The unintended consequences of increasing returns to scale in geographical economics

Abstract

1. Introduction

The varying scale of economic activity is a striking characteristic of economic geography. People and businesses cluster in ever larger cities. Innovation, the engine of economic growth, is even more concentrated (Audretsch and Feldman, 1996; Bettencourt et al., 2007a, 2007b). It is unmistakable that scale has a fundamental role explaining local differences in productivity, innovation and growth. Understanding exactly how scale characteristics lead to regional differences requires explaining the explicit mechanisms that generate returns to scale. Returns to scale is the basis for many powerful results in economics and economic geography. Increasing returns to scale recognizes that production of outputs increases in greater proportion than required inputs. In economic geography and urban economics, returns to scale translates into forces of concentration, dispersion and spatial sorting that balance to determine the spatial organization and scale characteristics of economies (Proost and Thisse, 2019). Increasing returns to scale is also fundamental to endogenizing growth. First, it means that productivity depends on the stock of knowledge rather than its division between people (Romer, 1990). And second, endogenizing growth requires returns to some unit scale in the ideas production function, so that innovation is affected by effort (Jones, 1999).

But an approximation that knowledge has a single dimension leads to an implicit assumption that ideas production has returns to the aggregate scale, otherwise known as the ‘scale effect’. This is a well-known limitation of first-generation endogenous growth theories most publicized by Jones (1995b). It implies that an increase in any rival factor endowment in the economy results in a higher growth rate without any microfoundation for such an increase. If any rival factor is growing, it implies an ever-increasing growth rate and explosive output in finite time. Time series data on the growth of inputs to R&D for the USA are also not consistent with the functional form of ideas production in these first-generation models of endogenous growth (Jones, 1995b). Returns to scale is indeed a characteristic of innovation and endogenous growth, but the scale effect is widely recognized as an error in aggregation (Laincz and Peretto, 2006).

While the scale effect has been overwhelmingly rejected by scholars of (aspatial¹) endogenous growth (Bond-Smith, 2019), scale is widely recognized by scholars of innovation and geography as a key characteristic of innovation and growth (Audretsch, 2003). This is not a paradox. Instead, the mechanisms that generate increasing and decreasing returns to scale in geographical economics, which translate into spatial forces, determine a distribution of activity within the spatial economy but not necessarily an effect on the aggregate economy. In order to reconcile local, global and aggregate scale characteristics, researchers should carefully consider the specific appropriate spatial and economic scales in which returns to scale occurs. Avoiding the impact of the scale effect in these first-generation models requires a very limiting assumption of no population growth to avoid explosive growth rates. Unfortunately, growth models from (aspatial) economics have been used in economic geography without recognizing the spatial constraints of this limiting assumption, with unintended consequences.

Unsatisfied with the limitations of assuming no population growth, a dispute between researchers of (aspatial) economic growth has continued over three decades about the appropriate modeling technique to negate the scale effect (Bond-Smith, 2019). This dispute implies two types of solutions. First, Jones (1995a), Kortum (1997), and Segerstrom (1998) suppose that ideas become more difficult to find as the simplest ideas are found first. This eliminates the implicit assumption of returns to the aggregate scale in the long run, but allows returns to scale in the short run to endogenize growth in response to research effort. Unfortunately, in spatial models, the implicit returns to scale assumption that causes the scale effect and the diminishing returns to ideas assumption that is supposed to neutralize it, unintentionally translate into uneven forces of concentration and dispersion, respectively—as spatial models are supposed to do—but without any spatial foundation. This is because the limiting approximation that knowledge has a single dimension sustains the aggregate scale effect in the short run where spatial forces operate. Alternatively, Dinopoulos and Thompson (1998), Peretto (1998), Young (1998), and Howitt (1999) suppose that the important factor for endogenizing innovation is returns to relative scale in the product line. To achieve this, it is assumed that knowledge has two dimensions: variety and quality. As a result, increases in aggregate rival factors of production fragment across new product lines, removing the implicit limitation of increasing returns in the aggregate scale in both the short and long run without affecting returns to relative scale within product lines as required to endogenize growth. Hence, I describe these models as ‘scale-neutral’. In spatial models, this approach still eliminates the scale effect because returns to relative scale in the product line has no aggregate spatial implications. This allows deliberate spatial mechanisms included in the model to explicitly and intentionally define the spatial scales of returns to scale that directly define spatial forces in the spatial economy. Of course, knowledge is far more complex than these simplistic one and two-dimensional approximations, but the approximation of knowledge must be sufficiently sophisticated that returns to scale occurs at accurate spatial and economic scales.

While spatial economists attempt to describe regional growth with over-arching spatial equilibrium models, Storper (2010) describes in this journal that ‘much work from economic geographers is to go in the opposite direction from the spatial economists, emphasizing that spatial economic development is the result of unique, context-driven, place-specific combinations of forces that, as a consequence, can neither be modelled nor even subject to large-scale causal inquiry. At best, they can sometimes be captured in descriptive typologies’. There is an opportunity for a middle-ground, in which spatial growth models incorporate deliberate mechanisms that capture some elements of the context-driven and place-specific focus of geographers. Such models require otherwise scale-neutral models of growth that do not contain implicit biases regarding scale. Urbanization, Marshall-Arrow-Romer (MAR) and Jacobs’ externalities determine returns to scale mechanisms for a firm’s R&D facilities (Ciftci and Cready, 2011), the aggregate scale of a city’s R&D facilities (Bettencourt et al., 2007a, 2007b), the aggregate scale of a country’s R&D investment (Luintel and Khan, 2017,; ,Curtis et al., 2020) and the scale local or national portfolios of industries (Henderson, 1997,; ,de Groot et al., 2016). Such externalities also vary by industry and over time (Neffke et al., 2011,; ,Diodato et al., 2018,).

