Up to now, we’ve chosen indifference curves to illustrate points, and invented sample data to fit the curves. This is honest: the curves are all in our heads, anyway. If, however, we choose to accept some restrictions upon the form of the curves, then we can work in the opposite direction: fit a unique indifference curve to each observed combination of item quantities and their prices.
Suppose we accept the family of hyperbolas "y = m/(ax-h)" as the standard form for indifference curves. We must further stipulate that m > 0, a > 0, h is non-negative, and x is restricted to values greater than h/a. In other words, our attention is limited to that branch of the hyperbola which lies in the area x > h/a and y > 0. In introducing the factor a, we’ve complicated the measure of "necessity", for h/a, rather than just h, is now the economy’s minimum reservation of the "necessity" x. We’ll set a back to 1 after showing that it doesn’t alter anything that we can deduce about the indifference curve from observed data.
Let x1 and y1 be observed quantities of x and y
during some period, with prices Px1 and Py1. Let r
= Px/Py. From economic considerations, we require that dy/dx = -r,
so we can infer:
r = am/(ax1-h)^2 = (a/(ax1-h))*(m/(ax1-h)).
Multiply both sides by (ax1-h):
rax1-rh = am/(ax1-h) = ay1
<=> rx1-(r/a)h = y1
<=> (r/a)h = rx1-y1
<=> h/a = x1-(y1/r).
This is an important result. It tells us that the observed data will dictate the ratio h/a. We can pick any value we like for a and calculate h to fit the ratio, or vice-versa, but we can’t pick the ratio. This is disconcerting, because h/a has a supposed economic significance: it’s the economy’s minimum reservation of the "necessity" x. We don’t want it to wander around: necessity should remain necessity over moderate periods of time.
Be that as it may, once we pick an a and set h = a*(x1-(y1/r)),
the original formula of the hyperbola determines m:
m = y1*(ax1-h).
If we multiply both a and h by a common factor (preserving
the ratio h/a), m gets multiplied too, so it’s
the same curve.
Under the circumstances, we may as well choose a = 1, and forget
about it. The standard hyperbola which we’ll deduce from given data
will be:
y = m/(x-h),
and h can be read immediately as the minimum reservation of x.
We have to watch out, though, for one thing more: a self-consistent pair of indifference curves must not cross each other. If they did, every point on one curve (except at the intersection) would be equivalent to a point on the other with the same x but different y.
We can check fairly easily. It turns out[10] that two distinct hyperbolas y
= m/(ax-h) and y = M/(Ax-H)
are free of crossings provided
H = h = 0,
or
M/m = A/a,
or
H/h = A/a (h > 0)
or
M/m < A/a and H/h < A/a
(h > 0),
or
M/m > A/a and {H/h > A/a
or h = 0}.
If, as already suggested, we pick a = 1, and do the same with
A, these tests boil down to
H = h,
or
M = m,
or
M < m and H < h,
or
M > m and {H > h or h = 0}.
Mind you, two curves which fail all of these tests may be individually plausible; they just can’t make economic sense as part of the same indifference map. They imply a change of tastes.
With that said, let’s take on an example. There’s nothing special about this example, except that the hypothetical "observed data" are taken from an example used in the Boskin Report.[11] We deduce a hyperbola from the "data" offered for each period, and display it in the format of our previous examples. Additional points are shown for each hyperbola; the emphasized row in each case corresponds to the "observed" data point. Our x corresponds to Boskin’s "chicken", while y is "beef".
x1 ("chicken") |
y1 ("beef") |
Px1 |
Py1 |
1.00 |
1.00 |
1.00 |
1.00 |
From our formula, h/a = x1-(y1/r),
with r = Px1/Py1 = 1, and with a set to 1, we
get
h = 1.00-(1.00/1) = 0.00, and then
m = y1*(ax1-h) = 1.00*(1.00-0.00) = 1.00
in short, y = 1/x, the simplest hyperbola in our whole family
of candidates.
y = 1/x | ||
a |
m |
h |
1.0000 |
1.0000 |
0.0000 |
This is the same hyperbola as we used with our Example 1, but we’ll pick a different selection of points, to feature the "observed data" point in the middle:
x |
y |
dy/dx |
0.50 |
2.00 |
-4.0000 |
0.75 |
1.33 |
-1.7778 |
1.00 |
1.00 |
-1.0000 |
2.00 |
0.50 |
-0.2500 |
3.00 |
0.33 |
-0.1111 |
The emphasized row contains the "observed data" from which the hyperbola was deduced. No other hyperbola of form y = m/(x-h) passes through (1.00, 1.00) with a slope of -1.
