# City Size Does Not Predict City Growth

## Agglomeration, Smagglomeration!

There’s a demographics meme out there that is so persistent I don’t need to cite examples of it because, if you’re reading this blog, you’ve almost certainly seen it, and may believe it: “big cities are growing faster than small cities.” Or something like, “agglomeration causes the largest cities to outgrow the smallest ones.” Or “big cities are leaving the small cities behind.”

It turns out, this claim is extremely dubious. **The relationship between city size and growth rates is extremely weak, and what relationship does exist may have better explanations than agglomeration.**

### The Past Is Prologue

First, let’s think about how we might predict next year’s population growth rates. If I offered you $1,000 to predict a city’s population growth rate next year, how would you do it?

Well, there are tons of sources you might use. But let me make a recommendation: you may be best off just saying, “I bet it will be the same rate as this year.”

Here’s a chart of correlations over time between various growth rates for 382 US metro areas from 1969 to today:

As you can see, there’s a pretty strong correlation between current growth rates and past growth rates in most years. Sure, during the mid 1970s and the mid-2000s, metro-level growth rates showed some serious volatility. In 2006 especially, Hurricane Katrina radically added volatility, while around 2010 the housing market crash added a second wave of volatility. But, by and large, just knowing the last 1 year of population growth gives you a good guess at the next year. And if you use the gold or red lines instead, using a long-term average instead of just prior-year, or using a long-term growth rate to forecast a subsequent long-term growth rate, you can even more reliably get a fairly good forecast.

The best predictor of future growth is past growth.

This is easy to understand. If an area is growing, it’s usually because something good is happening. There are jobs, cost of living is low, the climate is nice, there’s a university, etc. These factors don’t tend to move wildly about each year; they’re pretty stable. The factors that determine where people want to live do not *usually* experience extreme year-over-year variation for most places. Indeed, many factors determining population growth can be extremely durable: presence of universities, presence of government centers, access to transportation networks, broad industry clusters, etc. These can be stable for *decades*.

### Population Is Accumulated Growth

To understand the argument I’m going to make here, you need to stop thinking of people as people. Think of them as just accumulated growth. At Year T, we add X growth, yielding P population. At year T+1, we can *reasonably expect* that we will add ~X growth, yielding ~P+X. That is, growth next year will probably, as we demonstrated, be fairly similar to growth this year, ultimately because most determinants of growth are not highly volatile. At some point way back in time, P=0, and all P that exists today is the result of some past X which, again, *tends to be stable*.

So when you hear the phrase “a large metro area,” don’t think “a metro area with lots of people.” Instead, think “a metro area where X has historically been high.” And then realize that, if X has historically been high, we can *probably bet* that X will be high in the future.

**Therefore, the basic mathematics of population combined with the general stability of demographically significant factors imply that we should always expect the most populous places to have high growth rates. **If they are populous, it means they had high growth in the past. And if they had high growth in the past, then at least relatively high growth is likely to continue, at least for the immediate future. At any given time, and for any given period, we should have a *normal assumption* that more populous areas will have fast growth rates. That is, the correlation between population growth and urban size should always be positive unless there is some sudden relative worsening of relevant factors in populous places, like, for example, a Hurricane or a credit crunch. Or a crime wave.

### Population Is Just Beginning to Predict Growth

The chart below shows various correlations between population size of a metro area and population growth.

The chart above shows several different things. Let’s start with the black line. For each year, I provide the correlation coefficient between a given metro area’s population *in that year*, and its population growth rate *in 2015.*

As you can see, it’s a smooth, linear slope upwards… from a basically 0 correlation, to just 0.12 correlation. In other words, **2015 population levels explain just 1.43% of the variation in 2015 growth rates. **And that is the absolute *best* that it gets! Holy cow!

The strongest population size-growth correlations I can find nonetheless have virtually no explanatory power for 2015.

But maybe things were different in other years! The green line shows how well the *prior year* population explains *current year* growth rates. So, in 1986, 1985 population levels had a 0.0543 correlation with 1986 population growth rates, meaning they could explain a whopping 0.3% of the variation in population growth rates. That’s not 30%. That’s not 3%. That’s 0.3%.

The yellow line shows how well population levels *ten years ago* explain growth rates in a given year, and the red line shows how well population levels ten years ago explain 10-year accumulated population growth.

