Lecture 2B: Alonso Model - University of Minnesota...

Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis

Econ 460 Urban Economics

Lecture 2B: Alonso Model

Instructor: Hiroki Watanabe

Spring 2011

© 2011 Hiroki Watanabe 1 /110


1 Land Consumption and LocationCheesecake and LandAssumptions

2 Alonso ModelLandscapeFeasible and Pareto Optimal Allocations

3 Edgeworth BoxStandard Edgeworth Box Doesn’t WorkEdgeworth Trapezoid

4 Normative AnalysisContract Curve in Alonso EconomyContract Curve ; Pareto OptimalExample: Quasilinear Preferences

5 Positive AnalysisEquilibriumEquilibrium RentExample: Cobb-Douglas Utility

6 Summary© 2011 Hiroki Watanabe 2 /110


Cheesecake and Land

Two models of residential location choice:1 Alonso model: discrete # of residents2 Monocentric city model (next lecture): city size

N ∈ R.Question: Who lives where? Do we need tointervene to correct suboptimal use of land?


Notes

Notes

Notes


Cheesecake and Land

How are the land in the city and the land in thesuburbs different from cheesecake and tea?Liz chooses the right amount of L = (L

C, L

T) where

1 Marginal willingness to pay for a slice of cheesecakein terms of tea

2 Marginal rate of substitution3 The slope of her indifference curve

coincide with1 The relative price of a slice of cheesecake in terms

of tea2 The opportunity cost of a slice of cheesecake in

terms of tea3 The slope of her budget line

Trinity meets trinity.



Cheesecake and Land

0 2 4 6 8 100

2

4

6

8

10

10

10

1010

20

20

20

2030

30

3040

40

40

50

50

60

6070

70

80

90

Land s (ft2)

Com

posi

te G

oods

z (

bask

ets)

The Way Liz Finds (s*, z*)

Indifference CurvesBudget Constraint



Cheesecake and Land

Does Liz choose the land consumption in the cityand the suburbs (C, S) in the same way?Recall Liz prefers (C, T) = (5,5) over = (8,2).What about (C, S)?


Notes

Notes

Notes


Cheesecake and Land

We can enjoy tea and cheesecake at the same time.We cannot simultaneously occupy two houses atdifferent locations.Land does not exhibit convexity.Consider an extreme example:

(C, S) =max{C, S}.



Cheesecake and Land

0 2 4 24/5 6 80

1

2

3

4

5

6

7

8

1 2

23

3

4

44

55

55

66

666

77

777

88

8888

Land in the City xC (ft2)

Land

in th

e S

ubur

b x S

(ft2 )

The Way Liz Finds (xC* , x

S* )

Utility Level u(x)=max{xC, x

S}

Budget Constraint 5xC+3x

S=24



Cheesecake and Land

We always get a corner solution (C,0) or (0, S),which does not tell us so much about urban landuse patterns.

We can’t take ∂φC(pC,pS,m)∂pS

for example.

Instead, we take land and other commodities andanalyze the location choice later.

You can live in a house and eat cheesecake at thesame time.You probably like a house and a cheesecake betterthan a very spacious house with no food.


Notes

Notes

Notes


Cheesecake and Land

0 2 4 6 8 100

2

4

6

8

10

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1010

20

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70

80

90

Land s (ft2)

Com

posi

te G

oods

z (

bask

ets)

The Way Liz Finds (s*, z*)

Indifference CurvesBudget Constraint



Assumptions

We assume that land is a normal good.



Assumptions

00

Land s (ft2)

Com

posi

te G

oods

z (

bask

ets)

Normal Land Consumption

Budget ConstraintOptimal BundleIncreased Budget Constraint


Notes

Notes

Notes


Assumptions

We also assume that a composite good is anuméraire.

i.e., pz = 1.Why can we do that?

