# The Dynamics of a Spatial Economic Model with Bounded Population Growth

Authors: Yue Zhong
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

We investigate a spatial economic growth model with bounded population growth to obtain the asymptotic behavior of detrended capital in a continuous space. The formation of capital accumulation is expressed by a partial differential equation with corresponding boundary conditions. The capital accumulation interacts with the morphology to affect the optimal dynamics of economic growth. After redrafting the spatial growth model in the infinite dimensional Hilbert space, we identify the unique optimal control and value function when the bounded population growth is considered. With nonnegative initial distribution of capital, the explicit solution of the model is obtained. The time behavior of the explicit solution guarantees the convergence issue of the detrended capital level across space and time.


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## Body

## 1. Introduction

Distribution of economic activities across space and time has been widely investigated in many literature [1–5], where economic geographers apply economic growth models to analyze the relationship between production agglomeration and location choices of people, reasons for migration flows, and formation of cities. In early studies of regional economic growth models, Beckmann [6] and Isard and Liossatos [7] discussed flows of labor and pollution, capital accumulation, and individuals welfare in discrete and continuous spaces, respectively. Recently, as one important issue of economic research studies, optimal market allocation has been considered in economic growth models with capital accumulation. Among relevant economic literature, Brito [8] has investigated the optimal dynamics of spatial capital accumulation in a continuous space. In the work of Brito [8], optimal consumption distribution is associated with the capital accumulation, which drives the dynamic processes of spatial capital growth and distribution. Maximizing an objective function in a continuous dimension, Boucekkine et al. [9] illustrated how and why policy makers choose optimal trajectories of the capital and consumption in a spatial Ramsey model. Considering a cyclic space, Boucekkine et al. [10] described characterizations of the optimal capital dynamics in the spatial Ramsey model with AK production function, which indicates that technology adoption decisions are closely tied to technological embodiment, capital, and goods. In a generic geographic structure, Fabbri [11] generalized the spatial AK economic growth model presented in [10] to obtain a generalized qualitative behavior of the economy.In global economic growth systems, several factors affect the formation of capital accumulation. These factors include population growth, environmental protection, and different production processes. For example, Guerrini [12] studied the one-dimensional Ramsey model with a population growth factor. The population growth rate is considered to be a variable, which follows the logistic growth law. La Torre et al. [13] discussed the interplay of pollution diffusion and capital accumulation in economic geography models, where the dynamics of spatial capital is driven by the pollution diffusion. In Kamihigashi and Roy [14], a discontinuous production function is applied to show the convergence of the capital stock when an optimal growth problem with the condition of discrete time is considered. Brianzoni et al. [15] considered the optimal dynamics of capital accumulation in a discrete economic model with one sector, in which nonconcave production factors lead to the complexity of capital stock in local and global economic systems. Capasso et al. [16] analyzed the relationship between industrial dynamics and economic geography to enrich the theory of spatial economy.In general, it is very hard to obtain the explicit solutions of spatial economic growth systems. Thus, the equilibrium distribution of spatial capital is widely considered in many research studies. Camacho and Zou [17] discussed existence of the solution for a spatial growth system with local diffusion and obtained convergence properties of capital distribution in the long run. Under assumption of a nonconcave production function, Capasso et al. [18] investigated the steady state of a spatial Solow model with technological diffusion. Brock et al. [19] applied a mathematical method to study spatial spillover in an economic growth model, in which capital mobility is excluded since the diffusion term disappears from the state equation. In the long run, Xepapadeas and Yannacopoulos [20] illustrated stable distribution of per capita capital across space. The stable capital distribution is related to the growth process of economy and shows nonhomogeneous properties. In autarkic and open regions, Yue and Wenyi [21] depicted the formation of capital distribution by applying the equilibrium solution of a generalized growth model, respectively. For more contributions to the study of growth models, the reader is referred to [22–29] and the references therein.In this paper, we introduce a bounded population growth factor into the spatial economic growth model presented in [10, 11] and analyze the optimal dynamics of the model in a continuous space. In our model, the optimal behavior of the economic growth is affected by the interaction between the morphology and the dynamics of spatial capital. The long-run convergence of the spatial capital is closely associated with the bounded population growth factor and geographical environment. After redrafting the spatial growth model in the infinite dimensional Hilbert space, we use the Hamilton-Jacobi-Bellman (HJB) equation and a linear production function to derive an explicit value function of the model. The value function is used to confirm the optimal control of the model in feedback form. Due to application of the optimal control, an explicit solution of the growth system is obtained in close form, which satisfies first-order optimality conditions to describe the time behavior of per capita capital. With nonnegative initial distribution of capital, we apply the explicit solution and Fourier series to express spatial detrended capital of our model. The asymptotic behavior of the spatial detrended capital distribution is proved to show that the capital distribution converges to a homogeneous result in the long run.In particular, we extend parts of works in [10, 11] by embedding the bounded population growth factor into the spatial growth model. In our discussion, the bounded population growth is one determinant of the spatial capital dynamics. When the bounded population growth rate is considered, we identify the unique optimal control and value function, which are used to derive the explicit solution of the model. Compared with the works in [10, 11], we apply the explicit solution and Fourier series to obtain an expression of the spatial detrended capital which helps to prove the long-run convergence of the spatial detrended capital. This is the main contribution in our work. In addition to referring the research methods in [10, 11], our work is also related to the study of Chatterjee [30], where the optimal control problem of a growth model with heterogeneous initial endowments is solved.This paper is organized as follows. Section2 states the growth model with bounded population growth and special spatial setting. In Section 3, the spatio-temporal capital dynamics of the growth model is analyzed by identifying the unique optimal control and the explicit value function. In Section 4, we apply the time behavior of per capita capital in close form to prove convergent properties of the spatial detrended capital distribution. Some numerical results are illustrated in Section 5, and our conclusions are presented in Section 6.


