Lecture 2B: Alonso Model

Transcription

1 Econ Urban Economics Lecture B: Alonso Model Instructor: Hiroki Watanabe Spring Hiroki Watanabe / Land Consumption and Location Cheesecake and Land Assumptions Alonso Model Landscape Feasible and Pareto Optimal Allocations Edgeworth Box Standard Edgeworth Box Doesn t Work Normative Analysis Contract Curve in Alonso Economy Contract Curve Pareto Optimal Example: Quasilinear Preferences Positive Analysis Equilibrium Example: Cobb-Douglas Utility Summary Hiroki Watanabe / Cheesecake and Land Two models of residential location choice: Alonso model: discrete # of residents Monocentric city model (next lecture): city size N R. Question: Who lives where? Do we need to intervene to correct suboptimal use of land? Hiroki Watanabe /

2 Cheesecake and Land How are the land in the city and the land in the suburbs different from cheesecake and tea? Liz chooses the right amount of L = ( L C, L T ) where Marginal willingness to pay for a slice of cheesecake in terms of tea Marginal rate of substitution The slope of her indifference curve coincide with The relative price of a slice of cheesecake in terms of tea The opportunity cost of a slice of cheesecake in terms of tea The slope of her budget line Trinity meets trinity. Hiroki Watanabe / Cheesecake and Land Composite Goods z (baskets) The Way Liz Finds (s *, z * ) Indifference Curves Budget Constraint 9 Land s (ft ) Hiroki Watanabe / Cheesecake and Land Does Liz choose the land consumption in the city and the suburbs ( C, S ) in the same way? Recall Liz prefers ( C, T ) = (, ) over = (, ). What about ( C, S )? Hiroki Watanabe /

3 Cheesecake and Land We can enjoy tea and cheesecake at the same time. We cannot simultaneously occupy two houses at different locations. Land does not exhibit convexity. Consider an extreme example: ( C, S ) = max{ C, S }. Hiroki Watanabe / Cheesecake and Land Land in the Suburb x S (ft ) The Way Liz Finds (x C *, xs * ) / Land in the City x C (ft ) Utility Level u(x)=max{x, x } C S Budget Constraint x C +x S = Hiroki Watanabe / Cheesecake and Land We always get a corner solution ( C, ) or (, S ), which does not tell us so much about urban land use patterns. We can t take φc(pc,ps,m) for example. p S Instead, we take land and other commodities and analyze the location choice later. You can live in a house and eat cheesecake at the same time. You probably like a house and a cheesecake better than a very spacious house with no food. Hiroki Watanabe 9 /

4 Cheesecake and Land Composite Goods z (baskets) The Way Liz Finds (s *, z * ) Indifference Curves Budget Constraint 9 Land s (ft ) Hiroki Watanabe / Assumptions We assume that land is a normal good. Hiroki Watanabe / Assumptions Normal Land Consumption Composite Goods z (baskets) Budget Constraint Optimal Bundle Increased Budget Constraint Land s (ft ) Hiroki Watanabe /

5 Assumptions We also assume that a composite good is a numéraire. i.e., p z =. Why can we do that? Tangency condition Mobility Hiroki Watanabe / Land Consumption and Location Cheesecake and Land Assumptions Alonso Model Landscape Feasible and Pareto Optimal Allocations Edgeworth Box Standard Edgeworth Box Doesn t Work Normative Analysis Contract Curve in Alonso Economy Contract Curve Pareto Optimal Example: Quasilinear Preferences Positive Analysis Equilibrium Example: Cobb-Douglas Utility Summary Hiroki Watanabe / Landscape Cf. Alonso [Alo] and Arnott and McMillen [AM] Ch.. households: Liz and Kenneth. A narrow strip (-ft wide for example) of land [, S) constitutes the urban residential area (a linear city). Preferences are identical L ( ) = K ( ) = (s, z). s L, s K are land consumption. z L, z K are composite goods consumption. L, K are front or driveway location. They commute to the city center. Commuting cost is t (baskets/mile) measured from driveway. Hiroki Watanabe /

