American Economic Review 2019, 109(1): 153–170
https://doi.org/10.1257/aer.20130249

A Spatial Knowledge Economy†
By Donald R. Davis and Jonathan I. Dingel*
Leading empiricists and theorists of cities have recently argued thatthe generation and exchange of ideas must play a more central rolein the analysis of cities. This paper develops the first system of c­ ities
model with costly idea exchange as the agglomeration force. The
model replicates a broad set of established facts about the cross section of cities. It provides the first spatial equilibrium theory of why
skill premia are higher in larger cities and how variation in thesepremia emerges from symmetric fundamentals. (JEL J24, J31, O31,
R12, R23)

In modern economies driven by innovation and ideas, local economic outcomesincreasingly depend on local idea generation. Empirically, the spatial distributionof human capital has consequences for productivity, prices, and inequality (Rauch
1993, Moretti 2004, Diamond 2016). Theoretically, however, the exchange ofideas in cities has often been treated as a special case of “black box” local external
economies.1
This paper introduces a model in which costly exchange of ideas is the agglomeration force driving a variety of spatial phenomena. Heterogeneous individuals, drawn
from a continuous distribution of ability, may produce tradables or ­non-tradables,* Davis: Department of Economics, Columbia University, 420 W 118 Street, New York, NY 10027, and NBER
(email: drdavis@columbia.edu); Dingel: Booth School of Business, University of Chicago, 5807 S Woodlawn
Avenue, Chicago, IL 60637, and NBER (email: jdingel@chicagobooth.edu). This paper was accepted to the AER
under the guidance of Debraj Ray, Coeditor. We thank four anonymous referees, Pol Antràs, Kerem Coşar, Arnaud
Costinot, Gilles Duranton, Jessie Handbury, Walker Hanlon, Sam Kortum, Corinne Low, Ben Marx, Joan Monras,
Suresh Naidu, Kentaro Nakajima, Stephen Redding, Holger Sieg, Daniel Sturm, Eric Verhoogen, Reed Walker,
David Weinstein, and seminar participants at the CESifo conference on heterogeneous firms in international
trade, Columbia applied micro and international trade colloquia, Conference on Urban and Regional Economics,
Empirical Investigations in International Trade, NBER ITI meeting, NYU, Urban Economics Association annual
meeting, Princeton IES Summer Workshop, Spatial Economic Research Centre annual conference, and Universityof Toronto for helpful comments on various drafts. We thank Yuxiao Huang, Paul Piveteau, and especially Meru
Bhanot for research assistance. We are grateful to Enrico Moretti and Stuart Rosenthal for sharing their housing-price measures with us. Dingel gratefully acknowledges resources provided by the University of Chicago
Research Computing Center and financial support from the Institute for Humane Studies, the Program for Economic
Research at Columbia University, and the Kathryn and Grant Swick Faculty Research Fund at the University of
Chicago Booth School of Business. The authors declare that they have no relevant or material financial interests that
relate to the research described in this paper.
†
Go to https://doi.org/10.1257/aer.20130249 to visit the article page for additional materials and author
disclosure statement(s).
1 ­Abdel-Rahman and Anas (2004, p. 2300): “One way to interpret this ­black-box model [of Marshallian externalities] is that the productivity of each worker is enhanced by the innovative ideas freely contributed by the labor
force working in close proximity.” Fujita, Krugman, and Venables (1999, p. 4), and Fujita and Thisse (2002, p. 129)
criticize the b­ lack-box version for being evanescent in empirical terms and close to assuming the conclusion intheoretical terms. Duranton and Puga (2004, p. 2065) describe “looking inside the black box… as one of the fundamental quests in urban economics.”
153

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THE AMERICAN ECONOMIC REVIEWJANUARY 2019and ­higher-ability individuals have comparative advantage in tradables. Tradables
producers divide their time between producing and exchanging ideas with each otherin order to raise their productivity. Cities with more numerous and h­ igher-ability
partners are better ­idea-exchange environments. ­Higher-ability individuals benefit
more from these conversations, so they locate in larger cities, paying higher local
prices to realize more valuable idea exchanges. In equilibrium, larger cities exhibit
better i­dea-exchange opportunities because their tradables producers are more talented, greater in number, and devote more time to exchanging ideas. Less skilledindividuals are employed in every city producing ­non-tradables, and larger cities
have higher n­ on-tradables prices to compensate them for their higher costs of living.
Our model replicates a broad set of empirical facts about the cross section of cities. First, while our model has symmetric fundamentals, ­idea-driven agglomeration
generates cities of heterogeneous sizes.2 Second, larger cities exhibit higher nominal
wages, housing prices, and productivity in equilibrium (Glaeser 2008). Third, larger
cities’ higher wages are partly attributable to h­ igher-ability individual sorting into
those locations, but this sorting is incomplete and individuals of many skill types arepresent in every city (Combes, Duranton, and Gobillon 2008; Gibbons, Overman,
and Pelkonen 2014; Carlsen, Rattsø, and Stokke 2016; de la Roca and Puga 2017).
This account of the spatial distribution of heterogeneous labor yields a novel
prediction about spatial variation in skill premia. Since ­higher-ability tradables producers locate in larger cities and raise their productivity by exchanging ideas while­non-tradables productivity does not vary across locations, the relative productivity of tradables producers is increasing in city size. This causes relative wages toincrease with city size when the productivity gap is only partially offset by higher­non-tradables prices. Empirically, Table 1 shows that the college wage premiumrises significantly with metropolitan population. This measure of the skill premium
ranges from about 47 percent in metros with 100,000 residents to about 71 percent
in metros with 10 million residents. This relationship is robust to controlling for twoother city characteristics that prior work has linked to cities’ skill premia: the fraction of the population possessing a college degree and housing prices.3 The positive
correlation between cities’ population sizes and skill premia is a robust, persistent,­first-order feature of the data.4
Theoretically linking together cities, ideas, and skill premia is n­ ontrivial. Unlike
temporal differences in wage premia, spatial differences in wage premia are disciplined by a ­no-arbitrage condition. As Glaeser (2008, p. 85) notes, when people
are mobile, differences in productivity “tend to show up exclusively in changes inquantities of skilled people, not in different returns to skilled people across space.”
The canonical s­patial-equilibrium model, in which there are two homogeneous
skill groups and preferences are homothetic, predicts that skill premia are spatially2
In classic models, heterogeneity across industries supports heterogeneous city sizes (Henderson 1974). In ourmodel, heterogeneity across individuals supports this outcome.
3
See Glaeser (2008); Glaeser, Resseger, and Tobio (2009); and Beaudry, Doms, and Lewis (2010) on college
shares and Black, Kolesnikova, and Taylor (2009) on housing prices.
4
Online Appendix C.2 reports regressions for 1990 and 2007 that also show a positive ­premia-size relationship.
More broadly, Wheeler (2001); Glaeser, Resseger, and Tobio (2009); Behrens and R­ obert-Nicoud (2014); andBaum-Snow and Pavan (2013) relate other measures of wage inequality to city size.

