Balanced Growth of Open Economies under Variable Degree of Homogeneity

Abstract

Production processes of closed economies which can be described by possibly non-linear systems of first order difference equations

$$\underline y \left( {t + 1} \right) = \underline H \left( {\underline y \left( t \right)} \right)\quad \left( {t = 0,1,2 \ldots } \right)$$

((0.1))

were analysed by R.M. Solow and P.A. Samuelson [7]; their results, concerning existence and behavior of so-called balanced growth solutions have been generalized by J.F. Muth [4], D.B. Suits [9], M. Morishima [3], H. Nikaido [5] and [6], and the author [8]. H. Nikaido was the first who examined production processes of open economies which can be described by a non-autonomous system of (possibly non-linear) first order difference equations

$$\underline x \left( {t + 1} \right) = \underline H \left( {\underline x \left( t \right)} \right) + K\left( t \right)\underline a \quad \left( {t = 0,1,2 \ldots } \right)$$

((0.2))

where aεℝⁿ ₊ is a constant vector, a̱ ≥ 0̱ ¹⁾, and K is a real-valued function satisfying K(t) > 0, K(0) = 1; x_i(t) is the income of sector i in period t. For a detailed representation of the underlying model, in which are involved assumptions about production techniques and consumption behavior, see Nikaido [3].