A LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES. 1. Introduction

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1 A LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ Abstract. In tis work we introduce a new land use equilibrium model wic is based on a goods and location excange process wit endogenous incomes. We define two possible market equilibrium conditions: one based on te auction teory and te oter on te utility maximization teory. We prove in te second part of tis paper tat bot approaces are equivalent under weak conditions. In te tird part we establis an existence result for te utility maximization equilibrium model. 1. Introduction In te istoric development of Land Use modelling, tere ave been two main approaces to understanding te economic agents decision making on location and te allocation of residential resources in an urban enronment. In te auction teory, described by Von Tünen (1966) and Alonso (1964), agents compete for land use wit teir willingness to pay. On te oter and, in te utility maximization approac, agents coose teir locations by maximizing teir utility given te set of location prices (McFadden, 1978; Anas, 1982). Bot approaces seek equilibrium prices tat allow all agents to locate following te corresponding process. In Martínez (1992) apparent differences between tese two approaces were acknowledged, and te equivalence between tem was discussed. Using linear utility functions, it was argued tat te two approaces produce te same distribution of ouseolds in space, and are complementary because tey prode complementary information on te land market clearing mecanism and indidual beaor. It was demonstrated tat troug te auction process, eac located agent is in fact maximizing is utility in equilibrium, owever, tere is no demonstration tat location based on maximum utility is consistent wit te auction process. As tere is no actual demonstration of tis equivalence, and as te one-sided demonstration was given only in te case of linear indirect utility functions, in tis work we formalize and generalize tis equivalency to te case of classical properties of direct utility functions. For tis purpose, it is necessary to compare bot equilibrium conditions defined in te same general economic context. In Section 2 a new economic framework for residential location models is introduced tat is flexible enoug to sustain te two approaces in land use modelling. We will define an Excange Economy wit Location. model wic describes an excange economy attaining general equilibrium on continuous goods and on te discrete 1

2 2 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ coice of residence location. Te equilibrium can be defined, following te two main frameworks in urban economics, as purely maximum utility equilibrium or as maximum bidding auction equilibrium ; an equivalence teorem merges tem into a consistent framework. Location attributes are defined eiter by te natural enronment or by te location pattern and te built enronment. Wile te former is exogenous to te urban market, te latter introduce a number of endogenous effects on te economy, generically called location externalities ; te best known are neigborood quality and agglomeration economies. Te interaction between agents is twofold: troug ouseold income tat depends on residences rents, defined by auction outputs, and troug location externalities tat define location attributes based on te result of te equilibrium location pattern. Te residential land use market offers an interesting setting to study an economy endowed wit an uncommon combination of features tat callenge state- of-te-art market equilibrium models. An additional feature differentiating tis model from oter land use models is tat in tis context we will seek equilibrium not only in te residential land use market, but also in consumption goods. Te set of goods includes te usual bundle of consumption goods and te residence; te former commonly considered as continuous, but location of residence introduces te case of discrete coices and te need for a discrete coice matematical framework. Additionally, ouseolds demand variable amounts of goods but only one residence. Te residential options are differentiated naturally by te difference in attributes defined by te spatial location, making eac residential option attribute unique. Tis feature calls for a trading process based on bid-auctions rules, as as been recognized in te urban economic literature since Alonso (1964). Te property of ouses is distributed among consumers, wit eac ouseold potentially playing te double role of consumer and owner simultaneously. Tis implies tat consumer income is naturally endogenous to te land use market equilibrium. Te urban economic model presented in tis paper takes into account all tese features assuming te ousing supply to be exogenous. Given te ig complexity added to modelling wit all te new features mentioned, only a static version of te model will be advanced ere suc tat te agent interaction a location externalities as been simplified, assuming tat tese externalities appen over time in a dynamic extension of te model. In tis case, agents only use te information about ow te urban enronment is at te moment tey make te location decision, but no inference about ow tis enronment may be in te future is considered. Neverteless, te teoretical economic framework proposed as te tools necessary to deal wit location externalities. In Section 3, wat is meant by using eac of te two approaces for our model is defined. To do tis, indirect utility and willingness to pay functions of ouseolds are introduced, wic lead to a formal definition of equilibrium regarding te two approaces. In Section 4 te conditions under wic equivalence properties of equilibrium points can effectively occur are stated, merging te two approaces into a consistent framework by an equivalence teorem for more general utility and willingness to pay functions. Finally, in Section 5, a result for te existence of equilibrium is demonstrated.

3 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 3 2. Te Model We begin by presenting our location model, in wic te property of all location options and available goods is distributed among te participating agents. Te model is based on an excange economy wit a finite number of goods, one of wic is location. Agents endogenous income consists of te value of teir endowment of goods and location possessions. Agents excange goods and decide were to locate. Consider a finite number of consumers denoted by { 1,..., }. Eac consumer is representative of a omogeneous cluster of agents, wic is a partition of total locating agents wit size N. In tis economy tere is a finite number of available location options partitioned into categories by building type v {1,..., v} and by zone i {1,..., ī} wit size f. Eac option is completely caracterized by a vector of attributes z IR a, were a is te number of attributes, including, among oters, te land lot size, a set of building caracteristics, index(es) of neigborood quality, accessibility and attractiveness advantages. Land use is described by a matrix H IR v ī + wose value for eac, H, indicates te number of agents of eac cluster located at. Some attributes describing dwelling caracteristics and te natural enronment are independent of te land use pattern, wile oters, like socioeconomic segregation, index of criminal records, access to local and city-wide serces, etc., are conditioned to te built enronment and so are dependent on te land use H. Terefore, te vector described above is actually: z = z (H) Tat is, te relevant information on location decisions, related to city structure, is considered in te attributes vector z. Houseolds decide teir consumption of goods and were to locate. Te total number of goods is l + 1. For an agent { 1,..., } te consumption quantities of te first l goods are given by a vector x IR l and for te location indexed by l + 1 te consumption quantity is equal to one. Tus, te consumption of an agent is a vector (x, x l+1 ) = (x 1,..., x j,..., x l, ) were te last term indicates agent s coice on te discrete set of location options, V := {1,..., v} {1,..., ī}. Te location of firms tat prode goods and serces is also affected by H, ence te space of feasible consumption for any locating agent in any zone i is also affected by H. Tis effect is modeled by te set-valued map X : IR + v ī IR +, l suc tat to eac tere is an associated set of feasible consumption X (H), assumed ereafter convex and closed, wic also includes tose goods tat are not affected by H and does not include location. Vector x can be described furter as te set of actities performed by te agent, including goods, serces and time. Te set-valued map X could include dependencies suc as time, spatial and/or tecnological constraints, among oters. Tis context would ten include te specifications described by Jara-Díaz and Martínez (1999) for te problem of te consumer tat faces tese kinds of decisions.