## 2. A Spatial Growth Model with Bounded Population Growth

### 2.1. A Model with Bounded Population Growth

LetΩ be an d-dimensional geographic structure (d = 2 or 3) and without boundary. We consider that economic activities are distributed homogeneously in geography Ω. Assume that x∈Ω is a generic spatial variable and kx,t denotes spatial capital stock of the individual located at x and time t≥0. The population stock Lx,t is considered to follow a logistic process:(1)L˙x,t=a−bLx,tLx,t,for all x,t∈Ω×0,T, where L˙x,t=∂Lx,t/∂t and a>b>0. The population growth rate is denoted by Nx,t=L˙x,t/Lx,t. Thus, logistic equation (1) implies the upper and lower limits to the population growth rate which satisfies a hypothesis of the bounded population growth rate [12].In open regions, we ignore intertemporal adjustment costs caused by the flows of capital and goods. When there is no capital depreciation, we have an aggregate balance equation in the geographyΩ, which is given by(2)∫Ω∂kx,t∂t−Akx,t+Nx,tkx,t+ix,t+cx,tdx=0.In equation (2), cx,t>0 denotes the consumption of the individual at point x and time t, and ix,t≠0 is the net trade balance of individual at x and time t. Note that, in this paper, the production function is considered as the linear function (A is a positive constant). The trade ix,t is symmetry of capital account balance that is matched by reallocations of the capital in different regions. In addition, we assume that there is no inter-regional arbitrage opportunities among regions. From equation (2), we obtain the budget constraint which is written as(3)∂kx,t∂t=Akx,t−Nx,tkx,t−ix,t−cx,t,Ω×0,T.Following the idea in [8], we assume that there are no institutional barriers when capital and goods flow from one region to others. In this case, the net trade balance ix,t is determined by different capital intensities among regions. Then, for a region E that is a bounded open subset of Ω, the trade balance ix,t satisfies(4)∫Eix,tdx=−∫∂E∂kix,t∂n^dx.Applying the divergence theorem to equation (4), we get(5)∫∂E∂kx,t∂n^dx=∫E∇⋅∇kx,tdx=∫E△xkx,tdx.Thus, we obtain(6)ix,t=−△xkx,t.Substituting equation (6) into equation (3), we have the following system:(7)∂kx,t∂t=△xkx,t+Akx,t−Nx,tkx,t−cx,t,x,t∈Ω×0,T,kx,0=k0x>0,x∈Ω.Note that, different from the model presented in [10, 11], the population growth factor (the term Nx,tkx,t) in system (7) describes the impact of the population growth on capital accumulation.In order to illustrate the optimal dynamics of economic activities, a controlc⋅,⋅ is chosen to maximize utility functional:(8)Jk0,c⋅,⋅≔∫0+∞e−ρt∫Ωcx,t1−1/σ1−1/σdxdt,where the constants ρ>0 and σ>1. For a given initial distribution k0⋅ of the capital, the value function of system (7) is defined as(9)Vk0≔supc⋅,⋅Jk0,c⋅,⋅.In equation (9), the supremum is associated with a suitable control c⋅,⋅ which ensures that spatial capital is nonnegative at any point x and time t.


