Lecture 1A: Monocentric City Model

Transcription

1 Econ 4935 Urban Economics Lecture A: Monocentric City Model Instructor: Hiroki Watanabe Fall 22 Watanabe Econ 4935 A Monocentric City / 9 Central Business District 2 Closed Monocentric City Model 3 Income/Geographic Stratification 4 Examples 5 Open Monocentric City Model 6 City Limit 7 Now We Know Watanabe Econ 4935 A Monocentric City 2 / 9 Central Business District Questions Where Do City Residents Work? The Rise of Monocentric Cities 2 Closed Monocentric City Model 3 Income/Geographic Stratification 4 Examples 5 Open Monocentric City Model 6 City Limit 7 Now We Know Watanabe Econ 4935 A Monocentric City 3 / 9

2 Questions Project. (Urban Design) Design your dream city on a map. Draw your worst city on the flip side. Consider the following: Decide your designing principle. What does your city does/doesn t achieve? 2 Your city attracts/keeps away people because...? 3 Your city is better/worse than St. Louis (MSA or city etc.) because...? 2 Compare your city with your neighbors. Are they similar to each other? If they are located side by side, how do they react to each other? Watanabe Econ 4935 A Monocentric City 4 / 9 Questions Question.2 (Agenda) Why does St. Louis City has a high population density than the rest of the counties in St. Louis MSA? 2 Why don t the rich live in the city? Does low income class suburbia exist? 3 How does the betterment of urban transportation affect the land use patterns? 4 Where does the city end and where does the farmland begin? What determines the city border? 5 How does Chicago suck population out of St. Louis? Watanabe Econ 4935 A Monocentric City 5 / 9 Where Do City Residents Work? Employment inside the central business district (Census): M M M Watanabe Econ 4935 A Monocentric City 6 / 9

3 Where Do City Residents Work? Employment within 3 and miles of the city center: Total Emp 3 mi (%) mi (%) Indianapolis 636K Portland 763K 3 76 Boston,52K 4 76 Minneapolis,295K 2 64 Atlanta,65K 4 43 Los Angeles 4,68K 8 28 Maps by O Sullivan [O S8] to follow. Watanabe Econ 4935 A Monocentric City 7 / 9 Where Do City Residents Work? Indianapolis, IN Watanabe Econ 4935 A Monocentric City 8 / 9 Where Do City Residents Work? Atlanta, GA Watanabe Econ 4935 A Monocentric City 9 / 9

4 CBD Closed City Stratification Examples Open City City Limit Where Do City Residents Work? Watanabe CBD Econ 4935 Closed City A Monocentric City Stratification Examples Open City / 9 City Limit Where Do City Residents Work? Watanabe CBD Econ 4935 Closed City A Monocentric City Stratification Examples Open City / 9 City Limit The Rise of Monocentric Cities Industrial revolution of 9th century generated scale economies. Firms benefit by co-locating Innovation in transportation: Omnibus Cable cars Electric trolley Subways wider exploitation of comparative advantage decrease in travel cost increased feasible radius of city resulting in heavy concentrations of employment in metro center. Watanabe Econ 4935 A Monocentric City 2 / 9

5 Central Business District 2 Closed Monocentric City Model Land in Urban Economic Theory Landscape Feasible Allocation & Equilibrium Bid-Rent Approach to Equilibrium 3 Income/Geographic Stratification 4 Examples 5 Open Monocentric City Model 6 City Limit 7 Now We Know Watanabe Econ 4935 A Monocentric City 3 / 9 Land in Urban Economic Theory You can enjoy tea and cheesecake at the same time. You cannot simultaneously occupy two houses at different locations. Land does not exhibit convexity. Consider an extreme example: (, 2 ) = max{2, 2 }, where is land in a city and 2 is land in a suburb. Watanabe Econ 4935 A Monocentric City 4 / 9 Land in Urban Economic Theory Indifference Curves Land in Suburb (sq ft) Land in City (sq ft) Watanabe Econ 4935 A Monocentric City 5 / 9