1 Tangency condition2 Mobility










Landscape

Cf. Alonso [Alo64] and Arnott and McMillen [AM08]Ch.7.2 households: Liz and Kenneth.A narrow strip (1-ft wide for example) of land [0, S)constitutes the urban residential area (a linear city).Preferences are identical

L(·) = K(·) = (s, z).

sL, sK are land consumption.zL, zK are composite goods consumption.L, K are front or driveway location.They commute to the city center.Commuting cost is t (baskets/mile) measured fromdriveway.


Notes

Notes

Notes


Landscape

30 Rock 1 3 4 8 S=9

xL xL+sL xK xK+sK

Linear City

Commuting distance (Liz)Commuting distance (Kenneth)



Feasible and Pareto Optimal Allocations

Definition 2.1 (Feasible Allocation)1 An allocation is a list (sL, sK , zL, sK , L, K).2 An allocation (sL, sK , zL, sK , L, K) is called feasible

ifzL + zK + tL + tK ≤ Z�

L, L + sL�

∩�

K , K + sK�

= ∅�

L, L + sL�

∪�

K , K + sK�

= [0, S).




Note that all the urban area needs to be distributedto be feasible.Otherwise some lot will be left unoccupied withoutbeing priced (a waste dump, for example).


Notes

Notes

Notes



Question 2.2 (Pareto Optimal Allocation)

Which one is more efficient? (Assume zK1= zK

2).

30 Rock m_2 m_1 S

Allocation 2

Allocation 1m

1

m2

LizKenneth




Does the allocation 21 leave both of them at least as well off as before and2 make at least one of them better off than the

allocation 1?




1 Kenneth (consumption bundle adjusted to have thesame utility level):

allocation 1 2

land consumption level sK sK1

sK2(= sK

1)

commuting cost tK tm1 0

baskets zK zK1

zK1+ tm1

baskets (after transfer) zK zK2= zK

1+ tm1 − tm1

☺ K (sK1, zK

1) = cK

1K (sK

2, zK

2) = cK

1


Notes

Notes

Notes



2 Liz:

allocation 1 2

land consumption level sL sL1

sL2(= sL

1)

commuting cost tL 0 tm2

baskets zL zL1

zL1− tm2

baskets (after transfer) zL zL2= zL

1− tm2 + tm1

☺ L(sL1, zL

1) = cL

1L(sL

2, zL

2) > cL

1




Utility level:1 They are both at least as well off as before.2 Liz is better off.

Conclude: allocation 2 Pareto dominates allocation1.Note that we cannot compare allocation 1 and 2without transfer.




Generalize the observation above as follows:

Theorem 2.3 (Pareto Optima (Berliant and Fujita[BF92]))

At an efficient allocation (sL, sK , zL, zK , L, K),1 K < L⇔ sK < sL.2 K < L ⇒ K(sK , zK) ≤ L(sL, zL).3 K(·) < L(·)⇒ K < L.4 K(·) < L(·)⇔mK <mL , where mX denotes

income level.


Notes

Notes

Notes



Sketch of the proof:1 Suppose K < L but sK ≥ sL. If they switch their

positions as follows, there will be extra baskets dueto reduced commuting cost:

Liz Kenneth

before t(K + sK) tK

after tK t(K + sL)

differential tsK −tsL

transfer −αt(sK − sL) +αt(sK − sL)

( sL

sK−sL ≤ α ≤sK

sK−sL ).




0 2 4 6 80

1

2

3

4

5

6

7

8

tsK−αt(sK−sL)

−tsK+α(sK−sL)

Land s (ft2)

Com

posi

te g

oods

(ba

sket

s)

Pareto Improvement

Liz: uL(zL, sL)=cL

Kenneth: uK(zK, sK)=cK




⇒ contradicts the claim that the original allocationis efficient.


Notes

Notes

Notes









Standard Edgeworth Box Doesn’t Work

How is the Alonso model represented in theEdgeworth box?




