# Economics with some altruism

Robert Waldmann

Arnold Kling states a widely held view succinctly

I see elite Democrats as exploiting “the poor” in order to pursue their own drive for power. People who genuinely care about the poor work with the poor and give their own money to the poor. Liberals mostly just pose about caring for the poor and take money from others (including many who are not well off) to burnish that pose.

Kling’s claim is a mathematical error as is shown after the jump by formal analysis of a model in which no one wants power and no one cares about his or her pose but people support tax and transfer programs even though they don’t give to the poor.

Consider a world in which people are slightly altruistic. So their happiness is not just the pleasure they get from consumption but the pleasure they get from consumption plus a tiny multiple of the pleasure everyone else gets from consumption.

Let’s say there is just one good. The pleasure is an ordinary concave utility function of consumption u(c) with u”(c)
u(c_i) + (sum over j not equal to i of u(c_j))alpha

alpha must be tiny.

Now assume that people have different endowments of this good and the world lasts one period. For simplicity say 2/3 are rich with a high endowment Ch and 1/3 are poor with a low endowment Cl Assume that u'(Ch)
update: typo corrected above thanks to anonymous, Cstar and Arnold Kling in person. Also on Kling’s suggestion I added a numerical example (watch out for typos)

People have 2 choices. They can give to others and they can vote on the level tau of a flat tax the proceeds of which are shared equally.

There will be no giving if, for the richest person i and the poorest person j
u'(Ch) = (u'(Cl))alpha. The rich person has lower marginal utility of consumption than the poor person, but alpha is much less than one so the optimal gift is zero

So what tax would there be ? In this case, by assumption, people can’t gain or lose power. They don’t have poses. They don’t feel pride or shame.

The tax rate is decided by the rich majority. It is not zero. Raising the tax above zero has 2 effects on rich person i — she has to pay the tax which is like giving to the poor. By assumption at tau=0, the rich person is indifferent about giving a little. The tax also transfers money from other rich people to the poor. This pleases rich person i as it increases average pleasure from consumption since u is concave. So all will support a tax and transfer program.

Kling’s claim that a person’s willingness to tax and transfer without a willingness to give shows that person doesn’t really care about the poor. He is wrong. In the model I have described people have no other motive to support a tax and transfer program and they do support it.

To be a bit more general, for any positive alpha, there can still be unanimous support for a positive tax even if rich people would like to take from the poor
u'(Ch)
Obviously the model is not realistic. People do feel shame so they care about something other than their consumption and the consumption of others. The point is that there is no need to introduce shame or pride or poses or the will to power to explain why some people who don’t give to charity support tax and transfer programs (or why others who have given all they plan to give then go and vote for politicians who implement tax and transfer programs).

The common claim is based on carelessly assuming that people are either perfectly altruistic or completely indifferent to the interests of others.

update: numerical example as requested by Arnold Kling himself in comments

A numerical example (likely more typos)

lets make Ch= 4 and Cl = 1 (before taxes and transfers). u(c) is the square root of c so

u'(Ch)= 1/4 and u'(Cl) = 1/2. there are 100 rich and 100 poor. Alpha = 0.1

1/4 > 0.1/2 so the rich don’t give to the poor.

The tax is returned equally to all so the lump sum is 2.5tau

The income of the rich after taxes is 4-4tau+2.5 tau = 4-1.5tau

The income of the poor after taxes is 1-tau+2.5tau = 1+1.5tau.

The example is extreme as there is no dead weight loss due to the tax. Income just happens there is no saving or labor supply or anything.

the utility of the rich is their own pleasure from consuming square root of (4-1.5tau) plus one tenth of the pleasure from consuming of the 99 other rich people or 9.9(4-1.5tau) plus one tenth of the pleasure from consuming of the poor or 10(1+1.5tau).

At tau equals zero the dereivative of the utility of the rich with respect to tau is – (10.9)(1.5)/4 + (10)(1.5)/2 = (9.1)(1.5)/4 >0.

Now alpha = 0.1 is really huge (on the other hand a population of 200 is really small)