Optimal consumption and investment in the economy with infinite number of consumption goods

In the article we present some extension for the classical problem of dynamic investment optimization. We take the neoclassical model of growth with one product and many consumption goods. The number of consumption goods can be infinite and the consumption bundle is defined on some abstract, measurable space. The instantaneous social utility of consumption is measured as the integral of individual utilities of the consumption goods. The process of transforming product into consumption goods is described by another measure. The performance of the economy is measured by current value of the total utility in some planning horizon. We show that the problem of choosing optimal consumption paths for each good can be decomposed into 1) problem of choosing optimal aggregate consumption, which can be solved using standard methods of optimal control theory, 2) problem of distribution aggregate consumption into consumption of specific goods.