Why climate change helps the rich — evidence of a nondoctrinaire kind

Although publications like Red Flag Magazine have long demonstrated that capitalism is entirely incapable by its very nature of ameliorating, let alone solving the climate crisis — and are perfectly effective even with deeply flawed and outdated geopolitical assumptions — Red Flag have now become truly open that the fight to prevent runaway climate change is purely and simply class struggle against the wealthy. In ‘Tropical Cyclone Alfred and the politics of ‘keeping politics out of it’’ and ‘Why Only Socialism Can Save Us from Climate Catastrophe’, alongside articles I could not find or were rewritten, Red Flag demonstrate that climate change is purely and simply class struggle on a global scale.

What is interesting is that Harvard University’s Branko Milanovic, Peter H. Lindert and Jeffrey G. Williamson, so early as 2007 in their thesis ‘Measuring Ancient Inequality’, made a strong implication that the capitalist class does have much to gain from global warming.

‘Measuring Ancient Inequality’ is a study of inequality in preindustrial societies. The critical issue the thesis notes is how the presence of a “subsistence minimum” below which human survival is impossible creates an Inequality Possibility Frontier (abbreviated to IPF) at a Gini index that appears low by modern industrial standards. Hence, in these ancient societies, even if extraction by the ruling class was the maximum possible, it only permitted seemingly modest inequality before the majority could not survive.

Milanovic, Lindert and Williamson use the symbol ε for the proportion of the population belonging to the ruling class, and s for the substance minimum in 1990 US dollars. The initially assumed value for the subsistence minimum is $PPP 400 — a value based upon the work of Angus Maddison in The World Economy: Historical Statistics from 2003.

Discussing their results, Milanovic, Lindert and Williamson note on pages 15 and 16 that:

“If we used Maddison’s subsistence level of $400, then four estimated Ginis would be significantly greater than the maximum Gini (at their level of income) implied by the IPF: three of these are based on data from India, and the fourth is from Nueva España.[The 1752 Old Castille is also slightly above the IPF.] Recalling our definition of the IPF, these four cases can only be explained by one or more of these five possibilities: (i) a portion of the population cannot even afford the subsistence minimum, (ii) the actual ε is much smaller than the assumed ε=0.001, (iii) inequality within the rich classes is very large, (iv) our estimate of inequality is too high, and/or (v) the subsistence minimum is overestimated. We have already analyzed and dismissed the first three possibilities. The fourth possibility is unlikely: since our estimates of inequality are based only on a few classes, they are likely to be biased downwards, not upwards. The last possibility offers the more likely explanation. It could well be that the subsistence minimum was less than $PPP 400 for some societies. In particular, this is likely to be the case for subtropical or tropical regions where calorie, housing and clothing needs are considerably less than those in temperate climates....”

“If the IPF is drawn under the S=$300 assumption, it shifts the frontier upwards enough to encompass at or below it all our estimated inequalities, with the possible (and modest) exceptions of Moghul India and Nueva España.”

If we invert what ‘Measuring Ancient Inequality’ says above about tropical and subtropical societies, we would logically conclude that in genuinely cold climates with extremely high food, shelter and clothing needs, subsistence minima might be substantially greater than $PPP 400. We might also suppose that in the very hottest regions with the most minimal such demands, the subsistence minimum could be still lower than $PPP 300. The initial ‘Measuring Ancient Inequality’ said nothing about how high or low subsistence minima could potentially be. A revised version from 2008, which incudes additional societies, does say it is unlikely that even in the hottest deserts the subsistence minimum could be less than $PPP 300. A 2011 version says:

...violations of the ‘maximum feasible’ definition of the IPF might be due to measurement errors or might reflect the possibility that some people can live below subsistence temporarily. The measurement errors could be of three types: mismeasuring national product per capita, mismeasuring inequality or applying the wrong (too high)  subsistence...[...made more likely by the fact that the extraction ratio is a ratio of two numbers, each calculated with significant amount of uncertainty: Gini, and maximum Gini which depends on estimate of mean income]. In the case of Moghul India and Nueva España, a portion of the population might have been expected to die from hunger, exhaustion due to forced and underpaid labour or lack of elementary shelter. Poor people’s income often does, in any given month, or even year, fall below the minimum and they survive by borrowing or selling their assets. Still, the same individuals cannot, by definition, stay below subsistence for long. The fact that the only two societies in our sample exhibiting a ratio higher than 100%, 1750 Moghul India and 1790 Nueva España, were notoriously exploitative seems consistent with this explanation

