Cube root law

The cube root law is an observation in political science that the number of members of a unicameral legislature, or the lower house of a bicameral legislature, is about the cube root of the population being represented.[1] The rule was devised by Estonian political scientist Rein Taagepera in his 1972 paper "The size of national assemblies".[2]

This article is about a trend observed in national legislature sizes relative to the country's population. For the scientific law, see Square–cube law.

The law has led to a proposal to increase the size of the United States House of Representatives so that the number of representatives would be the cube root of the US population as calculated in the most recent census.[3] The House of Representatives has had 435 members since the Reapportionment Act of 1929 was passed; if the US followed the cube root rule, there would be 693 members of the House of Representatives based on the population at the 2020 Census.

This proposal was endorsed by the New York Times editorial board in 2018.[4]

Subsequent analysis

It has been claimed by Giorgio Margaritondo that the experimental data, including the dataset originally used by Taagepera in 1972, actually fits better to a function with a higher exponent, and that there is sufficient deviation from the cube root rule to question its usefulness.[5] In this regard, analysis by Margaritondo gives an optimal formula of: , where A is the size of the assembly, P is the population, and E = 0.45±0.03.

Applying this formula to the U.S. House of Representatives as of the 2020 Census would give a House of between 379 and 1231 members, while using an exponent of 0.4507 gives 693 (the same result using the cube root rule).

Table comparing OECD nations in 2019 with EIU Democracy Index ranking
Further information: OECD and Democracy Index

Out of the countries listed, Lithuania is the only one to exactly match the cube root rule. Moreover, the countries of Denmark, Canada, and Mexico come close to matching the rule.

Some of these countries (eg Germany) have overhang seats in a mixed member proportional system, as a result the size of their parliaments can vary significantly between elections.

Country Lower or unicameral house Population (2019)[6] Lower house size (2019) Cube root of population (nearest person) Difference between lower house and cube root of population Difference between lower house and cube root of population (%) People per representative People per representative (cube root lower house) Democracy Index Ranking (2022)[7]
Australia House of Representatives 25,364,307 151 294 −143 −49% 167,976 86,327 15
Austria National Council 8,877,067 183 207 −24 −12% 48,509 42,873 20
Belgium Chamber of Representatives 11,484,055 150 226 −76 −34% 76,560 50,901 36
Canada House of Commons 37,589,262 338 335 +3 +1% 111,211 112,213 12
Chile Chamber of Deputies 18,952,038 155 267 −112 −42% 122,271 71,084 19
Colombia Chamber of Representatives 50,339,443 172 363 −191 −53% 303,250 136,334 53
Czech Republic Chamber of Deputies 10,669,709 200 220 −20 −9% 53,349 48,466 25
Denmark Folketing 5,818,553 179 180 −1 −1% 32,506 32,350 6
Estonia Riigikogu 1,326,590 101 110 −9 −8% 13,135 12,073 27
Finland Parliament 5,520,314 200 177 +23 +13% 27,602 31,235 5
France National Assembly 67,059,887 577 406 +171 +42% 116,222 165,060 22
Germany Bundestag 83,132,799 709 436 +273 +63% 117,254 190,480 14
Greece Parliament 10,716,322 300 220 +80 +36% 35,721 48,607 25
Hungary National Assembly 9,769,949 199 214 −15 −7% 49,095 45,701 56
Iceland Althing 361,313 63 71 −8 −11% 5,735 5,073 3
Ireland Dáil 5,100,000 166 173 −7 −5% 31,275 29,011 8
Israel Knesset 9,053,300 120 208 −88 −42% 75,444 43,438 29
Italy Chamber of Deputies 60,297,396 630 392 +238 +61% 95,710 153,768 34
Japan House of Representatives 126,264,931 465 502 −37 −7% 271,537 251,684 16
Korea, Republic of National Assembly 51,709,098 300 373 −73 −20% 172,384 138,796 24
Latvia Saeima 1,912,789 100 124 −24 −19% 19,218 15,409 38
Lithuania Seimas 2,786,844 141 141 0 0% 19,765 19,803 39
Luxembourg Chamber of Deputies 619,896 60 85 −25 −29% 10,332 7,270 13
Mexico Chamber of Deputies 127,575,529 500 503 −3 −1% 255,151 253,422 89
Netherlands House of Representatives 17,332,850 150 259 −109 −42% 115,552 66,975 9
New Zealand House of Representatives 4,917,000 120 170 −50 −29% 40,975 28,916 2
Norway Storting 5,347,896 169 175 −6 −3% 31,644 30,581 1
Poland Sejm 37,970,874 460 336 +124 +37% 82,545 112,971 46
Portugal Assembly of the Republic 10,269,417 230 217 +13 +6% 44,650 47,246 28
Slovakia National Council 5,454,073 150 176 −26 −15% 36,360 30,985 43
Slovenia National Assembly 2,087,946 90 128 −38 −30% 23,199 16,336 31
Spain Congress of Deputies 47,076,781 350 361 −11 −3% 134,505 130,378 22
Sweden Riksdag 10,285,453 349 217 +132 +61% 29,471 47,295 4
Switzerland National Council 8,574,832 200 205 −5 −2% 42,874 41,894 7
Turkey Grand National Assembly 83,429,615 600 437 +163 +37% 139,049 190,933 103
United Kingdom House of Commons 66,834,405 650 406 +244 +60% 102,822 164,690 18
United States House of Representatives 328,239,523 435 690 −255 −37% 754,574 475,840 30

