If system (1.1) is given in the form x1 = G11x1 ЦG12x2, & x2 = G21x1 ЦG22x2, & then an exponential growth less oscillations will be noted provided the correlation below is met (G11+G22)2>4(G11G22ЦG21G12).
For identical regions, this condition is determined by parameters ƒл€ идентичных регионов это условие определ€етс€ параметрами m1 and 2, in accordance with the expression m1/2>0, Which is always valid fro two regions, providing the absence of autarchy. On the qualitative level, it is possible to identify parameters of the balanced growth by means of a graphical illustration (see fig. 1.1), where H(y) = G11ЦG12yЦ1; F(y) = G22 Ц G21y, Y=x1/(1Цx1).
Fig. 1.1. Parameters of balanced growth on the qualitative level As seen from fig. 1.1, in the event the inclination to consumption in region 2 (2) falls, G22 and F(y) demonstrate growth. Because of the rise of 2, the parameter of balanced growth surges, as the share of region 1 in the total output does. Only in the event both regions equally increase their inclination to saving will their equilibrium growth rates surge in both regions, while the structure of output between them will remain unchanged.
The framework of the neoclassical analysis of growth suggests a less trivial model for the final number n of regions. In the absence of depreciation and savings of a domestic output and net imported offer, the dynamics of capital in each region will be determined by the following expression:
Kr = r(Qr ЦEr+Mr), where Kr Ц an instant change of capital in region r, r Ц inclination to savings in region r, Qr Ц GRP in region r, Er, Mr Ц export and import in region r.
Once we assume that the share of export in each regionТs output accounts for r, given that r>r>0, the model of growth of both regions will be then depicted by the following system of equations:
K1 = (1 Ц 1)Q1 + 2Q2, K2 = 1Q1+ (2Ц2)Q2.
The above models are based upon an analysis of value added and do not take into account differences in the production structure and possible associated effects. For the purpose of evaluation the interregional development, the interdependence of regional output in the models can be considered as follows: let us assume that xijflow of goods from region i into region j. The dynamics of flows of goods for each region and sector will consequently be defined with the following equations:
rs rs, xir xs + xs &j aij j bij sj sj r =1,Е,R; i =1,Е,N;
where xir Ц an aggregate production of good i in region r; aijrs Ц regional coefficients of production costs; bijrs Ц coefficient of investment costs of sector i per unit of output in sector j.
The balanced growth equilibrium, i.e. the equilibrium for which condition (1.2) I observed, implies the observance with the correlation:
rs rs xir = xs + xs, &j a b ij j ij sj sj where xir Ц production of good i in region r, bijrs Ц correlation between marginal capital and output (=Iijrs/xjs), Ц total coefficient of growth for which xir = xir.
If the production of good xir is determined by the Cobb-Douglas technology, i.e.
sr ln xir = ln air + ln xsr, (jisr1), ji ji (1.3) sj where xjisr Ц factor j imported from region s for the production of good i in region r, air, jisr Ц parameters of the technology, then the solution to the task of minimization of production costs under a preset level of output with account of production function (1.3) allows to show that all interregional coefficients of output costs form functions of only prices and transportation costs.
1.1.2. Agglomerations Theory An uneven distribution of production under equilibrium leads to the emergence of agglomerations. Their unfolding is attributed to a random factor or to the notions of increasing economies of scale6.
According to the theory, production activity concentrates in certain regions, because firms benefit from their expansion or positive externalities that emerge due to the presence of other firms on the market. The causes for the increasing economies of scale can vary.
For instance, they may result from a flow of knowledge, merger of labor markets, or due to the economy arising because of shorter dis Krugman (1991), Romer (1992).
tances between producers and consumers in the conditions of a trade that require costs.