4 4 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ In contrast to te classical model, endowment is split into two components. One associated wit te property of dwelling, denoted w [0, 1], and te oter associated wit all oter goods, denoted e IR ++, l tat is independent of location. Te term w gives te fraction of te value of a location, type, tat an agent of type obtains. We suppose tat eac location is completely owned by a set of participating agents so we must ave: w = 1. =1 Tese components are included in te ouseold budget constraint. Agent s income, wic is given by te value of is endowment relative to a pair of price vectors (p, r), is ten p e + r w, wic can be spent in a vector of goods x and location, given tat x IR + l and tat te value p x + r does not exceed tat income. Agent s consumption must be in te set: C (p, r) := { (y, ) IR l + V : p y + r p e + r w }. Following Discrete Coice Teory - were agents are modelled as deciding in two steps: first, conditional on location, tey decide teir consumption of te first l goods; ten, tey decide location. Terefore, in te first step anoter consumption set were location is fixed must be defined: Definition 1. We define te conditional on location Budget Set, C (p, r) IR l +, as: (1) C (p, r) := { y IR l + : p y + r p e + r w } = { y IR l + : p y p e + r w r (1 w ) }, were r is te usual notation for te price vector on all locations except for and, analogously, w is te location endowment vector of agent on all locations except for is endowment on. Eac ouseold as a utility function U : IR + l IR a IR tat depends on a vector of consumption goods x IR +, l wic does not include location, and te above mentioned vector of attributes z IR a. It is assumed tat agent prefers (strictly) a pair (x, ) IR + l V to (y, v i ) IR + l V if and only if U (x, z ) (>) U (y, z v i ). Ten, tis utility function represents a rational preference relation over IR + l V. As we are interested in following a Discrete Coice approac, we will study preferences conditional on location. In tis context, ten, we suppose tat utility functions U (, z ) are upper semi-continuous, quasi-concave, locally non-satiated and non-decreasing on eac coordinate x j of x, for any pair V. Using tese elements, te main framework for our model can be defined. Definition 2. An Excange Economy wit Location wit IN clusters of agents, v IN types of location, in a city wit ī IN zones, is denoted: (U, e, w, X ) =1, (z ) V, were U : IR l + IR a IR, X : IR v ī + IR l +, z : IR v ī + IR a, e IR l ++, w [0, 1] v ī. Eac cluster () of agents (locations) as size N (f ).

5 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 5 3. Equilibria Given an Excange Economy wit Location and a city structure H 0, we can obtain indirect utility, willingness to pay functions, and consumption correspondences for eac agent conditional on every location. Tese tools will be used to get a new city structure were location can be determined by eiter of two rules: 1. Te price of dwellings is defined as agents bid over all available location alternatives and an auction determines te location structure by te best bidder rule. 2. Given te prices of goods and dwellings, agents decide were to locate by coosing teir utility maximizing consumption and location alternative. In equilibrium tere must be a market-clearing condition on goods and land use. Tat is, in equilibrium, every agent must be located and demand on consumption must be met Functions and Correspondences Indirect Utility Functions. Given p IR l + and r IR v ī +, agent cooses a point d IR l + tat maximizes is utility on eac location. Ten d must be in te conditional on location Budget Set C (p, r) defined on (1) and in X (H). We define ten te Utility Maximization Problem, conditional on location. Definition 3. We call te Utility Maximization Problem, conditional on location, denoted (P ), te following: { (P ) sup x IR l U (x, z (H)) : x C (p, r) X (H) } Te value of (P ) is a function tat, to eac term (, p, r), associates: (2) V (p, r, z (H)) := sup x IR l { U (x, z (H)) : x C (p, r) X (H) }, and we call it te Indirect Utility Function, conditional on location. Te Utility Maximizing Consumption Correspondence is te set: { (3) D (p, r, z (H)) := argmax U (x, z (H)) : x C (p, r) X (H) } x IR l Remark 1. Te Indirect Utility Function, conditional on location, for agent, is te greatest utility tat can reac on given prices (p, r) and te city structure H. It is important to note tat dependence on comes not only from (can we omit tis:evaluating) te direct utility function U in z, but also from te budget constraint tat defines te set C (p, r) (see equation (1)). 1 Usually equilibrium conditions include not only tat every agent locates, but also tat all dwellings be occupied. In tis work we assume tat te ousing supply is fixed, so tere is no point in asking for optimal beaor on building prosion. Instead, and to allow consistency, we assume tat total ousing is equal to total locating agents.

6 6 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ Willingness to Pay Functions. Willingness to Pay Functions describe ow muc an agent is willing to pay for a certain location to obtain a specific utility level u given a vector of goods and location (but ) prices. In our context, in contrast to te Bid-Coice model (Martínez, 1992) in wic income is fixed and consumption goods are not considered, an optimal consumption correspondence will be obtained as a function of tese variables and conditional on location. Te following definition is closely related to te dual problem of Minimum Expenditure in te classical consumer teory. Following Jara (2003) Willingness to Pay is calculated as te most tat an agent can pay for location given te utility level and prices. Definition 4. Te Willingness to Pay Function for agent on location, denoted B (u, p, r, z (H)), is defined as: (4) B (u, p, r, z (H)) := sup b IR + : (b,x) D (u, p, z (H)) := argmin x p x + b(1 w ) p e + r w U (x, z (H)) u x X (H) We define te Expenditure Minimizing Consumption Correspondence, as: { } p x :. U (x, z (H)) u x X (H) Remark 2. In tis definition, agent cooses te ceapest consumption bundle tat delivers te utility level u. Condition b IR + ensures consistency in te sense tat: an agent will not be able to locate were e can not afford to (if te lowest bound for b is negative ten B = ) and it also assures feasibility of consumption on te budget constraint. Agents express a conditional demand as if tey were paying teir willingness to pay on eac location, wic is completely rational. If w = 1 ten te value of B depends only on te feasibility of x. If te problem is feasible ten B = +, and if it is unfeasible ten B =. Tus B is always a real extended valued function. Tis approac as an advantage wit respect to Sollow s (1973) and Rosen s (1974) approac were willingness to pay is te inverse in prices of te indirect utility function conditional on location because we avoid indetermination of bids wen w = 1. Remark 3. If te result of (4) is not B =, and w 1, ten te function B (u, p, r, z (H)) can be calculated as: B (u, p, r, z (H)) = p e + r w e (u, p, z (H)) (1 w ) were e (u, p, z (H)) is te minimum expenditure function defined by: { } (5) p x :. e (u, p, z (H)) := min x U (x, z (H)) u x X (H) Tat is, willingness to pay is income minus te value of te minimum expenditure 1 bundles, multiplied by te property correction factor. 1 w.