In order to explore this, I use a series of toy models to understand the role of assumptions about scale and growth in geographical economics. These ‘toy’ models are the most parsimonious specifications that generalize both endogenous growth and spatial growth theories. The models cited are of course far more nuanced and sophisticated than is articulated in this analysis, but the parsimonious model most clearly shows how implicit returns to scale assumptions about growth have unintended consequences for economic geography.²

This article serves a number of purposes. First, I highlight a fundamental misunderstanding of increasing returns to scale and the scale effect in endogenous growth theory when applied in a spatial context. I emphasize the importance of mechanisms that generate returns to scale in geographic research that translates into forces for concentration, diffusion and spatial sorting, such that spatial economists must be very careful about the scales in which increasing returns applies. In a series of toy models, I explore the spatial consequences of assumptions about scale effects, and techniques to remove scale effects. I review the broad economic geography literature for examples of the use and misuse of the implicit assumptions about returns to scale that are borrowed from aspatial endogenous growth but overlook its spatial limitations. I aim to reduce the risk that economic geographers or spatial economists inappropriately apply aspatial economic theory to spatial phenomena by offering techniques to avoid unintended spatial consequences. Lastly, I highlight that this article is but one example of the mistaken use of theoretical models and I encourage a best practice approach to economic theory that justifies all of its assumptions and notes its limitations.

2. Returns to scale assumptions and approximations

Assumptions matter. All theoretical models use assumptions to approximate relationships and avoid the complexity of a truly realistic but intractable model. Economists use theoretical mathematical models to organize their assumptions about the world and draw hypotheses about economic phenomena. Empirical research assumes an implicit underlying theory, even when a theory is not explicitly defined. Some assumptions are intended to approximate true facts about the world, but many are entirely unrealistic and used purely for simplicity. Assumptions eliminate specific variables so research can focus on what matters. Many assumptions do not matter. Some assumptions eliminate negligible factors or factors that are of no interest. And some assumptions are simply ‘modeling tricks’ to ensure the model is tractable, internally consistent or behaves in a way that reflects stylized facts. For example, the Constant Elasticity of Substitution (CES) function is often required to achieve tractable results in models of geography and trade or a balanced growth path in models of endogenous growth. As a result, the CES function has become a workhorse tool for models of growth, trade and geography. For all of these types of assumptions, realism is secondary to parsimony. Conclusions drawn from these assumptions are indirect because they need to account for the real-world context that is absent in the model. Theorists and empiricists alike, must understand the limitations of their assumptions to avoid drawing hypotheses and conclusions based on unintended effects. When assumptions are used to directly draw findings, those assumptions are crucial and should at least represent a realistic foundation on which to draw strong conclusions. In economic geography, such a foundation is also spatial.

In the most extreme case, the instrumentalist approach (e.g. Milton Friedman, 1953) values a theory by its predictive accuracy only, rather than whether its assumptions also correspond to the real world. Economic theory is often criticized that its assumptions are unrealistic (Martin and Sunley, 1998; Romer, 2015), but opposition to instrumentalism is not the main argument in this article. Instead, this article describes a contagion from implicit assumptions about growth that interacts with geographical economics—in ways that were never intended—to mistakenly draw false conclusions in both theoretical and empirical research about the spatial nature of economic growth. In this way, even the instrumentalist argument is defeated by its own standards when theories are applied in other contexts, such as the spatial application of endogenous growth theory. While the focus in this article is scale and geography, it is part of a broader narrative about the unintentional effects of overlooking the limitations of modeling assumptions to draw mistaken conclusions.

In its simplest form, increasing returns to scale is caused by a fixed cost of production or scale economies that are internal to the firm, but increasing returns can also create external economies. That is, returns to scale applies to specific (e.g. internal and external) scales such as the scale of a firm, industry, city or aggregate economy. Mechanisms that imply returns to scale are powerful tools to approximate real-world economic and spatial relationships. Like Blaug (1980) and Boland (1992), this is partly an argument for more realism in economic theory—specifically for these critical assumptions of returns to scale in geographical economics—in order to understand the distinct mechanisms that actually generate returns to scale and translate into forces in the spatial economy. While simplifying assumptions are necessary to approximate stylized facts or enhance tractability, they imply limitations that must be accounted for. In this way, if implicit assumptions about returns to scale and growth are used, their direct conclusions can be appropriately qualified, rather than used to define strong conclusions about relationships with scale. Before examining spatial growth models explicitly, I first explain the scales at which returns to scale assumptions apply to growth; geography and trade; and their combination.

2.1. Returns to scale in endogenous growth theory

Returns to scale is essential for understanding growth. Specifically, endogenous growth theory emphasized the role of nonrival knowledge spillovers for endogenous investment in innovation in return for temporary monopoly profits (Romer, 1990). Nonrivalry implies increasing returns to firm scale production because an idea can be used multiple times to produce outputs. A business can double production by duplicating its factory, but it can more than double production if its second factory uses a new design. The firm has constant returns to scale in its rival factors of production, but it has increasing returns to scale in the rival and nonrival factors combined. This is crucial for growth because per capita productivity depends on the stock of ideas, rather than its division between people. But for endogenous growth, returns to scale shows up twice: the scale of production of output and the scale of production of ideas. The ideas production function is explicitly assumed to have returns to firm scale research effort, which is crucial to endogenizing growth because it means that innovations are the result of deliberate research effort by firms to find new ideas. In this way, each business can make greater improvements by increasing its effort or investment in R&D.

Returns to scale is particularly unique in the ideas production function because ideas are both an output and nonrival input of innovation such that ideas multiply (Bond-Smith, 2019). While the ideas production function explicitly assumes returns to firm scale rival production inputs, the nonrival factor is increasing endogenously over time. In first-generation growth models (Romer, 1990; Grossman and Helpman, 1991; Aghion and Howitt, 1992), this means that growth in the scale of any rival input also makes use of the same endogenously growing pool of nonrival ideas, implying increasing returns for innovation to the growing aggregate scale of any rival input. In other words, first-generation models add an additional implicit assumption that innovations are produced with increasing returns to both the firm scale and the aggregate scale of any of the rival inputs.

