Let’s start with some standard definitions.

G(t) = government spending (purchases and transfers) in year t.

T(t) = government tax revenue in year t.

R(t) = gross nominal interest rate on government debt paid in year t+1.

D(t) = nominal government debt in year t (including interest-bearing central bank reserves).

Now let’s link these objects together using the following identity:

[1] G(t) + [R(t-1) – 1]*D(t-1) = T(t) + [D(t) – D(t-1)]

In words, the left-hand-side (LHS) of the identity measures the money needed to pay for government spending G(t) and the interest expense of the debt [R(t-1) – 1]*D(t-1), where [R(t-1) – 1] denotes the net nominal interest rate. The right-hand-side (RHS) of the identity measures the money collected by the government in the form of taxes T(t) and the money created through new nominal debt issuance [D(t) – D(t-1)].

I find it convenient to rewrite [1] as,

[2] G(t) – T(t) = D(t) – R(t-1)*D(t-1)

The LHS of [2] represents the primary government budget deficit. If the deficit is positive in period t, then the RHS of [2] tells us that the stock of debt in period t must be larger than the interest plus principal of the debt maturing from period t-1.

Next, define n(t) = D(t)/D(t-1), that is, the (gross) rate of growth of the nominal debt. Use this definition to write D(t-1) = D(t)/n(t) and substitute this expression into [2] to form,

[3] G(t) – T(t) = [1 – R(t-1)/n(t)]*D(t)

Let Y(t) denote the nominal GDP. Now define g(t) = G(t)/Y(t), τ(t)=T(t)/Y(t) and d(t) = D(t)/Y(t). Because I want to limit attention to “long run” scenarios, let me impose a stationarity restriction: g(t) = g, τ(t) = τ, d(t) = d, R(t-1) = R, and n(t) = n. Then we can write [3] as,

[4] g – τ = [1 – R/n]*d

Assuming d > 0, the identity [4] tells us that a sustained primary deficit is possible only if R < n. Recall that R represents the (gross) nominal interest rate on government debt and n represents the (gross) rate of growth of the nominal debt. Because of my stationarity assumption d = D(t)/Y(t), it follows that n also represents the (gross) rate of growth of the nominal GDP. A lot of mainstream thinking on the matter of “fiscal sustainability” is rooted, I think, in the assumption that R > n. In the standard DSGE model (which abstracts from financial market frictions), the “real” interest rate R/n is pinned down by time-preference and productivity growth. This real interest rate is typically estimated to be a positive number. If this is the view one adopts, then condition [4] implies that budget deficits cannot be sustained into the indefinite future. It’s not exactly made clear what might happen if deficit finance persists in such a case — maybe inflation and/or default. Bond vigilantes. Something like that.

But this view is, at best, seriously flawed. First of all, as just an empirical matter, R < n seems like a better approximation than R > n. Here the year-over-year growth rate of nominal GDP and the one-year Treasury-bill rate for the U.S. economy since 1961,

Secondly, the standard DSGE model ignores the role that U.S. Treasury debt plays as an exchange medium in financial markets. The growth in the demand for Treasury debt has come from many sources over the past few decades. It is used extensively as collateral in credit-derivative and repo markets. Foreign countries have clamored to accumulate U.S. Treasuries as a store of value. Its demand was further enhance as a “flight to safety” asset during the financial crisis. And more recently, changes in financial regulations (Dodd-Frank and Basel III) have further spurred the demand for Treasuries (for example, they can be used to satisfy the Basel III liquidity-coverage-ratio requirement for banks).

Because of the special role played by nominally safe government debt in financial markets, it can trade at a premium. That is, agents and agencies are willing to hold “monetary” objects for reasons other than their pecuniary rate of return. This is why the nominal (and real) interest rate on safe government securities can be set lower than the “natural” rate of interest. If R < n, then the RHS of [4] corresponds to seigniorage revenue. (Note: seigniorage is not limited to the purchasing power created by zero-interest cash.)
Some of this discussion seems related to what the MMT folks are talking about. I’m not an expert in that area (am still reading up on it), but see, for example, Scott Fullwiler’s article: The Debt Ratio and Sustainable Macroeconomic Policy. There’s also this nice piece by (the more mainstream) Neil Mehrotra: Debt Sustainability in a Low Interest Rate World and, of course, Olivier Blanchard’s AEA Presidential Address: Public Debt and Low Interest Rates.
Are there limits to how large a sustainable deficit might be? To answer this question, we need to go beyond the identities described above. Here’s a simple theoretical restriction: Assume that the demand for real debt d is increasing in its real yield R/n. In undergraduate money-macro textbooks, we might say “assume that the demand for money is increasing in the interest rate paid on money.” Note that for a given nominal interest rate R, this implies that the demand for *real* money balances is *decreasing* in the (expected) inflation rate. Let’s denote this theory of money demand by the behavioral equation:

[5] d = L(R/n), with L increasing in R/n.

Combine the theoretical statement [5] with the identity in [4] to form a *government budget constraint*:

[6] g – τ = [1 – R/n]*L(R/n)

Now, to discover the limit of how large the deficit can get, imagine that the government wants to maximize the sustainable deficit through its choice of R/n (all that matters here is the ratio). What are the limits to seigniorage revenue?

The answer to this question has a standard “Laffer curve” property to it. Increasing R (or decreasing n) is bad because doing so increases the interest expense of the debt. On the other hand, it increases the demand for debt. Think of [1 – R/n] as the tax rate and L(R/n) as the tax base. Increasing R/n has competing effects. So, for example, increasing n has the effect of increasing the inflation tax rate. This is good for revenue purposes. But it also has the effect of decreasing the tax base (as people substitute out of government debt into competing securities). This is bad for revenue purposes. The revenue (primary deficit) maximizing interest/inflation rate equates these two margins. In short, economic behavior places a restriction on how much the government can finance its operations through money/debt issuance.

This is a very simple theory and it can be extended in many different and interesting ways. But the point of this blog post was first, to demonstrate how government budget identities can be combined with economic theory to form a meaningful government budget constraint and second, to demonstrate that there’s nothing necessarily wrong or unsustainable about a government running a persistent budget deficit.

**Postscript: March 12, 2019**

I should have figured that Nick Rowe beat me to this post; see here. He also provides this nice Laffer curve diagram.

In the diagram above, r corresponds to my R and g corresponds to my n. I think I would have drawn the diagram with seigniorage revenue on the y-axis and the real interest rate (R/n) on the x-axis. Then (R/n)* would denote the seigniorage revenue maximizing real yield on government debt.

Nick points out that in the OLG model, the introduction of (say) land eliminates the possibility that R < n in equiilbrium. This is true only if government debt serves only as a store of value. My paper with Fernando Martin uses a standard macro model where debt has a liquidity role and coexists with a higher yielding alternative asset. It also has a diagram like Nick's (Figure 1). A final thought. One often hears MMTers say something like “we replace the government budget constraint with an inflation constraint.” I interpret this statement in the following way. Imagine setting the nominal interest rate to its lower bound R = 1 (I actually think it can go lower). Then the real rate of return on government debt (zero-interest money) is 1/n. If the real GDP is constant, then n represents the equilibrium inflation rate (in a model where we impose the additional market-clearing restriction). Assuming we are on the LHS of the Laffer curve, increasing the inflation rate increases the primary deficit. An inflation constraint n < n* then limits how large the primary deficit can be.