GEOGRAPHICAL ECONOMICS

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GEOGRAPHICAL ECONOMICS

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ELTE Faculty of Social Sciences, Department of Economics

Geographical Economics

week 3

VON THÜNEN MODELS Authors: Gábor Békés, Sarolta Rózsás

Supervised by Gábor Békés

June 2011

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week 3 Gábor Békés

Von Thünen Model Basis

The Von Thünen model more detailed Extension:

Neoclassical technology CBD: The urban land rent

Outline

Von Thünen (1826), Lösch (1954) Fujita Thisse 3.2.-3.3

The basic Von Thünen model Formal exposition

Extensions CBD models

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The basic Von Thünen model

R = rent

c = cost of production per unit Y = rate of return

p = price per unit F = transportation cost m = distance to the market R=_Y(_p−_c)−_Y ∗_F∗_m

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The basic Von Thünen model

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A Von Thünen example

Von Thünen (1826) monocentric city

The location of a certain activity depends on the transportation costs.

Vegetables/Fruits WoodWheat

Animals

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A Von Thünen example

Garlic cultivation

Lots of small enterprises Direct marketing Forests

Corn Grazing

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The same applied to a city

Von Thünen design game:

http://www.casa.ucl.ac.uk/software/vonthunen.asp

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Basis

An isolated town is a point in the Euclidean plane, each location r is identied by its distance r to the city

There are n dierent activities in the area each producing a dierent good denoted by i =1,2, ...n

An activity = a group of farmers: same product and technology

The production of one unit of good i requires a_i unit of land The quality and the density of land is unity: 2rπ annulus of innitely small diameter

independent of location CRS

Production function (q_i)

q_i(r) = ¹

a_i (1)

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Competition, prices

Perfectly competitive product and transport markets Prices of goods in the town are given, p_i

Transportation costs are also given, t_i

Land market is competitive, can only be used for agricultural purpose, rent R(r)

But: we can assume that in the land market producers are bidding

surplus by using one unit of land:

(_p_i−_t_i∗_r)_/a_i it is the base of the bidding process:

Ψ_i(r) = (p_i−t_i∗r)/a_i (2)

π_i(r) = (p_i−t_i∗r)q_i−R(r) =Ψ_i(r)−R(r) (3) If the prot related to good i at r is zero, the bid rent coincides with the market land rent

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Equilibrium

(non-negative) land rent function activity distribution

each activity's output is positive

R^∗(r)≡max

i=max1,2..nΨ_i(r),0

=max

i=max1,2..n(p_i−t_i∗r)/a_i,0 (4) The land rent function, R^∗(.), is the upper envelope of the bid rent function,Ψ_i(.): each location is occupied by the agent oering the highest bid

Theorem

If the transport cost function is linear in distance, then the equilibrium land rent is decreasing, piecewise linear, and convex.

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Equilibrium (2)

In other words, there is spatial specialization and segregation.

What determines the prices?

t_i/a_i we can arrange them in decreasing order:

t₁/a₁ ≥_t₂_/a₂ ≥...≥_t_n_/a_n

If the land demand is similar, then the fast decaying goods will be located closer to the town.

If the transportation costs are similar, then the land-intensive goods will be closer.

For all r, whereΨ_i(_r)<_R^∗(_r)the output is zero, but we assume that there is enough land for all the activities.

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Bid rent function and prices

t_i/a_i we can arrange them in decreasing order:

t₁/a₁ ≥_t₂_/a₂ ≥...≥_t_n_/a_n

The edge bid rents separating the adjecent lands equalizes Inner circle: Ψ₁(r₁^∗) =Ψ₂(r₁^∗) and outer circle:

Ψ₂(r₂^∗) =Ψ₃(r₂^∗)etc.

Generally:

Ψ_i(_r_i^∗) =_Ψ_i+1(_r_i^∗)→(_p_i−_t_i_r_i^∗)_/a_i = (_p_i+1−_t_i+1_r_i^∗)_/a_i+1 (5) r_i^∗= ^pⁱ^/aⁱ−p_i+1/a_i+1

t_i/a_i−_t_i₊₁_/a_i₊₁ ⁽⁶⁾ At World's end stands Thünen's wilderness: Ψ_n(_r_n^∗) =₀

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Social optimum

Is the competitive equilibrium socially optimal?

