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The existence of nontrivial equilibrium and poverty traps for a generalized Solow growth model with concave and nonconcave production functions is investigated. The explicit solutions of the growth model, which is expressed by a differential equation with corresponding boundary conditions, are employed to illustrate the spatial dynamics of the model in different economic regions. Numerical method is used to justify the validity of the theoretical analysis.

The distribution over space of spatial economic activities has been investigated in many literatures, in which economic geographers study how and why people make their location choices, consider the reasons of production agglomeration, and find the formation of cities and migration flows. Early regional economic growth models focus on capital, labor, pollution flows, individuals’ welfare, and the policymaker both in discrete and continuous cases. In recent years, the new economic geography emerges in economic analysis, where economic geographers employ a refined specification of whole market structures and several precise assumptions on the mobility of production factors. Fujita and Thisse [

Since the assumption of a continuous space structure in economic model fits better modern economies, several continuous space extensions of economic models have been discussed. Brito [

The classical Solow model describes the evolution in time of gross output which is based upon the input factors, labor, capital, and technology (see [

Precisely, the aim of this paper is to investigate the steady states of a generalized Solow model with both concave and nonconcave production functions and obtain the asymptotic properties of solutions for the model in continuous space and bounded time. Using the monotonic property of functions, we choose a function to show the steady state of the generalized Solow model with a concave production function. When production function is not concave, we introduce a nonconcave production function into the generalized Solow model to get the existence of nontrivial equilibrium and poverty traps. Our main contribution is to obtain the explicit solution of an ordinary differential equation which expresses the generalized Solow model in close regions and prove the existence and uniqueness of solution for a partial differential equation in open economy. The obtained result in our work shows the asymptotic capital distribution across space. This is different from those in [

This paper is organized as follows. Section

Assume that there is only one final good signed to be consumed or invested in one economy market. The generalized Solow model describes the evolution in time for gross output

The hypothesis of a concave production function has played a crucial role in many economic growth models based on intertemporal allocation. It describes the maximum output for all possible combinations of input factors and determines the way that the economic model evolves in time. Usually, a production function

Condition

By the change of capital per worker, or to be more precise, one conclusion of (

Assume that

Equation (

The steady state of the economy in Proposition

If

When

We focus on the existence of a poverty trap for (

At location

From (

Proposition

Assume that all regions are closed and there is are capital flows among the regions, which means that real transfers of goods between regions cannot be financed and there is no trade between regions. Furthermore, a mathematical representation of the assumption is that all locations have access to goods in modern economy (see [

Suppose that

By a change of variables

Assume that

Equation (

When capital and goods flow among open regions and there are no (intertemporal) adjustment costs, the aggregate balance equation for region

If we assume that the space interval is sufficiently large and there is no capital flow, we have the following Neumann boundary condition:

Suppose

Defining a sequence

In this section, numerical method is used to analyze problem (

Using classical difference quotient, we replace the second order space derivative by the following form:

we write (

In this paper, we have investigated a generalized Solow model with continuous space and bounded time. Introducing concave and nonconcave production functions into the generalized Solow model in close regions, we get the steady states of the model when several conditions are satisfied. The asymptotic properties of solutions for the generalized Solow model are proved. We obtain the explicit time path of capital per effective worker by solving an ordinary differential equation in close regions and prove the existence and uniqueness of the solution for the generalized Solow model in open regions. The obtained results show the asymptotic capital distribution across space. Discretizing the space variable, we employ numerical method for the system of partial differential equation to justify the validity of the theoretical analysis.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

The article is a joint work of two authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).