6 Land in Urban Economic Theory We will always get a corner solution. Not really informative. Instead, we take land and other commodities first and analyze the location choice later. Watanabe Econ 4935 A Monocentric City 6 / 9 Landscape References: Fujita Ch 2 [Fuj89] Arnott & McMillen Ch 6 [AM8] Brueckner Ch 2 & 3 [Bru] Wassmer Ch 8 [Was]. Watanabe Econ 4935 A Monocentric City 7 / 9 Landscape Assumption 2. (Landscape) One member of household commutes to employment area (at the center of the metropolitan area). Commuting cost is the only location factor People can switch their locations for free. Commodities ship free. Monetary a cost of commuting Noncommuting travel insignificant Ubiquitous public services, taxes, and amenities Land consumption is normal. a not time Watanabe Econ 4935 A Monocentric City 8 / 9

7 Landscape Land supply at distance r from central business district (CBD) is if the city area is a long strip of land: L(r) = linear city 2πr if the city is round: circular city Watanabe Econ 4935 A Monocentric City 9 / 9 Landscape Population N R +. Edge of the city (city limit) r. Endowment (amount of composite goods each consumer is endowed with). Total composite goods in economy: N. Watanabe Econ 4935 A Monocentric City 2 / 9 Landscape Utility (s, z). s(r) = land consumption by consumers distance r from CBD. Assumed normal. 2 z(r) = numéraire composite good consumption by consumers distance r from CBD. N.B. baskets can mean two things in our model: z, the number of baskets to be consumed 2 a unit of measure as in rent is 5 baskets. Jack Donaghy (absentee landlord) likes only z: D (z D, s D ) = z D. Watanabe Econ 4935 A Monocentric City 2 / 9

8 Landscape n(r) = population at distance r from CBD. A consumer at r has to incur tr in commuting to and from CBD. He reserves some of his endowment to pay for commuting. An allocation is a triplet z(r), s(r), z D. Watanabe Econ 4935 A Monocentric City 22 / 9 Landscape Cows Liz Kenneth Jenna Toofer Pete Watanabe Econ 4935 A Monocentric City 23 / 9 Landscape Land Supply L(r), Land Consumption s(r) (mi 2 ) pi 8pi 6pi 4pi 2pi Example Land Use Pattern Land Supply L(r) Land Consumption s(r) Population n(r) Distance r from 3 Rock (mi) Watanabe Econ 4935 A Monocentric City 24 / Population n(r)

9 Feasible Allocation & Equilibrium Definition 2.2 (Feasible Allocation) A feasible allocation is an allocation s(r), z(r), z D such that r n(r)dr = N = r L(r) s(r) dr. r 2 z(r)n(r)dr + zd + r n(r)trdr = N. r n(r)dr: Population at each location must add up to total population N. r L(r) L(r) dr: Location r accommodates s(r) s(r) residents. The total needs to come out to N. Watanabe Econ 4935 A Monocentric City 25 / 9 Feasible Allocation & Equilibrium 2 There are N composite goods in the economy, z(r)n(r) of which is consumed at location r. 2 z D of which is consumed by Jack. 3 tr n(r) of which is spent in commuting. ➊ and ➌ are summed over r (i.e., across the city). Watanabe Econ 4935 A Monocentric City 26 / 9 Feasible Allocation & Equilibrium Typical Consumption Pattern and Population Consumption, Pop (baskets, ft 2, persons) z(r) s(r) n(r) CBD Distance r from CBD (mi) Watanabe Econ 4935 A Monocentric City 27 / 9

