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# Urban Economics ECN 741

Syracuse

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This 43 page Class Notes was uploaded by Hannah Hahn on Wednesday October 21, 2015. The Class Notes belongs to ECN 741 at Syracuse University taught by John Yinger in Fall. Since its upload, it has received 40 views. For similar materials see /class/225647/ecn-741-syracuse-university in Economcs at Syracuse University.

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Date Created: 10/21/15

Notes on The Theory of Urban Residential Structure The Basic Urban Model John Yinger Professor of Economics and Public Administration The Maxwell School Syracuse University August 2002 The Basic Urban Model Page 1 1 Introduction These notes present the basic model that has been the foundation of the eld of urban economics This model contains the central features of the models developed by Alonso 1964 Muth 1969 and Mills 1967 1972 It is based on a series of strong assumptions about various aspects of urban structure These assumptions which are presented in the next section have served as an agenda for the eld of urban economics in the sense that scholars have searched for models with less restrictive versions of each assumption A list of selected studies that weaken these assumptions is provided at the end of these notes Urban models have two key features The first is that they are built around a household maximization problem in which a household decides how much housing to consume and where to live This problem is presented in Section 3 The key new analytical tool that emerges from this problem is the bid function which indicates the amount a household is willing to pay for housing at every location Bidfunctions are derived in Section 4 Another important element of an urban model housing production is introduced in Section 5 and competition between housing and other uses of land is considered in Section 6 The second key feature is that urban models inevitable are general equilibrium models in the sense that they involve more than one market The basic source of an urban model can be described in a simple sentence Households care about where they live because they must commute to work This simple statement contains elements of six markets housing because people live in housing land and capital because housing is built on land using capital transportation because people must use some form of transportation to get to work labor because people are commuting to a job and the market for some unstated product that is exported from the urban area because a firm will not hire people The Basic Urban Model Page 2 unless it has a product to sell Thus an urban model is not complete unless it includes some analysis however rudimentary of each of these six markets that is unless it is a general equilibrium model These notes begin by focusing on the basic household maximization problem The models in the rst six sections are all incomplete in the sense that they do not ensure either that all available land is used or that there is enough housing for all urban workers To put it another way they are incomplete because they do not ensure equilibrium in the land and labor markets Equilibrium conditions for these markets are introduced in Section 7 resulting in a complete urban model Two types of complete urban model have appeared in the literature closed models rule out migration between urban areas whereas open models assume that migration is costless The distinction between open and closed models is explored in Section 7 a closed model is solved in Section 8 and an open model is solved in Section 9 The last three substantive sections in these notes examine the conclusions that can be drawn from urban models Section 10 shows how population density and building height vary with location Section 11 shows how the results of the model change as key parameters change that is it presents some comparative static analysis and Section 12 compares the predictions of the model with some evidence about actual urban residential structure A series of exercises is provided at the end ofthese notes The Basic Urban Model Page 3 2 Key Assumptions This section presents the multipart assumptions that form the foundation of the standard urban model As noted earlier each of the assumptions is quite restrictive and each one has been relaxed in the literature See the references at the end of these notes Assumption 1 Housing Demand Household utility functions depend on a composite consumption good and housing and take the CobbDouglas form Let U stand for utility Z measure the consumption of the composite good and H measure the consumption of housing As explained more fully below H is expressed in units of housing services This assumption indicates that the utility function can be written U ocl lnZ x2 lnH where x1 and x2 are constants and In indicates a natural logarithm Remember that a household s decisions are invariant to any monotonic transformation of its utility function1 Equivalent expressions for this utility function therefore are U l alnZalnH where a a2 a1 a2 and U ZWWH 2 A note on notation Throughout