For starters, we’ll assume a very simple indifference curve:
y = 1/ x. [6]

This is only a practice run, to get a feel for the model. We note that:
dy/dx = -1/(x^2),
and that, for any point on the curve which is taken to be an observed equilibrium (where the transformation curve of the day touches the highest attainable indifference curve), money prices Px and Py fulfill the relationship
Px/Py = -dy/dx.
We put that together with the sum constituting total expenditures, E(t):
x*Px + y*Py = E(t).
Total expenditure, though, is plucked out of the air. (We imagine it to be set by the action of a monetary authority which manages our fiat currency.) To study the effects, we’ll apply a rate of monetary growth:
E(t) = E(0)*(1+i)^t, for t = 0, 1, 2, …
but except when otherwise noted, we’ll set i  = 0 (no monetary growth).

Taken together, these relationships allow us to calculate that, for any point on this curve which is taken to be the economic equilibrium at time t,
Px = E(t)/(2*x), and
Py = (x^2)*Px.

So let’s pick some points along the indifference curve y = 1/x, and see how the model behaves.

y = 1/ x
x	y	dy/dx
4.00	0.25	-0.0625
2.00	0.50	-0.2500
1.00	1.00	-1.0000
0.50	2.00	-4.0000
0.25	4.00	-16.0000

The dy/dx column indicates that as we move to less and less x, it takes more and more y to compensate us for giving up any more.

Now imagine that, for some reason, the transformation curve (i..e., the curve of productive possibilities) shifts from one period to another in just such a way as to keep the economy on this one indifference curve, nudging it along from one of the above points to the next. By whatever manipulation of the money supply may be needed, we’ll hold E(t) constant.

Here are the observed prices and expenditures (letting Ex and Ey represent expenditures on x and y):

x	y	Px	Py	Ex	Ey	E(t)
4.00	0.25	500.00	8000.00	2000.00	2000.00	4000.00
2.00	0.50	1000.00	4000.00	2000.00	2000.00	4000.00
1.00	1.00	2000.00	2000.00	2000.00	2000.00	4000.00
0.50	2.00	4000.00	1000.00	2000.00	2000.00	4000.00
0.25	4.00	8000.00	500.00	2000.00	2000.00	4000.00

Let Sx = Ex/E(t) and Sy = Ey/E(t) denote the shares of expenditure devoted to x and y. These show up as weights in the calculation of various indexes. In this instance, Sx = Sy = 0.5000 throughout the table. Since the weights in all of the price indexes which we’re studying depend on these shares, one might suppose, offhand, that all of them will be unperturbed by the above progression. Here, though, are the results:

Sx	Sy	Laspeyres	Paasche	Base- Weighted Geometric Mean	Chained Tornqvist	Fisher
0.5000	0.5000	1.0000	1.0000	1.0000	1.0000	1.0000
0.5000	0.5000	1.2500	0.8000	1.0000	1.0000	1.0000
0.5000	0.5000	2.1250	0.4706	1.0000	1.0000	1.0000
0.5000	0.5000	4.0625	0.2462	1.0000	1.0000	1.0000
0.5000	0.5000	8.0313	0.1245	1.0000	1.0000	1.0000

Even though the weights Sx and Sy are the same in every period, the mere fact that they enter differently into the calculations lead to dramatic differences. The Laspeyres index sees prices going up, Paasche sees them coming down, and the remaining indexes see a standoff between the rising price of x and the falling price of y. Why? Because the Laspeyres and Paasche indexes, which are arithmetic means, add the weighted price ratios, and thereby overlook the fact that they are (in this instance) exact reciprocals. For example, Px in the second row is twice the value in the first row, but Py is half. The Base-Weighted Geometric Mean and the Tornqvist index multiply these ratios, so the changes cancel out.

See Appendix 2 for the mathematical formulas which were used, and for a brief discussion of each index — what each is intended to measure, what economic theory can say about it, and how it figures in the report of the Boskin Commission.

If we divide the total expenditure E(t) in a period by any of these indexes, we come up with that index’s estimate of "real income":

"Real Income": E(t) / Index
Laspeyres	Paasche	Base-Weighted Geometric Mean	Chained Tornqvist	Fisher
$4,000.00	$4,000.00	$4,000.00	$4,000.00	$4,000.00
$3,200.00	$5,000.00	$4,000.00	$4,000.00	$4,000.00
$1,822.35	$8,500.00	$4,000.00	$4,000.00	$4,000.00
$984.62	$16,250.00	$4,000.00	$4,000.00	$4,000.00
$498.05	$32,125.00	$4,000.00	$4,000.00	$4,000.00

From our omniscient view, which sees into all hearts, it’s clear that the Laspeyres and Paasche indexes are wildly wrong, while the Base-Weighted Geometric Mean and the Tornqvist index are dead right: consumers have stayed all the while upon the same indifference curve. (The Fisher index lucks out, but only because it averages two exactly offsetting errors in each period.)

This is the ideal example to validate the Boskin Commission’s preference for Geometric Mean and Tornqvist indexes. It’s an extreme case of the relationship they envision between consumer preferences, changing relative prices, and calculated price indexes. Geometric means (of which the Tornqvist index is actually another type) are said to perform best when the elasticity of demand is unitary. We’ve chosen an exceptionally well-behaved indifference curve:

Elasticity of substitution = (dy/dx)*(x/y) = (-1/x^2)*(x/(1/x) = -1

for every point on the curve.

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