10 Feasible Allocation & Equilibrium How do we define the equilibrium? Who does what to whom in this economy? Watanabe Econ 4935 A Monocentric City 28 / 9 Feasible Allocation & Equilibrium Transportation Commuting tr w Kenneth at r p(r )s(r ) Commuting tr Liz at r p(r)s(r) w s(r ) Jack π r 2 s(r) Watanabe Econ 4935 A Monocentric City 29 / 9 Feasible Allocation & Equilibrium Definition 2.3 (Equilibrium) An equilibrium is a feasible allocation s(r), z(r), z D and rent p(r) such that consumers max r,s,z (s(r), z(r)) subject to = z(r) + s(r)p(r) + tr () at all r. Given the equilibrium rent p(r), a consumer picks up a location r and (z, s) that achieves the highest (s, z). Watanabe Econ 4935 A Monocentric City 3 / 9

11 Bid-Rent Approach to Equilibrium Question 2.4 (Equilibrium Utility Level) Can suburban residents achieve higher utility level than city residents in equilibrium in a monocentric city model? How about in reality? Watanabe Econ 4935 A Monocentric City 3 / 9 Bid-Rent Approach to Equilibrium Let s say Liz at r = 2 enjoys higher utility than Kenneth at r = 3. What will happen to p(2), p(3) and utility levels at two locations? Recall Assumption 2. : Land satisfies law of demand. 2 2 There is no moving cost. 2 Note that if land is normal, it satisfies LOD. Watanabe Econ 4935 A Monocentric City 32 / 9 Bid-Rent Approach to Equilibrium Let c be the equilibrium utility level (shared across the city). Since c is location independent but p(r) is not, we can pose the utility maximization problem () in this way: How much are you willing to spend for a unit of land at location r while maintaining the utility level at c. Watanabe Econ 4935 A Monocentric City 33 / 9

12 Bid-Rent Approach to Equilibrium Discussion 2.5 (Bid-Rent Trial) Suppose you have a monthly allowance of $2K and you have three locations to choose from: Downtown (r = ) 2 Central West End (r = 5) 3 Clayton (r = ) Commuting cost is $ per mile per month. What is the maximum unit rent you can spare at each location? 2 What factors do you take into consideration when you determine your bid rent? Watanabe Econ 4935 A Monocentric City 34 / 9 Bid-Rent Approach to Equilibrium Definition 2.6 (Bid-Rent Function) A bid-rent function is the maximum rent that a consumer can pay while achieving the equilibrium utility level c, i.e., p(r) := max s(r),z(r) tr z(r) s(r) : (s(r), z(r)) = c. Note p(r) is a unit rent (i.e. p(r) baskets/ft 2 or $/ft 2 ). What does a bid-rent function look like? Watanabe Econ 4935 A Monocentric City 35 / 9 Bid-Rent Approach to Equilibrium Recall at the optimal bundle. MRS(z(r), s(r)) = p(r) We know what value MRS takes at each r. We also know that everyone receives (s, z) = c in equilibrium. The next graph is the most important graph in this lecture. It shows how the bid-rent function is derived. Watanabe Econ 4935 A Monocentric City 36 / 9

13 Bid-Rent Approach to Equilibrium Composite Goods z (baskets) w z() w 5t z(5) Consumption Bundles at Different Locations u(s, z)=c w=z+p()s w=z+p(5)s+5t s() w/p() s(5) (w 5t)/p(5) Land s (ft 2 ) Watanabe Econ 4935 A Monocentric City 37 / 9 Bid-Rent Approach to Equilibrium Typical Bid Rent Function p() p(r)= MRS(s(r), z(r)) Bid Rent (baskets) p(5) 3 Rock 5 Distance r from CBD (miles) Watanabe Econ 4935 A Monocentric City 38 / 9 Bid-Rent Approach to Equilibrium Proposition 2.7 (Muth-Mills Condition) In equilibrium, the slope of the bid-rent function p(r) is given by dp(r) = t dr s(r). Proof. Use envelope theorem (describes the behavior of p(r) against the change in r). Watanabe Econ 4935 A Monocentric City 39 / 9

