Economist Dixit Stiglitz Model

Lecture 5

SPATIAL ECONOMY: THE DIXIT-STIGLITZ MODEL

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

1. Introduction

2. The Dixit-Stiglitz model of monopolistic competition: spatial implications.

3. Applications.

4. Conclusion

Outline

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3Gobalización, comercio internacional and economía geográfica

3

Figure 1.1 Cartogram of GNPAreas are NOT proportional to population

1. Introduction

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4Gobalización, comercio internacional and economía geográfica

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Figure 1.2 GDP per capitaHighland variable among countries

1. Introduction

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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 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.

1. Introduction

ACp

n

AC = average cost in the firm

= n CF/ S + c

PP = price in the industry

price = c + 1 / nb

E

N2 = n companies in equilibrium (with Profit=0)

p2 = CM2

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

P

c

N: countries / variety

P = c + 1 / bn

LARGER MARKET BECAUSE OF TRADE

C (S2) = n CF / S2 + c

CLOSED COUNTRY

C (S1) = n CF / S1 + c

p1

n1

p2

n2

1

2

1. Introduction

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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.

2. The Dixit-Stiglitz Model

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


μ1μAMU

μ1

1

μM

UA

M

A

μ1

μRMSMA

1ρ0dim(i)M1/ρn

0

ρ

• 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-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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1. Consumers Behavior:


σ

11ρ;

ρ1

1σ

We define sigma (σ) as:

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 chosen according to the relative prices of both goods.

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1. The Consumers Behavior: the budget constraint


n

0A p(i)m(i)diApY• 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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Mdi)(..

)()(

1/ρρn

0

0

imts

diimipMinn

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:1. The first-order condition establishes the equality of marginal rates

of substitution MRS to price ratios:

p(j)

p(i)

m(j)

m(i)1

1

1. Consumer Behavior: Phase I:

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p(j)

p(i)

m(j)

m(i)1

1

σji

ρ1

1

m(j)pp(i)

p(j)m(j)m(i)

• For any pair i, j that leads to:

1. Consumer behavior: Phase I:

• And bringing the common term outside the integral-11

m(j)p(j)

• Substituting this into the original constraint: Mdim(i)1/ρn

0

ρ

M

dip(i)

p(j)m(j) 1/ρ

n

0

1ρ

ρ

11/ρ

We have that:

• 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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Mdip(i)p(j)m(j)dj

1/ρ-ρn

0

1ρ

ρn

0


• 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 G*M=total expenditure in manufactures

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)11/(n

0

1

1)/ρ-(ρn

0

1ρ

ρ

dip(i)dip(i)

G

• 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.


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

MG

p(j)M

G

p(j)m(j)

)1/(1

• We substitute G [4.7] in equation [4.5]:

M

dip(i)

p(j)m(j) 1/ρ

n

0

1ρ

ρ

11/ρ

[4.7]

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• 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. Consumer behavior: Phase II:

• This maximization gives: (MRS=price ratio)

Y..

1

ApGMts

AMUMaxA

• 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).

G

YμM

Ap

Yμ)(1A

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)1(

p(j)m(j)

G

Y

• Pulling the stages together, we obtain the following uncompensated consumer demand functions :

1. Consumer behavior: Phase I + Phase II:

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

Ap

Yμ)(1A • For agriculture:

• For manufactured products:

For n][0,j

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• 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:


μ)(1Aμμ1μ )(pYGμ)(1μU

μ)(1Aμ )(pG Cost-of-living index in the economy

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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:

-11/-11/n

0

-1 dip(i)G npM

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

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2. The Dixit-Stiglitz Mode

-11/G npM• The relationship between G and n depends on the elasticity of

substitution between varieties σ• 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:


• 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)

)1(

p(j)m(j)

G

Y

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• 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 of the original unit actually arrives. The rest is “lost” (melted) in the way as a transportation cost.

• The constant represents the amount of the agricultural good dispatched per unit received in s.

2. Multiple locations and transportation cost: iceberg costs

ArsT/1

ArsT

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2. Multiple locations and transportation cost: CIF prices

Mrs

Mr

Mrs Tpp

• 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:

R1,...,s,TpnGσ1

1R

1r

σ1Mrs

Mrrs

1σs

σMrs

Mrs G)T(pμY

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. 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:

Mrs

R

1s

1σS

σMrs

MrS

Mr TG)T(pYμq

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.

I have to produce Tmrs in r, knowing that a

portion 1/ Tmrs is lost during the trip

(transportation cost)

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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:

MMM qcFl • 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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Mr

MMr wc/σ(p )11

3. Producer Behavior: Profit maximization

)qc(Fwqpπ Mr

MMr

Mr

Mrr

• Firms maximize profits with a given income (sales) and with given costs (according to the wages)

Costs: F+V (given the wages wr)Revenues (sales)

• 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:

ρwc

elasticityelastCMg

pMr

MMr

1

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• 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:

F

1σ

cqwπ

MMrM

rr

M/c1-σFq*

Fσq*cFl* M

Fσ

L

*l

Ln

Mr

Mr

r

• 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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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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3. Producer Behavior: wages

1σs

σ1Mrs

σMr

R

1S G)(TpYμq*

s

R

1

1σs

σ1MrsS

σMr G)(TY

*q

μp

s

1/σR

1s

1σs

σ1MrsSM

Mr G)(TY

*q

μ

σc

1σw

• 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:

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

• Using the price rule (*) we get:ρ

wcp

Mr

MMr (*)

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3. Producer Behavior: wages


μ)(1Ar

μr )(pG

μ)(1Ar

-μr

Mr

Mr )(pGw

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

M is

• Real wages: real income at each location is proportional to nominal income deflated by the cost-of-living index,

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ρ)(σ

1σCM

M

rMr wp σ

μF

μ

Ln

Mr

r

μl*q*

σ)1/(1R

1s

σ)(1Msr

Ms

Ms

σ)1/(1R

1s

σ)(1Msr

Mssr )T(wL

μ

1)T(pnG

1/σR

1s

1σs

σ1Mrss

1/σR

1

1σs

σ1MrssM

Mr G)(TYG)(TY

*q

μ

σc

1σw

s

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).

3. Producer Behavior: normalizations

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2.The Dixit-Stiglitz Model

• 5. The price index effect and the Home Market Effect

22σ)(1

11-12 wL)T(wL

μ

1G

• We consider an economy with 2 regions, that produce 2 manufacturing varieties:

σ)(12211

-11 )T(wLwL

μ

1G

1σ22

σ11σ112

σ11σ22

1σ111

GYTGYw

TGYGYw

• 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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• 5. The price index effect and the Home Market Effect


w

dwσ)(1

L

dL)T(1

w

G

μ

L

G

dGσ)(1 σ1

1σ

G

dG1σ

Y

dY)T(1

w

G

w

Y

w

dwσ σ1

1σ

• [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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σ1

σ1

T1

T1Z

Y

dY

L

dLZ

w

dwσ1Z

Z

σ

• Now , let us consider how relative demand affects the location of manufacturing. It is convenient to define a new variable, Z,

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

• 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 nominal wage (qualified labor agglomeration).

• 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.

5. The price index effect and the Home Market Effect

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• 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:

μρσ

1σ

6. The “No-Black-Hole” Condition

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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

Ω= technology matrixV= factor endowments of country c

Xngc= output of product g in industry n in country c.

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3. Applications

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Economist Dixit Stiglitz Model