# Capital taxes: who do they really affect?

When Mitt Romney’s effective tax rate was released, the common reaction was that this is preposterous and that capital gains taxes should be increased. This is an arguably understandable reaction. However, this post shows that basing such decisions as whether a tax should be increased on intuition or gut feelings can lead to the wrong conclusions.

As it turns out, in many cases the effects of a tax spread across the economy and determining who is actually impacted by it may need some more careful analysis. This post shows what the effect of a capital gains tax really is and who is really affected by them.

Assume that we have two economies, a small open economy (denoted by 1) and the rest of the world (denoted by 2). Assume further that capital (K) is perfectly mobile and that labor (L) is immobile. Capital’s mobility means in practice that if someone wants to invest their money in another country, they can easily do so without any barriers; a reasonable approximation. Labor’s immobility means that workers do not just move freely across countries to chase higher wages, therefore wage rates differ across countries, which is clearly true.

The two economies have the production functions

$F_1(K_1, L_1),$

$F_2(K_2, L_2).$

Now the interest rate (i.e. the price of / return on capital) will just be the marginal product of capital. Since capital is perfectly mobile, this rate has to be equal across the two economies. Otherwise, one could just pull money out of the economy with the lower return and invest it in the other one (this is like arbitrage in finance). This would be done until the two returns are equal. Therefore, in equilibrium the interest rate is

$r_1 = r_2 = F_{1K} = F_{2K} = r^*.$

Similarly, the wage rates are the marginal product of labor, but they differ between the two economies because of immobility:

$w_1 = F_{1L}(K_1, L_1),$

$w_2 = F_{2L}(K_2, L_2).$

Now, suppose the small economy (1) introduces a tax (tau) on capital. The effect will be that returns on capital drop, but since capital is perfectly mobile, capital holders can just pull out their money from this economy and invest it in the rest of the world. They will do this until post-tax returns in the small economy and returns in the rest of the world are the same, and thus equilibrium is restored:

$r^* = F_{2K} = (1 - \tau) F_{1K}.$

However, since the small economy is so small relative to the rest of the world, this adjusment will be so minimal, that for all intents and purposes worldwide return on capital will not change. In other words, a capital tax increase in a small country won’t affect worldwide interest rates.

But in any case, capital will be pulled out of the small economy because now the marginal product of capital in country 1 (F_1K) will have to equal the return before the tax divided by 1 minus the tax rate (r*/(1-tau)). Therefore, only those projects/investments will be funded in country 1 whose return is equal to or higher than the new increased F_1K; money from investments with lower returns will be pulled out of the country.

Now given that labor is fixed (because it’s immobile), we will have the same amount of workers who will have to work with less capital. Relatively speaking, labor will become more abundant and thus wage rates are going to decrease. The net effect is, therefore, that while capital holders who were originally supposed to be worse off just took their capital out of the country and were pretty much unaffected, while workers got their wages cut.

Let us see how this would work with a Cobb-Douglas production function. Assume that the production function of the first economy takes the form

$F_1 = K_1^{\alpha}L_1^{\beta},$

where alpha and beta are of course positive.

Recall that the wage is the marginal product of labor, which is

$w_1 = \beta K_1^{\alpha}L_1^{\beta - 1}.$

Then if the tax is introduced, as we have already established, capital will move out of the small economy because equilibrium requires this. Knowing this, what we have to see is what the effect of a change in capital levels on wage is:

$\frac{\partial w_1}{\partial K_1} = \alpha \beta K_1^{\alpha -1}L_1^{\beta - 1} > 0.$

Therefore, wage and capital move in the same direction, just as predicted. A decrease in capital will generate a decrease in the wage rate.

The set of assumptions needed to establish this result is basically the mobility of capital and immobility of labor. Also, the economy in question must be small relative to the rest of the world. Smaller European countries could more or less satisfy these conditions: capital is mobile within Europe, but labor not so much. And the small economies of Europe are negligible in size relative to the whole of Europe. But one may argue that in general the whole world is approaching a time when this will be applicable to any (relatively small) country. Capital is becoming more and more mobile as economies become more integrated, while labor will probably stay immobile because of cultural and language barriers.

The main lesson here is that capital taxes may not serve their intended purpose of “hurting” capital owners. Instead they may very well hurt workers. So while Mitt Romney’s tax rate may seem unjust to some people, increasing the capital gains tax may not really make the situation more just. There may of course be some room for intervention, as the U.S. is not a relatively small economy, so the above results will not apply completely. Since the U.S. is a large economy, a drop in return on capital will also influence worldwide returns, so the situation may be more complicated. But unless capital is perfectly immobile and the U.S. is the whole world economy (neither of which is true obviously), wages will go down. The closer we are to the post’s original assumptions, the more wages will take a hit instead of capital owners. At least because of this, it is worth considering alternatives to a capital gains tax hike in the U.S.