7 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES Equilibrium Definition. We consider four types of correspondences for eac ouseold cluster { 1,..., } and eac location option V: Te Conditional Indirect Utility Function, V (p, r, z (H)), is te maximum utility tat agent can obtain in location given te price vector (p, r) and te city structure H. Tat is, if agent were to locate in and beave in an optimal way, tis would be is utility level. Te Utility Maximizing Consumption Correspondence, D (p, r, z (H)), is te utility-optimal consumption set for agent in location, given te price vector (p, r) and te city structure H. If agent were to locate in is consumption bundle would be an element of tis set. Te Willingness to Pay Function, B (u, p, r, z (H)), is te maximum tat agent is willing to pay to locate in to reac a utility level u, given te price vectors p of consumption goods and r of all oter location options and city structure H. Tat is, if agent were to locate in seeking a utility level u, tis is te quantity tat e would be willing to pay for it. Te Expenditure Minimizing Consumption Correspondence, D (u, p, r, z (H)), is te expenditure-optimal consumption set for agent in location. Again, if agent were to locate in seeking a utility level u, given te price vectors p of consumption goods and r of all oter location options, and city structure H, is consumption bundle would be an element of tis set. Agents in te economy must locate and, given teir location, te aggregate demand for goods must be met. For consistency, and due to location options being fixed, we assume tat: (6) f = N. Land use will be described by te matrix H. In equilibrium all agents must be located and all available dwellings must be occupied. So te first equilibrium condition is: (7) (8) H =N H =f Aggregate consumption of cluster is te sum of elements of te Consumption Correspondences multiplied by te number of located agents. Elements of tis demand correspondence are denoted d and we calculate tem as: (9) d := H d were d is an element of te Consumption Correspondence conditional on.

8 8 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ Total supply of consumption goods is e = e N so te excess supply correspondence is obtained as all te points s IR l tat can be calculated as: s := e N d. for any vector d obtained as above. Te second equilibrium condition is ten written as te existence of a point s in te excess supply correspondence suc tat: s IR l + Equilibrium is determined by matrix H, prices (p, r) and utility levels u. Depending on te cosen approac, we ave different rules to obtain H and an excess of supply points s Location Auction Equilibrium. Following Alonso (1964), dwelling prices are determined by auctions in wic bids are ouseolds Willingness to Pay Functions. To allocate location we use te best bidder rule. Tat is, in te f available locations of type, only ouseolds belonging to cluster(s) tat present te igest willingness to pay will be allocated. Te equilibrium allocation and prices must satisfy a Walrasian Equilibrium condition on consumption goods. In tis approac suppliers of dwellings coose teir inabitant troug te auction process and ouseolds compete to locate using teir Willingness to Pay Functions. To describe dwelling allocation we use a matrix µ M v ī {[0, 1]} wose coefficients µ indicate te proportion of ouseolds belonging to cluster cosen by dwellings of type by te best bidder rule. Eac column of matrix µ, µ, can be interpreted as a probabilistic distribution over te set { 1,..., }. Tat is, µ could be understood as te probability tat agent is te best bidder at location. For eac location, we would ten ave a distribution over te set of ouseolds. Ten, eac column µ of µ satisfies µ were: { } := µ IR + : µ = 1 So te allocation condition is: (10) µ > 0 = B = max g {1,..., } B g and can be written as a complementarity condition: µ (B max g {1,..., } B g) = 0. Te number of agents allocated to type dwellings is: H := µ f ; and te price of type dwellings is te value of te auction: r = max g {1,..., } B g.

9 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 9 Note tat as µ, condition (8) is already satisfied. Hence using tis rule, it is enoug to ask for condition (7). In tis approac, suc tat H > 0, te demand vectors d, tat appear in (9), must belong to te consumption correspondence set D (u, p, r, z (H 0 )). Definition 5. A vector (ū, p, r, µ, d) IR IR l + IR v ī + IR l v ī + is a Location Auction Equilibrium if it satisfies: (i) 0 = µ (B (ū, p, r, z (H 0 )) r ) (ii) µf N (iii) 0 e N µ f d (iv) r = max B (ū, p, r, z (H 0 )) (v) µ > 0 = d D (ū, p, r, z (H 0 )) Condition (i) stands for condition (10). Condition (ii) ensures tat all agents locate 2. (iii) is a market-clearing condition on consumption goods and (iv) states tat dwelling prices are obtained by auctions. Remark 4. It is important to note tat in te determination of aggregate demand of cluster in condition (iii), only locations for wic H > 0, are considered Location Utility-maximizing Equilibrium. In te line of Discrete Coice Teory, agents decide te consumption of one unit of a good available on a finite set of options caracterized by its attributes, maximizing teir utility in two steps. In tis context, eac agent solves is conditional on location utility maximization problem for eac location option and ten cooses te alternative tat yields a maximum utility, given a price vector. Equilibrium prices and allocations must again satisfy tat demand in all markets. As above, te description of ousing allocation is given by a matrix µ M v ī {[0, 1]} in wic te columns of µ satisfy µ v ī, wit: { } v ī := Te allocation condition is: µ IR v ī + : V µ = 1 µ (V max V () ) = 0. () Wit tis, te number of agents tat coose type dwellings is calculated: H := µ N. Now condition (7) is automatically satisfied, tus it is necessary to meet only condition(8). In (9), d D (p, r, z (H 0 )) for all suc tat H > 0. Equilibrium in tis case is defined as follows.. 2 Condition (6) ensures tat, in equilibrium, condition (ii) is satisfied wit equality and all agents will be located.

