A Note on Capital and Output Aggregation in a General Equilibrium Model of Production - Sato, Kazuo (Econometrica, 1979)
BROWN AND CHANG [2] have sought the exact conditions of aggregation within and across sectors in a general equilibrium model of production. Complete intrasector aggregation of nonlabor inputs requires all gross rental rates to change in the same proportion, while complete intersector aggregation requires all prices of products and rental rates on capital aggregates to vary in the same proportion. Then, the full aggregation is to be established when all prices change in the same proportion, provided that all nonlabor inputs depreciate at the same rate [2, Theorems 1 and 8]. The necessary and sufficient condition for it is that labor's shares in total value of product are equal in equilibrium across sectors. This condition fails when nonlabor inputs depreciate at different rates because gross rental rates no longer change in the same proportion even if all prices change in the same proportion. Brown and Chang have derived a condition for the equiproportionate changes in gross rental rates [2, Theorem 8]. Though the condition allows intraand intersector aggregation of nonlabor inputs, it no longer permits intersector output aggregation because product prices do not vary at the same rate.