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Table 1—Skill Premia and Metropolitan Characteristics, 2000
Population (log)Rent (log)0.031(0.0037)

College ratio (log)
R2

0.151

0.029
(0.0053)
0.019(0.033)0.034(0.0042)−0.036(0.018)0.026(0.0049)
0.097(0.035)−0.069(0.018)

0.153

0.171

0.199

Notes: Robust standard errors in parentheses. Each column reports an OLS regression with 325
observations. The dependent variable is a metropolitan area’s difference in average log hourly
wages between college and high school graduates. See online Appendix C for details.

invariant (Black, Kolesnikova, and Taylor 2009). Our departure from these standard
assumptions yields a novel prediction that matches the data.
Our modeling of heterogeneous abilities, cities, and skill premia in a setting with spatially symmetric fundamentals distinguishes our theory from recent
work that engages these topics by assuming either asymmetric fundamentals or­talent-homogeneous cities. A number of recent contributions have sought to explain
differences in outcomes for skilled and unskilled workers across cities by appealing
to exogenous differences in fundamental characteristics of those cities.5 A recent
paper by Behrens, Duranton, and R­ obert-Nicoud (2014) assumes symmetric fundamentals and a continuum of abilities, as we do, but they focus on equilibria withheterogeneous cities in which each city is populated by individuals of only one
ability. A theory of ­talent-homogeneous cities cannot explain spatial variation inskill premia.6
The role of cities in facilitating idea exchange has been noted by economists
since at least Marshall (1890). Empirical studies suggest that larger cities rewardcognitive and people skills rather than motor skills or physical strength (Bacolod,
Blum, and Strange 2009; Michaels, Rauch, and Redding forthcoming). Physicalproximity is associated with increased communication and intellectual interaction(Jaffe, Trajtenberg, and Henderson 1993; Gaspar and Glaeser 1998; Audretsch andFeldman 2004; Charlot and Duranton 2004; Arzaghi and Henderson 2008). Sincemuch knowledge is tacit and requires ­face-to-face transmission, we treat cities as theloci of idea exchange.
Our model unites two strands of theoretical literature on the exchange of ideas.
One focuses on individuals’ spatial choices when knowledge spillovers are exogenous externalities (Henderson 1974, Black 1999, Lucas 2001). Another focuses
on choices of learning activities within a single location of exogenous population(Jovanovic and Rob 1989; Helsley and Strange 2004; Berliant, Reed, and Wang
2006; Berliant and Fujita 2008). In our model, locational choices shape idea5
For example, Glaeser (2008) and Beaudry, Doms, and Lewis (2010) use ­skill-segmented housing markets
and s­kill-biased housing supplies to explain spatial variation in skill premia. These neoclassical models do not
relate skill premia to city sizes. Gyourko, Mayer, and Sinai (2013) model exogenous differences in housing supply
elasticities.
6
In their model, all ­within-city inequality is due to exogenous shocks that individuals experience after selecting
their location. Educational attainment is neither randomly assigned nor ­location-specific.

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exchanges because learning opportunities are heterogeneous and depend uponthe ­time-allocation decisions of local participants.7 Our characterization of idea
exchanges is simple compared to the second strand of literature, but this allows us totractably model endogenous exchanges of ideas in a system of cities.
We focus on the exchange of ideas between rather than within firms. Idea
exchange within firms is surely important, but it does not motivate firms to locate incities, since i­ntrafirm idea exchange may occur in geographic isolation. Our model
describes ­interfirm interactions because these are the idea exchanges that may
underpin urban agglomeration.8
I. A Spatial Knowledge Economy

The economy consists of a continuum of individuals of mass ​L​ , whose heterogeneous abilities are indexed by ​z​and distributed with density ​μ(z)​on connectedsupport on ​​핉​+​​​. There are a number of homogeneous sites that may be cities withendogenous population and ability composition.
A. Preferences and Production
Individuals consume three goods: tradables, ­non-tradable services, and(­non-tradable) housing. Services and housing are strict necessities; after consuming ​ ​n̅​​units of n­ on-tradable services and one unit of housing, consumers spend allof their remaining income on tradables, which we use as the numéraire.9 Therefore,
the indirect utility function for a consumer with income ​y​facing prices ​​p​n,c​​​and ​​ph​,c​​​in city c​ ​ is(1)	​V( ​p​n,c​​, ​ph​,c​​, y) = y − ​p​n,c​​ ​ n̅​ − ​p​h,c​​.​Individuals are perfectly mobile across cities and jobs, so their locational and occupational choices maximize V​ ( ​p​n,c​​, ​ph​,c​​, y)​.
An individual can produce tradables (​t​) or n­ on-tradables (​n​). ­Non-tradables canbe produced at a uniform level of productivity by all individuals, which we normalize
to 1 by choice of units. Tradables, by contrast, make use of the underlying heterogeneity in ability. An individual’s tradables output is ​​z̃ ​(z, ​Zc​​​  )​ , which depends on bothindividual ability ​z​and learning opportunities available through local i­nteractions,​​
7
Glaeser (1999) is an important precursor to our approach. His model specifies two locations, a city and a rural
hinterland. In contrast to our approach, the fundamental difference between the two locations is exogenous, sincelearning is possible only in the city.
8
Recent research suggests that physical proximity facilitates such activities. Allen, Raz, and Gloor (2010)
examine i­nterfirm communication amongst individual scientists at biotech firms in the Boston area and find thatgeographic proximity and firm size are both positively associated with ­interfirm communication on the extensive and intensive margins. Inoue, Nakajima, and Saito (2015) examine ­interfirm collaboration on Japanese patentapplications and find that it is more geographically concentrated than ­intrafirm collaboration and this localizationis stable over the last two decades.
9
The merit of this stark specification is tractability. Section IIC shows that our assumption of perfectly inelastic demand for housing and services generates a compensation effect that in fact works against finding a positive­premia-population relationship. Section IID shows that this inelastic specification is nonetheless consistent withrealistic housing expenditure shares. Assuming unit demand for housing is common in urban theory (e.g., Moretti
2011; Behrens, Duranton, and ­Robert-Nicoud 2014).