10 10 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ Definition 6. A vector (ū, p, r, µ, d) IR IR l + IR v ī + v ī IR l v ī + is a Location Utility-maximizing Equilibrium, if it satisfies: (i) 0 = µ (V ( p, r, z (H 0 )) ū ) (ii) µ T N f (iii) 0 e N µ N d (iv) ū = max V ( p, r, z (H 0 )) (v) µ > 0 = d D ( p, r, z (H 0 )) 4. Auction v/s Utility Maximization In Rosen (1974) and Martínez (1992) utility maximization relative to attributes of consumption goods is discussed. In te case of te Bid-Coice Model (Martínez, 1992), consumer surplus maximization is used to state an equivalence between te auction (or Bid ) and utility maximization (or Coice ) approaces on Land Use Models. However, tis treatment is limited to Indirect Utility Functions tat are linear or affine on utility level, wic does not fit te classical properties of tese functions nor te properties of Willingness to Pay Functions, as seen in Jara (2003). Furtermore te demonstration is one-sided in te sense tat utility maximization equilibria do not necessarily assure tat agents are best bidders in te implicit auctions eld in teir allocated dwellings. Tis section focuses on finding conditions to obtain equivalence between te two approaces in te context of our model, comparing te equilibrium conditions described in Definitions 5 and 6. Te next Proposition states tat in a Location Auction Equilibrium, te utility level reaced by ouseolds on teir allocated dwellings is greater tan te utility level tey would obtain if tey were to locate in any of te oter location options at equilibrium prices. Proposition 5. Let (ū, p, r, µ, d) be a Location Auction Equilibrium. Ten: µ > 0 = V ( p, r, z (H 0 )) max () µ () =0 V () ( p, r, z (H 0 )) Proof. Take ouseold and location option. Ten: B (ū, p, r, z (H 0 )) r. Now let us look at indirect utility as a function of te price in : V ( p, b, r, z (H 0 )) := (11) sup x IR l { U (x, z (H 0 )) : px + b(1 w ) pe + r w x X (H 0 ) Te utility tat would reac if e were located at at equilibrium prices, is calculated as V ( p, r, z (H 0 )); tat is, b = r. If B (ū, p, r, z (H 0 )) < r, ten, from (4), px + r (1 w ) pe + r w x X (H 0 ) suc tat, U (x, z (H 0 )) ū }

11 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 11 ence, because of te upper semi-continuity of U we get: { V ( p, r, z (H 0 )) = U(x, z (H 0 )) : < ū max x IR l px + r (1 w ) pe + r w x X(H 0) } If B (ū, p, r, z (H 0 )) = r, ten, as r 0, tere exists an element y X(H 0 ) suc tat, px + b(1 w ( r, y) argmax (b,x) b IR ) pe + r w + : U (x, z (H 0 )) ū x X(H 0 ) because if not, B (ū, p, r, z (H 0 )) =. Ten V ( p, r, z (H 0 )) ū, because y is feasible in (11) for b = r, and U (y, z (H 0 )) ū. Ten if in equilibrium, is te best bidder in and not in () : V ( p, r, z (H 0 )) ū > V () ( p, r, z () (H 0 )) wic gives te stated result. Toug tis property states tat, at Auction Equilibrium prices, te utility an agent would get if e is located in a place were e is te best bidder, is iger tan te utility e would obtain in a place were e is not, we can not assure tat tis equilibrium utility level ū will be te actual utility tat type agents ave in equilibrium wen allocated. Tis is because in some of te places were an agent is te best bidder, e could actually get an even iger utility tan te equilibrium utility, and furtermore, te utility level need not be te same in all allocated places. To assure tat equilibrium utility is te actual utility, te following elements need to be defined: u min (p) := max x x min (p) U (x, z (H 0 )), is te igest utility an agent can obtain by consuming is ceapest feasible bundle. We state te following proposition. Proposition 6. If te functions U (, z (H 0 )) are continuous, let (ū, p, r, µ, d) be a Location Auction Equilibrium, suc tat { 1,..., }, (12) ū max { : µ >0} {umin ( p)} Ten, µ > 0 = V ( p, r, z (H 0 )) = max () V () ( p, r, z (H 0 )) Proof. Due to Proposition 5, it is enoug to prove: µ > 0 = V ( p, r, z (H 0 )) = ū.

12 12 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ As ū u min ( p) it must be tat d D (ū, p, z (H 0 )) U (d, z (H 0 )) = ū, if not, convexity of X (H 0 ) and te level sets of U and continuity of U allow us to move along a segment [d, x], wit x x min ( p), diminising te value of te consumption bundle and utility. Moreover, if µ > 0, ten we ave: r = B (ū, p, r, z (H 0 )) = pe + r w e (ū, p, z (H 0 )) 1 w = pe + r w p d 1 w Suppose tere exists x X (H 0 ) C ( p, r) suc tat U (x, z (H 0 )) > ū. Ten, Terefore, p e + r w p x + r e (ū, p, z (H 0 )) + r = p e + r w p x = e (ū, p, z (H 0 )) wic means tat x D (ū, p, z (H 0 )) and ten U (x, z (H 0 )) = ū, wic is a contradiction. So, for any x X (H 0 ) C ( p, r), it must be U (x, z (H 0 )) ū Wit te preous proposition we can identify Location Auction Equilibria in wic locating agents maximize teir utility. Next we study maximum utility allocation conditions to establis wen te Maximum Utility Equilibrium prices and allocation come from an auction process. Suppose tere is a price vector ( p, r) tat satisfies te Location Utility-maximizing Equilibrium conditions. It may be necessary to know if dwelling prices are te result of an auction on eac location. To ascertain tis, we must ceck tat allocated ouseolds are best bidders. Willingness to Pay Functions must, terefore, be calculated for all ouseolds in all locations, and evaluated in equilibrium points. Tis produces te following result: Proposition 7. Let (ū, p, r, µ, d) be a Location Utility-maximizing Equilibrium. Defining: ū := V ( p, r, z (H 0 )) ten: (i) ū = ū = B (ū, p, r, z (H 0 )) r (ii) ū < ū = B (ū, p, r, z (H 0 )) < r

