Transcript The Spatial Dixit

Lecture 5
SPATIAL ECONOMY:
THE DIXIT-STIGLITZ MODEL
By Carlos Llano,
References for the slides:
• Fujita, Krugman and Venables: Spatial Economy. Ariel Economía, 1999.
• Brakman S., Garretsen H. van Marrewijk C. (2009): The New Introduction to Geographical Economics. Cambridge
University Press.
• The World Bank (2008): “Reshaping economic geography”. WB report.
1
Outline
1. Introduction
2. The Dixit-Stiglitz:
1. Descriptions and assumptions.
2. Demand.
3. Transport costs and multiple locations.
3. Conclusion
2
1. Introduction: Geography-Economy in 3D
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• TWB (2009): Reshaping
Economic Geography.
1. Density
2. Distance
3. Division
• BGM (2010): The New
introduction to Geographical
Economics. Lesson 1:
1. Economic Agglomeration
2. Economic Interaction.
1. Introduction: density = spatial agglomeration
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1. Introduction
5
1. Introduction. Density: I-Current situation
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• Country level: the concentration of economic activity also
occurs within countries, and increases with the “income level”.
This agglomeration is not an artifact of the spatial unit used.
Administrative
areas
By Statistical
areas
By Land
areas
GDPpc # of administ. Areas
324
21
Concentration
15
Italy
France
Sweden
19,480
22,548
31,197
21
22
22
21
29
29
5
5
5
5
5
227,540
230,800
311,888
267,990
304,280
30.2
34.6
43.9
51.6
64.7
0.48
0.48
0.52
0.55
0.64
Tajikistan
204
Mongolia
406
El Salvador 1,993
Brazil
3,597
Argentina 7,488
Ghana
211
Lao
231
Poland
3,099
New Zealand 11,552
Norway 27,301
% household
%GDP in the consumption in Spatial Gini
leading area the leading
coefficient
area
By
Country
Tanzania
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1. Introduction. Density: II-Dynamics
Country level: Spatial inequality of regions within countries rose
and remained high before slowly declining, following an invertedU relationship. WB, pp86.
II-Dynamics
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1. Introduction. Density: II-Dynamics
Country level:
• Economic
development, in its
early stages, is
accompanied by a
rapidly rising spatial
concentration in a
country.
• Leading areas benefit
most from this
compression and
growth. TWB, pp86.
2. The Dixit-Stiglitz Model
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1. The Dixit-Stiglitz model is the most used framework for describing
a monopolistic competition models (Dixit-Stiglitz, 1977).
2. Since the 70’s, its use in the field of international trade has been
fundamental. It is the starting point for the New Economic
Geography (NEG): agglomeration, economies of scale,
transportation cost.
3. Fujita, Krugman and Venables (1999) present a spatial version of
the DSM:
•
•
2 regions; 1 mobile production factor (L= labor).
2 products:
•
•
Agriculture: residual sector, perfect competitive, constant returns to scale; no
transportation costs.
Manufacturing: differentiated goods (n varieties); scale economies; monopolistic
competition; transportation costs.
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2. The Dixit-Stiglitz Model
Structure of the Dixit-Stiglitz spatial model:
1. Solution to the consumer’s problem
2. Multiple Locations and Transportation Costs
3. Producer Behavior
4. The Price Index Effect and the Home Market Effect
5. Equilibrium
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2. The Dixit-Stiglitz Model
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1. Consumer Behavior: Utility function
–
Every consumer shares the same Cobb-Douglas tastes for the two type of goods (M, A).
1μ
UM A
μ
•
• M= composite index of the manufactured goods.
• A= consumption of the agricultural good.
• Mu (μ): constant: expenditure share in manufactured goods.
M is a sub-utility function defined over a continuum of varieties of manufactures:
– m(i): consumption of each available variety (i),
– n: range of varieties.
M   m(i) ρ di
 0

