Space and technology in catching-up economies: “the city as a laboratory for innovation”

Abstract

Successful catching-up economies tend to exhibit a set of properties: (i) rapid increases in skilled labor and R&D investment, (ii) industrial structure upgrading from technology adoption toward technology creation sectors, and (iii) spatial concentration on the rise. The paper accounts for them by combining the mechanisms of skill-city complementarity and knowledge spillover in the setting of catching-up economies where ideas are more challenging to find along the growth path. With the diminishing possibility of adopting existing ideas, catching-up economies gradually rely more on creating ideas, which requires efficient knowledge spillover through urban spatial concentration based on ability sorting. We highlight that (i) catching-up economies choose proper city size and structure to benefit from growth effects through fostering innovation and spillovers, and (ii) rents here are not just an economic rent but play a productive role of allocating innovative abilities over space in an incentive-compatible manner. We discuss the policy on urban rents.

1 Introduction

Economic growth performance is diverse, ranging from the traditional advanced economies that evolved over several centuries of industrialization, to a few catching-up economies, and to a majority of economies with little sign of convergence. Interestingly, fast catching-up economies tend to exhibit not only (i) rapid skill upgrading, (ii) active R&D investment, and (iii) industrial structure upgrading toward technology-intensive sectors, but also (iv) high urban concentration and industrial clusters. Viewing fast catching-up economies as those getting closer to the technology frontier, this paper attempts to account for these facts.

Taking East Asian catching-up economies as an example, their growth path has exhibited rapid human capital accumulation and a rising proportion of the population in large cities and industrial clusters.¹ Large-scale investment in R&D occurs to narrow down technology gaps from the world frontiers and to build up technology-intensive industries, as exemplified by a high share of technology-related spending and innovation in this region. Studying the city and technology structure through the lens of fast catching-up economies gives an opportunity to better understand the micro-mechanisms and dynamics of the following key elements: urbanization, skills upgrading, spatial ability sorting, knowledge spillover and R&D investment, and economic growth.

Historically, urbanization played an important role in the developing stages of modern economies, although the degree of agglomeration may be weak for countries where regionally balanced growth has been pursued or human capital/technology and resources have already been abundant. Not to mention the volume of urban agglomeration economies,² we also see the skilled workforce growing more disproportionately in productive cities and nearby industrial clusters. Moreover, fast catching-up countries often experience continuous urbanization even after the early industrialization stages and without high urban unemployment. For instance, even after reaching the urbanization rate of 80% in the 1980s, the major cities and industrial clusters of South Korea not only expanded despite regulations against regional concentration but absorbed highly educated workers in fast-growing technology and R&D-intensive industries, creating a skill gap across regions without severe urban unemployment.³ Rents and housing prices appear unreasonably high in those areas, causing a severe social problem. Given all these features, we think that there are non-trivial mechanisms unexplained by traditional growth models, Harris-Todaro (1970) models, and urban/regional economics literature.⁴

The paper accounts for the aforementioned observations by combining the mechanisms of skill-city complementarity and knowledge spillover in catching-up economies where ideas are more challenging to find along the growth path. The mechanisms self-enforce, generating an endogenous growth path: with the diminishing returns to adopting existing ideas (e.g., falling technology gaps and more stringent enforcement of intellectual property rights), catching-up economies gradually rely more on creating ideas, which requires efficient knowledge spillover through urban spatial concentration based on ability sorting. While part of these mechanisms are already known or implied by existing studies, we show that combining them in an integrated setting opens up a new possibility to account for the facts listed above.

This paper contributes to the literature in the following ways: (i) it provides a more detailed micro-foundation of how cities facilitate innovation through higher knowledge spillovers, given information decay over space, (ii) it analyzes the dynamic incentives in utilizing knowledge spillovers for technology creation that shape urbanization and technology structure of catching-up economies, and (iii) it demonstrates how a geographic concentration of skilled workers occurs endogenously through self-selection and how knowledge spillovers are more important for human capital in technology creation sector.

Given their spatial structure, cities allow more active knowledge spillovers than non-cities since knowledge spillovers diminish rapidly with distance. When more able workers learn and create more profitable ideas, we can obtain the following self-selection mechanism: (a) high-ability workers invest more in human capital to become skilled workers (selection), (b) they sort themselves into cities with a more able worker residing closer to the city center despite paying higher rents (sorting),⁵ and (c) they conduct technology creation activities in cities because there they can more efficiently conduct R&D activities using more knowledge spillovers based on their residential choice (learning)—i.e., “the selection, sorting, and learning (SSL) mechanism.” This self-selection process continues along the growth path, accounting for the aforementioned observed correlations among skill investment, urbanization, and economic performance in fast catching-up economies.

The process of catching-up gradually redirects the driving force of growth from technology adoption to technology creation. The latter process requires new basic R&D investment, which cannot be replaced by adopting the existing pool of ideas from advanced countries and therefore the demand for skilled labor rises. In line with Romer’s (1990) variety expansion model, we posit that technology improves by combining a variety of blueprints as in endogenous growth models but we add skill-city complementarity and knowledge spillover in a catching-up economy context.⁶

The SSL mechanisms describe how a city attracts skilled workers who raise their productivity of basic R&D activities through inner-city knowledge spillovers. The raised productivity in turn attracts skilled workers into the city, reinforcing agglomeration and economic growth. This process leads to the complementarity between urbanization and the basic R&D activities of skilled workers. More importantly, we demonstrate the inner-city residential sorting mechanism, describing how knowledge spillovers among urban skilled workers are structured and affect their R&D productivity and rents: more able workers can generate more knowledge spillovers, and knowledge spillovers attenuate with the distance between givers and takers of knowledge. As a result, a location closer to the city center allows for higher knowledge spillovers, attracting more able workers despite higher rents. As we will see later, a combination of rents and information processing capacities over space provides incentive compatibility conditions for our spatial sorting growth equilibrium.

We characterize the role of rents in our innovation economy. As will be shown later, rents here are not just economic rent but play a productive role of allocating innovative abilities over space in an incentive-compatible manner. The equilibrium rent of a location falls with its distance from the urban center, supporting spatial ability sorting equilibrium. This spatial pattern of rents depends on the magnitude of urban knowledge spillovers, in contrast to the usual commuting (time) cost argument (e.g., Abdel-Rahman and Anas 2004; Behrens et al. 2010). The nature of returns from a fixed input combined with the scarce resource allocation role of rents warrants a proper rent policy design from social welfare and growth perspectives.

1.1 Literature

The mechanism behind catching-up draws on the interaction between human capital and R&D investment in the economic growth context. Aghion and Howitt (1996) investigate the composition between research and development in a Schumpeterian growth model. Vandenbussche et al. (2004) examine the composition of various types of human capital in a model of technology adoption and innovation. Using the notion of absorptive capacity, Aghion and Jaravel (2015) argue that knowledge spillovers can induce complementarities in R&D efforts. Ha et al. (2009) present empirical evidence that a catching-up economy tends to increase the demand for skilled labor and basic R&D activities. While retaining the main elements of the literature (e.g., R&D, technology adoption vs. creation, and heterogeneity in human capital), our paper focuses on how catching-up economies take advantage of these elements from spatial and dynamic perspectives. As in Romer (1990), creativity in our paper comes from various blueprints of ideas. On top of this, we show spatial sorting provides an environment for creating new ideas more efficiently.

Skill-city complementarity leads to spatial sorting, which has been addressed in the urban wage premium literature as an essential element (e.g., Combes et al. 2008; Mion and Naticchioni 2009; Matano and Naticchioni, 2011; Bacolod et al. 2009; Kim 2014; Baum-Snow and Pavan 2012). While sharing a similar spirit with the literature, we look at spatial sorting in the context of taking advantage of knowledge spillover and creation possibilities in catching-up economies with a low stock of knowledge. The resulting equilibrium wage and rent at each point in urban space need to be interpreted differently than the classic views of Roback (1982) and/or Tiebout (1956), where amenities are exogenous and agents are homogeneous. We present that spatial ability sorting within a city fosters the efficiency of knowledge creation, enhancing economic growth.

The vast literature on knowledge spillover studies human capital externalities: the interactions among various workers lead to sharing of ideas, which raises their productivity and becomes a driving force of economic growth (e.g., Marshall 1890; Lucas 1988; Lucas and Moll 2011; Glaeser 1999, Palivos and Wang, 1996 among others). Subsequent studies (e.g., Rauch 1993; Acemoglu and Angrist 2000; Moretti 2004a and 2004b, and De-la Roca and Puga 2017 among others) provide evidence of knowledge spillover effects on wages. More direct empirical evidence shows that cities facilitate innovation, diffusion, and sharing of knowledge and skills (e.g., Jaffe et al. 1993; Audretsch and Feldman 1996).⁷

We advance the literature (i) by providing a more micro-foundation of city space and distance: city space is contested among workers and those who utilize urban knowledge spillovers more efficiently (e.g., R&D workers) will take the city space based on comparative advantages. The chance of coming across new ideas is highest at the city center, given the decay of information over space combined with spatial ability sorting we will analyze later. While a more productive place for idea creation requires higher rents, new ideas yield high wages and a greater variety of new ideas boosts technology and production, a source of growth. (ii) In addition, we add to related literature by analyzing the dynamic incentives that catching-up economies shape their spatial and industrial technology structure at a point in time to benefit more from urban knowledge spillover for technology creation than from the comparable spillover from the rest of the world for technology adoption. (iii) We confirm that a city with high average human capital yields high wages (e.g., Rauch 1993), while relaxing the assumption that human capital is an exogenous regional productive amenity. That is, a geographic concentration of skilled workers occurs endogenously along the growth path of catching-up economies, still accounting for the wage premium. (iv) Our view is also consistent with Abel et al. (2012) showing evidence that the external effect of human capital is greater in knowledge-intensive industries. In addition, we can also account for empirical evidence that rising skill premium is spatially limited to cities (Chung et al. 2009).

Finally, despite the vast related literature, there is little research that examines the interactions among (i) knowledge spillovers, (ii) agglomeration, (iii) R&D, (iv) spatial ability sorting, and (v) economic growth of catching-up economies in an integrated setting. Our paper fills this gap. In particular, this study analyzes the incentives that catching-up economies with a low initial stock of knowledge choose proper city size and structure to benefit from growth effects through fostering innovation and spillovers in a heterogeneous agent model.

The rest of the paper is organized as follows. Section II presents the model and derives the demand and supply of different types of human capital and R&D investment. Section III characterizes urban spatial sorting, the equilibrium structures of R&D, human capital investment and rents, and conducts some comparative statics analyses for establishing our main arguments. We also offer a numerical example illustrating the model’s dynamics. Section IV concludes.

