Before we advance to less well-behaved indifference curves, let’s
look at the same curve writ larger:
y = 4/x
This blows up every point on the first curve by a factor of 2 —
for example, it cuts the 45-degree line from the origin at (2,2) instead
of (1,1). We can choose points on it corresponding to those we chose on
the first curve:
x |
y |
dy/dx |
8.00 |
0.50 |
-0.0625 |
4.00 |
1.00 |
-0.2500 |
2.00 |
2.00 |
-1.0000 |
1.00 |
4.00 |
-4.0000 |
0.50 |
8.00 |
-16.0000 |
For corresponding points, the quantities of x and y are both doubled. As we might expect from the shape of the curves, derivatives dy/dx for corresponding points are the same. If our imaginary monetary authority continues to hold total expenditures at $4,000, all prices must be cut in half:
x |
y |
Px |
Py |
Ex |
Ey |
E(t) |
8.00 |
0.50 |
250.00 |
4000.00 |
2000.00 |
2000.00 |
4000.00 |
4.00 |
1.00 |
500.00 |
2000.00 |
2000.00 |
2000.00 |
4000.00 |
2.00 |
2.00 |
1000.00 |
1000.00 |
2000.00 |
2000.00 |
4000.00 |
1.00 |
4.00 |
2000.00 |
500.00 |
2000.00 |
2000.00 |
4000.00 |
0.50 |
8.00 |
4000.00 |
250.00 |
2000.00 |
2000.00 |
4000.00 |
Unsurprisingly, the results of moving from point to point along this new curve, as long as we pick points corresponding to those on the first curve, are identical to moves on the first curve:
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 |
Once again the elasticity of substitution is identically equal to -1
— as, indeed, it would be for any value of m we might choose
in a wider family of hyperbolas:
y = m/x
Elasticity of substitution = (dy/dx)*(x/y) = (-m/x^2)*(x/(m/x))
= -1.
What’s more interesting is to look at the results of moving between the curves, from any point on the first to the corresponding point on the second:
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 |
These moves generate the following reportage by various price indexes:
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 |
All of the indexes agree: the price level has fallen by half. The Laspeyres and Paache indexes are not confused by any of these moves. But then, why would they be? In every case, the base period and end period proportions between x and y quantities are unchanged. There has been no substitution between them, only an equi-proportional substitution of more for less.
Since the monetary authority maintained total expenditure at $4,000,
prices were forced down, and the indexes dutifully report a deflation of
50 percent. If you divide the $4,000 total expenditure in the second period
by any of the indexes, each reports a "real income" in the second
period of $8,000.
If the monetary authority, in the face of this growth, were to let total expenditures rise to $8,000, then all prices would hold steady, all the indexes would read 1.0000, and calculated "real income" would still be $8,000 in the second period, as compared with $4,000 in the first.. Such a case is as close as we can come to defining a "pure income effect": equi-proportional changes in the quantities of all items, while money prices remain fixed. As we’ll see when we tackle a more general family of indifference curves, there is usually no satisfactory definition of substitution-free "corresponding" points on lower and higher indifference curves. Changing from one indifference curve to another in the same family almost always involves some change in the proportions consumed.