n
•
1/ρ
0  ρ 1
M is defined by a constant-elasticity-of-substitution (CES):
– Rho (ρ): intensity of the preference for variety (love for variety)
– If ρ=1, differentiated goods are nearly perfect substitutes (low love for variety)
– If ρ=0, the desire to consume a greater variety of manufactured goods increases.
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2. The Dixit-Stiglitz Model
1. Consumers Behavior:
We define sigma (σ) as:
1
1
σ
;ρ  1
1 ρ
σ
σ: elasticity of substitution between any 2 varieties
The consumer’s problem: maximize utility defined by the function U
subject to the budget constraint.
We solve it in 2 steps:
1. First, the consumption of varieties will be optimized:
• The ideal consumption of each variety will be given by the
combination that ensures utility with the minimum cost (given
the relative prices of each variety).
2. Once the consumption of varieties in generic terms has been
optimized (for every M), then, the desired quantity of A and M will be
chosen according to the relative prices of both goods.
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2. The Dixit-Stiglitz Model
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1. The Consumers Behavior: the budget constraint
n
Y  p A A   p(i)m(i)di
0
•
•
•
•
PA= Price of the agricultural goods.
A= consumption of the agricultural good.
p(i)= price of each variety (i) of
manufacturing product.
m(i)= quantity of each variety (i).
To maximize the utility U subject to the budget constraint Y, there are
2 steps:
1. Whatever the value of the manufacturing composite (M), each
m(i) needs to be chosen so as to minimize the cost of attaining de
M (Phase I).
2. Afterwards, the step is to distribute the total income (Y) between
agriculture (A) and manufactures (M) in aggregate (Phase II).
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2. The Dixit-Stiglitz Model
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1. Consumer Behavior: Phase I:
1. Minimize expenditure for any given M :
n
Min  p (i )m(i )di
•
•
•
0


s.t.    m(i ) di 
0

ρ
n
1 /(  1)
 p(j) 
m(j)  

 G 
1/ρ
M
•
PA= Price of the agricultural goods.
A= consumption of the agricultural good.
p(i)= price of each variety (i) of
manufacturing product.
m(i)= quantity of each variety (i).

 p(j) 
M
 M
 G 
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2. The Dixit-Stiglitz Model
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1. Consumer behavior: Phase II:
•
Now we have to divide the total income (Y) between the two goods,
M and A. We will do it by maximizing U constrained to the optimal
expenditure derived from minimizing the cost of attaining M.


1 
Max U  M A

s.t.  GM  p A A  Y
•
•
•
•
PA= Price of the agricultural goods.
G= Manufactures’ Price Index
A= consumption of the agricultural good.
p(i)= price of each variety (i) of manufacturing
product.
• m(i)= quantity of each variety (i).
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2. The Dixit-Stiglitz Model
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1. Consumer behavior: Phase I + Phase II:
Now FKV introduce a variation of the DS Model:
• They make that the range of manufactures on offer becomes an
endogenous variable.
•
•
If ↑n → ↓G (manufactures’ price index), because consumers value
variety.
Therefore ↓ Cost of attaining a given level of utility.
• To prove it, we assume that all manufactures are available at the same
price, pM . Then, the price index G becomes:
1/ 1- 
G   p(i) 1- di
 0

n
 p M n1/ 1- 
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2. The Dixit-Stiglitz Model
1. Consumer behavior: Phase I + Phase II:
•
The relationship between G and n depends on the elasticity of
substitution between varieties σ
•
•
1/ 1- 
G p n
M
The lower is σ (the more differentiated are varieties) → the greater is the
reduction in G caused by an increase in the number of varieties.
Changing the range of products available also shifts demand curves for
existing varieties.
•
To prove it, we look at the demand curve for a single variety:
p(j) 
m(j)  Y ( 1)
G
• When Δn → ↓G , the demand m(j) shifts downward,
• Important: it allows us to know the equilibrium n:
• If Δn → Δ competition → shifts downward the
existing products m(j) and reduces the sales of those
varieties (evolution to more firms with profit=0)
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2. The Dixit-Stiglitz Model
2. Multiple locations and transportation cost: iceberg costs
•
We consider the existence of R possible discrete locations.
•
Each variety is produced in only one location and all varieties produced
in a particular location are symmetric in technology and price.
•
•
nr= number of varieties in location r.
•
pmr= FOB price of manufacturing in location r.
Agricultural and Manufactured products can be shipped between
locations incurring in transport costs:
•
Iceberg costs: if a unit of a good is shipped from a location r to another location
s, only a fraction 1 / TrsA of the original unit actually arrives. The rest is “lost”
(melted) in the trip (transport cost).
•
The constant Trs represents the amount of the agricultural good dispatched per
unit received in s.
A
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2. The Dixit-Stiglitz Model
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2. Multiple locations and transportation cost: CIF prices
•
If pmr is the FOB price of the manufacturing product in location r, and there
are iceberg transport costs, the CIF price when delivered to location s is
given by:
prsM  prM TrsM
•
Then, the manufacturing price index (Gs) may take a different value in each
location according to the location s where it is consumed:

