Deficit spending in growing economies

investing in your future self

This is me thinking out loud about deficit spending, corrections welcome.

Let’s say you know for certain that you’ll get $1000 a year from now, that means that you can borrow money today and pay it off later with your future income.

Governments do the same thing; they know they will get tax revenue in the future, take out debt, and pay it back with future tax revenue.

How much can they reasonably borrow? If each citizen’s income grows at a rate i, the income tax rate is r and the discount rate is gamma, then the net present value of someones taxes after t years in the future is:

\(V_{ind.} = \gamma^t \cdot r \cdot i^t \cdot I_0\)

Where I_0 is the citizens initial income.

If there is a population of N_0 income-earners growing at rate n then the equation becomes:

\(V = \gamma^t \cdot r \cdot i^t \cdot I_0 \cdot n^t \cdot N_0\)

Let’s imagine an economy where tax rate, income growth, and population growth are steady. Starting “today” (at time 0), the government spends some revenue to pay off old debt and borrows some new debt. Specifically, it borrows on fixed fraction of future tax receipts and pays off debts using the same fixed fraction on its current tax receipts. Call that faction f.

So the difference between the borrowed money and the debt payments is:

\(\Delta(revenue) = f \cdot \gamma^t \cdot r \cdot i^t I_0 \cdot n^t N_0 - f \cdot r \cdot I_0 \cdot N_0\)

\(\Delta(revenue) = ((\gamma \cdot i \cdot n)^t - 1) \cdot f \cdot r \cdot I_0 \cdot N_0\)

Where I_0 and N_0 are the incomes and population size today. So to reiterate, the first term is the net present value of the government’s tax revenue t years into the future multiplied by a fraction f. At that future date, the government will use that fraction of their revenues to pay off the original debt from t years ago.

The debt as a fraction of current tax receipts is:

\(d = \frac{\Delta(revenue)}{revenue} = ((\gamma \cdot i \cdot n)^t - 1) \cdot f \)

So for the government to take on any debt (d > 0), we have to assume that:

\(\gamma \cdot i \cdot n > 1\)

Let’s plug some numbers in. Incomes in the U.S. have grown by 3% per year, with 0.5% population growth recently. With a 2% discount rate that means that:

\(d = ((1.014)^t - 1) \cdot f\)

How much is the U.S. government actually borrowing? I can’t find direct estimates for this, but spending as a percent of GDP is around 20% and the deficit to GDP ratio is around 6% which corresponds to a d of 30%. The fraction of federal tax receipts relative to interest payments (f) is about 35%, implying a time horizon t of 43 years¹.

In other words, the government’s borrowing behavior looks similar to a borrower that expects population growth and income growth to continue for over 40 years². That’s a long time!

In general, if the government is confident that growth will continue forever (and i*gamma*n is greater than 1), they can borrow an arbitrary amount of money each year in perpetuity by setting t high enough; never fully collecting the taxes it needs to fund itself³. However, uncertainty about future growth creates risk and should lower the borrowing time horizon.

The weird thing is that the government could spend borrowed money boost growth. Investments in education, healthcare, research, automation, and population growth could increase the growth rate of the economy and make it easier to pay off the debt.

Strong institutions can ensure long-term growth and raise the safe time horizon⁴, allowing the government to invest over longer periods. Is it possible that these investments could be self-reinforcing? More concretely, could the government just invest its debt in index funds and make a profit⁵?

1. These numbers are kind of ad-hoc, I invite suggestions on other ways to estimate these numbers.

2. In general the time horizon can be approximated by (assuming d=f):

\(t \approx \frac{\ln 2}{\gamma \cdot n \cdot i - 1}\)

3. Spreading spending over different tax years is also a good idea because its less vulnerable to yearly variation in revenue and allows governments to spend more in recessions.

4. And potentially lower time discounting.

5. This is suspiciously similar to John Roemer’s idea in A future for socialism where the government owns all stocks and gives citizens an equal share.