Now we take off the training wheels. The simple indifference curves of form y = m/x, with their elasticity of substitution everywhere equal to -1, violate a nagging intuition: that some substitutions are more equal than others. Common language makes a distinction between "necessitites" and "luxuries". Can our indifference curves really be of a shape which omits that distinction?
There’s a pretty way to deal with this. We introduce a broader family
of hyperbolas:
y = m/(x-h), with m > 0 and h > 0.
With h > 0, this shifts either of the previous hyperbolas h
units to the right. The new hyperbola is asymptotic to the vertical line
x = h, rather than to the y axis. In economic terms, the meaning
is that consumers insist upon a minimum reservation of item x, before
they will consider giving up any of it in exchange for y. Once that
minimum is achieved, additional units of x begin to trade
for y. We have converted x, in the common parlance, into a
"necessity", while y is an unaffordable "luxury"
until we get our minimum fix of x.
For any curve in this family (with x > h > 0),
dy/dx = -m/((x-h)^2).
We’ll start with the curve
y = 0.0625/(x-1.0) — i.e., m = 0.0625, h
= 1.0, x > h,
and a sequence of points on it which crowd us closer and closer to the unacceptable
x = h.
x |
y |
dy/dx |
1.25 |
0.25 |
-1.0000 |
1.11 |
0.50 |
-4.0000 |
1.06 |
1.00 |
-16.0000 |
1.03 |
2.00 |
-64.0000 |
1.02 |
4.00 |
-256.0000 |
The rate of substitution, dy/dx, gets wilder and wilder as we approach the minimum reservation of necessity x.
From
Px/Py = -dy/dx = -m/((x-h)^2) and xPx
+ yPy = E(t),
we can calculate that
Px = E/(2x-h) and
Py = (((x-h)^2)/m)*Px.
So, if the monetary authority again holds total expenditure constant while
changes in the transformation curve move us along from point to point on
this indifference curve:
x |
y |
Px |
Py |
Ex |
Ey |
E(t) |
1.25 |
0.25 |
2666.6667 |
2666.6667 |
3333.33 |
666.67 |
4,000.00 |
1.11 |
0.50 |
3200.0000 |
800.0000 |
3600.00 |
400.00 |
4,000.00 |
1.06 |
1.00 |
3555.5556 |
222.2222 |
3777.78 |
222.22 |
4,000.00 |
1.03 |
2.00 |
3764.7059 |
58.8235 |
3882.35 |
117.65 |
4,000.00 |
1.02 |
4.00 |
3878.7879 |
15.1515 |
3939.39 |
60.61 |
4,000.00 |
Notice how desperately the price of y is falling, in order to induce consumers to substitute any more of it for x. Meanwhile, the price of x is rising, and more is spent to get a lesser total amount. This is what we would predict from the elasticities of substitution:
x |
y |
Elasticity of Substitution: |
Elasticity of Substitution: |
1.25 |
0.25 |
-5.0000 |
-0.2000 |
1.13 |
0.50 |
-9.0000 |
-0.1111 |
1.06 |
1.00 |
-17.0000 |
-0.0588 |
1.03 |
2.00 |
-33.0000 |
-0.0303 |
1.02 |
4.00 |
-65.0000 |
-0.0154 |
Since x is a "necessity", the demand for it is increasingly inelastic as the quantity approaches its lower limit — and then, as y is the only other item in the economy, the demand for it must become increasingly elastic. (They’re reciprocals.)
Here’s how the various price indexes register this case of pure substitution:
Sx |
Sy |
Laspeyres |
Paasche |
Base- Weighted Geometric Mean |
Chained Tornqvist |
Fisher |
0.8333 |
0.1667 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
0.9000 |
0.1000 |
1.0500 |
0.9231 |
0.9524 |
0.9975 |
0.9845 |
0.9444 |
0.0556 |
1.1250 |
0.7273 |
0.8399 |
0.9950 |
0.9045 |
0.9706 |
0.0294 |
1.1801 |
0.4948 |
0.7059 |
0.9933 |
0.7642 |
0.9848 |
0.0152 |
1.2131 |
0.2991 |
0.5772 |
0.9923 |
0.6023 |
As in earlier examples, the Laspeyres and Paasche index take opposite directions. Since this example was constructed with neither monetary inflation nor income change, we know that the effects are due to substitution alone: by rights, the index should be 1.0000. The Tornqvist index comes closest to a correct result, while the base-weighted geometric mean and the Fisher index are surprisingly wide of the mark. Splitting the difference between Laspeyres and Paasche doesn’t work for the Fisher index in this instance, because the figures are such that a "preference" for the end-period market basket leads Paasche to a lopsided error, greater than that of Laspeyres.
Were we to use these indexes in the usual estimate of real income, we’d get a confusing mix of opinions:
Laspeyres |
Paasche |
Base-Weighted Geometric Mean |
Chained |
Fisher |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$3,809.52 |
$4,333.33 |
$4,199.74 |
$4,010.08 |
$4,063.00 |
$3,555.56 |
$5,500.00 |
$4,762.20 |
$4,019.97 |
$4,422.17 |
$3,389.41 |
$8,083.33 |
$5,666.67 |
$4,026.96 |
$5,234,28 |
$3,297.42 |
$13,375.00 |
$6,929.57 |
$4,031.13 |
$6,641.01 |
There is a lesson in this example, as far as we’ve taken it: the Boskin Commission is quite right in its preference for the Tornqvist index, when that becomes available. The hitch (as they agree) is that it doesn’t become available until too late, because of its dependence on end-period data that take time and resources to collect. They therefore recommend that an initial estimate be made with the base-weighted geometric mean, and a follow-up study should publish the Tornqvist index a year or two later.
