Wednesday, February 5, 2020

New Paper -- the fiscal roots of inflation

I recently finished drafts of a few academic papers that blog readers might find interesting. Today, "The Fiscal Roots of Inflation."

The government debt valuation equation says that the real value of nominal debt equals the present value of surpluses. So, when there is inflation, the real value of nominal debt declines. Does that decline come about by lower future surpluses, or by a higher discount rate? You can guess the answer -- a higher discount rate.

Though to me this is interesting for how to construct fiscal theory models in which changes in the present value of government debt cause inflation, the valuation equation is every bit as much a part of  standard new-Keynesian models. So the paper does not take a stand on causality.

Here is an example of the sort of puzzle the paper addresses. Think about 2008. There was a big recession. Deficits zoomed, through bailout, stabilizers, and deliberate stimulus. Yet inflation.. declined. So how does the government debt valuation equation work? Well, maybe today's deficits are bad, but they came with news of better future surpluses. That's hard to stomach. And it isn't true in the data. Well, real interest rates declined and sharply. The discount rate for government debt declined, which raises the value of government debt, even if expected future surpluses are unchanged or declined. With a lower discount rate, government debt is more valuable. If the price level does not change, people want to buy less stuff and more government debt. That's lower aggregate demand, which pushes the price level down. Does this story bear out, quantitatively, in the data? Yes.

If you don't like discount rates and forward looking behavior, you can put the same observation in ex-post terms. When there is a big deficit, the value of debt rises. How, on average, does the debt-GDP ratio come back down on average? Well, the government could run big surpluses -- raise taxes, cut spending to pay off the debt. That turns out not to be the case. There could be a surge of economic growth. Maybe the stimuluses and infrastructure spending all pay off. That turns out not to be the case. Or, the real rate of return on government bonds could go down, so that debt grows at a lower rate. That turns out to be, on average and therefore predictably, the answer.


OK, to work. The paper starts by developing a Cambpell-Shiller type identity for government debt. This works also for arbitrary maturity structures of the debt. Corresponding to the Campbell-Shiller return linearization, $$ \rho v_{t+1}=v_{t}+r_{t+1}^{n}-\pi_{t+1}-g_{t+1}-s_{t+1}. $$ The log debt to GDP ratio at the end of period \(t+1\), \(v_{t+1}\), is equal to its value at the end of period \(t\), \(v_{t}\), increased by the log nominal return on the portfolio of government bonds \(r_{t+1}^{n}\) less inflation \(\pi_{t+1}\), less log GDP growth \(g_{t+1}\), and less the real primary surplus to GDP ratio \(s_{t+1}\). Surpluses, unlike dividends, can be negative, so I don't take the log here. This surplus is scaled to have units of surplus to value, so a 1% change in "surplus" changes the log value of debt by 1%. I use this equation to measure the surplus.

Iterating forward, and imposing the transversality condition, we have a Campbell-Shiller style present value identity, $$ v_{t}=\sum_{j=1}^{\infty}\rho^{j-1}s_{t+j}+\sum_{j=1}^{\infty}\rho^{j-1}g_{t+j} -\sum_{j=1}^{\infty}\rho^{j-1}\left( r_{t+j}^{n}-\pi _{t+j}\right). $$ Take innovations \( \Delta E_{t+1} \equiv E_{t+1}-E_t \) and we have $$ \Delta E_{t+1}\pi_{t+1}-\Delta E_{t+1} r_{t+1}^{n}= -\sum_{j=0}^{\infty} \rho^{j} \Delta E_{t+1}s_{t+1+j} -\sum_{j=0}^{\infty} \rho^{j} \Delta E_{t+1} g_{t+1+j}+\sum_{j=1}^{\infty} \rho^{j} \Delta E_{t+1}\left( r_{t+1+j}^{n}-\pi_{t+1+j}\right) $$ Unexpected inflation devalues bonds. So it must come with a decline in surpluses, a rise in the discount rate, or a decline in bond prices. Notice the value of debt disappeared, which is handy.