14 Bid-Rent Approach to Equilibrium Interpret Proposition 2.7 as: dp(r) dr s(r) = t Δp(r) s(r) = tδr. } {{ } LHS: RHS: Watanabe Econ 4935 A Monocentric City 4 / 9 Central Business District 2 Closed Monocentric City Model 3 Income/Geographic Stratification 4 Examples 5 Open Monocentric City Model 6 City Limit 7 Now We Know Watanabe Econ 4935 A Monocentric City 4 / 9 Source: O Sullivan [O S8]. Watanabe Econ 4935 A Monocentric City 42 / 9

15 Why do the poor occupy land near the center? Why does household income generally increase as we move outward? Watanabe Econ 4935 A Monocentric City 43 / 9 What if endowment is different among households? Who lives near the CBD? Suppose we have two classes: rich & poor. Muth-Mills condition implies: dp R (r) dr = t s R (r) and dpp (r) dr = t s P (r). Suppose somewhere in the city, p R (r ) = p P (r ). Watanabe Econ 4935 A Monocentric City 44 / 9 Land is a normal good. If both live at r, s P (r ) < s R (r ), i.e., t s P (r) = dpp (r) < dpr (r) = t dr dr s R (r) (2) at r. p P (r) is steeper at r. (2) holds everywhere. p R (r) intersects p P (r) at most once, and if it does, it crosses p P (r) from northwest to southeast. Watanabe Econ 4935 A Monocentric City 45 / 9

16 Income Stratification p P (r) p R (r) Bid Rent (baskets or $) CBD r Distance r from CBD (miles) Watanabe Econ 4935 A Monocentric City 46 / 9 Other explanations: deteriorating central housing quality, new suburban housing Fleeing central-city problems (negative neighborhood externality) Suburban zoning excludes low-income household The existence of racial preferences Tiebout mechanism Watanabe Econ 4935 A Monocentric City 47 / 9 Central Business District 2 Closed Monocentric City Model 3 Income/Geographic Stratification 4 Examples Example: Quasilinear Preferences Comparative Statics Example: Perfect Complements 5 Open Monocentric City Model 6 City Limit 7 Now We Know Watanabe Econ 4935 A Monocentric City 48 / 9

17 Example: Quasilinear Preferences Example 4. (Quasilinear Preferences) Consider a long strip of land (L(r) = from r = to r = r). Suppose that a consumer s preferences are represented by (s, z) = log(s) + z. The marginal rate of substitution at (s, z) is s. What is the budget constraint? 2 What is the tangency condition? Watanabe Econ 4935 A Monocentric City 49 / 9 Example: Quasilinear Preferences Indifference Curves 2 u(s, z)=c Composite Goods z (baskets) Land s (ft 2 ) Watanabe Econ 4935 A Monocentric City 5 / 9 2 Example: Quasilinear Preferences Budget constraint: = z(r) + p(r)s(r) + tr Relative price = p(r). MRS at (s(r), z(r)) is s(r). Tangency condition: s(r) = p(r). Watanabe Econ 4935 A Monocentric City 5 / 9

18 2 Example: Quasilinear Preferences Indifference Curves and Budget Constraints Composite Goods z (baskets) (w 5t)/p(5) w/p() Indifference Curves w=z+p()s w=z+p(5)s+5t w p()s() w p(5)s(5) 5t Land s (ft 2 ) Watanabe Econ 4935 A Monocentric City 52 / 9 Example: Quasilinear Preferences s(r)p(r) = 2 z(r) = tr. 3 3 ( tr) + log(s) = c 4 s(r) = e c ++tr. 5 n(r) = s(r) = e c+ tr 6 dn(r) dr /n(r) = t (population drops t% every one ft, called population gradient in empirical literature). 3 need > to avoid corner solutions. Watanabe Econ 4935 A Monocentric City 53 / 9 Example: Quasilinear Preferences 7 N = r n(r)dr c = log N + log t + log( e t r ). 8 p(r) = s(r) = e c+ + tr. 9 dp(r) dr = s (r)s(r) 2 = t (Muth-Mills condition). s(r) z D = r p(r)l(r)dr = N. 4 Feasibility condition for composite goods is met: r (z(r) + tr)n(r)dr + zd = N. 4 The landlord consumes the sum of whatever the residents pay for their rent Watanabe Econ 4935 A Monocentric City 54 / 9

