Inferring an entire indifference curve from a single point and a slope, and then exploring the consequences of moving to other points on the curve, or between it and another such curve, is literally an exercise in going around the bend. It reveals, however, that if hypothetical data are cooked up without considering what the underlying indifference curves might be, a discussion that is seemingly about monetary and substitution effects alone may actually draw upon income effects as well.
We want to study this third influence under more controlled conditions. Our remaining examples therefore go back to "reverse engineering": we pick the curves to depict a situation, and fit "data" to the curves.
In the preceding example, force-fitting a curve to pretended data obliged us to overlook an awkwardness: we passed from a curve with h = 0 to one with H = .040. As noted at the time, this signifies a change in consumer preferences. How can item x go from being perfectly substitutable with y to being a "necessity" with a minimum reservation?
In general, we prefer to dodge the question of changing consumer preferences: the "well-being" which we attempt to measure is ill-defined enough without that complication. However, there is one kind of change with which we can cope, at least roughly. Remember that the quantities in our model are aggregates for the whole economy. So in case of population growth or decline (and assuming homogeneous tastes), the new preference map should look like a scaled-up or scaled-down image of the original. The minimum reservation of x and the equivalent of any given curve in the original map can be calculated, with no change in the preferences of individuals.
Let k be a factor by which population grows or declines. Then for any given aggregate indifference curve y = m/(x-h) in the old map, there is a curve Y = (k^2)m/(X-H) in the new map representing a new aggregate curve which furnishes the same per capita satisfaction. For with Y = ky, X = kx, and H = kh, it can be written ky = (k^2)m/(kx-kh) — not indeed the original curve, but one which, when deflated by the population index, yields the original aggregate.
In this next example, we start from a comfortable-looking point on one indifference curve, y = 64/(x-1), and examine alternative moves from it to five different points on a much lower curve, Y = 16/(X-0.5). The lower curve, though we can plot it in the same coordinate system, lies in a new set of indifference curves, after the population has been cut in half (k = 0.5, m = 64). Per capita, it represents the same real income as the first. If we want an allegory, we can imagine that x and X are water, and a prolonged drought killed off half the population. The new minimum reservation of water is 0.5. The die-off was what was signified by the "minimum reservation" of 1.0 in the original curve. When we say "necessity", we mean it.
We start, then, with the curve
y = 64/(x-1.0) — i.e., m = 64, h = 1.0,
x > h,
and a comfortable-looking point upon it:
x |
y |
dy/dx |
12.00 |
5.82 |
-0.5289 |
With the usual understandings as to the total level of expenditure and the behavior of prices,
x |
y |
Px |
Py |
Ex |
Ey |
E(t) |
12.00 |
5.82 |
173.9130 |
328.8043 |
2086.96 |
1913.04 |
4,000.00 |
We’ll examine five alternative "crisis" moves to a lower curve, y = 16/(x-0.5). First, though, a check of reasonableness: we examine a point on the lower curve in which x and y have both been cut in half, exactly in proportion with the fall in population and in h:
x |
y |
dy/dx |
6.00 |
2.91 |
-0.5289 |
Notice that the slope at this point is the same as that at the original point on the higher curve. The analysis proceeds in the usual way:
X |
Y |
Px |
Py |
Ex |
Ey |
E(t) |
6.00 |
2.91 |
347.8261 |
657.6087 |
2086.96 |
1913.04 |
4000.00 |
Quantities x and y are halved, but with total money expenditure held at $4,000, prices Px and Py have doubled, so the expenditures Ex and Ey are unaltered, as are the expenditure shares Sx and Sy.
Sx |
Sy |
Laspeyres |
Paasche |
Base- Weighted Geometric Mean |
Tornqvist |
Fisher |
.5217 |
.4783 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
As expected, all of the indexes agree that prices have doubled. (How could it be otherwise, when Px and Py have changed in the same proportion?) And therefore, when applied to the total expenditure $4,000 as deflators, they all agree that the new real income is $2,000. Remember, that’s an aggregate for the whole economy. Divided by the halved population, it will come out to the same per capita income as the original.
Of course, this point must represent an equilibrium after the drought has ended. By hypothesis, it was unattainable at the peak of the drought, or why would half the population have died? Let us therefore examine five alternative "crisis" moves between a single point on the higher curve and five different points on the lower one, with increasingly radical substitution towards y. These are points at which the post-drought indifference curve might have been attained during recovery, if it were possible to provide the stipulated amounts of x and y. The points will be:
x |
y |
dy/dx |
1.00 |
32.00 |
-64.0000 |
0.90 |
40.00 |
-100.0000 |
0.80 |
53.33 |
-177.7778 |
0.70 |
80.00 |
-400.0000 |
0.60 |
160.00 |
-1600.0000 |
Holding total expenditures constant, prices and expenditures will work out to:
x |
y |
Px |
Py |
Ex |
Ey |
E(t) |
12.00 |
5.82 |
173.9130 |
328.8043 |
2086.96 |
1913.04 |
4000.00 |
12.00 |
5.82 |
173.9130 |
328.8043 |
2086.96 |
1913.04 |
4000.00 |
12.00 |
5.82 |
173.9130 |
328.8043 |
2086.96 |
1913.04 |
4000.00 |
12.00 |
5.82 |
173.9130 |
328.8043 |
2086.96 |
1913.04 |
4000.00 |
12.00 |
5.82 |
173.9130 |
328.8043 |
2086.96 |
1913.04 |
4000.00 |
Comparing these points with the pre-drought situation, various indexes report the resulting price level changes as follows:
Sx |
Sy |
Laspeyres |
Paasche |
Base- Weighted Geometric Mean |
Tornqvist |
Fisher |
.5217 |
.4783 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
.5217 |
.4783 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
.5217 |
.4783 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
.5217 |
.4783 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
.5217 |
.4783 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
1.0000 |
The upshot is a variety of estimates for the change in real income:
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 |
Yet these are on the same indifference curve which we evaluated at $2,000 when it could provide x and y in their pre-drought proportions.
Should, we then, redraw the indifference curves? Probably so, but not to redeem the price indexes, and not by altering the increasing desperation for x which they depict, as it draws near its minimum reservation h. Rather, we need to think about the underlying transformation curves. As the economy is plunged into crisis by the disappearance of x, it is hardly likely that it can expand the production of y as dramatically as shown. If anything, the production of y is likely to suffer as well. The portions of lower indifference curves which compensate for diminished x by grotesquely enlarged servings of y are likely never to be attained.
What we might expect, then, is a series of transitional indifference curves in which h diminishes as population is killed off, but meanwhile, per capita income is not held constant. People are forced to indifference curves in which they are not fully compensated for diminishing x. In terms of our scheme, M falls faster than (k^2)m. We have seen enough, though, to trust that the various indexes will indeed notice the fall, and that whatever substitution effects accompany the fall, they will be least valued by the Laspeyres index, most valued by Paasche, and only moderately noticed by the Tornqvist index. The base-weighted geometric mean tends to be quirky when substitutions are extreme.