We often summarize that the fiscal theory is a theory of the price level: The price level adjusts so that the real value of government debt equals the present value of surpluses. That characterization seems to leave it to a secondary role. But with any even tiny price stickiness, fiscal theory is really a fiscal theory of inflation. The following two parables should make the point, and are a good starting point for understanding what fiscal theory is really all about. This point is somewhat buried in Chapter 5.7 of Fiscal Theory of the Price Level.
Start with the response of the economy to a one-time fiscal shock, a 1% unexpected decline in the sum of current and expected future surpluses, with no change in interest rate, at time 0. The model is below, but today's point is intuition, not staring at equations. This is the continuous-time version of the model, which clarifies the intuitive points.
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Response to 1% fiscal shock at time 0 with no change in interest rates |
The one-time fiscal shock produces a protracted inflation. The price level does not move at all on the date of the shock. Bondholders lose value from an extended period of negative real interest rates -- nominal interest rates below inflation.
What's going on? The government debt valuation equation with instantaneous debt and with perfect foresight is
With flexible prices, we have a constant real interest rate, so
But that's not how the simulation in the figure works, with sticky prices. Since now both
In this sticky-price model, the price level cannot jump or diffuse because only an infinitesimal fraction of firms can change their price at any instant in time. The price level is continuous and differentiable. The inflation rate can jump or diffuse, and it does so here; the price level starts rising. As we reduce price stickiness, the price level rise happens faster, and smoothly approaches the limit of a price-level jump for flexible prices.
In short, fiscal theory does not operate by changing the initial price level. Fiscal theory determines the path of the inflation rate. It really is a fiscal theory of inflation, of real interest rate determination.
The frictionless model remains a guide to how the sticky price model behaves in the long run. In the frictionless model, monetary policy sets expected inflation via
It is a better characterization of these dynamics that monetary policy---the nominal interest rate---determines a set of equilibrium inflation paths, and fiscal policy determines which one of these paths is the overall equilibrium, inflating away just enough initial debt to match the decline in surpluses.
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Response to 1% deficit shock at time 0 with no change in interest rate |
This second graph gives a bit more detail of the fiscal-shock simulation, plotting the primary surplus
To see how initial bondholders end up financing the deficits, track the value of those bondholders' investment, not the overall value of debt. The latter includes debt sales that finance deficits. The real value of a bond investment held at time 0,
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Response to interest rate shock at time 0 with no change in surpluses |
The third graph presents the response to an unexpected permanent rise in interest rate. With long-term debt, inflation initially declines. The Fed can use this temporary decline to offset some fiscal inflation. Inflation eventually rises to meet the interest rates. Most interest rate rises are not permanent, so we do not often see this long-run stability or neutrality property. The initial decline in interest rates comes in this model from long-term debt. As the dashed line shows, with shorter-maturity debt inflation rises right away. With instantaneous debt, inflation follows the interest rate exactly.
Again, in this continuous-time model the price level does not move instantly. The higher interest rate sets off a period of lower inflation, not a price-level drop.
With long-term debt the perfect-foresight valuation equation is
How does the price level not jump or diffuse with sticky prices? Now
Again, the frictionless model does provide intuition for the long-run behavior of the simulation. The three year decline in price level is reminiscent of the downward jump; the eventual rise of inflation to match the interest rate is reminiscent of the immediate rise in inflation. But again, in the actual dynamics we really have a theory of \emph{inflation}, not a theory of the \emph{price level}, as on impact the price level does not jump at all. Again, the valuation equation generates a path of inflation, of the real interest rate, not a change in the value of the initial price level.
The general lessons of these two simple exercises remain:
Both monetary and fiscal policy drive inflation. Inflation is not always and everywhere a monetary phenomenon, but neither is it always and everywhere fiscal.
In the long run, monetary policy completely determines the expected price level. As the inflation rate ends up matching the interest rate, inflation will go wherever the Fed sends it. If the interest rate went below zero (these are deviations from steady state, so that is possible), it would drag inflation down with it, and the price level would decline in the long run.
One can view the current situation as the lasting effect of a fiscal shock, as in the first graph. One can view the Fed's option to restrain inflation as the ability to add the dynamics of the second graph.
