Asymptotic Behaviors Analysis of the Spatial AK Model with Trade Costs

We consider a spatial economic growth model with 
trade costs whose spatial-temporal dynamics of the capital stock are governed 
by a parabolic partial differential equation and analyze the effects of the 
trade costs on the asymptotic behaviors of the capital in the framework of 
spatial AK model, where the technology is time-varying. Taking advantage of the 
good properties of the Green function, we formulate an explicit solution to the 
auxiliary partial differential equation of the original problem and derive the 
sufficient conditions on the consumption function that guarantee the existence 
and the uniqueness of the solution and obtain the convergence properties of the 
capital in the long run for the spatial AK model with trade costs.


Introduction
Models of optimal development dealing jointly with time and space, which are regarded as a suitable vehicle for studying economic growth in a geographical context, were proposed in the seventies in (Isard & Liossatos, 1979). The dynamic spatial economic theory that incorporates space into the dynamic analysis of economic growth is developed in recent years. (Zou & Camacho, 2004) present a spatial extension of the Solow model in which the evolution of capital is governed by a parabolic partial differential equation and prove the existence and convergence of the solution to the equation. (Brito, 2004) extends the traditional neoclassical growth theory to a spatial growth theory which would provide models incapable of approximating spatial heterogeneity across regions and studies the asymptotic distribution and the local stability properties of the solution of How to cite this paper: Hu, H. L. (2021). Asymptotic Behaviors Analysis of the Spa-H. L. Hu DOI: 10.4236/tel.2021.113040 604 Theoretical Economics Letters the corresponding optimal control problem. (Camacho et al., 2008) develop a numerical algorithm to study the dynamics of capital accumulation across space. (Boucekkine et al., 2009) discuss the spatial Ramsey model with the linear utility case, which reconciles growth and geographical economics. (Boucekkine et al., 2013) formulate and analyze an optimal control problem for an AK model with spatial variations by using the dynamic programming approach, and prove that spatio-temporal dynamics lead to the convergence of time-detrended capital stocks across space in the long run. (Juchem Neto & Claeyssen, 2015) present stability analyses and numerical simulations on the spatial Solow model with capital-induced labor migration. (Fabbri, 2016) investigates the role of geography in the evolution of the spatial growth model with a general geographical structure modeled in Riemannian manifold and gives the conditions that guarantee convergence of the detrended capital across locations in the long run. (Ballestra, 2016) shows that the Pontryagin maximum principle is capable of analyzing the spatial AK model under a Michel-type transversality condition. (Aldashev et al., 2014) propose a complementary approach to analyze the asymptotic convergence of the spatial AK model, which possesses the time-varying technology parameter. More literature on spatial dynamics is referred to a survey by (Desmet & Rossi-Hansberg, 2010).
Factor mobility and transport costs are two key ingredients that set apart the new economic geography from the traditional trade theory. The implications of factor mobility for trade and the spatial structure of the economy have been analyzed in (Brito, 2004;Boucekkine et al., 2009;Boucekkine et al., 2013). However, many researchers assume that perfect mobile holds. The transport costs or other potential trade barriers, which are related to spatial heterogeneities, have always been neglected. The simplifying assumptions on trade costs in models of new economic geography causing serious consequences for the regional specialization and economic activity agglomeration are shown in (Behrens & Picard, 2011).
Inspired by the work in (Juchem Neto et al., 2018), which introduces transport costs into the spatial Solow model and discusses the effects on the dynamic behavior, we consider an extension of the spatial AK model presented in (Boucekkine et al., 2013). The mobile frictions like the transport costs or trade barriers on the mobility of the capital are added in our model, which is different from that in (Boucekkine et al., 2013). Another difference is that the technology is assumed to depend on time t. Due to the heterogeneity of the geographic distribution of capital, the time-space connection is modeled by assuming that the capital flows in the opposite direction of the gradient of the capital distribution across space, which derives that the spatio-temporal evolution of capital is expressed by a parabolic partial differential equation. The purpose of the paper is to find a non-negative classical solution of the partial differential equation that describes the dynamic of capital stock. Through formulating the explicit solution to an auxiliary parabolic partial differential equation by the approach in (Aldashev et  The rest of the paper is organized as follows. Section 2 introduces the spatial AK model with trade costs. Section 3 develops our analytical results. Section 4 concludes and gives perspectives for future research.

The Model
Following the works of (Boucekkine et al., 2013), we assume that the population is non-growing and spatially homogeneous distributed on unit circle where  is the set of spatial parameters θ under polar coordinates. Capital stock moves from locations of high concentration to locations of low concentration, which is consistent with diminishing returns to capital since the high concentration implies low marginal productivity. According to the literatures (Brito, 2004), the spatial movements of the stock of capital across locations are modeled through a trade balance approach.

At a given point ( ) [ )
, c t θ and study the asymptotic behavior of the stock of capital. Assuming that the consumption function ( ) , c t θ is smooth, bounded, concave with respect to the spatial variable for any time, we have the following result.
Assume that the consumption function and initial capital distribution satisfy respectively. The problem (7) admits a unique non-negative classical solution in Ω .

Proof of Theorem 1
Before solving the problem (7), we consider an auxiliary problem in the follow- Then the equation , is equivalent to system (7).
Noting that if we choose 0 b = , there will be no trade costs in the economy.
In this case, the auxiliary problem is exactly the system considered in (Aldashev et al., 2014). We intend to remove the term ( ) ( ) , A t h t θ in Equation (10) by using the following Lemma.
Lemma 3. Let ( ) , g t θ be a positive function defined in Ω , which is related with initial condition ( ) ( ) Combining with Equation (10) which implies that Equation (15) holds.
Proposition 4. For any positive integer n, define the Green function which is a formal solution of Equation (15).
Proof. Observing that the Green function ( ) , , G t θ ξ satisfies the following homogeneous partial differential equation  Considering the above-mentioned results and using Lemmas 2 and 3, we conclude that is the unique classical solution of Equation (7).
Proposition 6. Let ( ) , g t θ be the classical solution of Equation (15) with initial condition ( ) ( ) Then the capital  Thisse and Fujita (2002) who believe that agglomeration is a result of the trade-off between transport costs and increasing returns.
If b → +∞ , the discounted capital is exhausted in the long run and resources can not move in because of the high trade costs. If the trade costs coefficient 0 b = , the initial spatial inequality of the capital disappears and the growth dynamics which satisfies the condition (9) lead to spatial convergence in capital over time. If 0 b < , the trade cost is negative, we can take the situation as there exist preferential policies, that the local governments provide subsidies to attract investments and promote the economic development.

Conclusion
In the present work, we introduce the trade costs into the spatial AK model whose spatial-temporal dynamics of the capital stock are governed by a parabolic partial differential equation. Taking advantage of the good properties of the Green function, we formulate an explicit solution to the auxiliary partial differential equation of the original problem and derive the sufficient conditions on the consumption function that guarantee the existence and the uniqueness of the solution. We also analyze the effects of the trade costs on the asymptotic behaviors of the capital in the framework of spatial AK growth model, where the technology is time-varying. In addition, the technology might depend not only on time but also on space. The spatial heterogeneities of technology A could be taken into account for future research, which may reflect spatial externality associated with location.