the economics of space 433: lectures 3 and 4 a brief ......i we will discuss the von thunen model of...

The Economics of Space 433: Lectures 3 and 4

A Brief History of Space in Economics

Costas Arkolakis1

1Yale University

26 February 2020

The Economics of Space. Lecture 3 and 4 c© Costas Arkolakis 1

In Economics, Space has History

I Spatial economics has been one of the oldest and most celebrated fields ineconomicsI Models as old as the work of Ricardo 1817, Von Thunen 1826, Hotelling 1929 has

fascinated scholars of economics since the 19th century

I We will illustrate some of the key ideas in these theories to understand the role ofspace in traditional modelsI We will discuss the Von Thunen model of the Central Business District, the

Hotelling model of location choice.I We will also discuss the workhorse spatial setup of Rosen Roback ’82I Instead of the traditional Ricardian theory of comparative advantage we will go

over the Armington theory of product differentiation (Anderson ’79)

The Economics of Space. Lecture 3 and 4 Introduction c© Costas Arkolakis 2

In Economics, Space has History

I Spatial economics has been one of the oldest and most celebrated fields ineconomicsI Models as old as the work of Ricardo 1817, Von Thunen 1826, Hotelling 1929 has

fascinated scholars of economics since the 19th century

I We will illustrate some of the key ideas in these theories to understand the role ofspace in traditional modelsI We will discuss the Von Thunen model of the Central Business District, the

Hotelling model of location choice.I We will also discuss the workhorse spatial setup of Rosen Roback ’82I Instead of the traditional Ricardian theory of comparative advantage we will go

over the Armington theory of product differentiation (Anderson ’79)


Main Elements of a Spatial Model

I We discussed the main elements of a spatial modelI Topography: Productivities (Ai ), Amenities (ui ), and Spatial Links (τij)I Geography: economic fundamentals

(Ai , ui

)that determines these overall

productivities/amenities, and transportation network, tijI We will return to a more detailed discussion of those fundamentals later on in the

course

I We will now give a brief history of how the key ideas in spatial economics wereformed


Roadmap

I The Monocentric City Model

I The Rosen-Roback Spatial Framework

I Hotelling location model

I The Armington model

The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 5

The Monocentric City Model

I The remarkable first approach to spatial analysis was pioneered by Von Thunen1826I The model was designed to capture the specialization of production in citiesI A farming sector that locates in the suburbs produces to sell to the downtownI The downtown produces manufacturing that is shipped to the suburbs

I How further out does the city extend?I If transportation is costly, with a homogeneous product, there is a finite threshold

that defines the border of the cityI We will set this up in the line as the original model


Setup

I Sectors: manufacturing M and agriculture A

I Space:I j = c is the city center. f is the agricultural frontier location, c, f ∈ SI τj,s is the iceberg trade cost from city center c to location j and from j to c of

good s = M,AI We assume that τj,s is a continuous and monotonically increasing in the distance

from c


Production (Manufacturing)

I M producer:I Only locates in city center c and produces a homogeneous output.I Inputs one unit of labor and outputs one unit. Thus wc,M = pc,M .


Production in Agriculture

I A producer (landlord):I Produces a homogeneous agricultural output.I Inputs 1/Ai,A units of labor and one unit of land and outputs one unit.I There is only trade between c and each agriculture location. No trade across

agriculture locations

I The resulting price is

pi,A =pc,Aτi,A


Production in Agriculture

I Landlords cannot move

I Landlord’s problem at i to maximize rents

ri,A = max

{0, pi,A −

wi,A

Ai,A

}= max

{0,

pc,Aτi,A− wi,A

Ai,A

}.

I Agriculture takes place at i ∈ S where S ≡ {i |ri,A ≥ 0}.

I At the endogenous agricultural frontier f , it must be that rf ,A = 0 =⇒

pc,Aτf ,A

=wf ,A

Af ,A

. (1)


Consumer Demand

I Consumers are homogeneous workersI Free mobility across locations

I Consume M and A to maximize a Cobb-Douglas aggregate of the twoI µ ∈ (0, 1) share of manufacturing in consumption

I Workers choose the location j and sector s.I There are no amenity differences so ui = 1.I Welfare of the worker at location j and sector s is Wj,s =

wj,s

(pj,M)µ(pj,A)1−µ.


