The Economics of Space 433: Lectures 3 and 4 A Brief ......I We will discuss the Von Thunen model of...
Transcript of 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)
The Economics of Space. Lecture 3 and 4 Introduction c© Costas Arkolakis 3
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
The Economics of Space. Lecture 3 and 4 Introduction c© Costas Arkolakis 4
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
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 6
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
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 7
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 .
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 8
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
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 9
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)
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 10
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.
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 11
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
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 12
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
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 13
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
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 14
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
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 15
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
The Economics of Space. Lecture 3 and 4 The Monocentric City Model c© Costas Arkolakis 16
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 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
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 18
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
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 19
Producers
I Perfectly competitive producers with labor productivity Ai
I Given no trade costs we have price equalization
p = wi/Ai =⇒ wi = Ai
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 20
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
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 21
Welfare and Population in Our Baseline Model
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 22
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
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 23
Evidence from Glaeser Gottlieb ’08: Welfare Equalization
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 24
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 Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 25
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
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 26
Evidence from Glaeser Gottlieb ’08: Wages and Amenities
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 27
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
The Economics of Space. Lecture 3 and 4 Rosen-Roback Framework c© Costas Arkolakis 28
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 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
The Economics of Space. Lecture 3 and 4 Hotelling Location Model c© Costas Arkolakis 30
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
The Economics of Space. Lecture 3 and 4 Hotelling Location Model c© Costas Arkolakis 31
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.
The Economics of Space. Lecture 3 and 4 Hotelling Location Model c© Costas Arkolakis 32
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 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
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 34
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.
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 35
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.
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 36
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/)
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 37
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
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 38
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
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 39
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
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 40
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
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 41
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)
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 42
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
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 43
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)
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 44
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
Representative Consumer Demand Function: Levels and Logarithm (log)
log purchasesPurchases of good 1The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 46
Demand Function: Slope
Representative Consumer Demand Function: Levels and Logarithm (log)
log purchasesPurchases of good 1
slope: -σ
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 47
Demand Function: Changes with Income
Representative Consumer Demand Function: High and Low Income
Purchases of good 1
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 48
Demand Function: Prices of Other Goods
Representative Consumer Demand Function: High and Low Price of the Second Good
Purchases of good 1
, ,
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 49
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
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 50
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
The Economics of Space. Lecture 3 and 4 Armington Model c© Costas Arkolakis 53