Closed-Form Solution for the Solow Model with Constant Migration

Abstract

ABSTRACT. In this work we deal with the Solow economic growth model, when the labor force is ruled by the Malthusian law added by a constant migration rate. Considering a Cobb-Douglas production function, we prove some stability issues and find a closed-form solution for the emigration case, involving Gauss' Hypergeometric functions. In addition, we prove that, depending on the value of the emigration rate, the economy could collapse, stabilize at a constant level, or grow more slowly than the standard Solow model. Immigration also can be analyzed by the model if the Malthusian manpower is declining.

An Economic Model for OECD Economies with Truncated M-Derivatives: Exact Solutions and Simulations

“…Here, t represents time, Y is the national production, K and L are state variables that denote, respectively, the quantities of capital and labour factors used in production (both measured with appropriate units). Meanwhile, A represents a technological constant that is usually interpreted as the total productivity of all factors [10]. Obviously, the SM indicates that national production is the result of combining capital, labor and certain technological constant, for any period of time t. Moreover, the function Y must satisfy properties which are typical in a production function, as required in the following definition.…”

“…Definition 4. A function Y : (0, ∞) → R is a production function if it satisfies the following properties [10], which are known in Economics as the Inada conditions:…”

“…In this section, we analyse the case without migration and with a Malthusian law, considering the population as labor force. Our approach will be similar to that followed in [10].…”

“…where the parameters s and δ are the savings constant and the rate of depreciation of capital, respectively. Neoclassically, sY can be taken as the gross investment and δK is the capital depreciation of the entire economy [10]. For the remainder of this work, we will use the notation L…”

“…In addition to the classical SM, factors such as migration have been considered to determine the dynamics of capital and production. In [10], a workforce governed by a Malthusian law adding a constant migration is proposed. Under a Cobb-Douglas production function and depending on the migration value, it can be observed that the economy can collapse, stabilize or grow more slowly than in the traditional SM.…”