The prices and money expenditures come out as follows:
x |
y |
Px |
Py |
Ex |
Ey |
E(t) |
0.50 |
2.00 |
2.0000 |
0.5000 |
$1.00 |
$1.00 |
$2.00 |
0.75 |
1.33 |
1.3333 |
0.7500 |
$1.00 |
$1.00 |
$2.00 |
1.00 |
1.00 |
1.0000 |
1.0000 |
$1.00 |
$1.00 |
$2.00 |
2.00 |
0.50 |
0.5000 |
2.0000 |
$1.00 |
$1.00 |
$2.00 |
3.00 |
0.33 |
0.3333 |
3.0000 |
$1.00 |
$1.00 |
$2.00 |
If we now study the effects of moving from point to point along this indifference curve, with the top row taken as the starting point, we get:
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 |
1.0833 |
0.9231 |
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 |
3.0833 |
0.3243 |
1.0000 |
1.0000 |
1.0000 |
Qualitatively, these are the same results we studied in Example 1, and for the same reason: the hyperbola 1/x is as perfect as an indifference curve could be found for illustrating the Boskin arguments concerning the shortcomings of the Laspeyres index.
Our present example, though, has another hyperbola on tap, deduced from "Period 2" data of the Boskin example:
x2 ("chicken") |
y2 ("beef") |
Px2 |
Py2 |
2.00 |
0.80 |
0.80 |
1.60 |
From our formula, h/a = x2-(y2/r),
with r = Px2/Py2 = 0.5, and with a set to 1,
we get
h = 2.00-(0.80/0.5) = 0.40, and then
m = y2*(ax2-h) = 0.80*(2.00-0.40) = 1.28
in short, y = 1.28/(x-0.40), a rather more interesting
indifference curve than for Period 1. We’ll designate its parameters
as A, M, and H, to distinguish them from the a,
m, and h of Period 1.
y = 1.28/(x-0.40) | ||
A |
M |
H |
1.0000 |
1.2800 |
0.4000 |
Routinely, we check that the two curves don’t cross each other: we have M > m and H > h, with A = a = 1, so they don’t.
Notice that x, "chicken", has now become a "necessity", upon our iterpretation, with a minimum reservation of 0.40. Substitution of "beef" for "chicken" begins only after that reservation is satisfied.
This is hard to defend: can we really account for a change in the minimum reservation of x, while assuming for the sake of argument that preferences haven’t changed? (If they have changed, intertemporal comparisons of utility are even more of a mess than usual.)
Of course, the difficulty is ours, rather than that of the Boskin authors: we’ve superimposed a discussion of indifference curves and stipulated strong restrictions upon their form, with a novel interpretation of the parameter "h" in y = m/(ax-h). Nonetheless, it’s an illuminating exercise: these curves resemble the ones the Boskin authors would sketch if they did get into a discussion of indifference curves, and simulate rather well the relationships they have in mind.
It is possible to argue that an indifference curve is stitched together from a series of arc segments, and can be approximated by a hyperbola only in a small region around the observed data. Fair enough, but even in an immediate neighborhood of the "observed data", the combination of quantities and prices may imply a much more insistent turning away from the y axis in one period than another, which does suggest a change in preferences.
Admittedly, real-life data would never in this world fit such nice curves — if we could even formulate reasonable curves for them to fit in a multi-dimensional market basket. We’re just exploring what unintended implications can lie hidden in innocent-looking concocted data, according to the very relationships they’re founded upon.
Anyhow, accepting this second hyperbola, let’s run it through the usual exercises.
x |
y |
dy/dx |
0.50 |
12.80 |
-128 |
1.00 |
2.13 |
-3.5556 |
2.00 |
0.80 |
-0.5000 |
2.50 |
0.61 |
-0.2902 |
3.00 |
0.49 |
-0.1893 |
The emphasized row contains the "observed data" from which the hyperbola was deduced. No other hyperbola of form y = m/(x-h) passes through (2.00, 0.80) with a slope of -0.5.