0 2 4 6 8 10 120

1

2

3

4

5

6

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Liz′s Indifference Curve

Liz: uL(sL, zL)=cL


Notes

Notes

Notes



0246810120

1

2

3

4

5

6

Land sK (ft

2)

Com

posite Goods z

K (baskets)

Kenneth′s Indifference Curve

Kenneth: uK(s

K, z

K)=c

K




0 2 4 6 8 10 120

1

2

3

4

5

6

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Liz: u(xLC, xL

T)=cL

Kenneth: u(xKC, xK

T)=cK

0246810120

1

2

3

4

5

6

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)




This is not an Edgeworth box.Some of the allocations in the box is not feasible.Suppose t = 1/6 and consider an allocation(sL, sK , zL, sK , L, K) = (2,10,1,5,0,10).

There are Z = 6 baskets in total.6 baskets are allocated as follows:

zL + zK + tL + tK

= 1+ 5+ 16 · 10+ 0

> Z. ☹

The previous box ignores the location (i.e., itrepresents an aspatial economy).

So, give up the Edgeworth box altogether?


Notes

Notes

Notes


Edgeworth Trapezoid

Don’t even. Slice off some part to make theEdgeworth box a trapezoid.Consider the following two cases:

1 L > K : Liz lives farther away from 30 Rock.2 K > L: Kenneth lives farther away from 30 Rock.



Edgeworth Trapezoid

1 L > K .If Liz consumes sL, she has to spendtL = tsK = t(S− sL) on commuting.i.e., t(S− sL) is deducted from her basket.(sL, zL) with zL < t(S− sL) is not affordable(otherwise she won’t be able to commute).



Edgeworth Trapezoid

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Liz′s Consumption Set When xL>xK

0 2 4 6 8 10 120

1

2

3

4

5

6Not feasible zL<txL=t(S−sL)


Notes

Notes

Notes


Edgeworth Trapezoid

Define net composite good zL by

zL︸︷︷︸

net consumption

:= zL︸︷︷︸

gross consumption

−t(S− sL).

t(S− sL) is part of her basket zL (grossconsumption) but it is not for her to consume.Liz’s net consumption is smaller than her grossconsumption level if L > 0.



Edgeworth Trapezoid

Added commuting cost alters her utility functionL(sL, zL) to:

L(sL, zL) ≡ L(sL, zL − t(S− sL)).



Edgeworth Trapezoid

0 2 4 6 8 10 120

1

2

3

4

5

6

Land sL (ft2)

Net

Consm

pti

on

Lev

elz

L(b

ask

ets)

Liz′s Indifference Curve When xL>xK

Liz: uL(sL

, zL) = c

L


Notes

Notes

Notes


Edgeworth Trapezoid

And this was the last time you see anythingmeasured in net consumption levelzL = zL − t(S− sL) on a graph.In what follows, everything’s measured in grossconsumption level zL on a graph.I.e., Liz’s indifference curve will be shifted upwards.



Edgeworth Trapezoid

Land sL (ft2)

Gro

ss C

onsu

mpt

ion

Leve

l zL (

bask

ets)


0 2 4 6 8 10 120

1

2

3

4

5

6uL(sL, zL)=cL (Aspatial)

uL(sL, zL−t(S−sL))=cL

N/A. zT<txL=t(S−sL)



Edgeworth Trapezoid

Observe that indifference curves are skewedupwarads in the following:


Notes

Notes

Notes


Edgeworth Trapezoid

0 2 4 6 8 10 120

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3

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6

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4

44

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Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Liz′s Indifference Curves (Aspatial)

Liz: uL(sL, zL)=sLzL



Edgeworth Trapezoid

−2.99998−2.99998

2.22427e−005

2.22427e−005

2.22427e−0052.22427e−005

3.00002

3.00002

3.00002

3.00002

6.00002

6.00002

6.00002

6.00002

9.00002

9.00002

9.00002

9.00002

12

12

12

1215

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1518

18

1821

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2124

24

2427

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2730

30

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36

36

3939

4242

4545

4848

5154576063

66

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Liz′s Indifference Curves (Spatial)

0 2 4 6 8 10 120

1

2

3

4

5

6

Liz: uL(sL, zL−t(S−sL))=sL[zL−t(S−sL)]



Edgeworth Trapezoid

Confirm that any allocation in the followingtrapezoid is feasible (take sL = 6 for example).