If the subsistence minimum were greater than $PPP 400, this would however be difficult to prove. One would need to demonstrate inequality as sufficiently restricted relative to income that a higher requirement becomes exceedingly probable. Even then there always exists the possibility of lesser extraction above a lower minimum. No society in any climate colder than temperate oceanic Western Europe is analysed in ‘Measuring Ancient Inequality’, and partly for reasons I will discuss below, candidates for such are few. [China in 1880 would lie partially within climates likely to have subsistence minima above $PPP 400, but the average subsistence minimum for the whole society would be dominated by the hot southern regions — at least judging by the result of the thesis.]

The above facts acquire deep relevance in light of runaway global warming. Milanovic, Lindert and Williamson imply that a hotter and hotter world will, ipso facto, increase the profits of the capitalists by reducing the absolute minimum wage they are required to provide to workers. Thus, the international capitalist class not merely has zero incentive to ameliorate global warming, but has vast potential gains from making it as bad as possible and permitting lower wages and greater profits. The results also explain the rapid movement in many countries from cooler to hotter regions: capitalist classes were seeking areas where they could gain greater profits, and the lower subsistence minima in hotter regions fit this goal perfectly.

Another fact explained very clearly by ‘Measuring Ancient Inequality’ is why civilisations evolved predominantly in hot deserts. The establishment of a stratified society requires that a surplus be produced, and logically this surplus will initially be small. Thus, it would be critical for the development of a surplus that the society’s subsistence minimum be as low as possible. Hot deserts — especially as they are the most likely region after extremely high mountains to evolve animals with the hierarchical social structures required for domestication — are ideal as the would-be elite is required to give such limited wages. Even more dramatically, the absence of even one origin of civilisation or even of crop agriculture in any latitude beyond the subtropics — European agriculture came via migration from the Middle East — suggests very low subsistence minima were critical for the evolution of stratified societies.

Did Barrington Moore write too early?

Re-downloading and re-reading Ron Rogowski’s Commerce and Coalitions, I was struck by a quite interesting and revealing quotation:

“In one of the classic works of modern comparative sociology, Barrington Moore, Jr. (1967) [Social Origin of Dictatorship and Democracy] focused attention on a particularly malignant — indeed, protofascist — developmental coalition, namely the protectionist one of capitalists and landowners against labor. If the present [Commerce and Coalitions, 1989] approach is correct, such an alliance was likeliest to arise in the formative nineteenth century in countries where land and capital were both scarce and only labor was abundant — that is, virtually all of Europe save its economically advanced northwestern corner, and all of eastern and southern Asia. There capital and land could be expected to unite in support of protection and imperialism; only labor, and the most labor-intensive agricultural and manufacturing enterprises, will normally have supported free trade and a less expansive foreign policy.

What Moore saw — rightly, in my [Rogowski’s] estimation — as the far more hopeful coalition of capital and labor should have arisen, according to the present theory, principally in two quite different circumstances: where both of those factors were abundant, and only land was scarce (essentially northwestern Europe, our first case[)]; and where both labor and capital were scarce, and only land was abundant (the “frontier” societies of the third case). In the former case, workers and capitalists alike will have favored free trade and a foreign policy of restraint; in the latter, both will have embraced protection and imperialism. In either of the two cases, however, the fatal alliance of land and capital is circumvented and, in Moore's perceptive telling, the path to a tolerably free society remains open.”

In light of last year’s US presidential election, the rise of Peter Dutton, and the much longer-term rise of geopolitically powerful petro-dictatorships, I get the feeling that Moore wrote his book too early, indeed much too early.

