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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 25

where Z = [WOW] and h = {b* b**}, The least-squares
estimator, h,, of h is

(IV.49) h, =(Z', 27! Z'w

Given %;, we can return to (IV. 38), replace b*, and b**,
by b*, and b**, and calculate the next approximation to g,
namely g,= {}c*,:c**,{. If we continue in this way until the
estimates cease to change, we shall have reached a solution.
We can see from (IV. 44) than W;, is a scalar matrix, and
50 in estimating » the system of equations breaks down into a
set of single equations. From (IV. 37) we can see that Y , is not
a scalar matrix, and so in estimating g we are, in effect, obtaining
 average values derived from all the equations. From
‘IV, 20) we can see that bp appears as a separate term on the
right-hand side, and so, since p’e=p, it follows from the addingup
 theorem that the constraint ’b=1 is automatically satisfied
by the more complicated form in (IV. 21).
Further details of this procedure and of the results and
orojections obtained by it for the eight expenditure groups are
given in [39] [45]. We are at present working on combining
these estimates with those obtained from family budgets and
on analysing the components of each main group by the same
procedure.
Until this work is completed we have to use more rough and
ready methods. What we do is to estimate the levels of expenditure
 on the components of each main group by reference
to their changing relative importance within the group; for
example, within the food group the proportion spent on bread
and cereals tends to fall with time, whereas the proportion spent
on meat, fruit and vegetables increases at a rate well above
the group’s average. We also try to allow subjectively for
the tempo of substitutions, such as an acceleration of the substitution
 of electricity and oil for coal as domestic fuels.

1] Stone - pag. 64