Transcript The Spatial Dixit

Lecture 5
SPATIAL ECONOMY:
THE DIXIT-STIGLITZ MODEL
By Carlos Llano,
References for the slides:
• Fujita, Krugman and Venables: Economía Espacial. Ariel Economía, 2000.
Outline
1. Introduction
2. The Dixit-Stiglitz model of monopolistic competition:
spatial implications.
3. Applications.
4. Conclusion
Tema 5 -EE
1. Introduction
Figure 1.1 Cartogram of GNP
Areas are NOT proportional to population
Gobalización, comercio internacional and
economía geográfica
3
Tema 5 -EE
1. Introduction
Figure 1.2 GDP per capita
Highland variable among countries
Gobalización, comercio internacional and
economía geográfica
4
Tema 5 -EE
1. Introduction
1. The external and internal economies of scale can act as an
engine for international trade in addition to the existence of CA or
differences in factor endowments.
2. The internal ES require to develop an imperfect competition
model. The perfect competition model is the most used since the
70’s. (Dixit-Stiglitz, 1977).
3. With the ES the distinction between inter-industrial and intraindustrial trade arises.
4. The advantages of the external economies are less clear, and
give rise to several arguments that are commonland for
protectionism in international trade.
5
1. Introduction
AC
p
AC = average cost in
the firm
E
p2 = CM2
= n CF/ S + c
PP = price in the industry
price = c + 1 / nb
n
N2 = n companies in equilibrium (with Profit=0)
3. Number of companies in equilibrium:
•
PP Curve: the more + firms in the industry + competition and - price.
•
CC Curve: the more + firms in the industry + average cost in each firm.
•
E: long run equilibrium in the industry(n2 firms producing with CM2)
1. Introduction
CLOSED COUNTRY
P
C (S1) = n CF / S1 + c
c
p1
LARGER MARKET
BECAUSE OF TRADE
1
C (S2) = n CF / S2 + c
2
p2
P = c + 1 / bn
n1
n2
N: countries / variety
Tema 5 -EE
2. The Dixit-Stiglitz Model
1. The Dixit-Stiglitz model is the starting point for the of the
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.
8
Tema 5 -EE
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
9
2. The Dixit-Stiglitz Model
Tema 5 -EE
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 manufactured
goods:
μ A
RMSMA 
– m(i): consumption of each available variety (i),
1 μ M
– n: range of varieties.
1
n

M   m(i)ρ di 
 0

•
1/ρ
0  ρ 1
 U  1μ
A μ 
M 
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)
10
– If ρ=0, the desire to consume a greater variety of manufactured goods increases.
Tema 5 -EE
2. The Dixit-Stiglitz Model
1. Consumers Behavior:
We define sigma (σ) as:
σ
1
1
;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:
1. 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), the desired quantity of A and M will be
11
chosen according to the relative prices of both goods.
2. The Dixit-Stiglitz Model
Tema 5 -EE
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).
12
2. The Dixit-Stiglitz Model
Tema 5 -EE
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

n
ρ
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).
To minimize:
1. The first-order condition establishes the equality of marginal rates
of substitution MRS to price ratios:
m(i) 1 p(i)

 1
m(j)
p(j)
13
Tema 5 -EE
2. The Dixit-Stiglitz Model
m(i) 1 p(i)

 1
m(j)
p(j)
1. Consumer behavior: Phase I:
 p(j) 

m(i)  m(j)
 p(i) 
•
For any pair i, j that leads to:
•
Substituting this into the original constraint: 
•
1
1ρ
 
And bringing the common term m(j)p(j)
We have that: m(j) 
p(j)1/ρ1
ρ
 n

ρ 1
0 p(i) di 


1/ρ
1
1- 
M
n
0
 m(j)pji
σ
m(i)ρ di 

1/ρ
M
outside the integral
• m(j)= this is the
compensated demand
function (Hicks demand:
compensation for the price
variation; constant utility in all
the curve;) for the jth variety
14
of manufactures
Tema 5 -EE
2. The Dixit-Stiglitz Model
1. Consumer behavior: Phase I:
• We can also derive an expression for the minimum cost of attaining
M:
• Since the expenditure on the jth variety is p(j)m(j), if we use the
previous equation and integrating over all j we get:

n
0
 n

p(j)m(j)dj  p(i) di 
 0

ρ
ρ 1
•
•
ρ -1/ρ
M
Now we want to express this term as the
manufactures price index (G)
So G*M=total expenditure in
manufactures
15
Tema 5 -EE
2. The Dixit-Stiglitz Model
1. Consumer behavior: Phase I:
• The price index G measures the minimum cost of purchasing a unit
of the composite index M of manufacturing goods,
• If M is thought as a utility function, G would be the expenditure
function.
 n