Returns to the aggregate scale of the economy is problematic because population growth implies an ever-increasing growth rate to explosive levels of output in finite time. In first-generation endogenous growth models, this is known as the scale effect: that any rival factor increasing in scale results in an ever-increasing growth rate because the nonrival factor is also increasing endogenously through this circularity of ideas. This illogical result has been widely recognized as a limitation of first-generation endogenous growth models that requires an assumption of no population growth. The scale effect also implies that a larger economy grows faster than a smaller economy but involves no spatial or microfoundation for why this would be true: it is only a result of this implicit assumption that biases growth in favor of scale. Economic geographers and urban economists can strongly argue that such a result may well be true, due to agglomeration economies for innovation, but agglomeration externalities are the result of well-known microfounded mechanisms rather than an implicit assumption. This implies a further limitation of not examining scale differences if the underlying growth model contains the scale effect because it would mistakenly apply the benefits of agglomeration economies to scale differences defined by arbitrary unit scales rather than to their causal mechanisms. To be clear, returns to firm scale rival inputs to innovation is an approximation that is essential for endogenous growth, but requires appropriate limiting assumptions in the aggregate scale to avoid scale differences or scale growth. Clearly, this has implications for its use in economic geography.

The scale effect is eliminated in subsequent generations of semi-endogenous (Jones, 1995a; Kortum, 1997; Segerstrom, 1998) and Schumpeterian (Dinopoulos and Thompson, 1998; Peretto, 1998; Young, 1998; Howitt, 1999) growth models (See Bond-Smith, 2019 for a summary). Semi-endogenous growth models counteract its effect by assuming diminishing returns to cumulative ideas. For innovation, this still implies returns to scale in the short run so that ideas are produced endogenously, but eliminates returns to scale in the long-run aggregate scale where returns to scale and the ever-increasing flow of nonrival ideas cause the scale effect. In the balanced growth path, the net effect implies constant returns because the scale effect is matched by diminishing returns to cumulative ideas. Schumpeterian models of endogenous growth without scale effects allow innovation to expand in two dimensions (Dinopoulos and Thompson, 1998; Peretto, 1998; Young, 1998; Howitt, 1999) such that aggregate increasing returns are tempered by the additional dimension. This implies constant (or increasing; (Peretto, 2018)) returns for innovation at the relative scale of the firm so that idea production is endogenous. But it eliminates the scale effect from aggregate productivity growth because nonrivalry in the vertical dimension of innovation no longer interacts with aggregate scale in the horizontal dimension of the ideas production function. Instead, nonrivalry only interacts with the relative scale of research effort as required to endogenize growth. There is increasing returns to the scale of the firm, but there is now no aggregate scale relationship with productivity growth as greater scale is spread across additional varieties. By clearly understanding the scales at which returns to scale should apply, Schumpeterian models enable increasing returns to the scale of the firm or industry to achieve stylized facts about endogenous innovation while eliminating returns to scale in aggregate. Rather than eliminating increasing returns to scale, these techniques are additional assumptions that change the scale in which returns to scale appears in the ideas production function.

2.2. Returns to scale in trade and geography

New trade theory and new economic geography recognize that increasing and decreasing returns to scale occur over both aspatial scales (firm; industry; market; aggregate) and spatial scales (local; regional; urban; international; global). Krugman (1979) and Helpman and Krugman (1985) showed how this tool could be embedded into a general equilibrium model of trade that influenced the scale and location of production in order to explain that trade of similar goods between countries was the result of increasing returns to scale by differentiated producers. Krugman (1991) extended his trade model to economic geography by enabling the mobility of workers such that increasing returns to scale also had implications for the location of people and firms between cities or regions.

In these models of trade or geography, a fixed cost and otherwise constant returns to scale in production, imply increasing returns to the scale of the firm’s local production. That is, the firm’s local output increases by more than its required inputs simply by increasing the scale of production without having to incur additional fixed costs. With trade, this enables a larger factory in each location to produce differentiated varieties, rather than requiring multiple factories in each location. In this way, increasing returns to the scale of the local factory has implications for the scale and spatial distribution of economic activity. Transport costs are a pecuniary externality related to distance that result in decreasing returns to the scale of the local factory as any increase in the scale of production must be exported to customers further away. For an exporting firm, trade costs also imply increasing returns to the scale of the local market, as less production needs to be exported since local sales are more profitable. Competition also leads to decreasing returns to the scale of aggregate local and global export market that balances increasing returns to the scale of the local factory. While transport costs imply decreasing returns at the local scale of the firm, with vertical linkages it also implies increasing returns to the scale of local relative aggregate production as the firm makes greater use of local suppliers. Venables (1999) disaggregated production such that industries divide into clusters based on these vertical linkages, implying increasing returns to the scale of local supply chains. Other mechanisms related to city scale such as public goods (Andersson and Forslid, 2003), amenities (Zhang, 2007) and congestion costs (Ottaviano et al., 2002; Fujishima, 2013) also generate increasing or decreasing returns to scale that implies additional concentration and dispersion forces.³

In these models, increasing returns to aggregate industry or economy scales implies productivity benefits for density or concentration at those scales while decreasing returns to aggregate scales implies a congestion cost and dispersion force. Notably, all of these instances of returns to scale in theories of trade and geographical economics (excluding growth) are the result of deliberately assumed (and empirically supported) mechanisms that imply these returns to scale relationships at very deliberate spatial scales. Carefully applied to the correct scales, mechanisms that lead to increasing and decreasing returns to appropriate scales offer powerful insights on the spatial economy.

2.3. Returns to scale in geography and growth

The spatial nature of knowledge spillovers (Audretsch and Feldman, 1996) and similarities in modeling techniques implied a suitable marriage between endogenous growth and new economic geography (Bond-Smith and McCann, 2014). The continued development of endogenous growth theories over four generations (Bond-Smith, 2019) is repeatedly followed by spatial versions to understand the geographic implications of endogenous growth. By understanding how returns to scale applies in either growth or geographical economics, assumptions from endogenous growth theory must be carefully examined when they are applied in spatial models. Models that combine growth and geography contain a labyrinth of implicit and explicit assumptions about increasing and decreasing returns to scale.

