Kelly’s price index calculator

The formula

P₀₁ (Kelly) = Σp₁q / Σp₀q × 100

Here q is a set of fixed quantities, one per commodity. Σp₀q is what the fixed basket costs at base-year prices and Σp₁q what it costs now. The formula has the same shape as Laspeyres and Paasche; the difference is only where the basket comes from.

The fixed-weight idea

Laspeyres ties the weights to the base year and Paasche to the current year, so changing the base year means changing the weights, and a Paasche series needs new quantities every period. Kelly separates the two decisions. The basket q is fixed once, typically from the average quantities of a run of “normal” years or from a household survey, and prices from any pair of years are compared using that same basket. Because the weights never move:

• the base year for prices can be shifted without recomputing weights;
• no fresh quantity data are needed each period, only prices;
• indices for different years measure the same basket, so they can be compared with each other directly, and the index passes the time reversal test.

A worked example

The calculator's default data uses fixed quantities q = 18, 15, 32, 10 and 28 (say, the average bought over the last three years):

1. Σp₀q = 180 + 120 + 160 + 200 + 112 = 772.
2. Σp₁q = 270 + 150 + 192 + 240 + 112 = 964.
3. P₀₁ = 964 / 772 × 100 = 124.87.

With the same prices, the base-year quantities (Laspeyres) give 125.97 and the current-year quantities (Paasche) give 123.33. The average of q₀ and q₁ as the fixed basket gives 124.67, the Marshall–Edgeworth figure.

Does it over- or under-state inflation?

That depends entirely on the basket. A fixed basket never responds to price changes, so Kelly's index has the same kind of substitution bias as Laspeyres, measured against whatever year the basket represents. If q comes from years before the base, it overweights goods that have since become relatively dear and overstates inflation, and the bias grows the older the basket gets. If q is an average spanning the base and current periods, it usually lands between Laspeyres and Paasche and the bias is small, as in the example (and with q = (q₀ + q₁)/2 exactly, it always lies between them). The advantage is that the bias is stable and known, rather than changing with every choice of base year.

Common mistakes

• Using q₀ in one sum and q₁ in the other. Kelly uses the same q in the numerator and the denominator; that is what makes it a fixed-weight index.
• Assuming Kelly always equals Marshall–Edgeworth. It does only when q is exactly (q₀ + q₁)/2.