Steve Landsburg Asks For An Example

20 Mar

In a recent post on his blog, Steve Landsburg critiques an article by Uwe Reinhart. Steve Landsburg believes strongly that tax on capital gains should be zero. Uwe Reinhart argues that sometimes it’s easy to disguise earned income as capital gains, hence capital gains should be taxed at the same rate as earned income. Steve Landsburg, in his critique, asks :

Can you please write down the optimization problem which has as its solution “tax the doctor’s income and the homeowner’s capital gain at the same rate”?

However, it’s actually not difficult to construct an example exactly like this.

So, here’s my example. What follows is quite technical, but here’s a non-technical summary :

Imagine there’s a rich guy who doesn’t get much of a kick out of money, and a poor guy for whom money is very important. The tax rates on earned income and investment income don’t affect the rich guy’s happiness very much, but they greatly affect government revenues. However, the choice of tax rates greatly affect the poor guy’s happiness, without much affecting revenue.

If the poor guy likes spending up-front, it makes sense for the government to pay its bills by taxing capital gains. On the other hand, if the poor guy has more of a preference for savings, it can make sense to tax earned income instead. After all, the rich guy in this scenario doesn’t care. In other words, the optimal tax rates depend very much on the poor guy’s preferences. It can easily be the case that investment income and earned income should be taxed equally. In fact, it can go way off the scale in either direction, with one of wages or investment income heavily taxed in order to subsidise the other.

Now for the technical bit….

Imagine two people. They earn incomes I0 and I1, get taxed at a rate of p, spend X0 and X1, then save the rest. Some years later, they get taxed on they capital gains and spend the rest.

Converting all future dollar amounts to today’s dollars, they earn incomes I0 and I1 today, get taxed at a rate of P, spend X0 and X1, then save the rest. Some years later, their savings are taxed at a rate of Q, leaving them with Y0 and Y1 dollars respectively, which they spend.

The two people get to choose how much (X0 or X1) they spend today. This determines Y0 and Y1, assuming they know Q, so X0, X1, Y0 and Y1 are all functions of P and Q.  The individuals make their choices to maximise their happiness (the technical term is “utility”), which I’ll give a formula for. Let U0 = A0 * X0^B0 * Y0^C0, and U1 = A1 * X1^B1 * Y1^C1. Now the government must choose, in advance, the tax rates P and Q. The constraint is that the government’s total tax take Ttot (in today’s dollars) is fixed, but apart from that, the government will try to maximise the total happiness of the population. So, here’s the optimisation problem:

Maximise U(P,Q) =  A0 * X0(P,Q)^B0 * Y0^C0 + A0 * X0(P,Q)^B0 * Y0^C0, subject to
T(P,Q) = P * (I0 + I1) + Q * ((1-P) * (I0 + I1) – (X0 + X1)) = Ttot

That’s a bit hard to solve in general, so let’s look at a special case, B0=1, C0=2, B1=2, C1=1. Here, person 0 is a saver – he or she gets more happiness out of Y than X. Person 1 is the opposite, a spender. I’ll also let Ttot be 1/10 of I0+I1, just for the sake of having a specific number. For the same reason, assume I1 is twice I0.

If A0 is four times A1, the optimal value of Q turns out to be 0, in other words, savings (that is, interest and dividends) are not taxed, and all the tax burden falls on the initial earned income. The tax rate P is 10%, matching Ttot.

On the other hand, we can make all the tax burden fall on savings by assuming instead that A0 = 1280/713 A1, that is, A0 is about 1.795 times A1. Then, the interest on earned income is nil, and the present value of savings are taxed at 22.5% before you get to spend them.

But Should The Government’s Tax Policy Aim To Optimise Total Utility?

One objection to trying to tax utility is that we can’t measure utility. However, Steve Landsburg wont make that objection, because he dismisses it when Uwe Reinhart makes it – Uwe says “we can’t always tell whether income is investment or earned” and Steve says “So what? That doesn’t mean treating them equally is good policy”.

So if you say “we can’t really tell in real life if one person gets more satisfaction out of spending than another”, I can say “so? That doesn’t mean it’s good policy to assume they don’t and treat them so.”

Another objection might be raised on a point of fairness – why consider only one person’s happiness when choosing the tax rates? The answer is that in fact, in this example, both person’s happiness was considered, but (in the example) one of them really doesn’t care much how things are taxed, and they happen to be the person with all the moola. If their preferences are so different, why would you insist on pretending they aren’t?

Maximising total utility may not be a practical policy, but any alternative means that you’re insisting on imposing policies on people that they don’t actually (in aggregate) like. How could that possibly be good?



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  1. Steven E Landsburg

    March 20, 2012 at 8:05 pm

    If your agents have different utility functions, you don’t need separate rates on income and capital gains — you can just tax income at different rates for the two different people.

    I haven’t worked out the math, but this has got to lead to a better outcome than your proposed policy, because it gets the distributional issues right without distorting the saving choice, so the rich guy achieves a better allocation of consumption across time.

    You have assumed that the tax rates P and Q are the same across people, which is to say that you’ve ruled out progressive taxation. But much of the point of Chamley-Judd is that if you want to transfer income from the rich to the poor, you can do it more effidiently with a progressive tax code than by taxing capital.

  2. Mike H

    March 21, 2012 at 6:42 am

    However, surely taxing people at different rates depending on their preferences would create horribly strong incentives to lie about those preferences? In practice, is this not likely to create more deadweight loss than an approximately optimised system of uniform taxes? Or is there some clever system that optimises each person’s individual tax rate, yet keeps them all honest?