GEOGRAPHICAL ECONOMICS

GEOGRAPHICAL ECONOMICS

Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics, Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE)

Department of Economics, Eötvös Loránd University Budapest Institute of Economics, Hungarian Academy of Sciences

Balassi Kiadó, Budapest

Authors: Gábor Békés, Sarolta Rózsás Supervised by Gábor Békés

June 2011

ELTE Faculty of Social Sciences, Department of Economics

GEOGRAPHICAL ECONOMICS

week 7

Two-region Krugman model

Gábor Békés, Sarolta Rózsás

1 Krugman model

1.1 Production structure

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

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: N₁firms inR1,N₂firms inR2. 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

Transportation costs

• Transportation cost – a necessary element

• Samuelson (1952) iceberg transportation costs – a part melts. Cost = what does not arrive

• = von Thünen – wheat falling off from the wagon

• T>1 units of good need to be shipped to ensure that 1 unit arrives, e.g.TAB=T^DAB, whereDAB

is the distance betweenAandB. IfD=_0,T=₁

• Advantage: there is no separate transportation sector

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

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 inter- national models)

– manufacturing firms are also mobile between the two regions

• it is possible that all the manufacturing firms and laborers are located in one region

1.2 Geography steps in: two regions

Two regions

• BGM Chapters 3.7-3.9

• Two regions,

– demand and supply side, – transportation costs.

• Question: who is where?

Two regions

• Laborers:γin the manufacturing, 1−γin the food sector

• the distirbution ofLwithin the food sector:φ₁,φ2, within the manufacturing sector: λ₁,λ2

Region 1: production

• The mass of laborers in the food sector:φ1(1−γ)L – = output of food sector (1:1)

– = wage income in the food sector

• Manufacturing: there can be different conditions in the two regions:

– Wages:W1andW2

– Prices: let’s consider one product: p₁=βW₁/ρandp2=T p₁ – The size of manufacturing sector:N₁=l₁/αe=λ₁γL/αe – Within a region: the number of firms = f(laborers)

1.3 Short-run equilibrium

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 flow of laborers – describing dynamics (transition)

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 profit (food sector: CRS, manufacturing: free entry)

• Income = wage for the manufacturing and food sector workers

Y₁=λ₁W₁γL+φ₁(1−γ)L (1)

• Prices: productions costs, transportation costs

• Region 1: p1, region 2:T p1

• orp₁is the f.o.b. (factory gate) price,T p₁is the c.i.f. (import) price

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

Region 1: price-level

• Size of manufacturing:Ns=λsγL/αe

• The prices of goods inrproduced ins:(βWs/ρ)Tsr⇒ ^β

ρWsTrs

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

• Is =

∑

p¹⁻_i ^e

!1/(1−e)

⇒ Nsp^1−e_s 1/(1−e)

• Altogether there ares=1...Rregions. The price-level,Ir, in reigonr:

• Ir =

∑

λsγL αe (^β

ρWsTrs)^1−e

!1/(1−e)

= ^β

ρ(^γL

αe)^1/(1−e)

∑

λs(WsTrs)^1−e

!1/(1−e)

Equilibrium

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

I₁= ^β ρ(^γL

αe)^1/(1−e)λ₁W₁^1−e+λ₂(W₂T)^{1−e1/(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 thatIis an indicator of utility, it is increasing inN)

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

Equilibrium

• The wages are determined by the product market equilibrium.

• There is a demand from both regions, the demand curve – the demand of product iin 1: c_i1 = p^−e_i1 I₁^e−1δY1, the price: p1=βW1/ρ.

• Supply = aggregate demand:

x₁= (δβ^−eρ^e)(W₁^−eI₁^e−1Y₁+T^−eW₁^−eI₂^e−1Y2) (3)

• The elasticity of demand with respect to the price – (p1andp2=T p1) – is constant (e)

• The supply,x₁, is not exactly the same as the demand. Why?