Historical US House sizes

The following table describes how the US House of Representatives would have looked historically under the cube root rule according to the Huntington–Hill method.

Census, YearSize ALAKAZARCACOCTDE DCFLGA HIIDILINIAKSKYLAMEMD MAMIMNMSMOMTNENVNHNJ NMNYNCNDOHOKORPARISC SDTNTXUTVTVAWAWVWIWY
1st, 1790158 103 313206814161831034
2nd, 1800175 82 571219672016202124530
3rd, 1810194 72 71111196727156222127627
4th, 1820213 362 8131337912256311413242119524
5th, 1830235 651 1036134781133563514172521113522
6th, 1840258 9251 11710125871136646371223262913419
7th, 1850286 103151 1111012212677125884638112529281234184
8th, 1860316 104451 2111714712767138281237391024129271163168
9th, 1870338 94551 2102215113126671310471511383910241312611731149
10th, 1880369 966251 212231512712757131268163138381124132271212311510
11th, 1890398 1078351 312125141291274714138817171293910124234272111421125111
12th, 1900426 1078351 2313127141381284716141091816121141112242362821117221135121
13th, 1910453 11812461 24132281311811846171410916261212451132383383731119221066121
14th, 1920475 112815461 241322913118118361716118153612142461132694393831021221067121
15th, 1930500 112723461 2612231131081183717201081526121625113327104393731123221067121
16th, 1940512 112827471 371223113107119371720119152512162521432794393731125211077121
17th, 1950536 113738571 3101223114971010381723118142512172531422885373821227211287121
18th, 1960566 1014649681 216122232159710103101625117142412193531423176363821130311296121
19th, 1970590 1015658692 220132232158791131117261161424222135315231763438211323113105131
20th, 1980612 1117664882 2261533311586101131116251171324232044816229873238212384114115131
21st, 1990631 1019675882 233163329147691131215241171324332044617227873039212434216125121
22nd, 2000657 102126791082 13719 33291476910312 152311713245320 44419227882929 213495217144131
23rd, 2010677 102146821182 14121 332814761010313 142212713246319 543212258828210 214556118154121
24th, 2020695 112156831282 24522 34271476910313 152112613247319 442222258927211 215617118164121
Census, YearSize ALAKAZARCACOCTDE DCFLGA HIIDILINIAKSKYLAMEMD MAMIMNMSMOMTNENVNHNJ NMNYNCNDOHOKORPARISC SDTNTXUTVTVAWAWVWIWY

See also

References
  1. Lutz, Donald S. (2006). Principles of Constitutional Design. Cambridge University Press. ISBN 9781139460552.
  2. Taagepera, Rein (1972). "The size of national assemblies". Social Science Research. 1 (4): 385–401. doi:10.1016/0049-089X(72)90084-1.
  3. Kane, Caroline; Mascioli, Gianni; McGarry, Michael; Nagel, Meira (January 2020). Why the House of Representatives Must Be Expanded and How Today's Congress Can Make It Happen (PDF) (Report). Fordham University School of Law. Retrieved 17 September 2020.
  4. "America Needs a Bigger House". New York Times. 9 November 2018. Retrieved 17 September 2020.
  5. Margaritondo, Giorgio (2021). "Size of National Assemblies: The Classic Derivation of the Cube-Root Law is Conceptually Flawed". Frontiers in Physics. 8: 606. Bibcode:2021FrP.....8..606M. doi:10.3389/fphy.2020.614596. ISSN 2296-424X.
  6. "Population, total - OECD members | Data". data.worldbank.org. Retrieved 2020-09-19.
  7. "EIU Report: Democracy Index 2022". Economist Intelligence Unit. 2023. Retrieved April 24, 2023.
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