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Z​c​​​. An individual working in sector ​σ​earns income equal to the value of her output,
which is​p​n,c​​if σ = n ​​.​​​  ​ (2)	​y = ​ ​  {​z ​̃(z, ​Zc​​​) if σ = t
Tradables productivity depends both on an individual’s ability and participation
in idea exchanges. Tradables producers can raise their productivity by exchanging
ideas with other tradables producers in their city.10 Each person has one unit oftime that they divide between interacting and producing. Exchanging ideas is an
economic decision, because time spent interacting trades off with time spent producing output directly. The production function for tradables B​ ​depends on timespent exchanging ideas (​1 − β​), time spent producing (​β​), own ability (​ z)​ , and local
learning opportunities ​(​Z​c​​)​. When time is allocated optimally, the output of a tradables producer of ability ​z​ is(3)	​​z ​̃(z, ​Zc​​​)  = ​ max​​​  B(1 − β, z, ​Zc​​​).​
β∈[0, 1]The value of local idea exchanges, ​​Z​c​​​ , is determined by the ­time-allocation decisions of all the tradables producers living in city c​ ​. In particular, it is a function ofboth the time they devote to exchanges and their abilities.11 We denote the time
devoted to idea exchange by individuals of ability ​z​in city ​c​ by ​1 − ​βz​ ,c​​​and the
population of individuals of ability z​ ​in city c​ ​by L​ ⋅ μ(z, c)​ , where μ​ (z, c) /μ(z)​ is
the share of z​ ​-ability individuals who live in ​c​. The value of the local i­ dea-exchange
environment ​​Zc​​​​is a functional of the ­time-allocation decisions and the population
composition,(4)	​​Z​c​​  = Z( { 1 − ​βz​ ,c​​},  { L ⋅ μ(z, c)}).​Key to the tractability of our model is that these local learning opportunities are summarized by a scalar. To further summarize behavior, denote the total time devoted tolearning by tradables producers in city c​ ​by ​​Mc​​​​ , which is defined as ​​ ​(1 − ​β​z,c​​) μ(z, c ) dz.​	​​Mc​​​  = L ​∫z:σ(z)=tThis depends on the city’s total population of tradables producers, their ability composition, and their ­time-allocation choices.
We make three assumptions about the production of tradables and exchange ofideas. First, tradables output is increasing in both ability ​z​and the ­idea-exchange
environment ​Z​ c​​​​. In the absence of i­dea-exchange opportunities (​​Z​c​​  = 0​) or time10
Our static model focuses on the location of idea exchange and abstracts from dynamic accumulation ofknowledge. Lucas and Moll (2014) and Perla and Tonetti (2014) study knowledge accumulation while abstractingfrom the spatial dimension. We hope that future work might unify these topics.
11
These elements distinguish our agglomeration mechanism from a “black box” function that depends on alocation’s population size. Heterogeneous individuals’ choices of locations and time allocations are economic decisions that both have opportunity costs and determine local opportunities for idea exchange.

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devoted to them (​β = 1​), tradables output is the product of time spent producing
and ability, β​ z​.
ASSUMPTION 1: The production function for tradables B​ (1 − β, z, ​Zc​​​)​is continuous, strictly concave in 1​ − β​, strictly increasing in z​​, and increasing in ​Z​ c​​​​.​
B(1 − β, z, 0)  = βz​ and ​B(0, z, ​Zc​​​) = z​ ​∀ z​.
Second, individual ability and local learning opportunities are complements.12
When exchanging ideas, the output gain from greater ability is increasing in ​​Zc​​​​.
ASSUMPTION 2: Tradables output ​​z̃ ​(z, ​Zc​​​)​is supermodular and is strictly supermodular on ⊗​ ≡ {(z, Z ) : ​z̃ ​(z, Z ) > z}​.
Third, the ­idea-exchange environment ​​Zc​​​​is better when those devoting time toidea exchange are of higher ability and when all tradables producers devote moretime to idea exchange.
ASSUMPTION 3: The ­idea-exchange functional ​Z( { 1 − ​βz​ ,c​​  },  { L ⋅ μ(z, c)})​ is
continuous, equal to 0 if ​​Mc​​​  = 0​, and bounded above by s​up  { z : 1 − ​βz​ ,c​​  > 0,{(1 − ​β​z, c​​  ) μ(z, c)}​stochastically dominates
μ(z, c )   > 0}​. If ​​Mc​​​  > ​M​​c′ ​​​​and ​​{(1 − ​β​z,​c′ ​​​  )μ(z, ​c′ ​  )}​, then ​Z( { 1 − ​βz​ ,c​​  },  { L ⋅ μ(z, c)}) > Z( { 1 − ​βz​ ,​c′ ​​​  },  { L ⋅ μ(z, ​c′ ​  )})​.
Assumption 3 implies that knowledge has both horizontal and vertical dimensions. There is horizontal differentiation in the sense that individuals can learnsomething from anyone and are therefore better off when all tradables producers
devote more time to exchange. Vertical differentiation means that they learn morefrom more able counterparts.
For some of our analysis, we focus on particular functional forms for B​ ( ⋅ )​ and​Z( ⋅ )​:(5)	​B(1 − β, z, ​Zc​​​) = βz(1 + (1 − β) A ​Zc​​​  z),(6)Z( { 1 − ​β​z, c​​  },  { L ⋅ μ(z, c)}) = ​(1 − exp (−ν ​Mc​​​  ))​ ​​  z̅c​​ ​​  ,⎧z(1 − ​β​z,c​​) μ(z, c)​ z:σ(z)=t∫ ​​ ​ ​___________________      ​ dz if  ​Mc​​​  > 0​​  z ̅c​​ ​​  = ​⎨​     ​ ​​​​.
​∫z:σ(z)=t ​​ ​(1 − ​β​z,c​​  ) μ(z, c ) dz ​ ⎩0otherwise⎪⎪

Online Appendix A.6 shows that these functions satisfy Assumptions 1–3.
In this special case, productivity gains are the product of random matches between
individuals devoting time to idea exchange. The term A​ ​indexes the scope for gains
12
Assumption 2 is stated in terms of ​​z̃ ​(z, ​Zc​​​  )​ , which incorporates the optimal choice of ​1 − β​. In online
Appendix A.5, we identify sufficient conditions for B​ (1 − β, z, ​Z​c​​  )​that imply Assumption 2. This Assumption ​2​ ′ ​​ issufficient but not necessary. For example, equation (5) satisfies Assumption 2 but not Assumption ​​2′ ​​.