### 2.2. The Hilbert Space Setting

For convenience, we state the Hilbert space setting to rewrite the optimal control problem (8). Let L2Ω denote the Hilbert space, in which we describe the dynamics of spatial capital by the adjusted model. In L2Ω, the scalar product of f and g is defined as(10)f,g=∫Ωfxgxdx,and the norm of the function is given by(11)fL2Ω≔f,f1/2=∫Ωf2xdx1/2.In addition, the distance between two elementsf,g∈L2Ω is denoted by f−gL2Ω. For time t∈0,T, space variables k⋅,t and c⋅,t are considered as elements of the Hilbert space L2Ω. In order to depict the second derivative of functions in L2Ω, we express operator G as(12)Gf=△xf,where G is the generator of the heat semigroup, and it is also a C0 semigroup on L2Ω [31, 32]. We denote a subset of L2Ω by DG that is the domain of the operator G. DG is denoted by the Sobolev space H2Ω, which is the completion for the norm fH2≔∑i=02∫Ω∇ifx2dx1/2 in the space of the C∞ functions [31]. Thus, the functions in DG have first and second derivatives in the domain L2Ω.Since the population growth rateNx,t satisfies the logistic equation allowing the limits to the population growth, for simplifying calculation, we denote the logistic population growth rate Nx,t by n which is independent of t, and we verify the lower and upper limits of the population growth rate in Proposition 1 and Theorem 3, respectively. Based on the previous analysis, in the Hilbert space L2Ω, system (7) is rewritten as the following form:(13)∂kt∂t=Gkt+Akt−nkt−ct,t≥0,k0=k0,where ct and kt are elements of L2Ω for any t≥0. Note that, ct and kt are considered as the functions of space variable x and expressed by ctx and ktx, respectively. That is to say, ct and kt still denote consumption and capital at time t with space point x, respectively [10, 11]. Therefore, we write ctx≡cx,t and ktx≡kx,t in the sequel.We apply the notationLloc2L2Ω;0,+∞ to denote a function space, which reads(14)Lloc2L2Ω;0,+∞≔f:Ω×0,+∞⟶R:∫0T∫Ωfx,t2dxdt<∞,∀T>0,and express the set of admissible controls as(15)Uk0≔c∈Lloc2L2Ω;0,+∞:ctx,ktx≥0,∀x,t∈Ω×0,+∞.Thus, value function (9) is rewritten as(16)Vk0=supc⋅∈Uk0Jk0,c⋅.Remark 1.
Choosingc⋅∈Uk0 for an initial capital k0∈L2Ω, we know that kx,t is the solution of system (7) if the solution of system (13) is denoted by kt. In addition, it is not necessary to ensure that the initial distribution and the control are regular. In this case, the solution of system (7) is not required to be C2 for the space variable and C1 for the time variable. Therefore, the unique solution of system (7) is obtained in other form, i. e., it belongs to CL2Ω;0,+∞ [32, 33]. We define function 1:Ω⟶R in L2Ω, which is a constant equal to 1 in Ω, that is,(17)1x≡1.
Thus, utility functional (8) is rewritten as the form(18)Jk0,c⋅,⋅≔∫0+∞e−ρt∫Ωcx,t1−1/σ1−1/σdxdt,=∫0+∞e−ρt1,Uctdt,where function Uθx=θx1−1/σ/1−1/σ for a given θ∈L2Ω.