Fiani (1984) suggests a model of the economies of two regions (North and South), wherein the existence of an increasing return in the production from a non-tradable factor (services) entails a rise in differences between the regionsТ growth rates. The model suggests that initially the regions are identical, i.e. they enjoy the access to identical technologies. As far as tradable goods are concerned, the production is described by function F:
QT = F[Val(LT, KT), QN], where QT Ц output of a given tradable, Val(Ј) Ц value added as a labor and capital function, LT, KT, QN Ц production factors (labor, capital and a non-tradable intermediate factor).
The market for non-tradables is monopolistically competitive, and the output therein is driven by function G(Ј) that bears a constant elasticity by production factors:
QN = G(LN, KN), where QN Ц output of the non-tradable, LN, KN Ц production factors.
The increasing economies of scale in this model arises due to the fact that the market for non-tradables is represented by monopolistically competitive producers that face identical demand curves.
The prices for the non-tradables are computed according to the rule PN=waN(1+q), where w Ц wages, аN Ц оlabor to capital ratio, q Ц the monopolyТs extra.
In this model, a special attention is paid to export and investment in regions with different development levels. Whilst saving in one region can be invested in the other, this, with a due account of costs associated with establishment of investments, would allow to fulfill the following balance correlation:
rNKN + rSKS = IN + V(IN) + IS + V(IS), where V(Ј) Ц function of investment costs.
It can be demonstrated that under static expectations, the effective discounted marginal investment in region i will be computed according to the formula j R j/iЦV (I j)ЦpT=, where i Ц the total discount rate.
While studying into dynamics of the model, its authors showed that the increasing economies of scale entail divergence of growth rates between different regions within a given economy. The expansion of the model by adding the third region allowed to demonstrate that each region would be keen to center on the production of a sole tradable. The authors showed that even in the assumptions of a cost-free production and instant inter-regional capital flows there would be noted a specialization in production of a given good, which would prove to be more intense in terms of a non-tradable factor (services) in the North. By contrast, the South would see a specialization of the production of a good, which will be less intense in the above terms. Accordingly, the authors assume that a high level of substitution between capital and service factors necessitates encouragement of investment in the Southern region, with the greatest emphasis to be put on sectors with low demand for service factors and, accordingly, a low impact of the mutliplier on the local economy.
1.1.3. УNucleusЦPeripheryФ Models It was Murdal (1957) and Hirschman (1958) that laid down and pioneered studies into these particular models. MurdalТs analysis implied that the start of a given regionТs development is related to a random factor, such as, for instance, a discovery of mineral deposits or development of the exportation of food stuffs. A rise in real wages coupled with a high return on capital generate increasing economies of scale and give a rise to a spatial external economy agglomeration that manifests itself in the growing productivity of labor and capital as the function of the regional output growth rate.
Ottavanio and al. (2002) consider various aspects of agglomeration and trade by using the УnucleusЦperipheryФ model. The authors show that their model can be used particularly to study the welfare effect associated with the rise of agglomerations. Their model suggests there exist two regions H и F and two factors A и L. The immobile factor ј, which is farmers, is distributed evenly between the two regions. Ц the proportion of mobile factor L in region H. It is consequently assumed that there exist two kinds of goods in the economy Ц homogenous (q0) and differentiated (qi)7.
Under this model individualsТ preferences are identical and can be desribed by a quasi-linear utility function with quadratic correlations that take the following form:
NN N - 2 U(q0;q(i),i[0,N])= q(i)di - 2 [q(i)] di - 2 [q(i)di] + q0 ;
0 0 where q(i) Ц amount of differentiated good i, i[0,N]; >0 Ц parameter that characterize the level of preference of the differentiated product;, Ц utility parameters. Notably, with the preset, characterizes the substitution feature between differentiated goods.
It is assumed that consumer are more apt to consume a differentiated product, i.e. >>0.
An individualТs budget constraint takes the form:
It is assumed that the market for the differentiated good is monopolistically competitive.
N p(i)q(i)di + q0 = y + q0, where y Ц the individualТs labor income, q0 Ц initial allotment with a good of the first type (it price is standardized by unit).