6 Landscape Linear City x L x L +s L x K x K +s K Commuting distance (Liz) Commuting distance (Kenneth) Rock S=9 Hiroki Watanabe / Feasible and Pareto Optimal Allocations Definition. (Feasible Allocation) An allocation is a list (s L, s K, z L, s K, L, K ). An allocation (s L, s K, z L, s K, L, K ) is called feasible if z L + z K + t L + t K Z L, L + s L K, K + s K = L, L + s L K, K + s K = [, S). Hiroki Watanabe / Feasible and Pareto Optimal Allocations Note that all the urban area needs to be distributed to be feasible. Otherwise some lot will be left unoccupied without being priced (a waste dump, for example). Hiroki Watanabe /

7 Feasible and Pareto Optimal Allocations Question. (Pareto Optimal Allocation) Which one is more efficient? (Assume z K = zk ). Liz Kenneth Allocation m Allocation m Rock m_ m_ S Hiroki Watanabe 9 / Feasible and Pareto Optimal Allocations Does the allocation leave both of them at least as well off as before and make at least one of them better off than the allocation? Hiroki Watanabe / Feasible and Pareto Optimal Allocations Kenneth (consumption bundle adjusted to have the same utility level): allocation land consumption level s K s K s K (= sk ) commuting cost t K tm baskets (after transfer) z K baskets z K z K K (s K, zk ) = ck z K + tm z K = zk + tm tm K (s K, zk ) = ck Hiroki Watanabe /

8 Feasible and Pareto Optimal Allocations Liz: allocation land consumption level s L s L s L (= sl ) commuting cost t L tm baskets (after transfer) z L baskets z L z L L (s L, zl ) = cl z L tm z L = zl tm + tm L (s L, zl ) > cl Hiroki Watanabe / Feasible and Pareto Optimal Allocations Utility level: They are both at least as well off as before. Liz is better off. Conclude: allocation Pareto dominates allocation. Note that we cannot compare allocation and without transfer. Hiroki Watanabe / Feasible and Pareto Optimal Allocations Generalize the observation above as follows: Theorem. (Pareto Optima (Berliant and Fujita [BF9])) At an efficient allocation (s L, s K, z L, z K, L, K ), K < L s K < s L. K < L K (s K, z K ) L (s L, z L ). K ( ) < L ( ) K < L. K ( ) < L ( ) m K < m L, where m X denotes income level. Hiroki Watanabe /

9 Feasible and Pareto Optimal Allocations Sketch of the proof: Suppose K < L but s K s L. If they switch their positions as follows, there will be extra baskets due to reduced commuting cost: Liz Kenneth before t( K + s K ) t K after t K t ( K + s L ) differential ts K ts L s L transfer αt(s K s L ) +αt(s K s L ) ( α sk s K s L s K s L ). Hiroki Watanabe / Feasible and Pareto Optimal Allocations Composite goods (baskets) Pareto Improvement Liz: u L (z L, s L )=c L Kenneth: u K (z K, s K )=c K ts K αt(s K s L ) ts K +α(s K s L ) Land s (ft ) Hiroki Watanabe / Feasible and Pareto Optimal Allocations contradicts the claim that the original allocation is efficient. Hiroki Watanabe /

10 Land Consumption and Location Cheesecake and Land Assumptions Alonso Model Landscape Feasible and Pareto Optimal Allocations Edgeworth Box Standard Edgeworth Box Doesn t Work Normative Analysis Contract Curve in Alonso Economy Contract Curve Pareto Optimal Example: Quasilinear Preferences Positive Analysis Equilibrium Example: Cobb-Douglas Utility Summary Hiroki Watanabe / Standard Edgeworth Box Doesn t Work How is the Alonso model represented in the Edgeworth box? Hiroki Watanabe 9 / Standard Edgeworth Box Doesn t Work Liz s Indifference Curve Liz: u L (s L, z L )=c L Composite Goods z L (baskets) Land s L (ft ) Hiroki Watanabe /

11 Standard Edgeworth Box Doesn t Work Land s K (ft ) Composite Goods z K (baskets) Kenneth: u K (s K, z K )=c K Kenneth s Indifference Curve Hiroki Watanabe / Standard Edgeworth Box Doesn t Work Composite Goods z L (baskets) Land s K (ft ) Liz: u(x L C, xl T )=cl Kenneth: u(x K C, xk T )=ck Composite Goods z K (baskets) Land s L (ft ) Hiroki Watanabe / Standard Edgeworth Box Doesn t Work This is not an Edgeworth box. Some of the allocations in the box is not feasible. Suppose t = / and consider an allocation (s L, s K, z L, s K, L, K ) = (,,,,, ). There are Z = baskets in total. baskets are allocated as follows: z L + z K + t L + t K = > Z. The previous box ignores the location (i.e., it represents an aspatial economy). So, give up the Edgeworth box altogether? Hiroki Watanabe /