## 2.1. A Model with Bounded Population Growth

LetΩ be an d-dimensional geographic structure (d = 2 or 3) and without boundary. We consider that economic activities are distributed homogeneously in geography Ω. Assume that x∈Ω is a generic spatial variable and kx,t denotes spatial capital stock of the individual located at x and time t≥0. The population stock Lx,t is considered to follow a logistic process:(1)L˙x,t=a−bLx,tLx,t,for all x,t∈Ω×0,T, where L˙x,t=∂Lx,t/∂t and a>b>0. The population growth rate is denoted by Nx,t=L˙x,t/Lx,t. Thus, logistic equation (1) implies the upper and lower limits to the population growth rate which satisfies a hypothesis of the bounded population growth rate [12].In open regions, we ignore intertemporal adjustment costs caused by the flows of capital and goods. When there is no capital depreciation, we have an aggregate balance equation in the geographyΩ, which is given by(2)∫Ω∂kx,t∂t−Akx,t+Nx,tkx,t+ix,t+cx,tdx=0.In equation (2), cx,t>0 denotes the consumption of the individual at point x and time t, and ix,t≠0 is the net trade balance of individual at x and time t. Note that, in this paper, the production function is considered as the linear function (A is a positive constant). The trade ix,t is symmetry of capital account balance that is matched by reallocations of the capital in different regions. In addition, we assume that there is no inter-regional arbitrage opportunities among regions. From equation (2), we obtain the budget constraint which is written as(3)∂kx,t∂t=Akx,t−Nx,tkx,t−ix,t−cx,t,Ω×0,T.Following the idea in [8], we assume that there are no institutional barriers when capital and goods flow from one region to others. In this case, the net trade balance ix,t is determined by different capital intensities among regions. Then, for a region E that is a bounded open subset of Ω, the trade balance ix,t satisfies(4)∫Eix,tdx=−∫∂E∂kix,t∂n^dx.Applying the divergence theorem to equation (4), we get(5)∫∂E∂kx,t∂n^dx=∫E∇⋅∇kx,tdx=∫E△xkx,tdx.Thus, we obtain(6)ix,t=−△xkx,t.Substituting equation (6) into equation (3), we have the following system:(7)∂kx,t∂t=△xkx,t+Akx,t−Nx,tkx,t−cx,t,x,t∈Ω×0,T,kx,0=k0x>0,x∈Ω.Note that, different from the model presented in [10, 11], the population growth factor (the term Nx,tkx,t) in system (7) describes the impact of the population growth on capital accumulation.In order to illustrate the optimal dynamics of economic activities, a controlc⋅,⋅ is chosen to maximize utility functional:(8)Jk0,c⋅,⋅≔∫0+∞e−ρt∫Ωcx,t1−1/σ1−1/σdxdt,where the constants ρ>0 and σ>1. For a given initial distribution k0⋅ of the capital, the value function of system (7) is defined as(9)Vk0≔supc⋅,⋅Jk0,c⋅,⋅.In equation (9), the supremum is associated with a suitable control c⋅,⋅ which ensures that spatial capital is nonnegative at any point x and time t.


## 2.2. The Hilbert Space Setting

For convenience, we state the Hilbert space setting to rewrite the optimal control problem (8). Let L2Ω denote the Hilbert space, in which we describe the dynamics of spatial capital by the adjusted model. In L2Ω, the scalar product of f and g is defined as(10)f,g=∫Ωfxgxdx,and the norm of the function is given by(11)fL2Ω≔f,f1/2=∫Ωf2xdx1/2.In addition, the distance between two elementsf,g∈L2Ω is denoted by f−gL2Ω. For time t∈0,T, space variables k⋅,t and c⋅,t are considered as elements of the Hilbert space L2Ω. In order to depict the second derivative of functions in L2Ω, we express operator G as(12)Gf=△xf,where G is the generator of the heat semigroup, and it is also a C0 semigroup on L2Ω [31, 32]. We denote a subset of L2Ω by DG that is the domain of the operator G. DG is denoted by the Sobolev space H2Ω, which is the completion for the norm fH2≔∑i=02∫Ω∇ifx2dx1/2 in the space of the C∞ functions [31]. Thus, the functions in DG have first and second derivatives in the domain L2Ω.Since the population growth rateNx,t satisfies the logistic equation allowing the limits to the population growth, for simplifying calculation, we denote the logistic population growth rate Nx,t by n which is independent of t, and we verify the lower and upper limits of the population growth rate in Proposition 1 and Theorem 3, respectively. Based on the previous analysis, in the Hilbert space L2Ω, system (7) is rewritten as the following form:(13)∂kt∂t=Gkt+Akt−nkt−ct,t≥0,k0=k0,where ct and kt are elements of L2Ω for any t≥0. Note that, ct and kt are considered as the functions of space variable x and expressed by ctx and ktx, respectively. That is to say, ct and kt still denote consumption and capital at time t with space point x, respectively [10, 11]. Therefore, we write ctx≡cx,t and ktx≡kx,t in the sequel.We apply the notationLloc2L2Ω;0,+∞ to denote a function space, which reads(14)Lloc2L2Ω;0,+∞≔f:Ω×0,+∞⟶R:∫0T∫Ωfx,t2dxdt<∞,∀T>0,and express the set of admissible controls as(15)Uk0≔c∈Lloc2L2Ω;0,+∞:ctx,ktx≥0,∀x,t∈Ω×0,+∞.Thus, value function (9) is rewritten as(16)Vk0=supc⋅∈Uk0Jk0,c⋅.Remark 1.
Choosingc⋅∈Uk0 for an initial capital k0∈L2Ω, we know that kx,t is the solution of system (7) if the solution of system (13) is denoted by kt. In addition, it is not necessary to ensure that the initial distribution and the control are regular. In this case, the solution of system (7) is not required to be C2 for the space variable and C1 for the time variable. Therefore, the unique solution of system (7) is obtained in other form, i. e., it belongs to CL2Ω;0,+∞ [32, 33]. We define function 1:Ω⟶R in L2Ω, which is a constant equal to 1 in Ω, that is,(17)1x≡1.
Thus, utility functional (8) is rewritten as the form(18)Jk0,c⋅,⋅≔∫0+∞e−ρt∫Ωcx,t1−1/σ1−1/σdxdt,=∫0+∞e−ρt1,Uctdt,where function Uθx=θx1−1/σ/1−1/σ for a given θ∈L2Ω.


