## GEOGRAPHICAL ECONOMICS

ELTE Faculty of Social Sciences, Department of Economics

### Geographical Economics

week 7

TWO-REGION KRUGMAN MODEL Authors: Gábor Békés, Sarolta Rózsás

Supervised by Gábor Békés

June 2011

week 7 Gábor Békés

Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics

### Outline

1 Krugman model Production structure

Geography steps in: two regions Short-run equilibrium

Long-run equilibrium Dynamics

week 7 Gábor Békés

Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics

### Basis

Krugman model (1991)

http://www.koz-gazdasag.hu/images/stories/4per2/13- krugman.pdf

For now BGM Chapter 3.3

Topics for today: Two-region model Production structure

Short-run equilibrium Long-run equilibrium Basis of dynamics

week 7 Gábor Békés

Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics

### Krugman model basis

Two regions: 1, 2: R1, R2

Two sectors: food and manufacturing

Laborers in the food sector, CRS, region 1 they sell in region 1 or 2. There are no transportation costs.

Manufacturing: N_{1} rms in R1, N_{2} rms in R2. Monopolistic
competition (as we have seen)

In the case of manufacturing goods there are transportations costs if the good produced in one region is not sold there

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### Transportation costs

Transportation cost a necessary element

Samuelson (1952) iceberg transportation costs a part melts.

Cost = what does not arrive

= von Thünen wheat falling o from the wagon

T >1 units of good need to be shipped to ensure that 1 unit
arrives, e.g. T_{AB} =_{T}^{D}^{AB}_{, where D}_{AB} is the distance
between A and B. If D=_{0, T} =_{1}

Advantage: there is no separate transportation sector

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### Consumers

Consumers: food and manufacturing good Food is homogeneous:

Consumers don't care whether they consume domestic or import wheat

Provided that there are no transportation costs prices are the same

Consumption of manufacturing goods: variety matters domestic and if they are available import goods as well The same porduct if imported would be more expensive transportation costs

Because of liking for variety, they would like to consume some units of all varieties

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### The source of dynamics

nominal vs real value

wage wage expressed in the numeraire

real wage price-level adjusted = purchasing power mobile sector (manufacturing) vs immobile sector (food)

laborers in the food sector are immobile

laborers in the manufacturing sector are mobile between the two regions (regional vs international models)

manufacturing rms are also mobile between the two regions it is possible that all the manufacturing rms and laborers are located in one region

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### Two regions

BGM Chapters 3.7-3.9 Two regions,

demand and supply side, transportation costs.

Question: who is where?

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### Two regions

Laborers: *γ*in the manufacturing, 1−*γ* in the food sector
the distirbution of L within the food sector: *φ*_{1},*φ*_{2}, within the
manufacturing sector: *λ*_{1},*λ*_{2}

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### Region 1: production

The mass of laborers in the food sector: *φ*_{1}(_{1}−*γ*)_{L}

= output of food sector (1:1)

= wage income in the food sector

Manufacturing: there can be dierent conditions in the two regions:

Wages: W_{1}and W_{2}

Prices: let's consider one product: p_{1}=_{βW}_{1}/ρand
p_{2}=_{Tp}_{1}

The size of manufacturing sector: N_{1}=l_{1}/αe=*λ*_{1}*γL/αe*
Within a region: the number of rms = f(laborers)

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### Equilibrium

The point is regional mobility Equilibrium, dynamics

The essence of Economic Geography Equilibrium

short-run: the distribution of laborers is given

long-run: long-run equilibrium under endogeneous ow of laborers

describing dynamics (transition)

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### Short-run equilibrium

Assumptions:

food sector laborers' market is in equilibrium the amount of food

manufacturing sector laborers' market is in equilibrium the amount of products

zero prot (food sector: CRS, manufacturing: free entry) Income = wage for the manufacturing and food sector workers

Y_{1}=*λ*_{1}W_{1}*γL*+*φ*_{1}(1−*γ*)L (1)
Prices: productions costs, transportation costs

Region 1: p_{1}, region 2: Tp_{1}

or p_{1} is the f.o.b. (factory gate) price, Tp_{1} is the c.i.f.

