## GEOGRAPHICAL ECONOMICS

ELTE Faculty of Social Sciences, Department of Economics

### Geographical Economics

week 8

KRUGMAN (1991) MODEL: DYNAMICS AND SIMULATION Author: Gábor Békés, Sarolta Rózsás

Supervised by Gábor Békés

June 2011

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Outline

1 Krugman model 2: dynamics Equilibrium and simulations Equilibrium

Results and history

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Equilibrium

Krugman (1991) model - continuation Dynamics, equilibrium

BGM Chapter 4.2-4.4 BGM Chapter 4.5 in part

Krugman's slogan: geographical economics model =

1 Dixit-Stiglitz core

2 + icebergs

3 + evolution

4 + a computer

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Equilibrium

Very dicult, non-linear model

How can we calculate an equilibrium for a given distribution
of*λ?*

1 Determining the exogenous parameters

2 and using a computer for simulations. . .

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### The model

The model, if *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*) _{(1)}
I_{1}= (*λ*_{1}W_{1}^{1}^{−}* ^{e}*+

*λ*

_{2}W

_{2}

^{1}

^{−}

*T*

^{e}^{1}

^{−}

*)*

^{e}^{1}

^{/(}

^{1}

^{−}

*; (2) I*

^{e)}_{2}= (

*λ*

_{1}T

^{1}

^{−e}W

_{1}

^{1}

^{−e}+

*λ*

_{2}W

_{2}

^{1}

^{−e})

^{1}

^{/(}

^{1}

^{−e)}(3)

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

_{2}

^{e}^{−}

^{1}]

^{1}

^{/e}; (4) W

_{2}= [

_{Y}

_{1}

_{T}

^{1}

^{−}

^{e}_{I}

_{1}

^{e}^{−}

^{1}+

_{Y}

_{2}

_{I}

_{2}

^{e}^{−}

^{1}]

^{1}

^{/e}

_{(5)}

w_{1} =_{W}_{1}_{I}_{1}^{−δ};w_{2} =_{W}_{2}_{I}_{2}^{−δ} _{(6)}

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Parameters

How should we choose the values of parameters for the simulation?

Empirical observations
Round numbers
Usefulness. . .
*δ*=_{0}.4

L=1

*λ*_{1}+_{λ}_{2}=_{1}
*φ*_{1}=*φ*_{2}=_{0}.5
*e*=5

*ρ*=_{1}−_{1/5}=_{0}.8
1/(_{1}−*e*) =−0.25
T =_{1}.7

T^{1}^{−}* ^{e}*=

_{0},12

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### The model

After normalization and simplications

Y_{1}=0.4λ_{1}W_{1}+0.3;Y_{2} =0.4λ_{2}W_{2}*δ*+0.3 (7)
I_{1}= (*λ*_{1}W_{1}^{−}^{4}+_{0},12λ_{2}W_{2}^{−}^{4})^{−}^{0}^{.}^{25}; (8)
I_{2}= (0,12λ_{1}W_{1}^{−}^{4}+*λ*_{2}W_{2}^{−}^{4})^{−}^{0}^{,}^{25} (9)

W_{1} = [_{Y}_{1}_{I}_{1}^{4}+_{0},12Y_{2}I^{4})^{0}^{,}^{25}; (10)
W_{2} = [0,12Y_{1}I_{1}^{4}+Y_{2}I_{2}^{4}]^{0}^{,}^{25} (11)

w_{1}=W_{1}I_{1}^{−}^{0}^{,}^{4};w_{2} =W_{2}I_{2}^{−}^{0}^{,}^{4} (12)

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Procedure

Sequential iteration

Denition: W_{1},5:=the value of W_{1}after the fth iteration
(_{it})

Guess an initial solution for the wage rate in the two regions
(W_{1},0=_{W}_{2}_{,}_{0}=1), where 0 indicates the number of
iterations

Calculate the income levels (Y_{1},0Y_{2},0) and price indices (I_{1},0

I_{2},0)

Substitute and determine a new possible solution for the
wage rates (W_{1},1,W_{2},1)

Repeat these steps until a solution is found: when W barely changes

(W_{r},_{it}−W_{r},_{it}_{−}_{1})/W_{r},_{it}_{−}_{1}<*σ,* for each r =1,2
*σ*:=_{0}.0001