2 Model

2.1 Environment

2.1.1 Individuals’ heterogeneity

We consider an economy with income maximizers who are mobile, live for a period and leave ideas (e.g., skills and knowledge as social capital) to their descendants. In each period (e.g., a generation), a measure one of a continuum of individuals is born with heterogeneous abilities ${a}_{j}$ that is uniformly distributed on the interval $\left[\underline{a},\overline{a}\right]$:

$$a_{j} = j \in \left[ {\underline {a} ,\overline{a}} \right].$$

(1)

The ability distribution is identical across generations but future generations inherit a greater stock of knowledge, which leads to higher productivity. We abstract from population growth for brevity.

As essential economic activities, these individuals invest in education, work, consume, and choose their residence, which is also identical to their workplace for the sake of analytical simplicity. Education cost is paid with part of their future wages. There are three essential activities of our model—(i) creating new ideas, (ii) commercializing created ideas, and (iii) adopting world frontier ideas. To perform these activities, individuals choose one of the three types of human capital investment. Without education, they become ordinary (unskilled) workers U conducting activity (iii) above; with some education, they become skilled workers S, conducting activity (ii); and with more education, they become highly skilled workers H, conducting activity (i). The workforce at time t (or generation t) is expressed as:

$$U_{t} + S_{t} + H_{t} = \overline{a} - \underline {a} .$$

(2)

For the sake of exposition, we set $\overline{a}-\underline{a}=\overline{L }$ with $\underline{a}=0 {\text{ and }} \overline{a}=\overline{L },$ where $\overline{L }$ is normalized at unity in our theory section.

The more able a worker, the less he pays for educational attainment, to be specified later. Given her human capital, a domestic worker holds a job in the knowledge sector that consists of (i) technology adoption and (ii) technology creation sub-sectors. First, in the adoption sector, the R&D activities of ordinary workers U produce various intermediate goods (blueprints) for technology adoption by using the ideas available from advanced economies and inherited ideas from ancestors. Unsurprisingly, these activities become less lucrative as the country gets closer to the world technology frontier, which is considered to be external for analytical convenience.

Second, in the creation sector, highly skilled workers H engage in creating new basic ideas by conducting basic R&D activities (e.g., Babina et al. 2023). By residing closer to the city’s center, they can benefit more from knowledge spillovers, leading to a higher productivity of basic R&D activities. The inventor of each new basic idea is modeled to monopolize the profit during her life only. Third, using both freely available creative ideas from domestic R&D and the inherited ideas from ancestors, skilled workers S conduct R&D (commercialization) activities to produce various new intermediate goods. These intermediate goods are traded at monopolistically competitive markets as in Romer (1990).

The intrinsic complementarity in production between H and S workers arises due to the dependence of S workers on new ideas created by H workers. However, unlike H workers, S workers do not rely much on the face-to-face interactions that are essential for knowledge creation and spillovers. For this reason, U and S workers can be said to be independent of space for production. The firm produces the final output by using technology that combines these types of knowledge labor U, S, and H. Meanwhile, workers supply labor inelastically. Finally, an externally given population of landlords lives on rental incomes without working.

2.1.2 Spatial structure and new concepts

Now, we discuss the spatial structure of the model: city vs. non-city.⁸ As shown in Fig. 1, a line represents space with a center denoted by location $x=0$. Within the city, we assume that population density is uniform with a certain high-density value normalized at 1. Therefore, when more individuals move to the city, the city boundary expands. Given their (dense) spatial nature, only cities provide externalities of knowledge spillovers. More able workers can produce greater ideas and hence emit greater positive urban externalities. At the same time, they can absorb a greater amount of ideas (knowledge spillovers) as urban residents. Some remarks are in order.

Fig. 1

City with size H. Notes: (i) The city center is at the middle, i.e., point 0. (ii) The population is uniformly distributed with the density equal to 1 over the left and right boundaries {− H/2, H/2}, which yields the city size equal to H. (iii) In the model’s spatial sorting equilibrium, the highest-ability worker resides at the center $x=0$, and the lowest-ability worker resides at the boundaries

First, the space of city yields the well-known knowledge spillovers, and therefore it is contested among workers based on comparative advantages. Following Abel et al. (2012), we take knowledge spillovers as more essential for creating new ideas than adopting existing ideas or doing routine work. As will be shown later in the spatial equilibrium analysis, benefiting from urban knowledge spillovers, highly skilled workers H can more efficiently undertake idea creation and can afford to pay high urban rents as well, and therefore, H workers reside in the city based on the comparative advantage; the equilibrium city size is therefore H. Of course, we may allow for part of other types of workers (e.g., U and S workers) as urban residents, but our simplifying assumption does not affect main results.⁹ Landowners supply fixed housing services and take rents from urban workers H. To simplify the story, we normalize non-city rents at zero.

Second, creation of new ideas diminishes with distance in our spatial ability sorting equilibrium to be shown later.¹⁰ For a worker with ability $a$ residing at the equilibrium location of the city denoted by x with $x\in \left[-H/2,H/2\right]$ and $x=0$ at the city’s center (see Fig. 1), the equilibrium distance from the city’s center is denoted by z (i.e., $z=|x|$ by definition) and satisfies $z(a)=(1-a)/2$ under ability sorting and the uniform distribution of residence (see Fig. 2). A worker with ability a creates new ideas $I\left(z(a)\right)$ that are shared among city residents through knowledge spillovers,¹¹ but idea creation declines geometrically with distance z due to ability sorting (see Fig. 3)¹²:

$$I\left(z(a)\right)=I\left(0\right){\text{exp}}\left(-\theta \cdot z(a)\right)=I\left(0\right){\text{exp}}\left(-\frac{\theta \left(1-a\right)}{2}\right)$$

(3)

where $z=(1-a)/2$ in our ability sorting equilibrium, $I(0)$ is the knowledge spillover generated by the highest-ability worker with $a=1$ residing at $z=0$ (city center), and $\theta$ reflects that idea creation becomes increasingly difficult for low-ability workers living far away from the center.

Fig. 2

Spatial distribution of ability, and threshold points. Notes: The horizontal axis measures ability of individuals from 0 to 1. There are two threshold points ${a}_{S}^{*}$ and ${a}_{H}^{*}$. Given the uniform density function for ability, workers whose types are either U, S, or H are measure 1: $U+S+H=1$

Fig. 3

Created ideas over urban space under ability sorting equilibrium. Notes: (i) Ability sorting causes workers to be distributed from the center to both ends of the city in the order of their ability. The resulting urban ideas that are freely available diminish from the center to the peripheries. (ii) The specification for I reflects that ideas creation becomes increasingly costly for low ability and therefore declines at the rate of $\theta.$

Third, we express the heterogeneity in benefiting from knowledge spillovers using the information absorption capacity $c(a)$: $c^{\prime}(a)>0$ and $c(a=0)=0$. Later, we will specify $c(a)$, using an exponential form similar to (3).

Fourth, knowledge spillovers attenuate with distance at the rate of $\delta$ (e.g., Rosenthal and Strange 2008): with a wider distance d between the sender and receiver of knowledge spillovers, a receiver obtains less information by a factor of ${e}^{-\delta \cdot d}$. The geographic attenuation rate in our knowledge economy is comparable to the notion of infrastructure congestion in conventional urban economics.¹³

Fifth, the within-city sorting and rents are based on the property that high-ability workers can adopt a high level of knowledge spillovers to improve their basic R&D productivity and thus wages. The following two micro-features lie behind this. (i) First, a location closer to the center can provide a higher information flow (spillover). (ii) Second, higher ability helps adopt a higher amount of information. In contrast, relatively low-ability workers among H are forced to move away from the city’s center. This is not only because low ability limits adopting knowledge spillovers, but also because locations closer to the center entail higher rents.

Finally, reflecting the differential idea creation and flows across locations, we assume that the information processing capacity at each location develops differently following residential workers’ needs for information flow, which depends on ability.¹⁴ Therefore, a high-ability worker cannot capture information flows as much as he wants in a remote location for a low-ability worker, where the information processing capacity is underdeveloped for a high-ability worker. Using the parameter $\omega$ with $0\le \omega \le 1$, we can express a worker’s effective absorptive capacity (e.g., Aghion and Jaravel 2015) as follows: a worker with high-ability a in a low-ability region x adopts information not as much as $c(a)$ but $\omega c({a}_{x})+(1-\omega )c(a)$, where $c({a}_{x})$ is the amount of information extracted by the equilibrium ability worker at location $x$: ${a}_{x}$, and $\omega$ is the degree of being constrained at a location. Conversely, a low-ability worker in a location for a high-ability worker cannot extract the large volume of information that a high-ability worker does, so the information absorption constraint is not binding for the low-ability worker: she adopts simply $c(a)$.

This feature supports our spatial sorting equilibrium (see Fig. 2), which matches ability with location in an incentive-compatible manner. That is, workers do not move away from their equilibrium location to others with lower rents since doing so leads to a net loss: the information-handling capacity there is too low for a high-ability individual and the productivity/wage loss (–) dominates the rents saving ( +).¹⁵ This assumption is comparable to the zoning regulation in the Tiebout (1956) equilibrium. As the homogeneous agents model of Tiebout needs the zoning regulation, our heterogeneous ability sorting model needs a differential location-specific information processing capacity for equilibrium.

2.2 Technology structure

The technology sector consists of technology adoption and technology creation, and their levels are measured by functions ${A}_{t}$ and ${C}_{t}$, respectively. Overall technology level ${T}_{t}$ is posited to depend on the levels of technologies A and C with a constant elasticity of substitution $\sigma$ between them:

$$T_{t} = \left[ {A_{t}^{{\frac{\sigma - 1}{\sigma }}} + \kappa C_{t}^{{\frac{\sigma - 1}{\sigma }}} } \right]^{{\frac{\sigma }{\sigma - 1}}} ,$$

(4)

where $\kappa$ implies the relative importance of creation. For analytical simplicity, we assume that the gross substitutability between technology adoption and technology creation processes is infinite, $\sigma =\infty$¹⁶:

$$T_{t} = A_{t} + \kappa C_{t} .$$

(4′)

We posit that technology improves with the variety of ideas.¹⁷ Invoking Romer’s (1990) variety expansion model, we describe technologies A and C that are determined by combining various intermediate goods ${x}_{{A}_{t}}(i)$ and ${x}_{{C}_{t}}(i)$ as many as the number of “blueprints” G_A and G_C, respectively:

$${A}_{t}={\left[{\int }_{0}^{{G}_{At}}{x}_{At}(i{)}^{\frac{\alpha -1}{\alpha }}di\right]}^{\frac{\alpha }{\alpha -1}}$$

(5)

and

$$C_{t} = \left[ {\mathop \int \limits_{0}^{{G_{Ct} }} x_{Ct} (i)^{{\frac{\alpha - 1}{\alpha }}} di} \right]^{{\frac{\alpha }{\alpha - 1}}} ,$$

(6)

where G_At represents the pool of all basic ideas appropriate for technology adoption at time $t$, G_Ct is the comparable counterpart for technology creation, and $\alpha >1$ indicates the usual high elasticity of substitution among various intermediate goods as in the literature. The sum of G_At and G_Ct will determine the total level of knowledge of this economy G_Tt, which will be inherited to the next generation $t+1$ as social capital. We now discuss how the pool of basic ideas for adoption and creation, G_A and G_C, evolves over time.

