Equivalence between input-output and value-added economies

I show that, under standard assumptions, input-output (or network) economies are equivalent to value-added ones. Using a generalized version of the model in Acemoglu et al. (Econometrica, 2012), I show that the degree of influence of a given sector is equal to its value added share. This occurs because - by using the input-output network - the output of a given sector indirectly contributes to the production of the final consumption of the rest of the sectors, which constitutes the source of its value. Thus, value-added economies deliver the same aggregate response to sectoral shocks than input-output ones. Despite this equivalence, the Leontief multiplier, which applies to sales and gross output, is intact.