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Relative real wage

Real wages are the incentive to move

When we get the short-run equilibrium setting (I,Y,W ) ⇒
we can calculate the ratio w_{1}/w_{2}

Figure on real wages

Simulations - x a given value of*λ*_{1}and seek the equilibrium
values of variables to this

Execute this program several times, varying*λ*_{1}between zero
and one

Plotting the relative real wage in region 1 against the value of
*λ*_{1}

Equilibrium, if

w_{1}/w_{2}=_{1 and 0}<_{λ}_{1}<_{1 or}
complete agglomeration (λ_{1}=_{1 or 0)}

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Figure on the relative real wage

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Figure on the relative real wage (2)

There are three types of equilibrium

A,E - complete agglomeration of manufacturing production C - spreading of manufacturing production over the two regions

B,D - manufacturing production is partially agglomerated Total of ve long-run equilibria

3 equilibria `nding' them analytically (guessing) (A,E,C) 2 equilibria nding them with simulations (B,D)

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Stability

Stability (on the basis of w_{1}/w_{2})

Suppose, e.g., that we are in point F ; w_{1} is greater than w_{2},
therefore it is worth moving to R1 (λ_{1}increases), and get to
point C.

It is valid for any arbitrary point between points B and C When the economy is located somewhere between point B and D, it reaches the spreading equilibrium sooner or later.

This point is the basin of attraction for the spreading equilibrium.

Similar reasonings hold for the segments between points A and B and between points D and E. They are called the basin of attraction for the agglomeration equilibrium.

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Instability of equilibria

There are two points (B and D), that are equilibria, but unstable.

If the economy `falls' exactly in these points, it will stay there (real wages are equal)

Any arbitrarily small perturbation of this equilibrium will set in motion a process of adjustment. . .

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Figure on transport costs

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### The eect of transport costs

Recall: transport (transaction) costs are the `heart' of the model, the most important exogenous factor

Repeating the previous procedure for T ={1.3,1.5,1.7,1.9,2.1}

If transport costs are large (T =_{1}.9 or T =_{2}.1), the
spreading equilibrium is the globally (unique) stable
equilibrium

When the two regions are too far away from each other, it is not worth producing in either of them and shipping to the other.

If transport costs are smaller (T =1.3 or T =1.5), the agglomerating equilibria are stable

If the two regions are very close to each other, the one that has a production cost-advantage (lower wage), will be the

`winner' (complete agglomeration).

The spreading equilibrium exists but unstable!

T =_{1}.7 - there exist more equilibria. How special is this
settings?

Not so frequent

But it always exists (for each parameter setting can be assigned a particular T )

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Figure on transport costs

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### The eect of changes in transport costs

Put the equilibrium distribution of mobile workforce *λ*on the
vertical axis and transport costs T along the horizontal axis
S sustain point - until which complete agglomerations are
equilibria

B break point - from which the spreading is equilibrium The segment between points B and S may be arbitrarily small or even a point.

>The tomahawk diagram

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### The `tomahawk' diagram (a)

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Agglomeration equilibrium (Reminder)

All manufacturing workers are located in one of the regions.

Agglomeration in region 1 (λ_{1}=_{1},*λ*_{2} =_{0)}
W_{1} =1

Then I_{1}=_{1},I_{2}=_{T}

and Y_{1}= (1+*δ*)/2,Y_{2}= (1−*δ*)/2
W_{1} =_{1},w_{1} =_{1}

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

_{1}+

*δ*)

_{/2}]

_{T}

^{1}

^{−}

^{e}^{−}

*+ (*

^{eδ}_{1}−

*δ*)

_{/2}]

_{T}

^{e}^{−}

^{1}

^{−}

^{eδ}If T is nor too large (but T >_{1), w}_{2}<1, i.e. nobody wants
to move

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Sustain point (S)

What does the assumption `T is not too large' mean? We can determine a sustain point (S):

w_{2}* ^{e}* =f(T) = [(1+

*δ*)/2]T

^{1}

^{−}

^{e}^{−}

*+ (1−*

^{eδ}*δ*)/2]T

^{e}^{−}

^{1}

^{−}

*=1*

^{eδ}⇒_{S}(_{T})'_{1}.81

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Sustain point (S)

