National and Regional Economic Growth with Fiscal Policies : Congested Public Goods and Endogenous Labor Supply of a Small-Open Economy

The purpose of this study is to build a two-regional growth model with capital accumulation, endogenous time distribution between leisure and labor, and regional public goods with fiscal policies. We emphasize dynamic interactions among capital accumulation, externalities, supply of public good with different fiscal policies, congestion of public good, endogenous time, and economic geography. The economy consists of two regions and each region consists of the industrial sector and public sector. First, we develop the two-region growth model with public goods and fiscal policies. Second, we show how to find equilibrium values of the dynamic system. Then, we simulate model with specified parameter values. Finally, we carry out comparative statics analysis with regard to parameter changes in tax rates and congestion. 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).


Introduction
Interactions between economic growth, public good supply and environmental changes have caused a great attention from economists.Nevertheless, it is argued that a few theoretical models have been proposed to deal with these interactions within a interregional framework.As argued by [1,2], a main obstacle to properly modeling regional economic dynamics is that the traditional approaches to consumer behavior over time makes it analytically intractable to model interregional growth on the basis of profit-and utility-maximization.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 [3].This paper generalizes the previous model in introducing different fiscal policies and endogenous time distribution between leisure and work into the regional dynamic model.We also take account of congestion into consideration.This paper is organized as follows.Section 2 defines the two-region model with capital accumulation, endogenous time distribution, and public goods.Section 3 shows how to determine equilibrium of variables.Section 4 examines effects of changes in some parameters upon long-term national economic growth and economic geography.
Section 5 concludes the study.

The 2-Region Trade Model with Capital Accumulation
This paper builds a dynamic one-commodity and two-region trade model to examine interdependence between regional trades and national growth with regional public goods and congestion.We analyze trade issues within the framework of a simple international macroeconomic growth model with perfect capital mobility.This model is influenced by the neoclassical trade theory with capital accumulation.Many one-commodity trade models with capital accumulation have been proposed, for instance, by [4][5][6][7][8][9][10].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.Exchanges take place in perfectly competitive markets.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 system consists of two regions, indexed by Each region has industrial and public sectors, indexed by i and respectively.Perfect competition is assumed to prevail in good markets both within each region and between the regions, and commodity is 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.We denote wage and interest rates by and respectively, in the th region.The interest rate is equal throughout the national economy, i.e., j where is the rate of interest fixed in the international economy.We assume a homogenous population, , N in the economy.A person is free to choose his residential location within the country.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 vary between the regions.Let

Behavior of Producers
We assume that each firm chooses two productive factors, capital, j ( ) , K t and labor, j 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 as follows the effect of congestion of public goods.Similar to [11], we interpret that when e c 0, there is no congestion and no 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, implying 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 (see [12]).Here we neglect possible congestion effects due to the region's population and consumption activities.

K t
Markets are competitive; thus labor and capital earn their marginal products, and firms earn zero profits.The rate of interest, and wage rates, are determined by markets.Hence, for any individual firm and  , are given at each point of time.According to the neoclassical growth theory as in the Solow model [1], the marginal conditions are given by where k  is the depreciation rate of physical capital and

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.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 We assume that utility level that the consumers obtain is dependent on the lot size, the leisure time,  , hj T t the consumption level of commodity,  , j c t and the saving,  .
The utility level of the consumer in region is specified as follows , j

 
and h  propensities to consume lot size, to consume goods, and to hold wealth (save), respectively.We assume that households would like to have more public goods with the other things fixed, that is,  is called region 's amenity level.In this study, we specify and are parameters.We don't specify the sign of as the population may have either positive or negative effects on regional attractiveness.

b b
Maximizing   j U t subject to the budget constraints (4) yields in which , , , , According to the definitions of  , j s t the wealth accumulation of the representative person in region 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.,    .
The public sector maximizes the level of public services by choosing capital, Let   pj Y t stand for government 's tax income.Then we have where , and wj j j w N  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 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, ( ), K t 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 j L is the given (residential) area of region We also have The total wealth of the national economy is the sum of the wealth owned by all the households We introduce as the value of the economy's net foreign assets at The income from the net foreign assets, which may be either positive, zero, or negative, is equal to 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 is the change in the value of its net claims over the rest of the world -the change in its net foreign assets.If the economy as a whole is lending (in this case we say that the current account balance is in surplus); if 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 conduct dynamic analysis, we are only concerned with steady states.In equilibrium the change rates of all the variables are equal to zero.By (7) where / From the definition of  , j t  (8) and ( 16), we can express the production functions as follows where From ( 1), ( 16) and ( 17), we have From the marginal conditions for capital in (18), we have 2 , From the definition of j y and / , From in ( 14) and we have 0 , We thus determined the time distribution.Inserting / .
hj j wj T h From ( 19), (20) and 1 U U 2  , we have 2 We see that the labor distribution is uniquely determined as functions of 1 and From the definition of We see that the capital distribution is uniquely determined as functions of and 1 From ( 11) and ( 16), we have From the definition of we have , where we use / , and the equation for ij in (16).From the marginal conditions for labor markets in (18) and Substituting this equation and ( 23) and ( 25) into (24) The above equations determine the equilibrium values of .16) The rate of interest is fixed at 5 percent in international market.The total population is 10 with human capital level being unit.The productivity parameter, 1 i A , is higher than region 2, the productivity parameter of the public sector in region 1 is higher than that in region 2. We consider that region 1 is technologically more advanced than region 2. Region In (28), the variables, ij f and pj , are respectively the output level per worker and the expenditure on public goods per resident in region defined as Less than half of the national population but more than half of the total 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 capita in region 1 is higher than in region 2. We 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.