12 Don t even. Slice off some part to make the Edgeworth box a trapezoid. Consider the following two cases: L > K : Liz lives farther away from Rock. K > L : Kenneth lives farther away from Rock. Hiroki Watanabe / L > K. If Liz consumes s L, she has to spend t L = ts K = t(s s L ) on commuting. i.e., t(s s L ) is deducted from her basket. (s L, z L ) with z L < t(s s L ) is not affordable (otherwise she won t be able to commute). Hiroki Watanabe / Liz s Consumption Set When x L >x K Not feasible z L <tx L =t(s s L ) Composite Goods z L (baskets) Land s L (ft ) Hiroki Watanabe /

13 Define net composite good ẑ L by }{{} ẑ L := }{{} z L t(s s L ). net consumption gross consumption t(s s L ) is part of her basket z L (gross consumption) but it is not for her to consume. Liz s net consumption is smaller than her gross consumption level if L >. Hiroki Watanabe / Added commuting cost alters her utility function L (s L, z L ) to: L (s L, ẑ L ) L (s L, z L t(s s L )). Hiroki Watanabe / Liz s Indifference Curve When x L >x K Liz: u L (s L, ẑ L ) = c L Net Consmption Level ẑ L (baskets) Land s L (ft ) Hiroki Watanabe 9 /

14 And this was the last time you see anything measured in net consumption level ẑ L = z L t(s s L ) on a graph. In what follows, everything s measured in gross consumption level z L on a graph. I.e., Liz s indifference curve will be shifted upwards. Hiroki Watanabe / Gross Consumption Level z L (baskets) Liz s Indifference Curve When x L >x K u L (s L, z L )=c L (Aspatial) u L (s L, z L t(s s L ))=c L N/A. z T <tx L =t(s s L ) Land s L (ft ) Hiroki Watanabe / Observe that indifference curves are skewed upwarads in the following: Hiroki Watanabe /

15 Liz s Indifference Curves (Aspatial) Composite Goods z L (baskets) Liz: u L (s L, z L )=s L z L Land s L (ft ) Hiroki Watanabe / Liz s Indifference Curves (Spatial).e.e e.e Composite Goods z L (baskets) Liz: u L (s L, z L t(s s L ))=s L [z L t(s s L )] Land s L (ft ) Hiroki Watanabe / Confirm that any allocation in the following trapezoid is feasible (take s L = for example). Hiroki Watanabe /

16 Composite Goods z L (baskets) Land s K (ft ) N/A. z L <tx L =t(s s L ) u L (s L, z L t(s s L ))=c L u K (s K, z K )=c K Composite Goods z K (baskets) Land s L (ft ) Hiroki Watanabe / Note that we do not have to shift Kenneth s indifference curves (why?) Feasibility of baskets becomes: In gross terms z L + z K ẑ L + t(s s L ) + z K = Z Ẑ + t(s s L ) In net terms ẑ L + z K z L t(s s L ) + z K = Z t(s s L ) Ẑ when L > K. Hiroki Watanabe / L < K. Same argument in reverse. Hiroki Watanabe /

17 Land s K (ft ) N/A. z K <tx K =t(s s K )) u K (s K, z K )=c K (Aspatial) u K (s K, z K t(s s K ))=c K Kenneth s Indifference Curve When x K >x L Hiroki Watanabe 9 / Composite Goods z K (baskets) Land s K (ft ) Composite Goods z K (baskets) N/A. z K <tx K =t(s s K ) u K (s K, z K t(s s K ))=c K u L (s L, z L )=c L Land s K (ft ) Hiroki Watanabe / Composite Goods z K (baskets) Land Consumption and Location Cheesecake and Land Assumptions Alonso Model Landscape Feasible and Pareto Optimal Allocations Edgeworth Box Standard Edgeworth Box Doesn t Work Normative Analysis Contract Curve in Alonso Economy Contract Curve Pareto Optimal Example: Quasilinear Preferences Positive Analysis Equilibrium Example: Cobb-Douglas Utility Summary Hiroki Watanabe /

