Regional Economic Geography with Externalities , Congestion , and Fiscal Policies in a Small-Open Growth Economy

This paper develops a two-regional growth model with amenity, capital accumulation and regional public goods with public goods and fiscal policies. The economy consists of two regions and each region consists of the industrial sector and public sector. The industrial sector provides goods in perfectly competitive markets. The public sector, which is financed by the regional government’s tax incomes, supplies regional public goods. The public goods affect both firms and households. We show how to find equilibrium values of the dynamic system and simulate model. Then, we carry out comparative statics analysis with regard to parameter changes in tax rates, congestion and amenity. Our comparative statics analysis provides some important insights. For instance, a main difference between the effects of increasing the two regions’ tax rates on the output is that as the technologically advanced region’s (the other region’s) tax rate on the industrial sector is increased, the national industrial output, national capital employed by the economy, and the national wealth are increased (reduced). In the region which increases the tax rate, the wage rate, consumption and wealth per capita, output per labor force, the population, and land rent are increased, and the corresponding variables in the other region are reduced.


Introduction
The purpose of this study is to propose a model of interactions among economic growth, environmental changes, production externalities, public good supply with different fiscal policies.Regional scientists and regional economists have proposed various kinds of regional dynamic models [1][2][3][4][5].It should also be remarked that there are many other models developed by regional scientists.For instance, some models based on the input-output system or/and the gravity theory are proposed to examine interregional trade patterns [6][7][8][9][10][11][12].However, most of these models do not integrate the spatial factor satisfactorily from the theoretical point of view.This paper builds a dynamic model of interregional economies to show a way of analyzing the total spatial array of economic activities within a consistent framework.
This study analyzes a perfectly competitive economy with public goods and regional differentials in living co-nditions and productivity.The productivity advantages of one region may be offset to some extent by the higher wages that must be paid in a system where people are free to choose where they work and live.Higher wages are often associated with some kinds of disamenities (such as noise, pollutants, and densely populated neighborhood) and high living costs.Labor and capital are easily mobile between regions in industrialized economies.As capital mobility becomes high and costs associated with capital movement among regions become low, it is reasonable to assume that capital movement tends to equalize marginal productivities of capital among regions within a national economy.But there are different principles for analyzing temporary equilibrium conditions for labor movement in a dynamic regional framework.In this study, we determine population distribution by the condition of equalizing utility level.This paper is a generalization of the two-region growth model proposed by Zhang [13].This paper generalizes the previous model in

The Two-Region Trade Model with Capital Accumulation
This paper is concerned with a dynamic one-commodity and two-region trade model.It is assumed that the regions produce a homogenous commodity and public goods.It is assumed that there is only one (durable) good in the national economy under consideration.Households own assets of the economy and distribute their incomes to consume and save.Industrial sectors or firms use capital and labor.Industrial sectors sell their product to households or to other sectors and households sell their labor and assets to industrial sectors.Factor markets work well; factors are inelastically supplied and the available factors are fully utilized at every moment.Saving is undertaken only by households, which implies that all earnings of firms are distributed in the form of payments to factors of production.We omit the possibility of hoarding of output in the form of non-productive inventories held by households.All savings volunteered by households are absorbed by firms.We require saving and investment to be equal at any point of time.
The two regions are indexed by . 2 , 1  j Perfect competition is assumed to prevail in good markets both within each region and between the regions, and commodities are traded without any barriers such as transport costs or tariffs.The labor markets are perfectly competitive within each region and between the regions.Let prices be measured in terms of the commodity and the price of the commodity be unity.where * r is the rate of interest fixed in the international economy.We assume a homogenous population.A person is free to choose his residential location.We assume that any person chooses the same region where he works and lives.Each region has fixed land.Land quality, climates, and environment are homogenous within each region, but they may vary among the regions.We neglect transportation cost of commodities between and within regions.As amenity and land are immobile, wage rates and land rent may not be equal between regions.
We introduce N  the given population of the economy; i and p  subscript index for the industrial sector and public sector, respectively; j L  the given (residential) area of region j, j = 1, 2 at each point of time to maximize its profit, with the level of public goods in the region as given.The production functions are given by where is a function of externalities, public service and congestion.We specify measures the effect of public service on the region's productivity, the effect of externalities, and the effect of congestion of public goods.Similar to [14], we interpret that when there is neither congestion nor externality.The nonrival and nonexcludable public service is available equally to each agent, independent of the usage of others.Obviously this is a limited case as most of public services are subject to some degree of congestion.We take account of congestion effects by the term, , which implies that for a fixed level of public capital, a rise in the private capital tends to reduce the efficiency of public services.There are different ways of describing congestion [15].Here we neglect possible congestion effects due to the region's population and consumption activities.
The marginal conditions are given by ., Behavior of consumers Each worker may get income from land ownership, wealth ownership and wages.In order to define incomes, it is necessary to determine land ownership structure.It can be seen that land properties may be distributed in multiple ways under various institutions.This study assumes the absentee land ownership.It should be noted that another two popular assumptions in the literature of urban economics are the equally shared landownership and the public ownership.For instance as accepted in Kanemoto the city government rents the land from the landowners at certain rent and sublets it to households at the market rent, using the net revenue to subsidize city residents equally [16].Hochman considers a mixture of absentee and equally shared ownerships [17].Different ownerships may have different effects on interregional dynamics.The topic about how different land ownerships may affect interregional dynamics will be examined in the future.Land is owned by absentee landlords who spend their land incomes outside the economic system.This study uses the approach to consumers' behavior proposed by Zhang in the early 1990s.As demonstrated systematically in [5,18], this approach makes it possible to solve many important (national) economic problems, such as growth problems with heterogeneous households, which are analytically intractable by the traditional approaches in economics.Let   t k j stand for the per capita wealth in region j.Each consumer of region j obtains income The budget constraint is given by .ˆ* We assume that utility level that the consumers obtain is dependent on  , t l j  , t c j and  .