Equilibrium

I Set the numeraire pc,M = 1. Thus, wc,M = 1I We are looking for the price of the good at the center pc,A and the frontier f

I The equilibrium conditions boil down to free mobility and agriculture marketclearing for regions j :

Wc,M = Wj,A, (2)

(1− µ)Yc =∑i∈S

Xic,A. (3)

where Xic,A is the imports of agriculture goods at the city center from region iand Yc is total income in the city center (from manufacturing sales)

I We will now use (2) along with (1) to solve for the location of the frontier


Equilibrium Characterization: Welfare Equalization

I In particular, the free mobility condition between the manufacturing workers inthe city center and the agriculture workers in the frontier agriculture region fimplies

Wc,M =wc,M

(pc,A)1−µ=

wf ,A

(τf ,M)µ (pc,A/τf ,A)1−µ= Wf ,A

I We can now derive the prices.I Combine this equation with (1), which gives

pc,Aτf ,A

=wf ,A

Af ,A, and the normalization

which gives wc,M = 1 we obtain

pc,A =(τf ,M)µ (τf ,A)µ

Af ,A

. (4)

I Recall that the price at any other point is pi,A =pc,Aτi,A


Rents in Space

0.1 0.2 Agricultural Frontier 0.5

Distance From City Center

Note: The graph shows agricultural goods price and agricultural wage.The area between the lines show the rent. = 0.5,

M = 2, and

A=1.5.

0

0.2

0.4

0.6

0.8

1

Pric

e of

Agr

icul

tura

l Goo

ds

Rent

Agricultural goods priceAgricultural wage


Linear City

0.1 Agricultural Frontier 0.3 0.4 0.5

Distance From City Center

Note: The graph shows free mobility condition and market clearing condition.With = 0.5,

M = 2, and

A=1.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5P

rice

of A

gric

ultu

ral G

oods

Free Mobility ConditionMarket Clearing Condition


Modern Reformulations with Commuting

I The model has been recently reformulated by Alonso ’64 and Muth ’69 as acommuting modelI Cost of commuting to travel downtown in terms of a time endowmentI Analysis of the modern city

I A recent quantitative application of this reformulation was developed by Ahlfeldt,Redding, Sturm, Wolf ’15I To analyze the actual commuting patterns and the city economic activity with a

wealth of data from Berlin


Roadmap





The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 17

Baseline Urban Model

I We now study the Rosen-Roback ’82 model.I In this model a single homogenous agricultural good freely tradedI This means that prices are the same everywhere

I A mere trick to price labor: there is no trade in equilibriumI We will relax this assumption down the road

I Denote by j or i the location of the worker and the firm


Workers

I Worker utilityWj = Cjuj

where Cj is the consumption in location j and uj the amenity

I Agent has a budget constraintpjCj = wj

pj is the price of agricultural good, Cj the consumption and wj the wage in jI Good freely traded so that pj = p. We can normalize one price, pj = p = 1I Thus, Cj = wj


Producers

I Perfectly competitive producers with labor productivity Ai

I Given no trade costs we have price equalization

p = wi/Ai =⇒ wi = Ai


Productivities and Amenities

I Let us now consider the case that

Ai = AiLαi

ui = uiL−βi

I Thus, taking all into account

Wj = wjuj = Ajuj = Aj ujLα−βj

and assume β > α so that the model is ‘well-behaved’ and welfare decreases withpopulationI If α > β we end up with a ’black hole’!I Thus, β < α is also sometimes referred to as a no-’black hole’ condition


Welfare and Population in Our Baseline Model


Labor Supply and Welfare

I We assume that workers are freely mobile and thus in equilibrium Wj = W

I Now notice that we have,

Wj = Ajuj = Aj ujLα−βj =⇒ Lj = (

Wj

Aj uj)

1α−β (5)

I Since labor is mobile we will have Wj = W ∀ j . Thus we have,

L =∑j

Lj = W 1/(α−β)∑j

(1

Aj uj)

1α−β

I Dividing the two expressions with Wj = W we obtain the population in terms offundamentals

Lj =(Aj uj)

1β−α∑

j′(Aj′ uj′)1

β−αL


Evidence from Glaeser Gottlieb ’08: Welfare Equalization


Baseline Urban Model: Welfare

I Furthermore we can write welfare as

W =

[∑j(Aj uj)

1β−α

]β−αLβ−a

(6)

I You can combine the expressions for welfare and labor to finally obtain

Lj = (Aj uj)1

β−αW1

α−β (7)

I Taking logs of equation (7) gives,

log Lj =1

β − αlog Aj +

1

β − αlog uj +

1

α− βlogW


The Rosen-Roback ’82 model: Predictions

I More workers in i if:I More productive ⇐⇒ Ai higherI Better amenities ⇐⇒ ui higher

I Higher wages in i if:I More productive ⇐⇒ Ai higherI Worse amenities ⇐⇒ ui lower


