Optimal Growths from Two Perspectives, with Conditional Macro- and Micro-consistency: Dynamic Summation of Economic Variables Under Differential Regional Representative Production Functions

Although the representative theory of consumer/firm is widely used in particular forms of utility and production functions, is the optimal growth model adequately supported by a multi-regional microeconomic base in a "general form of utility and production functions"? Is the "optimization of macro-aggregate functions" consistent with the "summation of micro-optimal outcomes"? Economists have explored the problem of summation of macro-variables in a special form of production and utility function setting to find some theoretical foundations of macroeconomics in the case of heterogeneous representative consumers. However, heterogeneous representative producers have not been studied enough, and the conclusions are too dependent on the functional form. This study searches for the conditions for macroeconomics to reach micro-foundations in the framework of heterogeneous producers and general functions. Linking the entire macroeconomy through inter-regional capital flows, we distinguish between using and equity capital, construct a multi-regional dynamic optimal growth model in a homogeneous representative consumer and heterogeneous representative manufacturers, compare the summation of the regional dynamics with the aggregate macro-dynamics, and find that (A) the regional and macro-perspectives are generally inconsistent in their optimization results; (B) if the population growth rate is the same across regions, the macro-steady state is a weighted summation of the steady states; and (C) if the inverse of the absolute risk aversion is linearly additive to consumption, the model is reliable. In particular, CARA and CRRA-type utility functions are eligible. The eligible utility function can only be an exponential utility function or a power function utility function (the logarithmic utility function can be considered a degenerate power function utility).