t s j
The utility level of the consumer in region j,  , where ) 0 ( j  are positive parameters and b is a parameter.We don't specify the sign of b as the population may have either positive or negative effects on regional attractiveness.Maximizing According to the definitions of  , t s j the wealth accumulation of the representative person in region j is given by .
As households are assumed to be freely mobile between the regions, the utility level of people should be equal, irrespective of in which region they live, i.e.
We now describe the public sector.In this model, we assume that the public sector is financially supported by the regional government's tax income.The capital stocks and workers employed by the public sector are paid at the same rates that the private sector pays the services of these factors.We assume that the regional governments behave effectively in the sense that they use the tax income to maximize public services.The production of public services is to combine capital,  , stand for government j's tax income.Then we have where , are respectively the tax incomes from the production sector's output, the households' interest payments and the households' wage incomes.
The public sector in region j is faced with the following budget constraint Maximization of public services under the budget constraint yields in which ., The total capital stock employed by the economy is equal to the total capitals employed by all the regions.That is where The assumption that labor force and land are fully employed is represented by where The total wealth of the national economy is the sum of the wealth owned by all the households We introduce   According to the definitions of the national wealth, the capital stocks employed by the economy and the net foreign assets, we have . A country's current balance at time t is the change in the value of its net claims over the rest of the world -the change in its net foreign assets.If   , 0  t B  the economy as a whole is lending (in this case we say that the current account balance is in surplus); if   t B  0  , the economy as a whole is borrowing (the current account balance is in deficit); and if   , the economy as a whole is neither borrowing nor lending (the current account balance is in balance).We have thus built the model with endogenous capital accumulation and regional capital and labor distribution.We now examine spatial equilibrium and effects of changes in different conditions upon the economic geography.

Economic Equilibrium
As it is difficult to examine all the dynamic properties of the regional economy, we are only concerned with steady states.By (9) From ( 2) and ( 14), we solve and , From the definition of , j  (11) and ( 18), we can express the production functions as follows where ., , , From (1), ( 18) and ( 19), we have From the definition of j y ˆ and , / ˆ j j k y  we solve , From the marginal conditions for capital in (20), we have , where we use ( 7), (11) and the full labor employment condition, we have 1 2 , . 1 1 From the definition of K  and (23) From ( 14) and ( 18), we have From the definition of , where we use , and the equation for ij N in (18).Substituting ( 27) into (26 The above equations determine the equilibrium values of .~j k The following lemma describes a procedure to determine the equilibrium values of all the variables.