M
G s   n r p M
T
r
rs
r

1

R


1σ



1
1σ
, s  1,..., R
M σ
σ 1
μYs (p M
T
)
G
r
rs
s
Price index in s of
manufactures produced in r
Consumption demand in location s for a
product produced in r
• Ys= income for location s: this gives the consumption of the variety in s.
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2. The Dixit-Stiglitz Model
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2. Multiple locations and transportation cost: CIF prices
•
As a consequence, summing across locations in which the product is
sold, the total sales of a single location r variety is:
R
q
M
r
M σ
σ 1 M
 μ  YS (p M
T
)
G
r
rs
S Trs
s 1
I have to produce Tmrs in r, knowing that a
portion 1/ Tmrs is lost during the trip
(transportation cost)
Important consequences:
• Sales depend on: income and the price index in each location, on the
transportation costs and the mill price.
• Because the delivered prices of the same variety at all consumption locations
change proportionally to the mill price, and because each consumer’s demand
for a variety has a constant price elasticity sigma (σ), the elasticity of the
aggregate demand for each variety with respect to its mill price is also
sigma (σ), regardless of the spatial distribution of consumers.
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2. The Dixit-Stiglitz Model
3. Producer Behavior:
•
The agricultural goods is produced with constant returns;
•
Manufacturing involves economies of scale at the level of the variety (internal).
• Technology is the same for all varieties and in all locations:
• The only input is labor L, the production of a quantity qM of any variety at any
given location requires labor input lM , given by:
l M  F  cM qM
•
With increasing returns to scale, consumer’s preference for variety, and the
unlimited number of potential varieties of manufactured goods, no firm will
choose to produce the same variety supplied by another firm,
•
Each variety is produced in only one location by a single specialized firm,
•
The number of manufacturing firms is the same as the number of
available varieties.
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2. The Dixit-Stiglitz Model
3. Producer Behavior: Profit maximization
•
Firms maximize profits with a given income (sales) and with
given costs (according to the wages)
π r  p Mrq Mr  w Mr (F  cMq Mr )
Revenues (sales)
Costs: F+V (given the wages wr)
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2. The Dixit-Stiglitz Model
3. Producer Behavior: wages
•
•
•
R
If firms in “r” make profit =0, they produce q*.
With this q* they satisfy the demand of products
from “r”:
We can turn this equation around and say that
active firms break even if and only if the price
they charge satisfies:
s 1
p 
M σ
r
Using the price rule (*) we get:
 σ  1  μ
M 1σ
σ 1 
wM

Y
(T
)
G
 M 
 S rs
r
s 
 σc   q * s1

R
 
q*  μ  YS p Mr
σ
(TrsM )1σ G sσ 1
μ R

YS (TrsM )1σ G sσ 1

q * s 1
(*) p M ( 1  1/σ )  c M wM
r
r
1/σ
This is the wage equation: it gives the manufacturing wage at which firms in each location
break even, given the income levels and price indices in all locations and the costs of
shipping into these locations:
•
The wage increases with the income (Ys) at location s, the access to location s from location r
(Tmrs), and the less competition the firm faces in location s (G decreases with n)
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2. The Dixit-Stiglitz Model
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3. Producer Behavior: wages
• Real wages: real income at each location is proportional to nominal
income deflated by the cost-of-living index,
G μr (p Ar ) (1μ)
• This means that the real wage of manufacturing workers in location r,
denoted by ωrM is
  w G (p )
M
r
M
r
-μ
r
A  (1 μ)
r
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2. The Dixit-Stiglitz Model
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•
5. The price index effect and the Home Market Effect
•
We consider an economy with 2 regions, that produce 2 manufacturing varieties:


