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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
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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?
© 2011 Hiroki Watanabe 3 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 4 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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
© 2011 Hiroki Watanabe 5 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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)?
© 2011 Hiroki Watanabe 6 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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}.
© 2011 Hiroki Watanabe 7 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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
© 2011 Hiroki Watanabe 8 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 9 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Cheesecake and Land
0 2 4 6 8 100
2
4
6
8
10
10
10
1010
20
20
20
20
30
30
30
40
40
40
50
50
60
60
70
70
80
90
Land s (ft2)
Com
posi
te G
oods
z (
bask
ets)
The Way Liz Finds (s*, z*)
Indifference CurvesBudget Constraint
© 2011 Hiroki Watanabe 10 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Assumptions
We assume that land is a normal good.
© 2011 Hiroki Watanabe 11 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Assumptions
00
Land s (ft2)
Com
posi
te G
oods
z (
bask
ets)
Normal Land Consumption
Budget ConstraintOptimal BundleIncreased Budget Constraint
© 2011 Hiroki Watanabe 12 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Assumptions
We also assume that a composite good is anuméraire.
i.e., pz = 1.Why can we do that?
1 Tangency condition2 Mobility
© 2011 Hiroki Watanabe 13 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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 14 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 15 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Landscape
30 Rock 1 3 4 8 S=9
xL xL+sL xK xK+sK
Linear City
Commuting distance (Liz)Commuting distance (Kenneth)
© 2011 Hiroki Watanabe 16 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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).
© 2011 Hiroki Watanabe 17 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
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).
© 2011 Hiroki Watanabe 18 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
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
© 2011 Hiroki Watanabe 19 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
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?
© 2011 Hiroki Watanabe 20 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
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
© 2011 Hiroki Watanabe 21 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
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
© 2011 Hiroki Watanabe 22 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
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.
© 2011 Hiroki Watanabe 23 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
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.
© 2011 Hiroki Watanabe 24 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
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 ).
© 2011 Hiroki Watanabe 25 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
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
© 2011 Hiroki Watanabe 26 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Feasible and Pareto Optimal Allocations
⇒ contradicts the claim that the original allocationis efficient.
© 2011 Hiroki Watanabe 27 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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 28 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Standard Edgeworth Box Doesn’t Work
How is the Alonso model represented in theEdgeworth box?
© 2011 Hiroki Watanabe 29 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Standard Edgeworth Box Doesn’t Work
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
© 2011 Hiroki Watanabe 30 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Standard Edgeworth Box Doesn’t Work
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
© 2011 Hiroki Watanabe 31 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Standard Edgeworth Box Doesn’t Work
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)
© 2011 Hiroki Watanabe 32 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Standard Edgeworth Box Doesn’t Work
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?
© 2011 Hiroki Watanabe 33 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 34 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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).
© 2011 Hiroki Watanabe 35 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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)
© 2011 Hiroki Watanabe 36 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 37 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Edgeworth Trapezoid
Added commuting cost alters her utility functionL(sL, zL) to:
L(sL, zL) ≡ L(sL, zL − t(S− sL)).
© 2011 Hiroki Watanabe 38 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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
© 2011 Hiroki Watanabe 39 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 40 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Edgeworth Trapezoid
Land sL (ft2)
Gro
ss C
onsu
mpt
ion
Leve
l zL (
bask
ets)
Liz′s Indifference Curve When xL>xK
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)
© 2011 Hiroki Watanabe 41 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Edgeworth Trapezoid
Observe that indifference curves are skewedupwarads in the following:
© 2011 Hiroki Watanabe 42 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Edgeworth Trapezoid
0 2 4 6 8 10 120
1
2
3
4
5
6
4
4
4
44
8
8
8
8
12
12
12
12
16
16
16
16
20
20
20
24
24
24
28
28
28
32
32
32
36
36
40
40
44
44
48
48
52
56
60
64
68
Land sL (ft2)
Com
posi
te G
oods
zL (
bask
ets)
Liz′s Indifference Curves (Aspatial)
Liz: uL(sL, zL)=sLzL
© 2011 Hiroki Watanabe 43 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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
15
1518
18
1821
21
2124
24
2427
27
2730
30
33
33
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)]
© 2011 Hiroki Watanabe 44 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Edgeworth Trapezoid
Confirm that any allocation in the followingtrapezoid is feasible (take sL = 6 for example).