18 Contract Curve in Alonso Economy How do we find the efficient allocations on the Edgeworth trapezoid? Recall in an aspatial economy, the contract curve satisfies: MRS L (sl, z L ) = MRS K (S s L, Z z L ). Hiroki Watanabe / Contract Curve in Alonso Economy Composite Goods z L (baskets) Land s K (ft ) Liz: u L (s L, z L ) Kenneth: u K (s K, z K ) Contract Curve Composite Goods z K (baskets) Land s L (ft ) Hiroki Watanabe / Contract Curve in Alonso Economy In the spatial economy with L > K, the contract curve satisfies: MRS L (sl, ẑ L ) = MRS K (s K, z K ) MRS L s L, z L t(s s L ) = MRS K (S s L, Z z L ). Hiroki Watanabe /

19 Contract Curve in Alonso Economy Composite Goods z L (baskets) Land s K (ft ) Liz: u L (s L, z L t(s s L )) Kenneth: u K (s K, z K ) Contract Curve Composite Goods z K (baskets) Land s L (ft ) Hiroki Watanabe / Contract Curve in Alonso Economy What does MRS L ( ) look like? To begin with, we need to find: ( in parentheses indicates that the variable is fixed) L (s L, ẑ L ) = L (sl, ) + L (sl, ẑl ) ẑl s L s L z L s L = L (s L, ) + L (sl, ẑl ) [zl t(s sl )] s L z L s L = L (s L, ) s L + L (sl, ẑl ) z L t. Hiroki Watanabe / Contract Curve in Alonso Economy Then, MRS L (sl, ẑ L ) = L (s L, ẑ L )/ s L L (s L, ẑ L )/ z L = L (s L, )/ s L L (s L, ẑ L )/ z L = MRS L (sl, ẑ L ) t. + L (sl, ẑl )/ zl ẑl L (s L, ẑ L )/ z L s L () Hiroki Watanabe /

20 Contract Curve in Alonso Economy Question. (Tangency Condition in Alonso Economy) What does () mean? MRS L (sl, ẑ L ) = MRS L (sl, ẑ L ) t To make things easy, write everything in positive terms: MRS L (sl, ẑ L ) = MRS L (sl, ẑ L ) t = MRS L (sl, ẑ L ) +t Hiroki Watanabe / Contract Curve in Alonso Economy Aspatial Liz (or Liz ) is willing to reduce s L by one unit if she gains (or compensated with) MRS L (sl, z L ) baskets in return. Spatial Liz (or Liz ) is willing to reduce s L by one unit if she gains MRS L (sl, ẑ L ) +t baskets in return. Why does she need extra t baskets for compensation? Hiroki Watanabe 9 / Contract Curve in Alonso Economy Liz is willing to reduce s L by one unit if she gains (or compensated with) MRS L (sl, z L ) baskets in return. Liz is willing to reduce s L by one unit if she gains MRS L (sl, ẑ L ) +t baskets in return. Why does she need extra t baskets for compensation? t L = ts K = t(s s L ) grows as s L gets smaller. Consider a change from s L = to s L = in the following graph: Hiroki Watanabe /

21 Contract Curve in Alonso Economy Gross Consumption Level z L (baskets) Liz s Indifference Curve When x L >x K u L (s L, z L )=c L (Aspatial) u L (s L, z L t(s s L ))=c L N/A. z T <tx L =t(s s L ) Land s L (ft ) Hiroki Watanabe / Contract Curve in Alonso Economy In conclusion, an allocation (s L, s K, z L, z K ) is on the contract curve if MRS L (s L, z L ) + t = MRS K (s K, z K ) when L > K MRS L (s L, z L ) = MRS K (s K, z K ) + t when L K. () Hiroki Watanabe / Contract Curve Pareto Optimal Some allocations on the contract may not be efficient. Recall Theorem. : K ( ) < L ( ) K < L. Hiroki Watanabe /