Notes

Notes

Notes


Edgeworth Trapezoid

N/A. zL<txL=t(S−sL)


uK(sK, zK)=cK

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

0 2 4 6 8 10 120

1

2

3

4

5

6

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)

0246810120

1

2

3

4

5

6



Edgeworth Trapezoid

Note that we do not have to shift Kenneth’sindifference curves (why?)Feasibility of baskets becomes:

1 In gross terms

zL + zK ≡ zL + t(S− sL) + zK = Z ≡ Z + t(S− sL)

2 In net terms

zL + zK ≡ zL − t(S− sL) + zK = Z − t(S− sL) ≡ Z

when L > K .



Edgeworth Trapezoid

2 L < K .Same argument in reverse.


Notes

Notes

Notes


Edgeworth Trapezoid

Land sK (ft

2)

Com

posite Goods z

K (baskets)

Kenneth′s Indifference Curve When xK>x

L

0246810120

1

2

3

4

5

6u

K(s

K, z

K)=c

K (Aspatial)

uK(s

K, z

K−t(S−s

K))=c

K

N/A. zK<tx

K=t(S−s

K))



Edgeworth Trapezoid

N/A. zK<txK=t(S−sK)

uK(sK, zK−t(S−sK))=cK

uL(sL, zL)=cL

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)

024681012

0

1

2

3

4

5

6

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)

0246810120

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2

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5

6









Notes

Notes

Notes


Contract Curve in Alonso Economy

How do we find the efficient allocations on theEdgeworth trapezoid?Recall in an aspatial economy, the contract curvesatisfies:

MRSL401(sL, zL) = MRSK(S− sL, Z − zL).




10

10

10

1020

20

2030

30

30

40

4050

50

60

10

10

10

10

20

20

20

3030

30

40

40

50

50

60

Liz: uL(sL, zL)

Kenneth: uK(sK, zK)Contract Curve

0 2 4 6 8 10 120

1

2

3

4

5

6

10

10

10

1020

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2030

30

30

40

4050

50

60

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10

10

10

20

20

20

3030

30

40

40

50

50

60

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)

0246810120

1

2

3

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5

6




In the spatial economy with L > K , the contractcurve satisfies:

MRSL460(sL, zL) = MRSK(sK , zK)

⇔ MRSL460

�

sL, zL − t(S− sL)�

= MRSK(S− sL, Z − zL).


Notes

Notes

Notes



0

0

0

0

10

10

10

10

20

20

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30

30

40

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50

60

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10

10

10

20

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20

30

30

30

40

40

50

50

60

Liz: uL(sL, zL−t(S−sL))


0

0

0

0

10

10

10

10

20

20

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30

30

40

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50

60

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10

10

10

20

20

20

30

30

30

40

40

50

50

60

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

0 2 4 6 8 10 120

1

2

3

4

5

6

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)

0246810120

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2

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What does MRSL460(·) look like?

To begin with, we need to find: (� in parenthesesindicates that the variable is fixed)

∂L(sL, zL)

∂sL=∂L(sL,�)

∂sL+∂L(sL, zL)

∂zL∂zL

∂sL

=∂L(sL,�)

∂sL+∂L(sL, zL)

∂zL∂[zL − t(S− sL)]

∂sL

=∂L(sL,�)

∂sL+∂L(sL, zL)

∂zLt.