2.2.1 Basic ideas for creation

An innovator, a highly skilled worker j of generation t with ability ${a}_{j}$, creates new basic ideas as many as $B({a}_{j};{H}_{t})\cdot {G}_{{T}_{t-1}}$ by combining the product of basic R&D efforts for new ideas $B({a}_{j};{H}_{t})$ with existing ideas ${G}_{{T}_{t-1}}$, as in Romer (1990), where (i) $B({a}_{j};{H}_{t})$ is the productivity of basic R&D activities for ideas creation of a highly skilled worker j with ability ${a}_{j}$ residing in the city with size H (see Appendix A for details), and (ii) ${G}_{{T}_{t-1}}$ is the total number of existing ideas accumulated in the economy at time $t-1$ (inherited from generation $t-1$) that are freely accessible for creation activities, i.e., spillover effects. In our ability sorting equilibrium to be specified later, the properties of $B({a}_{j};{H}_{t})$ include: (i) it increases with city size $H$ and the workers’ ability ${a}_{j}$.¹⁸ This is because knowledge spillovers from agglomeration effects increase with size $H$, and because higher-ability workers adopt more from knowledge spillovers; and (ii) under some regularity conditions, $B({a}_{j};{H}_{t})$ exhibits properties consistent with data or common sense.¹⁹ We adopt the Romer specification of new knowledge creation, using the existing knowledge stock like “standing on the shoulders of giants” combined with a new input of basic R&D investment. The new basic ideas invented by the technology creation sector $\Delta {G}_{c}$ at time t is therefore described by the following dynamic equation:

$$\Delta G_{Ct} = f_{C} \left( {\mathop \int \limits_{{a_{H}^{*} }}^{1} B\left( {a_{j} ;H_{t} } \right)dj} \right) \cdot G_{Tt - 1} \Rightarrow \Delta G_{Ct} = f_{C} \left( {\overline{B}_{t} H_{t} } \right) \cdot G_{Tt - 1} ,$$

(7)

where ${\Delta G}_{Ct}={G}_{Ct}-{G}_{Ct-1}$, $\overline{B }$ is the average R&D productivity, $H$ is the equilibrium number of highly skilled workers ($H=1-{a}_{H}^{*}$) (i.e., city size), $\overline{B }H$ is the stock of total ideas created in the urban sector, and ${f}_{C}$ is a monotone function. Here, workers with ability greater than ${a}_{H}^{*}$ become highly skilled workers and migrate to the city.

2.2.2 Basic ideas for adoption

We posit that, benefiting from knowledge spillovers from the world frontier, the new ideas available in the adoption sector $\Delta {G}_{A}$ evolve over time by:

$${\Delta }G_{At} = f_{A} \left( {\tau_{t} } \right) \cdot G_{Tt - 1} ,$$

(8)

where ${\Delta G}_{At}={G}_{At}-{G}_{At-1}$, and ${\tau }_{t}=\tau \left(\overline{T }-{T}_{t-1}\right)$ with ${\tau }_{t}>0$ and $\tau ^{\prime}>0$. Technically, our specification of technology adoption reflects the reality that ideas are challenging to find along the growth path. For an economy at the stage of catching-up, a greater technology gap $\overline{T }-{T}_{t-1}$ from the world frontier $\overline{T }$ permits productivity improvement for the adoption sector through knowledge spillovers; and ${f}_{A}$ is a monotone function of ${\tau }_{t}$. Agents do not optimize on the choice of $\tau$, and they take the technology gap $\tau$ to be given externally.²⁰ The intuition behind (8) is that by combining the knowledge spillovers ${\tau }_{t}$ from abroad with the existing domestic ideas ${G}_{Tt-1}$, the catching-up economy creates new adoption ideas, an analogue to Romer’s (1990) idea creation.

2.2.3 Evolution of overall ideas

We can obtain the economy’s change in the overall pool of ideas ${\Delta G}_{T}$ that evolves as follows:

$${\Delta }G_{Tt} = \left[ {f_{C} \left( {\overline{B}_{t} H_{t} } \right) + f_{A} \left( {\tau_{t} } \right)} \right] \cdot G_{Tt - 1} .$$

(9)

where ${\Delta G}_{Tt}={G}_{Tt}-{G}_{Tt-1}$ and ${\Delta G}_{T}={\Delta G}_{At}+{\Delta G}_{Ct}$. At this point we can see that the growth rate of total pool of ideas is ${f}_{C}\left({\overline{B} }_{t}{H}_{t}\right)+{f}_{A}\left({\tau }_{t}\right)$, where $\tau$ falls with development toward zero while $\overline{B }H$ rises with development, converging to a certain steady-state value corresponding to the long-run urbanization level. Reviewing the structure of dynamics in Eqs. (7), (8), and (9), we can see that the evolution of ${G}_{T}$ determines ${G}_{A}$ and ${G}_{C}$.

2.2.4 Intermediate goods production

Now, we describe the production of intermediate goods to be used in technology adoption and creation sectors, respectively. The simple linear production technology below produces the intermediate good (e.g., blue print) ${x}_{jt}$ using the j-sector (j = A or C) human capital ${l}_{jt}$:

$$x_{jt} = l_{jt} {\text{ for }}j = A, C.$$

(10)

Next, the aggregate human capital levels of technology adoption and creation sectors are defined as ${\int }_{0}^{{G}_{At}}{l}_{At}(i)di$ and ${\int }_{0}^{{G}_{Ct}}{l}_{Ct}(i)di$, respectively. They are embodied in unskilled (U) and skilled (S) workers, respectively, and satisfy the following resource constraints:

$$\mathop \int \limits_{0}^{{G_{At} }} l_{At} \left( i \right)di = D_{At} \cdot U_{t} ,$$

(11)

$$\mathop \int \limits_{0}^{{G_{Ct} }} l_{Ct} \left( i \right)di = D_{Ct} \cdot S_{t} ,$$

(12)

where ${D}_{At}$ and ${D}_{Ct}$ represent the scale parameters for specific knowledge embodied in the workers of technology adoption and creation sectors, respectively. Here, learning-by-doing (LBD) mechanisms apply in human capital accumulation such that ${D}_{At}$ and ${D}_{Ct}$ are simply modeled to grow over time externally.²¹ Other things being constant, we can see that with a greater variety of ideas G_A and G_C, the workforce of U and S can be more thinly engaged in the production of each intermediate good x_j for j = A or C if LBD occurs actively.

Meanwhile, the intermediate goods market of the adoption sector is posited to be competitive, whereas the creation sector counterpart is monopolistically competitive. This is based on the empirical observation that profits arise on the goods applying newly created ideas that are usually patented, whereas profits on the goods adopting existing ideas tend to be trivial. Based on the discussions above, we now can reexpress technology (4′) as follows:

$$T_{t} = \underbrace {{G_{Tt - 1}^{{\frac{1}{\alpha - 1}}} \left( {(1 + \tau_{t} )^{{\frac{1}{\alpha - 1}}} D_{At} U_{t} + \kappa (\overline{B}_{t} H_{t} )^{{\frac{1}{\alpha - 1}}} D_{Ct} S_{t} } \right)}}_{{ = A_{t} + \kappa C_{t} }}.$$

(14′′)

The above equation presents the complementarity between H and S in production, implying their comovement tendency.

2.2.5 Production of output

The final output is produced by applying the current technology as shown above in (4′′) to a given non-human resource that is normalized at unity:

$$Y_{t} = T_{t} .$$

(13)

As usual, the final output market is competitive and consists of a unit measure of identical firms.

Reviewing (4′′) and (13) gives the following implications. (i) As in Romer (1990), the equilibrium allocation differs from the Pareto optimal allocation, because of market failing features: monopolistic competition at the technology creation sector, knowledge spillovers, and the social capital nature of basic ideas. (ii) The engine of economic growth is the accumulation of ideas, ${\Delta G}_{Tt}=\left[{f}_{C}\left({\overline{B} }_{t}{H}_{t}\right)+{f}_{A}\left({\tau }_{t}\right)\right]\cdot {G}_{Tt-1}$ which boosts final output in future. (iii) Finally, the growth of a catching-up economy depends on (a) its own knowledge creation and spillovers within the city and (b) knowledge spillovers from the world frontier as well.

2.2.6 Dynamics

The state variables of the model are the publicly available pool of ideas, ${G}_{Tt}, {G}_{At}, {\text{and }} {G}_{Ct}$. Technically speaking, since all these state variables depend on the total pool of ideas of the past period ${G}_{Tt-1}$, knowing the dynamics of ${G}_{T}$ is sufficient to describe the model’s dynamics. Individuals living a period do not optimize on the pool of ideas (i.e., taken externally), so the evolution of ${G}_{T}$ will determine equilibrium over time: (i) at a given value of ${G}_{Tt-1}$ at time t (or generation t) individuals optimize their choice variables of skill investment (${U}_{t},{S}_{t}, {\text{and }} {H}_{t})$ and residence to be determined below; in addition, wages and R&D productivities are determined accordingly; (ii) in the next period $t+1$, the new $t+1$ generation faces a new value of ${G}_{Tt}$ and optimizes their choices accordingly; (iii) and repeat the similar updating for later periods. The model produces endogenous growth because the aggregate pool of ideas grows over time. We will show a possible transition path in a later subsection on numerical analysis.