© 2011 Hiroki Watanabe 45 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Edgeworth Trapezoid
N/A. zL<txL=t(S−sL)
uL(sL, zL−t(S−sL))=cL
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
© 2011 Hiroki Watanabe 46 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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 .
© 2011 Hiroki Watanabe 47 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Edgeworth Trapezoid
2 L < K .Same argument in reverse.
© 2011 Hiroki Watanabe 48 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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))
© 2011 Hiroki Watanabe 49 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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
1
2
3
4
5
6
© 2011 Hiroki Watanabe 50 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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 51 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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).
© 2011 Hiroki Watanabe 52 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
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
20
2030
30
30
40
4050
50
60
10
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
4
5
6
© 2011 Hiroki Watanabe 53 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
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).
© 2011 Hiroki Watanabe 54 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
0
0
0
0
10
10
10
10
20
20
20
30
30
40
40
50
60
10
10
10
10
20
20
20
30
30
30
40
40
50
50
60
Liz: uL(sL, zL−t(S−sL))
Kenneth: uK(sK, zK)Contract Curve
0
0
0
0
10
10
10
10
20
20
20
30
30
40
40
50
60
10
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
1
2
3
4
5
6
© 2011 Hiroki Watanabe 55 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
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.
© 2011 Hiroki Watanabe 56 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
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)
© 2011 Hiroki Watanabe 57 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
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
© 2011 Hiroki Watanabe 58 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
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?
© 2011 Hiroki Watanabe 59 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
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:
© 2011 Hiroki Watanabe 60 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
Land sL (ft2)
Gro
ss C
onsu
mpt
ion
Leve
l zL (
bask
ets)
Liz′s Indifference Curve When xL>xK
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)
© 2011 Hiroki Watanabe 61 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve in Alonso Economy
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)
© 2011 Hiroki Watanabe 62 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve ; Pareto Optimal
Some allocations on the contract may not beefficient.Recall Theorem 2.3 :
4 K (·) < L(·)⇒ K < L.
© 2011 Hiroki Watanabe 63 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve ; Pareto Optimal
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
© 2011 Hiroki Watanabe 64 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve ; Pareto Optimal
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
© 2011 Hiroki Watanabe 65 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve ; Pareto Optimal
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
© 2011 Hiroki Watanabe 66 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve ; Pareto Optimal
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
© 2011 Hiroki Watanabe 67 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Contract Curve ; Pareto Optimal
uL(sL, zL−t(S−sL))=c
uK(sK, zK)=c
uL(sL, zL)=c
uK(sK, zK−t(S−sK))=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
© 2011 Hiroki Watanabe 68 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Quasilinear Preferences
Example 4.2 (Quasilinear Preferences)
Consider quasilinear preferences represented by
L(sL, zL) =p
sL + zL.
© 2011 Hiroki Watanabe 69 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Quasilinear Preferences
1
2
2
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
8
8
8
9
1
2
2
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
8
8
8
9
Liz: uL(sL, zL)
Kenneth: uK(sK, zK)PO Allocations
0 2 4 6 8 10 120
1
2
3
4
5
6
1
2
2
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
8
8
8
9
1
2
2
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
8
8
8
9
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
© 2011 Hiroki Watanabe 70 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Quasilinear Preferences
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)|.
© 2011 Hiroki Watanabe 71 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Quasilinear Preferences
−1
0
1
1
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
7
7
7
8
8
9
1
2
2
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
8
8
8
9
Land sL (ft2)
Com
posi
te G
oods
zL (
bask
ets)
Contract Curve (Spatial, xL>xK)
0 2 4 6 8 10 120
1
2
3
4
5
6Liz: uL(sL, zL−t(S−sL))
Kenneth: uK(sK, zK)Contract Curve
© 2011 Hiroki Watanabe 72 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Quasilinear Preferences
Some allocations on the contract curve are notefficient.