these notes curly brackets and are used to enclose the argument of a function whereas parentheses and square brackets are used to clarify algebraic expressions The expression lnZ for example indicates that Z is the argument of the In function Assumption 2 Housing supply Housing services are produced with land and capital according to a CobbDouglas production function with constant returns to scale and housing is owned by absentee landlords The first part of this assumption can be expressed by the equation H K 139 L where K stands for capital and L stands for land This production function focuses on the production of 1 This result is proved in most microeconomic texts The Basic Urban Model Page 4 housing services from existing structures which require little labor input Note that this approach ignores the role of maintenance rehabilitation and conversion activities in the provision of housing Assumption 3 The Transportation Network Households commute between place of residence and place of work in a straight line at a constant transportation cost per mile using a single transportation mode The traditional phasing of this assumption has been that the urban area is located on a featureless plain In addition however most urban models ignore any complexities in commuting patterns that arise because of the character of the transportation network There is no distinction between commuting arteries and side streets between radial streets and a street grid or between various transportation modes such as cars buses and subways and tra ic congestion is not considered Instead the implicit transportation network is an abstract one in which everyone travels along the shortest distance between home and work at a constant cost per mile Assumption 4 Why Location Matters In selecting a place to live distance to work is the only locational characteristics households care about This assumption rules out neighborhood amenities Households are assumed not to care about access to shopping or recreation or about air quality about the characteristics of their neighbors or about any other feature of the location where they live Assumption 5 Types of Households All households are alike According to this assumption all households have the same income family composition and utility function 2 Note that the symbols an N are used to distinguish among various forms of the utility function For convenience the last form has no distinguishing symbol The Basic Urban Model Page 5 Assumption 6 Labor Market Income is fixed and all households have a single worker with a job in the central business district CBD These assumptions deal with the labor market The first part of this assumption which is built on an implicit view of the labor market that is discussed in Section 7 greatly simpli es the analysis of an urban model Combined with Assumption 2 the second part of this assumption implies that a household s commuting cost depends only on the distance between its residence and the CBD Assumption 7 Household Mobility All households are perfectly mobile within an urban area that is if they have an opportunity to improve their utility they will take it This assumption is fundamental to the logic of urban models As we will see it implies that all households achieve the same utility level Assumption 8 Local Governments There are no local governments The standard urban model leaves out the local government sector 3 The Basic Household Maximization Problem An urban model is built around a household maximization problem in which the household decides how much housing and how much of the composite good to consume as well as where to live This section presents this problem with general utility and transportation cost functions Section 4 derives specif1c solutions with the functions in Assumptions 1 and 2 The two goods in a household s utility function are a composite good and housing The composite good labeled Z is measured in units that have a price of unity It consists of all consumption goods except housing Housing is measured in units of housing services H which are the annual services provided by a house or apartment The quantity of housing services in a given housing unit is a The Basic Urban Model Page 6 function of the characteristics of that unit including its size number of rooms quality of construction and so on The basic urban model presented here like most urban models focuses on housing services and does not explicitly consider housing characteristics The annual price per unit of housing services is P As we will see this price depends on household location measured in miles from the CBD u that is P Pu We will also see that P39u the derivative of Pu with respect to u is negative In other words the price of housing services is lower at greater distances from the CBD Note that throughout these notes a prime is used to indicate the derivative of a single valued functions for example P39u dPu du Although according to Assumption 2 the models in these notes focus on the case of renters this approach to housing can be applied to owners or renters In the rental market an observed apartment rent equals the housing services in that apartment multiplied