Next, let us introduce the bulk of the requisite notation. Let

i

signify the HH (i = 1, . . . N),

j

the purchase occasion,

Ji0

the set of purchase occasions by the ith HH in base period 0,

Jit

the set of purchase occasions by the ith HH in comparison period t,

Qgij0

the number of units of good g purchased by the ith HH, jth purchase occasion, during base period 0,

Qgijt

the number of units of good g purchased by the ith HH, jth purchase occasion, during comparison period t,

N

the number of households in the universe

pgij0

price per unit (of good g) paid by the ith HH, jth purchase occasion, during base period 0, and

pgijt

price per unit (of good g) paid by the ith HH, jth purchase occasion, during comparison period t.

In these definitions, we use the convention

for nonbuyers in the base period and

for nonbuyers in the comparison period. We assume there is at least one buyer, Q0 > 0 and Qt > 0, in each period.

Average unit volumes, and , and average prices per unit, and , are defined in the obvious way. The decomposition of the period-to-period trend in total dollar volume is now given by

where TN is the trend in the total HH count, Tq is again the trend in average units per HH, and Tp is again the trend in average price per unit. As above, Tp may be called the price index and Tq the unit volume index.

We can next further extend the work to a still more realistic world in which a static set of goods is available in the market at both time periods. Let subscript g signify a good, and to simplify the notation let G represent both the set and the number of goods. Total dollar volumes are now defined by



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