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from such interactions. With random matching, the expected value of devoting amoment of time to idea exchange in a city is the probability of encountering anotherindividual during that moment times the expected ability of the individual encountered.13 Since idea exchanges are instantaneous and individuals devote an interval oftime to idea exchange, every individual devoting time to exchanging ideas realizes
the expected gains from these exchanges, ​​Z​c​​​.
The probability of encountering someone during each moment of time spentseeking idea exchanges is 1​ − exp (−ν ​M​c​​  )​. As in Diamond (1982), exchange is easier when there are more potential exchange partners. Thus, one potential benefit oflarger cities is that idea exchanges may occur with greater frequency there (Glaeser
1999).14 The population of individuals available for such encounters is determined
endogenously by tradables producers’ t­ime-allocation choices.
The average ability of the individuals encountered in these matches is ​​​  z ̅c​​ ​​​. This isa weighted average of the abilities of local tradables producers, in which the weights
are the time each type of individual devotes to interactions. Conditional on meeting
another learner and one’s own ability, conversations with more able individuals are
more valuable.
The agglomeration mechanism described by Assumptions 1–3 trades off with asimple congestion force. Each individual in a city of population ​​Lc​​​​pays a net urban
cost (in units of the numéraire) of(7)	​​ph​,c​​  = θ ​L​  γc​  ​  , ​with θ​ , γ > 0​. We will refer to ​​ph​,c​​​as the price of housing in city c​ ​ , though thisobject incorporates both land rents and commuting costs when given standard
microfoundations.15

13
Random matching is not particularly realistic, but it is tractable. Related work in growth theory, Lucasand Moll (2014) and Perla and Tonetti (2014), also assumes random meetings. In those models, ­lower-ability
agents spend time in order to observe a random competitor and improve their productivity through imitation of­higher-ability agents. In our model, both participants in a meeting opt to spend time exchanging ideas. The benefitof the meeting can be interpreted as combining both what one can learn from the other and one’s ability to learn
from any encounter. Thus, even when participants in a meeting have different skill levels, both benefit. By contrast, those g­ rowth-theory papers feature meetings that benefit only the ­lower-ability participant, who imitates the­higher-ability participant. Such perfectly asymmetric accounts of idea exchange are poorly suited as a foundation
for spatial models. The ­highest-ability agent would never pay to be in a city where meetings occurs because idea
exchanges with l­ower-ability agents yield zero benefit. By iteration, every agent would want to move to locations
where ­higher-ability agents reside, while the latter would leave locations populated by l­ower-ability agents. Our
approach in which idea exchange is mutually beneficial provides a framework consistent with spatial equilibrium.
14This ­city-level scale effect embodies the horizontal dimension of knowledge in Assumption 3 and implies anupper bound on the number of heterogeneous cities. See online Appendices A.2 and A.6.1. Most empirical evidence
on matching processes describes job search, which is distinct from idea exchange in numerous dimensions. Early­job-search studies, while noisy, were often interpreted to suggest constant returns (Petrongolo and Pissarides 2001).
More recent studies have found results more favorable to increasing returns to scale (Petrongolo and Pissarides
2006, Di Addario 2011, Bleakley and Lin 2012).
15
Behrens, Duranton, and R­ obert-Nicoud (2014) provide microeconomic foundations for this functional form,
which they derive from a standard model of the internal structure of a monocentric city in which commuting costs
increase with population size as governed by the technological parameters ​θ​and γ​ .​ See online Appendix A.1 fordetails.

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B. Equilibrium
Individuals choose their locations, occupations, and time allocations optimally.
Since individuals are perfectly mobile, two individuals with the same ability ​z​ willobtain the same utility in equilibrium wherever they are located.
An equilibrium for a population ​L​with ability distribution ​μ(z)​in a set of loca​ ​​  }​and populations μ​ (z, c)​such that workers maxtions {​ c}​is a set of prices {​  ​p​h,c​​  , ​pn,cimize (1) by their choices of ​c, σ​ , and β​ ​and markets clear.16 Markets clear when​​β​z,c​​  = β(z, ​Zc​​​  )​ , and equations (4), (7), and the following conditions hold:(8)	​μ(z) = ​∑​ ​​  μ(z, c)  ∀ z,c(9)	​Lc​​​   = L ​∫​  ​​μ(z, c) dz  ∀ c,(10)	​ n̅​ ​L​c​​   = L ​∫z:σ(z)=n ​​ ​μ(z, c) dz  ∀ c​.
The equilibrium value of local idea exchanges ​Z​ c​​​  = Z( { 1 − ​βz​ ,c​​},  { L ⋅ μ(z, c)})​ in
equation (4) is a fixed point, since individuals’ choices of location and ​​β​z,c​​​ depend
on local learning opportunities ​​Z​c​​​. Equation (7) defines the ­market-clearing housing
price in each city. Equations (8) and (9) are ­adding-up constraints for worker types
and city populations. Equation (10) equalizes demand and supply of n­ on-tradable
services within each location. The tradables market clears by Walras’ Law.
II. The Cross Section of Cities in Equilibrium

We now characterize equilibrium outcomes. By comparative advantage,h­igher-ability individuals produce tradables and l­ower-ability individuals produce ­non-tradables. Since individual ability and ­idea-exchange opportunities arecomplements, there is spatial sorting of tradables producers by ability. When cities are heterogeneous, tradables producers are sequentially segmented so that the­lowest-ability tradables producers locate in the smallest city and the ­highest-ability
tradables producers locate in the largest city.17 For the marginal tradables producer,
a larger city’s ­idea-exchange benefits are exactly offset by its higher cost of living.
Together, sorting and idea exchange cause larger cities to exhibit higher skill premia.
A. Equilibrium Occupations and Prices
Occupational choices are governed by comparative advantage. ­High-ability individuals produce tradables since labor heterogeneity matters in that sector.
16
In this exposition, we define equilibrium where each member of the set ​{c}​is populated, ​​L​c​​  > 0​. In online
Appendix A.2, we discuss the endogenous number of cities that make up this set, since not all potential city locations must be populated.
17
Spatial models with a continuum of heterogeneous individuals sorting across a finite number of locations date
to at least Westhoff (1977), who studied conditions for the existence of an equilibrium in a model of local publicfinance.

