SPATIAL ECONOMY:

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Lecture 5

• SPATIAL ECONOMY
• THE DIXIT-STIGLITZ MODEL

• By Carlos Llano,
• References for the slides
• Fujita, Krugman and Venables Economía Espacial.
  Ariel Economía, 2000.

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Outline

1. Introduction
2. The Dixit-Stiglitz model of monopolistic
   competition spatial implications.
3. Applications.
4. Conclusion
   

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Figure 1.1 Cartogram of GNPAreas are NOT
proportional to population
1. Introduction
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Figure 1.2 GDP per capitaHighland variable among
countries
1. Introduction
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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 70s. (Dixit-Stiglitz,
   1977).
3. With the ES the distinction between
   inter-industrial and intra-industrial 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.

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1. Introduction
AC average cost in the firm n CF/ S c
AC p
E
p2 CM2
PP price in the industry price c 1
/ nb
n
N2 n companies in equilibrium (with Profit0)

• 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)

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1. Introduction
CLOSED COUNTRY C (S1) n CF / S1 c
P c
LARGER MARKET BECAUSE OF TRADE C (S2) n CF / S2
c
1
p1
2
p2
P c 1 / bn
N countries / variety
n1
n2
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2. The Dixit-Stiglitz Model

• The Dixit-Stiglitz model is the starting point
  for the of the monopolistic competition models
  (Dixit-Stiglitz, 1977).
• Since the 70s, 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.
• 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
• Solution to the consumers problem
• Multiple Locations and Transportation Costs
• Producer Behavior
• The Price Index Effect and the Home Market Effect
• Equilibrium

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2. The Dixit-Stiglitz Model

• Consumer Behavior Utility function
• Every consumer shares the same Cobb-Douglas
  tastes for the two type of goods (M, A).

• 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
• m(i) consumption of each available variety (i),
• n range of varieties.

• M is defined by a constant-elasticity-of-substitu
  tion (CES)
• Rho (?) intensity of the preference for variety
  (love for variety)
• If ?1, differentiated goods are nearly perfect
  substitutes (low love for variety)
• If ?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 (s) as
s elasticity of substitution between any 2
varieties

• The consumers problem maximize utility defined
  by the function U subject to the budget
  constraint.
• We solve it in 2 steps
• 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).
• Once the consumption of varieties in generic
  terms has been optimized (for every M), 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

1. The Consumers Behavior the budget constraint

• 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
• 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).
• 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
1. Consumer Behavior Phase I
1. Minimize expenditure for any given 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
• The first-order condition establishes the
  equality of marginal rates of substitution MRS to
  price ratios

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2. The Dixit-Stiglitz Model
1. Consumer behavior Phase I

• 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 of manufactures

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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

• Now we want to express this term as the
  manufactures price index (G)
• So GMtotal expenditure in manufactures

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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.

4.7

• Now we can write the demand for m(i) more
  compactly

• We substitute G 4.7 in equation 4.5

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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.

• 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 (MRSprice ratio)

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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

For

• If Gconstant, the price elasticity of demand for
  every available variety is constant and equal to
  (s).

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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

Cost-of-living index in the economy
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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

• The relationship between G and n depends on the
  elasticity of substitution between varieties s

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2. The Dixit-Stiglitz Mode
1. Consumer behavior Phase I Phase II

• The relationship between G and n depends on the
  elasticity of substitution between varieties s
• The lower is s (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

• When ?n ? ?G , the demand m(j) shifts downward,
• Important it allows us to know the equilibrium
  n
• If ?n ? ? competition ? shifts downward the
  existing products m(j) and reduce the sales of
  those varieties (evolution to more firms with
  profit0)

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2. The Dixit-Stiglitz Model
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2. The Dixit-Stiglitz Model
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

• Then, the manufacturing price index (Gs) may take
  a different value in each location according to
  the location s where it is consumed

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
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

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 consumers demand for a variety has a
  constant price elasticity sigma (s), the
  elasticity of the aggregate demand for each
  variety with respect to its mill price is also
  sigma (s), 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

• With increasing returns to scale, consumers
  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)

Costs FV (given the wages wr)
Revenues (sales)

• Each firm accept the price index G as given.
  Thus, the perceived elasticity of demand is
  therefore s, and the profit maximization (Img
  CMg) implies that

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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

• Therefore, the zero-profit condition, implies
  that the equilibrium output is

• 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

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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.

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2. The Dixit-Stiglitz Model

• 3. Producer Behavior wages

• 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

• Using the price rule () we get

()
Put in PCMg/? and clear s

• 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 (Tmrs), and the less competition the
  firm faces in location s (G decreases with n)

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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,

• This means that the real wage of manufacturing
  workers in location r, denoted by ?rM is

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2. The Dixit-Stiglitz Model

• 3. Producer Behavior normalizations

For selecting the units we have to notice the
requirement so that the marginal labor satisfies
the next equation

• Now, the price index and the wage equation
  becomes

IMP with these normalizations we have shifted
attention from the number of manufacturing firms
and product prices (n/G) to the number of
manufacturing workers and their wages rates.
(L/W).
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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

• These pairs of equations are symmetric, and so
  its solutions.
• So, if L1L2 Y1Y2, then there is a solution
  with G1G2 and with w1w2.

• 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 dGdG1-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

• Eq 1 Price Index Effect We suppose that the
  supply of labor is perfectly elastic, so that
  dw0. Bearing in mind that 1-s lt0 and that Tgt1,
  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 regions 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,

• Z is sort of an index of trade cost, with value
  between 0-1
• Z0, if trade is costless
• Z1, if trade is impossible.

• Using the definition of Z and eliminating dG/G,
  we have

• If dw0, supply of labor is perf. elastic Home
  market effect A 1 change in demand for
  manufactures (dY/Y) causes a 1/Z (gt1) 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 dwgt0, 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 nominal wage (qualified
  labor agglomeration).

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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

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3. Applications
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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

•  

•  

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3. Applications
Xngc output of product g in industry n in
country c.

•  

•  

O technology matrix
V factor endowments of country c
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3. Applications