The bond return comes from future expected returns or inflation, so it's nice to get rid of that too. With a geometric maturity structure in which the face value of bonds of \(j\) maturity is \(\omega^j\), a high bond return today must come from lower bond returns in the future. $$ \Delta E_{t+1}r_{t+1}^{n} = -\sum_{j=1}^{\infty}\omega^{j}\Delta E_{t+1} r_{t+1+j}^{n} =-\sum_{j=1}^{\infty}\omega^{j}\Delta E_{t+1}\left[ (r_{t+1+j}^{n}-\pi_{t+1+j})+\pi_{t+1+j}\right] $$ Substitute and we have the last and best identity $$ \sum_{j=0}^{\infty}\omega^{j} \Delta E_{t+1}\pi_{t+1+j} = -\sum_{j=0}^{\infty} \rho^{j} \Delta E_{t+1}s_{t+1+j} -\sum_{j=0}^{\infty} \rho^{j} \Delta E_{t+1}g_{t+1+j} +\sum_{j=1}^{\infty} (\rho^{j} -\omega^{j})\Delta E_{t+1}\left( r_{t+1+j}^{n}-\pi_{t+1+j}\right) . $$ With long-term debt a weighted sum of current and future inflation corresponds to changes in expected surpluses and discount rates. A fiscal shock can result in future inflation, thereby falling on today's long term bonds. Equivalently, a surprise deficit today \(s_{t+1}\) must be met by future surpluses, by lower returns, or by devaluing outstanding bonds, so that the debt/GDP ratio is reestablished.


I ran a VAR and computed the responses to various shocks.

Here is the response to an inflation shock - -an unexpected movement \(\Delta E_1 \pi_1\). All other variables may move at the same time as the inflation shock.

Inflation is persistent, so a 1% inflation shock is about a 1.5% cumulative inflation shock, weighted
by the maturity of outstanding debt.

So, where is the 1.5% decline in present value of surpluses? Which terms of the identity matter?
Inflation does come with persistent deficits here. The sample is 1947-2018, so a lot of the inflation shocks come in the 1970s. You might raise three cheers for the fiscal theory, but not so fast. The deficits turn around and become surpluses. The sum of all surpluses term in the identity is a trivial -0.06, effectively zero. These deficits are essentially all paid back by subsequent surpluses.

Growth declines by half a percentage point cumulatively, accounting for 2/3 of the inflation. And the discount rate rises persistently. Two thirds of the devaluation of debt that inflation represents comes from higher real expected returns on government bonds, which in turn means higher interest rates that don't match inflation. (More graphs in the paper.)

Growth here is negatively correlated with inflation, which is true of the overall sample, but not of the story I started out with. What happens in a normal recession, that features lower inflation and lower output? Let's call it an aggregate demand shock. To measure such an event, I simply defined a shock that moves both output and inflation down by 1%. Here are the responses to this "recession shock."
 Inflation and output go down now, by 1%, and by construction. That's how I defined the shock. This is a recession with low growth, low inflation, and deficits. Not shown, interest rates all decline too.

So where does the low inflation come from in the above decomposition. Do today's deficits signal future surpluses? Yes, a bit. But not enough -- the cumulative sum of surpluses is -1.15% On its own, deficits should cause 1% inflation, the fiscal theory puzzle that started me out in this whole business. Growth quickly recovers, but is not positive for a sustained period. Like 2008, we see a basically downward shift in the level of GDP. That contributes another 1% inflationary force. The discount rate falls however,  so strongly as to raise the real value debt by almost 5 percentage points! That overcomes the inflationary forces and accounts for the deflation.

Here is a plot of the interest rates in response to the same shock. i is the three month rate, y is the 10 year rate, and rn is the return on the government bond portfolio. Yes, interest rates at all maturities jump down in this recession. Sharply lower rates mean a one-period windfall for the owners of long term bonds, then expected bond returns fall too.

The point 

Discount rates matter. If you want to understand the fiscal foundations of inflation, you have to understand the government debt valuation equation. Inflation and deflation over the cycle is not driven by changing expected surpluses. If you want to view it "passively," inflation and deflation over the cycle does not result in passive policy accommodation through taxes, as most footnotes presume. The fiscal roots (or consequences) of inflation over the cycle are the strong variation in discount rates -- expected returns.