Then,

MRSL460(sL, zL)

=−∂L(sL, zL)/∂sL

∂L(sL, zL)/∂zL

=−∂L(sL,�)/∂sL

∂L(sL, zL)/∂zL+−∂L(sL, zL)/∂zL

∂L(sL, zL)/∂zL∂zL

∂sL

= MRSL401(sL, zL)− t.

(1)


Notes

Notes

Notes



Question 4.1 (Tangency Condition in Alonso Economy)

What does (1)

MRSL460(sL, zL) = MRSL

401(sL, zL)−t

mean?

To make things easy, write everything in positiveterms:

|MRSL460(sL, zL)| = |MRSL

401(sL, zL)−t|

= |MRSL401(sL, zL)|+t




Aspatial Liz (or Liz 401) is willing to reduce sL byone unit if she gains (or compensated with)|MRSL

401(sL, zL)| baskets in return.

Spatial Liz (or Liz 460) is willing to reduce sL by oneunit if she gains |MRSL

401(sL, zL)|+t baskets in

return.Why does she need extra t baskets forcompensation?




Liz 401 is willing to reduce sL by one unit if shegains (or compensated with) |MRSL

401(sL, zL)|

baskets in return.Liz 460 is willing to reduce sL by one unit if shegains |MRSL

401(sL, zL)|+t baskets in return.

Why does she need extra t baskets forcompensation?

tL = tsK = t(S− sL) grows as sL gets smaller.

Consider a change from sL = 12 to sL = 11 in thefollowing graph:


Notes

Notes

Notes



Land sL (ft2)

Gro

ss C

onsu

mpt

ion

Leve

l zL (

bask

ets)


0 2 4 6 8 10 120

1

2

3

4

5

6uL(sL, zL)=cL (Aspatial)


N/A. zT<txL=t(S−sL)




In conclusion, an allocation (sL, sK , zL, zK) is on thecontract curve if

|MRSL(sL, zL)|+ t = |MRSK(sK , zK)| when L > K

|MRSL(sL, zL)| = |MRSK(sK , zK)|+ t when L ≤ K .(2)



Contract Curve ; Pareto Optimal

Some allocations on the contract may not beefficient.Recall Theorem 2.3 :

4 K (·) < L(·)⇒ K < L.


Notes

Notes

Notes



uL(sL, zL−t(S−sL))=c

uK(sK, zK)=cContract Curve (PO)Contract Curve (non PO)

0 2 4 6 8 10 120

1

2

3

4

5

6

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)

0246810120

1

2

3

4

5

6




uK(s

K, z

K−t(S−s

K))=c

uL(s

L, z

L)=c

Contract Curve (PO)Contract Curve (non PO)

0246810120

1

2

3

4

5

6

Land sK (ft

2)

Com

posite Goods z

K (baskets)

Land sL (ft

2)

Com

posite Goods z

L (baskets)

0 2 4 6 8 10 120

1

2

3

4

5

6




uK(sK, zK−t(S−sK))=c

uL(sL, zL)=cContract Curve (PO)Contract Curve (non PO)

0 2 4 6 8 10 120

1

2

3

4

5

6

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)

0246810120

1

2

3

4

5

6


Notes

Notes

Notes



uL(sL, zL))=c

uK(sK, zK−t(S−sK))=cContract Curve (PO)Contract Curve (non PO)

0 2 4 6 8 10 120

1

2

3

4

5

6

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)

0246810120

1

2

3

4

5

6





uK(sK, zK)=c

uL(sL, zL)=c


PO (xL>xK)

PO (xL<xK)

0 2 4 6 8 10 120

1

2

3

4

5

6

Land sL (ft2)

Com

posi

te G

oods

zL (

bask

ets)

Land sK (ft2)

Com

posi

te G

oods

zK (

bask

ets)

0246810120

1

2

3

4

5

6



Example: Quasilinear Preferences

Example 4.2 (Quasilinear Preferences)

Consider quasilinear preferences represented by

L(sL, zL) =p

sL + zL.