2.3 Labor demand and wages

We derive the demand for ordinary workers by differentiating output with respect to U. After derivation, we obtain:

$$w_{Ut} = \frac{{\partial Y_{t} }}{{\partial U_{t} }} = G_{Tt - 1}^{{\frac{1}{\alpha - 1}}} (1 + \tau_{t} )^{{\frac{1}{\alpha - 1}}} D_{At} .$$

(14)

Because the market for intermediate good i of the creation sector is monopolistically competitive, the usual mark-up pricing of ${p}_{C}(i)=\frac{\alpha }{\alpha -1}\frac{1}{{D}_{C}}{w}_{S}$ holds as in typical monopolistic competition models. Using this and another equilibrium condition ${p}_{C}(i)=\frac{\partial Y}{\partial {x}_{C}(i)}=\frac{1}{{D}_{C}}\frac{\partial Y}{\partial S}$, we derive:

$$\begin{aligned} w_{St} = & D_{Ct} \frac{\alpha - 1}{\alpha }p_{Ct} \left( i \right) \\ = & \frac{\alpha - 1}{\alpha }\frac{{\partial Y_{t} }}{{\partial S_{t} }} \\ = & \frac{\alpha - 1}{\alpha }G_{Tt - 1}^{{\frac{1}{\alpha - 1}}} \kappa \left( {\overline{B}_{t} H_{t} } \right)^{{\frac{1}{\alpha - 1}}} D_{Ct} \\ \end{aligned}$$

(15)

We obtain intuitive results: (i) ${w}_{U}$ and ${w}_{S}$ rise with the stock of knowledge ${G}_{T}$; (ii) catching-up economies face a diminishing technology gap ($\tau$) from the world frontier, which will put downward pressure on ${w}_{U}$. Therefore, workers are better off with investing in more education to be skilled workers; and (iii) ${w}_{S}$ rises with the complementary input $H$, $\overline{B}\left( H \right)$, and ${G}_{T}$.

Meanwhile, highly skilled workers contribute to the creation of new ideas, which are used to produce intermediate goods in the creation sector. Innovators are modeled to hold the property right on new ideas during the lifetime, i.e., a period. Then the profit per idea $\pi$ is given by $\pi =\frac{1}{\alpha -1}\frac{{w}_{S}}{{D}_{C}}\frac{{D}_{C\cdot }S}{{G}_{C}}$, and therefore, the demand for highly skilled workers with ability ${a}_{j}$ is expressed by:

$$w_{Ht} \left( {a_{j} } \right) = \left[ {B\left( {a_{j} ;\,H_{t} } \right)G_{Tt - 1} } \right] \cdot \pi_{t} = \frac{S}{\alpha - 1}\frac{{B\left( {a_{j} ;H_{t} } \right)}}{{\overline{B}_{t} H_{t} }}w_{St} ,{\text{ for all }}j{\text{ in }}a_{H}^{*} \le a_{j} \le 1.$$

(16)

Equation (16) implies that ${w}_{H}$ is positively associated with $B\left( {a_{j} ;H} \right)/\overline{B},\;S$, and the skilled workers’ wage ${w}_{S}$ as well, i.e., complementarity between H and S.

2.4 Education decision: self-selection

Individuals make their education choice decision to become a U, S, or H type of labor by considering ability, education cost, and wage rates: ${w}_{U}$, ${w}_{S}$ and ${w}_{H}$. The demand and supply for each type of labor jointly determine equilibrium wage rates, education investment, and thus technology and output growth rates.

For the sake of tractability, we express education cost in terms of a fraction of output: $n(a)\cdot {w}_{i}$ for i = S and H workers, where $n(a)$ is posited to decrease in $a$: $n^{\prime}(a)<0$. Without education, individuals become ordinary workers U with basic skills (i.e., those with compulsory education only) who simply copy the world frontier ideas and produce intermediate goods for technology adoption.

With these simplifying assumptions, we write down the condition for making education investment to become an S worker, from which we can derive the critical level of ability ${a}_{S}^{*}$:²²

$$w_{St} - n\left( a \right) \cdot w_{St} > w_{Ut} \Rightarrow a_{t} > n^{ - 1} \left( {1 - \frac{{\alpha D_{At} (1 + \tau_{t} )^{{\frac{1}{\alpha - 1}}} }}{{\left( {\alpha - 1} \right)\kappa D_{Ct} (\overline{B}_{t} H_{t} )^{{\frac{1}{\alpha - 1}}} }}} \right) \equiv a_{St}^{*} ,$$

(17)

where ${n}^{-1}(\cdot )$ is the inverse function of $n(a)$. The worker whose ability is greater than or equal to ${a}_{S}^{*}$ will make education investment to become a skilled (S) worker while paying the education cost $n(a){w}_{S}$. Note that either a narrower technology gap $\tau$ from the world frontier or more new ideas $\overline{B}H$ lowers the threshold ability for S-type workers, boosting the creation sector relative to the adoption sector. Since knowledge spillovers are essential for urban workers creating new ideas, both skilled and ordinary workers (whose tasks are not idea creation) self-select to reside outside the city to save rent.²³

Similarly, we write down the condition for making further education investment to become an H worker:²⁴

$$w_{Ht} \left( {a_{j} } \right) - n\left( {a_{j} } \right) \cdot w_{Ht} \left( {a_{j} } \right) - R_{t} \left( {a_{j} } \right) > w_{St} ,$$

(18)

where $R({a}_{j})$ is the equilibrium rent that a worker with ability ${a}_{j}$ pays for her urban residence. From this, we can define a bottom threshold level for highly skilled workers ${a}_{H}^{*}$ with ${a}_{H}^{*}>{a}_{S}^{*}$ given in (17). A worker with ability ${a}_{j}$ that is greater than or equal to ${a}_{H}^{*}$ will invest in further education to become a highly skilled worker based on a similar self-selection criterion and reside in the urban sector.

2.5 Determination of rents²⁵

We consider the following ability sorting equilibrium where H workers determine their residential location in a descending order of ability from the city center: more able workers live closer to the city’s center. We describe the city with size $H$ using a line $[-H/2,H/2]$, as in Fig. 1.

Landowners are viewed as those with suppliers of fixed input – land, and therefore they can extract “rents” as in the context of rent-seeking: without land, it is impossible to generate urban R&D productivity improvement through knowledge spillovers. To model rents, we take the liberty of choosing an analytically tractable form as follows. By taking the spirit of the Nash bargaining solution, we posit that landowners take away a fraction $\psi$ (i.e., bargaining power or the rent share for landlords) of the net urban knowledge spillover benefits and the rest $1-\psi$ goes to highly skilled workers. Given these, the equilibrium rent $R({a}_{H})$ for workers with ability ${a}_{H}$ is determined by dividing the productivity gain net of education cost $\left(1-n\left({a}_{H}\right)\right)\left({w}_{H}\left({a}_{H}\right)-{w}_{0}\right)$ between landlords and H workers through the Nash bargaining:

$$\begin{aligned} R_{t} \left( {a_{Ht} } \right) = & \psi \cdot \underbrace {{\left( {1 - n\left( {a_{Ht} } \right)} \right)\left( {w_{Ht} \left( {a_{Ht} } \right) - w_{0t} } \right)}}_{{\text{net urban knowledge spillover benefit}}} \\ = & \psi \left( {1 - n\left( {a_{Ht} } \right)} \right)\left( {B\left( {a_{Ht} ;\,H_{t} } \right) - B_{0t} } \right) \cdot G_{Ct - 1} \pi_{t} \\ = & \psi \left( {1 - n\left( {a_{Ht} } \right)} \right)\frac{{B\left( {a_{Ht} ;\,H_{t} } \right) - B_{0t} }}{{\overline{B}_{t} H_{t} }}\frac{{S_{t} }}{\alpha - 1}w_{St} \\ \end{aligned}$$

(19)

where ${w}_{0}$ represents a “threat point”: if highly skilled workers were to reside outside the city, then they could not benefit from knowledge spillovers and therefore ${B}_{0}$ is the comparable basic R&D productivity without urban knowledge spillovers. We can rearrange (19) as a ratio of rent to net earnings:

$$\frac{{R_{t} \left( {a_{Ht} } \right)}}{{\left( {1 - n\left( {a_{Ht} } \right)} \right)w_{Ht} \left( {a_{Ht} } \right)}} = \frac{{\psi \cdot \left[ {w_{Ht} \left( {a_{Ht} } \right) - w_{0t} } \right]}}{{w_{Ht} \left( {a_{Ht} } \right)}} = \psi \left( {1 - \frac{{B_{0t} }}{{B\left( {a_{Ht} ;H_{t} } \right)}}} \right),$$

(20)

where the term $\left(1-n({a}_{Ht})\right){w}_{H}({a}_{H})$ represents the net (of the education cost) earnings. Here Eq. (20) implies that the rent ratio of a worker depends positively on the bargaining power of landlords $\psi$ and negatively on the relative productivity ratio $\frac{{B}_{0t}}{B({a}_{Ht};{H}_{t})}$.

3 Characterizing equilibrium

For the sake of exposition, we first discuss urban spatial sorting equilibrium as part of the entire dynamic spatial equilibrium and then demonstrate the macroeconomic equilibrium supporting the spatial sorting equilibrium. Below we sketch the underlying story while leaving technical details to Appendix A. To obtain analytical results, we introduce some simplifying assumptions as follows.

Assumptions (i) to (iii).

• (I) More able individuals acquire more ideas at a cheaper cost: (1) ideas absorption capacity is set by $c\left(a\right)=D(0){e}^{-\phi z}$ rising convexly with ability:${c}^{\prime}\left(a\right)>0$ and ${c}^{{\prime}{\prime}}\left(a\right)>0$ where $z=(1-a)/2$ and $\phi$ refers to the limitation in adopting ideas; and (2) the education cost share is posited by $n\left(a\right)=1-{e}^{-\lambda z}$ with ${n}^{\prime}\left(a\right)<0$ and ${n}^{{\prime}{\prime}}\left(a\right)<0$.

• (II) Equal “depreciation” rates $\theta =\phi$ apply to idea-creation and idea-absorption in the forms of $I\left(a\right)=I\left(0\right){e}^{-\theta z} {\text{ and }} c\left(a\right)=D\left(0\right){e}^{-\phi z}$, respectively.

• (III) We normalize the baseline R&D productivity: ${B}_{0}=0$.

3.1 Spatial ability sorting

Given the equilibrium wages and rents described above, highly skilled workers move to the city, and they do not move from their equilibrium location to other locations. We call the conditions enforcing this spatial separation “incentive compatibility conditions,” which work as follows. (i) If a worker with ability $a$ migrates from her equilibrium location $x$ to a location closer to the center (i.e., the location for a higher-ability worker): $a\le {a}_{x}\equiv 1-2x$, the rent increase is greater than the increase in basic R&D productivity, i.e., the wage increase arising from the higher knowledge spillovers. This is because her (relatively) low ability to absorb knowledge spillovers in that location deters her wage from rising as much as the wage of a worker properly matched with the location. (ii) Conversely, if a worker with ability $a$ migrates from her equilibrium location $x$ to a more remote location: $a>{a}_{x}\equiv 1-2x$, the rent saving falls short of the wage loss. This is because the information processing capacity at each location has been developed up to a level that its residents can adopt. A remote location taken by a low-ability worker has a low information processing capacity, so even a high-ability worker cannot fully extract the amount of information comparable to her ability and is constrained by the location’s equilibrium capacity to a degree of $\omega$: $\omega c\left({a}_{x}\right)+\left(1-\omega \right)c(a)$. We summarize spatial sorting equilibrium using Proposition 1.

