The debate launched by Thomas Malthus is now almost two centuries old, and still it rages. According to Malthus, human population tends to increase faster than the means of its support. In the absence of intentional population control, therefore, it must ultimately be controlled by nature’s traditional methods: in particular, famine.

In the 1950’s, economists tended to dismiss this argument. Events had disproven it time and time again: since Malthus made his prediction, human population had multiplied many times over, but agricultural production had grown even more, and the average per capita income was far more comfortable than that of 1798. The attribution of a slower growth rate to production was gratuitous, argued the economists: it could be accelerated as needed by investment in new technology, exploration, capital equipment, and education.

Then, in the anxious 1970’s, the Club of Rome revived the debate. Its computer model showed population again outstripping resources, and dire consequences were predicted.

We won’t resolve the issue here, but it’s interesting to see how various price indexes, and the resulting estimates of real income, would report the matter if growth did follow the pattern predicted by Malthus.

Once more we’ll launch from a point on one indifference curve, y = 8/(x-1), and explore alternative moves to a second curve, this time y = 74/(x-16). The second curve is higher, but we note from the discussion in the preceding example that since h has been multiplied by k = 16, and this is assumed to be caused by a 16-fold increase in population, m would have to be multiplied by k^2 = 256 to depict a curve which could furnish the same per capita satisfaction as the lower one. The second curve would have to be y = 2048/(x-16), instead of a measly y = 74/(x-16). The transformation curve is setting us down far short of where we’d like to be.

Here’s the original springboard:

y = 8/(x-1.0)
x	y	dy/dx
12.00	0.73	-0.0661

with prices and expenditures:

x	y	Px	Py	Ex	Ey	E(t)
12.00	0.73	173.9130	2630.4348	2086.96	1913.04	4000.00

And here are five alternative points on the higher curve:

y = 74/(x-16.0)
x	y	dy/dx
18.00	37.00	-18.5000
17.50	49.33	-32.8889
17.00	74.00	-74.0000
16.50	148.00	-296.0000
16.25	296.00	-1184.0000

Both x and y have grown in comparison with the initial situation, but x has grown with more difficulty, and is growing uncomfortably close to what is now its minimum, h = 16.

The comparative price and expenditures schedules for the two periods will this time be:

x	y	Px	Py	Ex	Ey	E(t)
12.00 18.00	0.73 37.00	173.9130 200.0000	2630.4348 10.8108	2086.96 3600.00	1913.04 400.00	4000.00 4000.00
12.00 17.50	0.73 49.33	173.9130 210.5263	2630.4348 6.4011	2086.96 3684.21	1913.04 315.79	4000.00 4000.00
12.00 17.00	0.73 74.00	173.9130 222.2222	2630.4348 3.0030	2086.96 3777.78	1913.04 222.22	4000.00 4000.00
12.00 16.50	0.73 148.00	173.9130 235.2941	2630.4348 0.7949	2086.9633882.35	1913.04 117.65	4000.00 4000.00
12.00 16.25	0.73 296.00	173.9130 242.4242	2630.4348 0.2048	2086.96 3939.39	1913.04 60.61	4000.00 4000.00

We are led to the following price indexes:

Sx	Sy	Laspeyres	Paasche	Base- Weighted Geometric Mean	Tornqvist	Fisher
0.5217 0.9000	0.4783 0.1000	1.0000 0.6020	1.0000 0.0398	1.0000 0.0777	1.0000 0.2255	1.0000 0.1548
0.5217 0.9211	0.4783 0.0789	1.0000 0.6327	1.0000 0.0301	1.0000 0.0621	1.0000 0.2146	1.0000 0.1380
0.5217 0.9444	0.4783 0.0556	1.0000 0.6672	1.0000 0.0202	1.0000 0.0445	1.0000 0.1962	1.0000 0.1162
0.5217 0.9706	0.4783 0.0294	1.0000 0.7060	1.0000 0.0102	1.0000 0.0243	1.0000 0.1601	1.0000 0.0849
0.5217 0.9848	0.4783 0.0152	1.0000 0.7273	1.0000 0.0051	1.0000 0.0129	1.0000 0.1244	1.0000 0.0610

None of which is especially intelligible, until we inspect the comparative estimates of "real income":

"Real Income": E(t) / Index
Laspeyres	Paasche	Base-Weighted Geometric Mean	Tornqvist	Fisher
$4,000.00 $6,644.90	$4,000.00 $100,456.52	$4,000.00 $51,475.78	$4,000.00 $17,734.78	$4,000.00 $25,836.47
$4,000.00 $6,321.68	$4,000.00 $132.811.59	$4,000.00 $64,392.19	$4,000.00 $18,638.45	$4,000.00 $28,975.73
$4,000.00 $5,995.09	$4,000.00 $197,608.70	$4,000.00 $89,905.69	$4,000.00 $20,388.64	$4,000.00 $34,419.21
$4,000.00 $5,665.51	$4,000.00 $392,173.91	$4,000.00 $164,780.32	$4,000.00 $24,976.65	$4,000.00 $47,136.65
$4,000.00 $5,499.72	$4,000.00 $781,434.78	$4,000.00 $310,371.69	$4,000.00 $32,141.72	$4,000.00 $65,556.63

Bearing in mind that the population has supposedly exploded by a factor of 16, the second member of each pair in this table, just to stand still, would need to be $64,000. It would then represent an aggregate real income adequate in the second period to furnish the same per capita income as the first member of the pair. We see that, as usual, the Tornqvist index correctly senses a shortfall of growth, though not without questionable results as we progress to heavier and heavier emphasis upon y in the end-period market basket. Supposedly, all of the end-period combinations lie upon the same indifference curve, but even the Tornqist index almost doubles its evaluation from $17,734.78 to $32,141.72 between the first alternative and the last. As for the Paasche index, it goes absolutely bonkers, and the Fisher index is dragged into dubious territory with it.

That leaves the base-weighted geometric mean, which as we’ve noticed before, tends to lose its bearings when substitution is extreme. Perhaps it bears repetition here that the Boskin Commission does not recommend a base-weighted geometric mean except grudgingly, in the interest of timely publication of the CPI, and then only in the calculation of lower-level price indexes.[13]

[Proceed to Conclusions]

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