## An Assertion About a Conjecture

As I previously noted, I am way out on a limb here mathematically. I could either keep my mouth shut while I double-check my work, or double down. I have decided to double down (perhaps I could say that my account was hacked if my proof does not work?).If we look at the (asserted) proof of the conjecture, interest rate aficionados would have noted that a locked nominal rates at zero, pretty much as an afterthought. Even so, we ended up with a situation in which every model with a steady state ends up with a steady state with an inflation rate of zero. Obviously, that steady state assumption could be challenged; its existence would require a lot more effort. That said, most standard classes of economic models do converge to a steady state under such conditions (no investment, growth, disturbances); in fact, pretty much all workhorse DSGE models are based on linearisations around some (unknown) steady state.

By applying basic arithmetic, we see that the real (and nominal) rate of interest is zero in every single one of the steady state for every single model. This by itself should not be disturbing to neoclassical economists; it seems to be a standard result that the natural rate of interest is zero in a no-growth economy.

However, if the government wants to torment neoclassical economists, it could institute a policy of indexation of the Job Guarantee wage, and the progressive income tax brackets, with the indexation factor fixed. For example, the government may have read on the internet that a 2% rate of inflation is optimal, and have all nominal values in policy parameters grow by 2% per year.

It is clear that the inflation rate can no longer be zero under these circumstances. If there is a steady state with a rate of wage inflation of less than 2%, the average private sector wage would drop below the minimum reservation wage (defined versus the Job Guarantee wage) in finite time. If the steady state wage inflation were greater than 2%, the average tax rate would rise to 100% (or whatever draconian marginal tax rate is imposed to preserve price stability). (It is clear that the proof of non-explosive wage growth is non-trivial, and I would need to formalise things more to make it acceptable.)

Since the nominal interest rate was still fixed at 0%, we end up with a steady state real rate of interest that is less than or equal to -2% (assuming a steady state exists). I am unaware of a rigorous standard definition of the natural rate of interest, but I assume that it would have difficulties with such a thought experiment. A change to fiscal policy parameters -- which may be very difficult to discern from just the time series of spending and taxation -- has moved the real rate of interest.

Of course, we can use a statistical procedure to tell us what the average real rate of interest is in a steady state. For example, we know that the observed real rate of interest in any steady state will converge to the 12-month moving average of the real rate of interest. However, that observation tells us nothing about the real rate in any other steady state (unless the steady state value is coincidentally the same!). Therefore, it is clear that if we define the "natural rate of interest" to be the output of a statistical procedure, it will likely exist, but has no predictive power for other steady states.

It is clear that this result would contradict the interpretation of most mainstream economic models. The reason for the contradiction is that someone's mathematics is wrong (possibly mine), or it is an artefact of poorly-specified fiscal policy in mainstream models. In mainstream economics, tax rates and spending are indexed to the price level (pure price-taking behaviour), and so it accommodates

*any*steady state inflation rate. As my assertion suggests, indexation policy is a key determinant of long-run inflation trends. If we assume that fiscal policy is actively destabilising price level stability, it is clear via a process of elimination that only monetary policy can control inflation.

## Definition Difficulty

We see that it is difficult to come up with a definition of a natural rate that will apply to all of these models. Any attempt to argue that the steady state rate of interest will be constant can be as easily shot out of the water as J*.We then need to say that the observed rate of interest in steady state is equal to some "natural level," but then modified by "other factors." However, if we do not pin down those other factors, the statement is trivial.

*Any*steady state value of

*any*variable is zero plus "other factors."

Unfortunately for the attempt to define the natural rate of interest, the value is being determined by the change in tax rate schedule in these models. At any given point in time, the tax schedule is driven by a pair of vectors (tax rates, bracket levels). We need to somehow incorporate these vector-valued variables into our statistical estimation procedure. It is unclear whether such a procedure exists.

We cannot just use the observed average tax rates to replace these vectors. The observed tax rate will rise and fall with the cycle, and it makes it appear that a purely passive tax policy is active. This is misleading, and it would be near impossible to extrapolate historical behaviour to a new policy regime.

## Back to the Real World

What's happening in the real world? In the high-inflation era of the 1960-1970s, fiscal policy was actively indexed to inflation. The highest income tax brackets featured a high marginal tax rate, but only fools and the unlucky paid taxes at those rates. (Entertainers are essentially employees and unable to structure their affairs to avoid taxes, and thus the "hippies against high taxes" sub-genre of popular culture appeared.) For the bulk of the income distribution, spending and tax brackets accommodated inflation.In the current era of price stability, tax brackets and government spending has been stable. Unsurprisingly, we have been able to sit in a steady state where inflation sticks around 2% (outside of Japan, where the steady state has been closer to 0%).

In any event, my discussion here is theoretical, and is not meant to be policy advice.

(c) Brian Romanchuk 2017

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