Handbook of Regional Science pp 1-25 | Cite as

# Incorporating Space in the Theory of Endogenous Growth: Contributions from the New Economic Geography

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## Abstract

We describe how endogenous growth theory has now incorporated spatial factors. We also derive some of the policy implications of this new theory for growth and economic integration. We start by reviewing the product variety model of endogenous growth and discuss similarities with modeling techniques in the new economic geography. Both use Dixit-Stiglitz competition. Increasing returns provide an incentive for innovation in endogenous growth theory, and in combination with transport costs, increasing returns provide an incentive for firm location decisions in the new economic geography. Since innovation is the engine of growth in endogenous growth models and knowledge spillovers are a key input to innovation production, we also explore how innovation and knowledge have distinctly spatial characteristics. These modeling similarities and the spatial nature of knowledge spillovers have led to space being incorporated into the theory of endogenous growth. We guide the reader through how space is modeled in endogenous growth theory via the new economic geography. Growth by innovation is a force for agglomeration. When space is included, growth is enhanced by agglomeration because of the presence of localized technology spillovers. We consider the many other spatial factors included in models of space and growth. We explore the spatial effects on economic growth demonstrated by these theoretical models. Lastly, we consider policy implications for integration beyond lowering trade costs and discuss how lowering the cost of trading knowledge is a stabilizing force and is growth enhancing.

## Keywords

Transport cost Real wage Knowledge spillover Endogenous growth Unskilled worker## 1 Introduction

Theoretical models of endogenous growth explain the engine of economic growth with intentional investments in innovation motivated by monopolistic competition. But these theories have typically ignored space. Endogenous growth and the new economic geography (NEG) have grown quite separately despite similarities in modeling using Dixit and Stiglitz (1977) preferences. Within the literature on innovation, contributions on systems of innovation and the geography of innovation (Audretsch and Feldman 1996) have the potential for a number of spatial aspects to also be incorporated into the theory of growth. More recently, endogenous growth theory has been combined with the NEG and provided insights on how geographic space can influence economic growth.

There are persistent differences in growth rates and incomes between even highly integrated regions such as the European Union or the United States. Endogenous growth theory offers some explanations for varying growth rates. Firms invest in research and development (R&D) to design new innovations, whereby knowledge of existing products is an integral input to R&D. Profits provide an incentive for investment and are protected by patents. The theory implies that varying rates of economic growth may be caused by regions specializing in different sectors with varying rates of productivity or rates of innovation and by differing institutions that protect patents. The theory fails to provide an adequate explanation of varying growth rates because it does not explain differences in levels of innovation when regions have similar institutions or innovations that are not protected by institutions (e.g., process innovations, firm structure). Spatial factors offer some explanation for these differences, but economic growth theory typically ignores the role of space in determining economic growth outcomes.

Kaldor (1970) explains how trade can drive apart even identical regions as industry agglomerates in a single location. Some contributions to endogenous growth theory include this trade mechanism (Lucas 1988; Grossman and Helpman 1991b, 1995) but still ignore the role of space (distance-related factors) in economic growth. Despite the increasing use of space in economic theory through developments in new trade theory (Krugman 1979), new economic geography (Krugman 1991), and similarities in modeling, it has only been a recent development to incorporate geographic space into growth theory to create spatial models of endogenous growth (Martin and Ottaviano 1999). Developments from the NEG have now led to the incorporation of spatial factors related to both production and knowledge into theoretical growth models. These types of models may help explain varying growth rates between even highly integrated regions with similar institutions. For example, McCann (2009) suggests an economic geography perspective of New Zealand might help explain the difference in growth rates with Australia.

Hence, the new economic geography and growth (NEGG) literature incorporates space into the theory of growth by combining endogenous growth theory with the NEG. This chapter starts by describing the basic theory of endogenous growth (Romer 1990; Grossman and Helpman 1991a) followed by a typical NEGG approach where the theory accounts for the spatial factors of transport costs, migration, and imperfect knowledge spillovers. We review the contribution of these types of spatial models and variations in the use of spatial parameters and discuss the consequences for regional growth policy.

## 2 A Simple Model of Endogenous Growth

Endogenous growth theory uses increasing returns as an incentive for firms to make intentional investments to develop innovations. In all theoretical models of growth, the accumulation of capital (physical and human capital) is the engine of growth. Romer (1990), Lucas (1988), and Aghion and Howitt (1992) treat investment in innovation as investment in an additional type of capital, with increasing returns. While accumulation of physical and human capital suffers from diminishing returns, returns to investment in innovation are not diminishing and growth is sustained in the long run. These theoretical models are separated into two groups: Grossman-Helpman-Romer models (Romer 1990; Grossman and Helpman 1991a) use a love of variety with Dixit and Stiglitz (1977) competition and an increasing number of varieties as the source of growth. Alternatively, Schumpeterian growth models (Aghion and Howitt 1992) use creative destruction or quality ladders where higher-quality products replace existing varieties.

In this section, we present a simple product variety model of endogenous growth through research and development (Romer 1990; Grossman and Helpman 1991a). In subsequent sections, we explore contributions that add space to this model of endogenous growth. To focus on innovation, the models here overlook factor accumulation (such as investments in physical and human capital) so that all investment is in the form of creating new technologies (innovations). In the product variety model, growth comes from an expanding variety of goods. We treat these as final goods using Dixit-Stiglitz preferences as in Grossman and Helpman (1991a). In contrast, Romer (1990) has an expanding variety of intermediate goods which are used to make a final good with a Dixit-Stiglitz production function. Grossman and Helpman (1991a) acknowledge the alternative Dixit-Stiglitz specification of the production function rather than the utility function. The outcomes of the model are essentially the same. We use the final goods version for consistency with the global and local spillover models in Baldwin et al. (2003).