As I have repeatedly emphasised over the past two years, during the era discussed by Moore there existed no example of Rogowski’s “fourth case” — where both capital and land were abundant and only labour was scarce. Under such a situation, which today prevails in:

1. North America
2. continental Oceania [Australia, New Zealand, New Caledonia]
3. the Persian Gulf oil states
4. the more advanced countries within North and Central Asia

capital and land are expected to unite in favour of free trade and a (theoretically) less expansive foreign policy, while labour and more labour-intensive enterprises will favour protectionism and (theoretically) imperialism. The problem, which Rogowski does hint at, is that the recent history of the United States and the East Slavic and Turkic “core USSR”, alongside the entire history of the Gulf States since oil was discovered, demonstrate the capital and land are never satisfied with whatever suppression of labour they achieve, because their goal, as emphasised here and here by Red Flag Magazine, is always to increase profit. Moreover, the scarcity of labour encourages ruling classes to import labour in order to lower wages in the labour-intensive nontradable sector. This has several critical effects:

1. it virtually eliminates the ability of labour in those labour-abundant countries [South Asia, Tajikistan, the Levant, Central America and the Caribbean, non-continental Oceania] from which labour is imported to gain from exporting
2. it reduces the ability of the countries noted in 1) to offer competition for the capital- and land-intensive production via labour-intensive production
3. it thus eliminates the political power of labour in the countries noted in 1) and further increases the control of capital and land in countries rich in both
4. as Christopher Allen Culver of Pennsylvania State University and the US Air Force Academy has noted in his ‘Remittances and Autocratic Regime Stability’ and ‘Manipulating Remittances’, remittances sent back to the countries noted in 1) allow these labour-abundant countries stronger currencies, which further:
1. weakens export competitiveness
2. strengthens local capital and land against labour
3. strengthens the global hand of the ruling elites in capital- and land-rich nations

As discussed most explicitly by the late Sophie Body-Gendtrot in her 2002 The Social Control of Cities: A Comparative Perspective, the ruling elites of this free-trading coalition favour the most extreme possible social control of the domestic underclass. Although it is not explicitly discussed, the fact that the wealthy favour the most intense policing is, plainly, because political activity — requiring international class solidarity — is the only exit route for the ghetto underclass. Mass incarceration is a direct reflection of the political hegemony of America’s very rich, whose primary goal is closing any route to a challenge from the bottom ninety percent. Placing the poorest and potentially most dangerous under direct control is a huge step towards this goal, and the Gulf States do the same thing via their contract system with expatriate workers. Labour, contrariwise, will favour a much more open and free domestic policy, illustrating Sigmund Freud’s principle that economic freedom is opposed to political and sexual freedom.

Free-trading capital and land under expanding trade must ensure trade continues to expand. Externally driven declining trade [via export controls by labour-rich nations, as with Cold War-era Southern Africa] will in these labour-scarce trade-open economies empower previously powerless labour, as discussed by Ron Rogowski, and Texas Tech’s David Letzkian and Dennis Patterson. A free-trading coalition of capital and land must thus ensure labour-rich nations cannot restrict exports to them. Consequently, the theoretically less expansive foreign policy of the free-trading coalition of capital and land becomes much less restrained than Stolper–Samuelson theory predicts. This factor explains the existence of the IMF, World Bank and WTO, whose strict and enforceable trade rules are exceedingly critical to the global power of the United States and its Gulf Cooperation Council allies. Alexander Etkind in his Russia Against Modernity (reviewed here) similarly suggests that Russia’s war against Ukraine is aimed to prevent foreign states decarbonising and Russia losing its fossil fuel profits. Whilst Etkind’s assessment is debatable, massive and influential lobbying by the Gulf States in the West is much more definitively aimed at preventing the loss of profits through reduced global oil and gas use. US wars against Central America (between 1944 and 1996) undoubtedly had the same aim of preventing those nations controlling trade.

Returning to Social Origins of Dictatorship and Democracy, Moore did note the importance of the alliance between Southern planters and northern industrialists to US politics — transformed today into an alliance of fossil fuel producers and giant agribusinesses — since the great globalisation following Reconstruction. He noted the possibility of this coalition evolving earlier, which presumably would have shaped the US into the de facto oligarchy it became during the Jim Crow era and more decisively after the expansion of trade that began according to Rogowski in 1963 — but minus the democratising reforms of Reconstruction. In Saudi Arabia, power-sharing between the Al Saud and Al Ash-Sheikh meant an alliance between the rentier capital of the state and the Wahhabi clergy (land) had forged a free-trading coalition even before capital became abundant after oil was discovered. Similar links between imams and sheikhs developed in other Gulf states. This has prevented any possibility of a tolerably free society for anyone except capitalists and the traditional nomadic pastoralists, and, as noted in the preceding paragraph, is potentially much more dangerous in the long term than the better-known protectionist alliance of land and capital.

