1. Indices

2. Definitions
Paasche Quantity Index:
Laspeyres Quantity Index:

3. Can we make any welfare statement if we know that the Paasche Quantity Index or the Laspeyres Quantity Index has gone up or down?
This depends on:
If Pq > 1 then
Since the original bundle (xb, yb) is still affordable we could buy it  but have not. Thus,

4. If Pq < 1 then
No ranking of bundles is possible, xt, yt chosen when xb, yb is not available: so cannot compare.
If Lq < 1 then

5. (xt, yt) not available when (xb, yb) chosen.
If Lq > 1 then
(xt, yt) not available when (xb, yb) chosen.
Possible Pq < 1 < Lq

6. Original Budget Constraint: Pxb xb + Pyb yb = Mb
In time t, Prices, Income and Quantities have changed.
We want to know if we are better or worse off when Pxt xt + Pyt yt = Mt

7. We are not able to rank using the revealed preference.
Price Indices
Paasche Price Index weights are quantities:
Laspeyres Price Index:
We are not able to rank using the revealed preference.

8. Need to define expenditure index:
If Pp > M then
Rearranging gives:
pxb xb + pyb yb > pxb xt + pyb yt
Thus:

9. pxt xb + pyt yb < pxt xt + pyt yt
If Lp < M then
pxt xb + pyt yb < pxt xt + pyt yt
Problems with Price Indices
Choice of ‘representative weights’
CPI Base year weights: Laspeyres
Used to index pensions and social security over-estimates effect of price changes as this does not take account of substitution possibilities.

10. Overall real income down
One price up, other down
Overall real income down y0
Original Prices x0

11. Income after Pension rise based on CPI/Laspayres
So CPI compensation overestimates effect of inflation since does not allow for substitution of y for x y0 U1 U0 x0

12. By contrast, the Paasche index under-estimates the effect of price change.