The purpose of this paper is to construct a prototype model for multiregional economy, and to discuss most important policy issues like (a) factor-price-equalization (FPE) and (b) maintenance of regional income share (MIS).
We assume that each of n regions has common linear homogeneous COBB-DOUGLAS production function with m factors, and growth rate of each factor in each region has two components: common natural growth rate and growth rate by social movement, which is specified as elasticity of interregional movement (EIM) multiplied with difference between logarithms of marginal productivity of this region and of national average. Thus the prototype model consists of 3nm+2m+n equations.
We compile the results in form of theorems without lengthy rigorous mathematical proofs, which are shown in another forthcoming paper Fukuchi (3), but by confirming by empirical results based upon actual data. For this purpose 3 factors (private capital, public capital, labor) and 9 regions model was estimated with Japanese data (1956-80).
First theorem states the condition for FPE of each factor in long-run, requiring the inequality between natural growth rate and combination of elasticities of movement. Second theorem states that if this condition is met for (m-1) factors, FPE is guaranteed for all factors. In case of Japan, private and public capitals satisfied the condition, so FPE is secured for all factors although labor does not satisfy the condition. Third theorem suggests the increasing income share of j-th region when j-th region has highest productivity in h-th factor with highest elasticity of movement. In Japan elasticities of movement was 0.0473, 0.0110, and 0.0058 for labor, public capital and private capital respectively. Actually Kanto or Tokyo region had highest labor productivity at 1956 and raised her income share.
Nextly an auxiliary variable is defined as a function of initial endowments of factors and parameters of production function and EIM's. It is proved that regional shares of this variable are preserved overtime, and lon-run income share is expressed by this variable. Thus other theorems result by expressing lon-run income share in terms of initial endowments and parameters, and partially differentiating by these parameters. Theorem four calculates elasticity of long-run income share to initial endowment of each factor. Thus elasticity of income share to private capital in Kanto was calculated as 0.6744, and approximately confirmed by simulation. Theorem five states that when EIM of k-th factor becomes bigger income share of j-th region increases if k-th factor is bigger than average and j-th region is k-th factor using overtime. Result is confirmed by an experiment of transfering initial capital in Kanto to other regions. Theorem six discusses the effect of self-financing policy of taxing a factor and subsidizing another factor, and concludes that long-run income share increases if subsidized factor has bigger EIM than taxed factor. Result is confirmed by an experiment of taxing private capital and subsidizing labor in Kanto region.
Thus these discussion suggests a taxonomy of multiregional economy into different types based upon absolute values and relations of EIM's. In each type, different trend of FPE emerges, and different policy is required for MIS. Thus discussion throws some lights to clarify similarities and dissimilarities of n countries international economy and n regions domestic economy.