## 3. Spatio-Temporal Capital Dynamics

In this section, we use the dynamic programming method to analyze spatio-temporal dynamics of the capital inL2Ω. Since optimal control problem (8) is equivalent to maximizing (18) subject to system (13), we concentrate our attentions on HJB equation related to system (13) to discuss the optimal control and explicit value function of system (13). The HJB equation of system (13) is given by(19)ρvk=k,G▽vk+Ak,▽vk−nk,▽vk+supc∈L2T:R+−c,▽vk+1,Uc,where the function v:L2Ω⟶R, i.e., for any k∈L2Ω.In equation (19), the term k,G▽vk equals Gk,▽vk since G is self-adjoint. According to the concept of Gateaux derivative, the gradient of the function vk (denoted by ▽vk) is an element of L2Ω. Then, the gradient ▽vk at a certain point x is computed in k∈L2Ω. Furthermore, since ▽vk∈L2Ω, we obtain G▽vk at the point x when ▽vk is in the domain DG [34]. Similar results appear in the standard dimensional case. This fact allows us to expect that the solution vk of the HJB equation equals the value function of the infinite dimensional problem.LetM≔k∈L2Ω:k,1>0 be an open set. We consider a continuous function v∈C1M with continuous differential, which means that the gradient ∇v:M⟶L2Ω is continuous in L2Ω. Now, in order to obtain the optimal control of system (13) in feedback form, we identify an explicit solution (denoted by v) of equation (19) on M in the following proposition.Proposition 1.
On the half space≔k∈L2Ω:k,1>0, if the population growth rate n satisfies(20)n>A+σρ1−σ,then there exists an explicit solution of HJB equation (19), which is given by(21)vk=μk,11−1/σ,where(22)μ=σσ−1σρ−A−nσ−1volΩ−1/σ,for volΩ≔∫Ω1dx.Proof.
Consideringvk=μk,11−1/σ for a positive real number μ, we have(23)▽vk=μ1−1σk,1−1/σ1.
Since▽vk belongs to the domain DG and G▽vk:M⟶L2Ω is continuous for any k∈M, then substituting vk=μk,11−1/σ into equation (19), we obtain (24)ρμk,11−1/σ=μ1−1σk,1−1/σk,G1+Aμ1−1σk,1−1σk,1,−nμ1−1σk,1−1/σk,1+supc≥0−μ1−1σk,1−1σc,1+1,Uc.
Due toG1=0, the supremum is obtained if c=μ1−1/σ−σk,11 in equation (24). Thus, we write equation (24) as (25)ρμk,11−1/σ=Aμ1−1σk,1−1/σk,1−nμ1−1σk,11σk,1−volΩμ1−1σμ1−1σ−σk,11−1/σ+volΩσμ1−1/σ−σk,11−1/σσ−1,where volΩ≔∫Ω1dx. To simplify μk,11−1/σ (a nonzero factor) in equation (25), we obtain(26)ρ=A−n1−1σ−volΩ1−1σμ1−1σ−σ+volΩμ1−1σ−σ,which reads(27)μ=σσ−1σρ−A−nσ−1volΩ−1/σ.
Then, we obtain an explicit solution of HJB equation (19), which is of the form(28)vk=σσ−1σρ−A−nσ−1volΩ−1/σk,11−1/σ.
On the contrary, the trajectoryk⋅ related to a control c⋅ in set (15) is the solution of the following system:(29)k˙t=Gkt+Akt−nkt−ct,k0=k0,where k˙t≔∂kt/∂t. At every time t and point x, we observe that k⋅ remains below:(30)k^t=eA−nteGtk^0,where k^t is the solution of the following system:(31)k^˙t=Gk^t+Ak^t−nk^t,k^0=k^0.