Another missing sequence

Today, looking again through OEIS, I noticed I could create the following sequence:

5, 67, 5, 13, 7, 17, 11, 37, 11, 31, 13, 29, 17, 61, 17, 37, 19, 41, 23, 127, 23, 139, 31, 53, 29, 109, 29, 61, 31, 71, 97, 199, 37, 73, 37, 83, 41, 157, 41, 167, 43, 89, 47, 181, 47, 97, 151, 101, 53, 307, 53, 109, 61, 113, 59, 229, 59, 127, 61, 131, ...,

Tabulated, this sequence is:

n	k	Representation
2	5	101two
3	67	2111three
4	5	101four
5	13	23five
6	7	11six
7	17	23seven
8	11	13eight
9	37	41nine
10	11	11
11	31	2911
12	13	1112
13	29	2313
14	17	1314
15	61	4115
16	17	1116
17	37	2317
18	19	1118
19	41	2319
20	23	1320
21	127	6121
22	23	1122
23	139	6123
24	31	1724
25	53	2325
26	29	1326
27	109	4127
28	29	1128
29	61	2329
30	31	1130
31	71	2931
32	97	3132
33	199	6133
34	37	1337
35	73	2335
36	37	1136
37	83	2937
38	41	1338
39	157	4139
40	41	1140
41	167	4341
42	43	1142
43	89	2343
44	47	1344
45	181	4145
46	47	1146
47	97	2347
48	151	3748
49	101	2349
50	53	1350
51	307	6151
52	53	1152
53	109	2353
54	61	1754
55	113	2355
56	59	1356
57	229	4157
58	59	1158
59	127	2959
60	61	1160
61	131	2961

Each member of the sequence [the second column] is the smallest prime greater than n whose base-n expansion is also a valid decimal expansion of a prime.

Without the requirement to be bigger than the base, every member for n greater than 2 would be 2 itself.

Although it is normally difficult to write bases larger than 35 (if it be assumed O and 0 are not distinct as I have always done) and it is not easy to establish a standard convention for them, the numbers in the above table can be written easily without differences of convention. This is because the decimal digits are a subset of the digits for any larger base, so that a basic decimal representation is also possible for any larger base, and this is the objective behind this sequence.

Looking at the list, one notices a clear pattern by which odd bases give larger k than even bases — the opposite of the pattern noted for smallest weakly prime number at OEIS A186995. However, the reasoning is the same as that for A186995 — that in an odd base there exist more possibilities for the last digit of a prime, although certain digit combinations which yield decimal expansions of primes cannot do so in many odd bases.

Comparative base maximum periods

(* indicates the expansion of the reciprocal terminates)

