Paul Krugman has been explaining (very slowly and clearly) that if the US attracts foreign investment by cutting taxes on profits, then it will have to pay the foreign investors. The Tax Foundation appears not to have noticed that loans are not gifts.

This is the ultra-static effect of the cut in which its effects on behavior aren’t considered. Krugman went on to note that if there is a huge inflow of foreign cash (as promised by supporters of the bill) then there will be a huge increase in US payments to the foreigners. He calls this Leprechaun economics, because it is a very important reason that Irish GDP is much greater than their gross national income. He wrote

GDP is actually the wrong measure. If you’re going to be pulling in foreign capital, you’re going to be paying more investment income to foreigners; so gross national income – income accruing to domestic residents – is going to go up by less. And surely that’s the measure we care about.

and

There are really two bottom lines here. One is that the true growth impacts of Cut Cut Cut would be even more pathetic than the numbers you’ve been hearing. The other is that if you’re going to make international capital flows central to your arguments, you really need to think about the implications for future investment income.

Krugman raises a question

In fact, when you bear in mind the reduced taxes collected on foreign investors who are already here, GNI could actually go down, not up.

It is interesting. I think that somewhere he explains that the answer depends on another debated issue — the true incidence of taxes on profits. Enthusiasts for the tax cuts assert that, in the long run, all of the benefits will go to workers. People who look at data, estimate that about one quarter of the benefits of a reduction of taxes on profits go to workers

(before going on, there are no free lunches — the benefits are at the expense of the Treasury so other taxes will have to be raised or programs will be cut.)

This matters for the discussion of Gross National Income vs GDP, because roughly 35% of shares of US firms are owned by foreigners. So if the money goes to investors, 35% goes to foreigners. This is true both of the old foreign investment in the USA and the new investment attracted by the low taxes. After the jump I will try a lot of horrible pain ascii formulas attampting to answer Krugman’s question of whether a profits tax cut causes higher or lower domestic gross national income. The key parameters are the current tax rate, the incidence on workers, and the share of capital.

Doing the algebra, I conclude that unless more than half of the incidence of profits tax falls on labor, cutting the rate of taxes on profits below 35% reduces gross national income.

“Half” is embarrassingly close to a whole number, but it is what came out of the horrible algebra. Also 35% is coincidentally the current statutory rate. I regret the fact that messy calculations gave such a suspiciously simple result. I don’t totally trust my algebra and don’t think my effort added much to Krugman’s. The point is that, for plausible parameters, cutting the tax on capital income reduces US gross national income.

Warning horrible horrible algebra after the jump

OK always following Krugman, I assume that capital is paid it’s marginal product. If there were no tax on capital income, then new foreign investment of deltaK would not change gross national income (GNI) because it would raise GDP by (detaK)r and also reduce net capital income by (deltaK)r where r is the marginal product of capital. However, if there is a tax tau on capital income, the IRS gets (deltaK)(tau)r so the inflow causes larger GNI.

Krugman notes that if the new statutory tax rate is 20%, then even ignoring loopholes and such, the increase in GNI due to the inflow is only 20% of the increase in GDP. But he also stresses the 35% of existing shares owned by foreigners and suggests that this might imply a reduction of US GNI due to a reduction in Tau. To find out, I need to use a model

First GDP is a standard cobb douglas function of capital and labor. I choose units so

1) GDP = Y = K^alpha L^(1-alpha) so the share of capital is alpha.

2) rK = (alpha)Y

3) WL = (1-alpha)Y

I will choose units of labor so L= 1

so W = WL = (1-alpha)Y

With Tau = 0.

4) GNI = Y – 0.35rK = Y(1-0.35alpha)

OK now imposing Tau will cause lower investment and change K down to K_tau and r up to r_tau. also investors will get (1-tau)(K_tau)r_tau

Workers will get wage (1-alpha)(K_tau)^alpha

Investors will get (1-tau)(K_tau r = (1-tau)alpha(K_tau)^alpha

Tax cut enthusiasts assume that K changes so much that (1-tau)r_tau doesn’t depend on tau. This is extreme.

