Wednesday, January 4, 2023

Fun Fisherian Graph

In working on a revision to fiscal theory of the price level chapter 5 on sticky price models, and a revision of "Expectations and the neutrality of interest rates" I came up with this fun impulse-response function. It  has an important lesson about interpreting impulse response functions.

It's a response to the indicated interest rate path, with no change in fiscal policy, in a simple new-Keynesian model with short-term debt. 

Rational expectations new-Keynesian models have the implication that higher interest rates raise inflation in the long run. They also tend to raise inflation in the short run. I've been looking for better mechanisms by which higher interest rates might lower inflation in the short run in these models, without adding a contemporaneous fiscal austerity as standard new-Keynesian models do. Fiscal theory explores a model based on long-term debt that does the trick, but has a lot of shortcomings. So I'm looking for something better. 

This graph has only short term debt. I generate the pretty interest rate response by hand. It follows \(i_t=30e^{-1.2t}-29.5e^{-1.3t}-0.05.\) Then I compute inflation and output in response to that interest rate path. 

Wow! Higher interest rates lead to high real interest rates,  send inflation down, and create a little recession. Once inflation is really lowered, the central bank can lower interest rates. The price level (not shown) falls nearly linearly, as we often see in VARs.   

Doesn't this look a lot like the standard story for the 1980s? A big dose of high real rates lowers inflation, and then the Fed can follow inflation downward and get back to normal at a lower rate. 

That analysis is totally wrong!  In this model, a higher interest rate always leads to higher inflation in both the short and the long run.  Inflation is a two-sided moving average of interest rates with positive coefficients. Inflation declines here in advance of the protracted interest rate decline starting in year 2. Lower future interest rates drag inflation down, despite, not because of the rise in interest rate from year 0 to year 2, and despite, not because of the high real interest rates of that period. Those high real rates add interest costs on the debt and are an inflationary force here.  If the central bank wants a disinflation in this model, it will achieve that sooner by simply lowering interest rates immediately. The Fisherian effect will kick in faster, and it will not be fighting the fiscal consequences of higher interest costs on the debt. 

Beware facile interpretations of impulse-response functions! It would be easy to read this one as saying high interest rates bring down inflation and cause a recession, and then the central bank can normalize. But that intuition is exactly wrong of the model that produces this graph.  

The model is \[ \begin{align*} E_t dx_{t} & =\sigma(i_{t}-\pi_{t})dt\\ E_t d\pi_{t} & =\left( \rho\pi_{t}-\kappa x_{t}\right) dt \\ dv_{t} & =( rv_{t}+i_t-\pi_{t}-\tilde{s}_{t}) dt  \end{align*}\] Parameters are \(\kappa = 0.1, \  \sigma = 0.25,\   \rho = 0.1,\ r = 0.01.\) I used a lot of price stickiness and an unrealistically high \(\rho\) to make the graph prettier.  

Update: For Old Eagle Eye. I'm plotting an impulse response function. Variables start at zero, there is one shock, then we solve the deterministic version of the model. The system has two variables with expectations, and two unstable eigenvalues. So we solve forward to determine the initial conditions uniquely. All explained in FTPL, see especially the new Chapter 5 and pointer to the Online Appendix with formulas. 


  1. The first and second equations of the model stand alone; dropping the expectation operators and adding two correlated but distinct Wiener processes, one to each of the first two equations allows the equations to be integrated and the distributions in probability of x(t) and pi(t) to be found, given the specification of i(t) as described in the article, and assumptions of the respective volatility parameters modifying the Weiner process terms.

    The third equation, the one for v(t), could also be solved to find the distribution in probability of v(t) if the process represented by (s~)(t) is known; otherwise, little of use can be said for v(t). Since (s~)(t) is elsewhere defined as S(t)/B(t) (nominal) or s(t)/b(t) (inflation-adjusted, or "real") and v(t) is a function of B(t) or b(t) in magnitude, one would ideally want to have some idea of the process characteristics of the primary surplus, S(t) (nominal), to close the model.

    Given that the three-equation model is implicitly stochastic, the best that can be achieved is a description of the variables, x(t), pi(t), and v(t), probability as a function of time (given constant parameters). The charts in the article present deterministic paths for x(t) and pi(t) which might represent the mean or central limit of the pathwise evolution of those variables.

    The models are interesting as examples, but there is still much that we don't know and as such it is difficult, in my view, to infer sweeping real-life generalizations based on such examples. On the other hand, the time-tested adage, "nothing ventured, nothing gained," comes to mind.

    1. Yes, I've followed all of that. The SDE for v(t) takes for its inputs the rate of interest, i(t) (presumably represented by the FFR), and the primary surplus sp(t) scaled by the divisor y(t)∙exp(v*), where v* is the steady-state value of ln(V*/P*/y*). * indicates the steady-state value of the variable. Definitions: V* = M* + ∑Q*B*, where P* is the price level, y* is the national income, Q* is the price of one bond, and B* is the quantity of bonds issued and outstanding.