Which, though, should be used for adjusting the incomes of people who receive cost-of-living adjustments?[7] Politically, it would be wildly impracticable to set a tentative rate one year, and take it back or add more a year or two later. Alternatively, the rate could be set with a one-or-two year lag in the first place. That might fly, once it got established, but there would be some initial heartburn in making the transition. In times of steep inflation, people would soon start to demand an advance upon expected price changes.
More likely, the index using a geometric mean will be the one to which we settle down, if we depart from the Laspeyres index at all. Our table above suggests that the geometric mean can go badly wrong under some circumstances. In fairness, though, there are two points to be added. First, the Boskin Commission has recommended the geometric mean only for the lower stages of the calculation — those in which samples of very similar items such as Granny Smith apples and Macintosh apples are aggregated to estimate the change of prices for "apples". These very similar items are close substitutes, so the elasticity of substitution is close to -1, and the geometric mean is on its best behavior. (They would also employ the geometric mean for an initial, provisional estimate of price changes at the higher levels of aggregation, though, and as just argued, this is the one which would probably stick for the setting of COLA’s.)
Secondly, the behavior suggested by this example is pretty extreme. Surreptitiously, it introduces unrealistic assumptions about the ways in which transformation curves can behave. Who says that the economy can compensate people for the loss of x with anything like the enhanced production of y shown by some of the combinations in the example? We’re only exploring some extremes in the logic which recommends one index over another.
We’re not quite done with this example. Thus far, we’ve only examined its performance under pure substitution. What if there were a change from one such curve to another in the same family?
The economy as depicted by the above curve is in a precarious situation: it’s desperately short of x, a "necessity". So let’s start instead with a point on a higher curve, and study alternative descents from it to each of our five points on the lower curve. Inevitably, there will be subsitutions between x and y, as well as the lowering of income.
In the beginning, there will be the curve
y = 1/(x-1.0) — i.e., m = 1, h = 1.0, x
> h.
This differs from the preceding curve only in its numerator, 1 instead 0.0625.
(In general, a higher numerator in this family of curves designates a higher
real income, for given h.) The starting position will be:
x |
y |
dy/dx |
1.50 |
2.00 |
-4.0000 |
With the usual understandings as to the total level of expenditure and the behavior of prices,
x |
y |
Px |
Py |
Ex |
Ey |
E(t) |
1.50 |
2.00 |
2000.00 |
500.00 |
3000.00 |
1000.00 |
4,000.00 |
We consider, then, the following alternative moves between a single point on the higher curve and five different points on the lower one, with increasingly radical substitution towards y:
x |
y |
Px |
Py |
Ex |
Ey |
E(t) |
1.50 |
2.00 |
2000.0000 |
500.0000 |
3000.00 |
1000.00 |
4000.00 |
1.50 |
2.00 |
2000.0000 |
500.0000 |
3000.00 |
1000.00 |
4000.00 |
1.50 |
2.00 |
2000.0000 |
500.0000 |
3000.00 |
1000.00 |
4000.00 |
1.50 |
2.00 |
2000.0000 |
500.0000 |
3000.00 |
1000.00 |
4000.00 |
1.50 |
2.00 |
2000.0000 |
500.0000 |
3000.00 |
1000.00 |
4000.00 |
The various indexes report the resulting change in the price (per util of well-being) as follows:
Sx |
Sy |
Laspeyres |
Paasche |
Base- Weighted Geometric Mean |
Tornqvist |
Fisher |
0.7500 |
0.2500 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
0.7500 |
0.2500 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
0.7500 |
0.2500 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
0.7500 |
0.2500 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
0.7500 |
0.2500 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
Since total expenditures were held constant at $4,000, the ratio E(1)/E(0) = 1.0000, and the fact that both the Laspeyres index and the Paasche index exceed this ratio (except in the last row) implies that real income has fallen.[See the inequalities reproduced from Thomas Carroll in Appendix 2.] Behold, though, how the base-weighted geometric mean index goes astray beginning in the fourth pair. The effects upon a calculation of real income are lamentable:
Laspeyres |
Paasche |
Base-Weighted Geometric Mean |
Tornqvist |
Fisher |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
$4,000.00 |
All of the indexes agree, in the first three cases, that real income is falling, though by different amounts. Beginning with the fourth case, however, the base-weighted geometric index does a turnaround, and estimates an increased real income. The Paasche index, usually noted for its overly optimistic evaluation of the end period, is an unenthusiastic booster compared to the geometric mean in cases 4 and 5. Curiously, the erring indexes are most optimistic as the situation becomes most critical: closer and closer to the bare minimum supply of x.[9]
The total agreement in case 2, by the way, is fortuitous: the ratios between second-period and base-period prices were identical for Px and Py, so it didn’t matter which weights were used.
Incidentally, the inflation reported by the indexes (except when they err) has occurred while our supposed monetary authority has held total expenditures constant. This is the type of inflation which we might picture in a nineteenth-century mining camp, after all trade with the outside world and mining operations have been cut off by an unexpectedly severe winter. A fixed stock of gold chases a dwindling supply of goods.
We might wonder what happens if a panicky government, as desperate as everyone else to lay hold of the dwindling supply of goods, prints up some dollar bills, giving us monetary inflation as well as scarcity inflation. The short answer is that in our model, with perfectly flexible prices, all of the indexes are multiplied by r, the factor by which total dollar expenditures have grown. Since both E(t) and any index we may care to apply have risen by the same factor, the "real income" estimated by that index will remain unchanged. Each index sticks resolutely to its original opinion.