by the price per unit of housing services at that location In the sales market we observe the price of a house which is an asset not the annual value of the services from that asset which is the implicit rent However the amount someone would pay for a house that is expected to yield H units of housing services per year at price Pu per unit is the present value of the stream PuH over the lifetime of the house Let V stand for the market value let 139 be the household s real discount rate and let M be the house s expected lifetime Then the 3 appropr1ate present value express1on 1s 1 3 This expression leaves out several complications including the tax treatment of owneroccupied housing mortgage terms and conditions risk and uncertainty It implicitly assumes static expectations and requires a real interest rate For more on these issues see Yinger et al 1988 The Basic Urban Model Page 7 Because summations are dif cult to work with it will prove convenient to use an alternative formulation of this equation namely 4 V 2 z where t 2 3 l 1 1z39 M Note that f can be interpreted as the infinite horizon discount rate that is as the discount rate assuming an in nite horizon that is equivalent to a discount rate 139 with a horizon M If M is large above about 40 years then f is a close approximation for 139 and 2 simpli es to V 4 Moreover the annual ow associated with this asset value is simply 139 V PuH which is the same as apartment rent Under some conditions namely no downpayment no taxes and a mortgage rate equal to the opportunity cost of investing in housing this annual ow is observed in the form of the household s annual mortgage payment The worker in a household also must commute to work Let Tube the annual round trip cost of commuting from residential location u to the CBD A more distant location always implies greater commuting costs so that T39u gt 0 On the basis of this discussion the problem facing a household with daily income Y is to select ZH and u so as to 4 To derive equation 2 simply divide equation 1 by 1 i and subtract the result from equation 1 The Basic Urban Model Page 8 Maximize U Z H Subjectto YZPuHTu 5 The Langrangian expression for this problem is UZH1Y Z PuH Tu 6 where l is the Langrangian multiplier The rstorder conditions are 540 7 6U E Apuo 8 and lP39uHT39u0 9 Combining 7 and 8 leads to a standard result in microeconomics 6U6H2 6U6Z 10 In words the marginal rate of substitution between H and Z must be set equal to the ratio of the price of Hto the price of Z It will prove convenient to develop an alternative interpretation of this condition In a household maximization problem the Lagrangian multiplier associated with the budget constraint 1 can be interpreted as the marginal utility of income5 According to Equation 7 therefore the household sets 6U BZ equal to the marginal utility of income Thus the left side of 10 is the marginal utility of H divided by the marginal utility of income which is the same The Basic Urban Model Page 9 as the marginal value of H in dollar terms Another interpretation of 10 therefore is that the household sets the marginal dollar bene t from H equal to its marginal dollar cost The third condition is not found in standard microeconomics texts however It is the condition that introduces us to the logic of urban models The Lagrangian multiplier obviously can be canceled leaving P39uHT39u0 11 The first term is the marginal change in the cost of housing at the value of H selected by the household from moving one mile farther from the CBD The second term is the increase in transportation cost from moving one mile farther from the CBD Since T39 and H are positive this condition can only be satisfied at a point where P39 is negative that is where Pu declines as one moves away from the CBD More specifically it is satis ed at the location where the housing cost savings from moving a little farther out is just offset by the transportation cost increase Now comes the twist that drives urban models which is the fundamental insight provided by Alonso 1964 According to Assumptions 5 and 7 all households are alike and perfectly mobile If one location is best for one household it also is best for every other household However everyone obviously cannot live in the same spot Households compete for housing in a desirable location driving up the price there This competition ends only when the price has been driven up enough so that no household prefers that location to any other The same competition occurs at every location in the metropolitan area Locational equlibrium exists only when no household has an incentive to move that is only when households are indifferent across locations Because Equation 1 l is a condition for household 5 Again see a microeconomics text The Basic Urban Model Page 10 equilibrium this condition must hold at every location Thus another way to state the locational equilibrium condition is that P must satisfy the equation P39u JAM 12 A P function that satis es this equation indicates the maximum amount a household is willing to pay at one location relative to another and is therefore known as the equilibrium bid function As written Equation 12 is