19 Example: Quasilinear Preferences Example: r =, = 5, t =, N =. Consumption Bundles at Different Locations z(r) s(r) p(r) n(r) Distance r from CBD (miles) Watanabe Econ 4935 A Monocentric City 55 / 9 Comparative Statics What if t drops (installation of Metrolink extension or revamped I-64 for example)? What if population N or city limit r change? What does those environmental changes do to bid rent, lot size or commodity consumption in equilibrium? Watanabe Econ 4935 A Monocentric City 56 / 9 Comparative Statics Definition 4.2 (Comparative Statics) Variables predetermined outside the model are called exogenous variables. Variables whose values are determined in the model are called endogenous variables. Comparative statics analyzes the change in equilibrium values (z(r) etc, endogenous variables) corresponding to the change in one particular parameter (t etc, exogenous variables). Watanabe Econ 4935 A Monocentric City 57 / 9

20 Comparative Statics It s not that hard actually. Our result is given as a function of parameters rather than a numeric value. All we need to do is compare two equilibria with different parameters: z(r) t= = r to z(r) t=2 = 2r. Which one is larger? And what does this imply? How doe we interpret the result? Watanabe Econ 4935 A Monocentric City 58 / 9 Comparative Statics Let s say t = rises to t = 2. 5 Take Example 4. for example. s(r) = e c ++tr p(r) = e c+ tr z(r) = tr z D = N, where c := log N + log t + log( e t r ) is a constant. 5 Metrolink permanently reduced the service frequency for example. Watanabe Econ 4935 A Monocentric City 59 / 9 Comparative Statics Comparative statics compares the following: s(r) t= = e c ++r vs s(r) t=2 = e c ++2r p(r) t= = e c+ r vs p(r) t=2 = e c+ 2r z(r) t= = r vs z(r) t=2 = 2r z D t= = N vs z D t=2 = N Or, if you know how to take partial derivatives, that works too. Watanabe Econ 4935 A Monocentric City 6 / 9

21 Comparative Statics Example: r =, = 5, t =, N = compared to r =, = 5, t = 2, N = Consumption Bundles z(r) t= s(r) t= p(r) t= (=n(r) t= ) z(r) t=2 s(r) t=2 p(r) t=2 (=n(r) t=2 ) Distance r from CBD (miles) Watanabe Econ 4935 A Monocentric City 6 / 9 Example: Perfect Complements Example 4.3 (Perfect Complements) A resident in a circular city has preferences represented by (s(r), z(r)) = min{s(r), rz(r)} and endowed with baskets of composite goods. Find s(r) and z(r). 2 What is the population at r? 3 What is the efficient allocation? 4 Does the equilibrium exist? Watanabe Econ 4935 A Monocentric City 62 / 9 Example: Perfect Complements Composite Goods z() (baskets) Indifference Curves at r= Indifference Curves z=s/r Land s() (ft 2 ) Watanabe Econ 4935 A Monocentric City 63 / 9

22 Example: Perfect Complements Composite Goods z(2) (baskets) Indifference Curves at r=2 Indifference Curves z=s/r Land s(2) (ft 2 ) Watanabe Econ 4935 A Monocentric City 64 / Example: Perfect Complements Recall that a consumer s preferred ratio of commodities is s(r) : z(r) = r :. 2 Then the optimal bundle satisfies z(r) = s(r)/r. (3) 3 Together with the budget constraint, (3) implies tr = z+prz+tr (z(r), s(r)) = + p(r)r, tr + p(r)r r. Watanabe Econ 4935 A Monocentric City 65 / 9 Example: Perfect Complements 4 n(r) = L(r) +p(r)r = 2π s(r) tr 5 Suppose that everyone achieves (z, s) = c in the end: c = rz = tr + p(r)r r p(r)r = r ( tr) c (4) p(r) = c ( tr) r Watanabe Econ 4935 A Monocentric City 66 / 9