(import) price

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### Conditions of the equilibrium

Dominant factors of the equilibrium

1 the price of local products is a function of local wage

2 the prices of imported goods are higher because of transportation costs

3 the number of local products depends on the number of local workers

week 7 Gábor Békés

### Region 1: price-level

Size of manufacturing: N_{s} =*λ*_{s}*γL/αe*
The prices of goods in r produced in s:

(_{βW}_{s}/ρ)_{T}_{sr} ⇒ ^{β}

*ρ*W_{s}T_{rs}

If the prices of goods within a region are identical, but dier across regions the price-level is:

I_{s} =

### ∑

N i=1p^{1}

_{i}

^{−e}

!_{1}/(1−e)

⇒ _{N}_{s}_{p}_{s}^{1}^{−}^{e}^{}^{1/(1}^{−e)}

Altogether there are s =1...R regions. The price-level, I_{r}, in
reigon r:

I_{r} =

### ∑

^{R}

s=1
*λ*_{s}*γL*

*αe* (^{β}

*ρ*W_{s}T_{rs})^{1}^{−e}

!_{1}/(1−*e)*

=

*β*
*ρ*(^{γL}

*αe*)^{1/}^{(}^{1}^{−}^{e}^{)}

### ∑

R s=1*λ*_{s}(_{W}_{s}_{T}_{rs})^{1}^{−e}

!_{1}/(1−*e)*

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### Equilibrium

In the case of two regions, the price-level of the rst region:

I_{1}= ^{β}*ρ*(^{γL}

*αe*)^{1/}^{(}^{1}^{−}^{e}^{)}^{}*λ*_{1}W_{1}^{1}^{−}* ^{e}*+

*λ*

_{2}(W

_{2}T)

^{1}

^{−}

^{e}^{}

^{1/(}

^{1}

^{−e)}(2) What determines the price-level of region 1?

It is a weighted average of domestic and import products' prices

market size (do not forget that I is an indicator of utility, it is increasing in N)

external factors (e.g. production function, preferences)

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### Equilibrium

The wages are determined by the product market equilibrium.

There is a demand from both regions, the demand curve
the demand of product i in 1: c_{i1}=_{p}_{i1}^{−}^{e}_{I}_{1}^{e}^{−}^{1}_{δY}_{1}, the price:

p_{1}=_{βW}_{1}_{/ρ.}

Supply = aggregate demand:

x_{1} = (*δβ*^{−e}*ρ** ^{e}*)(

_{W}

_{1}

^{−e}

_{I}

_{1}

^{e−}^{1}

_{Y}

_{1}+

_{T}

^{−e}

_{W}

_{1}

^{−e}

_{I}

_{2}

^{e−}^{1}

_{Y}

_{2})

_{(3)}The elasticity of demand with respect to the price (p

_{1}and p

_{2}=

_{Tp}

_{1}) is constant (

*e*)

The supply, x_{1}, is not exactly the same as the demand. Why?

Because the transportation cost is a loss (it melts on the way)
x_{1}= (*δβ*^{−e}*ρ** ^{e}*)(

_{W}

_{1}

^{−e}

_{I}

_{1}

^{e−}^{1}

_{Y}

_{1}+

_{T}(

_{T}

^{−e}

_{W}

_{1}

^{−e}

_{I}

_{2}

^{e−}^{1}

_{Y}

_{2}))

week 7 Gábor Békés

### Wages equilibrium

We already know the supply (zero prot): x =*α*(*e*−_{1})/β
We are looking for the equilibrium in the wages, not in the
prices

*α*(*e*−_{1})/β=

(*δβ*^{−}^{e}*ρ** ^{e}*)(W

_{1}

^{−}

*I*

^{e}_{1}

^{e}^{−}

^{1}Y

_{1}+T(T

^{−}

*W*

^{e}_{1}

^{−}

*I*

^{e}_{2}

^{e}^{−}

^{1}Y

_{2})) remembering that

*e*=

_{1/}(

_{1}−

*ρ*)

_{,}

W_{1}=*ρβ*^{−ρ}
*δ*

(*e*−1)*α*
_{1/e}

[_{Y}_{1}_{I}_{1}^{e−}^{1}+_{Y}_{2}_{T}^{1}^{−e}_{I}_{2}^{e−}^{1}]^{1/e} _{(4)}
Wages in region 1 are higher if the market size is greater
(local and other market), the transportation cost is lower