The fiscal process

Notice that the response of primary surpluses in all these graphs is s-shaped. Primary surpluses do not follow an AR(1) type process. In response to today's deficits, there is eventually a shift to a long string of surpluses that partially repay much though not all of that debt. This seems completely normal, except that so many models specify AR(1) style processes for fiscal surpluses. Surely that is a huge mistake. Stay tuned. The next paper shows how to put an s-shaped surplus process in a model and why it is so important to do so.

Comments on the paper are most welcome.


  1. Question:

    "These deficits are essentially all paid back by subsequent surpluses."

    Could this be used to justify that MMT could actually work? The fiscal side essentially takes over the role of the monetary side - the roles are flipped. Print your way to victory?

    Alternatives - Monetary Adjustment to the Fiscal: Force the interest rate down with an uptick in demand for government debt (bonds)? The price of bonds increase, however, in theory. The Fed did this essentially - they unleashed their unconventional monetary policy in 2008 by buying up long term government debt/bonds. Then there's IOR, too, effectively limiting supply of borrowable capital from the retail banking section to consumers, because banks get a better return on IOR than any other return opportunity - sterilization helped prevent inflation due to huge capital injections by the Fed into banks, to heal their liquidity problems [ability to loan out surpluses to get a return] (880B -> 4.5 T on their balance sheet due to rounds of QE).

    So I see forces working against one another to balance things out...

    How does this model affect stuff like demand-pull and cost-push inflation? Those are separate forces that work in conjunction or against fiscal forces that contribute to inflation, the way cost-push and demand-pull can work with one another (bad) or push against one another?

    More to read I suppose. Very interesting stuff.


  2. Well, real interest rates declined and sharply. The discount rate for government debt declined, which raises the value of government debt, even if expected future surpluses are unchanged or declined. With a lower discount rate, government debt is more valuable. If the price level does not change, people want to buy less stuff and more government debt. That's lower aggregate demand, which pushes the price level down. Does this story bear out, quantitatively, in the data? Yes.---John C.

    Well, it has been a few decades since I passed calculus.

    But my question is related to the comment also seen above. What happens if a government extinguishes debt, say through quantitative easing, but the bonds purchased are simply disappeared.

    Or if it government chooses to stimulate through money-financed fiscal programs?

    Would heavy globalization of real and financial markets affect your observations?

    If the US Federal Reserve purchased $10 trillion of foreign sovereign bonds, would that change your outlook?

  3. "the real value of nominal debt equals the present value of surpluses. So, when there is inflation, the real value of nominal debt declines. Does that decline come about by lower future surpluses, or by a higher discount rate?'

    1) Is money valued like stock or like bonds? If it is like stock, then changes in the backing assets will cause corresponding changes in the value of money. But if it's like bonds, the backing assets only affect the value of the money when the government's net worth is negative.

    2) The history of American colonial currency sheds light on how government assets back money. Specifically the works of Bruce Smith, Curtis Nettels, Leslie Brock, John McCusker, Joseph Ernst, and Andrew McFarland Davis.

    3) There is a problem of inflationary feedback. Inflation reduces the value of the dollar, which reduces the value of the Fed's (dollar-denominated) bonds, which reduces the value of the dollar still more, which reduces the value of the Fed's bonds still more, etc.

    1. Yes yes and maybe. In the end the value of the dollar comes from the treasury's willingness to tax. In the end, if Fed insolvency causes a problem, the Treasury and will recapitalize it (same with ECB). I can think of no more too big to fail institution than the Fed. So in the end, it all comes down to whether the treasury has the resources to repay debt.

  4. The first equation appearing in the blog post above is ρ∙vₜ₊₁ = vₜ + rⁿₜ₊₁ - πₜ₊₁ - gₜ₊₁ - sₜ₊₁. This is evidently a log-linear approximation. The coefficient, ρ, is defined in the associated paper as exp(-r) and is apparently intended to be a one-period discount factor to bring the variable vₜ₊₁ to a common time date with the variable vₜ .

    I contend that inclusion of the coefficient ρ is unnecessary, and leads to an error in the models subsequently evolved in the blog. The proper log-linear model is simply vₜ₊₁ = vₜ + rⁿₜ₊₁ - πₜ₊₁ - gₜ₊₁ - sₜ₊₁, where it is impliedly understood that the variable sₜ₊₁ includes an error term εₜ₊₁ in addition to the quotient Sₜ₊₁/Yₜ₊₁ (where Sₜ₊₁ is the real primary surplus, and Yₜ₊₁ is the real gross domestic product, both in period t + 1). In keeping with models of this type, the expectation of εₜ₊₁ is nil, and the second moment is positive.

