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: N1 rms in R1, N2 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
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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. TAB =TDAB, where DAB is the distance between A and B. If D=0, T =1
Advantage: there is no separate transportation sector
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Two regions
BGM Chapters 3.7-3.9 Two regions,
demand and supply side, transportation costs.
Question: who is where?
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Two regions
Laborers: γin the manufacturing, 1−γ in the food sector the distirbution of L within the food sector: φ1,φ2, within the manufacturing sector: λ1,λ2
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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 dierent conditions in the two regions:
Wages: W1and W2
Prices: let's consider one product: p1=βW1/ρand p2=Tp1
The size of manufacturing sector: N1=l1/αe=λ1γL/αe Within a region: the number of rms = f(laborers)
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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)
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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
Y1=λ1W1γL+φ1(1−γ)L (1) Prices: productions costs, transportation costs
Region 1: p1, region 2: Tp1
or p1 is the f.o.b. (factory gate) price, Tp1 is the c.i.f.
(import) price
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Region 1: price-level
Size of manufacturing: Ns =λsγL/αe The prices of goods in r produced in s:
(βWs/ρ)Tsr ⇒ β
ρWsTrs
If the prices of goods within a region are identical, but dier across regions the price-level is:
Is =
∑
N i=1p1i−e!1/(1−e)
⇒ Nsps1−e1/(1−e)
Altogether there are s =1...R regions. The price-level, Ir, in reigon r:
Ir =
∑
Rs=1 λsγL
αe (β
ρWsTrs)1−e
!1/(1−e)
=
β ρ(γL
αe)1/(1−e)
∑
R s=1λs(WsTrs)1−e
!1/(1−e)
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Equilibrium
In the case of two regions, the price-level of the rst region:
I1= β ρ(γL
αe)1/(1−e)λ1W11−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 that I is an indicator of utility, it is increasing in N)
external factors (e.g. production function, preferences)
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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: ci1=pi1−eI1e−1δY1, the price:
p1=βW1/ρ.
Supply = aggregate demand:
x1 = (δβ−eρe)(W1−eI1e−1Y1+T−eW1−eI2e−1Y2) (3) The elasticity of demand with respect to the price (p1 and p2=Tp1) 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−eρe)(W1−eI1e−1Y1+T(T−eW1−eI2e−1Y2))
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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−eρe)(W1−eI1e−1Y1+T(T−eW1−eI2e−1Y2)) remembering that e=1/(1−ρ),
W1=ρβ−ρ δ
(e−1)α 1/e
[Y1I1e−1+Y2T1−eI2e−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
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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)λ1W11−e+λ2W21−eT1−e1/(1−e)
(6)
W1=ρβ−ρ δ
(e−1)α 1/e
[Y1I1e−1+Y2T1−eI2e−1]1/e (7)
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Real wage
What is novelty:
w1 =W1I1−δ (8)
Long-run equilibrium = where
w1 =w2 (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.
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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 γinstead of δ this is not a question of measure, but does not change a lot)
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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= (λ1W11−e+λ2W21−eT1−e)1/(1−e); (11) I2= (λ1T1−eW11−e+λ2W21−e)1/(1−e) (12)
W1 = [Y1I1e−1+Y2T1−eI2e−1]1/e;W2= [Y1T1−eI1e−1+Y2I2e−1]1/e (13) w1 =W1I1−δ;w2 =W2I2−δ (14)
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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+T1−e)1/(1−e) and Y1=Y2=0.5
Now, plugging Y , I to the previous equations: W1=1=W2 real wages are also identical, w1 =w2: this is an equilibrium
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Agglomeration
Every manufacturing laborer is in one region. Agglomeration is in region 1: λ1 =1,λ2=0
W1 =1
This implies that I1=1,I2=T and Y1= (1+δ)/2,Y2= (1−δ)/2
Now, plugging Y , I to the previous equations: W1=1 and w1 =1
W2 =?in reality it does not exist, since there is no one in region 2. How much would it be for the rst departer?
W2 =[(1+δ)/2]T1−e+ (1−δ)/2]Te−1 1/e
w1 =1, for small values of T , w2<1, thus no one wants to relocate
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium 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: rms/region (R1,R2)
4 transportation cost (T12)
5 mobility for factors of production (labor mobility because of real wage)
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
The source of dynamics
Manufacturing workers move according to real wages Letη denote the speed of adaptation and w the weighted average wage (w =λ1w1+λ2w2)
The motion of laborers in R1 is described by the following dynamic equation:
dλ1
λ1 =η(w1−w) (15) The long-run equilibrium if
1 the distribution of laborers is such that w1=w2 =w,
2 all the workers are in one region
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Dynamics around the symmetric equilibrium
The index of trade costs is determined by:
Z := (1−T1−e)/(1+T1−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
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Deduction of the price index eect
Remember, that I1= (λ1W11−e+λ2W21−eT1−e)1/(1−e) I11−e=λ1W11−e+λ2W21−eT1−e (16) Totally dierentiate:
(1−e)I1−edI1 = [W11−edλ1] + [(1−e)λ1W1−edW1]+
[W21−eT1−edλ2] + [(1−e)λ2W2−eT1−edW2 ] + [(1−e)λ2W21−eT−edT]
Around the spreading equilibrium the changes are symmetric:
dI :=dI1=−dI2,dW :=dW1=−dW2,dλ:=dλ1=
−dλ2
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Deduction of the price index eect (cont.)
Multiplying and dividing in order to get relative changes ex :=dx/x
(1−e)I1−edI
I = (1−T1−e)λW1−edλ λ + + (1−T1−e)(1−e)λW21−edW
W + [(1−e)λ2W21−eT1−edT
T ] Remember that at the equilibrium the two regions are identical
λ1=λ2=0.5 és W1=W2=1
I1=I2=λ1/(1−e)(1+T1−e)1/(1−e)⇒I1−e=λ(1+T1−e)
⇒dividing by (1−e)(1+T1−e)λ dI
I = 1
1−e
1−T1−e 1+T1−e
dλ
λ +1−T1−e 1+T1−e
dW W eI =ZWf−[Z/(e−1)]eλ
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Price index eect
eI =ZWf−[Z/(e−1)]eλ
The optimal price of manufacturing goods (p=βW/ρ⇒Wf=ep)
Because of the proportional change of labor force/number of goods, (eλ=N):e
eI =Zep−[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.)
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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
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 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
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
The extent-of-competition eect
The demand for the products in region 1 from the two regions:
c1=p1−e(I1e−1δY1+T1−eI2e−1δY2) The demand of a rm (in R1): ci1=pi1−e(.)
As we've seen in the bigger market the prices are lower We've also seen that pi1depends on external factors Lower price index (I1)⇒lower demand (xi1)
Fiercer competition (larger variety of products) reduces demand for certain goods through lower price index. This is the extent-of-competition eect.
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
Simple D-S eects
Demand: x =p−e(Ie−1δY1)MC =βW , MR= e−1
e p⇒ MR=MC
In equilibrium: x= α(e−1)
β and p= e−e1βW
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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.
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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
week 7 Gábor Békés
Krugman model Production structure Geography steps in: two regions Short-run equilibrium Long-run equilibrium Dynamics
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