• Because the transportation cost is a loss (it melts on the way)

• x₁= (δβ^−eρ^e)(W₁^−eI₁^e−1Y₁+T(T^−eW₁^−eI₂^e−1Y₂)) Wages – equilibrium

• We already know the supply (zero profit):x=α(e−1)/β

• We are looking for the equilibrium in the wages, not in the prices

• α(e−1)/β= (δβ^−eρ^e)(W₁^−eI₁^e−1Y1+T(T^−eW₁^−eI₂^e−1Y2))

• remembering thate=1/(1−ρ), W1=ρβ^−ρ

δ (e−1)α

1/e

[Y1I₁^e−1+Y2T^1−eI₂^e−1]^1/e (4)

• Wages in region 1 are higher if the market size is greater (local and other market), the transporta- tion cost is lower

1.4 Long-run equilibrium

Long-run equilibrium

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

• We’ve already got till this point:

Y₁=λ₁W₁γL+φ₁(1−γ)L (5) I₁= ^β

ρ(^γL

αe)^1/(1−e)λ₁W₁^1−e+λ₂W₂^1−eT^1−e1/(1−e)

(6)

W₁=ρβ^−ρ δ

(e−1)α 1/e

[Y₁I₁^e−1+Y₂T^1−eI₂^e−1]^1/e (7)

Real wage

• What is novelty:

w₁=W₁I₁^−δ (8)

• Long-run equilibrium = where

w1=w2 (9)

Theorem 1 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.

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 define the fixed labor requirement)

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

• A tiny cheat: γ = δ(or we useγinstead ofδ– this is not a question of measure, but does not change a lot)

The model

• After the normalization, assuming that the distribution of food sector workers is even:φ1=φ2= 0.5

Y1=λ₁W1δ+0.5(1−δ);Y2=λ₂W2δ+0.5(1−δ) (10) I₁= (λ₁W₁^1−e+λ2W₂^1−eT^1−e)^1/(1−e); (11) I2= (λ₁T^1−eW₁^1−e+λ₂W₂^1−e)^1/(1−e) (12)

W₁= [Y₁I₁^e−1+Y₂T^1−eI₂^e−1]^1/e;W₂= [Y₁T^1−eI₁^e−1+Y₂I₂^e−1]^1/e (13)

w₁=W₁I₁^−δ;w2=W2I₂^−δ (14) Equilibriate distributions

• Agglomeration in region 1: λ₁=1,λ₂=0

• Agglomeration in region 2: λ₁=0,λ₂=1

• Spreading, the two regions are completely identical:λ₁=λ₂=0.5

Spreading

• Spreading: the two regions are completely identical, λ₁ = λ₂ = 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.

• W1=W2=1

• Now,I1=I2= (0.5)^1/(1−e)(1+T^1−e)^1/(1−e)

• andY1=Y2=0.5

• Now, pluggingY,Ito the previous equations:W1=1=W2

• real wages are also identical,w1=w2: this is an equilibrium

Agglomeration

• Every manufacturing laborer is in one region. Agglomeration is in region 1:λ1=1,λ2=0

• W1=1

• This implies thatI1=1,I2=T

• andY1= (1+δ)/2,Y2= (1−δ)/2

• Now, pluggingY,Ito the previous equations:W1=1 andw1=1

• W₂=? in reality it does not exist, since there is no one in region 2. How much would it be for the first departer?

• W₂=[(1+δ)/2]T^1−e+ (1−δ)/2]T^{e−1 1/e}

• w₁=1, for small values ofT,w2<1, thus no one wants to relocate

1.5 Dynamics

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: firms/region (R1,R2) 4. transportation cost (T12)

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

The source of dynamics

• Manufacturing workers move according to real wages

• Letηdenote the speed of adaptation andwthe weighted average wage (w=λ₁w₁+λ2w2)

• The motion of laborers inR₁is described by the following dynamic equation:

• dλ₁

λ₁ =η(w₁−w) ₍₁₅₎

• The long-run equilibrium if

1. the distribution of laborers is such thatw1=w2=w, 2. all the workers are in one region