Proposition 1

Spatial ability sorting equilibrium

Under the condition of $\frac{\delta }{\phi +\delta }<\psi <\omega$, spatial ability sorting equilibrium arises in which (i) the city as the space for knowledge spillovers is taken by H workers, and (ii) workers choose their location in incentive compatibility manners. (iii) Spatial ability sorting boosts economic growth. Finally, (iv) rents play an essential role in ability sorting.

Proof

See Appendix A.⃞

The intuition behind the incentive compatibility condition $\frac{\delta }{\phi +\delta }<\psi <\omega$ is this. (i) The first inequality of $\frac{\delta }{\phi +\delta }<\psi$ implies this. $\frac{\delta }{\phi +\delta }$ represents the (relative) benefit of moving closer to the city center, i.e., receiving more ideas with less information decay. If this marginal benefit is less than the cost of rents implied by $\psi$, moving does not occur. Therefore, in equilibrium, lower-ability workers voluntarily reside at the urban periphery, considering both their lower capacity to adopt knowledge spillovers and lower rents for locations far from the center. (ii) The second inequality of $\psi <\omega$ implies that the benefit of rents saving from moving away from the city center $\psi$ should be less than the cost of being constrained by the information absorption capacity at a remote location, $\omega$. In short, higher-ability workers can afford to pay more for locations closer to the city’s center, because they can adopt more knowledge spillovers there, increasing their basic R&D productivity and wage; i.e., a complementary relationship between location and ability/skills arises. Note that (i) rental structure combined with (ii) the information processing capacity of each location provides a self-selection mechanism for aligning ability with location in a descending order of ability from the city’s center.²⁶

3.2 Allocation of skills

3.2.1 Equilibrium conditions

We can define equilibrium at period (generation) t by using the following three labor allocation equations. First, combining (18) with (19) yields an equation for H:²⁷

$$\begin{aligned} H_{t} = & 1 - a_{Ht}^{*} \\ = & 1 - n^{ - 1} \left( {1 - \frac{{\left( {\alpha - 1} \right)\overline{B}_{t} H_{t} }}{{S_{t} \left( {1 - \psi } \right)B\left( {a_{Ht}^{*} ;H_{t} } \right)}}} \right) \\ = & - \frac{2}{\lambda } \cdot {\text{ln}}\left( {\frac{{\left( {\alpha - 1} \right)\overline{B}_{t} H_{t} }}{{S_{t} \left( {1 - \psi } \right)B\left( {a_{Ht}^{*} ;H_{t} } \right)}}} \right) \\ \end{aligned}$$

(21)

The last line has made use of Assumption (i). Equation (21) implies a complementarity between $H$ and $S$. Second, using (17) and (18), we obtain an equation for S:

$$\begin{aligned} S_{t} = & a_{Ht}^{*} - a_{St}^{*} \\ = & 1 - H_{t} - n^{ - 1} \left( {1 - \frac{{\alpha D_{At} (1 + \tau_{t} )^{{\frac{1}{\alpha - 1}}} }}{{\left( {\alpha - 1} \right)\kappa D_{Ct} (\overline{B}_{t} H_{t} )^{{\frac{1}{\alpha - 1}}} }}} \right) \\ = & - H_{t} - \frac{2}{\lambda }\ln \left( {\frac{{\alpha D_{At} (1 + \tau_{t} )^{{\frac{1}{\alpha - 1}}} }}{{\left( {\alpha - 1} \right)\kappa D_{Ct} (\overline{B}_{t} H_{t} )^{{\frac{1}{\alpha - 1}}} }}} \right) \\ \end{aligned}$$

(22)

Third, (17) yields the equilibrium levels of skilled and ordinary workers, U:

$$\begin{gathered} U_{t} = a_{St}^{*} \hfill \\ = n^{ - 1} \left( {1 - \frac{{\alpha D_{At} (1 + \tau_{t} )^{{\frac{1}{\alpha - 1}}} }}{{\left( {\alpha - 1} \right)\kappa D_{Ct} (\overline{B}_{t} H_{t} )^{{\frac{1}{\alpha - 1}}} }}} \right) \hfill \\ = 1 + \frac{2}{\lambda }\ln \left( {\frac{{\alpha D_{At} (1 + \tau_{t} )^{{\frac{1}{\alpha - 1}}} }}{{\left( {\alpha - 1} \right)\kappa D_{Ct} (\overline{B}_{t} H_{t} )^{{\frac{1}{\alpha - 1}}} }}} \right) \hfill \\ \end{gathered}$$

(23)

Given the shape of $B({a}_{j};H)$ combined with ability sorting, Eqs. (21), (22), and (23) can technically determine the equilibrium relationships among $H$, $S$, and $U$. The system of equations is so general that we cannot obtain definitive comparative static results at this point. Nevertheless, we can derive some interesting results by imposing to the model minimum assumptions and restrictions implied by the data.

3.2.2 Characterizing equilibrium

As shown in (17), at the threshold ability ${a}_{S}^{*}$, investing in education to become a skilled worker is equal in net value to remaining as an unskilled labor:

$$w_{Ut} = w_{St} - n\left( {a_{St}^{*} } \right)w_{St} { } \Leftrightarrow { }(1 + \tau_{t} )^{{\frac{1}{\alpha - 1}}} D_{At} = \kappa (\overline{B}_{t} H_{t} )^{{\frac{1}{\alpha - 1}}} D_{Ct} \left( {1 - n\left( {a_{St}^{*} } \right)} \right).$$

(24)

While it looks complex, Eq. (24) finds the threshold point ${a}_{S}^{*}(=U)$ where at a certain level of H, whether to invest in education to become a S–type worker or to remain as a U–type worker is indifferent. Given the facts of ${a}_{S}^{*}=U$ and $U=1-(S+H)$, choosing U is identical to choosing the rest of workers, $S+H$. Defining $X\equiv S+H$ (i.e., the size of workers in the technology creation sector), we can view Eq. (24) technically as an equation determining X (or S) when H is given: $X={X}_{r}(H)$, where ${X}_{r}(\cdot )$ can be viewed as a response function in the game theory context.

Similarly, at a value of U (i.e., essentially at a given X since $X=1-U)$, we define the threshold point ${a}_{H}^{*}(=1-H)$ where investing in higher education to become an H–type worker is identical in net value to remaining as an S–type worker. This is achieved by equating ${w}_{S}$ and the marginal H worker’s net wage (net of additional education and housing costs) using (18) and (19) as follows:

$$w_{St} = w_{Ht} \left( {a_{Ht}^{*} } \right) - n\left( {a_{Ht}^{*} } \right)w_{Ht} \left( {a_{Ht}^{*} } \right) - R_{t} \left( {a_{Ht}^{*} } \right) \Leftrightarrow \left( {1 - \psi } \right)\left( {1 - n\left( {a_{Ht}^{*} } \right)} \right)\frac{\Delta }{{\overline{B}_{t} H_{t} }}\frac{{S_{t} }}{\alpha - 1} = 1,$$

(25)

where $\Delta$ denotes the R&D productivity gap $B({a}_{Ht}^{*};{H}_{t})-{B}_{0t}$ based on locational choice. Technically, we can view (25) as an equation determining H when X is given: $H={H}_{r}(X)$ where ${H}_{r}$ is a response function.

Now, by invoking the Nash equilibrium concept, we can define a Nash equilibrium as a pair of strategies ${\{X}^{*},{H}^{*}\}$, such that two response functions [${X}^{*}={X}_{r}({H}^{*})$ and ${H}^{*}={H}_{r}({X}^{*})$] are jointly satisfied under a plausible set of assumptions to be listed below. We will assume the existence of equilibrium to account for the reality where both skilled and unskilled workers coexist in usual economies. While leaving technical details to Online Appendix B, we present some notable results.

Lemma 1

Declining R&D productivity with city expansion

When city size is above a threshold of $\mathit{max}\left\{\frac{2}{\theta +\delta },\frac{1}{\theta }\right\}$, the basic R&D productivity of the marginal worker declines with city size: $dB({a}_{H}^{*};H)/dH<0$.

Proof

See Online Appendix B.⃞

This lemma characterizes the condition for a realistic equilibrium: while the total ideas pool expands with an incoming marginal city mover, he is less able than existing workers and resides further away from the center. Therefore, he adopts fewer ideas than earlier movers, with a resulting declining R&D productivity of a marginal worker. Looked at differently, when city size is “too small,” an incoming marginal worker tends to be still highly able and her R&D productivity can be even higher than that of the existing workers since she benefits from the earlier highly able workers’ ideas with little depreciation (due to geographic proximity at a small city). If so, R&D productivity keeps rising, generating a further mobility flow into city, which eventually leads to a pattern of diminishing R&D productivity: $dB\left({a}_{H}^{*};H\right)/dH<0$.

Proposition 2

Multiple equilibria (one stable and the other unstable).

Suppose the set of assumptions (AS (i) to (iii)) including $H>max\left\{\frac{2}{\theta +\delta }, \frac{1}{\theta }\right\}$ holds. Then we obtain: (i) one response curve is convex while the other is concave, and therefore, (ii) two equilibria can arise with one stable equilibrium and the other unstable under a set of parameter values {$\psi$, $\lambda$, $\kappa$, and $\tau$} (see Fig. 5).

Proof

See Online Appendix B.⃞

We can show whether an equilibrium is stable or unstable using the slopes of the two response functions {${X}_{r}(H)$ and ${H}_{r}(X)$}. As we see in Fig. 4, a stable equilibrium arises when the slope of ${H}_{r}$ is greater than that of ${X}_{r}$. An unstable equilibrium arises under the opposite condition. Given the convexity of ${H}_{r}$ and the concavity of ${X}_{r}$, we can obtain multiple equilibria. An example of the multiple equilibria is given in Fig. 5. (i) We have a stable equilibrium in the {$H$, $X$} plane where the slope of the response curve ${H}_{r}$ is greater than ${X}_{r}$ (see the point E₂ in Fig. 5). (ii) An unstable equilibrium arises where the slope of ${H}_{r}$ is lower than the slope of ${X}_{r}$ (see the point E₁ in Fig. 5).