Then, we havek^t,1≥kt,1 for all t>0. Since(32)k^t,1=eA−nteGtk^0,1=eA−ntk^0,eGt1=eA−ntk^0,1,we have a bounded utility:(33)e−ρtvkt=e−ρtμkt,11−1/σ≤e−ρtμk^t,11−1/σ≤e−ρteA−n1−1σtμk^0,11−1/σ.
Thus, the bounded utilitye−ρtvkt⟶0 if(34)A−n1−1σ−ρ<0.
Inequality (34) yields(35)n>A+σρ1−σ,and otherwise, e−ρtvkt⟶+∞.Remark 2.
In Proposition1, condition (20) is applied to guarantee the finiteness of the objective function. Indeed, we also obtain the finiteness of the value function with condition (20) because of explicit expression (21). In the following proposition, we illustrate that the feedback control is admissible and prove that the solution of HJB equation (19) obtained in Proposition 1 is the value function. Furthermore, the optimal control of problem (18) is derived by employing the value function.Proposition 2.
Assume that the hypotheses of Proposition1 are satisfied. Then, the feedback control c=μ1−1/σ−σk,11 is the unique optimal control of system (13) and the value function of system (13) computed at k0 is(36)vk0=σσ−1σρ−A−nσ−1volΩ−1/σk0,11−1/σ.Proof.
The feedback control linked to the solution of HJB equation (19) is(37)Φ:M⟶L2Ω,Φk∈argmaxc∈L2Ω−μ1−1σk,1−1σc,1+1,Uc=μ1−1σ−σk,11=σρ−A−nσ−1volΩk,11.
Choosingk0∈L2Ω and replacing ct by the feedback control Φkt in system (29), we obtain(38)k˙t=Gkt+Akt−nkt−σρ−A−nσ−1volΩk,11,k0=k0.
According to the definition of mild solution in Bensoussan et al. [32], we have a mild solution of system (38), which is given by(39)kΦt=eG˜tk0−∫0teG˜t−sσρ−A−nσ−1volΩkΦs,11ds,where G˜≔G+A−n is the identity operator.
Note that, due to examples in [32], we know that equation (39) has a unique solution even if it is not an explicit form. We denote the unique solution of equation (39) by kΦ∗t, which is the related trajectory driven by the feedback control c∗ with c∗t≔ΦkΦ∗t for all t>0. Under the hypothesis of positive kΦ∗tx, c∗t is positive and admissible.
In general, we know thatc∗⋅ is the optimal control if we obtain(40)Jk0,c∗⋅≥Jk0,c¯⋅.
In equation (40), c¯⋅ denotes any other admissible control of (15). Supposing that k¯⋅ is the trajectory linked to the control c¯⋅ and considering function wt,k:R×L2Ω⟶R with wt,k≔e−ρtvk, we have (41)vk0−wT,k¯T=wt,k¯0−wT,k¯T=−∫0Tddtwt,k¯tdt=∫0Te−ρtρvk¯t−Gk¯t+A−nk¯t−c¯t,▽vk¯tdt.
Regularizing properties of the heat semigroup, we know thatk¯t∈DG for any t>0. Using inequality (33), we know that wT,k¯T⟶0,T⟶∞. Thus, we obtain (42)vk0=∫0+∞e−ρtρvk¯t−A−nk¯t−c¯t,▽vk¯t−k¯t,G▽vk¯tdt.
Considering the solutionvk of equation (19), we have (43)vk0−Jk0,c¯⋅=∫0+∞e−ρtρvk¯t−Ak¯t,▽vk¯t+nk¯t,▽vk¯t−k¯t,G▽vk¯t+c¯t,▽vk¯t−1,Uc¯tdt=∫0+∞e−ρtsupc∈L2T:R+1,Uc−c,▽vk¯t−1,Uc¯t−c¯t,▽vk¯tdt≥0,which results in(44)vk0−Jk0,c¯⋅≥0.
Expression (44) implies that there exists(45)vk0−Jk0,c∗⋅=0,where c∗⋅ is a control defined with the feedback in (37). Therefore, for all admissible control c¯⋅, we obtain(46)vk0−Jk0,c¯⋅≥vk0−Jk0,c∗⋅,which leads to(47)Jk0,c¯⋅≤Jk0,c∗⋅.
Inequality (47) shows that c∗⋅ is the optimal control.