Parameter Changes and Economic Geography
It is important to ask questions such as how one region may affect the national economy as its technology or amenity is improved; or how the regional trade patterns may be affected as the propensity or the total population to save is increased.This section examines impact of changes in some parameters on the national economy and regional economic structures.As we have explicitly provided the procedure to simulate the motion, it is straightforward to make comparative analysis.First, we examine effects of change in the total productivity of region 2's industrial sector, 2 .
i A We increase where  stands for the change rate of the variable in percentage due to changes in parameter value.As region 2's total productivity is increased, the national output, wealth and capital are increased.The trade balance is also improved.Hence, the national economy is improved as a whole.However, there are interregional differences in the impact due to the technological change.The region whose technology is improved attracts more households to the region.Region 2's rate is increased and the region becomes more attractive.People immigrate to region 2 from region 1.The redistribution leads to fall in region 1's land rent and rise a region 2's land rent.The region experiencing the technological change will increase its wage rate, while the wage rate in the other region falls.
As the work time is not affected by the technological change, the changes in the national labor supply are due to regional reallocation of the households.As region 2 has more population, its lot size per household faces.We thus see that as region 2 increases its technology, at a new equilibrium its household will get more wage per work hour and increase the consumption and wealth, but the household will live in a smaller space.In our dynamic system each region may conduct different fiscal policies.It is important to compare effects of change in different taxes upon the economic system.First, we increase the tax rate on region 1's output from 0.04 to 0.05.The national output and wealth are increased.The capital stock employed by the country is increased.The trade balance is deteriorated.As region 1's tax rate is increased, the expenditure on public good is increased (given all the other conditions).The improved infrastructure makes the region more attractive.The region's population is increased and productivity per work hour is increased.Region 1's wage rate and consumption are increased in association of rise in land rent; region 2's wage rate and consumption are reduced in association of fall in land rent.
0.60 1.28 0.63 0.58 0.17 , 0.63 , , 0 14.00 0.63 0 We now increase the tax rate on region 1's wage from 0.04 to 0.05.Different from the increase in the tax rate on output, the national output and the capital stock em-ployed by the country are reduced.The trade balance is improved and wealth is increased.As region 1's tax rate on the wage rate is increased, the expenditure on public good is increased and work time is reduced (given all the other conditions).The improved infrastructure makes the region more attractive.Although the population of region 1 is increased, its labor force The region's population is increased and productivity per work hour is increased.Region 1's wage rate and consumption are increased in association of rise in land rent; region 2's wage rate and consumption are reduced in association of fall in land rent.As the tax rate on wage is increased, the national supply of labor is reduced.Although more people immigrate to more productive region, the national output is still reduced.Because the output and wage are reduced in region 2, the regional public service and per capita expenditure on public good are reduced.
We now study the impact of changes in the congestion parameter, c  ． We increase the parameter value from 0.05 to 0.06.The effects are listed in (32).The time distribution is not affected.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 increase the externality parameter value, , e  from 0.03 to 0.04.The effects are listed in (33).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.As region 2 loses labor and the time distribution is not affected, the region's output and capital stocks are reduced.Although region 1's output is increased, as its change rate is low, the net effect on the national industrial good falls.Also the labor of each sector in region 1 is increased.The wage rate, consumption level and wealth per capita in the two regions are all reduced.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 goods 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.As the model is structurally general, it is possible to deal with various national as well as regional issues.Given the contemporary literature on (spaceless) economic growth, we may extend and generalize the model in different ways.We may analyze behavior of the model with other forms of production or utility functions.Other important extensions include incorporating transportation costs and endogenous population growth into the model.We may also refine our model by dividing public expenditures on capital into maintenance and new investment.As empirically demonstrated in [14,15], maintenance expenditures on public capital can account to 2 to 3 percent of GDP in some economies like USA and Canada.As in [16] in which the composition of public capital expenditures under congestion is an important factor for optimal fiscal policies, we may consider possible effects of public maintenance expenditures on the depreciation of public capital.
the marginal conditions in labor market in (18), we get

jk 1
The following lemma describes a procedure to determine the equilibrium values of all the variables.Lemma 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: j k  (26) Itshould 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, ,  in the Cobb-Douglas production is h  The tax rates on the output level, the income from wealth and wage income are the same within each region and the tax rates in region 1 are higher than the tax rates region 2. The externality and congestion parameters, e  and , c  are positive.