**As you can see, the relationship between population growth and population size has varied over time, but most specifications show it getting more positive over time. **In other words, in the 1970s, there was a *negative* association between population and growth! Holy cow!

Remember what we said earlier: that implies that the 1970s must have seen some kind of serious negative shock to the population-relevant factors in populated places in the 1970s.

We see these values go positive and negative several times, but, today, the relationship between metro-area population growth and metro-area population size is near the most positive it has been since my data begins in 1969.

Thus, what we are seeing today with a *very weak *correlation between metro size and metro growth is (1) the strongest that relationship has ever been but also (2) exactly what we should expect, assuming no powerfully negative shocks to large cities. **The growth of large cities should be as natural as gravity pulling you down, but we have only begun to see that relationship develop in recent years. **This trend does show lots of volatility, and so there is no way we can know for sure if it will continue into the future.

### A Weak Relationship

I want to show (1) how weak this relationship is and (2) why it nonetheless shows up so much.

Here’s a bar graph of every metro’s 2005–2015 population growth, organized by 2005 population.

Okay, so, nobody should be looking at that chart and saying, “Yep, totally a clear trend upwards!” I mean, just… no. It’s not there. The highest growth rate is in a small city, and even excluding that outlier, the within-size-range volatility is *way* bigger than any possible trend!

But I can take the same data and present it this way:

And *that* looks rather more convincing. The largest and smallest seem quite different! And the crazy thing is, if I take the largest 50 metros and the smallest 50 metros and do a formal difference-of-two-means test, I’ll turn up a result telling me they are, if not quite statistically significant, then very close to it!

That’s the problem with clusters like this. It makes it easy to hide variation. And of course, if we test along the whole spectrum, it turns out that *not a single cluster of 50 metros* is statistically significantly different from its neighboring clusters.

PS- I’ve used clusters myself, in my article on higher education. I used a clustered line graph. I had qualms about doing so, because it does hide variation. Ultimately, I did include it, but was careful to describe what I’d done in clear terms, offered caveats, admitted the hiding of variation, and also showed that the relationship held up quite well within meaningful sub-groups. Ergo, my use was pretty reasonable. Plus, when I used clustering to make my point about higher education, the underlying distribution of graduate degree holders relative to population growth looked like this:

In that series, grouping off along the spectrum makes some sense because it’s fairly clear that the high end really is different, and there is a somewhat more clearly visible trend. No surprise then that the raw correlation is about 0.3, more than double that on display in the city size/population growth relationship.

### So The Big Cities Stay Big?

Well, no. See, even though city growth *should* be fairly stable, it turns out American cities have been hit by many shocks. So, across the 46 years of my sample, there have been some changes. Just 5 of the 1969 Top 10 cities remain in the Top 10. Detroit, San Francisco, Pittsburgh, and St. Louis have been displaced by Dallas, Miami, Houston, and Atlanta. Just 17 of the Top 25 cities are still in the top 25. About 44 cities in the Top 50 remain, while 86 of the Top 100. **A total of 31 metro areas actually have fewer total people than they had in 1969. **Pittsburgh, Cleveland, Buffalo, Detroit, and Youngstown have all lost more than 100,000 residents. And that’s not due to suburbanization either — these are metro-level populations, and I’m using stable metro-area definitions.

Overall, it ought to be the case that big cities stay big and grow fast: but there really are shocks, and for much of the 20th century, those shocks have come so fast that large cities lost ground. In recent years, it may be that a mixture of long stability and more city-positive shocks have boosted cities, though I’m not sure that trend will continue. The history of the population/growth relationship is volatile enough that we could easily be back at a negative relationship in a few years time.

**In the long run, it’s likely that large population centers will keep getting bigger. That’s just how the world works. They’re big for a reason,and those reasons tend to be fairly durable. But we don’t need fanciful arguments about agglomeration to explain this, and the actual relationship turns out to be highly volatile and frequently violated.**

*Check out my Podcast about the **history of American migration**.*

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*I’m a native of Wilmore, Kentucky, a graduate of Transylvania University, and also the George Washington University’s Elliott School. My real job is as an economist at USDA’s Foreign Agricultural Service, where I analyze and forecast cotton market conditions. I’m married to a kickass Kentucky woman named Ruth.*

*My posts are not endorsed by and do not in any way represent the opinions of the United States government or any branch, department, agency, or division of it. My writing represents exclusively my own opinions. I did not receive any financial support or remuneration from any party for this research.*