23 Example: Perfect Complements 6 Plug (4) in 4: n(r) = 2π tr 7 Population n(r) adds up to N: + rc ( tr) = 2π r c. r n(r)dr = N π r2 c = N c = π r2 N. (How is city area size and total population related to the equilibrium utility level?) Watanabe Econ 4935 A Monocentric City 67 / 9 Example: Perfect Complements 8 For any r( r), (z(r), s(r)) = c = π r2 N z(r) = π r2 Nr s(r) = π r2 N (Do they seem reasonable? Carry out comparative statics.) 9 Bid-rent function: p(r) = tr c r = N( tr) π r 2 r. Watanabe Econ 4935 A Monocentric City 68 / 9 Example: Perfect Complements Example: r =, N =, =, t =. Bid Rent Function Bid-Rent Function p(r) = N(w tr)/(π r 2 ) / r.5 Bid Rent p(r) ($) Distance r from CBD (miles) Watanabe Econ 4935 A Monocentric City 69 / 9

24 Example: Perfect Complements If you live near CBD, you get some commodities back (no matter how large the endowment is). There is no equilibrium for any. Second welfare theorem fails. WT2: Under certain conditions, with convex preferences, if endowments can be redistributed, then any efficient allocation is an equilibrium allocation. Watanabe Econ 4935 A Monocentric City 7 / 9 Central Business District 2 Closed Monocentric City Model 3 Income/Geographic Stratification 4 Examples 5 Open Monocentric City Model Open/Closed Monocentric City Model Comparative Statics on City Size 6 City Limit 7 Now We Know Watanabe Econ 4935 A Monocentric City 7 / 9 Open/Closed Monocentric City Model Discussion 5. (City Size) Have a listen to NPR Clip. While Niagara Falls seems determined to attract residents, New York City, for example, does not directly incentivize people to move in or kick people out. What determines the size of a city? Watanabe Econ 4935 A Monocentric City 72 / 9

25 Open/Closed Monocentric City Model Let s see what monocentric city model has to say. So far, we did not allow interurban migrations. If you were born in St. Louis, you get your job in St. Louis. N is constant and equilibrium utility level c is determined within each city. This type of model is called a closed monocentric city model. Watanabe Econ 4935 A Monocentric City 73 / 9 Open/Closed Monocentric City Model What happens if we allow interurban migrations? For simplicity, assume that there are many identical cities in the US. Since people come and go as they see fit, N is no longer exogenous. Question 5.2 (Equilibrium Utility Level with Interurban Migration) What about c? Watanabe Econ 4935 A Monocentric City 74 / 9 Open/Closed Monocentric City Model The analogous argument to Question 2.4 in interurban context than in intraurban. If this c is different from what other cities have to offer (c USA ), people move in if c > c USA. 2 move out if c < c USA This changes N and we have to revise the equilibrium to find a new c. Watanabe Econ 4935 A Monocentric City 75 / 9

26 Open/Closed Monocentric City Model Our model will not give definite predictions unless c = c USA. c is predetermined at the national level and we take c as given (exogenous variable). This type of model is called an open monocentric city model. Watanabe Econ 4935 A Monocentric City 76 / 9 Open/Closed Monocentric City Model model type city size N equilibrium utility level c closed exogenous endogenous open endogenous exogenous Watanabe Econ 4935 A Monocentric City 77 / 9 Open/Closed Monocentric City Model Open monocentric city model can address: Does ameliorating congestion help attract people? If so, by how much? 2 How many households will move in if the city expand its boundary? 3 How is intraurban land use pattern affected by interurban relationships? Watanabe Econ 4935 A Monocentric City 78 / 9