G   p(i) di 
0


ρ
ρ 1
•
(ρ -1)/ρ
  p(i)1 di 
 0

n
1/(1 )
[4.7]
Now we can write the demand for m(i) more compactly:
1/(  1)
 p(j) 
m(j)  

 G 

 p(j) 
M
 M
 G 
• We substitute G [4.7] in equation
[4.5]:
m(j) 
p(j)1/ρ1
 n

0 p(i) di 
16


ρ
ρ 1
1/ρ
M
Tema 5 -EE
2. The Dixit-Stiglitz Model
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 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).
This maximization gives: (MRS=price ratio)
Mμ
Y
G
A  (1  μ)
Y
pA
17
Tema 5 -EE
2. The Dixit-Stiglitz Model
1. Consumer behavior: Phase I + Phase II:
•
•
•
Pulling the stages together, we obtain the following uncompensated
consumer demand functions :
For agriculture:
For manufactured
products:
•
Y
A  (1 μ) A
p
p(j)
m(j)  Y ( 1)
G
For
j [0,n]
If G=constant, the price elasticity of demand for every available variety is
constant and equal to (σ).
18
Tema 5 -EE
2. The Dixit-Stiglitz Model
1. Consumer behavior: Phase I + Phase II:
•
We can now express maximize utility as a function of income, the
price of agricultural output, and the manufactures’ price index, giving
the indirect utility function:
U  μ μ (1 μ)1μ YGμ (pA ) (1μ)
μ
A (1μ)
Cost-of-living index in the economy G (p )
19
Tema 5 -EE
2. The Dixit-Stiglitz Model
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.
• Therefore it is important to understand the effects on the consumer of
changes in n the number of varieties.
•
•
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:
n

G   p(i)1- di 
 0

1/ 1- 
•
 p M n1/ 1- 
The relationship between G
and n depends on the elasticity
of substitution between
varieties σ
20
2. The Dixit-Stiglitz Mode
Tema 5 -EE
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 reduce the sales of those
varieties (evolution to more firms with profit=0)
21
Tema 5 -EE
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 way as a transportation cost.
•
A
The constant Trs represents the amount of the agricultural good dispatched per
unit received in s.
22
2. The Dixit-Stiglitz Model
Tema 5 -EE
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:
p p T
M
rs
•
M M
r
rs
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 (pM
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.
23
2. The Dixit-Stiglitz Model
Tema 5 -EE
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 (pM
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.
24
2. The Dixit-Stiglitz Model
Tema 5 -EE
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.
25
Tema 5 -EE
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)
πr  pMrq Mr  w Mr (F  cMq Mr )
Revenues (sales)
•
Costs: F+V (given the wages wr)
Each firm accept the price index G as given. Thus, the
perceived elasticity of demand is therefore σ, and the profit
maximization (Img = CMg) implies that:
prM ( 1 1/σ )  c M wrM
M M
CMg
c
wr
prM 

elast  1
ρ
elasticity
26
Tema 5 -EE
2. The Dixit-Stiglitz Model
3. Producer Behavior: Profit maximization
•
If there is entry and exit in the industry, the profits of a firm at
location r are:
M M


q
M
r c
πr  w r 
 F
 σ 1

•
Therefore, the zero-profit
condition, implies that the
equilibrium output is:
q*  Fσ - 1/cM
l*  F  cMq*  Fσ
•
Both q* and l* are constants common to every active firm in the economy.
•
Thus, if LrM is the number of manufacturing workers at location r, and nr is the
number of manufacturing firms (=number of varieties) at r, then:
LMr LMr
nr 