There are two sectors – final goods and the R&D sector. Labor is either employed in producing final goods or in R&D which produces new designs. Workers are free to choose the sector in which they are employed and supply their labor inelastically. Consumers have a taste for diversity and are made better off by an expanding number of varieties. For each new good, there is a sunk cost of innovation that occurs once, when the product is developed. Each firm must first obtain a design from the R&D sector, but once the design is obtained, the firm can produce that variety forever at a constant marginal cost. The presence of fixed costs leads to monopolistically competitive markets. The up-front cost is financed by monopoly profits that are later earned in sales.

### 2.1 Demand

*C*

_{t}is the consumption index of goods,

*ρ*is the rate of time preference, and time is indexed by

*t*(for simplicity, the subscript

*t*will be dropped hereafter where the time dimension is clear). Consumers have constant elasticity of substitution (CES) preferences over the continuum of final goods [0,

*K*]:

*c*

_{i}is the consumption of variety

*i*and

*σ*is the constant elasticity of substitution. Consumers have a taste for diversity over an infinite set of products

*i*∈ [0, ∞] where at any point in time, a subset

*K*is available in the marketplace. Consumers allocate income between consumption and savings and distribute consumption across available varieties. Intertemporal utility optimization implies that expenditure changes over time according to the Euler equation:

*E*is consumer expenditure, \( \dot{E} \) is expenditure differentiated with respect to

*t*, and

*r*is the risk-free rate of return on savings. In equilibrium, we have

*r*=

*ρ*, ∀

*t*. Subject to the budget constraint

*p*

_{i}is the price of variety

*i*, so that

*E*=

*CP*. It can be shown (e.g., Appendix 2.A of Baldwin et al. 2003) that the demand function facing an individual firm is

Hence, demand is inversely related to relative price.

### 2.2 Production

*L*workers/consumers. Each worker produces one unit of labor per period and supplies its labor inelastically. We assume that each firm takes other firms’ prices as given, and with a large number of varieties, firms ignore the effect of their own price on the price index. Some portion of labor

*L*

_{M}<

*L*is employed in the manufacturing sector. Remaining labor is employed in the innovation sector

*L*

_{I}such that

*L*

_{I}+

*L*

_{M}=

*L*. Firms choose optimal prices in order to maximize profit

*π*

_{i}= (

*p*

_{i}−

*βw*)

*c*

_{i}, where

*β*describes the marginal units of labor per unit of production and

*w*describes the wage, such that

*βw*is the marginal cost of production. This implies that optimal prices are a constant markup over marginal cost:

*p*

_{i}=

*w*. We avoid this normalization here so that the reader can identify in the formulas how the productivity of labor in both manufacturing and R&D is important for the allocation of labor between sectors. As there are symmetric demands for all varieties, every firm in the manufacturing sector yields the same price, production, and operating profit per period. We normalize expenditure

*E*= 1 ∀

*t*. With

*E*= 1 and all firms being otherwise identical, individual firm profit in each period is given by

### 2.3 Research and Development

*a*

_{I}units of labor per unit of capital produced. We follow Grossman and Helpman (1991a) and Romer (1990) using a learning curve such that the marginal cost of new knowledge capital

*a*

_{I}declines as cumulative knowledge output increases. Romer (1990) rationalizes this assumption by referring to the non-rival nature of knowledge, emphasizing the role of knowledge spillovers. Labor and the stock of knowledge (equal to the number of varieties) are used to develop new innovations. Each innovation adds to the stock of knowledge that can be used for developing future innovations. Innovation production is given by

*L*

_{I}is the labor employed in the innovation sector,

*F*is the fixed cost of innovation to develop a new variety in the R&D sector, and \( {a}_I=\frac{1}{K_t} \) describes the productivity of the R&D sector increasing with cumulative output (i.e., the fixed cost of each innovation decreases over time). The model is based on Grossman and Helpman (1991a), but for consistency, the functional form used here is adapted from Baldwin and Forslid (2000). From Eq. (8), it follows that the rate of growth in the number of varieties is equal to

*L*

_{I}, which may be scaled by a constant to calibrate the model as described in Grossman and Helpman (1991a).

*i*forever at constant marginal cost. Equity owners are paid the infinite stream of profits from the firm. Free entry into the research sector implies that labor is hired such that its wage equals its marginal product. At time

*t*and with constant interest rates, the present value of the future stream of profits,

*v*

_{t}, is

This can also be written as a rate of return, \( \frac{{\dot{v}}_t}{v_t}+\frac{\pi_t}{v_t}=r \). The “no arbitrage” condition describes that in the interval between *t* and *t* + *dt*, the owners of the patent (equity holders) receive a return (made up of the profit rate \( \frac{\pi_t}{v_t} \) and the rate of capital gain (loss) \( \frac{{\dot{v}}_t}{v_t} \)) equal to the yield on a riskless loan. In other words, for a manufacturing firm to purchase a patent (or investors to hold equity/debt), the payoff must exceed the opportunity cost.

The cost of research that yields \( \dot{K}=\frac{l_I}{a_I} \) incremental varieties is *wa*_{I} and has the value \( {v}_t\dot{K}={v}_t\frac{l_I}{a_I} \), where *l*_{I} is the labor input by a typical entrepreneur. Continuous growth, \( \dot{K}>0 \), involves an active research sector, and free entry requires the research costs to be equal to the value of research for all *t*. If the costs of research are greater than the value of R&D, no research would occur in equilibrium. A situation where the cost of research is less than the value of R&D will never occur in equilibrium because it would cause an unbounded demand for research labor. Equilibrium therefore requires *v*_{t} ≤ *wa*_{I} with equality when \( \dot{K}>0 \).