13 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 13 Proof. (i) Take d D ( p, r, z (H 0 )), so tat ū = V ( p, r, z (H 0 )) = U (d, z (H 0 )). Ten d is feasible in te following problem: (13) B (ū, p, r, z (H 0 )) := sup b IR + : togeter wit b = r. Terefore, B (ū, p, r, z (H 0 )) r. px + b(1 w ) pe + r w U (x, z (H 0 )) ū x X(H 0 ) (ii) As ū < ū, x C ( p, r) X (H 0 ) suc tat U (x, z (H 0 )) ū. Hence r is not feasible in (13). If te supremum is finite, it is met, so: B (ū, p, r, z (H 0 )) < r. Te preous proposition states tat in places were ouseolds maximize teir utility, teir willingness to pay is equal to or iger tan te equilibrium price, wereas in locations were teir utility is not maximal, teir willingness to pay is strictly lower tan te equilibrium price. However, in a Location Utility-maximizing Equilibrium it is possible tat ouseolds maximize utility in a location option witout being allocated to it. We want to prode a condition tat allows us to conclude tat te price tey are willing to pay for suc places is at most equal to te equilibrium price. Tis would mean tat dwelling prices are a result of an auction. Proposition 8. Let (ū, p, r, µ, d) be a Location Utility-maximizing Equilibrium. If it satisfies te following condition: (14) Ten: ū = ū = D ( p, r, z (H 0 )) intx (H 0 ) (i) ū = ū = B (ū, p, r, z (H 0 )) = r (ii) ū < ū = B (ū, p, r, z (H 0 )) < r Proof. We ten must prove only(i). Suppose tat B > r ten tere exist x X (H 0 ) and b > r 0, suc tat: U ( x, z (H 0 )) ū, p x pe + r w b(1 w ) < pe + r w r (1 w ). Tat is x C ( p, r) and U ( x, z (H 0 )) ū. Ten x D ( p, r, z (H 0 )) and because of condition (14), tere exists a neigborood V x of x suc tat V x X (H 0 ) C ( p, r). Locally non satiation of U gives te existence of an element y V x suc tat U (y, z (H 0 )) > U ( x, z (H 0 )) wic is a contradiction.

14 14 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ Corollary 9. If X (H 0 ) IR l + and (ū, p, r, µ ) is a Location Utility-maximizing Equilibrium, wit p >> 0, ten prices r are te result of an auction. Given te results and comments tus far, we can state an equivalence result between te two approaces wit te following teorem: Teorem 10 (Equivalence). If te utility functions U (, z (H 0 )) are continuous, X (H 0 ) IR l +, and te following conditions are satisfied: (i) z IR a, inf x IR l + U (x, z) = U (0, z) = u min (ii) p >> 0. for some u min IR Ten, (ū, p, r, µ, d) is a Location Utility-maximizing Equilibrium if and only if (ū, p, r, µ, d) is a Location Auction Equilibrium wit ū u min, were ū := max V ( p, r, z (H 0 )); and te following relation olds: µ N = µ f A sufficient condition guaranteeing te equivalence of tese two approaces is for example te strict monotonicity on utility and on Willingness to Pay Functions. Indeed, te conditions of Teorem 10 allow us to invert Indirect Utility Functions to obtain Willingness to Pay Functions evaluated at equilibrium points wic leads to te conclusion tat suc a condition is enoug to demonstrate equivalence. Tis conclusion implies tat maximizing utility is consistent wit te auction process under te above condition, extending te results of te Bid-Coice Model (Martínez, 1992), were income is exogenous. 5. Existence In tis section we are interested in presenting an existence result for Equilibrium points. For tis purpose we will use te Utility-maximizing approac developed in te preous section. Suppose tat X (H 0 ) X wit X := { x IR l + : x e } were e is total endowment on consumption goods in te economy, ence X is a compact subset of IR l +. Clearly, e intx. Consider now positive and continuous utility functions U (x, z (H 0 )); in addition to te caracteristics mentioned in Section 2. Assume tat te utility function of eac agent satisfies: j suc tat x j = 0 U (x, z (H 0 )) = 0 x >> 0 U (x, z (H 0 )) > 0. Tis means, no matter te location option, an agent prefers a strictly positive bundle to any wit some null coordinate. In addition : u max = U (e, z (H 0 )) > U (x, z (H 0 )) x X In wat follows we make te following surval assumption:

15 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 15 (15) (p, r) suc tat I (p, r) > 0. wit I (p, r) defined as: I (p, r) := p e + r w r. Assumptions made over utility functions imply tat if income is strictly positive in some location, ten conditional indirect utility is strictly positive. If I (p, r) = 0 and p 0, agent can only consume in goods j suc tat p j = 0, ence p j > 0 implies d j = 0 and ten V (p, r, z (H 0 )) = 0. Teorem 11 (Existence). Under te surval assumption (15), if utility functions are strictly concave, ten tere exists a Location Utility-maximizing Equilibrium. To demonstrate Teorem 11, we generate a correspondence tat associates a consumption set, for goods and location, wit eac pair (p, r). For tis purpose consider te conditional on location problem (P ): sup { U (x, z (H 0 )) : x C (p, r) X }. From tis problem we obtain te Utility Maximizing Consumption Correspondence D (p, r, z (H 0 )) and te Conditional Indirect Utility Function V (p, r, z (H 0 )). Wit all te Conditional Indirect Utility Functions, we state te second step maximization problem, in wic agents coose teir location: max {V (p, r, z (H 0 ))} As tis problem is not convex and can ave more tan one solution, we consider te following convexified problem over te simplex of IR v ī + : { } max µ V (p, r, z (H 0 )) µ v ī V From te second step convexified problem, we obtain its solution set Γ (p, r). Note tat, { } Γ (p, r) = µ v ī : V (p, r, z (H 0 )) < max V (p, r, z (H 0 )) µ = 0. Recall tat λ > 0, C (λ(p, r)) = C (p, r) = D (λ(p, r), z (H 0 )) = D (p, r, z (H 0 )) V (λ(p, r), z (H 0 )) = V (p, r, z (H 0 )) = Γ (λ(p, r)) = Γ (p, r). Terefore we can restrict te price set to: l := (p, r) IRl+ v ī + : p j + r = 1. j=1 V