The limitations of first-generation endogenous growth models discussed in Section 2.1 imply limiting assumptions of no population growth and no examination of scale differences. These limitations still apply to spatial applications but are typically overlooked by geographical economists. Specifically, the ideas production function in which ideas have a single dimension is applied directly to a spatial model, without any consideration for its appropriate spatial scale. The key modification to the ideas production function is typically to assume that nonrival knowledge spillovers do not transfer perfectly between locations to determine the spatial distribution of knowledge inputs to research. If there is an implicit assumption of returns to the aggregate scale in the ideas production function (as in the scale effect in first generation and in the short run in semi-endogenous growth models), then these imperfect spatial externalities also unintentionally determine the degree of aggregate returns to scale for innovation. But, there is no explicit microfoundation to believe that the degree of aggregate returns to scale in the ideas production function is specifically related to the spatial nature of knowledge spillovers: it is assumed implicitly by overlooking the approximation in first-generation growth models that knowledge has a single dimension only and its corresponding limitations. As a result, the assumption of returns to the aggregate scale unintentionally interacts with spatial assumptions about knowledge spillovers. This is in addition to the intended direct impacts of the imperfect spatial externality mechanism for knowledge spillovers.

When implicit assumptions about returns to scale from aspatial growth theories are mistakenly applied to inappropriate spatial scales in geographical economics, they unintentionally imply consequences for the spatial economy. In spatial models, an ideas production function with vertical returns to the aggregate scale unintentionally implies greater local innovation productivity for larger regions, which amplifies any spatial factor that affects innovation productivity as a larger group of researchers make greater use of available local knowledge. This means spatial models, which contain a first-generation or semi-endogenous growth model imply agglomeration economies for innovation without specifying a causal mechanism. Second-generation semi-endogenous growth models eliminate increasing returns to the aggregate scale in the long run only, but still sustain aggregate returns to scale in the short run and therefore unintentionally imply agglomeration economies for innovation in the short run. In the long run, this translates to an unintended level effect on incomes in larger scale cities. However, there are additional unintended consequences. These models negate the scale effect by assuming decreasing returns to the scale of cumulative ideas. Although this leads to the desired result of constant growth in an aspatial growth model, despite a growing population, in geographic space the assumption of decreasing returns to the cumulative scale of innovation unintentionally assumes decreasing returns to the local scale of ideas by interacting with nonrivalry while not changing the assumption of increasing returns to the local scale of innovation. Instead of eliminating the scale effect, in geographic models, this implies a balance of concentrating research effort due to the scale effect implying a level effect on incomes and diminishing innovation implying ideas congestion from cumulative local knowledge, without any spatial foundation for such forces.

The assumptions about increasing returns to the aggregate scale amplify—and assumptions about decreasing returns to remove the scale effect unequally diminish—any spatial mechanisms in the model because they are applied to inappropriate spatial scales without a spatial- or microfoundation, leading to mistaken conclusions about the spatial nature of growth. If such scale relationships can be logically justified then this implies that such mechanisms could be modeled explicitly, rather than implicitly embedded in the ideas production function.

Schumpeterian growth models without scale effects have the potential for a scale-neutral approach (Bond-Smith et al., 2018; Bond-Smith and McCann, 2019) that enables only the intentionally-included spatial mechanisms to affect the spatial equilibrium rather than implicit assumptions about growth and returns to scale. These models allow the inclusion of mechanisms that enable returns to scale for innovation at various scales, but require deliberate mechanisms to determine the spatial scales of returns to scale. For example, imperfect knowledge spillovers only determine the location of knowledge and its spatial dispersion as intended, rather than the degree of returns to local and global scale.

2.4. Returns to scale mechanisms

Although the scale effect has been overwhelmingly rejected by scholars of (aspatial) endogenous growth (Bond-Smith, 2019), scale is increasingly recognized by scholars of innovation and geography as a key characteristic of innovation and growth (Audretsch, 2003). While I argue that underlying growth models must be scale neutral in geographical economics, I am not dismissing scale as a characteristic of geography, innovation or growth. There is strong evidence that scale is an important factor in the spatial distribution of economic activity, and innovation in particular. Indeed, the World Intellectual Property Organization (2019) cites a number of economic forces that explain the concentration of innovation in urban hotspots related to pools of skilled labor, market scale and knowledge spillovers, as well as weaker dispersion forces. Scale has crucial but varied roles in economic geography (Krugman, 1991; McCann and Acs, 2011), firm size or industry structure (Acs and Audretsch, 1988; Tether et al., 1997). These roles are determined by many factors and mechanisms that economists and geographers both seek to understand. The extent that scale might also explain aggregate relationships at a country level, such as the recent experience of economic growth in China, or urban and regional level, such as the concentration of innovation in metropolitan areas (WIPO, 2019), would also invoke spatial mechanisms at appropriate nation or supra-regional scales that facilitate scale differences. It is these mechanisms, which explain scale phenomena in economic geography and represent the opportunity in geographical economics if underlying growth models are scale neutral. To understand these mechanisms spatial research should avoid implicit assumptions embedded in the economic growth model by translating these known forces into distinctive spatial mechanisms that determine returns to scale at appropriate spatial scales, which translate into concentration, dispersion and spatial sorting.

Implicit assumptions about ideas production with increasing returns to the aggregate scale of the local economy are often rationalized as ‘agglomeration economies’. However, agglomeration economies are not magic. Agglomeration economies are the result of positive externalities related to scale exceeding negative externalities related to scale. Such positive externalities relate to specialization, diversity and competition (Glaeser et al., 1992; de Groot et al., 2016), as well as the amenities or infrastructure that can be provided with economies of scale. When negative externalities of scale exceed the positive externalities, there are agglomeration diseconomies. Such negative externalities relate to land rents from competition for space, congestion, pollution, commuting time and overcrowding. For geographers and economists to truly understand appropriate spatial scales for returns to scale, these mechanisms should be assumed explicitly. Implicit assumptions about returns to local scales are not an explanation of agglomeration economies, but a control variable for unknown or unobserved agglomeration externalities at those scales.