    Taking the antilogarithm of the revised log-linear equation gives the underlying (non-linear) model and demonstrates that the coefficient ρ equals 1 (i.e., r ≡ 0 identically). Using upper case letters for variables, the model is given by Vₜ₊₁ = Vₜ∙Rⁿₜ₊₁∙(Pₜ / Pₜ₊₁)∙(Yₜ / Yₜ₊₁)∙exp(-Sₜ₊₁/Yₜ₊₁)∙exp(-εₜ₊₁). In this expression, (Pₜ / Pₜ₊₁) = exp(- πₜ₊₁) and (Yₜ / Yₜ₊₁) = exp(- gₜ₊₁).

    If one lets Bₜ = Vₜ∙Yₜ , where B = the value of government bonds outstanding, in real dollars, then the variable gₜ₊₁ drops out of the log-linear model and the quotient Yₜ / Yₜ₊₁ drops out of the corresponding antilog expression. If one converts the equations to nominal dollar bond values, the quotient Pₜ / Pₜ₊₁ disappears from the nominal dollar antilog expression which becomes
    Bⁿₜ₊₁ = Bⁿₜ∙Rⁿₜ₊₁∙exp(-Sₜ₊₁/Yₜ₊₁)∙exp(-εₜ₊₁). Dividing both sides of this expression by Bⁿₜ yields the expression for the holding period gross return on the bond portfolio (1 + aₜ₊₁) = Rⁿₜ₊₁∙exp(-Sₜ₊₁/Yₜ₊₁)∙exp(-εₜ₊₁), where the variable Rⁿₜ₊₁ is interpreted as the gross expected return on the nominal bond portfolio with Sₜ₊₁/Yₜ₊₁ identically zero.

    From the foregoing workup, it is evident that ρ ≡ exp(-r) = 1 (i.e., r ≡ 0 for all t) in the first expression appearing in the blog post.

    1. Eagle eye has an eagle eye. Yes it is a loglinear approximation and yes rho = e^ - r is the point about which one takes the approximation. Also yes it works just fine with r = 0 and rho = 1. However, the campbell shiller linearization uses a rho less than one, and people at early workshops were confused with sums that didn't have the rho. So I use rho > 1 in the exposition but the calcluations use rho =1. This is the sort of thing clear in papers but not in blog posts!

    2. Yes, it was clear in the paper that rho = 1 is assumed, but it wasn't clear (to this reader) why that might be so. The workup in my comment demonstrates that rho is identically unity in the log-linear model, and, furthermore, it can take on no other value.

      The expression rⁿₜ₊₁ - πₜ₊₁ (the log nominal rate of return less the log rate of change in the price index) estimates the discount rate (real) and serves the same purpose that rho would serve if rⁿₜ₊₁ - πₜ₊₁ was absent from the log-linear model.

      I haven't taken this onto the next step in the model evolution (forward expectations and forward innovations). However, E(rⁿₜ₊₁ - πₜ₊₁|t) should, in principle, be nil for risk-free government bills, notes and bonds. Conversely, E((rⁿₜ₊₁ - πₜ₊₁)²|t) should be non-negative and probably positive for all rⁿₜ₊₁ - πₜ₊₁t in order that the model robustly mirrors observations of day-to-day market action.

      If rⁿₜ₊₁ - πₜ₊₁ is taken to be a martingale, would that materially alter the model output? This would likely have significance to the interpretation of the innovations expectations operator, ΔE(rⁿₜ₊₁ - πₜ₊₁|t). The short answer could well be that rⁿₜ₊₁ - πₜ₊₁ is not a martingale, and E(rⁿₜ₊₁ - πₜ₊₁|t) is not identically nil, i.e., the expected real rate of return is non-zero, and possibly negative most, if not all, of the time. Ideally, one would like to explore the possibilities of both alternative views.

  5. A quibble:
    " the treasury's willingness to tax."
    should say "the combined assets of the money-issuer" . For example, if the government owned a bunch of land, enough to retire all the dollars it issued, then willingness to tax would not matter.


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