b	2-1	3-1	4-1	5-1	6-1	7-1	8-1	9-1	10-1	11-1	12-1	13-1	14-1	15-1	16-1	17-1	18-1	19-1	20-1	21-1	22-1	Max.	#>4	#>5	#>6	#>8	#>9	#>10
2	*	2	*	4	2	3	*	6	4	10	2	12	3	4	*	8	6	18	4	6	10	18	8	8	5	4	4	2
3	1	*	2	4	1	6	2	*	4	5	2	3	6	4	4	16	1	18	4	6	5	18	7	5	2	2	2	2
4	*	1	*	2	1	3	*	3	2	5	1	6	3	2	*	4	3	9	2	3	5	9	4	2	1	1	0	0
5	1	2	1	*	2	6	2	6	1	5	2	4	6	2	4	16	6	9	1	6	5	16	9	7	2	2	1	1
6	*	*	*	1	*	2	*	*	1	10	*	12	2	1	*	16	*	9	1	2	10	16	5	5	5	5	4	2
7	1	1	2	4	1	*	2	3	4	10	2	12	1	4	2	16	3	3	4	1	10	16	4	4	4	4	4	2
8	*	2	*	4	2	1	*	2	4	10	2	4	1	4	*	8	2	6	4	2	10	10	4	4	3	2	2	0
9	1	*	1	2	1	3	1	*	2	5	1	3	3	2	2	8	1	9	2	3	5	9	4	2	2	1	0	0
10	*	1	*	*	1	6	*	1	*	2	1	6	6	1	*	16	1	18	*	6	2	18	6	6	2	2	2	2
11	1	2	2	1	2	3	2	6	1	*	2	12	3	2	4	16	6	3	2	6	1	16	5	5	2	2	2	2
12	*	*	*	4	*	6	*	*	4	1	*	2	6	4	*	16	*	6	4	6	1	16	5	5	1	1	1	1
13	1	1	1	4	1	2	2	3	4	10	1	*	2	4	4	4	3	18	4	2	10	18	3	3	3	3	3	1
14	*	2	*	2	2	*	*	6	2	5	2	1	*	2	*	16	6	18	2	2	5	18	6	4	2	2	2	2
15	1	*	2	*	1	1	2	*	1	5	2	12	1	*	2	8	1	18	2	1	5	18	5	3	3	2	2	2
16	*	1	*	1	1	3	*	3	1	5	1	3	3	1	*	2	3	9	1	3	5	9	3	1	1	1	0	0
17	1	2	1	4	2	6	1	2	4	10	2	6	6	4	1	*	2	9	4	6	10	10	7	7	3	3	2	0
18	*	*	*	4	*	3	*	*	4	10	*	4	3	4	*	1	*	2	4	3	10	10	2	2	2	2	2	0
19	1	1	2	2	1	6	2	1	2	10	2	12	6	2	4	8	1	*	2	6	10	12	7	7	4	3	3	1
20	*	2	*	*	2	2	*	6	*	5	2	12	2	2	*	16	6	1	*	2	5	16	6	4	2	2	2	2
21	1	*	1	1	1	*	2	*	1	2	1	4	1	1	4	4	1	18	1	*	2	18	1	1	1	1	1	1
22	*	1	*	4	1	1	*	3	4	*	1	3	1	4	*	16	3	18	4	1	*	18	2	2	2	2	2	2
23	1	2	2	4	2	3	2	6	4	1	2	6	3	4	2	16	6	9	4	6	1	16	6	6	2	2	1	1
24	*	*	*	2	*	6	*	*	2	10	*	12	6	2	*	16	*	9	2	6	10	16	8	8	5	5	4	2
25	1	1	1	*	1	3	1	3	1	5	1	2	3	1	2	8	3	9	1	3	5	9	4	2	2	1	0	0
26	*	2	*	1	2	6	*	2	1	5	2	*	6	2	*	8	2	3	1	6	5	8	6	4	1	0	0	0
27	1	*	2	4	1	2	2	*	4	5	2	1	2	4	4	16	1	6	4	2	5	16	4	2	1	1	1	1
28	*	1	*	4	1	*	*	1	4	10	1	12	*	4	*	16	1	9	4	1	10	16	5	5	5	5	4	2
29	1	2	1	2	2	1	2	6	2	10	2	3	1	2	4	16	6	18	2	2	10	18	6	6	4	4	4	2
30	*	*	*	*	*	3	*	*	*	10	*	6	3	*	*	4	*	3	*	3	10	10	3	3	2	2	2	0
31	1	1	2	1	1	6	2	3	1	5	2	4	6	1	2	16	3	6	2	6	5	16	7	5	1	1	1	1