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LEMMA 1 (Comparative Advantage): Suppose that Assumption 1 holds. There isan ability level ​​zm
​ ​​​such that individuals of greater ability produce tradables andindividuals of lesser ability produce ­non-tradables:
t if z > ​zm
​ ​​
​  ​ 
​​​.​
	​
σ(z) = ​ ​   
{n if z < ​zm​ ​​The proofs of Lemma 1 and subsequent results appear in online Appendix A.6. By
Lemma 1 and equations (8) through (10), the ability level of the individual indifferent between producing tradables and n­ on-tradables, ​​zm
​ ​​​ , is given by(11)	​​ 0​  ​​  μ(z ) dz = ​ n̅​.​
∫
​zm
​ ​​

Since individual ability and local learning opportunities are complements, there isspatial sorting of tradables producers engaged in idea exchange. ­Higher-ability tradables producers locate in cities with better i­dea-exchange environments.
LEMMA 2 (Spatial Sorting of Tradables Producers Engaged in Idea Exchange):
Suppose that Assumptions 1 and 2 hold. For z​ > ​z′ ​ > ​z​m​​​, if ​μ(z, c) > 0​, ​μ(​z′ ​, ​c′ ​   )> 0​, ​β(z, ​Zc​​​) < 1​, and ​β(​z′ ​, ​Z​c​′ ​​​) < 1​, then ​​Zc​​​  ≥ ​Z​​c′ ​​​​.
As a result, individuals of ability ​​z​m​​​producing tradables will be located in the citywith the lowest value of ​Z​ ​c​​​. Label cities in order of the value of their idea exchanges,​​Z​c​​  ≥ ​Z​c−1​​​ , so that ​​Z1​​​  = ​min​c​​  { ​Zc​​​  }​. Indifference between producing tradables and­non-tradables implies that ​​pn​,1​​​ satisfies​ ​​, ​Z1​​​).​(12)	​​pn​, 1​​  = ​z̃ ​(​zm
There is a population of ­non-tradables producers located in each city. In spatial equilibrium, each of these individuals obtains the same utility, so equation (1)implies that spatial differences in n­ on-tradables prices exactly compensate for spatial differences in housing prices:(13)	​(1 − ​ n̅​) ​p​n,c​​  − ​p​h,c​​  = (1 − ​ n̅​) ​p​n,​c′ ​​​  − ​p​h,​c′ ​​​  ∀ c, ​c′ ​​.
All equilibria exhibit this pattern of occupations and prices. We now distinguishbetween equilibria based on whether cities vary in size.
B. Equilibrium Systems of Cities
There are two classes of equilibria for this economy: equilibria in which all cities
have the same population size and equilibria with heterogeneous cities. The latterare the empirically relevant class. We analyze the properties of equilibria with heterogeneous cities after describing why systems of ­equal-sized cities are only stableequilibria if the marginal gains from idea exchange are too small relative to marginal

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congestion costs to break the symmetric arrangement. When idea exchange is sufficiently rewarding, a system of heterogeneous cities is an equilibrium configuration.18
Systems of ­Equal-Sized Cities.—Given symmetric fundamentals, systems ofe­ qual-sized cities are possible equilibria. By equations (7) and (13), ­equal-sized
cities have equal local prices. To be in equilibrium, they must also have equal­idea-exchange benefits for the marginal tradables producer.
When idea exchange occurs nowhere, ​Z​ ​c​​  = 0 ∀ c​ , these benefits are equal
because every tradables producer devotes zero time to idea exchange. This is individually rational when others do the same. While not the focus of our paper, the­no-idea-exchange equilibrium illustrates an important aspect of the economic mechanism: ideas are not manna from heaven but the outcome of a costly allocation oftime by those acquiring knowledge. Though not the empirically relevant case, thispossibility highlights the relevant economic ­trade-off.
A system of e­ qual-sized cities in which idea exchange occurs can only be astable equilibrium when the marginal benefits of a better ­idea-exchange environment are small relative to marginal congestion costs. Denote the city with the best
­idea-exchange environment by C
​ ​. Given identical prices, living in C
​ ​is optimal for
all individuals of ability such that ​β​(z, ​ZC​ )​​ ​  < 1​. If ​C​is the only city with idea
exchanges, this is an equilibrium only if both ​​Z​C​​​and the ability distribution are so
low that all those possibly gaining from idea exchange fit in this single city and themarginal tradables producer’s ­idea-exchange benefit is zero. If another city also has
idea exchanges, the value of its ­idea-exchange environment must be the same. An
equilibrium with two cities with equal i­ dea-exchange environments is locally stable
only if the marginal gains from idea exchange are small relative to marginal congestion costs. Otherwise, the movement of some ­high-ability tradables producers
from one city to the other would improve the latter’s ­idea-exchange environment,
thereby drawing in more tradables producers and breaking the symmetric arrangement.19 In particular, if any improvement in the ­idea-exchange environment is sufficiently valuable to the ­highest-ability tradables producers, then an equilibrium of­equal-sized cities cannot be stable.20
Thus, a system of ­equal-sized cities is a stable equilibrium in the trivial case thatno one exchanges ideas or if marginal congestion costs are large relative to the benefits of a better i­ dea-exchange environment. We henceforth focus on the empirically
relevant equilibrium configuration, a system of heterogeneous cities.
Systems of Heterogeneous Cities.—Equilibria with heterogeneous cities exhibitc­ ross-city patterns that can be established independent of the number of cities thatarise.21 Proposition 1 characterizes the characteristics of heterogeneous cities inequilibrium.
18
We provide sufficient conditions for the existence of a stable equilibrium with two heterogeneous cities inonline Appendix A.3. While we do not provide conditions for the existence of equilibria with more than two cities,
we have numerically computed equilibria with many more cities. Section IID reports an example with 275 cities
and online Appendix B summarizes many more numerical results.
19
See online Appendix A.4 for our definition of local stability and the relevant argument.
20
See Proposition 3 in online Appendix A.4 for the formal result.
21
Since these patterns characterize all equilibria with heterogeneous cities, we do not address issues of uniqueness or determine the equilibrium number of cities. See online Appendix A.2 for further discussion.

