Production theory for constrained linear activity models

The purpose of this paper is to generalize the framework of activity analysis discussed in the paper by Antonio Villar without requiring any dimensional requirements on the activity matrices and by introducing a model of activity analysis in which each activity may (or may not) have a capacity constraint. We follow the usual nomenclature of input-output analysis for “the quantity of a good supplied to the consumers outside the production (or manufacturing) sector” and refer it as “final demand”. We obtain results similar to those in Villar concerning solvability, non-substitution and existence of efficiency prices. We apply our analysis and results to the two-period multisector activity analysis model with capacity constraints. The activity matrix is the difference between a non-negative output coefficient matrix and a non-negative input coefficient matrix, with the coefficients being measured in money units for each activity. Almost all the results obtained thus far get replicated in this macroeconomic context. However, some reformulations are required for issues related to existence of equilibrium price vector and as a consequence, issues related to efficiency prices via the non-substitution theorems. The corresponding concepts in this application refer to “inflation rate” vectors.