Agglomeration externalities for innovation are typically categorized by MAR (Glaeser et al., 1992; Henderson et al., 1995), Jacobs’ (Jacobs, 1969) and urbanization externalities. Urbanization externalities refer to city-wide internal and external economies such as spatial sorting of highly educated workers (Eeckhout et al., 2014; Redding, 2020; Verginer and Riccaboni, 2021), centers of R&D (Bettencourt et al., 2007), national and international connectivity (Simmie, 2003) and infrastructure (Blind and Grupp, 1999; Aghion et al., 2013) that all facilitate innovative activity. MAR or Jabobs’ externalities refer to internal and external economies within cities based on their industrial structure. Specifically, MAR externalities refer to the benefits associated with specialized cities such as labor market pooling to reduce matching costs (Duranton and Puga, 2004), vertical linkages (Venables, 1996, 1999) and knowledge spillovers (Combes and Duranton, 2006) within industries or along supply chains (Isaksson et al., 2016). Businesses located close to their suppliers can benefit from shared knowledge through frequent interactions or from knowledge spillovers through imitation and skills transfers between competitors. Jacobs’ externalities refer to the benefits of industrial diversification or the benefits from combining knowledge across different industries. Serendipitous interactions between workers with different knowledge give rise to new product combinations, and stable diversified demand reduces risk from the volatility of input and output prices. Similarly, the concept of relatedness (Hidalgo et al., 2018) captures that diversification is a spectrum of related and unrelated varieties (Frenken et al., 2007; Boschma and Frenken, 2009). Furthermore, complex economic activities also concentrate in cities (Balland et al., 2020).

These examples of mechanisms and externalities that generate scale characteristics are not inconsistent with the rejection of scale effects by growth theorists. Instead, this multitude of mechanisms ultimately implies a spatial distribution within the economy but not an effect across the aggregate economy.

3. Two-region endogenous growth models

Endogenous growth theories can be characterized into generations by the scale effect (Jones, 1999; Bond-Smith, 2019). Each generation is followed by its application to core-periphery or trade models to understand the spatial implications of growth. Using toy models, this section disentangles the unintended spatial consequences of assumptions about returns to scale and growth: (i) in first-generation endogenous growth models; (ii) in semi-endogenous growth models; and (iii) in Schumpterian models without scale effects. Microdetails vary, but this analysis distils the essential elements. These toy models are the most parsimonious specifications that generalize both endogenous growth and spatial growth theories in order to clearly show the relationship between assumptions about returns to scale for innovation and conclusions in economic geography.

3.1. A simple growth framework

For each of the following models, I first present the standard (aspatial) endogenous growth framework. I then add a spatial dimension by extending the model to two regions. For simplicity, I add only one spatial mechanism, localized knowledge spillovers that weight knowledge spillovers by its original location. Other spatial mechanisms affecting the production of knowledge can be easily included and realistic models would obviously require multiple mechanisms. To keep the scope of the analysis focused, I only specify the portions of these models that are necessary to the main issue of concern.

In these models, final output is a composite good $Yt=AtαLYt$ where L_{Y t} is labor used in production, A_t represents technology or the stock of ideas and α represents the preference for technology improvement or the degree of increasing returns to scale in production. This function is usually the result of a CES aggregator of multiple intermediate varieties that is required to generate ‘balanced growth’ in which all sectors grow at the same rate, but other aggregators are also possible. The CES aggregator is useful in these toy models, but not essential to the argument about unintended spatial forces in the model. A variable elasticity of substitution function would generate unbalanced growth paths in which some sectors or factors grow faster than others as an additional source of growth (Palivos and Karagiannis, 2010). But the argument regarding unintended spatial consequences would still apply. Physical and human capital are left aside in these toy models in order to focus on technological change.

New ideas are the result of research effort that builds on the stock of ideas to increase productivity. There is free entry for entrepreneurs to develop new ideas. Productivity growth from the flow of new ideas is given by the function $At˙=f(LAt,At)$ where the dot indicates change in technology over time. This ideas production function assumes that new ideas are a function of the existing stock of ideas A_t (i.e. intertemporal knowledge spillovers) and the research effort to discover new ideas L_At. Along the balanced growth path, a constant share of labor (s < 1) is employed in research such that L_At = sL_t where $Lt=LYt+LAt$⁠.

These toy models are used to understand the effect of assumptions about returns to scale and mechanisms that generate returns to local scale on the spatial forces affecting growth and the location of economic activity. Migration and transport costs are not specified, but the toy models here are flexible enough to accommodate many alternatives.

3.2. First-generation endogenous growth

3.2.1. The aspatial model

In first-generation models of endogenous growth (Romer, 1990; Grossman and Helpman, 1991; Aghion and Howitt, 1992), the function

$At.=γLAtAt$

(3.3)

describes the flow of productivity improvements where $γ>0$ is a parameter for calibration. In this model, productivity A can be interpreted as product variety (Romer, 1990) or quality (Grossman and Helpman, 1991; Aghion and Howitt, 1992). Output per capita is proportional to the stock of knowledge $yt=YtLt=At(1−s)$⁠. Taking the time derivative of output and rearranging, growth of per capita output is ⁠. The growth rate of technology $gA=At˙At$ and per capita output are proportional to scale:

$gAt=γsLt$

(3.3a)

and $gyt=αγsLt$⁠. The scale factor L in Equation (3.3a) is the well-known ‘scale effect’ limitation in first-generation endogenous growth models, where per capita growth and technology growth are an increasing function of the scale of the economy L_t such that a growing population implies an ever-increasing growth rate.⁴ Models with a scale effect require a limiting assumption that population is constant to generate a constant growth rate.

3.2.2. The regional model

While the spatial knowledge spillover diminishes the use of spatially dispersed knowledge as intended, the scale effect unintentionally amplifies the impact of the spatial knowledge spillover.

Spillover factors result from assumptions about the economic geography of knowledge spillovers and scale factors result from assumptions about growth. It is easy to see that an increase in scale (L_t and $L˜t$⁠), holding all else constant, results in a higher growth rate. This is the original scale effect that a growing population results in an ever-increasing growth rate and a well-acknowledged limitation of first-generation endogenous growth models. Now consider an increase in scale in only the home region (L_t). This causes three types of changes in the growth rate. First, there is an increase in growth in both regions due to its effect over time on increasing spillovers in the home region via the distribution of economic activity (n) which is itself a dynamic function of the relative scale of research. Second, there is a smaller decrease in growth due to the effect of decreasing spillovers in the foreign region over time also via the distribution of economic activity. The third effect is the local scale effect in which increasing home region scale increases home region innovation by greater than its proportional increase in population due to the implicit assumption about increasing returns to the aggregate scale that is now applied at some arbitrary local level. The first two changes are the intentional result of assumptions about economic geography when the distribution of research effort varies between regions. But the third is the unintentional result of the assumption about increasing returns to the aggregate scale applied at a local level without a spatial foundation. The scale effect unintentionally amplifies local knowledge spillovers as a larger group of researchers make use of available knowledge.

