Lars Nesheim. 17 January Last lecture solved the consumer choice problem.

Transcription

1 Lecure 4 Locaional Equilibrium Coninued Lars Nesheim 17 January 28 1 Inroducory remarks Las lecure solved he consumer choice problem. Compued condiional demand funcions: C (I x; p; r (x)) and x; p; r (x)) : Also, derived expression for opimal locaion L (I x; p; r (x)) = Also, compued uiliy as a funcion of locaion: L (C ; L ; ; x) = u (C (x) ; L (x)) Use hese o sudy locaional equilibrium De niion of locaional equilibrium 1. All consumers maximise uiliy. 2. Locaional equilibrium (a) No one wans o move. (b) Land goes o highes bidder. 3. Equilibrium in land marke. 1

2 1.1 Equilibrium condiion 1: consumer s opimise 1. This deermines consumer demand for C and L and consumer locaion choice. 2. Condiional demand funcions. If a consumer chooses locaion x; hen his demand for C and L condiional on ha choice of x can be expressed as: C (p; r (x) ; I x) and L (p; r (x) ; I x) 3. These are called condiional demand funcions, because hey express he demand condiional on choice of locaion. 4. Given he condiional demand funcions, he consumer s locaion choice sasi es x; p; r (x) + = : (a) Each person chooses opimal locaion x so ha marginal cos of moving equals marginal bene. (b) This deermines consumer s opimal locaion choice x : (c) This does no deermine r (x) : Need addiional condiions o deermine r (x) : 2 Equilibrium condiion 2: locaional equilibrium 1. Locaional equilibrium condiion. (a) In equilibrium, no one wans o move. (b) An equilibrium ren funcion sais es he condiion ha no one wans o move. 2. Implicaions. (a) N idenical people. 2

3 (b) If all idenical, and none wan o move, hen all locaions ha are inhabied mus yield he same uiliy. Oherwise, if some live a locaion 1 and some live a locaion 2 bu locaion 1 provides lower uiliy, hen all he people a locaion 1 would wan o move o locaion 2. (c) In an equilibrium in which all people are idenical and some people live a all locaions x x B, all locaions x x B yield idenical uiliy. (d) People spread ou across locaions x x B. (e) For his o be rue, i mus be he case ha (1) holds a every locaion x x B. This means = x; p; r (x)) for all x 2 [; x B] : (f) Locaional equilibrium condiion resuls in a condiion on he slope of he ren funcion. (g) x B is he boundary of he ciy. The value of x B is ye o be deermined. 3. Equilibrium condiion is a di erenial = x; p; r (x)) : (2) (a) Under sandard condiions, a soluion exiss. calculaed using a compuer. Usually mus be (b) The slope of he ren funcion depends on ranspor coss and on he demand for land. i. If ranpor coss are high, he slope will be seep. ii. If demand for land is low, he slope will be seep. 4. Le r be he ren a he cenre. A soluion o (2) will be of he form r (x; r ) = r + = r Z x Z x ds s; p; r (s; r )) ds:

4 This equaion saes ha he ren a a locaion x is equal o he ren a he cenre plus he change in ren beween and x: Because he change in ren is negaive, r (x) < r for all x: The ren funcion will depend on r. The variable s is a dummy inegraion variable ha akes on values beween and x: 5. Furher implicaion of locaional equilibrium. (a) Recall ha x B is he boundary of he ciy. (b) Locaional equilibrium implies ha he level of ren a he boundary equals r A : Why? (c) Since he urban boundary is x B : The ren a he boundary mus saisfy r (x B ) = r A : r A = r Zx B s; p; r (s; r )) ds: 6. In hese equaions, p, ; I and r A are known parameers whose values are xed ouside he model. The funcion L is a known funcion derived from he soluion of he consumer s maximisaion problem. The variable r is an unknown variable whose value is o be deermined in equilibrium. Once r is known, he funcion r (x; r ) can be calculaed using a compuer. In his class we will ask qualiaive quesions like: 1) How do changes in (p; ; I; r A ; N) a ec r (x) and r? 2) Given values for (p; ; I; r A ; N) ; wha is he welfare ha consumers obain? 2.1 Wha we have so far. 1. Locaional equilibrium among = : x;p;r(x)) (b) r = r (x; r ) = r (c) r A = r (x; r ) : ds: s;p;r(s;r )) 2. Consumer choices (maximisaion). 4

