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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, obtain- 
ing 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 adding- 
up 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 expen- 
diture 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 substi- 
tution of electricity and oil for coal as domestic fuels. 
1] Stone - pag. 64