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PROPOSITION 1 (Heterogeneous Cities’ Characteristics): Suppose that Assumptions 1 and 2 hold. In any equilibrium, a larger city has higher housing
prices, higher ­non-tradables prices, a better ­idea-exchange environment, and​ ​​  > ​p​h,​c′ ​​​​,­higher-ability tradables producers. If ​​Lc​​​  > ​L​​c′ ​​​​in equilibrium, then ​​ph,c​ ​​  ⇒​ ​μ(z, c ) μ(​z′ ​, ​c′ ​ )  ≥ μ(z, ​c′ ​  ) μ(​z′ ​, c) = 0​.
​​pn​,c​​  > ​p​n,​c′ ​​​​, ​​Zc​​​  > ​Z​​c′ ​​​​, and ​z > ​z′ ​ > ​zm
The mechanics of Proposition 1 are straightforward. Larger cities have higher
housing prices due to congestion, so ­non-tradables producers require higher wages
in these locations. Larger cities attract tradables producers because the benefitsof more valuable idea exchanges offset their higher housing and ­non-tradables
prices. More able tradables producers benefit more from participating in better idea
exchanges, so there is spatial sorting of tradables producers.22 This spatial sorting supports equilibrium differences in i­dea-exchange environments because these­high-ability individuals are better i­ dea-exchange partners, conditional on population
size and time allocations.
Equilibria with heterogeneous cities match the fundamental facts that cities differ
in size and these size differences are accompanied by differences in wages, housing prices, and productivity (Glaeser 2008). Empirically, larger cities exhibit higher
nominal wages in industries that produce tradable goods, which means that productivity is higher in these locations (Moretti 2011). Our model of why larger cities
generate more p­ roductivity-increasing idea exchanges is a microfounded explanation of these phenomena. Having matched these w­ ell-established facts, we now
describe the novel implication that skill premia will be higher in larger cities.
C. Skill Premia with Heterogeneous Cities
We define a city’s skill premium as the average income of its tradable producers
relative to the wage of its ­non-tradable producers. Our model typically predicts thatskill premia are higher in more populous cities. After discussing the mechanisms
contributing to spatial variation in skill premia, we formally state this prediction
for two cities in Proposition 2. Numerical analysis, detailed in online Appendix B,
suggests that this prediction generalizes from two cities to a large number of heterogeneous cities.
The nominal wages of both n­ on-tradables and tradables producers are higher inlarger cities. For ­non-tradables producers, higher nominal wages in larger cities arecompensation for higher housing prices that keeps their utility constant across cities,
per equation (13).
Differences in tradables producers’ wages across cities can be expressed as thesum of three components: composition, learning, and compensation effects. First,
due to spatial sorting, tradables producers in larger cities have higher innate abilities that generate higher incomes in any location. Second, since one’s own ability
complements others’ abilities in idea exchanges, these tradables producers realize
22
Our prediction of sorting among the more able and indifference among the less able is consistent with thelimited evidence available. Using Norwegian administrative data, Carlsen, Rattsø, and Stokke (2016) estimate
unobserved ability with individual fixed effects in wage regressions and find sorting among ­college-educated workers but none among those with primary and secondary education.

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larger income gains in larger cities’ better ­idea-exchange environments. Third, producers who are indifferent between two cities realize learning gains in the larger citythat exactly compensate for its higher n­ on-tradables and housing prices. For convenience, let ​​z​b​​​denote the ability of this boundary tradables producer who is indifferent, define inframarginal learning ​Δ(z, c, c′ )​as the ­idea-exchange gains accruing toa producer of ability z​ ​from locating in environment ​Z​ c​​​  > ​Z​​c′ ​​​​compared to thosegains for ability ​z​ b​​​​ , and define the density of tradables producers’ abilities in city c​ ​by ​​μ̃ ​(z, c)​.23 When a tradables producer of ability ​​zb​​​​is indifferent between cities c​ ​and ​​c′ ​​ , the difference in the cities’ average tradables wages can be expressed as​∫z:σ(z)=t ​​ ​z​ ̃ ​(z, ​Z​c​′ ​​​  ) μ(z, ​c′ ​) dz∫​ z:σ(z)=t ​​ ​z​ ̃ ​(z, ​Zc​​​)μ(z, c) dz _________________
  
  
  
 ​
 ​
− ​   ​​​w̅ ​​c​​  − ​​w̅ ​​​c′ ​​​   ≡ ​ _________________​∫z:σ(z)=t ​​ ​μ(z, ​c′ ​) dz​∫z ​​ :σ(z)=t​μ(z, c ) dz= ​​​∫  ​  ​​  [​μ̃ ​(z, c)  − ​  μ̃ ​  (z, ​c′ ​)​ ​ ]​z̃ ​(z, ​Z​​c′ ​​​  ) dz​​ + ​​​∫  ​    ​​​μ ​(z, c ) Δ(z, c, ​c′​ ​  ​) dz​​ + ​​​p
​n,c​​  − ​p​n,​c′​ ​  ​​​ .​​z​ ​​​z​ ​​ ̃
∞

∞

m


composition

m


inframarginal learningcompensation­ ross-city variation in skill premia can also be expressed in terms of these threeCcomponents. We define a city’s observed skill premium as its average tradables wage
divided by its (common) ­non-tradables wage, ​​​w̅ c​​ ​​/​pn​, c​​​. When a tradables producerof ability ​z​ b​​​​is indifferent between cities c​ ​and ​​c′ ​​ , this skill premium is higher in c​ ​ if
and only if  ​  ​​  [​μ̃ ​(z, c )  − ​  μ̃ ​  (z, ​c′ ​  ​ ​  ) ]​z̃ ​(z, ​Z​​c′ ​​​  ) dz​​ + ​​​∫  ​    ​​​μ ​(z, c ) Δ(z, c, ​c′​ ​  ​) dz​​(14)  ​​​​∫​zm​ ​​​z​m​​ ̃
∞

∞



composition

inframarginal learning​​w​​ ​​≥ ​​​(  ​p​n,c  ​​  − ​p​n,​c′ ​)​​   ​ ​(_​  ​p​̅ ​c′ ​  ​ ​​ −​ ​​
1).​​​
n,​c′ ​


relative compensation

The composition and inframarginal learning effects yield higher nominal incomes for
tradables producers in larger cities. These raise tradables producers’ wages relative
to n­ on-tradables producers’ wages in larger cities and therefore generate a positive­premium-population relationship. The compensation effect that reflects differencesin local prices makes the nominal wages of both tradables and ­non-tradables producers in larger cities higher by the same amount. Since ­higher-ability individuals
earn higher incomes, this compensation is a larger proportion of the ­non-tradables
producers’ incomes and therefore pushes toward a negative ­premium-population
relationship. When the composition and learning effects dominate this implicationof the compensation effect, the skill premium is higher in the larger city.
The sizes of these three effects depend on the distribution of abilities, ​μ(z)​  , thestrength of the complementarity between ​z​and ​​Zc​​​​in ​​z̃ ​(z, ​Zc​​​  )​ , and equilibrium differences in cities’ sizes. The composition and inframarginal learning effects ­necessarily