Notes

Notes

Notes



1

2

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9

Liz: uL(sL, zL)

Kenneth: uK(sK, zK)PO Allocations

0 2 4 6 8 10 120

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Land sL (ft2)

Com

posi

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oods

zL (

bask

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Land sK (ft2)

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zK (

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If L > K ,

L(sL, zL) =p

sL + zL ≡p

sL + zL − t(S− sL).

Allocations on the contract curve satisfies:

|MRSL(sL, zL)| = |MRSK(sK , zK)|⇔ |MRSL�

sL, zL − t(S− sL)�

|+t = |MRSK(S− sL, Z − zL)|.




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Land sL (ft2)

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Contract Curve (Spatial, xL>xK)

0 2 4 6 8 10 120

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6Liz: uL(sL, zL−t(S−sL))



Notes

Notes

Notes



Some allocations on the contract curve are notefficient.




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Liz: uL(sL, zL−t(S−sL))


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uK(sK, zK)=c

PO Allocations (xL>xK)

uL(sL, zL)=c


PO Allocations (xL<xK)

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Land sL (ft2)

Com

posi

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oods

zL (

bask

ets)

Land sK (ft2)

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oods

zK (

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Notes

Notes

Notes









Equilibrium

ωL, ωK are endowment of composite good.ωL +ωK = Z.Donaghy Real Estate (the absentee landlord) isendowed with [0, S).Their utility is given by

D(sD, zD) = zD.



Equilibrium

Definition 5.1 (Feasible Allocation (with Jack Donaghy))

A feasible allocation is a list(sL∗, sK∗, zL∗, zK∗, L∗, K∗, zD) such that

zL + zK + zD + tL + tK = Z

[L, L + sL) ∩ [K , K + sK) = ∅

[L, L + sL) ∪ [K , K + sK) = [0, S).


Notes

Notes

Notes


Equilibrium

Definition 5.2 (Equilibrium)

An equilibrium is a feasible allocation(sL∗, sK∗, zL∗, zK∗, L∗, K∗, zD∗) and a price densityp(y) such that

1 The bundle (sL∗, zL∗) solves

maxL,sL,zLL(sL, zL)

subject to ωL ≥ zL + tL +∫ L+sL

Lp(y)dy.

2 Analogous condition for Kenneth.



Equilibrium

In equilibrium, Liz satisfies

|MRSL(sL∗, zL∗)| = p(L∗ + sL∗) (3)p(L∗) = p(L∗ + sL∗) + t. (4)

(See Appendix ).

Eastbound (3) Liz’s willingness to pay for an additionalsq ft of land at the back of her lot is equal tothe cost of obtaining it in equilibrium.

Westbound (4) Adding one more sq ft to the front of herlot should cost exactly t baskets more thanadding one more sq ft to the back. If not,she will move forward to save oncommuting cost.



Equilibrium

Unfortunately, we can’t draw the Edgeworth box inthis economy.

Two consumers plus Jack Donaghy.The box shrinks from top to bottom (why?)

We can still find the rent as a function of distance.The box won’t shrink lengthwise (why not?)


Notes

Notes

Notes


Equilibrium Rent

Suppose K∗ < L∗ in equilibrium.What does p() look like?We know K∗ = 0 and L∗ = sK∗. Then (3) implies

p(sK∗) = |MRSK(sK∗, zK∗)|p(L∗ + sL∗ = S) = |MRSL(sL∗, zL∗)|.

And (4) implies

p(K∗ = 0) = p(sK∗) + tp(L∗ = sK∗) = p(S) + t.



Equilibrium Rent

In conclusion,

Proposition 5.3 (Equilibrium Rent)

In an equilibrium with K∗ < L∗, equilibrium pricedensity functions satisfy

p(L∗) = |MRSK(sK∗, zK∗)|p(S) = |MRSL(sL∗, zL∗)|p(L∗) = p(S) + t.