Data suggests that the change in wages is one fourth the change in (1-tau)(r_tau)K_tau. I want to play with incidence as a parameter so I assume

5) d(W)/dtau = (beta)K_tau(d((1-tau)r_tau)/dtau)

Now I can do the algebra which depends on beta, alpha and tau. I am sometimes going to leave out the _tau of K_tau (sorry)

I want to figure out d(K_tau/d_tau) using equations 2, 3 and 5

6) (1-alpha)(alpha)K^(alpha-1) dK_tau/dtau =

[(alpha(alpha-1))(1-tau)K^(alpha-1)(dK_tau/dtau)- K_tau(alpha)K^(alpha-1)]beta

Oh look I can divide both sides by (alpha)K^(alpha-1) (which happens to be equal to r) so

7) (1-alpha)dK_tau/dtau = (beta)(alpha-1)(1-tau)dKtau/dtau – Ktau_a(beta)

8) (1-alpha)(1+beta(1-tau))dK_tau/dtau = – Ktau_a(beta)

9) dK_tau/dtau = -(K_tau)(beta/[(1-alpha)(1+beta(1-tau))])

Now I can calculate the effect of changing tau on GNI

10) d(GDP)/dtau = (alpha)K^(alpha-1)(dK/dtau)= (alpha)(K^alpha)(dK/dtau)/K

11) d(GNI)/dtau =

(alpha)(K^(alpha-1))(dK/dtau) – 0.35 (k_tau)d[(1-tau)r_tau]/dtau] – (1-tau)[(alpha)(K^(alpha-1)(dK/dtau)]

The two terms I subtract are the increased after tax payments on old capital an

d the new after tax payments on new capital.

12) d(GNI)/dtau = (tau)(alpha)(K^(alpha-1))(dK/dtau) – 0.35 (K_tau)d[(1-tau)r_tau]/dtau

r_tau = alpha(K^(alpha-1) so

13 d(GNI)/dtau = (tau)(alpha)(K^(alpha-1))(dK/dtau) + 0.35 (k_tau)alpha(K^(alpha-1) –

0.35(K_tau)(1-tau)(alpha-1)(alpha)(K^(alpha-2))(dK/dtau)

collecting terms

13) d(GNI)/dtau = [(tau + 0.35(1-tau)(1-alpha))((dK/dtau) + 0.35(K_tau)]alpha(K^(alpha-1)

plugging in dK/dtau

14)dGNI/dtau=

= [-(tau + 0.35(1-tau)(1-alpha))(K_tau)(beta/[(1-alpha)(1+beta(1-tau))]))+0.35(K_tau)]alpha(K^(alpha-1)

= [0.35 – [tau + 0.35(1-tau)(1-alpha)]beta/[(1-alpha)(1+beta(1-tau))](alpha)K_tau^alpha

The lower is beta, the better high tau looks. For beta = 0 (so no effect of taxes on investment) national income clearly increases in tau (of course it just means taxing money before sending it to foreigners). Beta = 1/3 is the empirical estimate. I will be kind to tax cutters and assume beta = 1.

For alpha = 1/3, and beta = 1 this is

15) d(GNI)/dtau = [0.35 – [tau + 0.35(1-tau)(2/3)]/[(2/3)(2-tau)](alpha)K_tau^alpha =

[0.35 – [tau + 0.35(1-tau)]/(2-tau)](alpha)K_tau^alpha =

= [0.35 – (0.35+0.65tau)/(2-tau)](alpha)K_tau^alpha

Which implies GNI is maximized at tau such that

0.35(2-tau) = 0.35 + 0.65tau

so tau = 0.35

Oh my that happens (by pure coincidence) to equal the current statutory rate (which by another pure coincidence happens to equal the fraction of shares own by foreigners).

The effective tax rate is lower than 35% and beta is less than 1 so cutting taxes on profits would reduce US gross national income.