32	*	2	*	4	2	3	*	6	4	2	2	12	3	4	*	8	6	18	4	6	2	18	6	6	3	2	2	2
33	1	*	1	4	1	6	1	*	4	*	1	12	6	4	1	2	1	18	4	6	1	18	5	5	2	2	2	2
34	*	1	*	2	1	2	*	3	2	1	1	4	2	2	*	*	3	18	2	2	1	18	1	1	1	1	1	1
35	1	2	2	*	2	*	2	2	1	10	2	3	1	2	4	1	2	9	2	2	10	10	3	3	3	3	2	0
36	*	*	*	1	*	1	*	*	1	5	*	6	1	1	*	8	*	9	1	1	5	9	5	3	2	1	0	0
37	1	1	1	4	1	3	2	1	4	5	1	12	3	4	4	16	1	2	4	3	5	16	4	2	2	2	2	2
38	*	2	*	4	2	6	*	6	4	5	2	2	6	4	*	4	6	*	4	6	5	6	7	5	0	0	0	0
39	1	*	2	2	1	3	2	*	2	10	2	*	3	2	2	16	1	1	2	3	10	16	3	3	3	3	3	1
40	*	1	*	*	1	6	*	3	*	10	1	1	6	1	*	16	3	18	*	6	10	18	7	7	4	4	4	2
41	1	2	1	1	2	2	1	6	1	10	2	12	2	2	2	16	6	18	1	2	10	18	7	7	5	5	5	3
42	*	*	*	4	*	*	*	*	4	5	*	3	*	4	*	8	*	9	4	*	5	9	4	2	2	1	0	0
43	1	1	2	4	1	1	2	3	4	2	1	6	1	4	4	8	3	9	4	1	2	9	3	3	2	1	0	0
44	*	2	*	2	2	3	*	2	2	*	2	4	3	2	*	16	2	9	2	6	*	16	3	3	2	2	1	1
45	1	*	1	*	1	6	2	*	1	1	1	12	6	*	4	16	1	3	1	6	1	16	5	5	2	2	2	2
46	*	1	*	1	1	3	*	1	1	10	1	12	3	1	*	16	1	6	1	3	10	16	5	5	4	4	4	2
47	1	2	2	4	2	6	2	6	4	5	2	4	6	4	2	4	6	9	4	6	5	9	8	6	1	1	0	0
48	*	*	*	4	*	2	*	*	4	5	*	3	2	4	*	16	*	18	4	2	5	18	4	2	2	2	2	2
49	1	1	1	2	1	*	1	3	2	5	1	6	1	2	1	8	3	3	2	1	5	8	4	2	1	0	0	0
50	*	2	*	*	2	1	*	6	*	10	2	12	1	2	*	2	6	6	*	2	10	12	6	6	3	3	3	1
51	1	*	2	1	1	3	2	*	1	10	2	2	3	1	4	*	1	18	2	3	10	18	3	3	3	3	3	1
52	*	1	*	4	1	6	*	3	4	10	1	*	6	4	*	1	3	18	4	6	10	18	6	6	3	3	3	1
53	1	2	1	4	2	3	2	2	4	5	2	1	3	4	4	8	2	18	4	6	5	18	5	3	2	1	1	1
54	*	*	*	2	*	6	*	*	2	2	*	12	6	2	*	16	*	9	2	6	2	16	6	6	3	3	2	2
55	1	1	2	*	1	2	2	1	1	*	1	3	2	1	2	4	1	9	2	2	1	9	1	1	1	1	0	0
56	*	2	*	1	2	*	*	6	1	1	2	6	*	2	*	16	6	18	1	2	1	18	5	5	2	2	2	2
57	1	*	1	4	1	1	1	*	4	10	1	12	1	4	2	16	1	*	4	1	10	16	4	4	4	4	4	2
58	*	1	*	4	1	3	*	3	4	5	1	12	3	4	*	16	1	9	4	3	5	16	5	3	3	3	2	2
59	1	2	2	2	2	6	2	6	2	5	2	12	6	2	4	8	6	18	2	6	5	18	10	8	3	2	2	2
60	*	*	*	*	*	3	*	*	*	5	*	4	3	*	*	8	*	18	*	3	5	18	4	2	2	1	1	1
61	1	1	1	1	1	6	2	3	1	10	1	3	6	1	4	16	3	9	1	6	10	16	7	7	4	4	3	1
62	*	2	*	4	2	2	*	2	4	10	2	6	2	4	*	16	2	9	4	2	10	16	5	5	4	4	3	1
63	1	*	2	4	1	*	2	*	4	10	2	12	1	4	2	16	1	18	4	*	10	18	5	5	5	5	5	3