Fig. 4

Equilibrium stability vs. instability: slopes of two response functions. Notes: (i) $X=S+H$, and the equilibrium point has to be above the 45 degree line by the definition of $X$; (ii) Both stable and unstable equilibria are possible but we focus on the stable one based on empirical relevance

Fig. 5

Multiple equilibria: one stable and the other unstable. Notes: (i) $X=S+H$, and the equilibrium points ${E}_{1} {\text{ and }} {E}_{2}$ have to be above the 45-degree line by the definition of $X$; (ii) Assumptions AS (i) to (iii) are used; (iii) Curves (B7) and (B8) are log versions of Eqs. (24) and (25), respectively, which are interpreted as response functions. (iv) Note that E₂ is a stable equilibrium while E₁ is an unstable one, but we focus on the stable one based on empirical relevance that the economy usually grows steadily. See Fig. 4 also for the concept of stability vs. instability. (v) E₂ moves to the north-east direction along the transition path until the long-run spatial equilibrium is reached with convergence to the world frontier technology

3.3 Transition path and policy implications: comparative static analysis

3.3.1 Getting closer to the technology frontier

Now, we examine our main question of how equilibrium changes when the distance from the technology frontier narrows with catching-up: a reduction of $\tau$ in our context. Individuals in our model live for a period and take the pool of knowledge as an external variable, so they view a change in $\tau$ as an exogenous event. Using this, we can conduct a comparative static analysis.

In our model, a decrease in $\tau$ shifts the response curve implied by Eq. (24) upward: ${X}_{r}$ shifts upward [also see (B7) in Online Appendix B]. In a stable equilibrium, we can see both X and H go up.²⁸ Given the complementarity between H and S workers, the demand for both types of workers rises along with their wages relative to U workers. These are the changes we can see within a period.

Using the stable equilibrium at a point in time, we can also determine the transition path of the economy as follows. In the next period $t+1$, technology T improves and therefore the technology gap $\tau$ goes down again: ${\tau }_{t+1}=\tau \left(\overline{T }-{T}_{t}\right)$. In response to a fall in $\tau$ in period $t+1$, a similar pattern of changes described earlier occurs. This adjustment continues until the technology gap disappears, which defines the long-run spatial equilibrium where (i) convergence is achieved and technology improvement arises from own innovations, and (ii) city size also stops growing since no further external changes affect the economy. A numerical example will be discussed in a later subsection.

In an unstable equilibrium, however, both X and H fall in response to a decrease in $\tau$, which is inconsistent with the data: (i) more technologically advanced economies do not lower investment in education; and (ii) since the equilibrium is unstable in nature, an unexpected change in $\tau$ would lead to a corner (crash) equilibrium with only the U-type workers after the transitional adjustment process, i.e., another reason for not taking this as a sensible equilibrium. For these reasons, henceforth, we confine our interest to the former stable equilibrium case. Before moving on, we summarize the results obtained above.

Proposition 3

Catching-up and urban agglomeration with industrial structure upgrading

(i) In getting closer to the technology frontier, the economy faces greater incentives to engage in technology creation, causing skill/industrial structure upgrading and city expansion. (ii) These changes continue along the endogenous growth path until convergence occurs. (iii) After reaching the world frontier, the economy grows using its own innovations without changes in the composition of skills and spatial structure.

3.3.2 Role of rent subsidy

We also consider the effect of taxing rents at the rate of $\eta$ and redistributing through a rent subsidy in a balanced budget manner, so $\eta$ is also the subsidy rate. Technically, an introduction of a rent subsidy is equivalent to multiplying the left-hand side of (25) by $1-\eta$.²⁹ It shifts the response curve implied by (25) downward (i.e., ${H}_{r}$ shifts downward) (see (B8) in Online Appendix B), leading to increases in both H and X. Again, the complementarity between H and S triggers the aforementioned updating process through a rise in ${w}_{S}$, arriving at a new Nash equilibrium.

While the cost of public fund was not considered, this result is intuitive. Rents here partly represent returns to fixed factors, so it is unsurprising that a certain extent of redistributing rental incomes from rent-seeking landlords to urban workers can boost the urban population and R&D activities without distortionary effects on the urban rental markets. In fact, many successful “technology parks” around the world tend to implement government subsidies to reduce rent burden for researchers, lending support to this result.

However, rents in our model also play the productive role of allocating innovative abilities over space in an incentive-compatible manner, so an arbitrary rent subsidy would destroy our spatial sorting equilibrium. Using the incentive compatibility condition, we can easily study the possible range of the rent subsidy rate. Intuitively, too a cheap rent fails to prevent low-ability workers from migrating to central locations, disrupting the sorting equilibrium.

Proposition 4

Productive urban rent subsidy

There exists an interval of the effective rent share for landlords $(1-\eta )\psi$ in which subsidizing rents for urban workers can foster agglomeration and growth in an incentive-compatible manner.

Proof

Using Proposition 1, we can derive the incentive compatibility condition under the subsidy rate $\eta$: $\frac{\delta }{\phi +\delta }<(1-\eta )\psi <\omega$. From this, we can find the bound for $\eta$: $1-\frac{\delta }{\left(\phi +\delta \right)\psi }>\eta >1-\frac{\omega }{\psi }$. ⃞

We can see that the admissible range of subsidy rates can go up with (i) a lower $\delta$, (ii) a higher $\phi$, (iii) a lower $\omega$, and (iv) a higher $\psi$. Changes in the first three parameter imply more efficient information flows within a city, boosting welfare. Therefore, a higher subsidy rate for innovators applies. The last one implies an excessive rent-seeking behavior, justifying a high tax and subsidy rate on rents.

3.4 A numerical example

This subsection gives a numerical example to demonstrate the model’s dynamics and some qualitative properties that are not examined in our theory section. Since this paper tries to give a seminal idea on technology and space in fast catching-up economies, we focus on characterizing the model’s qualitative behavior while leaving a more realistic calibration to future research. We set the parameters at a set of intuitive values as follows: $\{\mathrm{\alpha }=1.5,\upkappa =100,\uptheta =\phi =0.025, \delta =0.005, \psi =0.25, \lambda =0.0063, \omega =0.7\}.$³⁰ And the scale parameters are set as follows:³¹$\{I\left(0\right)=0.1, D\left(0\right)=0.1,{\tau }_{0}=1.0, \overline{T }={10}^{5}, {B}_{0}=0, {d}_{A}={d}_{C}=0.02, \overline{L }=8000\}.$ We also set the functional forms for ${\tau }_{t}, {f}_{A}\left(\tau \right),\mathrm{ and}\ {f}_{C}(\overline{B }H)$: $\{{\tau }_{t}={\tau }^{0}\frac{\left(\overline{T }-{T}_{t-1}\right)}{\overline{T} }+\left(1-{\tau }^{0}\right){\tau }_{t-1},\mathrm{ with }\,{\tau }^{0}=0.5;\, {f}_{A}\left(\tau \right)={s}_{A}\cdot \tau ;$ ${f}_{C}(\overline{B }H)={s}_{C}\cdot \overline{B }H$ with ${ s}_{A}=0.00015 {\text{ and }} {s}_{C}=0.00005\}$. Initial conditions are set as follows. At ${T}_{0}=10$ and ${A}_{0}=3.5$, we can obtain ${C}_{0}=0.065$ under $\upkappa =100$; and assuming the initial human capital levels for adoption and creation are ${l}_{{A}_{0}}={l}_{{C}_{0}}=1$, we can determine ${G}_{{A}_{0}}={{A}_{0}}^{(\alpha -1)/\alpha }=1.5183\mathrm{\ and}\ {G}_{{C}_{0}}={{C}_{0}}^{(\alpha -1)/\alpha }=0.4021.$ Now, we can determine all other values recursively to describe the model’s dynamics.

Figure 6 presents some notable findings from the numerical exercise that are consistent with the earlier analysis. First, we obtain an endogenous growth path where the economy grows to converge toward the long-run spatial equilibrium where no further urban concentration occurs (see Fig. 6e). The economic growth rate appears almost constant but it varies slightly depending on the speed of innovative ideas creation (see the slopes of curves in Fig. 6a). The time period is a generation in our model, so it takes about 12 generations [or $280 (=12\times 25)$ years] to complete the urbanization (see Fig. 6e). We also see an increasing demand for S workers who commercialize the new creative ideas from H workers, and the demand for U workers goes down over time as new ideas creation becomes more important than adopting from the frontier technologies (see Fig. 6d). During the transition period, skills and technology grow with spatial concentration and transition from technology adoption to creation (see Fig. 6a, b and c). These results confirm our theoretical predictions.

Fig. 6

A numerical example: the transition path with technology catching-up. Notes: The Y axis refers to the size of the variables denoted by legends in inner boxes of each graph a to f. a: log $T=$ log of total technology level; $A=$ adoption technology level; $C=$ creation technology level. b: ${G}_{A}=$ the # of adoption ideas; ${G}_{C}=$ the # of total creation ideas; ${G}_{T}=$ the # of total ideas. c: ${H*B}_{bar}=$ total urban creative ideas ($H\overline{B })$; ${B}_{center}=$ R&D at the city center; ${B}_{new\ entrant}={\text{R}}\&\mathrm{D\ of\ a\ new\ city\ entrant}$. d: S = skilled labor; U = unskilled labor. e: H = highly skilled labor conducting ideas creation. f: R = urban rents; ${w}_{H}$=wage for H labor; ${w}_{S}$=wage for S labor; ${w}_{U}$=wage for U labor

Second, since transition from technology adoption to creation happens gradually, the slope of technology stock logC is higher than that of logA. Growth rates stay almost constant around 4%, masking the structural changes (from technology adoption to creation) happening in the economy. Without such a transition, the growth rates would fall substantially. Industrial clusters and productive urbanization help catching-up economies grow faster by transiting toward a technology economy. Third, rents also rise continuously along with economic growth (see logR in Fig. 6f). Wages of H and S workers rise continuously, while that of U workers declines and remains stagnant. Overall, spatial concentration is part of the growth process for successful catching-up economies, which is consistent with Henderson (2003).

4 Summary and conclusion

In fast catching-up economies, we see not only (i) rapid increases in skilled labor and R&D investment with (ii) industrial structure upgrading toward knowledge-intensive sectors but also (iii) spatial concentration. The paper accounts for the facts using a model of “the city as a laboratory for innovation,” which combines the mechanisms of skill-city complementarity and knowledge spillover in catching-up economies where ideas are more challenging to find along the growth path. We highlight that (i) successful catching-up economies choose proper city size and structure so as to benefit from growth effects through fostering innovation and spillovers; (ii) rents here are not just an economic rent but play a productive role of allocating innovative abilities over space in an incentive-compatible manner; (iii) there exists a range of rent subsidy for booting innovation; and (iv) the long-run urbanization coincides with convergence to the world frontier technology.