## 4. Convergence Analysis of Capital Distribution

In the long run, the willingness of the planner induces the spatio-temporal dynamics to give a certain consumption which equalizes the capital level in different locations. Thus, the spatio-temporal dynamics eliminates the initial inequalities in capital endowments. This result on decreasing returns plays a key role in the traditional theory of convergence. In this section, we discuss the uniform convergence of the detrended capital distribution over space by applying the explicit solution of system (29).Theorem 1.
Assume that the hypotheses of Proposition1 are satisfied. Then, there exists a closed-form solution of system (29), which is given by(48)kt=k0eG+σA−σn−σρt,where k0=c0/σρ+A−n1−σ.The optimal consumption dynamics is an elementary factor of the dynamics programming method, which is used to maximize the utility function and obtain the behavior of optimal capital distribution. Thus, we apply the optimal consumption to complete proof of Theorem1.Proof.
According to the feedback controlc=μ1−1/σ−σk,11 obtained in Proposition 3.2, we have(49)c,1=μ1−1σ−σk,11,1=μ1−1σ−σk,11,1.
By calculation, we get(50)∫Ωctdx=∫Ωσρ+A−n1−σktdx.
Then, we have(51)ct=σρ+A−n1−σkt.
Plugging equation (51) into system (29) and using equation (39), we get the explicit solution of system (29), which is(52)kt=eGteAte−ntk0−∫0tet−sGet−sAe−t−sncsds=eGteAte−ntk0−c0σρ+A−n1−σeG+A−nt+c0σρ+A−n1−σeGt+σA−n−ρt=k0eG+σA−σn−σρt,where k0=c0/σρ+A−n1−σ. Equation (52) describes the time behavior of per capita capital in system (29).
According to the works in [10], we know that the behavior of spatial capital distribution implies composite transitional dynamics because of the second order term △xkx,t in system (7). Precisely, the aggregate level of capital distribution in the spatial model is equivalent to that in corresponding form of the one-dimensional model. In the following theorems, we discuss the characterization of aggregate capital with the optimal control obtained in Proposition 2.Theorem 2.
Under the assumptions of Proposition1, there exists per aggregate capital Kt with the optimal control, which is given by(53)Kt=eβtK0,where β≔σA−n−ρ and K0=∫Ωk0xdx.Proof.
Using closed-form solution (48), we obtain(54)Kt=kt,1=k0eG+σA−σn−σρt,1=eσA−n−ρtk0,eGt1=eσA−n−ρtk0,1=eσA−n−ρt∫Ωk0dx=eσA−n−ρtK0=eβtK0,where β≔σA−n−ρ and K0=∫Ωk0xdx.Remark 3.
According to the evolution of optimal aggregate capital stock in the spatial growth model, we find that the aggregate capital grows with an explicit growth rateβ from K0. By (51) and (53), we obtain the relationship between the optimal control c∗⋅ and the explicit growth rate β, which means that, for all individuals, the consumption level chosen by planners is independent of their location and generation. In the following theorem, we analyze the induced optimal spatio-temporal dynamics of detrended capital when the bounded population growth rate satisfies certain conditions.Theorem 3.
Suppose that the hypotheses of Proposition1 hold. Then, for a real number r1, if(55)n<A+ρσ1−σ+σr12σ−1,the detrended capital(56)kDx,t≔e−βtkx,t,converges uniformly to a constant K0/volΩ when t tends to infinity.In Theorem3, the optimal spatial dynamics of capital, which is induced by the optimal consumption obtained in Proposition 2, guarantees the asymptotic behavior of detrended capital. Inequalities in initial endowments are eliminated by the spatial dynamics of capital to equalize the level of capital across locations in the long run. This fact is due to the explicit expression of the capital in Theorem 1 and a threshold value of the population growth rate (n<n¯=A+ρσ/1−σ+σr12/σ−1). In the sequel, we provide the proof of Theorem 3 by employing corresponding form of Fourier series.Proof.
Firstly, we writekDx,t by using Fourier series. For j∈Z, we denote the eigenfunction of △x by a regular function ϕj:Ω⟶R, such that(57)△xϕjx=−rj2ϕjx.
In equation (57), rj is a real number eigenvalue in the set of eigenvalues which is discrete as form of the sequence 0=r0<r1<r2<⋅⋅⋅<rj<⋅⋅⋅ [35]. Therefore, for fixed x and considering φx,t=eGtϕjx, we obtain(58)dφx,tdt=eGtGϕjx=eGt△xϕjx=−rj2eGtϕjx=−rj2φx,t.
Equation (58) yields(59)x,t=e−rj2tφx,0,which implies(60)eGtϕjx=e−rj2tϕjx.
Then, we obtain(61)eAteGtϕjx=eA−rj2tϕjx.
In order to determinate the Fourier coefficients, using equation (48), we take the scalar product with ϕj for all j and obtain(62)kt,ϕj=k0,eG+σA−σn−σρtϕj=eσA−σn−rj2−σρtk0,ϕj,j≠0.
Particularly, ifj=0, we get(63)kDt,ϕ0=e−σA−n−ρtkt,ϕ0=k0,ϕ0=K0volΩ,where the unique normalized function is(64)ϕ0=1volΩ1.
Applying Fourier series to write functions onΩ, we have(65)k−∑j∈zϕjk,ϕjL2Ω=0,for every function k∈L2Ω [36]. In this case, we express kDt as(66)kDtx=∑j∈zkDt,ϕjϕjx=∑j∈ze−βtkt,ϕjϕjx=K0volΩ+∑j∈z,j≠0e−βteσA−σn−rj2−σρtk0,ϕjϕjx=K0volΩ+∑j∈z,j≠0eσA−σn−rj2−σρ−βtk0,ϕjϕjx.
UsingkL2T2=∑j∈zk,ϕj2 and expression of kDt as Fourier series, we have(67)kDtx−K0volΩL2Ω2=supx∈Ω∑j∈z,j≠0eσA−σn−rj2−σρ−βtk0,ϕjϕjx2≤e−2εt∑j∈z,j≠0eσA−σn−rj2−σρ−β+εtk0,ϕjsupx∈Ωϕjx2≤S2e−2εt∑j≥1e2σA−σn−rj2−σρ−β+εt∑j≥1k0,ϕj2=S2e−2εt∑j≥1e2σA−σn−rj2−σρ−β+εtk0L22,where S is a positive constant and satisfies supx∈Ωϕjx|≤S.
Since(68)A+ρσ1−σ<n<A+ρσ1−σ+σr12σ−1,we have(69)A−n1−1σ<ρ<A−n1−1σ+r12.
Consideringϵ∈0,A−nσ−1+σr12−σρ/σ, we obtain(70)σA−σn−rj2−σρ−β+ε<σA−σn−r12−σρ−β+A−nσ−1+σr12−σρσ=A−nσ−1−σρσ<0.
LetH≔∑j≥1e2σA−σn−rj2−σρ−β+εt. Then, we know H<+∞ due to condition (70). Therefore, we rewrite (67) as(71)kDtx−K0volΩL2Ω2≤e−2εtS2Hk0L22⟶0,which completes the proof.