27 Example 5.3 (Open Monocentric City Model) Consider an open monocentric city model, where the nationwide utility level is given by a constant c. Preferences are represented by a Cobb-Douglas function of the form (s, z) = sz. MRS at (s, z) is z. Suppose that everyone is endowed s with (> t r). What is the budget constraint of a typical resident? 2 What is the tangency condition? 3 Find the bid-rent function. Watanabe Econ 4935 A Monocentric City 79 / 9 Budget constraint: 2 Tangency condition: z(r) + p(r)s(r) + tr. (5) p(r) = z(r) s(r). (6) 3 (5) and (6) imply z(r) = tr 2 and s(r) = z(r) p(r). 4 In equilibrium, the city has to offer (s, z) = c: (s, z) = z z p = c p(r) = (z(r))2 tr 2 = c 2 c. (Bid-rent function). Watanabe Econ 4935 A Monocentric City 8 / 9 Example: r =, =, t =, c = (same parameterization for the following graphs as well). Bid Rent Function.25.2 Bid Rent p(r) ($).5. p(r) Distance r from CBD (miles) Watanabe Econ 4935 A Monocentric City 8 / 9

28 Bid Rent p(r).2 Distance r from CBD Distance r from CBD Watanabe Econ 4935 A Monocentric City 82 / 9 Bid Rent p(r) Distance r from CBD Distance r from CBD Watanabe Econ 4935 A Monocentric City 83 / 9 5 Verify Muth-Mills condition: dp(r) dr = dc dz2 dr z2 +c dr = c 2z t 2 = (zs) z( t) = t s(r). 6 s(r) = z(r) p(r) = 2c tr. 7 n(r) = L(r) s(r) πr( tr) =. c Watanabe Econ 4935 A Monocentric City 84 / 9

29 p(r), n(r), z(r), n(r)/l(r) (baskets, people, people/mi 2 ).5.25 Consumption at Different Locations p(r) n(r) z(r) n(r)/l(r) s(r) Distance r from CBD (miles) Watanabe Econ 4935 A Monocentric City 85 / 9 4 Land s(r) (ft 2 ) Population n(r).7.6 Distance r from CBD Distance r from CBD. Watanabe Econ 4935 A Monocentric City 86 / 9 Population n(r) Distance r from CBD Distance r from CBD Watanabe Econ 4935 A Monocentric City 87 / 9

30 St. Louis MSA (Log Population by County), Watanabe Econ 4935 A Monocentric City 88 / Population Denstiy n(r)/l(r).45.4 Distance r from CBD Distance r from CBD.5 Watanabe Econ 4935 A Monocentric City 89 / 9 Population n(r) Distance r from CBD Distance r from CBD Watanabe Econ 4935 A Monocentric City 9 / 9

31 St. Louis MSA (Log Population Density), Watanabe Econ 4935 A Monocentric City 9 / 9 22 Watanabe Econ 4935 A Monocentric City 92 / 9 Watanabe Econ 4935 A Monocentric City 93 / 9

32 CBD Closed City Stratification Examples Open City City Limit Watanabe CBD Econ 4935 Closed City A Monocentric City Stratification Examples Open City 94 / 9 City Limit Watanabe Econ 4935 Closed City CBD A Monocentric City Open City Examples Stratification 95 / 9 City Limit R r n(r)dr = 8 N (endogenous) = 9 Feasibility condition is met: π r 2 c 2 + t 3 r. R r R r R r z(r)n(r)dr + p(r)l(r)dr + trn(r)dr R r = (z 2 Lc + z 2 Lc + trzlc )dr R r = Lzc dr = N. Watanabe Econ 4935 A Monocentric City 96 / 9

33 Comparative Statics on City Size Comparative statics on N(, c, r, t) = π r2 c N(,, r, t) = π r2 2 + t r. 3 N(, 2, r, t) = π r t r. 3 If =, r =, t =, then N(c = ) = 6 π and N(c = 2) = 2 π. In summary, N( ) 2 + t r : 3 c t r N π r 2 2c > cn < πc r 3 3 < πc ( r t r 2 ) > Watanabe Econ 4935 A Monocentric City 97 / 9 Comparative Statics on City Size Example: =, r =, t =, or c =..6 N( r).55 N(c) City Size N.3 City Size N City Limit r Nationwide Utility Level c.6 N(t).4.2 City Size N Transportation Cost t Watanabe Econ 4935 A Monocentric City 98 / 9 Central Business District 2 Closed Monocentric City Model 3 Income/Geographic Stratification 4 Examples 5 Open Monocentric City Model 6 City Limit 7 Now We Know Watanabe Econ 4935 A Monocentric City 99 / 9

