Sekerka - Kyn - Hejl:  Price Systems Computable from Input-Output Coefficients



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 input-output table. Let Yc  be the vector of personal consumption. Now define

 G =    Yc/J'Yc      J' = (1, 1, ..., 1). (38)

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)




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

 P = (1 + n*) A’P + r*B' P + m*WG’P

If G is the group numeraire, G’P = 1, the simplified price formula becomes

P = (1 + n*)A'P + r*B'P + m*W,       (41)


G’P = 1.                                                (42)



P is a vector of price indices, designated as Pi 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. 

Thus, all types of prices can be calculated using a vector of wage coefficients W in place of matrix C, and with a nurneraire chosen so as to keep total personal consumption constant. Conditions governing m*,n*, and r* are unchanged, provided that both (41) and (42) hold.

The terms n* and r* must both lie within the limits given by eq. (24). Now m* has the following limits:

0 £ m* £ l/[G’ (I - A’)-1 W]  - 1.    (43)

If C is defined by eq. (40), limits for m* given by eq. (43) are identical with those of eq. (24). It is not possible to derive any upper bound for m* from eq. (41) alone, i.e., without the normalizing condition (42).

However, the assumption m* ³  0 assigns the upper bounds of eq. (24) to n* and r*. Even when we fix two of the three parameters 
m*,n*, r* in eq. (41), we do not uniquely predetermine the third without eq. (42).






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