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### Long-run equilibrium

The equations determining long-run equilibrium: income, price-level, wage (manufacturing) and real wage

We've already got till this point:

Y_{1}=*λ*_{1}W_{1}*γL*+*φ*_{1}(_{1}−*γ*)_{L} _{(5)}
I_{1}= ^{β}

*ρ*(^{γL}

*αe*)^{1/}^{(}^{1}^{−}^{e}^{)}^{}*λ*_{1}W_{1}^{1}^{−}* ^{e}*+

*λ*

_{2}W

_{2}

^{1}

^{−}

*T*

^{e}^{1}

^{−}

^{e}_{1/(}

_{1}−e)

(6)

W_{1}=*ρβ*^{−}^{ρ}*δ*

(*e*−1)*α*
_{1}/e

[Y_{1}I_{1}^{e}^{−}^{1}+Y_{2}T^{1}^{−}* ^{e}*I

_{2}

^{e}^{−}

^{1}]

^{1}

^{/e}(7)

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### Real wage

What is novelty:

w_{1} =_{W}_{1}_{I}_{1}^{−δ} _{(8)}

Long-run equilibrium = where

w_{1} =w_{2} (9)

### Theorem

In the long-run the labor force is mobile. The two-region world is in equilibrium, if the real wages in the two regions are identical. In this case there is no incentive to relocate.

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### Simplyfying the model

Simplyfying the parameters of the model Normalizations, the free choice of units

1 Population: L=1 (= million, thousand, ten million)

2 Labor force: *α*=*γL/e*(we could choose hour, day, year, but
instead we dene the xed labor requirement)

3 Output: *β*=*ρ*(we could choose kg, pieces, but instead we
dene marginal labor requirement)

A tiny cheat: *γ*=*δ*(or we use *γ*instead of *δ* this is not a
question of measure, but does not change a lot)

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### The model

After the normalization, assuming that the distribution of
food sector workers is even: *φ*_{1}=*φ*_{2} =_{0}.5

Y_{1}=*λ*_{1}W_{1}*δ*+0.5(1−*δ*);Y_{2}=*λ*_{2}W_{2}*δ*+0.5(1−*δ*) (10)

I_{1}= (*λ*_{1}W_{1}^{1}^{−e}+*λ*_{2}W_{2}^{1}^{−e}T^{1}^{−e})^{1}^{/(}^{1}^{−e)}; (11)
I_{2}= (*λ*_{1}T^{1}^{−e}W_{1}^{1}^{−e}+*λ*_{2}W_{2}^{1}^{−e})^{1}^{/(}^{1}^{−e)} _{(12)}

W_{1} = [_{Y}_{1}_{I}_{1}^{e−}^{1}+_{Y}_{2}_{T}^{1}^{−e}_{I}_{2}^{e−}^{1}]^{1/e};W_{2}= [_{Y}_{1}_{T}^{1}^{−e}_{I}_{1}^{e−}^{1}+_{Y}_{2}_{I}_{2}^{e−}^{1}]^{1/e}
(13)
w_{1} =W_{1}I_{1}^{−δ};w_{2} =W_{2}I_{2}^{−δ} (14)

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### Equilibriate distributions

Agglomeration in region 1: *λ*_{1}=1,*λ*_{2} =0
Agglomeration in region 2: *λ*_{1}=_{0},*λ*_{2} =_{1}

Spreading, the two regions are completely identical:

*λ*_{1}=*λ*_{2}=0.5

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### Spreading

Spreading: the two regions are completely identical,

*λ*_{1}=*λ*_{2}=_{0}.5. In this case, the nominal wages are identical,
too.

Proof: Suppose that the wages are identical, and check whether it is really an equilibrium.

W_{1} =_{W}_{2} =_{1}

Now, I_{1} =I_{2} = (0.5)^{1}^{/(}^{1}^{−}* ^{e)}*(1+T

^{1}

^{−}

*)*

^{e}^{1}

^{/(}

^{1}

^{−}

*and Y*

^{e)}_{1}=

_{Y}

_{2}=

_{0}.5

Now, plugging Y , I to the previous equations: W_{1}=1=W_{2}
real wages are also identical, w_{1} =_{w}_{2}: this is an equilibrium

week 7 Gábor Békés

### Agglomeration

Every manufacturing laborer is in one region. Agglomeration
is in region 1: *λ*_{1} =1,*λ*_{2}=0

W_{1} =1

This implies that I_{1}=_{1},I_{2}=_{T}
and Y_{1}= (_{1}+*δ*)_{/2},Y_{2}= (_{1}−*δ*)_{/2}

Now, plugging Y , I to the previous equations: W_{1}=1 and
w_{1} =1

W_{2} =?in reality it does not exist, since there is no one in
region 2. How much would it be for the rst departer?