### 2.4 Equilibrium

*L*=

*L*

_{I}+

*L*

_{M}, i.e., total employment is the sum of R&D employment and manufacturing employment. Substituting the interest rate

*r*=

*ρ*and the profit rate \( \pi =\frac{1}{K\sigma} \) into the no arbitrage condition, the change in firm value is a function of the value of a firm and the number of firms:

These two differential equations, Eqs. (11) and (12), describe the dynamic equilibria.

### 2.5 Balanced Growth

*g*

_{K}. If we define a new variable \( V=\frac{1}{Kv} \), representing the inverse of the economy’s aggregate equity value, the growth rate is

This is positive, so long as *L* > *ρ*(*σ* − 1); otherwise, growth is zero. Growth is positively related to the scale of the economy (*L*), which is a common property of these models. Innovation (and incentives for R&D investment) is sustained because there are offsetting forces of declining profits due to expanding varieties and falling product development costs due to research externalities.

This is not the overall growth rate of the economy. To understand macroeconomic growth, we are interested in the growth rate of the consumption index, \( C={\left[{\int}_0^K{c_i}^{\frac{\sigma -1}{\sigma }} di\right]}^{\frac{\sigma }{\sigma -1}} \). Since *E* = *CP* = 1 ∀ *t*, growth is also the rate at which the price index \( P={\left[{\int}_0^K{p_i}^{1-\sigma } di\right]}^{\frac{1}{1-\sigma }} \) declines. The growth rate of consumption *g*_{C} can be shown to be \( {g}_C=\frac{g_K}{\sigma -1} \). This is also not GDP growth. GDP is defined as the value added in both manufacturing and R&D. GDP grows at a rate equal to a weighted average of the growth rates of the index of manufacturing output/consumption and of research output. Since R&D is usually only a small percentage of a country’s GDP, the difference is negligible. See Grossman and Helpman (1991a, p. 63) for a discussion.

## 3 A Two-Region Model of Growth

Virtually all endogenous growth models rely on technical externalities such as knowledge spillovers and production externalities. Endogenous growth models usually assume a frictionless spillover of knowledge. The reality is that knowledge is not transferred so effortlessly. While some knowledge can be codified and transferred easily, much knowledge is at least partially tacit. Spillovers of tacit knowledge occur over space and time through face-to-face contact (McCann 2007) and migration (Faggian and McCann 2009). Eaton and Kortum (1999) show that knowledge spillover and production externalities are related to the geographic distribution of manufacturing and R&D. A better understanding of the economics of innovation (Nelson 1993) and its geographic characteristics (Audretsch and Feldman 1996) significantly improves our understanding of economic growth.

Innovation is a predominantly local event and is now included in economic geography. Acs and Varga (2002) note the similarities between modeling techniques of endogenous growth theory and the new economic geography, suggesting a new model of technology-led regional economic development that combines the two fields with insights from the economics of innovation. Knowledge and innovation also have space, time, and cost characteristics in their spillover between locations. This role of space and time in knowledge spillovers means economic growth also has spatial characteristics.

Given this understanding of innovation, the concentration of economic activity also results in greater knowledge spillovers between firms in concentrated locations. In endogenous growth literature, there is recognition of partial international knowledge spillovers. Grossman and Helpman (1991b) model foreign knowledge as an innovation input in a small economy where the availability of foreign knowledge is dependent on the level of trade, yet these models ignore the role of space. Space can be added to the theory of growth by including spatial characteristics in knowledge spillover inputs to innovation production.

Transport costs are also a key spatial parameter typically ignored in endogenous growth models. The new trade theory (Krugman 1979) and the new economic geography (NEG) (Krugman 1991) include transport costs and have Dixit-Stiglitz competition in common with many theoretical endogenous growth models. Transport costs can therefore be included easily within endogenous growth. The result of transport costs is the concentration of production in specific locations, when transport costs reach some low threshold. This is known as the core-periphery model. With low enough transport costs, firms choose to locate close to their customers to reduce transport costs. When models also allow for migration, workers choose to locate near producers to reduce their cost of living. These transport cost-related phenomena are known as the home market effect because it causes the concentration of firms and people.

Higher transport costs may induce firms to seek locations where there are fewer firms to compete with. This is known as the market crowding effect. It is the balance of these two effects that determines equilibrium and the steady state. Concentration occurs at low transport costs when the home market and cost of living effects dominate the market crowding effect, while dispersion occurs at higher transport costs, where market crowding dominates. The NEG suggests that imperfect integration may create regional winners and losers (Krugman 1991; Krugman and Venables 1995). A particularly interesting characteristic is that the economic conditions of two regions can be exactly the same yet yield dramatically different economic outcomes.

### 3.1 Incorporating Space in the Theory of Growth

*à la*Grossman-Helpman-Romer with the NEG (e.g., Baldwin et al. 2001; Baldwin and Forslid 2000; Martin and Ottaviano 1999; Fujita and Thisse 2003) predominantly due to the fundamental use of Dixit-Stiglitz competition. Different NEGG models vary assumptions on the mobility of capital, labor, and industry or consumer demand to influence the forward and backward linkages. Here, we describe a typical NEGG modeling approach (Baldwin and Forslid 2000) that includes the spatial factors:

Location

Migration

Transport costs

Local knowledge spillovers

Imperfect global knowledge spillovers

*L*

_{T}. Consumers have a taste for traditional goods such that

*C*=

*C*

_{M}

^{μ}

*C*

_{T}

^{1 − μ}, where

*C*

_{M}is the index of manufactured goods (similar to

*C*in the previous section) and

*C*

_{T}is the traditional goods sector. Foreign region variables are denoted by an asterisk (

^{∗}). The representative consumer is infinitely lived and has intertemporal preferences:

In what follows, the time subscripts will again be suppressed for simplicity. Transport costs are zero in the traditional goods sector, and workers in this sector cannot migrate between regions. In the real world, workers in the traditional goods sector are not necessarily unskilled or immobile. The important feature here is that the factor of production for traditional goods is immobile, and “unskilled” is the commonly used term in these models. The purpose of the additional sector in this model is that some residual demand remains in the periphery, even when there is full agglomeration, so that regions continue to trade.