## 5. Numerical Simulations

In the spatial economic model, the adjustment of optimal trajectory is not instantaneous when the spatial capital flows from rich regions to poor regions. In the rich area, the flow of capital brings a temporary production agglomeration, while a depressed condition appears in the poor area. After that, the spatial capital in the rich location moves to the poor location, and then, the optimal path is formed. In this section, we consider different values of relevant parameters to show the formation of the optimal path and illustrate results of the spatial capital distribution with exponential decays.We simulate two cases of initial capital distribution to illustrate approximate profiles of the physical capital across space and time when some values of relevant parameters are given. As shown in Figure1, the optimal trajectory of spatial capital distribution corresponds to A=0.1,n=0.03,ρ=0.05,σ=3, and r1=0.1, which satisfy conditions (20) and (55). In the initial stage, we observe that the spatial capital distribution is highly clustered in the center section of space interval, while it is very dispersed elsewhere. This case is consistent with the optimal spatio-temporal dynamics of the detrended capital since we know that, as a sum of different terms, the second term on the right of equation (66) is invisible because of the exponential factor eσA−σn−rj2−σρ−βt with condition (68). Precisely, the limit of detrended capital kDtx is the term K0/volΩ when t⟶∞. On the contrary, if we change values of the relevant parameters to satisfy n1−σ=A1−σ+ρσ−σr12, Figure 2 shows the corresponding profiles of the initial capital and spatial capital distribution. In this case, the detrended capital is not convergent to a spatially homogeneous distribution in the long run since the term ∑j∈z,j≠0eσA−σn−rj2−σρ−βtk0,ϕjϕjx is involved in the limit.Figure 1
Evolution of the spatial capital distribution corresponding toA=0.1,n=0.03,ρ=0.05,σ=3, and r1=0.1.
(a)(b)Figure 2
Evolution of the spatial capital distribution in the casen1−σ=A1−σ+ρσ−σr12.
(a)(b)


## 6. Conclusions

In this paper, we have investigated the optimal dynamics of a spatial economic growth model with bounded population growth in a continuous space. Considering the growth model redrafted in the infinite dimensional Hilbert space, we apply the HJB equation and linear production function to identify the unique optimal control and the value function of the growth model. Under assumption of the bounded population growth, the explicit solution of our model is derived in close form by using the optimal control, which satisfies the first-order optimality conditions to show the time behavior of per capita capital. With nonnegative initial capital distribution, we apply the explicit solution and Fourier series to express the detrended capital of our model and then prove the convergence of the detrended capital level across space and time.



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