a differential equation the solution to which namely P depends on specific forms for T and the utility function If differential equations are not your cup of tea don t worry As shown in Section 4 you do not have to solve a differential equation to find P Section 4 also shows however that given Assumptions 1 and 3 this differential equation can easily be solved directly Before we find a specific solution for P it is worth noting that two of the properties of this bid function follow directly from Equation 12 First because T39u and H are positive P39u must be negative that is housing bids must be a downwardsloping function of distance Consumers will substitute toward H as its price drops so that housing consumption will increase with the distance from the CBD Because 7quot is constant by Assumption 3 it follows that the second property of P is that its slope declines in absolute value as one moves away from the CBD In other words substitution between housing and the composite good insures that the second derivative of Pu is positive Figure 1 presents an example of a bid function with these two properties The Basic Urban Model Page 11 Finally note that the equilibrium de ned by Equation 12 does not require any particular household to be in any particular location In fact an urban model does not determine where a household lives instead it determines what the price of housing would have to be for the people who live at each location to be content to stay there Something outside the model such as idiosyncratic preferences for housing or location must control which households end up at each location6 4 Locational Equilibrium 01 Deriving Bid Functions Now let us derive a speci c bid function using the assumptions in Section 2 The Cobb Douglas utility function in Assumption 1 takes the form7 U 1 a1nZa1nH 13 It follows that 6U1 14 BZ Z and 031 15 6H H Moreover 139 to A J quot 3 a 39 39 39 39 must pay a constant cost per mile t to commute to work If a household lives u miles form the CBD therefore its T equals tu This I should be interpreted as the daily roundtrip cost which includes both the operating costs 6 In a multiclass urban model the model can determine the set of locations in which a class locates but it cannot determine which household within a class lives in which of the locations allocated to that class 7 A note on notation In equations 5 through 10 and the accompanying text U stands for a general utility function Elsewhere U or 7 stand for specific utility functions such as equation 13 The Basic Urban Model Page 12 for running a car or paying bus fare and the opportunity cost of time spent commuting which depends on travel speed MPH and the valuation of travel time as a fraction of the wage w In symbols tt0tyY 16 where to stands for roundtrip operating costs per mile and ty stands for roundtrip time costs as a fraction of income Moreover the hourly wage rate is W8 and the time it takes to travel one mile is 1MPH so roundtrip time costs can be written as 2wY 81MPH wY 4MPH We can now derive the demand functions for Z and H First substitute the two marginal utility expressions 14 and 15 into the firstorder conditions 7 and 8 Then rearrange the two conditions to isolate l on one side equate the two expressions for A and solve the resulting equation for Z These steps yield Z 1 aaPuH Substituting this result into the budget constraint makes it possible to solve for the demand function for H which is H M 17 P u The demand function for Z follows directly Z1 aY tu 18 The bid function P can now be found in two different ways both of which will be presented The two approaches are both worth examining because each of them is for convenient for solving some types of morecomplicated models 41 The Indirect Utility Function Approach The first method which was popularized by Solow 1972 makes use of a concept The Basic Urban Model Page 13 known as an indirect utility function The basic idea of an indirect utility function is straightforward Direct utility functions depend on quantities of commodities By substituting the demand functions for these commodities into the direct utility function one can obtain utility as a function of income and prices which is an indirect utility function For a general discussion ofthis concept see a microeconomics text Substituting the demand functions 17 and 18 into the direct utility function yields the indirect utility function for our case Remember that household choices are invariant with respect to monotonic transformations of the utility function To simplify the algebra therefore we apply the exponential transformation to the direct utility lnction 13 Thus the indirect utility function is U kY39t 19 P u where k l 00141 a 20 and U is the level of utility Because households are alike and mobile by Assumptions 5 and 7 U must take the same value say U for every household We cannot determine what this level is without further assumptions but we know that it is the same for all households that is it is a constant Thus we can solve 19 for the bid function P The result la Pu5j Y tufa 21 01 The Basic Urban Model Page 14 PM vYtu1 a 22 where y is an unknown constant In a later section