*L*

_{K}) are employed in either manufacturing or innovation (similar to workers in the previous section with subscript

*K*since they work in the knowledge sectors of manufacturing or innovation). The world population of skilled and unskilled workers is normalized to one such that

*L*=

*L*

_{K}+

*L*

_{T}= 1. Skilled workers and manufacturing firms have a choice of location. Skilled workers respond to wage pressure when making a decision to migrate between regions. If there are differences in real wages, there will be migration. The perfect price index describes the price index of utility and therefore includes traditional goods such that \( P\equiv {P_T}^{1-\mu }{P_M}^{\frac{\mu }{\sigma -1}} \). The change in skilled workers in the home region is given by the ad hoc migration equation in Fujita et al. (1999):

*s*

_{H}is the share of skilled workers in the home region, and

*ω*

_{K}is the real wage of skilled workers in the home region. Since the real wage is defined by means of the perfect price index, workers migrate to the region that provides the highest level of utility.

Manufactured goods transported between regions incur transport costs that take Samuelson’s “iceberg” form where transport costs are incurred in the good itself. The manufacturer produces more of the good than actually arrives because some portion of the good “melts” in transit. If *τ* represents the proportion of the final good that arrives at the destination, the remaining portion is used up during transportation. Hence, *τ* < 1 is a measure of the freeness of trade or an index of the inverse of transport costs. Transport costs for the traditional goods sector are assumed zero (*τ* = 1). Firms are incentivized to locate in the largest market to minimize transport costs. From the migration equation above, skilled workers try to locate in the region with more firms as this reduces their cost of living (since they have a taste for diversity) by increasing real wages.

*a*

_{I}units of labor for each unit of knowledge capital produced. Innovation production in the home region is given by

*L*

_{I}is the skilled labor employed in the innovation sector,

*λ*is the ability for foreign knowledge to be used in the home region, and \( {a}_I=\frac{1}{K+\lambda {K}^{\ast }} \) describes how productivity of the R&D sector increases with cumulative output. Hence, the model assumes perfect local knowledge spillovers but imperfect spillovers between regions. The parameter

*λ*represents how space affects knowledge production such that firms choose a location that considers how existing knowledge can be used for innovation. In this way, firms are attracted to regions where other firms are located because the cost of innovation is lower.

### 3.2 Model Description

*E*

_{w}= 1, ∀

*t*. Subject to the budget constraint, consumers allocate expenditure across varieties to maximize utility. Hence, in the home region,

*P*

_{M}

*C*

_{M}+

*P*

_{T}

*C*

_{T}≤

*E*, where

*P*

_{M}is the local price index of manufactured goods (the world equivalent is a weighted average price index such that \( \overline{P_M{C}_M}+{P}_T{C}_T\le {E}_w \) and

*P*

_{T}is the price of traditional goods. Consumers spend a constant portion of their expenditure on manufactured goods and the rest on traditional goods:

*μ*)

*E*

_{w}. The traditional goods sector is perfectly competitive, with 1:1 technology (one unit of unskilled labor input yields one unit of traditional goods output) and constant returns to scale. Total production of traditional goods is shared across both regions. Let

*L*

_{T}and \( {L}_T^{\ast } \) be the supply of unskilled workers in the home and foreign regions, respectively. We follow Krugman (1991) and set the worldwide stock of skilled workers to

*μ*and the stock of unskilled workers to (1 −

*μ*) shared equally between regions:

The choice of units (1 − *μ* unskilled workers and *μ* skilled workers) follows Krugman (1991) and ensures that prices and wages in the traditional goods sector are the numéraire and that the nominal wage rate of skilled workers equals that of unskilled workers. If the number of skilled workers was specified differently, the wages of skilled workers are a constant multiple of the wage rate of unskilled workers. We maintain simplicity by avoiding this additional multiple. A scaling factor could also be used to calibrate the model to any arbitrary growth or wage rate.

*μ*unskilled workers and 1:1 technology, the traditional goods sector is the numéraire:

*K*manufactured varieties and the foreign region produces

*K*

^{∗}varieties. Consumers have a CES preferences over the continuum of manufactured goods [0,

*K*+

*K*

^{∗}], such that

*c*

_{i}is the consumption of variety

*i*and

*σ*is the constant elasticity of substitution. Defining the local price index of manufactured goods as in the model of Sect. 2, \( {P}_M={\left[{\int}_0^{K+{K}^{\ast }}{p_i}^{1-\sigma } di\right]}^{\frac{1}{1-\sigma }} \) where

*p*

_{i}is the price paid by local consumers, the demand function in the home region facing an individual manufacturer is

*c*

_{j}=

*μEp*

_{j}

^{−σ}

*P*

_{M}

^{σ − 1}, and the equivalent demand function exists in the foreign region with the foreign region’s price index.

