Why Specifying r, g Makes No Sense Whatsoever

I ran across yet another article discussing how the r>g condition constrains fiscal policy. (For those who are not big fans of this stuff, that means that the long-term average real rate of interest is greater than the long-term real GDP growth rate.) I explain in this short post why any analysis that is premised on these concepts makes no sense.

Although some people might argue that I am targeting a straw man, I refuse to be drawn down the rabbit hole of discussing models that are obviously wrong.

I have probably written this out before, but I am doing this again with the absolute minimum of text.

I will denote the long-term steady states for growth, real interest rates, and inflation as: g*, r*, and i*. I am unconcerned about any other interpretation of those symbols.

I have three premises:

1. We assume that we are working with a model that admits a long-term steady state growth state. We could add realism (limit cycles, finite lifespan of the Earth, whatever), but that would only be a distraction from the premise, since we are likely able to find some kind of bounds for growth rates in an unchanged policy environment.
2. Maybe there are fundamental reasons that make long-term real growth (g*) a constant, or maybe it is dependent upon policy. This is not a concern.
3. My argument is that we cannot assert that there is a long-term real interest rate (r*) that is policy independent.

Why is #3 correct? We will do a proof by contradiction.

Let us assume the converse, that there is a policy-independent steady state real rate of interest r* (as well as a fixed real growth rate g*).

We then imagine a government policy framework (in a closed economy).

1. The nominal rate of interest is locked at 1%.
2. The growth rate of nominal expenditures is set in such a fashion that it will converge to a 5% nominal steady state growth rate. In steady state, they are "about" 20% of GDP.
3. Taxes are progressive, with an average tax take of about 20% at a 5% nominal steady state growth rate, and increasing (due to tax bracket creep) if growth is running above that level.

Points #2 and #3 are there just to create an argument that nominal growth rates cannot deviate much from 5% in steady state, as the government sector would either shoot to 100% or 0% of GDP, and the assumption of progressive taxes creates a stabilising force to prevent that. The interest rate of 1% is too low to create a debt spiral given a nominal growth rate of 5%. (Please see Technical Appendix for more details.)

Now what about r*? Since the nominal rate of interest is 1%, then that means that the inflation rate in steady state (i*) is  given by: i* = 1%-r*. Since the steady state for nominal growth is forced to be 5%, the real growth rate is g* = 5% - i* = 4% + r*. (I.e., g* > r* no matter what r* is.)

We then see that 5% was a policy choice. If the government decided to make the long-term nominal growth rate 5%, then we can re-run the numbers to see that g* = 5% + r*. In other words, real growth can be cranked up by increasing the nominal growth rate of spending, which implies that tending towards hyperinflation creates a free lunch. Furthermore, this contradicts the assumption that g* was fixed.

In summary, if you believe that r* is fixed, you have to accept that running increasingly high inflation rates will correspondingly increase real growth rates, and that g* is not fixed. Since nobody appears to believe that, we have a contradiction of the assumption that r* is fixed.

But Equilibrium!

It is possible that someone might object that equilibrium cannot exist if the nominal interest rate is fixed. That just tells us that the equilibrium-based model is an incorrect description of reality. There is no reason to believe that an economy following the policy structure above would fail to converge to a steady state, and assertions to the contrary would actually need to demonstrate why the mathematics proves there is no equilibrium. (I am not holding my breath waiting for that demonstration.)

Technical Appendix: Forcing Steady State Growth Rates With Fiscal Policy

My "verbal proof" relies on the ability to force a particular steady state growth rate with just fiscal policy. There's a lot hidden under those assertions, which I outline here.

My first point is that I am model-agnostic; the argument should apply to any "sensible" model. The only requirement is that there is some steady state growth rate.

If the steady state exists, nominal GDP has to grow at the pace of government expenditures, as otherwise we get outcomes that are precluded by the design of fiscal policy: government expenditures either going to 0% or 100%. We then turn to the question: will such a policy rule allow a steady state to exist?

Let's assume that the long-term population growth rate is 0%, in such a case, if we want 5% nominal GDP growth, we need to target 5% nominal per capita income growth. (If the population growth is 1%, which is a typical number for long periods in the post-war period, we need 4% per capita income growth.) We do this by building fiscal policy settings around that objective.

• Income tax brackets are set to rise at 5% per year.
• Things like governmental salaries, minimum wage, Job Guarantee wage, transfers to oldsters are set to rise at 5% per year.
• Governmental budget envelopes for non-wage expenses are similarly set to grow at 5%. This means that they need to buy less stuff if goods inflation rises.

The last bit will cause some consternation among neoclassical economists. The government is not setting a budget in real terms, and then paying the market price: it is acting like a currency monopolist (which it is) that cares about the value of the currency. Such rules might need to be set aside in wartime, but in a serious war, World War II demonstrated that the government does not need to act like a price taker.

The government just needs to keep the income tax system set up so that it punishes sustained rises beyond a 5% nominal growth rate; the government fiscal balance will zoom into surplus, and this will eventually strangle the private sector (remembering that this is a closed economy). One mechanism is that risk-free assets within private portfolios will dwindle, and so eventually a financial system will blow itself up if it keeps financing investment at an unsustainable pace.

Even if the government gets aggregate nominal wages to hit its target, this does not imply that GDP will do so at all times. Very simply, the profit share can rise and fall. However, unless the wage share either goes to 0% or (over) 100%, in steady state, it will have to fall in line with aggregate wage growth. Since we have pinned down nominal income growth in steady state, so is nominal GDP.

I am not saying that this is necessarily a good policy, all that matters is that it is feasible.

(c) Brian Romanchuk 2019