34 So far, the city limit r was given outside the model (exogenous). If not, how far does the residential area spread? Recall that bid-rent function is monotone decreasing. Suppose that land rent for agricultural purpose is p A. If p(r) > p A, then this location is part of the city. If p(r) < p A, then this location is part of farmland. Watanabe Econ 4935 A Monocentric City / 9 On the city border, we have p( r) = p A. Overall bid-rent function is p(r) if p(r) pa b(r) = max{p(r), p A } = p A if p(r) < p A. Watanabe Econ 4935 A Monocentric City / 9 Bid Rent Function Residential Bid Rent Function p(r) Agricultural Rent p A Bid Rent ($) 3 Rock City Limit Distance r from CBD (miles) Watanabe Econ 4935 A Monocentric City 2 / 9

35 Example 4. (closed city): Suppose pa = e +c e t. On the border, p( r) = e +c e t r = r A = e +c e t r =. Example 5.3 (open city): Suppose pa =. On the 6c border, p( r) = r = 2t. t r 2 2 c = r A = 6c Does new I-64 expand the city limit? Watanabe Econ 4935 A Monocentric City 3 / 9 Central Business District 2 Closed Monocentric City Model 3 Income/Geographic Stratification 4 Examples 5 Open Monocentric City Model 6 City Limit 7 Now We Know Watanabe Econ 4935 A Monocentric City 4 / 9 Closed / open monocentric city model Bid-rent function Muth-Mills condition Income/geographic stratification Comparative statics City limit Watanabe Econ 4935 A Monocentric City 5 / 9

36 CBD Closed City Stratification Examples Open City City Limit References [AM8] Richard J. Arnott and Daniel P. McMillen. A Companion to Urban Economics. Blackwell, 28. [Bru] Jan K. Brueckner. Lectures on Urban Economics. MIT Press, 2. [Fuj89] Masahisa Fujita. Urban Economic Theory: Land Use and City Size. Cambridge, 989. [O S8] Arthur O Sullivan. Urban Economics. McGraw-Hill, 28. [Was] Robert W. Wassmer. A Readings in Urban Economics: Issues and Public Policy. Blackwell, 2. Watanabe CBD Econ 4935 Closed City A Monocentric City Stratification Examples Open City 6 / 9 City Limit Map du Jour Source Watanabe CBD Econ 4935 Closed City A Monocentric City Stratification Examples Open City 7 / 9 City Limit Airline du Jour Today s color theme is provided by courtesy of Watanabe Econ 4935 Air New Zealand A Monocentric City 8 / 9

37 absentee landlord, 2 agricultural rent, allocation, 22 bid-rent function, 35 overall, b(r), see overall bid-rent function c, see equilibrium utility level circular city, 9 city limit, 2, closed city, 73 commuting, 22 comparative statics, 57 composite good consumption, 2 convexity, 4 c USA, see equilibrium utility level distance, 9 endogenous variables, 57 endowment, 2, 44 equilibrium, 3 equilibrium rent, 3 equilibrium utility level, 3, 33, 75 exogenous variables, 57 feasible allocation, 25 L(r), see land supply land consumption, 2 land supply, 9 law of demand, 32 linear city, 9 moving cost, 32 Muth-Mills condition, 39, 44 N, see population n(r), see population numéraire, 2 open city, 76 overall bid-rent function, p A, see agricultural rent population, 2, 22 population gradient, 53 p(r), see rent r, see distance r, see city limit rent, 3 s(r), see land consumption second welfare theorem, 7 t, see commuting Tiebout, 47 ( ), see utility utility, 2, see endowment z(r), see composite good consumption z D, see absentee landlord