16 16 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ Given prices (p, r), eac agent will ave is consumption correspondences Γ (p, r) and, for eac, D (p, r, z (H 0 )). Ten, aggregate demand of cluster for location, will be any vector of te form: (H ) V = (µ N ) V were µ = (µ ) V Γ (p, r). Consumption goods demand is generated in an analogous way as described in (9); tat is, d = V N µ d wit µ Γ (p, r) and d D (p, r, z (H 0 )) for eac suc tat µ > 0. Ten te demand correspondence associates wit eac pair (p, r) te set: {( ) } d D(p, r) := IR l+ v ī (H ) + : µ Γ (p, r), d D (p, r, z (H 0 )) V wic is equal to N ϕ (p, r) were ϕ (p, r) is given by: ϕ (p, r) := Consider te set: (d, (µ ) ) IR l+ v ī + : d = V µ d, µ Γ (p, r), d D (p, r, z (H 0)) P := { (p, r) IR l+ v ī + : I (p, r) 0 }, For te demand of agent being non-empty and utility greater tan we seek prices in te set P = V P. In addition we need prices (p, r) to be in all te sets P, so te price set is P, given by: P := {1,..., } V P Tus, correspondence D(p, r) takes non-empty values in te set P. Te surval assumption (15) implies P, tus restricting te solution to prices in. Considering assumption (1) and te utility functions, we can assume tat (for any price vector) agents always maximize utility in a location tat gives strictly positive utility and in wic income is also strictly positive. Te following Lemmas are useful for te proof of te next Proposition. Lemma 12. If I (p, r) > 0, te Indirect Utility Function V is continuous in (p, r) and te Conditional Demand Correspondence D is u.s.c. in (p, r). Lemma 13. Let (p ν, r ν ) (p 0, r 0 ), wit p 0 0, suc tat I (p ν, r ν ) 0 wen ν. Ten lim sup V (p ν, r ν, z (H 0 )) 0 ν Proposition 14. Under assumption (1), te correspondence ϕ : IR l+ v ī + is u.s.c..

17 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 17 To obtain te existence result, we look for a zero value for te excess supply correspondence ϕ defined by: ( ) e ϕ(p, r) := D(p, r) f ( ) e (16) = N ϕ (p, r) f Proof of Teorem 11. If tere exists a price vector ( p, r) suc tat ϕ( p, r) IR + l+ v ī, take a point (x, y) ϕ( p, r) IR + l+ v ī. Suc point (x, y) is of te form : ( ) ( ) x e = ( N µ ) d (p, r, z (H 0 )) (17) y f µ As (x, y) IR l+ v ī + conditions (ii) and (iii) of Definition 6 old. As µ Γ ( p, r) we ten ave tat (i), (iv) and (v) old. Tat is, te vector ( p, r, µ, d), tat appears in te rigt side of equation (17), generates a Location Utility-maximizing Equilibrium. So, we ave to prove te existence of suc a price vector ( p, r). To do tis we verify tat ϕ satisfies te ypotesis of te Debreu-Gale-Nikaïdo Teorem (see, Aubin, 1997). We already stated tat ϕ(p, r) as non-empty values in te simplex of IR l+ v ī +. Due to Proposition 14 and te fact tat every u.s.c. correspondence is upper emicontinuous we know ϕ(p, r) is upper emi-continuous. As utility functions are strictly concave, te sets D contain at most one element d (p, r, z (H 0 )). Terefore, for eac pair (p, r) te set ϕ (p, r) is convex. Indeed, take (d 1, µ 1 ),(d 2, µ 2 ) ϕ (p, r) and λ ]0, 1[. Ten λµ 1 + (1 λ)µ 2 =: µ λ Γ (p, r) since V (p, r, z) > V (p, r, z (H 0 )) implies µ 1 = µ2 = 0 wic in turn implies µ λ = 0. In addition, d 1 = d (p, r, z (H 0 ))µ 1 and d2 = d (p, r, z (H 0 ))µ 2 (recall tat in te case tat te conditional demand is empty for some µ = 0, so we can consider tis demand equal to zero), terefore, d λ :=λd 1 + (1 λ)d 2 =λ d (p, r, z (H 0 ))µ 1 + (1 λ) d (p, r, z (H 0 ))µ 2 = d (p, r, z (H 0 ))µ λ wit (d λ, µ λ ) ϕ (p, r).

18 18 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ As Γ (p, r) is compact, ten so is ϕ (p, r). Terefore, te set ϕ(p, r) is convex and compact. Tis implies tat ϕ(p, r) IR l+ v ī + is convex and closed. To complete te proof, we take (x, y) ϕ(p, r) and calculate (x, y) (p, r) = x p+y r. As (x, y) ϕ(p, r), we ave: ( ) ( ) x e = ( N µ ) d (p, r, z (H 0 )). y f µ Terefore, ( (x, y) (p, r) = e ) ( N µ d (p, r, z (H 0)) p + f ) N µ r = ( N e ) µ d (p, r, z (H 0)) p + f r N µ r. Note tat f r is te total value of location options, so we can calculate tis by adding values across all owners. Tat is, f r = wn r. Replacing, we get: (x, y) (p, r) = ( N e p µ d (p, r, z (H 0 )) p + N wr = ( N e p µ d (p, r, z (H 0 )) p µ r ) µ r + r w ) As µ Γ (p, r), µ = 1 so we get: (x, y) (p, r) = N ( µ e p + r w ) d (p, r, z (H 0 )) p r }{{} 0 0 Terefore σ(ϕ(p, r), (p, r)) 0. We ave sown tat ϕ(p, r) satisfies te ypotesis of te Debreu-Gale-Nikaido Teorem (see, Aubin, 1997), so we can conclude tat tere exists a price vector ( p, r) suc tat ϕ( p, r) IR l+ v ī +, wic completes te proof.

19 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES Conclusions In tis work we ave developed a new teoretical approac to model ow people make residential location decisions in an urban enronment. We ave defined wat we call an Excange Economy wit Location. As te property of all goods (consumption and location) is totally concentrated among te participating agents, teir income is endogenous to equilibrium and depends on te initial endowments and equilibrium prices. In addition to te usual budget constraints we incorporate several restrictions over consumption and actities, suc as time, capacity, spatial and tecnological constraints. Tese constraints are modelled by a set-valued map tat depends on te city structure and defines for eac agent is consumption possibilities associated wit is current location. Tis new feature does not affect te use of Willingness to Pay functions as we calculate tem, following Jara (2003), not as te inverse of te Conditional Indirect Utility Functions, but as te value of a minimum expenditure problem. Tis new metod extends all preous models in a coerent way allowing new constraints to te consumer s problem and terefore to te Conditional Indirect Utility Functions. Indeed, all te complexity added to te conditional on location consumer problem by te preously mentioned mapping, goes directly to te constraints of te minimum expenditure problem. In te case were te value of Willingness to Pay is finite, it is equal to te difference between income and te value of minimum expenditure on consumption goods. Two types of equilibrium conditions are defined. In te Location Auction Equilibrium, agents compete for location troug teir willingness to pay and location prices are determined by auctions. Te second type is te Location Utilitymaximizing Equilibrium. In tis case agents decide were to locate maximizing teir utility given goods and location prices. In bot equilibrium definitions marketclearing conditions are imposed on goods and locations determining all prices simultaneously. An important result in tis work gives conditions under wic te auction and utility approaces are equivalent, wic is defined as wen an equilibrium point satisfies conditions for bot Location Auction Equilibrium and Location Utility-maximizing Equilibrium. Tis result answers an important question in Urban Economics. In te last part of te paper, we prode an existence result using te Location Utility-maximizing Equilibrium approac. To obtain tis result we considered a version of te model wit budget constraints (only) in te consumer s problem and wit continuous, strictly concave utility functions. Tere are oter new features presented, altoug not fully explored in tis work. For example, we assume tat te interaction a location externalities appens in a dynamic version of te model. It remains for future researc to formulate suc an interaction and te notion of equilibrium, and to study issues suc as te equivalence of approaces and te existence of equilibrium in suc a dynamic setting. Anoter interesting extension of te model would be to study te location of firms and teir

