# [insert policy here] Won't Boost Growth Rates

Over at the Cato Institute, they hosted an online forum about reviving economic growth. There are lots of smart people involved. The web page has lots of big pictures of their heads, I guess to indicate that their brains are like, totally huge.

Anyway, each one wrote up some proposed policy reform that would help boost long-run growth prospects. Brad DeLong responded to many of the proposals here before his head exploded reading Doug Holtz-Eakin's essay.

I'm not going to quibble with any of the minutiae of the proposals. My point is going to be a general one on the possible growth effects of [insert policy here]. Short answer, there won't be any.

There are two ways to boost GDP growth. Either

• Actively raise current GDP through increased spending by some sector of the economy.
• Raise potential GDP and let transitional growth speed up.

The second one perhaps deserves a little explanation. Transitional growth is an extra boost to growth that occurs when current GDP is below potential GDP. Why does this occur? Bob Solow is why. In an economy with accumulable factors of production (physical capital, human capital, knowledge capital) being below potential GDP means that the return to these factors is relatively high, and hence more investment in those factors is done, boosting GDP growth. The wider is the gap between current and potential GDP, the stronger this transitional growth.

The issue is that [insert policy here] is a policy to raise potential GDP, not current GDP. But the transitional effects this encourages are inherently small. So even if [insert policy here] opens up a big gap between potential and actual GDP, this doesn't translate into much extra growth. In fact, the effects are likely so small that they would be unnoticeable against the general noise in growth rates year by year.

To give you an idea of how little an effect [insert policy here] will have on growth, let's play with math. Output in period ${t+1}$ can be written in terms of output in period ${t}$ this way

$$y_{t+1} = (1+g)[y_t + \lambda (y^{\ast}_t - y_t)].$$

This says that output in ${t+1}$ is equal to ${1+g}$ times current output. That is "regular" growth. The term with the ${\lambda}$ is the additional boost in growth we get from being below potential. ${y^{\ast}_t}$ is potential GDP in period ${t}$, and ${y^{\ast}_t - y_t}$ is the gap in GDP. ${\lambda}$ tells us how much of that gap we make up from period ${t}$ to ${t+1}$. If ${\lambda = 0}$, then we are stuck below potential (secular stagnation). If ${\lambda = 1}$, then immediately next period our GDP will be at potential again.

$$Growth = \frac{y_{t+1}-y_t}{y_t} = (1+g)\left[\lambda \frac{y^{\ast}_t}{y_t} + (1-\lambda)\right] - 1.$$

The growth rate from ${t}$ to ${t+1}$ depends on the ratio of potential to actual GDP today, period ${t}$. If that ratio were equal to one - meaning that we were at potential - then the growth rate just becomes ${g}$, the trend growth rate. The larger is ${y^{\ast}_t/y_t}$ - meaning the farther we are from potential - the higher is the actual growth rate.

Now we can go back to thinking about the possible growth impact of [insert policy here]. GDP today (${y_t}$) is about 16 trillion. Potential GDP today (${y^{\ast}_t}$) is probably about 17 trillion. You can get a lower estimate from the CBO, Robert Gordon, or John Fernald, or a higher estimate from older CBO forecasts. I'm going to err on the high side for potential because this will inflate the growth effect of [insert policy here].

We also need to know the value of ${\lambda}$, the percent of the GDP gap that is closed in a year. We've got lots of evidence that this value is about ${\lambda = 0.02}$, or 2% of the gap closes every year. This estimate goes back to the original cross-country convergence literature starting with Barro (1991), but consistently across samples (countries, US states, Japanese prefectures, Canadian provinces, etc..) economies converge to potential GDP at about 2% of the gap per year.

You get higher values of ${\lambda}$ if you assume that economies pursue optimal savings plans, like in the Ramsey model, meaning that they save at a higher rate when they are farther below steady state. But if there is an economy that saves according to the predictions of the Ramsey model, it is populated by unicorns.

Back to the calculation. The last thing we need is a value for ${g}$, trend growth. Let's call that ${g = 0.02}$, or trend growth in GDP is about 2% per year. Again, we can argue about whether that is higher or lower, but that's not going to be the important factor here.

Okay, so based on the fact that we are currently 1 trillion below trend, the growth rate today should be

$$Growth = (1+.02)\left[.02 \frac{17}{16} + (1-.02)\right] - 1 = .0213$$

or growth should be 2.13%. Growth will be about 0.13 percentage points higher than normal - that's a little over one-tenth of one percent - because we are below potential. The value of ${g}$ is really irrelevant. All the action is inside the brackets. Because ${\lambda}$ is small, there isn't much bite from transitional growth, even though we are $1 trillion below trend. But what about [insert policy here]? That will *raise* potential GDP, and therefore will induce faster transitional growth to the new, higher potential GDP. Okay. Let's say that [insert policy here] has an astonishingly positive impact on potential GDP. I mean massive. [insert policy here] adds a full$1 trillion to potential GDP, which is now $18 trillion. Now, growth under the [insert policy here] regime is $$Growth = (1+.02)\left[.02 \frac{18}{16} + (1-.02)\right] - 1 = .0225$$ Uh, wow? Growth will be an additional 0.12 percentage points higher thanks to [insert policy here]. This is not a massive change in growth. And the growth boost will *decline* over time as we get closer to potential. Fine, but what if [insert policy here] is truly revolutionary, and raises potential GDP by$2 trillion? Then growth will be 0.0238. This could be generously rounded to 0.025, meaning you added a half-point to the growth rate of GDP. But let's not kid ourselves that [insert policy here] is going to have that big of an effect on growth. \$2 trillion implies that [insert policy here] is raising potential GDP by about 12%. That would be an anomaly of historic proportions.

[insert policy here] will not generate any appreciable extra economic growth, even though in the very long-run [insert policy here] may be a net positive for the level of economic activity. The problem is that it takes a very, very, very long time for those positive effects to manifest themselves, and thus [insert policy here] won't do anything to fundamentally change GDP growth.

What about the exceptions I mentioned? Among the proposals, there are a few that could boost current GDP (and thus growth) directly and immediately by encouraging spending.

• Scott Sumner's NGDP targeting. The proposal speaks directly to raising current GDP, as opposed to raising potential GDP. I think of this as solving the balance sheet problems of households. Boost nominal spending and nominal incomes rise, while nominal debts like mortgages remain fixed, leading to extra spending.
• Brad DeLong's raising K-12 teacher salaries. If you could do it *now*, then it would raise incomes for these folks, and boost spending. The second part of the proposal, to tie this to teacher tenure changes, is more of a potential GDP changer. Question, how big of an impact would this really have on spending?
• A number of people mention infrastructure spending. Yes, if we would spend that money *now*, then it would materially boost GDP growth *now*, and as a bonus have long-run benefits for potential GDP.

Ultimately, the issue in the U.S. right now is not with potential GDP. We do not need policies to raise this potential GDP so much as we need policies to get us back to potential. That requires actively boosting immediate spending.