What does the assumption `T is not too large' mean? We can determine a sustain point (S):

w_{2}* ^{e}* =

_{f}(

_{T}) = [(

_{1}+

*δ*)

_{/2}]

_{T}

^{1}

^{−e−eδ}+ (

_{1}−

*δ*)

_{/2}]

_{T}

^{e−}^{1}

^{−eδ}=

_{1}

⇒T '1.81

As transport costs increase, however, the rst term in the above equation becomes arbitrarily small, while the second term becomes arbitrarily large if

1−*e*−*eδ*>0⇒1−*e*>*eδ*⇒(1−*e*)/e>*δ*⇒*ρ*>*δ*
*ρ*>*δ*= no-black-hole condition if this condition is not
fullled the forces working toward agglomeration would
always prevail (independently from transport costs), and the
economy would tend to collapse into a point.

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Symmetry break point (B)

Break point (B) - from which the spreading equilibrium is stable

Recall: Ha W_{1}=_{W}_{2}=_{1}

Then I_{1}=_{I}_{2}= (_{0}.5)^{1}^{/(}^{1}^{−e)}(_{1}+_{T}^{1}^{−e})^{1}^{/(}^{1}^{−e)}
and Y_{1}=_{Y}_{2}=_{0}.5

W_{1} =_{1}=_{W}_{2},és w_{1} =_{w}_{2} : this is an equilibrium
We can show, that the condition necessary to break the
spreading equilibrium, i.e., dw/dλ>0

g(T):= ^{1}−T^{1}^{−e}

1+_{T}^{1}^{−e} + [1−* ^{δ}*(

_{1}+

*ρ*)

*δ*^{2}+*ρ* ]<1 (13)
The rst term on the right-hand side is Z ∈(_{0},1) _{and is}
monotonically increasing in transport costs T, while the
second term is a constant fraction strictly in between zero
and one (if the no-black-hole condition is fullled). See the
gure above!

Now B(T)'1.63

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### Krugman-Fujita-Venables (1999) Theorem

### Theorem

Suppose the no-black-hole condition (ρ>*δ) holds in a*
symmetric two-region setting of the Krugman model, then (i)
complete agglomeration of manufacturing activity is not
sustainable for suciently large transport costs T, and (ii)
spreading is a stable equilibrium for suciently large transport
costs T.

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### History matters! (1)

An important implication of the model

Case A: Transport costs are large, e.g., T =_{2}.5, and the
spreading equilibrium is stable

Suppose that transport costs start to fall, T =_{1}.7 - as
B(T)=1.63, the spreading equilibrium remains stable
Case B: Transport costs are large, e.g., T =_{1}.3, then
agglomeration equilibrium is established in one of the two
regions

Suppose that transport costs start to rise, T =_{1.7 - as}
S(T)=1.81, nothing happens. Agglomeration of
manufacturing activity remains a stable equilibrium
That is, in the case of T =_{1}.7, the outcome equilibrium
depends on history.

= "Evolution"

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### History matters! (2)

Go back to the `tomahawk' diagram. Suppose that transport costs are large and we begin to reduce them (e.g.

technological progress).

week 8 Gábor Békés

Krugman model 2:

dynamics Equilibrium and simulations Equilibrium Results and history

### History matters! (2a)

Go back to the `tomahawk' diagram. Suppose that transport costs are large and we begin to reduce them (e.g.

technological progress).

Until a particular point there is symmetry, then the economy sharply renders to agglomeration

Recall: Let*η* be the speed of adjustment and w is the
weighted average of the real wages (w =*λ*_{1}w_{1}+*λ*_{2}w_{2})_{.}
Then the motion of manufacturing labor can be described by
this simple dynamic system: ^{d}_{λ}^{λ}_{1}^{1} =*η*(w_{1}−w)

Which of the regions?

The one to which the rst migrant decides to move or the outcome is solely the result of a historical accident Non-linear relationship!

Due to a small step the economy suddenly reaches one of the agglomeration equilibria

T falls until a particular point nothing happens T falls further sudden powerful change