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 effect

2. Home market effect, HME

• Spreading is stimulated by:

3. Extent-of-competition effect

• The balance between the three effects determine dynamics

Dynamics around the symmetric equilibrium

• The index of trade costs is determined by:Z:= (1−T^1−e)/(1+T^1−e)

• IfT =1, thenZ=0, ifT=2,e=5,Z=0.88, andT→∞⇒Z→1

• At the spreading equilibrium we can leave the sub-indices

• Let a tilde denote relative changes:xe:=dx/x Deduction of the price index effect

• Remember, thatI1= (λ₁W₁^1−e+λ₂W₂^1−eT^1−e)^1/(1−e)

•

I₁^1−e=λ₁W₁^1−e+λ₂W₂^1−eT^1−e (16)

• Totally differentiate:

•

(1−e)I₁^−edI₁= [W₁^1−edλ₁] + [(1−e)λ₁W₁^−edW₁]+

[W₂^1−eT^1−edλ₂] + [(1−e)λ₂W₂^−eT^1−edW₂ ] + [(1−e)λ2W₂^1−eT^−edT]

• Around the spreading equilibrium the changes are symmetric:

• dI:=dI1=−dI2,dW:=dW1=−dW2,dλ:=dλ1=−dλ2

Deduction of the price index effect (cont.)

• Multiplying and dividing in order to get relative changesxe:=dx/x (1−e)I^1−edI

I = (1−T^1−e)λW^1−edλ λ + + (1−T^1−e)(1−e)λW₂^1−edW

W + [(1−e)λ₂W₂^1−eT^1−edT

T ]

• Remember that at the equilibrium the two regions are identical – λ₁=λ₂=0.5 ésW1=W2=1

– I1=I2=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=ZWe −[Z/(e−1)]eλ Price index effect

• eI=ZWe −[Z/(e−1)]eλ

• The optimal price of manufacturing goods (p=βW/ρ⇒We =p)_e

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

eI=Zep−[Z/(e−1)]Ne (17)

• Suppose thatpe=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 effect of agglomer- ation.

– (The products are cheaper because less products have to be imported under given trans- portation costs.)

Home market effect (HME)

• It can be shown (a required HW), that

•

Ye=ZNe+ [e/Z+ (1−e)Z]We (18)

• IfWe =0 thenYe=ZNeand 0≤Z≤1

• eλ=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 manu- faturing laborers. This is the home market effect.

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

The extent-of-competition effect

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

• c1=p^−e₁ (I₁^e−1δY1+T^1−eI₂^e−1δY2)

• The demand of a firm (inR1):ci1= p^−e_i1 (.)

• As we’ve seen in the bigger market the prices are lower

• We’ve also seen thatpi1depends on external factors

• Lower price index (I₁)⇒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 effect.

Simple D-S effects

• – Demand:x= p^−e(I^e−1δY₁)MC=βW,MR= ^e−1

e p⇒MR=MC

– In equilibrium:x= ^α(e−1)

β and p= ^e

e−1βW

Effects

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

– This effect worksagainstagglomeration.

• Home market: Furthermore, the new firm raises new demand for laborers, which increases de- mand for local goods. (The demand and MR curve shifts upward.)

– This effect is self-reinforcing andstimulatesagglomeration.

• Price index effect: If the price index falls – cheaper living costs, real wages are increasing – nom- inal wages are decreasing. MC shifts downward, profitability grows, number of new firm entries grow.

– This effect is self-reinforcing andstimulatesagglomeration.

Effects 2

• The balance between the three forces determines the equilibrium.

• If a firm arrives from the spreading equilibrium

– If its profit grows, then the original equilibrium is not stable, more firms will come – If its profit falls, then it is worth returning, the original equilibrium is stable

Key terms

• iceberg transportation costs

• short-run and long-run equilibria

• elements of the model of economic geography

• price index effect

• home market effect

• extent-of-competition effect