Believe those who are seeking the truth. Doubt those who find it. Andre Gide

Monday, March 11, 2019

Sustainable deficits

There's been much welcome discussion of late concerning the sustainability of government budget deficits and whether the size of the public debt is anything to worry about. I'm not going to answer this question for you here today. But what I would like to do is describe a framework that economists frequently employ to help organize their thinking on the matter. I want to begin with some simple arithmetic and then move on to a bit of theory. I'll let you judge whether the framework has any merit.

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.


  1. What changes if you introduce the ability to print money? (Govt sells bonds to central bank which rolls over the debt in perpetuity).

    1. Luis, in this model I make no difference between government money and debt.

  2. I think I echo Enrique's sentiments. I am thinking of money-financed fiscal programs.

    This is a very recent paper from Larry Summers (see link). The short story is that monetary policy may be ineffective (or even risky, if negative interest rates are pursued) in next recession. This paper is an A-bomb, btw.

    But Summers seems iffy about large and chronic budget deficits, as seen in his comments re MMT.

    Seems to me money-financed fiscal programs are next. Japan is looking at near deflation again, even with 163 job openings for every job hunter, and after the central bank bought back 45% of their huge boatload of Japanese Government Bonds. I think JGBs outstanding equal something like 230% of Japan GDP.

    Maybe QE+federal deficits=money-financed fiscal programs. Michael Woodford seems to think so.

    I would like to see the macroeconomics profession to begin to explicitly treat money-financed fiscal programs.

    1. What is the difference between a digit recorded on a Fed account vs. a digit recorded on a Treasury account?

    2. I don't know.

      Suppose the Treasury simply prints money and buys goods and services. And records no digits anywhere. It paid for services in created cash. End of transaction.

      When the Fed buys Treasury bonds, the interest payments flow back to the Treasury and if the Fed "rolls-over" permanently on the debt (maintains the balance sheet), I guess that is similar to the Treasury printing money.

      (Although when the Treasury prints money and buys good and services, you avoid all the gobbledygook about fractional bank reserves, and IOER, and primary dealers, and re-pos and so on.)

      So...let's boil it down:

      Are money-financed fiscal programs a good idea to counter-act recessions? They strike me as better then the current Rube Goldberg-apparatus.

      Am I wrong?

  3. I am no expert on MMT either, but when I read their stuff I am reminded of the fiscal theory of the price level (FTPL). Do you have any thoughts on this connection, David?

    If I were to try to model MMT using mainstream tools, I would start with two building blocks: 1. the FTPL and 2. some theory of permanent output gaps/hysteresis (maybe like Roger Farmer's stuff).

    FTPL gives you the 'inflation constraint' part of MMT. Cochrane always says that the intertemporal government "budget" is not a constraint but a valuation equation that defines the equilibrium value of nominal government liabilities. (FTPL can also deliver some other things MMT people sometimes claim, e.g. that higher nominal interest rates can be expansionary due to income effects. See the discussion in Leeper and Keith's handbook chapter on the FTPL. By I don't think this is really central to MMT's theory of inflation determination. Another thing I think is not really central is the distinction between interest-bearing debt and 'cash,' which was mentioned by Blanchard in his recent tweet on MMT.)

    Under the FTPL, increases in nominal government liabilities that are not accompanied by the promise of higher future real surpluses are inflationary and generate no real government revenue. The second building block -- permanent output gaps -- opens the door for the increase in nominal liabilities to raise future surpluses by closing the output gap and thereby increasing tax revenues at given tax rates. Cochrane's 2005 JME paper 'Money as Stock' sort of anticipates this possibility; see the bottom of pp. 521. But in his case it is just a cyclical thing. The MMT people think fiscal policy can be used to close output gaps that would have no tendency to close 'naturally,' and that this will not lead to inflation. I think that can only be true if the output gains deliver sufficient increases in the path of real surpluses.

    1. Seems close to my way of thinking of things. But even so, this does not exhaust everything that concerns MMT or the different approaches they espouse to deal with concerns. So good to keep an open mind.

  4. The MMT version seems simple enough in concept without the math.

    In fact, without reference to inflation, there is no limit to the size of debt or deficits. This includes accumulated interest and its financing. Its all just the flow of funds, double entry booking, pricing bonds, and using the CB as bond buyer of last resort if deemed necessary. Nominal and real measures are not directly an issue. Sustainability is not an issue.

    The sustainability issue has directly to do not with the debt or the deficit, but with inflation. And it is only the policy response to inflation that may affect the path of the deficit and the debt.

    That's the inflation constraint.

    The budget constraint is just an accounting identity for the flow of funds. It's a constraint in math logic only.

  5. "The MMT version seems simple enough in concept without the math."

    Given the confusion that reigns out there, I'd say your assertion is highly questionable. If you'd give just a few minutes to work the through the simple arithmetic and application of theory, I think you'll see the benefit of my approach. Your last sentence tells me you did not read (or understand) my post.

  6. Why look for a fiscal rule that makes deficits "sustainable" that than on which maximizes growth in real income? This will imply thinks about the effects of spending on real income (do the activities being financed pass an NPV test?) and the dead weight loss of the taxation. I suspect that in such a regime deficits and debt would not appear explicitly although by making parametric assumptions about the costs and benefits of spending they could be made to reappear.

  7. This comment by Bob T (sent to me privately) is good, so I thought I'd post it here.

    Your logic in your post is impeccable, of course. I have a couple of observations. First on the data, the periods when nominal income growth has exceeded R in your chart are either when inflation was rising (unexpectedly) and the post-crisis period. So I dont think that is sustainable. Second, your discussion of the value of Treasuries as a safe asset ignores the ability of the private sector to create it's own safe assets as substitutes. Indeed, that was the whole problem in the 'wholesale deposit' market in 2007 in some people's view. Now lots of people would say that the government should therefore flood the world with govt safe assets. If that debt was being used to buy or finance revenue generating assets, that's fine. So I am not opposed to the infrastructure argument people advance, in principle. But I don't think govt is that disciplined; I'd welcome being proven wrong. Finally, I would say that your analysis, while totally solid, ignores risk in the broad sense of that term. Optimal public finance does say that one should treat conventional taxes and the inflation tax symmetrically in the sense that the marginal social costs of the two should be equalized. I just think that a country that shows it is willing to use the inflation tax will face a risk premium that it won't like, ex post. I dont think it is an accident that the countries that have used the inflation tax are countries that have had tax compliance issues (or have fought unfunded wars). I hope that helps because it's late here and I'm done for the day.