*βw*is the marginal cost of production. Firms ignore the effect of their own price on the index. Once again, optimal prices are a constant markup over marginal cost, and transport costs are passed on directly to consumers:

*p*

_{i}and \( {p}_i^{\ast } \) are the local and export prices of a home manufacturer. A foreign manufacturer has analogous prices, with transport costs on goods exported to the home region. Here, it is also possible to impose the same normalization \( \beta =\frac{\sigma -1}{\sigma } \) such that

*p*

_{i}=

*w*and \( {p_i}^{\ast }=\frac{w}{\tau }=\frac{pi}{\tau } \). While its distribution is subject to worker migration, by following Krugman’s (1991) choice of units where the worldwide stock of skilled workers is

*μ*, nominal skilled wages in equilibrium are

*w*= 1 or

*w*

^{∗}= 1 for the core-periphery outcome and

*w*=

*w*

^{∗}= 1 in the equal distribution outcome.

### 3.3 Long-Run Location

The equal distribution outcome

The core-periphery outcome

See Baldwin and Forslid (2000) for a more formal discussion of the conditions of the steady state in the NEGG model here and Baldwin et al. (2003) for a discussion of other NEGG models.

The equal distribution outcome is where both regions have half the skilled workers, half the manufacturing, and half the traditional goods production. The other steady state is the core-periphery outcome where all manufacturing concentrates in a single region (either home or foreign) known as the core and only unskilled workers (the traditional goods sector) remain in the other region known as the periphery. Traditional goods production is split equally between regions.

If there are asymmetric transport costs, it is not inevitable that the region with the lowest transport costs will be the core. The core region will be the one which has the higher share of varieties and where the difference in the number of varieties is large enough to trigger a switch from the equal distribution outcome to the core-periphery outcome. This could be for several reasons. Since every variety has a patent forever, hysteresis plays a large role in determining which region is the core. For example, an initial higher endowment of resources might lead to a greater number of manufacturers and innovators, or greater infrastructure investment at some stage (and temporarily freer trade) might also trigger agglomeration. Similarly, temporarily different policy settings between regions where one region has favorable policies for R&D could lead to initially higher rates of innovation, a greater share of varieties, and agglomeration. While not included in typical NEGG models, stochastic effects could mean one region gets “lucky.” In the model here, innovations are simply costs where each firm has to employ a certain amount of skilled labor in R&D in order to achieve an innovation. In reality, successful innovations are not so guaranteed. The inclusion of probabilistic outcomes in the R&D sector could mean one region achieves a higher rate of innovation by luck, resulting in it becoming the core.

*τ*

_{B}describes the level of trade freeness where the equal distribution outcome is no longer a steady state. The sustain point

*τ*

_{S}describes the level of trade freeness at which the distribution of firms and workers switches from the core-periphery outcome to the equal distribution outcome when trade freeness is declining (transport costs increase). The values of trade freeness between the sustain and break points represent situations which both the potential equilibrium outcomes are stable. As the level of knowledge spillovers

*λ*varies, so do both the break and sustain points. Figure 1 describes how the break and sustain points increase as knowledge spillovers increase. Alternatively, Fig. 1 describes the combinations of knowledge spillovers and trade freeness that result in stable (and unstable) equilibria for both the equal distribution outcome and the core-periphery outcome. There are three sections within the knowledge spillover (

*λ*) and trade freeness (

*τ*) space. In the top-left corner, the core-periphery equilibrium is unstable and the equal distribution is stable. In this situation, trade freeness is sufficiently low (high transport costs) that the market crowding effect means firms make a higher margin by locating away from other firms. There is very little trade (if any) between regions. Closer to the curve, regions will trade, but the market crowding effect always dominates the home market, cost of living, and innovation cost effects. In the middle section, both the equal distribution and core-periphery equilibrium are stable. If there is an equal distribution, regions will trade, but it is possible that with an external shock, the home market, cost of living, and innovation cost effects could dominate the market crowding effect and the system would switch to the core-periphery outcome. Similarly, if there is a core-periphery equilibrium, an external shock to the distribution could lead to the market crowding effect dominating the home market, cost of living, and innovation cost effects causing a switch to the equal distribution outcome. Lastly, the bottom-right section describes combinations of knowledge spillovers and trade freeness where only the core-periphery outcome is stable. In this situation, the home market, cost of living, and innovation cost effects always dominate the market crowding effect.

## 4 Spatial Consequences for Economic Growth

The incorporation of space in the theory of growth means the model recognizes the role of space through transport costs and through knowledge not transferring perfectly between locations. Let us consider world and regional growth in both the short and long run in the two possible types of equilibria: core-periphery and equal distribution. Because regions are able to trade, even a periphery region benefits from growth in the number of varieties produced in the core. Over time, the price index for manufactured varieties falls as more varieties are invented, and producers of traditional goods experience growth in real income because they trade for manufactured goods. We consider growth in terms of the number of manufactured varieties and growth in terms of the consumption bundles available to all consumers.

### 4.1 Integration

While traditional conceptions of integration refer to lowering of the cost of trading goods, Fig. 1 shows that incorporating space and growth gives a more detailed view of integration where we can also view integration as lowering the cost of trading ideas. Integration policies which focus solely on free trade may be destabilizing and result in a deindustrialization of the periphery region. That is, when we lower trade costs alone, the region that emerges as the periphery suffers is relative to the region that emerges as the core. Integration policies that also focus on knowledge spillovers (or entirely on knowledge spillovers) will be growth enhancing for both regions. The model shows how this form of integration is stabilizing, while pure trade cost integration can be destabilizing.

