# Debt and Taxes II

This is an extended post on the caveat to debt and taxes 1. It is joint work with Brad DeLong and Barbara Annicchiarico. The point is that, in his Presidential Address, Olivier Blanchard notes that the argument that higher debt causes increased welfare is weaker than the argument that it is feasible.

The Treasury can afford to increase debt D_t just by just giving bonds away and can pay interest and principal without ever raising taxes so long as the safe rate of interest which it has to pay is lower than the trend rate of growth of GDP. In that case it is possible to just roll over the debt forever and the debt (including the additional debt) shrinks to insignificance as a share of GDP.

Following Blanchard we assume zero trend growth and an economy which reaches a steady state. This really just simplifies notation a bit. Also Blanchard works with Rf_t which is 1 plust the safe rate of interest, so bonds mature in one period and pay Rf_t times the amount invested. In contrast investing in productive capital K_t gives a risky return R_t (includng the capital one still has so the ordinary notation is return r = R-1).

The time subscripts are there because, in general, both the safe and the risky return depend on D_t and K_t. This is a major nuisance and I will present a super absurdly simple example (due to Brad) here in which the state is constant.

The model is an overlapping generations model — the young work and get a wage W_t. They consume C_t^y and buy the bonds D_t and the capital K_t from the old. The old get capital income and also sell their bonds and capital to the young. They consume all of that so C_t^o = Rf_tD_t+R_K_t

Importantly, there is a technology shock which makes W_t and R_t stochastic. This is a crazy assumption, which is absolutely standard when one wants to put risk into a macro model and also wants to keep it super simple. The point is that bonds are safe and that ownership of capital (stock) is risky.

The way to make the model absurdly simple involves two steps. First assume that the policy is to increase D and to keep it at the new high level. If Rf_t<1 this involves an additional gift of D(1-Rf) to the young each period. Increased debt finances an increase in this universal basic income so long as Rf_t<1. The point of this is to keep things simple. It is not needed for the increase in debt to help all generations not just the first who get the windfall gift (see below).

Then the key extreme very convenient assumption just for a simple example.

Agents choose C_t^y to maximiz

1) C_t^y + beta ln(C_{t+1}^o)

The extreme extraordinary assumption is that utility is linear in consumption when young, so young people are risk neutral. The assumption that utility is logarithmic in consumption when old is fairly conventional but it is not innocent either. Together they make everything simple, because it means that people always save beta

so

2) K_{t+1}+D_{t+1} = beta

Then the assumption that the state keeps Debt constant implies that capital is constant.

This implies that Rf and E(R) and the stochastic distribution of R are all constant and everything is simple.

After the jump I will discuss other cases. Here I just note that the extreme assumptions don’t just make all the math simple. They actually matter, because constant Rf_t gaurantees that it stays below 1. In general (and certainly for independent technology shocks) Rf_t is sometimes greater than 1. I will put all this off to the after the jump appendix.

The super simple model shows three beneficial effects of increased debt. First there is the gift to the old at the time D is increased, second there is the gift of (1-Rf) to the young each period. Third debt causes higher Rf which is nice for the citizens so long as the new higher Rf is less than 1.

There is, however, a fourth effect which alarmed Blanchard. In the olg model debt crowds out capital. In the super simple model

3) K=beta-D

The reduction in K causes lower wages W_t and higher returns R_t (still with time subscripts because both depend on the technology shock).

This transfer from the young to the old is risky and it reduces expected welfare so long as the steady state risky rate is greater than 1. In the super simple model that means it reduces expected welfare s long as E(R) >1 when K = beta. Blanchard shows this at some length.

I am now going to do some algebra. I am going to set population + labor supply to 1 just to simplify notation.

With constant returns to scale, perfect competition implies

4) W_t + R_t K = Y_t

taking the derivative with respect to K

5) d W_t/dK + (dR_t/dK)K + R_t=R_t

so

6) d W_t/dK = – (dR_t/dK)K

7) K+D=Beta so dK/dD=-1 so

8) d W_t/dD = – (dR_t/dD)K

Now consider the following policy. D is increased by a small amount deltaD and a small tax on capital income of Tau is introduced so that

9) (1-tau)(R_t + deltaD dR_t/dD) = R_t

That is the tax is calculated so that the after tax income from ownership of risky capital of the old is unchanged.

The revenues tau(R_t + deltaD dR_t/dD)K are given to the workers. Equation 8 implies that their income is unchanged. The tax and transfer policy eliminates the fourth effect of incrased D.

This means that the policy of increasing debt and eliminating the effect on wages and the return on capital by taxing capital income and giving the proceeds to workers makes every generation better off and is a Pareto improvement.

In general in public economics a reform is said to increase efficiency if it is possible to combine the reform with taxes and transfers such that everyone is better off. So in the simple model, the standard use of the term implies that issuing debt and giving the proceeds to citizens would increase efficiency.

This is a very standard analysis of the absolutely standard model used to analyze the welfare effects of public debt. The conclusion really shouldn’t be controversial (but it will be).

OK so I warned that the crazy utility function is all too useful. For any ordinary or plausible utility function, the saving of the young will depend on their income and will fluctuate. This means that the capital stock K_t will fluctuate. This has two effects one of which is just a bother. The tax rate given by 9 changes as K_t changes. This means that the policy including taxes and transfers is complicated even in the extremely simple model. Also it would seem absurdly unfair. It is just not a politically plausible proposal.

We have made assumptions which seem relatively plausible (for always extremely stylized models) and checked that the tax given by our formula doesn’t fluctuate much and that a constant tax can cause an increase in long term average welfare (not necessarily helping each and every generation).

The other more serious problem is that the safe rate Rf_t fluctuates with K_t and can sometimes be greater than 1 even if it is less than one on average. This means for some generations the universal basic income (1-Rf_t)D will be negative — a poll tax.

This is really a problem with deciding to keep D constant to simplify things. Another policy is to increase debt once (with a big gift to the old) then roll it over with no net transfers and let it shrink to insignificance. This means that Tau_t has to vary with both K_t and D_t and becomes quite complicated for the simplest models.

Although D_t shrinks on average it is possible that a long string of terrible luck will cause high D_t which becomes unsustainable without using taxes to pay it back. The stochastic process for D_t is fairly complicated even for very simple models (except the absurdly simple one before the jump) so what we do is simulate for parameters which seem reasonable and note that the need to tax to pay the debt is very very rare and the long run average utility is increased even if some generations have to pay some taxes to repay some of the debt.

The policy does help all generations so long as none has to pay net taxes – transfers to the state. The reason is that increased D causes higher Rf (supply and demand of bonds) and this effect is not cancelled by the tax on capital income (supply and demand means the after tax return must be higher if the supply of bonds is higher). So all benefit, because they get the same income when young, face the same distribution of risky returns and get a higher safe return.

I can’t speak for co-authors, but I wonder if it is more embarrassing to present only the absurdly simple model or to get into this mess.