The prices and money expenditures come out as follows. (Our imaginary monetary authority allows total expenditures to rise to $2.88, in order to follow the Boskin example.)
x |
y |
Px |
Py |
Ex |
Ey |
E(t) |
0.50 |
12.80 |
4.8000 |
0.0375 |
$2.40 |
$0.48 |
2.88 |
1.00 |
2.13 |
1.8000 |
0.5063 |
$1.80 |
$1.08 |
2.88 |
2.00 |
0.80 |
0.8000 |
1.6000 |
$1.60 |
$1.28 |
2.88 |
2.50 |
0.61 |
0.6261 |
2.1571 |
$1.57 |
$1.31 |
2.88 |
3.00 |
0.49 |
0.5143 |
2.7161 |
$1.54 |
$1.34 |
2.88 |
If we study the effects of moving from point to point along this indifference curve, with the top row taken as the starting point, we get:
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 |
0.6250 |
0.3750 |
2.5625 |
0.5902 |
0.6814 |
0.9898 |
1.2298 |
0.5556 |
0.4444 |
7.2500 |
0.2991 |
0.4200 |
0.9827 |
1.4725 |
0.5435 |
0.4565 |
9.6957 |
0.2395 |
0.3599 |
0.9825 |
1.5240 |
0.5357 |
0.4643 |
12.1607 |
0.1997 |
0.3174 |
0.9825 |
1.5585 |
The Tornqvist index is by far the least confused by pure substitution.
Finally, we conduct the experiment of moving between the curves, from any of the points we chose on the first to our corresponding point on the second. (The correspondences are more or less arbitary, except that x ascends from sample to sample on each curve, thus moderating the radicalism of any one move, and the emphasized pair are from the Boskin example.)
x |
y |
Px |
Py |
Ex |
Ey |
E(t) |
0.50 |
2.00 |
2.0000 |
0.5000 |
$1.00 |
$1.00 |
$2.00 |
0.75 |
1.33 |
1.3333 |
0.7500 |
$1.00 |
$1.00 |
$2.00 |
1.00 |
1.00 |
1.0000 |
1.0000 |
$1.00 |
$1.00 |
$2.00 |
2.00 |
0.50 |
0.5000 |
2.0000 |
$1.00 |
$1.00 |
$2.00 |
3.00 |
0.33 |
0.3333 |
3.0000 |
$1.00 |
$1.00 |
$2.00 |
Notice that we can readily detect an income effect, besides whatever we may define to be the "inflationary" effect. For in the first move which meets our eye — the one in the first row, (0.50, 2.00) => (0.50, 12.80) —, an undiminished x is accompanied in the second period by a greater amount of y. But if that is an unambiguous move to a higher real income, then so are are all the rest, for they move between the same indifference curves.
As for inflation, in three cases of the five, Px and Py move in opposite directions, so we have no choice but to define "inflation" as some weighted average of the changes.[12] In the other cases, Px and Py both rise, so there’s no doubt about inflation, but they rise disproportionately, so we still have to settle upon a measure for it.
So here are the various indexes for the present example.
Sx |
Sy |
Laspeyres |
Paasche |
Base- Weighted Geometric Mean |
Tornqvist |
Fisher |
0.5000 |
0.5000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
0.5000 |
0.5000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
0.5000 |
0.5000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
0.5000 |
0.5000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
0.5000 |
0.5000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
The Laspeyres, Paasche, and Fisher indexes in the second period of the emphasized case are as given by the Boskin Commission in their example; the Tornqvist differs by what is probably a rounding error (they write 1.10).
If our use for the price index were solely to judge whether monetary policy were maintaining price stability, we would probably concur immediately: most of the time, the Tornqvist and Fisher indexes are less erratic than either the Laspeyres or the Paasche index taken alone. Most of the time, too, they agree on the right direction for monetary policy, if price stability is to be its goal.
However, if we are correct in our analysis that this example also involves changes in real income, we’ll want to compare their performance in measuring that.
Laspeyres |
Paasche |
Base-Weighted Geometric Mean |
Tornqvist |
Fisher |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
$2.00 |
All of the indexes correctly report an increase in real income, and in the Boskin example from which the curves were deduced, they do not differ widely upon the amount. The Tornqvist index, preferred by the Boskin Commission, is just a tad more optimistic than the Fisher index or the base-weighted geometric mean, while the Paache index values the end period most highly and the Laspeyres index values it least.
It’s encouraging that all of the indexes correctly discern a rise in real income. Putting a dollar amount on it, though, is questionable: there isn’t really any metric upon the "utility" measured by the indifference curves.
In any case, the behavior of the indexes is interesting in the other rows. All of them are in remarkable agreement in the fourth row. On the other hand, they are of decidedly different minds about the move in the first row. In that one, the quantity of x, which is uncomfortably near the minimum reservation of 0.40, is held constant at 0.50, while the entire increase in real income is taken in the form of additional y. Paasche and the base-weighted geometric mean are more or less ecstatic about the piling on of y, and even the Tornqvist and Fisher indexes are more enthusiastic than in any of the other cases. Only Laspeyres remains gloomy — as maybe it ought to, when the shortage of x hasn’t been addressed. In any case, it’s hard to argue that the move in the first row, between the same two indifference curves, should be so much more esteemed than others.