Lemma 1
For a given rate of interest in the global market, the national economy has a unique equilibrium point.The equilibrium values of all the variables are given by the following procedure: The rate of interest is fixed at 5 percent in international market.The total population is .10 The productivity parameter, ， A is higher than region .2 The productivity parameters of the public sectors in the two regions are equal.We consider that Region  [19,20].With regard to the technological parameters, what are important in our interregional study are their relative values.The presumed productivity differences among the regions are not very large.This similarly holds for the specified differences in the amenity parameters among regions.
Following the procedure in Lemma In (19), the variables, ij f and pj y , are respectively the output level per worker and the expenditure on public goods per resident in region j defined as More than half of the national population and capital are located in region .
1 The per-capita levels of wealth and consumption and wage rate in region 1 are much higher than the corresponding variables in regions . 2 The lot size of region 2 is larger than in region . 1 The consumption level per capita in region 1 is higher than in region . 2We see that although workers can earn more money in the advanced region than in the other region, they have to pay much higher rent for housing than the households in the other region.Expectably, the typical household in the advanced region consumes more goods and lives in a smaller house than the typical household in region .
2 Mainly because of its technological advantage, the advanced region attracts far more workers and capital than the other region.

Parameter Changes and Economic Geography
First, we examine effects of change in the total productivity of region 2 's industrial sector, .
We introduce a symbol, ,  by which a variable, , x  stand for the change rate of the variable, , x in percentage due to changes in parameter value.We increase As region 2 's total productivity is increased, the national output, wealth and capital employed are reduced.As region 2 's productivity is increased, the region's wage rate is increased and the region becomes more attractive.People immigrate to region 2 from region . 1 As region 2 's productivity after the change, , 1 .
the net result of the reallocation of the population reduces the national output level.
We now examine effects of change in the tax rate on the production sector.We increase 1 i  from 4 percent to 5 percent.As the tax rate is increased, the industrial output, total capital employed by the economy, and the total wealth are increased.The wage rate, consumption per capita, wealth per capita, output per labor force and public expenditure per capita in region 1 are increased.In region 2 the wage rate, consumption per capita, wealth per capita, and output per labor force are reduced.Although the region's output and total wealth are reduced, its public expenditure per capita is increased.This occurs because a large portion of the region population move to region 1 as region 1's population is increased, its land rent rises and lot size falls.
We increase 2 i  from 3 percent to 4 percent.As the tax rate is increased, the industrial output, total capital employed by the economy, and the total wealth are reduced.The wage rate, consumption per capita, wealth per capita, output per labor force and public expenditure per capita in region 2 are increased.In region 1 the wage rate, consumption per capita, wealth per capita, and output per labor force are reduced.As region 1's population is reduced, its land rent falls and lot size rises.A main differentce between the effects of increasing the two regions' tax rates is that as region 1's (2's) tax rate on the industrial sector is increased, the national industrial output, national capital employed by the economy, and the national wealth are increased (reduced).This occurs partly because the direction of labor migration is the opposite in the two cases.The effects are listed in (33).As the congestion effect of public goods becomes stronger, most of the economic variables are negatively affected.It should be noted that although region 1 's population is increased, the region's total capital and output are reduced.The economic loss caused by the strengthened congestion dominates the economic gain brought about by the labor increase.We now study the impact of changes in the externality parameter, ． e  We increase the parameter value from 0.03 to 0.04 The effects are listed in (34).As positive externalities become stronger, the industrial output, total capital employed by the economy, and the total wealth are increased.Region 1's population is increased.Also the labor of each sector in region 1 is increased.The wage rate, consumption level and wealth per capita in region 1 are increased.The lot size in region 1 is reduced and the region's housing rent is increased.The effects on those corresponding variables in region 2 are the opposite.We now study the impact of changes in the parameter, ． h v We increase the parameter value from 0.05 to 0.06 The effects are listed in (35).As households are more strongly affected by the supply level of public goods, like a rise in the value of positive externalities, the industrial output, total capital employed by the economy, and the total wealth are increased.Region 1's population is increased.Also the labor of each sector in region 1 is increased.The wage rate, consumption level and wealth per capita in region 1 are increased.The lot size in region 1 is reduced and the region's housing rent is increased.

Conclusions
This paper proposed a two-region growth model with capital accumulation, amenity and regional public good with congestion under assumptions of profit maximization, utility maximization, and perfect competition.We emphasize effects of congestion and various fiscal policies on long-term economic growth and economic geography.We simulated the model with the Cobb-Douglas production functions and demonstrated the existence of a unique equilibrium point.Our comparative statics analysis provides some important insights.For instance, a main difference between the effects of increasing the two regions' tax rates on the output is that as region 1's (2's) tax rate on the industrial sector is increased, the national industrial output, national capital employed by the economy, and the national wealth are increased (reduced).


now study the impact of changes in the congestion parameter, ． c We increase the parameter value from

,
, we have It should be remarked that although the specified values are not based on empirical observations, the choice does not seem to be unrealistic.For instance, some empirical studies on the US economy demonstrate that the value of the parameter, , c  are positive.