To be clear, this means that implicit assumptions about returns to the aggregate scale for innovation in first generation models of endogenous growth unintentionally bias conclusions by exaggerating agglomeration economies for innovation without any foundation in economic geography. This is in addition to any specific spatial mechanisms that actually create agglomeration economies such as increasing returns to the scale of production, transport costs or the mechanism shown here: spatial externalities on the transfer of knowledge.

To avoid the scale effect, the aspatial model requires an assumption of no population growth. But in the spatial model the impact of the scale effect is not eliminated by assuming no population growth if scale differences occur at the regional level. Holding the distribution of technology (n) fixed and differentiating with respect to time, setting to zero and rearranging implies that constant technology growth requires no population growth and no change in scale between regions.

But the model is also no longer capable of examining the intentional specific channel of spatial externalities on knowledge spillovers that create regional differences for innovation.

That is, to avoid the scale effect in the spatial model implies an even more restrictive assumption that there is zero growth in any rival factor of production in each region. This would prevent standard spatial mechanisms such as footloose labor or capital from being included in the model. While increases in effort (s and $s˜$⁠) increase growth as intended by endogenous growth models, shifts in any rival factor (an increase in one region and an equal decrease in the other) also still imply the unintentional scale effect at a local level, despite the restrictive assumptions. So a further restrictive assumption is required that s and $s˜$ are equal. This assumption is so restrictive that the model is now unable to examine relationships between growth and economic geography or trade. I have been unable to find a paper with a Romerian growth model that does not violate this restriction as soon as spatial mechanisms are included. The only option would be to clearly acknowledge that the scale effect is also a limitation on the conclusions that can be drawn regarding the spatial innovation economy and that this additional impact from scale should not be expected because it is an unintentional consequence of modeling assumptions. It can still be concluded that local scale might affect innovation and growth but only as a result of specific mechanisms to facilitate scale relationships. Such an unacknowledged weakness in these models is very concerning.

3.3. Semi-endogenous growth without scale effects

3.3.1. The aspatial model

3.3.2. The regional model

Assuming many of the same restrictive assumptions discussed in Section 3.2.2 above finally eliminates the impact of the scale effect at local scales. Population growth in aggregate is now required to sustain growth, but must apply equally to both regions. Unfortunately, these restrictions now also prevent the model from being a useful description of economic geography.

3.4. Schumpeterian endogenous growth without scale effects

3.4.1. The aspatial model

‘Schumpeterian’ models of endogenous growth allow ideas to expand in two dimensions: new varieties and variety-specific quality improvements. These models recognized that population growth leads to an increase in the variety of products, whereas productivity relates to the quality of individual products. This avoids scale in the ideas production function by sharing a larger population across additional varieties.

Technology growth is dependent on the share of labor devoted to research rather than the scale of factor components. The variety dimension of technology eliminates the implicit assumption of aggregate increasing returns to scale for innovation (the scale effect). Including a love of variety (⁠$θ>1$⁠) implies a so-called ‘weak’ scale effect on income levels but not a ‘strong’ scale effect on growth rates (Jones, 1999). Since variety is proportional to population, g_F is proportional to g_L, which implies that a portion of growth is a result of population growth but not population size or scale. This weak scale effect is equivalent to the income benefits from greater division of labor. As a result, increased research effort by firms or sectors increases the growth rate of average technology, but increases in the scale of population only increase the number of varieties eliminating the scale effect from first-generation models. Theoretical arguments (Peretto, 2018; Bond-Smith, 2019)⁶ and the weight of empirical research (Zachariadis, 2003; Laincz and Peretto, 2006; Ha and Howitt, 2007; Ulku, 2007; Madsen, 2008; Madsen et al., 2010; Ang and Madsen, 2011; Venturini, 2012; Greasley et al., 2013)⁷ now strongly support the Schumpeterian approach to modeling endogenous growth without scale effects.

3.4.2. The regional model

Spatial implications result directly from assumptions about the economic geography of knowledge spillovers. Effort factors result from assumptions about growth but are now neutral to scale. As intended by endogenous growth theories, increased effort (s and $s˜$⁠) increases the rate of productivity growth by increasing the rate of quality improvement per firm but changes in scale (L_t and $L˜t$⁠) have no effect on growth since any increases in L are matched by increases in F, represented by the calibration parameter η. Changes in local scale affect local shares leading to standard changes in the number of firms and associated spillover factors $((n+(1−n)λ) and (nλ+1−n))$ as intended from the assumptions about the geography of knowledge spillovers but do not imply any other changes in the growth rate. Effort factors differ from scale factors (and relative scale factors) in first-generation and semi-endogenous models in that there is no additional effect from scale because there are no implicit assumptions about increasing returns to the aggregate scale. The relationships between growth and scale in the model such as agglomeration economies or diseconomies for innovation are now only a result of the geographic mechanisms deliberately included in the model, not a result of implicit returns to scale assumptions about growth. This is an important distinction: In the scale-neutral model, all spatial implications resulting in scale differences are the result of deliberate, microfounded, assumptions about economic geography, whereas in first-generation and semi-endogenous models there are unintended spatial impacts from implicit assumptions about returns to scale and growth. As a result, the Schumpeterian branch of endogenous growth theory enables research in economic geography without the many strict restrictions that apply to earlier generations of growth theory.

3.4.3. Implicit assumptions in the Schumpeterian model

Davis and Hashimoto (2015) argue that product development costs may be higher in larger markets to justify these spatial implications. But spatial models are also the appropriate tool to directly model the spatial mechanisms that lead to differences in product development costs rather than rely on implicit assumptions in the innovation production function. Such a mechanism would specify the details about exactly how costs are affected. Taking the explanations proposed by Davis and Hashimoto (2015) for differences in product development costs, the cost of rent, wages and congestion or any other spatial product development cost difference can form specific spatial mechanisms that generate diminishing returns to scale for innovation without implicit assumptions. Each mechanism translates into different policy implications to optimize the innovation system. The explicit inclusion of spatial mechanisms leads to conclusions that are directly tied to the spatial mechanisms that actually cause spatial phenomena in the model. In such a model, the stylized facts targeted by Howitt (1999) become a result of spatial externalities.