As a catching-up economy grows closer to the technology frontier, knowledge spillovers from advanced countries decline while technology creation becomes more profitable than technology adoption, raising the demand for skilled workers with comparative advantages in technology creation. When this feature is combined with skill-city complementarity and urban knowledge spillover, the following observed correlations arise: (i) high-ability workers invest more actively in human capital, and sort into cities in descending order with the most able worker residing at the city center while paying the highest rent, and (ii) they efficiently conduct R&D activities benefitting more from urban ideas spillover externality. We also discussed some micro-policies for fostering innovation and growth. The model’s implications suggest that providing urban rent subsidies can foster economic growth in our knowledge economy setting.

While the current paper deals with a set of notable phenomena in catching-up economies, its framework can shed light on growth and inequality issues. We believe that the skill-city complementarity can provide useful explanations for regional income inequality and changing labor demand in recent decades of rapid technological innovations. A vast inequality literature documents that the rising inequality arises from skill-biased technological changes (SBTC), but it predicts little about regional inequality. We think that regional inequality can be reexamined from the perspective of the skill-city complementarity. More short-term projects would include (i) testing implications of the model using catching-up economies’ data and (ii) elaborating the micro-mechanism of knowledge spillover such that we can perform calibration exercises to explain related stylized facts.

Finally, we would like to mention that the main story of the paper holds only when agglomeration benefits steadily exist. Without agglomeration benefits, skill-city complementarity breaks down and the spatial sorting equilibrium cannot be defined. The future of agglomeration benefits seems an interesting empirical question in the era of the proliferation of modern communication technologies and global connectivity combined with the influential role of network effects in information transfer.

Appendix A

1.1 Proof of proposition 1

For the sake of exposition, we first prove statement (ii) of incentive compatibility for urban spatial ability sorting equilibrium, assuming statement (i) holds. And then we prove statement (i) and others later.

1.1.1 Proof of statement (ii)

We first define the value of R&D productivity: $V(a,x;H)$. The productivity of basic R&D activities of a worker with ability $a$ at location $x\in [0, H/2]$ should satisfy the incentive compatibility conditions (IC) to be shown below.³² Before addressing IC, we define the productivity $V(a,x;H)$ for a worker whose ability a is equal to or below the equilibrium ability at location x of city size H that is denoted by ${a}_{x}\equiv 1-2x\ge a$, i.e., “those under-qualified at location $x$”:

$$V\left(a, x;H\right)={B}_{0}+c\left(a\right)\cdot m\ for\ a\le {a}_{x}\equiv 1-2x$$

(26)

$${\text{where }} m \equiv \left\{ {\underbrace {{\mathop \int \limits_{0}^{x} I\left( 0 \right)\exp \left( { - \theta z} \right)\exp \left( { - \delta \left( {x - z} \right)} \right)dz}}_{{{\text{spillover from the center to location }} x}} + \underbrace {{\mathop \int \limits_{x}^{\frac{H}{2}} I\left( 0 \right)\exp \left( { - \theta z} \right)\exp \left( { - \delta \left( {z - x} \right)} \right)dz}}_{{{\text{spillover from }}x{\text{ to the city boundary H}}/2}} + \underbrace {{\mathop \int \limits_{0}^{\frac{H}{2}} I\left( 0 \right)\exp \left( { - \theta z} \right)\exp \left( { - \delta \left( {x + z} \right)} \right)dz}}_{{\text{spillover from the other side of the city}}}} \right\}$$

Here (i) ${B}_{0}$ refers to the productivity of basic R&D activities outside the city with no knowledge spillover; (ii) the bracket term {.} denoted by m refers to the total amount of knowledge spillovers from three different areas of the city, available to a worker at location x. Note that the density function for city residents is normalized at 1; and (iii) $c(a)$ is the information-absorptive power for a worker with ability a.

Now we consider the counterpart value for those whose ability a is above the equilibrium ability at location x with ${a}_{x}\equiv 1-2x<a$. The value $V(a,x;H)$ for those with ${a}_{x}\equiv 1-2x<a$, i.e., those “over-qualified at location x” is partially constrained by the low information processing capacity of location x, $c({a}_{x})$, as follows:

$$V\left(a, x;H\right)=V\left({a}_{x}, x;H\right) \;for\, a>{a}_{x}\equiv 1-2x={B}_{0}+\left[\omega c\left({a}_{x}\right)+\left(1-\omega \right)c(a)\right]\cdot m$$

(27)

where the combination of $c({a}_{x})$ and $c(a)$ reflects that the information processing capacity binds for those with ${a}_{x}<a$. We rearrange the three parts of m in (26, 27), respectively, as follows:

$$\begin{aligned} \int\limits_{0}^{x} I \left( 0 \right)\exp \left( { - \theta z} \right)\exp \left( { - \delta \left( {x - z} \right)} \right){\text{d}}z & = {\mkern 1mu} \int\limits_{0}^{x} I \left( 0 \right)\exp \left( { - \delta x - \left( {\theta - \delta } \right)z} \right){\text{d}}z \\ & = I\left( 0 \right){\text{exp}}\left( { - \delta x} \right)\int\limits_{0}^{x} {\exp } \left( { - \left( {\theta - \delta } \right)z} \right){\text{d}}z \\ & = {\mkern 1mu} I\left( 0 \right)\exp \left( { - \delta x} \right)\frac{1}{{\theta - \delta }}\left\{ {1 - \exp \left( { - \left( {\theta - \delta } \right)x} \right)} \right\} \\ & = I\left( 0 \right)\frac{1}{{\theta - \delta }}\left\{ {{\text{exp}}\left( { - \delta x} \right) - \exp \left( { - \theta x} \right)} \right\} \\ \end{aligned}$$

(28)

$$\begin{aligned} \int\limits_{x}^{{\frac{H}{2}}} I \left( 0 \right)\exp \left( { - \theta z} \right)\exp \left( { - \delta \left( {z - x} \right)} \right){\text{d}}z & = I\left( 0 \right){\text{exp}}\left( {\delta x} \right)\int\limits_{x}^{{\frac{H}{2}}} {\exp } \left( { - \left( {\theta + \delta } \right)z} \right){\text{d}}z \\ & = I\left( 0 \right){\text{exp}}\left( {\delta x} \right)\frac{1}{{\theta + \delta }}\left\{ {\exp \left( { - \left( {\theta + \delta } \right)x} \right) - {\text{exp}}\left( { - \frac{{H\left( {\theta + \delta } \right)}}{2}} \right)} \right\} \\ \end{aligned}$$

(29)

and

$$\begin{aligned} \int \limits_{0}^{\frac{H}{2}} I\left( 0 \right)\exp \left( { - \theta z} \right)\exp \left( { - \delta \left( {x + z} \right)} \right)dz =\;\, & I\left( 0 \right)exp\left( { - \delta x} \right) \int \limits_{0}^{\frac{H}{2}} \exp \left( { - \left( {\theta + \delta } \right)z} \right)dz \\ =\;\, & I\left( 0 \right)\exp \left( { - \delta x} \right)\frac{1}{\theta + \delta }\left\{ {1 - exp\left( { - \frac{{H\left( {\theta + \delta } \right)}}{2}} \right)} \right\} \\ \end{aligned}$$

(30)

Combining all three parts leads to simplified expressions for $V$:

$$\begin{gathered} V\left( {a,x;\,H} \right) = B_{0} + c\left( a \right) \cdot m\left( {x;\,H} \right)\;for\ a \le a_{x} \equiv 1 - 2x; \hfill \\ V\left( {a, x;\,H} \right) = B_{0} + \left[ {\left( {1 - \omega } \right)c\left( a \right) + \omega c\left( {a_{x} } \right)} \right] \cdot m\left( {x;\,H} \right)\ for{ }\,\,a > a_{x} \equiv 1 - 2x \hfill \\ \end{gathered}$$

(31)

where $m\left(x;\,H\right)$ is the urban knowledge spillovers given as follows:

$$m\left( {x;\,H} \right) = I\left( 0 \right)\left\{ {\frac{ - 1}{{\theta + \delta }}exp\left( { - \left( {\theta + \delta } \right)\frac{H}{2}} \right)\left( {\exp \left( { - \delta x} \right) + \exp \left( {\delta x} \right)} \right) + \frac{2}{{\theta^{2} - \delta^{2} }}\left( {\theta \exp \left( { - \delta x} \right) - \delta \exp \left( {\theta x} \right)} \right)} \right\}.$$

(32)

(31) implies that the productivity of basic R&D activities of a worker with ability $a$ at location $x$ of city size $H$ consists of two parts: (i) the worker’s absorptive capacity $c(a)$, and (ii) the total amount of information flows available at location $x$.

It is straightforward to show that $m(x;H)$ (and $V(a,x:H)$) decreases in {$x$, $\delta$, $\theta$} and increases in H. We can also infer that $\frac{{\partial }^{2}}{\partial {x}^{2}}m(x;H)<0$ (and $\frac{{\partial }^{2}}{\partial {x}^{2}}V(a,x:H)<0$).

1.1.2 Incentive compatibility conditions

Now, we show that the spatial ability sorting described above is supported by incentive compatibility conditions. If a worker with ability $a$ migrates from her equilibrium location $x=(1-a)/2$ to location $x^{\prime}$ closer to the center with a higher rent, the benefit of a wage gain caused by the increased knowledge spillovers should not exceed the cost of increased rental cost. Using (19), $\text{for all }x^{\prime}<x\in [\text{0,}{\text{ H}}\text{/2}]$ this condition is described by IC1:

$$\begin{aligned} & V(a_{x} ,x;H) - r(x) > V(a_{x} ,x^\prime ;H{\text{)}} - r(x^\prime ){\mkern 1mu} {\text{for}} > x^\prime \Leftrightarrow (1 - \psi )c(a_{x} )m(x) \\ & \quad > c(a_{x} )m(x^\prime ) - \psi c(a_{{x^\prime }} )m(x^\prime ) \Leftrightarrow c(a_{x} )\left[ {m\left( x \right) - m\left( {x^\prime } \right)} \right] \\ & \quad + \psi \left[ {c\left( {a_{{x^\prime }} } \right)m\left( {x^\prime } \right) - c\left( {a_{x} } \right)m\left( x \right)} \right] > 0 \Leftrightarrow c(a_{x} )m^\prime (x) \\ & \quad - \psi \left[ {\frac{{dc\left( {a_{x} } \right)}}{{dx}} \cdot m\left( x \right) + c\left( {a_{x} } \right)m^\prime \left( x \right)} \right] > 0 \Leftrightarrow (1 - \psi )m^\prime (x)/m(x) \\ & \quad - \psi \left[ {\frac{{\frac{{dc\left( {a_{x} } \right)}}{{dx}}}}{{c\left( {a_{x} } \right)}}} \right] > 0 \Leftrightarrow (1 - \psi )m^\prime (x)/m(x) > \psi \left[ {\frac{{\frac{{dc\left( {a_{x} } \right)}}{{dx}}}}{{c\left( {a_{x} } \right)}}} \right] \\ \end{aligned}$$

$$\therefore \left( {1 - \psi } \right)\left( { - \delta } \right) > \psi \left( { - \phi } \right) \Leftrightarrow \frac{\delta }{\delta + \phi } < \psi .$$

(33)

The fourth line is derived after dividing both sides by $\Delta x\equiv x^{\prime}-x$. We use the notations $m^{\prime}\left( x \right) = \underbrace {lim}_{{x^{\prime} \to x}}\frac{{m\left( x \right) - m\left( {x^{\prime}} \right)}}{{x - x^{\prime}}}$ and ${x}^{\prime}\cong x$. The last line of (33) uses definitions of the model’s parameters $\delta$ and $\phi$, i.e., information depreciation with distance and information absorption ability: $\frac{{\frac{{dc\left( {a_{x} } \right)}}{dx}}}{{c\left( {a_{x} } \right)}} = - \phi$ $\mathrm{and }$ is zero at the center and $-\delta$ at the boundary.