Evidence from Glaeser Gottlieb ’08: Wages and Amenities


Main Insights and Critiques

I The main insight of the Rosen-Roback framework is to put the agglomeration anddispersion forces in the forefront of the analysisI Due to supportive evidence of population sorting on productivities and amenities it

has been an extremely practical setup

I The main limitation that arises in that model is that space, at the end of the day,is modeled in a very limited wayI No cost of moving goods, no frictions of moving people or firmsI The next two models, developed independently, discussed the location of firms and

the costs of moving goodsI We will use insights from both setups when we develop the modern spatial

approach in future lectures


Roadmap





The Economics of Space. Lecture 3 and 4 Hotelling Location Model c© Costas Arkolakis 29

Hotelling Model: Linear City

I The Hotelling model studies the endogenous firm locationI Firms locate taking into account transportation cost to consumers

I ConsumersI Distributed uniformly on a line [0, 1] (the linear city)I We assume that consumers have a unitary demand and a linear transportation cost,

t

I FirmsI Firms sell a homogeneous good for freeI So consumers only pay the the transportation cost of getting to a firm


Hotelling Model: Planner Solution

I One way to solve the model is minimize the total transportation cost of thesystem with respect to the location of the firms on the lineI This is the ‘‘social planners’’ solutionI We will study the insights arising from this solution using a 2-firm example

I Intuition: why shouldn’t we locate firms in the middle or at the edge of the line?I Locating in the middle gives a firm access to the most consumersI But this isn’t Pareto optimal because we can lower the cost for consumers at the

extremes by spreading out firm locationsI That is, it is possible to choose locations of the firm in a way that makes everyone

better or at least as good as they were before


Hotelling Model: Linear City Solution

I The cost curve in the diagram gives the cost a consumer at x ∈ [0, 1] pays

I Summing over the costs yields the total transportation costI The social planner will minimize cost by choosing firm locations to minimize the

volume of the triangles

I Example with 2 firms in the figure located at a, b ∈ [0, 1]I x denotes the indifferent consumer.

I Those to the left will buy from a, while those to the right from b

I For example, T3, is total cost for consumers in (x , b)I Solution is a = 1

4, b = 3

4with total cost, t

8.


Roadmap





The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 33

Setup

I So far we have assumed that products are homogeneous

I This assumption typically implies absence of two-way trade or the absence oftrade altogetherI We observe a lot of two way trade in the data, most of it between nearby regions

I The Armington model answers the question why people across different locationstrade


Patterns of Trade Flows and Distance (World)

-10

-50

5Lo

g Tr

ade

flow

s

-3 -2 -1 0 1 2Log Distance

1950 2000

Source: Allen Arkolakis (2018). This figure plots the estimated coefficient of distancein a trade gravity regression with origin-year and destination-year fixed effects overtime using trade flows between locations.


Patterns of Trade Flows and Distance (Within USA)

-50

5Lo

g tra

de fl

ows

-3 -2 -1 0 1 2Log distance

2007 2012

Source: Allen Arkolakis (2018). The figure excludes trade flows within each state.The thick lines are from a non parametric regression with Epanechnikov kernel andbandwidth of 0.5 after partitioning out the origin-year and destination-year fixed effects.


Setup

I 2 locations denoted by i . Set of locations S = {1, 2}I Each location produces a differentiated commodityI E.g. Region-specific varieties of France (see http://winefolly.com/review/french-wine-exploration-map/)


Representative Consumer

I Consumer chooses location j and makes consumption choices

I We assume that the welfare of consumer in location j equals consumption timesamenity

Wj = Cj × uj

where Cj is per-capita real consumption in location j and uj is overall amenity


Representative Consumer: Consumption

I Next we characterize the representative consumers’ consumption choices

I Preferences: Constant Elasticity of Substitution (CES) utility function over twogoods, home and foreign

Cj(c1j , c2j) = ((c1j)(σ−1)/σ + (c2j)

(σ−1)/σ)σ/(σ−1)

I cj : consumption of location j = 1, 2I σ > 0: elasticity of substitution. For practical purposes we will henceforth assume

that σ > 1


Representative Consumer: Consumption

I Preferences: Constant Elasticity of Substitution (CES) utility function over twogoods, home and foreign

Cj(c1j , c2j) = ((c1j)(σ−1)/σ + (c2j)

(σ−1)/σ)σ/(σ−1)

I Budget constraint of consumer: p1jc1j + p2jc2j = wj

I pij : price of good from location i shipped to jI wj :wages in j


Consumer Optimization: Consumption

I For each location j consumer maximizes for each j

maxc1i ,c2i

((c1j)(σ−1)/σ + (c2j)

(σ−1)/σ)σ/(σ−1)

subject to p1jc1j + p2jc2j = wj

I It can be written as the Lagrangian optimization

L = maxc1i ,c2i

((c1j)(σ−1)/σ + (c2j)