Don't be too put off by the simple AR(1) dynamics. First, these are responses to a single, one-time shock. Historical episodes usually have multiple shocks. Especially when we pick an episode ex-post based on high inflation, it is likely that inflation came from several shocks in a row, not a one-time shock. Second, it is relatively easy to add hump-shaped dynamics to these sorts of responses, by standard devices such as habit persistence preferences or capital accumulation with adjustment costs. Also, full models have additional structural shocks, to the IS or Phillips curves here for example. We analyze history with responses to those shocks as well, with policy rules that react to inflation, output, debt, etc.
The model I use for these simple simulations is a simplified version of the model presented in FTPL 5.7.
I use parameters
Thanks much to Tim Taylor and Eric Leeper for conversations that prompted this distillation, along with evolving talks.
I am not sure that the word 'parables' in the first paragraph is really the best word for this article.
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ReplyDeleteBanks don't lend deposits. Deposits are the result of lending. So, all bank-held savings are unused and unspent. All this was explained in:
ReplyDelete“Should Commercial Banks Accept Savings Deposits?” Conference on Savings and Residential Financing 1961 Proceedings, United States Savings and loan league, Chicago, 1961, 42, 43.
This runs counter to your theory. It documents the rise and fall in velocity.
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ReplyDeleteI think it would be interesting to see the effect of a supply shock interpreted through fiscal theory. The standard technology shock would be sufficient. Under flex price, real interest rates and thus the price level should follow the dynamics of the shock, (negative supply shock = higher real interest rates relative to steady state = higher discount factor on surpluses = higher price level).
ReplyDeleteUnder sticky prices, it should get more complicated. Especially if the agents don't know the dynamics of the shock ex ante, and have to update their priors on how persistent the shock is. The more persistent the shock is, the more costly it is to not adjust prices and vice versa.
A nice feature here is that supply shocks will always feature a return to the trend in the price level, while fiscal shocks lead to a higher trendline.
I think it would be neat to see the math written down.
Nice. Thinking abt the ample reserves regime, a corridor system where repos set a ceiling, o/n reverse repos a floor so that the Fed funds settle between the administered rates, IOR and reverse repo RRP. So how much do these daily adjustments influence the real rate i-\pi. Is there a better question here? Or is this a minor effect, the larger effect the setting of the policy rate, the Fed Funds rate.
ReplyDeleteIf one works through the mathematics of the integral equation, it will be found that the effect is not on the rate of inflation, but rather on the rate of consumption by the representative household in period t, holding B(t)/P(t) constant. The real rate of interest is not defined by r(w) = i(w) - π(w), but by the expression r(t) = ρ + θ∙ċ(t)/c(t) where θ = 1/σ , as in the New Keynesian I.S. equation (predicated on an iso-elastic utility function of the household consumption {c(t)}). A negative 'shock' (e.g., zero-probability MIT shock, as postulated) increases consumption in period t. That increase in consumption is what we have experienced in the past 12 months, leading to an increase in the rate of inflation, π(t), in period t.
ReplyDeleteThe whole apparatus ties together in an interesting way that takes a bit of effort to disentangle. I was unsatisfied by the explanation in the body of the article, and it wasn't until I re-read David Romer's text, Advanced Macroeconomics 5th Ed. (2019, NY: McGraw-HIll Education), that the pieces began to fall into place. The key piece was finding the definition of the real rate of interest, namely, r(t) = ρ + θ∙ċ(t)/c(t) from the first-order conditions for the representative household's problem of optimization of the utility of consumption (the basis of the N-K IS equation listed in the model set of equations at the end of John's article). Then it clicked into place. The constraint B(t)/P(t) = constant, forces change in the current period household consumption to return the system to a state of equilibrium after the "MIT shock" occurs. This produces the change in the real rate of interest. The rate of inflation is the observable variable, and it is the topic of concern for the Central Bank. The state variables are P(t) and B(t). The manipulated variable is c(t), the consumption rate of the representative household. The 'shock' is felt as ∫ ds(t). A negative value results in a positive change in c(t), i.e., ċ(t) > 0. A positive vale of ∫ ds(t) would give a negative change in c(t), i.e., ċ(t) < 0, dπ(t) < 0.
Needless to say, John works at the non-obvious edges of macro. Frustrating at times, but in the end, very often, a satisfactory insight arises.