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 3

Von Thünen models

Gábor Békés, Sarolta Rózsás

1 Von Thünen Model

1.1 Basis

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

The basic Von Thünen model

A Von Thünen example

• Von Thünen (1826) – monocentric city

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

– Vegetables/Fruits – Wood

– Wheat – Animals

A Von Thünen example

• Garlic cultivation

– Lots of small enterprises – Direct marketing

• Forests

• Corn

• Grazing

The same applied to a city

• Von Thünen design game: http://www.casa.ucl.ac.uk/software/vonthunen.asp

1.2 The Von Thünen model more detailed

Basis

• An isolated town is a point in the Euclidean plane, each locationris identified by its distancerto the city

• There arendifferent activities in the area each producing a different good denoted byi=1, 2, ...n

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

• The production of one unit of goodirequiresa_iunit of land

• The quality and the density of land is unity: 2rπ– annulus of infinitely small diameter

• independent of location

• CRS

• Production function (qi)

qi(r) = ¹

a_i (1)

Competition, prices

• Perfectly competitive product and transport markets

• Prices of goods in the town are given,pi

• Transportation costs are also given,ti

• 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:

•

Ψi(r) = (p_i−t_i∗r)/a_i (2)

•

π_i(r) = (pi−ti∗r)qi−R(r) =Ψi(r)−R(r) (3)

• If the profit related to goodiatris zero, the bid rent coincides with the market land rent

Equilibrium

• (non-negative) land rent function

• activity distribution

• each activity’s output is positive

R^∗(r)≡max

i=1,2..nmax Ψi(r), 0

=max

i=1,2..nmax (p_i−t_i∗r)/a_i, 0

(4)

• The land rent function,R^∗(_.), is the upper envelope of the bid rent function,Ψi(_.): each location is occupied by the agent offering the highest bid

Theorem 1 If the transport cost function is linear in distance, then the equilibrium land rent is decreasing, piecewise linear, and convex.

Equilibrium (2)

• In other words, there is spatial specialization and segregation.

• What determines the prices?

• ti/ai– we can arrange them in decreasing order:t1/a1≥t2/a2≥...≥tn/an

• 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 allr, whereΨi(r)<R^∗(r)the output is zero, but we assume that there is enough land for all the activities.

Bid rent function and prices

• ti/ai– we can arrange them in decreasing order:t1/a1≥t2/a2≥...≥tn/an

• The edge bid rents separating the adjecent lands equalizes

• Inner circle:Ψ1(_r1^∗) =_Ψ2(_r^∗₁)and outer circle:Ψ2(_r^∗₂) =_Ψ3(_r^∗₂)_etc.

– Generally:

•

Ψi(r^∗_i) =_Ψi+1(r^∗_i)→(p_i−t_ir^∗_i)/a_i= (p_i+1−t_i+1r^∗_i)/a_i+1 (5)

•

r_i^∗= ^pi/ai−p_i+1/a_i+1

ti/ai−ti+1/ai+1 (6)

• At World’s end stands Thünen’s wilderness:Ψn(r^∗_n) =0

Social optimum

• Is the competitive equilibrium socially optimal?

•

S≡

∑

piQi−

∑

Ti (7)

• total surplus = aggregate land rent

• Computable:HW

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

1.3 Extension: Neoclassical technology

Beckmann (1972) – Neoclassical technology

• Von Thünen – classical economics: fixed technological coefficients

• 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 profit:

π_i(r) = (pi−tir)qi−wxi−R(r) (9)

•

x_i^∗(r) =

(pi−tir)α_i w

1/1−αi

aholr≤ ^pi

ti (10)

• ∂x^∗_i(r)/∂r=?

Results

• ∂x^∗_i(r)/∂r<0

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

• Pluggingx^∗_i(r)→π_i(r)and settingπ_i(r) =0 andR(r)_=Ψi(r):

Ψi(r) = (₁−α_i)(α_i/w)^αi/1−αi(p_i−t_i∗r)^1/1−αi ₍₁₁₎ Theorem 2 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 sufficient for compensation. Lower land rent means substitution between labor and land.

1.4 CBD: The urban land rent

Assumptions

• City model – trade-off 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)

• Nidentical workers commuting to the CBD

• IncomeY

• Utility: U(z,s), wherezdenotes the composite good, which price is pz=1,sdenotes the lot size of housing

• Uis strictly increasing in each good, twice continuously differentiable, and strictly quasi-concave;

bothzandsare essential goods,sis a normal good.HW: detailed explanation

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

• The expenditure constraint of workers atrdistance from the CBD:z+R(r)s+T(r) =Y

Utility

•

maxr,z,s U(z,s),z+sR(r) =Y−T(r) (12)

• Each worker is identical, thusU=u

• How it differs 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 distancer beside utilityu.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 distancerand consuming(z,s),Y−T(r)−zis the money available for land payment;^Y−T(r)−z_s represents the rent

Maximizing utility

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

• The equilibrium is the tangencypoint between the budget line of slopeΨ(r,u)and the indifference curve:S(r,u)is the equilibratory lot size inr:

Results

• What is the relationship between rent and distance?

• ^∂Ψ_∂r^(r,u) =−_S(r,u)^T0(r) <0

• Similarly ^∂S(r,u)_∂r >0

Theorem 3 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 flat and consumes less fromz.

• Close to the CBD the population density is higher.

CBD vs Thünen

• What is the difference between the Thünian and CBD models?

• von Thünen: the profit of each activity is zero

• CBD: everyone consumessland, utility is endogeneous (nonzero)

Key terms

• basic Von Thünen model

• bid rent function

• isolated town

• CBD