20 20 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ agglomeration economies in a general location enronment, comparing results to te work of Lucas and Rossi-Hansberg(2002)in a monocentric enronment. References Alonso, W. (1964). Location and Land Use. Cambridge, Harvard University Press. Anas, A. (1982). Residential Location Markets and Urban Transportation. Academic Press, London. Aubin, J.P.(1997). Optima and Equilibria. Springer Verlag. Berge, C. (1997). Topological Spaces. Dover Publications Debreu, G. (1959). Teory of Value. Yale University Press Jara, P. (2003). Un Modelo de Equilibrio en Uso de Suelos con Ingreso Endógeno. Tesis de Magíster y Memoria de Ingeniero, Universidad de Cile. Jara-Díaz, S. and Martínez, F.J. (1999). On Te Specification of Indirect Utility and Willingness to Pay for Discrete Residential Location Models. Journal of Regional Science, 39, Lucas, R. E., Jr. y Rossi-Hansberg, E. (2002). On te Internal Structure of Cities. Econometrica Vol. 70, N o 4, Martínez, F.J. (1992) Te Bid-Coice Land Use Model: an Integrated Economic Framework. Enronment and Planning A. Vol. 24, McFadden, D.L.(1978). Modelling te coice of residential location, in Karlqst et. al. (eds), Spatial Interaction Teory and Planning Models. Nort-Holland, Amsterdam, Rosen, S. (1974), Hedonic prices and implicit markets: product differentiation in pure competition. Journal of Political Economy Vol. 82 (1), Solow, R.M. (1973), On equilibrium models of urban location, in M. Parkin (ed.), Essays in modern economics, Barnes and Nobles, pp Von Tünen, J.H. (1863). See Von Tünen Isolated State, (ed) Peter Hall, Pergamon Press, London (1966). Appendix Proof of Teorem 10. We will ceck eac side of te statement separately. First implication (= )

21 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 21 Define µ as follows: so tat µ > 0 µ > 0. µ := µ N f Proposition 8 and Corollary 9 give: r = max B (ū, p, r, z (H 0 )) 0 = µ (B (ū, p, r ) r ) Besides: µ = µ N = 1 µ f f N = 1 f = 1 f and µ f = µ N = N. Let suc tat µ > 0. Because of Corollary 9 we get, B (ū, p, r, z (H 0 )) = r. Take d D ( p, r, z (H 0 )), wit wat Ten, d D (ū, p, z (H 0 )). p d + r = pe + rw. Terefore, (ū, p, r, µ, d) is a Location Auction Equilibrium. Second implication ( =) As p >> 0 and X (H 0 ) = IR l +, ū u min Due to Proposition 6, if we define, we get, and = u min ( p) for any pair µ := µ f N µ (ū V ( p, r, z (H 0 ))) = 0 µ = 1 µ N = f. Let s take now d D (ū, p, z (H 0 )) for suc tat µ > 0. Ten, again, r = pe + rw p d, wit d C ( p, r). From Proposition 6 we ave U ( d, z (H 0 )) = ū = V ( p, r, z (H 0 )), wit d D ( p, r, z (H 0 )). Tus, (ū, p, r, µ, d) is a Location Utility-maximizing Equilibrium.

22 22 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ Proof of Lemma 12. U (x, z (H 0 )) is continuous and I (p, r) > 0 = min{p x : x X}. applying (1) in 4.8 Debreu, , we ave tat te mapping (p, r) C (p, r) X is continuous. Applying Berge s Teorem (Berge, 1997) we get te result. Proof of Lemma 13. Consider te correspondence tat to eac (p, r) associates te set C (p, r) X. If we take x ν C (p ν, r ν ) X suc tat x ν x, we ave tat: and terefore, p ν x ν I (p ν, r ν ) p 0 x I (p 0, r 0 ) Besides x X. So, x C (p 0, r 0 ) X. Tat is, C X is upper semi-continuous in (p 0, r 0 ). As U (x, z (H 0 )) is continuous, we can apply Teorem 2 in Berge (1997) Capter IV 3 and ten, V is upper semi-continuous in (p 0, r 0 ). But C (p 0, r 0 ) = {x IR l + : p 0 x 0} because I (p 0, r 0 ) = lim ν I (p ν, r ν ) = 0, and tis set is given by : {x IR l + : p 0 j > 0 x j = 0}, terefore, if p 0 0, U (x, z (H 0 )) = 0 x C (p 0, r 0 ) X, ten: lim sup V (p ν, r ν, z (H 0 )) V (p 0, r 0, z (H 0 )) = 0 ν Proof of Proposition 14. Since we are cecking te upper semi-continuity of ϕ for any { 1,..., } we will omit te subscript, for te vectors µ, in tis demonstration. Let (p 0, r 0 ), and consider a sequence in, (p ν, r ν ) (p 0, r 0 ), and (d ν, µ ν ) (d 0, µ 0 ) wit (d ν, µ ν ) ϕ (p ν, r ν ). We want (d 0, µ 0 ) ϕ (p 0, r 0 ), tat is: 3 Proposition 15. If X is compact, convex, and if (p 0, I 0 ) is a point suc tat: I 0 > min{p 0 x : x X}. Ten γ is continuous in (p 0, I 0 )