In recent years I have had considerable interest in comparing various bases regarding certain features of representations of various fractions. Unless the denominator contains no prime factor which does not divide the base (in which case the fraction will terminate) the representation of a fraction in any base will be recurring with a period equal to the smallest repunit divisible by the denominator in question.

A fraction with denominator n can have a period of at most n-1 regardless of base, and a period of n-1 is possible only for prime numbers. Such prime numbers are referred to as full period primes or full repetend primes, and except for perfect powers they generally constitute about three-eighths of all primes.

In the table above, I have attempted to compare the periods of reciprocals of all numbers from 2 to 22 in all bases from 2 to 63 — although for bases beyond 35 it is very difficult to write numbers out in an easily understood notation. My aim is to see:

1. what base has the shortest maximum period for these reciprocals
2. what bases have the highest and lowest frequencies of long periods
• usually, a terminating expansion is taken as having period zero — effectively the shortest possible period

In the table above, I have compared:

1. the longest period in each base for the reciprocals of any number from 2 to 22
2. the number of numbers in that range with periods longer than 4, 5, 6, 8, 9 or 10
• no number smaller than 29 can have period 7 in any base, so the column “>7” was omitted after I experimented

Results:

The table above shows that for bases form 2 to 63:

• the shortest maximum period of reciprocal up to 22 is 6 for base 38, followed by 8 for bases 26 and 49 (all shaded dark green)
• the shortest maximum period of reciprocal up to 18 is 4 for bases 21, 34, and 55 (shaded light green)
• it is interesting to note that 21+34 = 55, and one wonders if there is a pattern involved?
• base 60, with all reciprocals up to 18 having periods of 5 or shorter, has also been shaded light green
• the most reciprocals up to 22 with periods longer than 4 is ten, for base 59
• the fewest reciprocals up to 22 with periods longer than 4 is one [in all cases the reciprocal of 19] for bases 21, 34 and 55

To more accurately consider the effect of base structure on period of reciprocal, I have compared bases by dividing them into five groups based upon factorisation:

1. prime bases
2. semiprime bases
3. nonsquare semiprime bases
1. I did this because square bases do not normally have any full period primes at all
2. thus, their maximum possible period is only half that of nonsquare bases
4. bases other than those in 1) or 2)
5. group 4) excluding square numbers

Comparative Results for Different Categories of Bases b
	Maximum	#>4	#>5	#>6	#>8	#>9	#>10
Prime	15.41176	6.352941	5.529412	2.882353	2.588235	2.235294	1.294118
Semiprime	14	4.045455	3.227273	2.272727	2.045455	1.681818	1.045455
Squarefree semiprime	15.16667	4.055556	3.5	2.444444	2.333333	2.055556	1.277778
Others	14.61905	4.761905	4.095238	2.428571	2.190476	1.952381	1.190476
Other nonsquare	15.21053	4.842105	4.315789	2.684211	2.473684	2.263158	1.315789

What appears to be the case is the prime bases have the greatest number of long periods, but semiprime bases on the whole have marginally fewer than bases with three or more factors. The difference between the three groups, though, is not large — indeed the prime base 47 is the smallest base where all reciprocals up to 24 have periods of nine or shorter. The one exception is that prime bases appear to have substantially fewer denominators with periods of 4 or shorter.

Further research could allow for investigation into questions like:

• how hard is it to find a base yielding consistently short periods for denominators of increasing size?
• how many such bases are there?
• can one calculate the smallest base for which all fractions up to n have periods of p or shorter with n and p arbitrary?
• what are the trends in frequency of denominators with short periods relative to size and number of factors in the base b as bases get bigger than studied here?

Another idea from ‘Commerce and Coalitions’: Why no tradition of private charity in most of the world?

Looking again at Ron Rogowski’s book Commerce and Coalitions: How Trade Affects Domestic Political Alignment, it has occurred to me that his theory might explain something I have long known about but have never even attempted to explain.

This question is about why traditions of private charity are so geographically restricted. Despite the commendation of private charity in all or almost all traditional religious systems, large-scale traditions thereof are exceedingly restricted. They are extremely well developed in the United States, marginally developed in the United Kingdom, well developed in the wealthier Muslim nations (zakat is a Muslim pillar), but almost absent everywhere else in the world.

Throughout my life, I have romanticised private charity, despite the fact that, as groups like Resistance [now part of Socialist Alliance] demonstrated to me a quarter of a century ago, its primary purpose has always been to prevent either:

1. rebellion by the oppressed and/or 
2. money moving into the control of the immense majority, as this quote from Socialist Worker demonstrates:

“Every dollar in Mark Zuckerberg's private charity is a dollar wrested from public coffers and from democratic control, writes Jason Farbman, in an article published [originally ]at Jacobin.”