l * Fσ
27
Tema 5 -EE
2. The Dixit-Stiglitz Model
3. Producer Behavior: Profit maximization
Conclusions:
• Odd results: the size of the market affects neither the markup of
price over marginal costs nor the scale at which individual goods are
produced. All scale effects work through changes in the variety of
goods available.
• Caveat: this is a strange result:, since normally the larger the
markets, + competition (- mark up), and + larger production in
scale.
• The Dixit-Stiglitz model says that all market-size effects work through
changes in variety.
28
Tema 5 -EE
2. The Dixit-Stiglitz Model
3. Producer Behavior: wages
R
• Nominal wages in the industry:
•
•
•
The production q* is the demand;
We can turn this equation around and say that
active firms break even if and only if the price
they charge satisfies:
 
q*  μ  YS p Mr
s 1
p 
M σ
r
σ
(TrsM )1σ G sσ 1
μ R

YS (TrsM )1σ G sσ 1

q * s 1
M M
(*) p M  c wr
r
Using the price rule (*) we get:
ρ
wM
r
 σ  1  μ R
M 1σ
σ 1 
  M 
Y S (Trs ) G s 

 σc   q * s1

1/σ
Put in P=CMg/ρ; and clear 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
29
(Tmrs), and the less competition the firm faces in location s (G decreases with n)
Tema 5 -EE
2. The Dixit-Stiglitz Model
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 (pAr ) (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
30
2. The Dixit-Stiglitz Model
Tema 5 -EE
3. Producer Behavior: normalizations
For selecting the units we have to notice the requirement so that the marginal labor satisfies the next
equation:
μ
σ 1
LMr
M
F

M
C 
( ρ)
nr 
p
σ
σ
μ
r 
wM
r
q*  l*  μ
• Now, the price index and the wage equation becomes:

M M (1σ) 
G r   n s (p s Tsr )

 s1

R
w
1
M
M M (1σ) 
   L s (w s Tsr )

μ
 s1

R
 σ  1  μ
M 1σ
σ 1 
  M 
Ys (Trs ) G s 

 σc   q * s 1

R
M
r
1/(1σ)
1/σ
1/(1σ)

M 1σ
σ 1 
  Ys (Trs ) G s 
 s1

R
IMP: with these normalizations we have shifted attention from the number of
manufacturing firms and product prices (n/G) to the number of manufacturing
31
workers and their wages rates. (L/W).
1/σ
Tema 5 -EE
•
2.The Dixit-Stiglitz Model
5. The price index effect and the Home Market Effect
•
We consider an economy with 2 regions, that produce 2 manufacturing varieties:
G
1-
2


1
L1w1  L 2 (w 2 T ) (1σ)
μ
1
  L1 (w1 T ) (1σ)  L 2 w 2
μ
G11- 

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:
32
2. The Dixit-Stiglitz Model
Tema 5 -EE
• 5. The price index effect and the Home Market Effect
dG L  G 
(1  σ)
  
G
μw
σ 1
(1  T
1σ
dw 
 dL
)
 (1  σ)
w 
L
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.
33
Tema 5 -EE
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 sort of an index of trade cost, with value between 0-1:
• Z=0, if trade is costless;
• Z=1, if trade is impossible.
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 (dL/L).
• The location with the larger home market has a more than + proportionately 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 is higher wages
instead of exports
• Locations with a larger home market (demand) tends to offer a higher nominal34
wage (qualified labor agglomeration).
Tema 5 -EE
2. The Dixit-Stiglitz Model
6. The “No-Black-Hole” Condition
•
We in general are not interested in economies in which increasing returns are
that strong, if only because, in such economies 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” theory, we usually impose what we call the
assumption of no black holes:
σ 1
ρμ
σ
35
Tema 5 -EE
3. Applications
36
Tema 5 -EE
3. Applications
n= # industries
g= # goods
c= # countries
ROW= rest of the world
Xngc= output of product g in industry
n in country c.
ROW= rest of the world
37
Tema 5 -EE
3. Applications
Xngc= output of product g in industry
n in country c.
Ω= technology matrix
V= factor endowments
of country c
38
Tema 5 -EE
3. Applications
39