(σ−1)/σ)σ/(σ−1)+

µ (wjLj − p1jc1j − p2jc2j)


Optimal Consumption

I Consumer optimization implies

(c1j)σ−1σ −1

(c2j)σ−1σ −1

=p1jp2j

=⇒ (c1j)− 1σ

(c2j)−1σ

=p1jp2j

=⇒

c1jc2j

=

(p1jp2j

)−σ(8)

I Using these derivations we can now define the elasticity of substitution

∂ ln (c1/c2)

∂ ln (p1/p2)= −σ


Aggregation: Trade Shares

I An essential derivation is that of the share of spending in each location’s good

I We define the share of location i on goods j = 1, 2,

λ1j =p1jc1j

p1jc1j + p2jc2j

and similar for λ2j

I Recall that from equation (8) we have, c1j/c2j = (p1j/p2j)−σ

, so that

λ1j =p1j (p1j/p2j)

−σ c2j

p1j (p1j/p2j)−σ c2j + p2jc2j

=p1j (p1j/p2j)

−σ

p1j (p1j/p2j)−σ + p2j

=p1−σ1j

p1−σ1j + p1−σ2j

(9)

I Define the Constant Elasticity Price (CES) index

P1−σj ≡ p1−σ

1j + p1−σ2j (10)


Demand Function

Representative Consumer Demand Function: Levels and Logarithm (log)

Purchases of good 1The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 45

Demand Function: Logarithms


log purchasesPurchases of good 1The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 46

Demand Function: Slope


log purchasesPurchases of good 1

slope: -σ


Demand Function: Changes with Income

Representative Consumer Demand Function: High and Low Income

Purchases of good 1


Demand Function: Prices of Other Goods

Representative Consumer Demand Function: High and Low Price of the Second Good

Purchases of good 1

, ,


Representative Firm

I We assume firm maximizes profits in a perfectly competitive environmentI The production function is given by

yij =Ai

τijlij

where yij is output of location i for j consumers, τij ≥ 1 the shipping cost from i toj , Ai > 0 the productivity, and lij ≥ 0 the corresponding labor used

I We assume (naturally) that local shipping costs are zero, τii = 1


Firm Optimization

I Because there is perfect competition the price equals to the marginal cost

pij = wiτij/Ai

I Prices increase with shipping costs, origin labor cost; decrease with originproductivity

I What are the relative prices for a given source?

pijpii

=τijτii

= τij

I We implicitly used this relationship (that preconditions arbitrage) in the previouslectures to derive shipping costs

I What are the relative prices for a given destination?

pi′jpij

=wi′τi′j/Ai′

wiτij/Ai=

Ai

wi× wi′

Ai′× τi′j

τij

I Now let us combine consumer and firm choicesThe Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 51

Trade Shares in Terms of Wages

I Let us look at location 1 ’s exports now.

I Use firm optimization, p1j = τ1jw1/A1 and p2j = τ2jw2/A2. Substitute in solutionfor consumer shares of each country’s product, equation (9):

λ1j =p1−σ1j

p1−σ1j + p1−σ2j

=(τ1jw1/A1)1−σ

(τ1jw1/A1)1−σ + (τ2jw2/A2)1−σ(11)

I Trade decreases with bilateral trade costs, τ1jI Elasticity of substitution, σ, plays key role to response of sales to costs τij , wi

I To see this you can think of the relative shares

λ1j

λ2j=

(τ1jw1/A1

)1−σ(τ2jw2/A2

)1−σ =⇒

ln

(λ1j

λ2j

)= (1− σ) ln

(τ1j

τ2j

)+ (1− σ) ln

(w1

A1

)− (1− σ) ln

(w2

A2

)The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 52

References

I Spatial Economics Primer, Allen and Arkolakis 2018. Mimeo.

I Trade and the Topography of the Spatial Economy, Allen and Arkolakis 2014. QuarterlyJournal of Economics, 2014, 129(3).

I Universal Gravity, Allen Arkolakis Takahashi 2017. Mimeo.

I The Welfare Effects of Transportation Infrastructure Improvement, Allen and Arkolakis 2017.Mimeo.

I Alonso, W., 1964. Location and Land Use. Harvard University Press, Cambridge

I Muth, R.F., 1969. Cities and Housing. University of Chicago Press, Chicago.

I Gabriel M. Ahlfeldt & Stephen J. Redding & Daniel M. Sturm & Nikolaus Wolf, 2015. ”TheEconomics of Density: Evidence From the Berlin Wall,” Econometrica, Econometric Society, vol.83, pages 2127-2189, November.

I Adao, Arkolakis, Esposito, Spatial Linkages, Global Shocks, and Local Labor Markets, 2018,Mimeo


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