5 (a) Demand for food and land. 3. Review. i. x; p; r (x; r )) ; C (I x ; p; r (x; r )) : ii. L and C can be shown on he sandard picure of consumer maximisaion subjec o a budge consrain. iii. Draw picure showing budge consrain, angency of indi erence curve, and opimal choice of C and L: (a) Each consumer maximises. i. Consumer chooses (C; L; x) o maximise uiliy subjec o budge consrain. ii. Condiional demand funcions A. C (I x ; p; r (x )) : B. x ; p; r (x )) : iii. Opimal locaion choice: x (I; ; p; r (x)) : (b) Equilibrium in urban spaial economy. i. Level and slope of ren funcion. ii. Number of people in ciy or level of welfare. iii. Radius of ciy or level of ren. (c) Locaional equilibrium. i. All idenical consumers obain same uiliy regardless of where hey choose o live dr (x) dx L (I x; p; r (x)) = dr (x) dx = x; p; r (x)) : ii. Land is used in highes value use. A. Land goes o highes bidder r (x) = r Z x s; p; r (s; r )) ds: 5

6 B. r (x; r ) > r A for x < x B ; urban (residenial and commuing). C. r (x; r ) = r A for x x B ; rural (farming). In equilibrium r (x; r ) r A for all x: Why? D. r (x) > r A land goes o housing, r (x) = r A land goes o farming. E. Hence, he value of r is deermined by r A = r Zx B s; p; r (s; r )) ds: iii. Wha s missing? A. Wha deermines x B? B. Supply and demand for land mus be equaed. 3 Equilibrium condiion 3: supply and demand for land are equal 1. Equilibrium in land marke. (a) Supply of land a disance x is S L (x) = 2x; draw picure. (b) How many people live a disance x? N (x) (c) Toal demand for land in housing a disance x is D L (x) = N (x) x; p; r (x; r )) if x x B : (d) N (x) is unknown bu if supply equals demand a locaion x hen N (x) is deermined by for all x x B : N (x) = 2x x; p; r (x; r )) (e) x B is sill unknown. Bu, we know ha he oal populaion in he ciy is N: 6

7 (f) In equilibrium, everyone mus live somewhere. Therefore, N = Zx B N (s) ds (3) = x B Z 2s s; p; r (s; r )) ds: (g) This nal equaion pins down x B : The variable x B mus saisfy equaion (3) (h) To compue an equilibrium: i. Guess a value for x B : ii. Compue he value of he righ side of equaion (3) : iii. If he righ side of (3) is less han he lef side, increase x B : Why? iv. If he righ side of (3) is larger han he lef side, reduce x B : Why? v. If he righ side of equaion (3) equals he lef side, hen x B is an equilibrium value. vi. Repea seps i.-, iv. unil v. is rue. 2. Equilibrium in consumpion good marke rivial. (a) Demand for he consumpion good is: D C = B N (s)c (I (b) Supply of consumpion good is in niely elasic a price p: Tha is supply of C adjuss so ha supply equals demand a price p: (c) Wha would he equilibrium condiion be if he supply were no in niely elasic? 3. Equilibrium equaions. s; p; r (s; r )) ds: (a) r (x; r ) = r (b) r A = r B ds: s;p;r(s;r )) ds: s;p;r(s;r )) 7

8 B (c) N = N (s) ds = B 2s ds: s;p;r(s;;r )) 3.1 Reprise of equilibrium condiions 1. Equilibrium. (a) Condiion 1: consumer maximizes uiliy. (b) Condiion 2: locaional equilibrium. i. For idenical people, changes in ren across space compensae exacly for changes in ranspor coss dr (x) dx = x; p; r (x)) : ii. For non-idenical people, land goes o he highes bidder. A. If r (x) > r A ; consumers bid more. B. If r A > willingness o pay of consumers; farmers bid more. C. r (x) = r A ; urban boundary. (c) Condiion 3: land marke equilibrium: A every locaion x; supply equals demand for land. (d) Condiion 4: consumpion good marke equilibrium: Supply equals demand for he consumpion good. 4 Summarise 1. Equilibrium. (a) Inpus o equilibrium: ; N; r A ; U (; ) ; I; p: These are he parameers of he problem. They are xed, speci ed ouside he model. (b) Oupus: C ; L ; r () ; r ; x B ; V = U (C ; L ) : These are deermined in equilibrium. (c) By choosing di eren values for he parameers, we can analyse how he oupus of he problem vary. 8