W_{2} =^{}[(_{1}+*δ*)_{/2}]_{T}^{1}^{−e}+ (_{1}−*δ*)_{/2}]_{T}^{e−}^{1} ^{1}^{/e}

w_{1} =1, for small values of T , w_{2}<1, thus no one wants to
relocate

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### The model of economic geography

The model of economic geography essential elements

1 increasing returns to scale (internal IRS wtihin manufacturing goods)

2 imperfect competition (D-S monopolistic competition)

3 location: rms/region (R1,R2)

4 transportation cost (T_{12})

5 mobility for factors of production (labor mobility because of real wage)

week 7 Gábor Békés

### The source of dynamics

Manufacturing workers move according to real wages
Let*η* denote the speed of adaptation and w the weighted
average wage (w =*λ*_{1}w_{1}+*λ*_{2}w_{2})

The motion of laborers in R_{1} is described by the following
dynamic equation:

dλ_{1}

*λ*_{1} =*η*(_{w}_{1}−_{w}) _{(15)}
The long-run equilibrium if

1 the distribution of laborers is such that w_{1}=w_{2} =w,

2 all the workers are in one region

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### The source of dynamics 2

What are the economic factors determining dynamics (motion of laborers)?

The model is complicated and non-linear...

But at the symmetric equilibrium we can identify the main factors:

The agglomeration is stimulated by:

1 Price index eect

2 Home market eect, HME Spreading is stimulated by:

3 Extent-of-competition eect

The balance between the three eects determine dynamics

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### Dynamics around the symmetric equilibrium

The index of trade costs is determined by:

Z := (_{1}−T^{1}^{−e})_{/}(_{1}+_{T}^{1}^{−e})

If T =1, then Z =0, if T =2,*e*=5, Z =0.88, and
T →∞⇒Z →1

At the spreading equilibrium we can leave the sub-indices Let a tilde denote relative changes: ex:=dx/x

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### Deduction of the price index eect

Remember, that I_{1}= (*λ*_{1}W_{1}^{1}^{−e}+*λ*_{2}W_{2}^{1}^{−e}T^{1}^{−e})^{1/(}^{1}^{−e)}
I_{1}^{1}^{−}* ^{e}*=

*λ*

_{1}W

_{1}

^{1}

^{−}

*+*

^{e}*λ*

_{2}W

_{2}

^{1}

^{−}

*T*

^{e}^{1}

^{−}

*(16) Totally dierentiate:*

^{e}(_{1}−*e*)_{I}_{1}^{−e}_{dI}_{1} = [_{W}_{1}^{1}^{−e}_{d}**λ**_{1}] + [(_{1}−*e*)*λ*_{1}W_{1}^{−e}dW_{1}]+

[W_{2}^{1}^{−}* ^{e}*T

^{1}

^{−}

*d*

^{e}

**λ**_{2}] + [(1−

*e*)

*λ*

_{2}W

_{2}

^{−}

*T*

^{e}^{1}

^{−}

*dW*

^{e}_{2}] + [(

_{1}−

*e*)

*λ*

_{2}W

_{2}

^{1}

^{−e}T

^{−e}dT]

Around the spreading equilibrium the changes are symmetric:

dI :=_{dI}_{1}=−_{dI}_{2},dW :=_{dW}_{1}=−_{dW}_{2},dλ:=_{d}*λ*_{1}=

−_{dλ}_{2}

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### Deduction of the price index eect (cont.)