1
G  L1w1  L 2 (w 2T ) (1σ)
μ
1
G1-2   L1(w1 T )(1σ)  L2w2 
μ
1-
1
w1  Y1G1σ 1  Y2G σ2 1T1σ
w 2  Y1G1σ 1T1σ  Y2G σ2 1
• These pairs of equations are symmetric, and so its’ solutions.
• So, if L1=L2; Y1=Y2, then there is a solution with G1=G2 and with w1=w2.
• We can explore the relationships contained in the price indices and wage equations by
linearizing them around the symmetric equilibrium:
• An increase in a variable in R1 is associated with a decrease in R2 but of equal
absolute magnitude.
• So letting dG=dG1=-dG2, and so on, we derive, by differentiating the price indices
and wage equations respectively, and we get:
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2. The Dixit-Stiglitz Model
5. The price index effect and the Home Market Effect
1)
2)
σ 1
Direct effect of the
dG L  G 
dw 
 dL
variation in the
(1  σ)
   (1  T 1σ ) 
 (1  σ)
G
μw
w 
L
location of firms on
the Price Index (G).
How relative
demand affects the
location of M.
σ 1
dw Y  G 
dG 
1σ  dY
σ    (1  T )   σ  1 
w ww
G
Y
• [Eq 1]: Price Index Effect: We suppose that the supply of labor is
perfectly elastic, so that dw=0. Bearing in mind that 1-σ <0 and that
T>1, the equation implies that a change dL/L in manufacturing
employment has a negative effect on the price index, dG/G.
• Conclusion: the location with a larger manufacturing sector also has a lower
price index for manufactured goods, simply because a smaller proportion of this
region’s manufacturing consumption bears transport costs.
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2. The Dixit-Stiglitz Model
5. The price index effect and the Home Market Effect
• Now , let us consider how relative demand affects the location of manufacturing.
It is convenient to define a new variable, Z,
1  T 1σ
Z
1  T 1σ
Z is an index of trade cost, with value between 0-1:
• Z=0, if trade is costless (T=1);
• Z=1, if trade is impossible (T=0).
dL dY
σ
 dw
• Using the definition of Z and eliminating dG/G, we have   Z1  σ 
Z

L
Y
Z
 w
• If dw=0, supply of labor is perf. elastic: Home market effect: A 1% change in
demand for manufactures (dY/Y) causes a 1/Z % (>1) change in the employment, and
the production of manufactures (dL/L).
• The location with the larger home market has a more than proportional larger
manufacturing sector (industrial agglomeration) and therefore also tends to export
manufactured goods.
• If dw>0, positive supply of labor : part of the home market advantages goes to higher
wages instead of exports
• Locations with a larger home market (demand) tends to offer a higher nominal
wage (qualified labor agglomeration).
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2. The Dixit-Stiglitz Model
6. The “No-Black-Hole” Condition
•
We are not interested in economies in which increasing returns are so strong
that the forces working toward agglomeration always prevail, and the economy
tends to collapse into a point. (Everyone to NY).
•
To avoid this “black-hole location” effect, we usually impose what we call the
“no black hole condition”:
σ 1
ρμ
σ
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In search of the Home Market Effect (HME)
 In classic Trade theory models, where industries are
suppose to have CRS, a larger demand of a product in
a country will imply being a net importer of such
product.
 In NTT, due to the Home Market Effect (HME):

Industries with Increasing Returns to Scale (IRS), and
larger demand in a country/region will imply more than
proportional production, and to be a net-exporter of that
product.

David & Wenstein test empirically the existence of HME:

DW 1996, 1997, 2003: 22 OECD countries; 26 industries.
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In search of the Home Market Effect (HME)
Journal of International Economics 59 (2003) 1–23
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In search of the Home Market Effect (HME)
n= # industries
g= # goods
c= # countries
ROW= rest of the world
Xngc= output of product g in industry
n in country c.
Dngc= demand of product g in
industry n in country c.
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In search of the Home Market Effect (HME)
Xngc= output of product g in industry n in country c.
Ω= technology matrix
V= factor endowments of country c
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In search of the Home Market Effect (HME)
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The spatial wage structure and real market potential
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1.
In neoclassical trade theory, there is no foundation for spatial
wage structure.
– In fact, H-O predicts a complete factor price equalization due to international
trade.
2.
3.
In New Trade Theory –without transport costs-, although in
autarky the larger market will pay higher wages, by trade, factor
price equalization is also predicted.
The basic NEG model (Krugman, 1991) predicts a spatial wage
structure in favor of the “core”.
– In other versions of the CP model, different stages are considered, with
increasing-stable-decreasing differences in terms of “real wages” along with
the reduction of transport costs.
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The spatial wage structure and real market potential
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Evidence for México (Hanson, 1997):
He analyzed the evolution of wage structure before and after the shift in
trade policy: from protectionism to NAFTA, testing 2 hypothesis:
• H1: the region´s wage relative to Mexico city are lower when transport
cost (distance) are higher.
• H2: Trade liberalization led to a compression of regional wage
differentials.
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– He found empirical evidence for Hyp1, but not for Hyp2.
Why? Trade liberalization shifted the weight center of the
economy north-wards, reducing the centrality of México-City.
Figure 5.5 Map of Mexico
Mexicali
Ciudad Juarez
Chihuahua
Torreon Monterrey
Mexico
Tampico
Leon
Guadalajara
Mexico City
Merida
Puebla
Veracruz
Acapulco
Source: BGM-2010:
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