22 Contract Curve Pareto Optimal Composite Goods z L (baskets) Land s K (ft ) u L (s L, z L t(s s L ))=c u K (s K, z K )=c Contract Curve (PO) Contract Curve (non PO) Composite Goods z K (baskets) Land s L (ft ) Hiroki Watanabe / Contract Curve Pareto Optimal Land s K (ft ) Composite Goods z L (baskets) u K (s K, z K t(s s K ))=c u L (s L, z L )=c Contract Curve (PO) Contract Curve (non PO) Land s L (ft ) Hiroki Watanabe / Composite Goods z K (baskets) Contract Curve Pareto Optimal Composite Goods z L (baskets) Land s K (ft ) u K (s K, z K t(s s K ))=c u L (s L, z L )=c Contract Curve (PO) Contract Curve (non PO) Composite Goods z K (baskets) Land s L (ft ) Hiroki Watanabe /

23 Contract Curve Pareto Optimal Composite Goods z L (baskets) Land s K (ft ) u L (s L, z L ))=c u K (s K, z K t(s s K ))=c Contract Curve (PO) Contract Curve (non PO) Composite Goods z K (baskets) Land s L (ft ) Hiroki Watanabe / Contract Curve Pareto Optimal Composite Goods z L (baskets) Land s K (ft ) u L (s L, z L t(s s L ))=c u K (s K, z K )=c u L (s L, z L )=c u K (s K, z K t(s s K ))=c PO (x L >x K ) PO (x L <x K ) Composite Goods z K (baskets) Land s L (ft ) Hiroki Watanabe / Example: Quasilinear Preferences Example. (Quasilinear Preferences) Consider quasilinear preferences represented by L (s L, z L ) = s L + z L. Hiroki Watanabe 9 /

24 Example: Quasilinear Preferences Composite Goods z L (baskets) Land s K (ft ) Liz: u L (s L, z L ) 9 Kenneth: u K (s K, z K ) PO Allocations Land s L (ft ) Hiroki Watanabe / 9 Composite Goods z K (baskets) Example: Quasilinear Preferences If L > K, L (s L, ẑ L ) = s L + ẑ L s L + z L t(s s L ). Allocations on the contract curve satisfies: MRS L (s L, ẑ L ) = MRS K (s K, z K ) MRS L s L, z L t(s s L ) +t = MRS K (S s L, Z z L ). Hiroki Watanabe / Example: Quasilinear Preferences Composite Goods z L (baskets) 9 Contract Curve (Spatial, x L >x K ) Liz: u L (s L, z L t(s s L )) Kenneth: u K (s K, z K ) Contract Curve Land s L (ft ) Hiroki Watanabe / 9

25 Example: Quasilinear Preferences Some allocations on the contract curve are not efficient. Hiroki Watanabe / Example: Quasilinear Preferences Composite Goods z L (baskets) Land s K (ft ) Liz: u L (s L, z L t(s s L )) 9 Kenneth: u K (s K, z K ) Contract Curve Land s L (ft ) Hiroki Watanabe / 9 Composite Goods z K (baskets) Example: Quasilinear Preferences Land s K (ft ) Composite Goods z L (baskets) u L (s L, z L t(s s L ))=c u K (s K, z K )=c PO Allocations (x L >x K ) u L (s L, z L )=c u K (s K, z K t(s s K ))=c PO Allocations (x L <x K ) Land s L (ft ) Hiroki Watanabe / Composite Goods z K (baskets)

26 Land Consumption and Location Cheesecake and Land Assumptions Alonso Model Landscape Feasible and Pareto Optimal Allocations Edgeworth Box Standard Edgeworth Box Doesn t Work Normative Analysis Contract Curve in Alonso Economy Contract Curve Pareto Optimal Example: Quasilinear Preferences Positive Analysis Equilibrium Example: Cobb-Douglas Utility Summary Hiroki Watanabe / Equilibrium ω L, ω K are endowment of composite good. ω L + ω K = Z. Donaghy Real Estate (the absentee landlord) is endowed with [, S). Their utility is given by D (s D, z D ) = z D. Hiroki Watanabe / Equilibrium Definition. (Feasible Allocation (with Jack Donaghy)) A feasible allocation is a list (s L, s K, z L, z K, L, K, z D ) such that z L + z K + z D + t L + t K = Z [ L, L + s L ) [ K, K + s K ) = [ L, L + s L ) [ K, K + s K ) = [, S). Hiroki Watanabe /

