8. On the matrix of worker’s consumption

To simplify our formulae, let us assume that matrices A, B and C are expressed in original prices and P are price indices. It is empirically difficult to estimate a matrix of workers’ consumption C. To do so, we introduce the simplifying assumption that the structure of consumption expenditures of workers in all branches of the economy has the same proportions as the vector of personal consumption in the second quadrant of the inputoutput table. Let Y_{c} be the vector of personal consumption. Now define
Let U be the vector of wage funds (the sum of wages paid during the year in the various branches). We define the vector of wage coefficients W = X^{^1} U. (39)

Let
C = GW’. (40) This matrix C is the matrix of coefficients of workers’ consumption, assuming that workers in all branches spend their wages in the same proportions as total personal consumption. 
Introducing eq. (40) into eq. (23), we obtain
If G is the group numeraire, G’P = 1, the simplified price formula becomes

P is a vector of price indices, designated as P^{i} above. Given eq. (38), we conclude that eq. (42) is equivalent to the requirement that the total volume of personal consumption in the newly computed prices be the same as in original prices. 