4. Discussion

The unintended consequences of implicit assumptions about returns to scale and growth extend beyond the parsimonious two-region toy models explained in Section 3. Any spatial models that rely on growth theories with implicit assumptions about aggregate returns to scale also derive unintended effects for spatial dynamics that result in mistaken conclusions misinterpreting the relationship between scale and growth. City-scale may well be a predictor of growth dynamics in various contexts, but urban economists and economic geographers limit their findings if implicit assumptions facilitate or amplify the spatial mechanisms that lead to scale economies. In this section, I explore how these implicit assumptions about scale effects and growth affect the broader economic geography literature and how research practices can be improved.

4.1. Broader impact of implicit assumptions about returns to scale on spatial research

To focus on trade, a number of trade models restrict migration and make other assumptions that limit the impact of the scale effect (Davis, 1998; Baldwin and Forslid, 2000b, 2010; Baldwin et al., 2001; Minerva and Ottaviano, 2010; Baldwin and Harrigan, 2011; Breinlich et al., 2014), but such limiting assumptions are not always distinctly defined or even sufficient. With labor mobility between sectors, economies of scale for innovation in the ideas production function still affect the spatial economy by amplifying any spatial mechanism for innovation when regions specialize.

Glaeser (2003) defines the subfield, the New Economics of Urban and Regional Growth focusing on empirical urban economics with spatial equilibrium equalizing utility across space. While linearizing an endogenous growth model implies that city scale is a determinant of productivity growth, spatial equilibrium assumes that the benefits of scale or density are offset by something else such as housing prices that would imply no role for scale in urban growth. Much of this research finds that urban growth of big cities tends to be the same as many small cities, but the result stems from an assumption of spatial equilibrium in which spatial forces have equalized, while retaining an implicit assumption of increasing returns to scale for innovation. As a result, assumed increasing returns to scale for innovation is immediately extracted by dispersion forces. Dispersion forces are modeled distinctly, but the concentration forces that determine the benefits of agglomeration economies for innovation are based on implicit assumptions. As a result, these models have no insight on the determinants of agglomeration economies for innovation at all.

Urban and regional economics models of growth typically combine first-generation endogenous growth theories into models of cities (e.g. see Duranton, 2006, 2007), in part because the implicit assumption of increasing returns to the aggregate scale of innovation (the scale effect) provides a useful modeling trick for assuming agglomeration economies for innovation. But first-generation models inadvertently predict that innovation (and productivity growth) increases with the aggregate scale of city-wide research effort without modeling any specific mechanisms for why such agglomeration economies occur. In this way, growth models in urban economics deny many mechanisms that support agglomeration economies for innovation such as sectoral clustering (Duranton and Puga, 2005; Bond-Smith and McCann, 2019) or skill-based sorting (Behrens et al., 2014; Davis and Dingel, 2019). To be clear, the scale effect is explicitly assumed to be the source of external economies associated with an agglomeration mechanism (as in Black and Henderson, 1999), without actually modeling these urban mechanisms. As a result, these models often draw strong conclusions about the benefits of agglomeration without understanding its causes. This implicit bias continues in very recent research. For example, Duranton and Puga (2019) use a Romerian ideas production function at the city scale to conclude that massive increases in the scale of major American cities, and declines in rural and regional city populations, would result in greater productivity growth. Of course, this is the conclusion when they assume it to be true but include no mechanism for why. Similarly, Arkolakis et al. (2020) model the role of immigrants in American growth by assuming a semi-endogenous model of growth applied at city scales, concluding that immigration restrictions hindered the benefits of scale effects. Indeed, this result confirms the finding above in Section 3.3.2 that scale effects in the spatial context are not eliminated by the semi-endogenous model yet they draw strong spatial conclusions based on implicit assumptions about increasing and diminishing returns in the growth model that have no foundation in the spatial economy at all.

Two recent articles are much more careful about the scales at which increasing and decreasing returns apply to innovation (Desmet et al., 2018; Aloi et al., 2021). Desmet et al. (2018) avoid the impact of the scale effect by not implementing any spatial externalities for knowledge spillovers and assuming no global population growth. While these are appropriate limiting assumptions, it means that the model is no longer useful for understanding many mechanisms from economic geography research on innovation. Aloi et al. (2021) instead define a scale-neutral Schumpeterian growth model so that the spatial mechanisms can be explicitly defined without any implicit scale bias, as I recommend in this article.

City scale could also still be predictive of innovation or growth in particular contexts. I am not arguing against its use as a model parameter, but its interpretation. City population may be used as a control variable for unobserved spatial factors that generate returns to scale at urban scales, which are unrelated to the variable of interest, but it is of limited use for drawing strong conclusions about specific agglomeration mechanisms or regional innovation production functions. Unfortunately, these strong conclusions are seen in the empirical urban economics literature wherever agglomeration economies for innovation are merely assumed with city size as a regressor (e.g. Glaeser and Gottlieb, 2009). Such conclusions are the consequences of implicit assumptions about increasing returns to scale in the growth model, which sidelines the externalities that cause agglomeration economies for innovation. Models in urban economics are the ideal tool for examining these causes, but only if appropriate assumptions are used as a basis for research.

Quantitative spatial economics as defined by Redding and Rossi-Hansberg (2017) offer a series of recent articles using dimensional space (i.e. along a line or a plane). While most of these models are static, and avoid modeling growth, a subset of this research led by Klaus Desmet and Esteban Rossi-Hansberg utilize a first-generation engine of endogenous growth (See e.g. Desmet and Rossi-Hansberg, 2009, 2010, 2012, 2014; Desmet et al., 2018; Nagy, 2020). The authors acknowledge that the scale effect is a limitation in terms of world population growth, but increasing returns to the aggregate scale for innovation is also an implicit agglomeration factor that amplifies spatial mechanisms at locations in space, rather than modeling the causes of agglomeration economies directly. Desmet et al. (2018) suggest that one solution is defining the cost of innovation as an increasing function of world population, which acts as a modeling trick so that the model is neutral to global scale, but does not neutralize implicit returns to local scale where the ideas production function applies. The scale effect would remain an implicitly assumed mechanism for agglomeration without modeling the externalities that cause increasing returns to agglomeration.