23

μ(z, c)That is, Δ​ (z, c, ​c′ ​  ) ≡ ​      ​​.
[​z ​̃(z, ​Zc​​​) − ​z ​̃(z, ​Z​c​′ ​​​  )]​ − ​[​z ​̃(​zb​​​, ​Zc​​​)  − ​z ​̃(​z​b​​, ​Z​c​′ ​​​  )]​​and ​​μ ​̃ (z, c) ≡ ​ ____________​∫z′:σ(​ ​​ z′ ​)=t​μ(​z′ ​, c) d​z′ ​VOL. 109 NO. 1

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depend on heterogeneity in tradables producers’ abilities.24 Inframarginal learning also depends on the degree to which ­higher-ability individuals experiencelarger gains from locating in a better i­dea-exchange environment. The relativecompensation effect is the product of ­size-related differences in costs of living(​​​p​n,c​​  − ​p​n,​c′ ​)​​ ​​and the level of the skill premium (​​​​w̅ ​c​​ ′ ​​​  / ​p​n, ​c′ ​)​​ ​​.25
Proposition 2 states three different sets of sufficient conditions under which,when the smallest city has population ​​L​1​​​and the ­second-smallest city has population​​L​2​​  > ​L​1​​​ , the skill premium is higher in the more populous city. These sufficientconditions depend jointly on assumptions about μ​ (z)​  , ​​z̃ ​(z, ​Zc​​​  )​ , and equilibrium city
sizes. For the production function, the relevant property concerns the ability elasticity of tradable output, ∂​ ln ​z̃ ​​(z, ​Zc​)​​ ​/  ∂ ln z​.
CONDITION 1: The ability elasticity of tradable output is ­non-decreasing in ​z​
and ​Z​ ​c​​​.
Condition 1 is satisfied by the production function in equation (5), as well asother production functions, as described in online Appendix A.5.
PROPOSITION 2 (Skill Premia): Suppose that Assumptions 1 and 2 hold. In an
equilibrium in which the smallest city has population ​L​ ​1​​​and the s­ econd-smallest
city has population ​​L​2​​  > ​L​1​​​(i) If the ability distribution is decreasing, ​​μ′ ​(z) ≤ 0​  , ​​z̃ ​(z, ​Zc​​​  )​ is ­log-convex in z​ ​  ,and ​​z̃ ​(z, ​Zc​​​  )​ is ­log-supermodular, then ​​​w̅ 2​​ ​​/​pn​,2​​  > ​​w̅ 1​​ ​​/​pn​,1​​​;​ in​​​ and k​ > 0​  ,(ii) If the ability distribution is Pareto, μ​ (z) ∝ ​z​​  −k−1​​ for z​ ≥ ​zm
and the production function satisfies Condition 1, then ​​​w̅ 2​​ ​​/​p​n,2​​  > ​​w̅ 1​​ ​​/​p​n,1​​​;​ ax​​)​​  , the production(iii) If the ability distribution is uniform, z​ ∼ U​(​z​min​​, ​zm​L​​​  − ​L​​​​L​  1​​ (1 − ​ n​) ( ​z​ ​​  − ​z​ ​​  )

min
̅ max
  
  
​​  , thenfunction satisfies Condition 1, and _____​​ 2 2 1 ​ > ​_L1 ​ ​____________​z​ ​​  + ​ n​( ​z​ ​​  − ​z​ ​​  )​​w̅ 2​​ ​/​p​n,2​​  > ​​w̅ 1​​ ​/​p​n, 1​​.

min

̅

max

min

These three cases trade off stronger assumptions about the production function with weaker assumptions about the ability distribution. In case (i), the­log-supermodularity of ​​z̃ ​​implies large inframarginal learning, and the log convexity of ​z​ ̃ ​​implies a large composition effect when h­ igher-ability individuals are notrelatively abundant. Together, these are sufficient to dominate the relative compensation effect, such that the skill premium is higher in the larger city. In case (ii),the Pareto ability distribution and the ability elasticity of tradable output jointly
24
In a model with only two skill types (i.e., homogeneous tradables producers and homogeneous ­non-tradables
producers), the skill premium is lower in the larger city. Homogeneity makes the composition and inframarginal learning components zero, leaving only the compensation term. This ­two-type case is the basis for the prediction by Black, Kolesnikova, and Taylor (2009) that skill premia will be lower in cities with higher housing
prices. Empirically, larger cities have both higher housing prices and skill premia.
25
Since ​p​ ​n,1​​  = ​z̃ ​​(​zm​ ​​  , ​Z1​)​​ ​​ , the level of the skill premium in city 1 depends on the heterogeneity in tradables
producers’ abilities, μ​ (z, 1)​ , and the complementarity between ​z​and ​​Z1​​​​in ​​z ​̃(z, ​Z1​​​  )​.

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generate composition and ­inframarginal-learning effects sufficient to dominate therelative compensation effect. In case (iii), the uniform ability distribution generates
weaker compensation and i­ nframarginal-learning effects for a given ability elasticityof tradable output. The sufficient condition in case (iii) is written in terms of endogenous equilibrium outcomes. It establishes a value of ​​L1​​​​small enough relative to theheterogeneity in tradables producers’ abilities such that the relative compensation
effect is less than these effects.26 Note that this condition is far from necessary, as
demonstrated in online Appendix B.1. The relative compensation effect approaches​ ​​​  − ​​pn,1​ ​​​  → 0.
0 as ​​L​1​​​  → ​​L​2​​​ because ​​pn,2Since the sufficient condition in case (iii) depends on endogenous city sizes, westudy the two-city, ­uniform-ability case further by numerically characterizing equilibria for the special case of equations (5) and (6) for a wide range of parametervalues in online Appendix B. We do find examples of parameter combinations thatyield equilibria in which a larger city has a lower skill premium, but they are rare(less than 0.3 percent of the parameter combinations for which equilibria exist).
These examples generate very large relative compensation effects (the ­right-handside of inequality (14)) by generating values of ​​​w̅ ​​1​​/​p​n,1​​​an order of magnitude larger
than those in the data. They also require values of ​γ​ , the congestion cost ­elasticity,that are implausibly large relative to empirical estimates, though n­ on-increasing
skill premia are atypical in equilibrium even for extreme parameter values.27
To extend the prediction of Proposition 2 to more than two cities, in online
Appendix B we numerically solve the special case of equations (5) and (6) for awide number of heterogeneous cities and wide range of parameter values for theuniform- and ­Pareto-ability cases. In the ­uniform-ability case, we again find thatskill premia are monotonically increasing in population size in almost all equilibria.
The exceptions to this pattern occur when ​​​w̅ ​​1​​/​p​n,1​​​and ​γ​are implausibly large relative to empirical values, consistent with the two-city case. In the ­Pareto-ability case,
we find that all equilibria examined exhibit monotonically increasing skill premia.28
Thus, our ­two-city result appears to generalize to many cities.
To summarize, our model typically predicts that skill premia are higher in more
populous cities, in line with the empirical pattern documented in Table 1. Proposition
2 analytically characterizes the pattern of premia for the two smallest cities in an
equilibrium, and numerical computations reported in online Appendix B show that
this pattern of premia generalizes across all cities in equilibrium for a wide range ofparameter values. This novel prediction distinguishes our model from the canonical­spatial-equilibrium model, which predicts spatially invariant skill premia.