S ≡

∑

ⁿ

i=1p_iQ_i−

∑

ⁿ

i=1T_i (7)

total surplus = aggregate land rent Computable: HW

The answer: yes, the market outcome ensures the largest social surplus . . .

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Beckmann (1972) Neoclassical technology

Von Thünen classical economics: xed technological coecients

Beckmann's model (1972): land and labor

Production function (q_i) Cobb-Douglas, x_i(r) =X/a labor/land

q_i(_r) =_f[_x_i(_r)] = [_x_i(_r)]^αⁱ ₍₈₎ where 0≤α_i ≤1 stands for the substitution parameter between labor and land. The marginal productivity of labor is positive and decreasing (HW).

The prot:

π_i(_r) = (_p_i−t_ir)_q_i−wx_i−R(_r) ₍₉₎

x_i^∗(_r) =

(_p_i−_t_i_r)α_i w

₁/1−α_i

ahol r ≤ ^pⁱ

t_i (10)

∂x_i^∗(_r)_/∂r =?

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Results

∂x_i^∗(_r)_/∂r <₀

The further an activity is located from the center the less labor it uses.

Plugging x_i^∗(_r)→π_i(_r) and settingπ_i(_r) =_{0 and R}(_r)

=Ψ_i(_r)_:

Ψ_i(_r) = (₁−α_i)(α_i/w)^αⁱ^/¹⁻^αⁱ(_p_i−t_i∗r)¹^/¹⁻^αⁱ ₍₁₁₎

Theorem

Each land rent function is decreasing and strictly convex in distance.

Although, not everythig is so simple now . . .

The employment function remains decreasing across rings only under strict assumptions (HW)

There is a more complex relationship between land rent and useage. If transportation costs grow, the reduced land rent may not be sucient for compensation. Lower land rent means substitution between labor and land.

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Assumptions

City model trade-o between accessibility and space in residential choice

Alonso (1964), Mills (1967), Muth (1969)

Monocentric city's one dimensional model, the center is called the central business district (CBD)

N identical workers commuting to the CBD Income Y

Utility: U(_z,s), where z denotes the composite good, which price is p_z =1, s denotes the lot size of housing

U is strictly increasing in each good, twice continuously dierentiable, and strictly quasi-concave; both z and s are essential goods, s is a normal good. HW: detailed explanation

R(_r) is the rent, T(_r)is the cost of transportation, which is strictly increasing in r.

The expenditure constraint of workers at r distance from the CBD: z+_R(_r)_s+_T(_r) =_Y

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Utility

maxr,z,sU(z,s), z+sR(r) =Y −T(r) (12) Each worker is identical, thus U =u

How it diers from the previous model?

The worker chooses the location (endogeneously)

This is the point: choice between lot size and transportation costs

The bid rent functionΨ(_r,u)is the maximum rent that a consumer is willing to pay at distance r beside utility u. Max bid rent, s.t. u

Ψ[Y −T(r),u] =max

z,s

Y −T(r)−z

s , U(z,s) =u (13) For a consumer residing at distance r and consuming(_z,s)_, Y −_T(_r)−z is the money available for land payment;

Y−T(r)−z

s represents the rent

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Maximizing utility

We get the rent by choosing a consumption bundle of(z,s), while U(z,s) =u.

The equilibrium is the tangencypoint between the budget line of slopeΨ(_r,u)and the indierence curve: S(_r,u)_{is the} equilibratory lot size in r:

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Results

What is the relationship between rent and distance?

∂Ψ(r,u)

∂r =−_S^T₍⁰_r⁽_,^r_u⁾₎ <0 Similarly ^∂S⁽_∂r^r^,^u⁾ >₀

Theorem

The bid rent function is continuously decreasing, while the lot size function is continuously increasing in the distance from the CBD.

Further results:

Each worker who resides further, lives in a bigger at and consumes less from z.

Close to the CBD the population density is higher.

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CBD vs Thünen

What is the dierence between the Thünian and CBD models?

von Thünen: the prot of each activity is zero

CBD: everyone consumes s land, utility is endogeneous (nonzero)

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Key terms

basic Von Thünen model bid rent function

isolated town CBD