Conversely, another condition IC2 should hold such that when a worker with ability $a$ migrates from her equilibrium location $x=(1-a)/2$ to a remote location $x^{\prime}$ with a lower rent, the benefit of a lower rental cost should be trivial compared to the cost of a large wage loss caused by the decreased knowledge spillovers $\text{for all }x^{\prime}>x\in [\text{0, H/2}]$:

$$\begin{aligned} V\left( {a_{x} ,x;H} \right) - r\left( x \right) > V\left( {a_{x} ,x^{\prime};H} \right) - r\left( {x^{\prime}} \right){\text{ for }}x < x^{\prime}{ } \Leftrightarrow & \left( {1 - \psi } \right)c\left( {a_{x} } \right)m\left( x \right) > \left[ {\left( {1 - \omega } \right)c\left( {a_{x} } \right) + \omega c\left( {a_{{x^{\prime}}} } \right)} \right]m\left( {x^{\prime}} \right) - \psi c\left( {a_{{x^{\prime}}} } \right)m\left( {x^{\prime}} \right)) \\ \Leftrightarrow & c\left( {a_{x} } \right)\left[ {m\left( x \right) - m\left( {x^{\prime}{ }} \right)} \right] + \psi \left[ {c\left( {a_{x^{\prime}} } \right)m\left( {x^{\prime}} \right) - c\left( {a_{x} } \right)m\left( x \right)} \right] + \left[ {\omega c\left( {a_{x} } \right) - \omega c\left( {a_{x^{\prime}} } \right)} \right]m\left( {x^{\prime}} \right) > 0 \\ \Leftrightarrow & - c\left( {a_{x} } \right)m^{\prime}\left( x \right) + \psi \left[ {\frac{{dc\left( {a_{x} } \right)}}{dx} \cdot m\left( x \right) + c\left( {a_{x} } \right)m^{\prime}\left( x \right)} \right] - \omega \cdot \frac{{dc\left( {a_{x} } \right)}}{dx} \cdot m\left( x \right) > 0 \\ \Leftrightarrow & - \left( {1 - \psi } \right)\frac{{m^{\prime}\left( x \right)}}{m\left( x \right)} + \left( {\psi - \omega } \right)\frac{{\frac{{dc\left( {a_{x} } \right)}}{dx}}}{{c\left( {a_{x} } \right)}} > 0 \\ \Leftrightarrow & \left( {1 - \psi } \right)\frac{m^{\prime}\left( x \right)}{{m\left( x \right)}} + \left( {\psi - \omega } \right)\phi < 0 \\ \Leftrightarrow & \left( {\psi - \omega } \right) < \frac{1}{\phi }\left( {1 - \psi } \right) \cdot \underbrace {{\frac{ - m^{\prime}\left( x \right)}{{m\left( x \right)}}}}_{{{\text{btn}}{. }0{\text{ and }}\delta }} \\ \therefore \psi - \omega < 0{ }\left( {\because {\text{the lowest value for }} - m^{\prime}\left( x \right)/m\left( x \right){\text{ is zero}}} \right) \\ \end{aligned}$$

(34)

Given $- \delta \le \frac{{m^{\prime}\left( x \right)}}{m\left( x \right)} < 0$, the sufficient condition for the inequality $(1-\psi )\frac{m^{\prime}(x)}{m(x)}+(\psi -\omega )\phi <0$ to hold is $\psi <\omega$. Combining IC1 and IC2 yields the joint condition of $\frac{\delta }{\delta +\phi }<\psi <\omega$. We can see that the information-related parameters $\delta$, $\phi$, and $\omega$ combined with rents-related parameter $\psi$ play essential roles in this IC condition.

1.1.3 Proofs of statements (i), (iii) and (iv)

For statement (i), we consider the general case where the city as the space for knowledge spillovers is contested among heterogeneous workers including U, S, and H. Since other types of workers U or S cannot afford to pay rents as much as H, the city space is taken only by H workers.

For statement (iii), note that in our spatial ability sorting equilibrium, the size of new ideas $\overline{B}$$H$ is larger than in other possible types of sorting, which more efficiently boosts economic growth. Statement (iv) has already been stressed in the proof of (ii).

1.1.4 Incentive compatibility condition (IC3) for multiple city equilibrium

Now, we discuss the condition for multiple city equilibrium, which is essentially identical to no inter-city migration equilibrium. Intuitively, multiple city equilibrium requires ability sorting within and across cities such that no further migration occurs between two cities.

To describe the two-city equilibrium, we introduce additional concepts. The smaller city B is assumed to have population density $d$ with $d<1$, while that of the large city A is normalized at unity. All previous equations still hold for the smaller city B with only one modification that $I(0)$, the productivity at the city center, is now replaced by $d\cdot I\left(0\right).$ We still can find that the first-order conditions (FOC’s) for the within-city B equilibrium are exactly identical to those already described in the text except for changing $I\left(0\right)$ to $d\cdot I\left(0\right)$, so cities A and B satisfy the within-city incentive compatibility conditions mentioned above, respectively. Changing $I\left(0\right)$ to $d\cdot I\left(0\right)$ for the city B problem does not affect the FOC’s, because economic agents treat $d\cdot I\left(0\right)$ as exogenously given when maximizing their utility. To economize the use of notation, we denote variables with superscript s as those for the smaller city.

Now, we prove that a decrease in the density lowers city size $H$. From the equations in Online Appendix B (B7 and B8), and (4-i) of Fig. 4, we can find that a low population density $d$ leads to a decrease in $H$. This is because in response to a decrease in density, the curve representing (B7) $X={X}_{r}\left(H\right)$ in Fig. 4 shifts downward, while the other curve representing (B8) stays unchanged. Note that (B2) and (B3) imply that both $\overline{B }\left(H\right)H$ and $B({a}_{H}^{*};H)$ increase strictly in proportion to $I(0)$. Using (32), we can find that $\frac{\partial m\left(x;H\right)}{\partial H}>0$ where m is a measure of knowledge spillovers available at location x. Therefore, the ratio of ideas externalities between cities $\frac{m\left(x;{H}^{s}\right)}{m\left(x;H\right)}$ is less than unity due to ${H>H}^{s}$ and decreases as $d$ falls.

To close the model, we need an additional inter-city equilibrium condition that the values of residing in cities A and B are equalized such that no further migration between two cities occurs for heterogeneous agents. This condition says that there is no benefit for a worker with ability $a$ living in one city to migrate to the other. We define the migration cost as a certain share ${c}_{m}$ of net-of-rent (cost-of-living adjusted) income at a reference city (e.g., the larger city): the migration cost is therefore ${c}_{m}\cdot \left(V\left({a}_{x},x;H\right)-r\left(x\right)\right)$, which is assumed identical in both cities for simplicity. Now, we can express the non-migration condition using the interval as follows:

$$\left( {1 - c_{m} } \right)\left( {V\left( {a_{x} ,x;H} \right) - r\left( x \right)} \right) \le V^{s} \left( {a_{x} ,x;H^{s} } \right) - r^{s} \left( x \right) \le \left( {{1 + }c_{m} } \right)(V\left( {a_{x} ,x;H} \right) - r\left( x \right){\text{) for }}x \in \left[ {0, H^{s} /2} \right].$$

The first inequality states that a large city dweller does not want to move to a smaller city B paying the migration cost ${c}_{m}\cdot \left(V\left({a}_{x},x;H\right)-r\left(x\right)\right)$. The second inequality states that a smaller city dweller does not want to move to a larger city A paying the migration cost ${c}_{m}\cdot \left(V\left({a}_{x},x;H\right)-r\left(x\right)\right)$.

These two inequalities can be further simplified as follows:

$$\left(1-{c}_{m}\right)\left(1-\psi \right)c\left({a}_{x}\right)m\left(x;H\right)\le d\cdot \left(1-{\psi }^{s}\right)c\left({a}_{x}\right)m\left(x;{H}^{s}\right)\le \left(1+{c}_{m}\right)\left(1-\psi \right)c\left({a}_{x}\right)m\left(x;H\right)\iff (1-{c}_{m})\left(1-\psi \right)\le d \left(1-{\psi }^{s}\right)\frac{m\left(x;{H}^{s}\right)}{m\left(x;H\right)}\le (1+{c}_{m})\left(1-\psi \right)$$

In short, the new inter-city incentive compatibility (IC) condition is summarized as:

$$\therefore \left( {1 - c_{m} } \right)\left( {1 - \psi } \right) \le d\frac{{m\left( {x;H^{s} } \right)}}{{m\left( {x;H} \right)}}\left( {1 - \psi^{s} } \right) \le \left( {1 + c_{m} } \right)\left( {1 - \psi } \right)$$

(35)

This non-migration incentive compatibility condition retains a nice intuition that a decrease in population density $d$ lowers the knowledge spillover ratio term $\frac{m\left(x;{H}^{s}\right)}{m\left(x;H\right)}$ while not affecting the terms at both ends of the inequality conditions, forcing the cost-of-living adjusted income ratio term $\left(1-{\psi }^{s}\right)$ to go up, resulting in a much lower value of the rent-to-income ratio ${\psi }^{s}$ of the smaller city B. This implies that city B has a much smaller rent because of the lower rent-to-income ratio combined with its lower income due to decreased knowledge spillovers.⃞