**REVISED 10/18/16, to report a new estimate of the Rahn curve after correcting a slight error in the previous estimate.**

**REVISED 10/20/16, to add a fourth explanatory variable, which improves the fit of the equation.**

The theory behind the Rahn Curve is simple — but not simplistic. A relatively small government with powers limited mainly to the protection of citizens and their property is worth more than its cost to taxpayers because it fosters productive economic activity (not to mention liberty). But additional government spending hinders productive activity in many ways, which are discussed in Daniel Mitchell’s paper, “The Impact of Government Spending on Economic Growth.” (I would add to Mitchell’s list the burden of regulatory activity, which grows even when government does not.)

What does the Rahn Curve look like? Mitchell estimates this relationship between government spending and economic growth:

The curve is dashed rather than solid at low values of government spending because it has been decades since the governments of developed nations have spent as little as 20 percent of GDP. But as Mitchell and others note, the combined spending of governments in the U.S. was 10 percent (and less) until the eve of the Great Depression. And it was in the low-spending, laissez-faire era from the end of the Civil War to the early 1900s that the U.S. enjoyed its highest sustained rate of economic growth.

In an earlier post, I ventured an estimate of the Rahn curve that spanned most of the history of the United States. I came up with this relationship (terms modified for simplicity (with a slight cosmetic change in terminology):

**Y _{g} = 0.054 -0.066F
**

To be precise, it’s the annualized rate of growth over the most recent 10-year span (*Y _{g}*), as a function of

*F*(fraction of GDP spent by governments at all levels) in the preceding 10 years. The relationship is lagged because it takes time for government spending (and related regulatory activities) to wreak their counterproductive effects on economic activity. Also, I include transfer payments (e.g., Social Security) in my measure of

*F*because there’s no essential difference between transfer payments and many other kinds of government spending. They all take money from those who produce and give it to those who don’t (e.g., government employees engaged in paper-shuffling, unproductive social-engineering schemes, and counterproductive regulatory activities).

When *F* is greater than the amount needed for national defense and domestic justice — no more than 0.1 (10 percent of GDP) — it discourages productive, growth-producing, job-creating activity. And because government spending weighs most heavily on taxpayers with above-average incomes, higher rates of *F* also discourage saving, which finances growth-producing investments in new businesses, business expansion, and capital (i.e., new and more productive business assets, both physical and intellectual).

I’ve taken a closer look at the post-World War II numbers because of the marked decline in the rate of growth since the end of the war:

Here’s the revised result (with cosmetic changes in terminology):

**Y _{g} = 0.0275 -0.347F + 0.0769A – 0.000327R – 0.135P
**

Where,

*Y _{g}* = real rate of GDP growth in a 10-year span (annualized)

*F* = fraction of GDP spent by governments at all levels during the preceding 10 years

*A* = the constant-dollar value of private nonresidential assets (business assets) as a fraction of GDP, averaged over the preceding 10 years

*R* = average number of Federal Register pages, in thousands, for the preceding 10-year period

*P* = growth in the CPI-U during the preceding 10 years (annualized).

The r-squared of the equation is 0.73 and the F-value is 2.00E-12. The p-values of the intercept and coefficients are 0.099, 1.75E-07, 1.96E-08, 8.24E-05, and 0.0096. The standard error of the estimate is 0.0051, that is, about half a percentage point. (Except for the p-value on the coefficient, the other statistics are improved from the previous version, which omitted CPI).

Here’s how the equations with and without P stack up against actual changes in 10-year rates of real GDP growth:

The equation with *P* captures the “bump” in 2000, and is generally (though not always) closer to the mark than the equation without *P*.

What does the new equation portend for the next 10 years? Based on the values of *F, A, R,* and *P* for the most recent 10-year period (2006-2015), the real rate of growth for the next 10 years will be about 1.9 percent. (It was 1.4 percent for the version of the equation without P.) The earlier equation (discussed above) yields an estimate of 2.9 percent. The new equation wins the reality test, as you can tell by the blue line in the second graph above.

In fact the year-over-year rates of real growth for the past four quarters (2015Q3 through 2016Q2) are 2.2 percent, 1.9 percent, 1.6 percent, and 1.3 percent. So an estimate of 1.9 percent for the next 10 years may be optimistic.

And it probably is. If *F* were to rise from 0.382 (the average for 2006-2015) to 0.438, the rate of real growth would fall to zero, even if *A, R,* and *P* were to remain at their 2006-2015 levels. (And *R* is much more likely to rise than to fall.) It’s easy to imagine *F* hitting 0.438 with a Democrat president and Democrat-controlled Congress mandating “free” college educations, universal “free” health care, and who knows what else.