Equilibrium Rent

Then how about this?

p() =�

|MRSK(sK∗, zK∗)| for 0 ≤ < L∗

|MRSL(sL∗, zL∗)| for L∗ ≤ < S.

(Note p(0) can be p(sK ) + t but = 0 is measure zero).


Notes

Notes

Notes


Equilibrium Rent

0 2 4 6 8 10 120

MRSL*

MRSK*

Distance x from 30 Rock (ft)

Ren

t (ba

sket

s)

Price Density Function

Proposed Price Density Function p(x)Differential = t



Equilibrium Rent

This p() actually won’t constitute an equilibrium.Since at = L, |MRSK(sK∗, zK∗)| > p(), Kennethhas an incentive to expand his lot further to theeast.p() has to be such that beyond = L, Kennethdoesn’t want to increase sK ,i.e., if p() is higher than |MRSK(, zK∗)|, he won’tincrease sK .



Equilibrium Rent

Let cK∗ := K(sK∗, zK∗) and ζK(sK , cK∗) be thenumber of baskets Kenneth has to get to maintainK(·) = cK∗ while consuming sK , i.e.,

K(sK , ζK(sK , cK∗)) = cK∗.

(Or, to put it in another way, (sK , ζK(sK , cK∗)) tracesthe indifference curve at cK∗).


Notes

Notes

Notes


Equilibrium Rent

Kenneth doesn’t want to expand sK as long as p()is higher than his MRSK(sK∗, zK∗) beyond L.Like the following for example:

p∗() =

|MRSK(sK∗, zK∗)| for 0 ≤ < sK∗

|MRSK(, ζK(, cK∗))| for sK∗ ≤ < s′

|MRSL(sL∗, zL∗)| for s′ ≤ < S,

where s′ is a location such thatMRSK(s′, ζK(s′, cK∗)) = MRSL(sL∗, zL∗).



Equilibrium Rent

0 2 4 6 8 10 120

MRSL*

MRSK*

s′


Ren

t (ba

sket

s)


Price Density Function p*(x)

MRSK(x, ζK(x, cK*))



Equilibrium Rent

Liz won’t want to shift her lot towards the west (i.e.,reduce L while maintaining the lot size sL) if

p() = |MRSL(sL∗, zL∗)|+ t.

Increased rent p() exactly offsets the savings fromreduced commuting cost t.


Notes

Notes

Notes


Equilibrium Rent

Once L reaches = 0 then she won’t want toreduce her lot size sL if

p() ≤ |MRSL(, ζL(, cL∗))|.

If p() > |MRSL(, ζL(cL∗, ))| at = sL, then sheWILL sell her lot at the east end to receive p().She only needs |MRSL(, ζL(cL∗, ))| to stay as welloff as before but she gets more than that (p()) byselling a lot. ⇒ it’s not an equilibrium.



Equilibrium Rent

The following works as an equilibrium price densityfunction for example:

p∗() =

|MRSK(sK∗, zK∗)| for 0 ≤ < s′′

|MRSL(, zL(cL∗, ))| for s′′ ≤ < sL∗

|MRSL(sL∗, zL∗)| for sL∗ ≤ < S,

where s′′ is a location such thatMRSL(s′′, zL(cL∗, s′′)) = MRSK(sK∗, zK∗).



Equilibrium Rent

0 2 4 6 8 10 120

MRSL*

MRSK*

s′′


Ren

t (ba

sket

s)


Price Density Function p*(x)

MRSL(x, ζL(x, cL*))


Notes

Notes

Notes


Equilibrium Rent

So, there is a continuum of equilibrium.Which one is most favorable to Donaghy RealEstate?Note

D(sD, zD) = zD =∫ S

0

p()d.

Note also: equilibrium allocations are efficient.Compare (3) and (4) to (2).



Example: Cobb-Douglas Utility

Example 5.4 (Equilibrium Rent)

Consider the following economy:Preferences are represented by

L(sL, zL) = log(sL) + log(zL)K(sK , zK) = log(sK) + log(zK).