27 Equilibrium Definition. (Equilibrium) An equilibrium is a feasible allocation (s L, s K, z L, z K, L, K, z D ) and a price density p(y) such that The bundle (s L, z L ) solves max L,s L,z L L (s L, z L ) subject to ω L z L + t L + L +s L p(y)dy. L Analogous condition for Kenneth. Hiroki Watanabe 9 / Equilibrium In equilibrium, Liz satisfies (See Appendix ). MRS L (s L, z L ) = p( L + s L ) () p( L ) = p( L + s L ) + t. () Eastbound () Liz s willingness to pay for an additional sq ft of land at the back of her lot is equal to the cost of obtaining it in equilibrium. Westbound () Adding one more sq ft to the front of her lot should cost exactly t baskets more than adding one more sq ft to the back. If not, she will move forward to save on commuting cost. Hiroki Watanabe / Equilibrium Unfortunately, we can t draw the Edgeworth box in this 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?) Hiroki Watanabe /

28 Suppose K < L in equilibrium. What does p( ) look like? We know K = and L = s K. Then () implies And () implies p(s K ) = MRS K (s K, z K ) p( L + s L = S) = MRS L (s L, z L ). p( K = ) = p(s K ) + t p( L = s K ) = p(s) + t. Hiroki Watanabe / In conclusion, Proposition. () In an equilibrium with K < L, equilibrium price density functions satisfy p( L ) = MRS K (s K, z K ) p(s) = MRS L (s L, z L ) p( L ) = p(s) + t. Hiroki Watanabe / Then how about this? MRS K (s K, z K ) for < L p( ) = MRS L (s L, z L ) for L < S. (Note p() can be p(s K ) + t but = is measure zero). Hiroki Watanabe /

29 Price Density Function Proposed Price Density Function p(x) Differential = t Rent (baskets) MRSK* MRSL* Distance x from Rock (ft) Hiroki Watanabe / This p( ) actually won t constitute an equilibrium. Since at = L, MRS K (s K, z K ) > p( ), Kenneth has an incentive to expand his lot further to the east. p( ) has to be such that beyond = L, Kenneth doesn t want to increase s K, i.e., if p( ) is higher than MRS K (, z K ), he won t increase s K. Hiroki Watanabe / Let c K := K (s K, z K ) and ζ K (s K, c K ) be the number of baskets Kenneth has to get to maintain K ( ) = c K while consuming s K, i.e., K (s K, ζ K (s K, c K )) = c K. (Or, to put it in another way, (s K, ζ K (s K, c K )) traces the indifference curve at c K ). Hiroki Watanabe /

30 Kenneth doesn t want to expand s K as long as p( ) is higher than his MRS K (s K, z K ) beyond L. Like the following for example: MRS K (s K, z K ) for < s K p ( ) = MRS (, ζ K (, c K )) for s K < s MRS L (s L, z L ) for s < S, where s is a location such that MRS K (s, ζ K (s, c K )) = MRS L (s L, z L ). Hiroki Watanabe / Price Density Function Price Density Function p * (x) MRS K (x, ζ K (x, c K* )) Rent (baskets) MRSK* MRSL* s Distance x from Rock (ft) Hiroki Watanabe 9 / Liz won t want to shift her lot towards the west (i.e., reduce L while maintaining the lot size s L ) if p( ) = MRS L (s L, z L ) + t. Increased rent p( ) exactly offsets the savings from reduced commuting cost t. Hiroki Watanabe 9 /

31 Once L reaches = then she won t want to reduce her lot size s L if p( ) MRS L (, ζ L (, c L )). If p( ) > MRS L (, ζ L (c L, )) at = s L, then she WILL sell her lot at the east end to receive p( ). She only needs MRS L (, ζ L (c L, )) to stay as well off as before but she gets more than that (p( )) by selling a lot. it s not an equilibrium. Hiroki Watanabe 9 / The following works as an equilibrium price density function for example: MRS K (s K, z K ) for < s p ( ) = MRS (, z L (c L, )) for s < s L MRS L (s L, z L ) for s L < S, where s is a location such that MRS L (s, z L (c L, s )) = MRS K (s K, z K ). Hiroki Watanabe 9 / Price Density Function Price Density Function p * (x) MRS L (x, ζ L (x, c L* )) Rent (baskets) MRSK* MRSL* s Distance x from Rock (ft) Hiroki Watanabe 9 /