### 4.2 Growth in Varieties

For simplicity, we drop the subscript *i* for *p*_{i} because home firms are symmetric and prices are equal for all home firms. Once a blueprint or variety is invented, manufacturers require *β* marginal units of labor per unit of production, so aggregate demand for labor in the manufacturing sector in the home region is \( \frac{\beta }{p} \). As in the endogenous growth model without space, equilibrium in the skilled labor market in the home region requires \( {L}_K={L}_I+{L}_M={a}_I\dot{K}+\frac{\beta }{p} \). In the equal distribution outcome, prices are higher than the core-periphery outcome because of the additional cost to transport goods between regions. A larger share of skilled labor is used in manufacturing because each producer has to produce a larger amount to cover the cost of transport. In other words, the cost of transport increases the marginal cost of production such that some labor is no longer available for innovation. When freeness of trade is greater, i.e., the cost of transport is lower, more labor is available for growth. As such, incorporating space in the theory of growth shows how trade liberalization and agglomeration are growth enhancing for world growth.

Trade liberalization and agglomeration (in the home region) are growth enhancing because they reduce the cost of transport. However, if transport costs induce the core-periphery outcome, there is no manufacturing in the periphery and therefore no growth in varieties produced by that region. That is, reducing transport costs means growth in varieties may be limited to a specific region(s). Therefore, trade liberalization is not growth enhancing for growth in varieties for the region that emerges as the periphery.

For both world and regional growth, the inclusion of space means firms face an innovation cost that is dependent upon location. The output of skilled workers in the innovation sector is greater when knowledge is more available. With *s* being the home region’s share of manufacturing, the rate of growth is \( \frac{{L_I}^w}{K+{K}^{\ast }}\left[s\left(K+\lambda {K}^{\ast}\right)+\left(1-s\right)\left(\lambda K+{K}^{\ast}\right)\right] \). That is, when *λ* is greater, both world and regional growth increase. Including space in the theory of growth shows how closer economic integration is growth enhancing for world and regional growth in varieties. Similarly, when one region has a greater share of manufacturing than the other region, growth increases for the agglomerated region. Agglomeration in either region is growth enhancing for world growth and for regional growth in the region where agglomeration occurs. However, in the core-periphery outcome, there is zero growth in the number of varieties in the periphery region as no varieties are manufactured there, no skilled workers are employed, and no innovation occurs.

### 4.3 Consumption Growth

While so far we have described the effect of space on the growth rate of the number of varieties, this is not the overall growth rate because we have ignored traditional goods. In considering the growth rate of the overall economy, we are interested in the growth rate of what people actually consume. In other words, we are interested in what the income to workers allows those consumers in each region to purchase which is measured by the growth rate of the consumption index, *C*, where *E* = *CP* = 1. This best describes how the well-being of consumers increases over time. While there is no growth in the number of varieties produced in the periphery, the ability to trade traditional goods for manufactured goods allows the unskilled workers to benefit from innovations in the core.

Since *E* = *CP*, the rate at which the consumption index grows is the rate at which the perfect price index declines. In the endogenous growth model of Sect. 2, the growth rate of consumption *g*_{C} was shown to be \( {g}_C=\frac{g_K}{\sigma -1} \). With the addition of the traditional goods sector, the overall perfect price index is to a power of \( \frac{\left(\sigma -1\right)}{\mu } \). The perfect price index is falling at a rate of \( {g}_C=\frac{\mu {g}_K}{\left(\sigma -1\right)} \). Notably, the growth rate of consumption is the same in both regions whether we have a symmetric outcome or the manufacturing concentration outcome. This is because the price index for both regions falls at the same rate, since consumers in both regions still spend the same portion of their earnings on traditional goods – in the steady state, the growth rate of consumption is equal in both regions. The inclusion of space does not explain the differences in growth rates between locations in the long run. Instead, space affects the world rate of growth and the share of wealth/earnings in each location.

In the short run, however, there can be different growth rates between locations if the regions are in transition between steady states. Given *τ* < 1, the price index will be permanently lower in a core location because core location consumers do not pay transport costs for manufactured goods. If the economies are shifting from an equal distribution to the core-periphery outcome, growth rates in the periphery will be temporarily lower (or even negative) as periphery consumers transition to paying transport costs on a greater share of the manufactured goods they consume (eventually all goods). Consumers in the core gradually pay transport costs on a smaller share of manufactured goods, and the core will have higher growth rates.

### 4.4 Agglomeration, Freeness of Trade, and Economic Growth

Agglomeration is growth enhancing in the long run through both transport costs and knowledge spillovers. Agglomeration minimizes the total cost of transport if all manufacturing and the majority of consumption is in one location. Agglomeration is also growth enhancing because it increases knowledge spillovers if all R&D occurs in one location.

Increased freeness of trade is growth enhancing in the long run, but in the short run, the outcome is ambiguous. Increased freeness of trade is always growth enhancing if there is no change in the distribution of economic activity. However, as described in Fig. 1, increased freeness of trade can lead to a switch from the equal distribution outcome to the core-periphery outcome. While this is significantly growth enhancing for the region that becomes the core, it is temporarily growth diminishing for the periphery, while the two regions transition to the new equilibrium.

### 4.5 Impact of Knowledge Spillovers upon Economic Growth

Knowledge spillovers are generally growth enhancing. Increased knowledge spillovers mean firms have a lower cost of innovating, and therefore, there is a greater growth rate in varieties and consumption. If we are in the core-periphery equilibrium, increasing knowledge spillovers has no effect on growth because knowledge is unaffected by space since all production is in a single location.