S = 60, 50 of which is sL∗ and 10 of which is sK∗.Z = 300, 236 of which is ωL and 64 of which is ωK .The observed equilibrium price density functionp() is piecewise linear (see next slide).t = 1.




0 10 22 50 600

?

?


Ren

t (ba

sket

s)

Equilibrium Rent

Equilibrium Rent


Notes

Notes

Notes



Question 5.5 (Equilibrium Rent)1 Find p(10) and p(60).2 Find zL∗.3 How many baskets does Jack get?4 Find the equilibrium.




1 Find p(10) and p(60).

First of all, who lives close to 30 Rock?Recall Theorem 2.3 .

From Proposition 5.3 ,

p(sK∗ = 10) = |MRSK∗(sK∗, zK∗)|.

zK∗ = ωK −∫ sK∗

0p(10)d = 64− 10p(10).

Note |MRSK(sK∗, zK∗)| = zK

sK. Then

p(10) = |MRSK�

sK∗, ωK − sK∗p(10)�

|

=64− 10p(10)

10

⇒ p(10) = 3.2.




Proposition 5.3 gives p(60) as follows:

p(60) = p(10)− t = 2.2.


Notes

Notes

Notes



0 10 22 50 600

2.2

3.2


Ren

t (ba

sket

s)

Equilibrium Rent

Equilibrium Rent

MRSK(x, ζK(x, cK*))=(320/x)/x

MRSL(x, ζL(x, cL*))=(5500/x)/x




Note nobody wants to relocate or change their lotsize under the prevalent equilibrium price density.




2 Find zL∗.

Liz pays

∫ S

L∗p()d =

1

2· 10 · 1+ 2.2 · 50 = 116

baskets in rent.Therefore,

zL∗ = ωL∗−∫ S

L∗p()d−tL∗ = 236−116−1·10 = 110.


Notes

Notes

Notes



3 How many baskets does Jack get?

Jack collects

zD = 10 · 3.2+ 116 = 148.

Then zL + zK + zD = 110+ 32+ 148 = 290(< Z).Where did the remaining 10 baskets go? Definition 5.1




4 Find the equilibrium.The equilibrium is

(sL∗, sK∗, zL∗, zK∗, L∗, zK∗, zD∗)= (50,10,110,32,10,0,148),

and

p() =

3.2 if 0 ≤ < 10

−112 +

12130 if 10 ≤ < 22

2.2 if 22 ≤ < 60.









Notes

Notes

Notes


Way the urban economists incorporate location.Allocations in Alonso model and graphicalpresentation.Disconnected contract curve and Pareto allocations.Equilibrium.



References

William Alonso.Location and Land Use.Harvard University Press, 1964.

Richard J. Arnott and Daniel P. McMillen.A Companion to Urban Economics.Blackwell, 2008.

Marcus Berliant and Masahisa Fujita.Alonso’s discrete population model of land use:efficient allocations and competitive equilibria.International Economic Review, 1992.



Map du Jour

Source http://www.commoncensus.org/


Notes

Notes

Notes

http://www.commoncensus.org/

http://www.commoncensus.org/


AppendixLiz solves

maxsL,zLL(sL, zL) s.t. ωL ≥ zL+

∫ L+sL

Lp(y)dy+tL.

Lagrangian

LL := L(·) + λL

ωL − zL −∫ L+sL

Lp(y)dy− tL

.



The first order conditions are:

∂LL

∂sL= L

s− λL

∂

∂sL

∫ L+sL

Lp(y)dy = L

s− λLp(L + sL) = 0 (5)

∂LL

∂zL= L

z− λL = 0 (6)

∂LL

∂L= −λL

∫ L+sL

Lp(y)dy+ t

= 0. (7)

(5) and (6) lead to (3).(7) leads to (4).


Notes

Notes

Notes

Lecture 2B: Alonso Model - University of Minnesota...

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