The other day I wrote about the potential impact of a wealth tax. In so doing, I wrote: “we can all agree that the wealth tax likely deters risk-free saving.” This was a paraphrase of a claim made by Larry Summers who then went on to say that it was unknown whether a wealth tax would encourage or discourage risky investment. But I did wonder what the impact of a wealth tax would be on various types of investments and in examining this I realized that the claim was incorrect. In fact, **a wealth tax is unlikely to have any change on the risk profile of investments** in contrast to an income (or even consumption tax) that will. I discovered later that this was a known result being contained in a paper from Joe Stiglitz (QJE, 1969).

To explain this, let’s consider a person who has wealth of *W*. That person can allocate that wealth between an investment that pays a risk-free return (in the future) of *r* per dollar investment and an investment that pays a return (in the future) of *R* per dollar invested with probability *p* (and otherwise you lose your money). I’m going to focus on really rich people and, therefore, will assume that their utility (in the relevant range) is risk-neutral (or, more generally, if you want, that the amount of wealth they have does not impact on their risk aversion). That means that absent any taxes, the person will either allocate all or none of their wealth to the risky investment over the risk-free one. The risky one will be favoured if:

p(1 + R)W > (1 + r)W or p(1 + R) > 1 + r

To make things interesting we assume that pR > r. Suppose there is a wealth tax of *T* placed on wealth both now (prior to investments) and in the future (if there is any left). In this case, the tax on wealth today means that the person can only invest (1 – T)W. Thus, the expected payoff from the risky investment is:

p(1 + R)(1 – T)(1 – T)W

while the payoff from the risk-free investment is:

(1 + r)(1 – T)(1 – T)W

Comparing these the risky one will be favour if:

p(1 + R)(1 – T)(1 – T)W > (1 + r)(1 – T)(1 – T)W

or p(1 + R) > 1 + r

In other words, while the total amount of wealth invested is reduced, the wealth tax does not change the margin between the risky and risk-free investment. There is no distortion in the risk profile.

Intuitively, what is going on here is that a wealth tax will only have real effects if wealth itself has real effects. As Stiglitz shows, if having more wealth makes you invest more in risky assets, then a wealth tax will reduce your incentive to do so. On the other hand, if the reverse is true, then a wealth tax will increase your incentives to make risky investments. When there are no wealth effects, there is no effect on riskiness from a wealth tax.

By contrast, consider an income tax, *t*, that is paid on realized income. In this situation, there is only one opportunity to earn income tax. I will assume that any dollar amount of investment made is deductable. So the payoff from the risk-free investment is (1+r(1-t))W while the payoff from the risky investment is p(1+R(1-t))W. Thus, the risky investment will be chosen if:

p(1 + R(1 – t))W > (1 + r(1 – t))W

or p(1 + R) > 1 + r + (pR – r)t

An income tax causes the person to invest less in the risky asset because the upside (*R*) is taxed only if it is realised while they have to bear all of the downside risk.

Notice that the expected tax revenue from the wealth tax is TW + Tp(1 + R)(1 – T)W which means that the optimal wealth tax (there’s a Laffer curve here) is T* = (1 + p(1 +R))/(2p(1 +R)). The expected tax revenue from the income tax is tpRW or trW as the case may be which is optimised at t such that the risky asset is chosen or t* = (p(1+R) – (1+r))/(pR -r). You can actually work out that at these optimal tax rates, the wealth tax raises more revenue than the income tax.

In summary, according to these calculations, despite reducing the amount of wealth that the private sector has to invest, the wealth tax does not distort the risk-margin of investment. By contrast, an income tax is constrained by not wanting to distort this margin and hence results in lower overall tax revenue.