However, as with agglomeration, the effect is ambiguous if there is a change in the steady state. A large enough increase in knowledge spillovers could lead to a switch from the core-periphery outcome to the equal distribution outcome (see Fig. 1). With a change in the location of production from one region to multiple regions, the knowledge spillover parameter now has an effect on growth when there was previously no effect. That is, firms initially had access to all knowledge because all manufacturing was in the same region, but in the new steady state, foreign knowledge is only partially available. While knowledge spillovers are generally growth enhancing, there is the possibility of knowledge spillovers being growth reducing in the former core region if it brings about the sharing of manufacturing.

If we consider the steady state where production is shared between locations, knowledge spillovers are growth enhancing. Furthermore, knowledge spillovers also make production in the equal distribution outcome more stable. That is, increasing knowledge spillovers means changes in trade costs are less likely to lead to a switch to the core-periphery outcome (see Fig. 1). With greater knowledge spillovers, production in both regions is a stable equilibrium for a greater range of trade freeness.

## 5 Variations to Incorporating Space in the Theory of Growth

In the NEGG literature, there are many variations of the model presented here. These include differences in the mobility of labor or capital, the inclusion of intermediate goods, heterogeneous firms, multiple labor types, and heterogeneous skill levels. Other areas of economics also incorporate space by using continuous space (rather than discrete regions), by defining location on an interval, by incorporating land as a factor of production, and by introducing congestion costs. All of these variations have different effects on the role of space, location, and geography on growth, but in general, incorporating space in the theory of growth has similar effects to those presented here.

### 5.1 Mobility of Labor and Capital

The model here describes the typical approach by NEGG scholars to incorporate space in the theory of growth with the inclusion of migration of skilled labor. The effect of footloose skilled labor can lead to catastrophic agglomeration, which means the model is unable to show other unequal internal steady states. We describe the model that includes skilled worker migration to demonstrate the role of firm and worker location choices and how migration influences innovation. Highly skilled workers and innovators are internationally mobile, so it is important to consider how this affects the location of innovation and subsequently economic growth.

Capital mobility is the ability for capital to shift between locations. In all endogenous growth models, growth comes from the accumulation of capital. Capital can come in a number of forms: human capital, physical capital, or knowledge capital. We think of labor and education as human capital, which is able to migrate between locations in the model above. Physical capital is the equipment used in production such as machinery and production plants. This has been excluded from the model above. Knowledge capital is the ideas generated in the innovation sector which are marketable and tradable through patents. This is the type of capital commonly modeled in endogenous growth and NEGG literature.

There are two options for the mobility of knowledge capital. With mobile capital, the owners of capital can decide where to locate production. If knowledge capital is mobile, the number of innovations produced (and owned) by one region may be different from the number of firms actually producing in that region. That is, the developer of a patent may choose to produce in a region other than their own. In this situation, the decision to accumulate capital is the same in all locations; the mobility of capital eliminates demand-linked causality such that the shifting of production does not shift the location of consumption or the earnings from owning a manufacturing firm. Alternatively with immobile capital, the owners of capital are only able to produce within the region where they are located. With immobile capital, any shift that favors production in one location leads to new capital in that region. Since owners are local, this also leads to expenditure shifting and further production shifting via the home market effect.

In many NEGG models, such as in Martin and Ottaviano (1999), Baldwin et al. (2001), and Baldwin and Martin (2004), migration is not allowed. In these models, workers are instead completely mobile between traditional manufacturing and innovation sectors but not between regions. These models require an extra assumption that a single country’s labor endowment must not be enough to meet global demand for traditional goods, to avoid complete specialization in manufacturing goods only.

In models with labor immobility and capital mobility, when we reach the steady state, the owners of capital are indifferent between producing in either region. With localized knowledge spillovers, however, innovators prefer to be located in the region with the highest level of manufacturing. Despite the differences, these models reach similar steady states to the model presented above. In particular, space has the same effects on growth because space is included using the same mechanisms with localized knowledge spillovers and transport costs. Agglomeration is growth enhancing due to localized knowledge spillovers, and knowledge spillovers are growth enhancing because they reduce the cost of innovation.

In models without labor or capital mobility, agglomeration is enabled by either vertical linkages in production or the spatial influence on knowledge creation and transfer. If NEGG models have immobile capital and mobile labor, these models have the same catastrophic agglomeration described by the model above (and most NEG models) because innovation occurs at a faster rate in a region with greater capital, and this is self-reinforcing as all new firms prefer to innovate in the location with the largest share of manufacturing. Whenever labor is mobile, agglomeration is catastrophic.

However, models with immobile capital and immobile labor offer an alternative advantage of unequal internal solutions. That is, various combinations of transport costs and knowledge spillovers yield steady states where one region has a larger share of manufacturing (but not all) than the other. As there is no migration, this means the region with the larger share of manufacturing has a share of traditional goods production smaller than the other region’s share of traditional goods production. Even though these models ignore the role of migration in economic activity and growth, it does allow us to consider the effect on knowledge spillovers and growth when there are unequal levels of agglomeration. The effect is very similar to the core-periphery outcome. Growth rates are equal in both regions because consumers in the low manufacturing region still benefit from innovations made in the high manufacturing region because of trade. Similarly, the growth rate in varieties is greater in the high manufacturing region because of localized knowledge spillovers. Real wages are also higher in a high manufacturing region. Without migration, there is no mechanism to equalize real wages between regions.

Another advantage of modeling with labor immobility between regions is there is no need for the modeling trick of the Krugman’s (1991) core-periphery model which fixes the share of skilled and unskilled workers. Instead, labor mobility between sectors equalizes real wages between manufacturing and traditional sectors within region, and zero transport costs in the traditional goods sector equalize nominal wages. Even though these alternative models have some features that are mathematically elegant, we chose to explore the model including migration because we view it as a more realistic description of spatial endogenous growth.