I have always imagined, without any sense of logic, that a society based on private charity would be softer, friendlier, more hospitable, and less selfish than one based upon taxation, despite the refutation of this argument by countless socialist groups over almost two centuries. Even among non-Marxists, Emory Paul alongside Jordan Weissman and Mike Konczal demonstrate how private charity has never been able to eliminate poverty, whilst Kathleen Wellman in her recent Hijacking History: How the Christian Right Teaches History and Why It Matters notes on page 286 that:

“private charity never provided an adequate social safety net before the Social Security Act of 1935 [Frances FitzGerald, The Evangelicals: The Struggle to Shape America (New York: Simon & Schuster, 2017), page 460]”

In fact, I will plainly confess that my romanticism noted above comes from the idea that private charity might provide a completely adequate social safety net — implicitly viewed possible by such books as the Politically Incorrect Guides.

If we follow from what socialist groups say, and from Rogowski, the geographically restricted nature of private charity becomes no surprise. Although the very rich will always prefer private charity to taxation, and most likely prefer private charity to paying rents to government, private charity still costs them a lot of money. Thus, something must be gained by the extremely rich from large-scale private charity before they will develop it. Under most economic conditions, very low wages for labour will cost the wealthy less than large-scale private charity — after all, the rich must exploit workers to become rich. The exception will occur when the profits from using labour are limited by scarcity and the rich can gain much more from owning and profiting from capital and/or land. This will occur when goods that use labour significantly can be more economically imported than produced locally, with the result that low-skill labour becomes restricted to a limited range of nontradable industries. Under these conditions — export of capital- and land-intensive goods and import of all even modestly labour-intensive ones — the ruling class can gain politically by demoting to private charity. Charity here:

1. prevents local labour from politically organising in the absence of employment opportunities
2. allows the ruling class to discriminate in whom it gives its donations to, so as to divide the lower classes by race and ethnicity and eliminate the possibility of a unified nonelite
3. serves to allow the local ruling class(es) to justify its views via philosophy, e.g.
• right-wing think tanks in the US
• jihadist groups funded by Saudi, Qatari, Emirati and Kuwaiti charity
• foreign aid to authoritarian regimes in labour-rich Levantine and South Asian countries
4. conseuqently prevents the governments of labour-abundant societies from becoming independent of capital- and land-abundant ones

Where labour is abundant, contrariwise, the ruling class will always profit most from employing it, while much less potential profits exist from labour-efficient production. Such a ruling class has neither the ability nor the incentive to encourage charity: it requires abundant labour to engage in the colonialism found when both capital and land lose from free trade and restrained foreign policies. Thus, enduring abundance of labour in Europe and monsoonal Asia is enough of itself to explain the absence of large-scale private charity in these societies — exactly as with anarcho-capitalism.

 

Large-scale private charity appears indeed to require extreme scarcity of labour. Commerce and Coalitions notes in its discussion of the long sixteenth century that in 1750 Latin America was ten times less sparsely populated than North America. This suggests that, although Latin America imported labour-intensive goods, there was not the same incentive to economise on labour and for owners of land — whose ownership was much more concentrated than among whites in North America — to engage in private philanthropy amongst the lower classes.

Another factor unrelated to Rogowski’s book that may explain the absence of traditions of private charity in Oceania and Latin America is that immigration was much more tightly controlled by colonial governments than in the United States. Private charity in the US was substantially developed by persecuted religious groups who migrated to North America to escape the oppressive influence of European governments. These governments often did not permit them to practice their faith or in some cases even to remain in their domains. Latin America’s and New Zealand’s much tighter control of immigration, and Australia’s antiquated and infertile soils that forbade these capital-scarce sects from establishing low-input farming communities, prevented the development of these networks of charity. This cannot be a full answer given that Canada, which saw the same type of settlement by persecuted religious sects as the United States, did not develop the same traditions of private charity.

 

Nonetheless, it is a very reasonable hypothesis that significant private charity will invariably be confined to societies that are, or historically were, exceeding land-abundant and labour-scarce.