© 2011 Hiroki Watanabe 73 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Quasilinear Preferences
−1
0
1
1
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
7
7
7
8
8
91
22
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
8
8
8
9
Liz: uL(sL, zL−t(S−sL))
Kenneth: uK(sK, zK)Contract Curve
−1
0
1
1
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
7
7
7
8
8
91
22
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
8
8
8
9
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
© 2011 Hiroki Watanabe 74 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Quasilinear Preferences
uL(sL, zL−t(S−sL))=c
uK(sK, zK)=c
PO Allocations (xL>xK)
uL(sL, zL)=c
uK(sK, zK−t(S−sK))=c
PO Allocations (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
© 2011 Hiroki Watanabe 75 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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 76 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 77 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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).
© 2011 Hiroki Watanabe 78 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 79 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 80 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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?)
© 2011 Hiroki Watanabe 81 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 82 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 83 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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).
© 2011 Hiroki Watanabe 84 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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
© 2011 Hiroki Watanabe 85 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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 .
© 2011 Hiroki Watanabe 86 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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∗).
© 2011 Hiroki Watanabe 87 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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∗).
© 2011 Hiroki Watanabe 88 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Equilibrium Rent
0 2 4 6 8 10 120
MRSL*
MRSK*
s′
Distance x from 30 Rock (ft)
Ren
t (ba
sket
s)
Price Density Function
Price Density Function p*(x)
MRSK(x, ζK(x, cK*))
© 2011 Hiroki Watanabe 89 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 90 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 91 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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∗).
© 2011 Hiroki Watanabe 92 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Equilibrium Rent
0 2 4 6 8 10 120
MRSL*
MRSK*
s′′
Distance x from 30 Rock (ft)
Ren
t (ba
sket
s)
Price Density Function
Price Density Function p*(x)
MRSL(x, ζL(x, cL*))
© 2011 Hiroki Watanabe 93 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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).
© 2011 Hiroki Watanabe 94 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 95 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Cobb-Douglas Utility
0 10 22 50 600
?
?
Distance x from 30 Rock (ft)
Ren
t (ba
sket
s)
Equilibrium Rent
Equilibrium Rent
© 2011 Hiroki Watanabe 96 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Cobb-Douglas Utility
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.
© 2011 Hiroki Watanabe 97 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Cobb-Douglas Utility
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.
© 2011 Hiroki Watanabe 98 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Cobb-Douglas Utility
Proposition 5.3 gives p(60) as follows:
p(60) = p(10)− t = 2.2.
© 2011 Hiroki Watanabe 99 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Cobb-Douglas Utility
0 10 22 50 600
2.2
3.2
Distance x from 30 Rock (ft)
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
© 2011 Hiroki Watanabe 100 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Cobb-Douglas Utility
Note nobody wants to relocate or change their lotsize under the prevalent equilibrium price density.
© 2011 Hiroki Watanabe 101 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Cobb-Douglas Utility
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.
© 2011 Hiroki Watanabe 102 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Cobb-Douglas Utility
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
© 2011 Hiroki Watanabe 103 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Example: Cobb-Douglas Utility
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.
© 2011 Hiroki Watanabe 104 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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 105 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Way the urban economists incorporate location.Allocations in Alonso model and graphicalpresentation.Disconnected contract curve and Pareto allocations.Equilibrium.
© 2011 Hiroki Watanabe 106 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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.
© 2011 Hiroki Watanabe 107 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
Map du Jour
Source http://www.commoncensus.org/
© 2011 Hiroki Watanabe 108 /110
Notes
Notes
Notes
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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
.
© 2011 Hiroki Watanabe 109 /110
Land Alonso Model Edgeworth Box Normative Analysis Positive Analysis
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).
© 2011 Hiroki Watanabe 110 /110
Notes
Notes
Notes