23 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 23 (i) V µ0 = 1 (ii) µ 0 > 0 = V (p 0, r 0, z (H 0 )) = max () V V () (p 0, r 0, z () ) (iii) d 0 = V µ0 d0, were d0 D (p 0, r 0, z (H 0 )) for all V suc tat µ > 0 We ave (d ν, µ ν ) ϕ (p ν, r ν ), so for eac V tere is a sequence d ν D (p ν, r ν, z (H 0 )) suc tat d ν = µν dν 4. Every sequence is in X so troug a subsequence we can get d 0. Ten we must sow tat µ0 Γ (p 0, r 0 ), and d 0 D (p 0, r 0, z (H 0 )) for all V suc tat µ 0 > 0. We will call V (p, r, z) to te maximum of te indirect utilities: First note tat: (18) V (p, r, z) := max V V (p, r, z (H 0 )). V µ ν = 1 = V µ 0 = 1 To prove (ii) take µ 0 > 0. We must consider two cases: p0 0 and p 0 = 0. Case p 0 0. As p ν p 0 we can suppose p ν 0. Consider te sets: I (p, r) := { : V (p, r, z (H 0 )) = V (p, r, z)} J (p, r) := { : I (p, r) > 0} Clearly, if p 0, I (p, r) J (p, r). As I is continuous, for ν sufficiently large J (p 0, r 0 ) J (p ν, r ν ) because I (p 0, r 0 ) > 0 I (p ν, r ν ) > 0 and tere is a finite number of location types. Ten Terefore, tat is (19) I (p 0, r 0 ) J (p 0, r 0 ) J (p ν, r ν ) ν > ν. µ 0 > 0 = µν > 0 ν > ν = V (p ν, r ν, z) = V (p ν, r ν, z (H 0 )) ν > ν, I (p ν, r ν ) ν > ν. If / J (p 0, r 0 ), as I (p ν, r ν ) > 0 ν > ν, ten we must ave I (p ν, r ν ) I (p 0, r 0 ) = 0. Because of Lemma 13, we ave: lim sup V (p ν, r ν ) 0 ν 4 If for some ν, D (p ν, r ν, z (H 0 )) = we can consider d ν = 0 since in tat case µν is equal to zero so tere is no arm doing tis.

24 24 PEDRO JARA, ALEJANDRO JOFRÉ, FRANCISCO MARTÍNEZ But we know tat V (p 0, r 0, z) > 0, wic means, () I (p 0, r 0 ), V () (p 0, r 0, z () ) = V (p 0, r 0, z) > 0. So, for ν sufficiently large V (p ν, r ν, z (H 0 )) < V () (p ν, r ν, z () ) V (p ν, r ν, z), because V () is continuous in (p 0, r 0 ), wic contradicts (19). So, J (p 0, r 0 ). Terefore V is continuous in (p 0, r 0 ). If ten, for ν large, and tis also contradicts (19). We conclude tat: (20) V (p 0, r 0, z (H 0 )) < V (p 0, r 0, z), V (p ν, r ν, z (H 0 )) < V () (p ν, r ν, z () ) µ 0 > 0 V (p 0, r 0, z (H 0 )) = max () V () (p0, r 0, z () ) Case p 0 = 0 Problem (P ) for (0, r) is: { X if I (0, r) 0 C (0, r) = if I (0, r) < 0. Terefore V (0, r, z (H 0 )) = { U (e, z (H 0 )) if I (0, r) 0 if I (0, r) < 0 D (0, r, z (H 0 )) = { {e} if I (0, r) 0 if I (0, r) < 0 Γ (0, r) = {µ v ī : µ = 0 suc tat I (0, r) < 0} Now take µ 0 > 0. As before, tis implies tat µ ν > 0 ν > ν and ten V (p ν, r ν, z) = V (p ν, r ν, z (H 0 )) ν > ν. Terefore I (p ν, r ν ) 0 ν > ν. In te limit, we get I (0, r 0 ) 0 and because of te preous results, V (0, r 0, z (H 0 )) = U (e, z (H 0 )) U (x, z (H 0 )), x X V. Terefore V (0, r 0, z (H 0 )) = V (0, r 0, z). So in tis case we also ave: (21) µ 0 > 0 V (0, r 0, z (H 0 )) = max () V () (0, r0, z () )

25 LAND USE EQUILIBRIUM MODEL WITH ENDOGENOUS INCOMES 25 To see tat (µ 0, d 0 ) ϕ (p 0, r 0 ) we still need tat d 0 as te form stated above, d 0 = µ0 d0. For tis we prove, µ 0 > 0 = d 0 D (p 0, r 0, z (H 0 )). Equation (20) tells us tat if p 0 0 te preous sum is only over elements of I (p 0, r 0 ) and lemma 12 gives te upper semi-continuity on (p 0, r 0 ) of te demand correspondences associated to tis set. Tus d 0 D (p 0, r 0, z (H 0 )) wic brings us to conclude tat (µ 0, d 0 ) ϕ (p 0, r 0 ). If p 0 = 0, ten, because of assumption 1, tere is at least one location for wic I (0, r 0 ) > 0. So, for tese locations we can still apply Lemma 12, to obtain upper semi-continuity of te conditional demand correspondences on (0, r 0 ) wit d 0 D (0, r 0, z (H 0 )). But, if I (0, r 0 ) = 0, we can not use Lemma 12 to assure te continuity. In tis case we ave to note tat, as µ ν > 0 ν > ν, we know tat we get a sequence d ν D (p ν, r ν, z (H 0 )) tat converges (a a subsequence if necessary) to d 0. If d0 = e, ten tere is no problem, since D (0, r 0, z (H 0 )) = {e}. If d 0 e, we ave te following: Since d ν D (0, r ν, z (H 0 )) and µ ν > 0, we get: U (d ν, z (H 0 )) = V (p ν, r ν, z (H 0 )), Due to te form of utility functions we get: U (d ν, z (H 0 )) < U (e, z (H 0 )). Wen we take te limit in te preous equations we get tat U (d 0, z (H 0 )) < U (e, z (H 0 )) = V (0, r 0, z (H 0 )). Te first inequality comes from te utility functions and te assumption tat d 0 e, wile te second is given above. As assumption 1 assures te continuity of V in (0, r 0 ) (due to te continuity of utility in te locations suc tat I (0, r 0 ) > 0), we get a contradiction since we would get V (0, r 0, z (H 0 )) = U (d 0, z (H 0 )) < V (0, r 0, z (H 0 )). Tus, eiter we get d 0 = e or I (0, r 0 ) > 0. In eiter of tese two cases we recover te fact tat d 0 D (0, r 0, z (H 0 )) and terefore, we get (µ 0, d 0 ) ϕ (p 0, r 0 ). Equations (18), (20) and (21) give te result.