32 So, there is a continuum of equilibrium. Which one is most favorable to Donaghy Real Estate? Note D (s D, z D ) = z D = S p( )d. Note also: equilibrium allocations are efficient. Compare () and () to (). Hiroki Watanabe 9 / Example: Cobb-Douglas Utility Example. () Consider the following economy: Preferences are represented by L (s L, z L ) = log(s L ) + log(z L ) K (s K, z K ) = log(s K ) + log(z K ). S =, of which is s L and of which is s K. Z =, of which is ω L and of which is ω K. The observed equilibrium price density function p( ) is piecewise linear (see next slide). t =. Hiroki Watanabe 9 / Example: Cobb-Douglas Utility? Rent (baskets)? Distance x from Rock (ft) Hiroki Watanabe 9 /

33 Example: Cobb-Douglas Utility Question. () Find p() and p(). Find z L. How many baskets does Jack get? Find the equilibrium. Hiroki Watanabe 9 / Example: Cobb-Douglas Utility Find p() and p(). First of all, who lives close to Rock? Recall Theorem.. From Proposition., p(s K = ) = MRS K (s K, z K ). z K = ω K s K p()d = p(). Note MRS K (s K, z K ) = zk s K. Then p() = MRS K s K, ω K s K p() p() = p() =.. Hiroki Watanabe 9 / Example: Cobb-Douglas Utility Proposition. gives p() as follows: p() = p() t =.. Hiroki Watanabe 99 /

34 Example: Cobb-Douglas Utility. Rent (baskets). MRS K (x, ζ K (x, c K* ))=(/x)/x MRS L (x, ζ L (x, c L* ))=(/x)/x Distance x from Rock (ft) Hiroki Watanabe / Example: Cobb-Douglas Utility Note nobody wants to relocate or change their lot size under the prevalent equilibrium price density. Hiroki Watanabe / Example: Cobb-Douglas Utility Find z L. Liz pays S p( )d = +. = L baskets in rent. Therefore, S z L = ω L p( )d t L = =. L Hiroki Watanabe /

35 Example: Cobb-Douglas Utility How many baskets does Jack get? Jack collects z D =. + =. Then z L + z K + z D = + + = 9(< Z). Where did the remaining baskets go? Definition. Hiroki Watanabe / Example: Cobb-Douglas Utility Find the equilibrium. The equilibrium is (s L, s K, z L, z K, L, z K, z D ) = (,,,,,, ), and. if < p( ) = + if <. if <. Hiroki Watanabe / Land Consumption and Location Cheesecake and Land Assumptions Alonso Model Landscape Feasible and Pareto Optimal Allocations Edgeworth Box Standard Edgeworth Box Doesn t Work Normative Analysis Contract Curve in Alonso Economy Contract Curve Pareto Optimal Example: Quasilinear Preferences Positive Analysis Equilibrium Example: Cobb-Douglas Utility Summary Hiroki Watanabe /

36 Way the urban economists incorporate location. Allocations in Alonso model and graphical presentation. Disconnected contract curve and Pareto allocations. Equilibrium. Hiroki Watanabe / References William Alonso. Location and Land Use. Harvard University Press, 9. Richard J. Arnott and Daniel P. McMillen. A Companion to Urban Economics. Blackwell,. Marcus Berliant and Masahisa Fujita. Alonso s discrete population model of land use: efficient allocations and competitive equilibria. International Economic Review, 99. Hiroki Watanabe / Map du Jour Source Hiroki Watanabe /

37 Appendix Liz solves max s L,z L L (s L, z L ) s.t. ω L z L + Lagrangian L +s L L p(y)dy+t L. L +s L L L := L ( ) + λ L ω L z L p(y)dy t L. L Hiroki Watanabe 9 / The first order conditions are: L L s = L L s λl s L L +s L () and () lead to (). () leads to (). p(y)dy = L s λl p( L + s L ) = () L L L z = L L z λl = () L L L +s L = L λl p(y)dy + t =. () L Hiroki Watanabe /