Production is represented by the number of firms nH and nF in regions H and F respectively, while their number equals the number pf differentiated products N. The authors show that a firmТs profit in region H can be presented as follows:
ѕH=pHHqHH(pHH)(A/2 + L) + (pHF Ц )qHF(pHF)[A/2+(1Ц)L] Ц wH, where pHH and qHH Ц price of, and demand of an individual in region H for an offer of a firm residing in region H, respectively, while pHF and qHF Ц price of, and demand of an individual in region H for an offer of a firm residing in region F respectively, Ц parameter that characterizes a reverse of the mas of firms, and Ц trade costs.
The demand is set by the following formulas:
qHH = a Ц (b + cN)pHH + cPH, qHF = a Ц (b + cN)pHF + cPF.
PH = nHpHH + nFpFH, PF = nHpHF + nFpFF.
In the equilibrium, prices are determined by the condition of maximization of the firmsТ profits, while equilibrium wages are set by the condition of zero profit as an effect of a free entrance to and exit from the market. The authors show that equilibrium prices depend on demand and distribution of firms between regions. More specifically, with the mass of local firms growing, both local and foreign firmsТ8 prices fall, while this downfall appears the lesser, In the model, this is conditioned by the drop in trade costs.
the smaller is. Within the limits9 the firmsТ moving to region H will not affect prices. They also show that with the rise in the amount of the mobile factor, a firm located in region H will see its profit on the one hand, decline, which can be explained to an intensifying competition, and grow on the other, as the number of consumers in the given region is growing. So an ultimate effect remains uncertain.
The most important result of Ottavanio et al.Тs model (2002) lies in the proof of stability of conclusions in relation to the selection of a specification of the model. More specifically, alternative assumptions in respect to preferences and transportation costs do not modify main conclusions, albeit the latter may change, providing a drastic change of the prerequisites.
While criticizing the УnucleusЦperipheryФ approach, Gilbert and Gagler (1982) argue that such models underestimate the role of international influence and pay insufficient attention to social regional aspects, such as poverty and income differentiation. As well, they these models do not consider the pre-colonial history of nations and groundlessly introduce an assumption of a government acting on the populationТs behalf.
1.1.4. The Random Growth Theory This particular theory forms an alternative explanation to the emergence of agglomerations. According to the theory, the latter emerges thanks to strong random shocks which provide impetus to the development of the economic activity in a given region. While employing a model of selection of location by plants, Allison and Glazer (1997) demonstrated that even if the plants are randomly dispersed in a given space and there exist no geographical preferences, the concentration of an industry will emerge at random.
When is infinitely small.
More than that, this process will lead to a positive correlation between the average size of the plant and concentration of the industry. The latter and the plantsТ average size will be growing over time in the location of a very huge plant.
Holmes (1999) argues that the enterpriseТs size should negatively correlate with the concentration of industrial activity, as the emergence of an expanded intermediary goods offer network in production concentration zones should encourage the rise of incentives to establish small, narrow-profile plants. Holmes and Stevens (2002) demonstrated that the plantsТ size on the whole grows in parallel with the rise in the concentration of production activity in all the sectors, except for the textiles. To explain the latter phenomenon, they argue that plants located in the zones of concentration of production activity wins because of production benefits vis-vis those located beyond such zones. That is why the former boost their size to capitalize on the benefits. Such benefits may take their roots in geographical differences or in the agglomeration benefits.
In their empirical study, Davis and Wainstein (2002) attempt to explain the distribution of economic activity within a single country by testing a model of increasing return from economies of scale along with two other theories, that is, the random growth theory and the location theory using Japanese cities as an example. The authors have arrived to the conclusion that the location theory could explain differences in regional concentration of economic activity, while the theory of increasing return from economies of scale was responsible for the level of a spatial differentiation of economic activity. The random growth theory has failed to find a proof against such a background.
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