Lastly, the empirical regional science literature uses spatial econometric models to understand spillovers between regions (Lesage and Fischer, 2008). By adding spillovers, the interpretation of ‘scale’ is somewhat different but could lead to misleading conclusions about the role of proximity to neighbors. In order to estimate the contribution of various factors to growth, a log transformation of first-generation endogenous growth models implies a bias that favors the size of a region and its neighbors combined as a significant explanatory variable without explaining a cause. Empirical analysis that favors this approach may either leave out or over-emphasize key predictors that are correlated with scale. Taking this approach, Izushi (2008) tests a spatial Romerian model that includes the stock of local R&D workers as an explanatory variable. As I would expect, Izushi finds that this is likely to be a mis-specification because the local stock of R&D workers is ultimately a result of how spatial units are defined. Instead, Izushi finds in favor of a scale-neutral spatial version of Lucas (1988).

4.2 Avoiding unintended consequences

Economic geographers and spatial economists often seek to examine the explicit spatial mechanisms that create internal and external economies for innovation as a key determinant of the geography of economic activity. Therefore, it is essential that the unintended internal and external economies caused by implicit assumptions about returns to scale are eliminated from underlying models or, as a minimum, are appropriately qualified as a limitation of the analysis.

Unintended spatial consequences can be avoided by carefully applying assumptions that cause increasing and decreasing returns to the appropriate scales. In this way, aspatial aspects can be modeled in a manner, which is neutral to space such that spatial consequences are only a result of intentional spatial mechanisms. Similarly, empirical work can be interpreted correctly in terms of intentional spatial mechanisms rather than implicit scale assumptions. The spatial or aspatial nature and analytical purpose of assumptions should be clearly defined. Assumptions used for analytical convenience should be examined closely to have no unintended impact on direct conclusions. Assumptions that do not meet these strict requirements should be avoided, and if they cannot be avoided, clear limitations must be specified regarding the conclusions that can be drawn.

Implementing a scale-neutral approach requires a careful examination of assumptions. Aspatial mechanisms (e.g. growth) must be carefully checked for their unintentional spatial implications. In this way, the conclusions drawn can be explicit about the source of spatial phenomena. Researchers should be explicit about the spatial mechanisms in the model and their microfoundation and spatial foundation. Any microfounded argument to assume a scale or inverse-scale relationship should be modeled directly as a microfounded spatial mechanism and should not be implicitly assumed in the knowledge production function.

Spatial conclusions should be directly connected with their causal spatial mechanisms. A requirement to identify the spatial mechanism(s) highlights any spatial consequence that is otherwise unintentionally caused by implicit assumptions. This provides an opportunity to revise the underlying model when unintended spatial consequences are identified. If such a revision is not possible, models with unintended spatial consequences do not have to be disregarded entirely, but affected conclusions can be treated appropriately as limitations of the particular model that is otherwise used for analytical convenience. This allows a focus on the intentional spatial conclusions and their implications.

5. Concluding remarks

Scale and growth interact through very specific spatial channels, but currently, research on growth and geography, is limited by implicit assumptions about returns to the aggregate scale for innovation that bias findings in favor of scale. Crucially, theorists and empiricists can use deliberate mechanisms and channels to carefully understand the different scales in which increasing and decreasing returns to scale apply and capture the specific externalities that cause spatial phenomena. There is a particular opportunity for interdisciplinary research to build from the nuanced descriptive typologies in the human economic geography literature by incorporating the many specific mechanisms that contribute to the uniquely local characteristics of innovation in the spatial economy.

While many of these mechanisms imply agglomeration economies, the disentanglement of the various mechanistic causes may overturn the findings of existing research supporting agglomeration. For example, it enables the possibility of greater productivity growth in many specialized cities and reductions in inter-city transport costs that would substantially alter conclusions in Duranton and Puga (2019) and Arkolakis et al. (2020). Rather than the recommended urbanization policy targeting settlement in only superstar cities and expanding immigration, this would promote a national and regional transport and connectivity infrastructure program and specialized regional priorities for innovation (Daboín et al., 2019; Escobari et al., 2019).

While the main argument in this article is about the foundations of spatial economic theory, perhaps the more important conclusion is with respect to empirical research. There may well still be a scale relationship where scale interacts with innovation in some way that implies growing regional disparities based on scale differences. But economic geography and urban economics are the perfect vehicles for examining the spatial mechanisms driving such scale relationships. Empirical research should justify assumptions about scale and test different assumptions about scale as a robustness check. In doing so, findings regarding agglomeration economies are far more nuanced than simply ‘bigger is better’. For example, the mechanisms that enable related industry clustering in peripheral regions (Bond-Smith and McCann, 2019) can also form key drivers of agglomeration economies and the clustering of complex economic activity (Balland et al., 2020) with nuanced implications for regional growth policy in both cities and the periphery (McCann and Ortega-Argilés, 2015; Balland et al., 2018).

The article’s conclusions are also broader than geographical economics and economic geography studies of innovation or growth. Overlooking the limitations of any approximations and assumptions in economic and spatial analysis can lead to unintended effects and mistaken conclusions. A best practice approach should require that no variable, model type and model form (linear, exponential or otherwise) be used without justification and/or acknowledging the limitations and potential unintended effects.⁸

Footnotes

Acknowledgements

This research was completed while I was a Senior Research Fellow at the Bankwest Curtin Economics Centre, Curtin University and I am very thankful for their support. For many helpful comments and suggestions, I am grateful to the editors Frédéric Robert-Nicoud and Harald Bathelt, two anonymous referees, Philip McCann, Jakob Madsen, Katsufumi Fukuda and seminar participants at the University of Western Australia. Your insights have clarified any misunderstandings and resulted in a much improved article.

References