# What size are national legislatures?

## Introduction

The other day, I came across something called the “cube root rule”, which is a proposed reform to the size of the U.S. House of Representative. Since 1913, the size of the U.S. House has remained constant at 435 seats (except for a few years when two seats were temporarily added to accommodate the addition of Alaska and Hawaii). Under the cube root rule, the size of the U.S. House would be increased to the cube root of the U.S. population (suitably rounded), adjusted every census. As I write this, the U.S. Census bureau estimates that the population is currently about 329.6 million, and so adopting the cube root rule would expand the U.S. House to 691 seats.

The cube root rule was first proposed by political scientist Rein Taagepera in a 1972 paper. In this paper, he examines the size of national legislative bodies and concludes that $n = \sqrt[3]{P}$ is a reasonable for the size $n$ of a nation’s legislature as a function of population $P$. Then he gives a theoretical argument for why $\sqrt[3]{P}$ is an optimal value for $n$.1

I do believe that we would benefit from having a larger U.S. House, for many of the reasons articulated by the New York Times in a 2018 editorial, and perhaps the Cube Root Rule would be a good way to accomplish that. However, the point of this post is not to make that argument, but rather, see what kind of models we can make for the actual observed sizes of legislative bodies.

## Disclaimer

I want to be clear here that I don’t think I’m doing anything novel here. I am pretty sure that many actual political scientists (which I am absolutely not) have tread this ground before. If I were writing an academic paper, rather than just messing around on my blog, I would do a proper literature review and put it here. This post is mostly about having fun programming than anything else.

## Regression for all countries

In order to investigate this question, I needed a data set that contained the size of legislative bodies and populations for a number of countries. If I were a real political scientist, I would probably use some carefully curated data set from some international organization, but because I’m just some guy messing around on his blog, I turned to Wikipedia, namely the article “List of legislatures by number of members”.2 Fortunately, Pandas has the capacity to scrape web pages and extract data frames from their tables.

For this analysis, I considered the size of the lower house of each legislature (in cases where the legislature has more than one chamber). Wikipedia’s table doesn’t just include sovereign nations, but also external territories of various degrees of autonomy (e.g. Puerto Rico and Hong Kong), as well as the European Union (which has an elected parliament). Note that countries are included even if their legislatures are not elected using what we would consider free and fair elections (such as North Korea).

I threw out countries where the size of the lower house was listed as “maximum” or “minimum” (because who knows what the actual size is). If it said “usually” or “normally”, I used the number listed. For Timor-Leste it gave a range (“52 to 65”), so I took the midpoint.

I took the logarithm3 of the population and lower house size and performed a least-squares linear regression, obtaining: $$\log(n) = 0.3482\log(P) - 0.5369, \; r^2 = 0.8482.$$ This corresponds to $n = 0.5846P^{0.3482}$.

Here is a scatterplot of the data along with $n = 0.5846P^{0.3482}$ and $n = \sqrt[3]{P}$:

And here is the same, but plotted in log scale (which makes more sense for this data):

(It should be noted that a very similar plot appears in Taagepera’s paper.)

## Regression for other subsets of countries

As mentioned before, the data set includes countries that are not democracies (e.g. North Korea) as well as non-countries (territories and the European Union). We might restrict our attention to various subsets of countries, such as the members of:

• The North Atlantic Treaty Organization
• The Organisation for Economic Co-operation and Development
• The European Union

The results are in this table:

Group of countriesRegression4$r^2$Plots
all$\log(n) = 0.3482\log(P) - 0.5369 \Rightarrow n = 0.5846P^{0.3482}$$r^2 = 0.8482linear log NATO\log(n) = 0.3990\log(P) - 1.0466 \Rightarrow n = 0.3511P^{0.3990}$$r^2 = 0.8764$linear log
OECD$\log(n) = 0.3850\log(P) - 0.8940 \Rightarrow n = 0.4090P^{0.3850}$$r^2 = 0.8026linear log EU\log(n) = 0.4477\log(P) - 1.8032 \Rightarrow n = 0.1648P^{0.4477}$$r^2 = 0.8817$linear log