar \(\alpha\) we will choose later so that \(\sum_i v_i=\sum_i U_i\) and adding altruism to the picture does not add utility to the overall system. In matrix form this is \[ \mathbf{U}=\alpha\mathbf{v}+\mathbf{W}\mathbf{U} \quad;\quad \mathbf{W}=\left[\begin{array}{cccc} 0 & w_{12} & \cdots & w_{1n} \\ w_{12}& 0 & \cdots & w_{2n} \\ \vdots&\vdots & \ddots & \vdots \\ w_{n1}& w_{n2} & \cdots & 0\end{array}\right] \] and therefore \( (\mathbf{I}-\mathbf{W})\mathbf{U}=\alpha\mathbf{v} \) and \(\mathbf{U}=\alpha (\mathbf{I}-\mathbf{W})^{-1}\mathbf{v} \equiv\mathbf{C}\mathbf{v}\). The matrix \(\mathbf{C}\) provides a link between everyone's "selfish" utility and everyone's total utility. Let's consider the simplest case where everyone feels equally altruistic so that \(w_{ij}=\beta/(n-1)\), and let us set \(\alpha=(1-\beta)\). That means a person allocates a share \(1-\beta\) to her own utility and a share \(\beta\) to all others. It is then possible to solve the above matrix equation as \[U_i=\frac{1-\beta}{1+\beta/(n-1)} v_i +\frac{\beta/(n-1)}{1+\beta/(n-1)}\sum_{j=1}^{n}v_j\] What that means is that when someone else's utility improves, this raises person \(i\)'s utility by a positive amount.

In the above framework it is most useful to use a quadratic utility function of the type \(v(y)=y/z-(y/z)^2/2\), with marginal utility given by \(v'(y)=(1-y/z)/z\gt0\) together with the range constraint \(y\lt z\). One can think of \(y/z\) as a form of income index from zero to one, and \(z\) as the highest possible income. When we add utility over all people in this economy, we obtain \[\sum_j v_j = n\left[v(\bar{y}) -\frac{1}{2}\sigma^2\right] \equiv n\Phi\] where \(\bar{y}=(1/n)\sum_j y_j\) is the mean income, \(\bar{v}\equiv v(\bar{y})\) is the utility derived from the mean income, and \(\sigma^2=\sum_j(y_j/z-\bar{y}/z)^2/n\) is the standard deviation of the income index. That is, \(\sigma^2\) is a measure of income dispersion in society. Thus, \[U_i=\frac{1-\beta}{1+\beta/(n-1)} v_i +\frac{n\beta/(n-1)}{1+\beta/(n-1)}\left[\bar{v} -\frac{1}{2}\sigma^2\right]\] What the above equation says is that a person's utility: (a) increases with her own utility; (b) increases with the utility of the average person in society; and (c) decreases with the income dispersion in society. In other words, when people feel atruistically in society, they care about a more equal society—a society with less income dispersion.

The above equation also reveals something else: the policy dilemma between making a society "more equal" and "more affluent". Suppose there was a public policy that raised the median income but led to a higher level of income dispersion. If the income gain is sufficient to compensate for the more unequal society, such a policy would still be desirable. Conversely, a policy that reduced income inequality but depressed mean income even more would not be desirable. The goldilocks policies are those that raise income and reduce income dispersion at the same time.

In the above model I had assumed that everyone cares equally about everybody else. That is, of course, not true. I care a lot more about the people dear and close to me: my family, my friends, my relatives than about people I have never met. Perhaps I care a bit more about the poor, the downtrodden, the disadvantaged people in our society than about the rich and powerful. When I see homeless children it makes me feel sad and ashamed that we can't do better in our society. When we see people fleeing violence and oppression in their home lands and seeking refuge in peaceful and prosperous places, should we not be more welcoming and supportive? The need for benevolence, individually and as a society, is enormous. As Christmas is approaching, it is to remind oneself that the most in need of gifts and benevolence are those that have little or nothing, not us who already enjoy most creature comforts modern society has to offer.

One last point about the importance of altruism. What happens if society becomes more altruistic? That is, what happens if we raise \(\beta\) by a very small amount? For large \(n\), we find that \(\partial U_i/\partial\beta\approx \Phi-v_i\). Because \(\Phi\) is constant, the sign of that expression depends on whether \(v_i\) is large or small. In a more altruistic society, more affluent people have a somewhat lower utility and poorer people have a somewhat higher utility. That is because richer people attach lower value to their direct utility (their own well-being, which is higher than average) and poorer people attach more value to their indirect utility (the well-being of others, who are richer on average). Perhaps these are not very deep insights, but the model world above can be tweaked in many different directions. It can be used to analyze how different policies affect different parts of society. For example, how would an income tax play out, and different versions of such taxes? One can experiment with heterogeneity of altruism across different groups in society. For example, what happens when altruism is correlated with income? Or what happens when altruism is correlated with societal characteristics such as urban vs. rural populations? Much to think about as we're going into the festive season this year.

Further readings:

Posted on Thursday, December 24, 2015 at 18:30 — #Economics
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