### 5.2 Vertically Linked Industry

Other types of NEGG models have vertically linked industry following the practice of some NEG models (Krugman and Venables 1995; Venables 1996). This is where goods are a factor of production. For example, final goods may be produced from a variety of manufactured intermediate goods (Yamamoto 2003), manufactured goods may be produced using a variety of manufactured goods (which have not been consumed), and/or the innovation sector could use manufactured goods as a factor of production (Martin and Ottaviano 1999).

If the vertical linkage is in the innovation sector, this generates a feedback between growth and agglomeration with a similar result to localized knowledge spillovers. Martin and Ottaviano (1999) do not use the localized knowledge spillover mechanism demonstrated here, and instead, their innovation sector uses manufacturing goods as an innovation input such that the location of manufacturing affects the cost of innovation through trade costs. Similarly, Yamamoto (2003) describes a model where final goods and innovation are produced using manufactured intermediate goods. This creates circular causation in growth and agglomeration because of the vertical linkages between intermediates and innovation.

### 5.3 Assumptions About Scale Effects

While modeling similarities have facilitated the addition of space to theories of endogenous growth using NEG techniques, aspatial theories of endogenous growth may also impose (possibly accidental) spatial assumptions in two region models. First-generation models (Romer 1990; Grossman and Helpman 1991a; Aghion and Howitt 1992) imply an empirically refuted scale effect that a larger population generates a higher rate of innovation. As a result, first-generation endogenous growth models combined with the NEG implicitly assume innovation benefits from agglomeration economies. The scale effect is typically eliminated by assuming diminishing innovation productivity for additional varieties to retain stylized facts about growth (Jones 1995; Howitt 1999). Yet when such inverse-scale assumptions are combined with the NEG model, it implies an opposite effect such that innovation is easier in peripheral regions. Many of these models derive conclusions about the spatial nature of growth directly from assumptions about scale effects or the inverse-scale assumptions that negate scale effects, without any spatial mechanism to drive these conclusions about the regional distribution of innovation. While agglomeration economies and other scale characteristics appear in regional growth analyses, spatial models can focus on the spatial causes of regional differences, rather than rely on implicit assumptions about growth.

Instead, the model in Bond-Smith et al. (2018) avoids the spatial implications of assumptions about scale effects by using the endogenous growth model of Young (1998) in a two-region model in order to focus on the regional mechanisms that lead to spatial economic phenomena. In this model, spatial mechanisms such as entrepreneur location, knowledge spillovers, and transport costs drive the regional distribution of economic activity. As a result, the location of economic activity is a balance of the home market, cost of living, market crowding, and technology clustering effects. Agglomeration is growth enhancing due to knowledge externalities from co-location rather than any implicit assumptions about scale effects in the model of growth. By avoiding the spatial impact of assumptions about scale effects, such models provide more relevant tools to examine the spatial mechanisms that drive regional disparities for innovation and growth.

### 5.4 Other Characteristics

There are many different factors which affect firm location decisions and subsequently space, innovation, and economic growth. Above, we have explored how these are dealt with in NEGG models by combining endogenous growth with the new economic geography and recognizing localized knowledge spillovers. But there are many more modeling choices for spatial factors which influence growth. For example, studying heterogeneous firms (Baldwin and Forslid 2010) helps describe the characteristics of which firms choose to locate in core or lagging regions. Other models include land requirements and continuous space (Desmet and Rossi-Hansberg 2009), whereby every firm is in a different location but willing to pay higher land rents to access more valuable locations. All of these have some influence on location choices for firms but ultimately demonstrate the same role of space in growth – that space is a barrier to knowledge transfer and technology diffusion which are inputs to innovation – and that policies or decisions by firms that reduce these spatial costs are growth enhancing.

## 6 Conclusions

We have described how NEGG models incorporate space into the Grossman and Helpman (1991a) product variety model of endogenous growth. Incorporating space into endogenous growth increases the complexity of these theoretical models. In all of these models with full local knowledge spillovers and partial global knowledge spillovers, space affects growth and growth affects location. The circular linked causality reinforces the core-periphery outcome of the NEG models. We show that integration between regions is more complex than is described by international trade models. In particular, we find that the cost of transferring knowledge between locations is important for firm location, stability, innovation, and growth.

From our discussion of the effect of space on growth through freeness of trade, agglomeration, and knowledge spillovers, there are a number of implications for economic policy in different locations. Agglomeration, freeness of trade, and knowledge spillovers are generally growth enhancing. The natural conclusion is that closer integration of economies will lead to increased growth rates. However, in these spatial models of growth, integration has two dimensions: trade costs and knowledge spillovers.

While traditional conceptions of integration refer to lowering of the cost of trading goods, Baldwin and Forslid (2000) show that combining theories of growth and space produces a more subtle view of integration where we can also view integration as lowering the cost of trading information. Integration policies which focus solely on free trade may be destabilizing and result in a deindustrialization of the periphery region. Alternatively, integration policies that also focus on knowledge spillovers (or entirely on knowledge spillovers) will be growth enhancing for both regions. The model here shows how this form of integration is stabilizing, while pure trade cost integration can be destabilizing.

While lowering trade costs induces uneven development, it also results in higher rates of economic growth. Alternatively, policies that improve knowledge spillovers improve stability of the location of economic activity. Growth policies should consider the effect of trade, knowledge spillovers, labor, and capital market integration.

## 7 Cross-References

## Notes

### Acknowledgments

We would like to thank Jacques Poot for excellent comments on earlier drafts of this chapter. Steven Bond-Smith would also like to thank the Royal Society of New Zealand Marsden Fund and the University of Waikato for financial support.

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