Multiplying and dividing in order to get relative changes
ex :=_{dx/x}

(1−*e*)I^{1}^{−}* ^{e}*dI

I = (1−T^{1}^{−}* ^{e}*)

*λW*

^{1}

^{−}

*d*

^{e}

**λ***+ + (*

**λ**_{1}−

_{T}

^{1}

^{−e})(

_{1}−

*e*)

_{λW}_{2}

^{1}

^{−e}

^{dW}

W
+ [(_{1}−*e*)*λ*_{2}W_{2}^{1}^{−e}T^{1}^{−e}dT

T ] Remember that at the equilibrium the two regions are identical

*λ*_{1}=*λ*_{2}=_{0}.5 és W_{1}=_{W}_{2}=_{1}

I_{1}=_{I}_{2}=*λ*^{1/(1}^{−e)}(_{1}+_{T}^{1}^{−e})^{1/(1}^{−e)}⇒_{I}^{1}^{−e}=*λ*(_{1}+_{T}^{1}^{−e})

⇒dividing by (1−*e*)(1+T^{1}^{−}* ^{e}*)

*λ*dI

I = ^{1}

1−*e*

1−T^{1}^{−e}
1+_{T}^{1}^{−e}

dλ

*λ* +^{1}−T^{1}^{−e}
1+_{T}^{1}^{−e}

dW
W
eI =ZWf−[Z/(*e*−1)]e*λ*

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### Price index eect

eI =ZWf−[Z/(*e*−1)]e*λ*

The optimal price of manufacturing goods
(p=* _{βW}*/ρ⇒Wf=

_{e}

_{p)}

Because of the proportional change of labor force/number of
goods, (e*λ*=N):e

eI =_{Z}_{e}_{p}−[_{Z}_{/}(*e*−1)]Ne (17)
Suppose that ep=0

What does this mean?

The price index falls if the market size (N) grows

Large market is advantageous because of lower prices. This is the price index eect of agglomeration.

(The products are cheaper because less products have to be imported under given transportation costs.)

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### Home market eect (HME)

It can be shown (a required HW), that

Ye =ZNe+ [*e/Z*+ (1−*e*)Z]fW (18)
IfWf=0 then Ye =ZN and 0e ≤Z ≤1

e*λ*=Ne

Under non-zero transportation costs the region with higher aggregate income (higher GDP) will have a more than proportional variety of products and a higher than

proportional rate of manufaturing laborers. This is the home market eect.

T =1.5,*e*=4⇒Z =0.5 thus if income grows by 10%,
then there will be 20% more products available

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### The extent-of-competition eect

The demand for the products in region 1 from the two regions:

c_{1}=_{p}_{1}^{−e}(_{I}_{1}^{e−}^{1}_{δY}_{1}+_{T}^{1}^{−}^{e}_{I}_{2}^{e−}^{1}_{δY}_{2})
The demand of a rm (in R1): c_{i1}=_{p}_{i1}^{−}* ^{e}*(.)

As we've seen in the bigger market the prices are lower
We've also seen that p_{i1}depends on external factors
Lower price index (I_{1})⇒lower demand (x_{i1})

Fiercer competition (larger variety of products) reduces demand for certain goods through lower price index. This is the extent-of-competition eect.

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### Simple D-S eects

Demand: x =_{p}^{−e}(_{I}^{e−}^{1}_{δY}_{1})_{MC} =* _{βW , MR}*=

^{e−}^{1}

*e* p⇒
MR=MC

In equilibrium: x= ^{α(e−}^{1}^{)}

*β* and p= _{e−}^{e}_{1}_{βW}

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### Eects

Competition: As a new rm enters, I falls and so does the demand, x. (The demand and MR curve shifts downward.) Consequently, prot falls.

This eect works against agglomeration.

Home market: Furthermore, the new rm raises new demand for laborers, which increases demand for local goods.

(The demand and MR curve shifts upward.)

This eect is self-reinforcing and stimulates agglomeration.

Price index eect: If the price index falls cheaper living costs, real wages are increasing nominal wages are

decreasing. MC shifts downward, protability grows, number of new rm entries grow.

This eect is self-reinforcing and stimulates agglomeration.

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### Eects 2

The balance between the three forces determines the equilibrium.

If a rm arrives from the spreading equilibrium

If its prot grows, then the original equilibrium is not stable, more rms will come

If its prot falls, then it is worth returning, the original equilibrium is stable

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### Key terms

iceberg transportation costs short-run and long-run equilibria

elements of the model of economic geography price index eect

home market eect

extent-of-competition eect