This sufficient condition can also be written as ​(​L2​​​  − ​L​1​​)/​L​  21​ ​  > (​zmax​ ​​  − ​z​m​​)/(L × ​z​m​​  )​.
For example, we find ­non-increasing skill premia in parameter combinations in which ​γ = 5​ , but lessthan 0.5 percent of parameter combinations with ​γ = 5​for which equilibria exist exhibit ­non-increasing skill
premia.
28
The numerical findings for hundreds of thousands of parameter values are strongly suggestive. Unfortunately,
our analytical proof of the t­wo-city result for a Pareto ability distribution does not extend naturally to an arbitrary
number of cities.
26
27

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167

D. An Illustrative Example with 275 Cities
To illustrate our model’s capacity to match empirical patterns linking cities’ sizes,
wages, and prices, we report an example of an equilibrium for the special case ofequations (5) and (6) that is consistent with 3 empirical moments of interest. First,
Zipf’s law says that a city’s size rank is inversely proportional to its size (Gabaix1999). Second, there is the positive correlation between skill premia and population
size documented in Table 1. Third, Davis and ­Ortalo-Magné (2011) document that
housing expenditure shares vary little across cities.29
Our illustrative example is a ­uniform-ability equilibrium with 275 heterogeneous
cities, akin to the number of US metropolitan areas.30 The exogenous parameter val​ ax​​  = 2​.
ues are A​ = 3, ​ n̅​  = 0.4, θ = 1, γ = 0.1, L = 2,062.5, ν = 50, ​zm​ in​​  = 1, ​zm
The large values of ν​ ​and L​ ​make the equilibrium matching rate m​ ​(​Mc​)​​ ​​exceed 0.99
in every city, so ­cross-city variation in ­idea-exchange environments is much more
due to variation in average ability ​​ ​z ̅​​c​​​than scale effects. The average time devoted toidea exchange ranges from 0.41 to 0.46 and is monotonically increasing in city size.
Regressing log population rank on log size yields a coefficient of −1.025, near the
typical empirical estimate of this ­power-law exponent. Regressing log skill premium
on log size yields a coefficient of 0.092, which is greater than those reported inTable 1 but plausible. Housing expenditure shares vary from 0.32 to 0.36 and havea population elasticity reasonably close to 0, −0.023. Thus, this illustrative example
exhibits properties consistent with empirical patterns.31
While our model does not yield ­closed-form comparative statics, this illustrative example exhibits local comparative statics consistent with economic shifts inrecent decades. Work in labor economics has emphasized ­skill-biased technicalchange, which we interpret as an increase in ​A​ , as one reason for growth in the(­economy-wide) skill premium (Acemoglu and Autor 2011). Around the illustrativeequilibrium, a 1­ 0 percent increase in A
​ ​leaves the ­power-law exponent virtuallyunchanged and increases both the ­economy-wide average skill premium and the
population elasticity of skill premia by about 7 to 8 percent. Table C.2 shows thatthe population elasticity of skill premia did increase from 1990 to 2007. Thus, ourmodel’s mechanics are qualitatively consistent with and introduce a spatial dimension to the leading explanation for changes in the skill premium in recent decades.
III. Conclusion

The presence of skyscrapers is a defining characteristic of cities’ central business
districts. These attest to an intense desire to concentrate large numbers of ­people in29
In light of their finding, Davis and O­ rtalo-Magné (2011) use C­ obb-Douglas preferences. Our results showthat such preferences are not necessary to obtain housing expenditure shares that are approximately spatially invariant, as previously established by Behrens, Duranton, and ­Robert-Nicoud (2014).
30
The geographic delineations used in Census 2000 publications define 280 (consolidated) metropolitan statistical areas, including 4 in Puerto Rico.
31
While this example matches empirical regularities well, our model does not generically generate these patterns. Other parameterizations can yield equilibria in which the p­ ower-law exponent is quite far from −1. In our
numerical explorations, small changes to the parameter values cause small changes in the equilibrium distributionof outcomes and all eight parameters are quantitatively important to the joint determination of outcomes such as
the p­ ower-law exponent.

168

THE AMERICAN ECONOMIC REVIEWJANUARY 2019a tiny geography. This extreme concentration is neither to exchange goods nor tofacilitate hiring. One benefit of this concentration is that it facilitates idea exchange.
While idea exchange within firms is surely of great importance, an individual firm
need not pay the costs to be in a central business district for this benefit. Idea exchange
outside the boundaries of the firm provides a foundation for agglomeration.
It is precisely this costly, voluntary interaction that we seek to capture in ourmodel of idea exchange. In our theory, individuals allocate their time according to the
expected gains from exchanging ideas in their city. The gains from idea exchange are
greater in places where conversation partners are more numerous, devote more time
to idea exchange, and are of higher ability. The h­ ighest-ability tradables producers
reap the most from such learning opportunities. This simple setup, designed to overcome the “black box” critique that has inhibited research in this crucial area, nonetheless yields a rich set of spatial patterns. Larger cities are places with more idea
exchanges between ­higher-ability participants, and they in turn exhibit higher wages,
productivity, housing prices, and skill premia—all prominent features in the data.
This account suggests important implications for various aspects of urban policythat affect the city as a locus of idea exchange. Transportation policy determines
the frequency with which meetings may feasibly occur. Zoning shapes not only the
population density of potential participants but the venues in which idea exchanges
may arise. And our model provides an account in which larger cities’ higher nominal
wage inequality does not imply that their ­lower-income residents have lower welfare
than their counterparts in other locations.
Our static model characterizes the cross section of cities resulting from the complementarity between individual ability and i­dea-exchange opportunities. We thus
provide a microfounded account of the spatial distribution of economic activity ina world in which cities are defined by the skills and ideas of those who choose tolive in them. Future theoretical work might also capture the dynamics of knowledge
accumulation and innovation, in light of the empirical evidence in Wang (2016);
de la Roca and Puga (2017); and Carlsen, Rattsø, and Stokke (2016) that larger
cities’ benefits to ­high-ability individuals accrue over time.
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