      If one draws a diagram of the system of the three equations, it is easily seen that the deviation of the rate of inflation, π(t), and the deviation of the output gap, x(t), are not influenced by the deviation v(t). The variable s̃(t), representing the deviation of the primary surplus, is the disturbance or "shock" variable, but it only affects the value of v(t) and does not affect the value of π(t). According to the FTPL, variations in v(t) should feed back into variations in π(t) and x(t). But, the system of equations shown above do not allow for this.

      I concur that there are two unstable eigenvalues in the system represented by the first two SDEs. But, the third SDE adds one more unstable eigenvalue, namely that related to the parameter r in the equation for dv(t), from the solution of the homogeneous equation dv(t) - r∙v(t) = 0. Taking the three-equation model as a system, the model is 3rd-order having one stable eigenvalue and two unstable eigenvalues. There is one manipulated variable, i(t), one disturbance variable (exogenous), s̃(t), and three state variables: π(t), x(t), and v(t).

      The model represents a linearized approximation to the non-linear system. Linearized approximations have validity over a narrow range of variable values, in this case in the close neighborhood of the steady-state values. The assumptions on the linearized IS curve are straight-forward. The assumptions underpinning the PC and V(t) equations are less so.

      Can the new Keynesian IS and PC equations be dispensed with in the FTPL? The V(t) curve appears to be a central element of the FTPL. Presumably, it cannot be dispensed with. I suppose, the question boils down to this: When, or, under what circumstances, does the FTPL determine the deviation in the price level?

  2. When you have a government deficit spending like a drunk sailor do interest rates have the desired effect?

  3. Very interesting results as far as the inflation response goes as time t increases vs interest rate. Would be interesting to see the response when a Dirac delta distribution is used as the input function(proper impulse response). From the model, I cannot tell if it would end up with a unique solution or converge but that would I think fully describe the transfer function between inflation and interest rate in the model if did so.

  4. This seems like a big departure from Friedman's moentary theory that has opposite policy implications. It may be true, but like MMT, it should be tried on a small sample (or country) not the world's largest economy and reserve currency

  5. Since June of 2022 through November of 2022, both CPI-U and PCE price indices have been increasing at an average annual compound growth rate of 2.42%. See the following chart at FRB St. Louis' FRED website: for details. Both indices in the chart are rescaled to 100 as at June 2022.

    To the extent that these indices are representative of the general price level, it would appear that the rate of inflation is within the FOMC's target range of 2% plus or minus 100 basis points. A rational central bank planner would be declaring 'victory in the "fight against inflation" and pause the interest rate hikes at this point in time. To continue hiking the Fed Funds Rate from here is to risk an unnecessary economic downturn. The change in focus of the rhetoric emanating from the FOMC members from headline inflation to labor market conditions suggests that the members lack confidence in the tools of their trade. That lack of confidence will see the FOMC bring on an unnecessary recession this year or next. Was it ever not thus?

  6. The level of interest rates is not going to follow the Fisher effect ("tendency for nominal interest rates to change to follow the inflation rate"). We will have higher structural levels of interest rates due to outsized government deficits and debts without QE. QE has its limits under the payment of interest on interbank demand deposits.

  7. So. Erdogen in Turkey is correct? And the IMF has been wrong all these years? We should lower rates to beat inflation?
    I think this analysis IS correct assuming a low debt overhang and a consistent, manageable deficit below the GDP growth.
    We have neither of those, and our seigniorage... is a decreasing quantity.
    More likely, our new inflation rate, absent QE, absent a Fed zero rate, will have a CAPE-SHILLER type average over the next ten years, equal to, or greater than, our annual deficit.

  8. If one plots the CPI urban consumers all-items price index (CPIAUCSL) and the PCE chain-type price (PCEPI) versus calendar date from June 2021 thru December 2022, with both data series re-scaled to 100 on June 2022, it is evident that the rate of inflation in the U.S. is now trending at an annual average compound rate of approximately 2.4%. This is within the 1% to 3% per annum target range that the FOMC has set as a criterion for 'price stability.

    From the chart below, the headline inflation rate of 6% for CPI and 5% for PCE is easily seen to be an artifact of taking the measure of inflation obtained by applying the secant method between data points separated by a 12 month interval. Using that approach, the headline figures for inflation overstate the trend inflation rate by a wide margin.

    See the FRED chart for details.

    What more can be said? Well, if we follow the principles of the Fiscal Theory of the Price Level, we might draw the inference, tentatively, that financial markets and producers are convinced federal authorities will repay the federal debt by running primary surpluses in the near future. Furthermore, we might infer that the central bank has demonstrated conviction and credibility in its monetary policy operations.

    That J. Powell has been talking about wage rate cost-push risks rather than the rate of inflation per se, and has warned against "letting down our guard" amid remarks about Arthur Burns's tenure during the 1970s and the then-current resurgence of inflation, may be taken as further support for the inferences postulated above.

    What of the Fisher equation? Does it follow that the rate of inflation will rise to the rate of interest over time? Such would be the case if the real rate of interest is pinned at 0%. This bears watching, but it is not a given.


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