we will solve for this constant which is equivalent to solving for the utility level U One might say that 21 gives the general shape of the bid function but does not reveal its height In fact there exists a family of bid functions each member corresponding to a different utility level As illustrated in Figure 2 higher utility levels correspond to lower bid functions This link has strong intuitive appeal higher housing prices use up more of a household39s resources and with a constant income lead to lower utility 42 The Differential Equation Approach The second approach which was first used by Alonso 1964 Mills 1967 1972 and Muth 1969 is to interpret Equation 12 as a differential equation and solve it for P In the case of CobbDouglas utility functions this derivation is straightforward The derivation begins by substituting the transportation cost function Tu tu and the demand for housing Equation 17 into the locational equilibrium condition 12 These steps yield 23 01 mzaW m 24 Equation 24 is known as an exact differential equation because all the expressions involving Pu are on the left side and all the expressions involving u are on the right side Exact differential equations can be solved by integrating each side separately In this case both sides The Basic Urban Model Page 15 take the same general form namely g39u gu On the left side the g function is simply Pu and on the right side it is Y tu Thus both sides ofthe equation can be solved with the following basic integral g39lu du ln u K 25 l g gi where K is a constant of integration Applying 25 to both sides of 24 we obtain lnPulnY tuK 26 Applying the exponential function to both sides of this equation yields the bid function 27 01 Pu7Ytul a 28 where y equot is an unknown constant This is of course exactly the same result we obtained using an indirect utility function 5 Housing Production 01 Bringing in Land The next step in developing an urban model is to add housing production With this The Basic Urban Model Page 16 addition we can determine the pattern of land prices in an urban area According to Assumption 2 housing services are produced with land and capital Although labor obviously is an important input in the building of houses and apartments it is not a major input in the production of housing services from existing dwellings and is not considered a housing input in most urban models With the CobbDouglas form of Assumption 2 the quantity of housing services produced at locationuHS u is given by HS uAKuHLua 29 where K u is the amount of capital used for housing at location u Lu is the square miles of land available for housing at location u and a and A are constants Profitmaximizing housing rms set the value of the marginal product of each input equal to its price Because the production function has constant returns to scale payments to factors by pro tmaximizing rms exactly exhaust total revenue and the rm earns zero economic pro ts The marginal products are found by differentiating Equation 29 so the conditions for pro t maximization can be written lt1 agtPuHu Ku 3 and aplulelulzRM 31 The Basic Urban Model Page 17 where r is the annual rental rate for a unit of capital and Ru is the rent per acre of land at location u With a national capital market the rental rate for capital does not depend on location and is not in uenced by events in one metropolitan area We assume therefore that r is constant In contrast Ru is endogenous to the model This is a key point Remember that factors receive the value of their marginal product total revenue is just exhausted and the rm earns zero profits Because the pricedistance function is exogenous to firms Equation 31 indicates that land rent adjusts so that land is paid its marginal productiand firms earn zero profitsiat every location Thus the form of the rentdistance function is determined by the form of the price distance functioninot vice versa It is not correct to say that housing prices are high in a given location because land is expensive there Exactly the opposite is true land rents are high in a given location because housing is expensive there To put it in more general terms land rents are high at a particular location because people are willing to pay a great deal there for housing or for some other economic activity that requires land Now we can solve Equation 30 for K M and 31 for Lu and substitute the results into the housing production function 29 These steps yield er 1322m 32 The Basic Urban Model Page 18 The supply of housing services H S u can be canceled from both sides of this equation After this cancellation and multiplying both sides by Ru this equation becomes Ru CPu 33 where c Aa in 34 With a little algebra Equation 33 can be rewritten to express Ru as a function of Pu or Vice versa Ru CPu1 35 Or Pu Ru 36 This relationship between rents and housing prices is at the heart of an urban model By combining 35 with 22 we can now determine how land rents vary with location In particular RuC1 yY tulala yY tu1 37 where y 1s an unknown constant This equation which describes the amount housing producers bid for land is also known The Basic Urban Model Page 19 as a bid function To distinguish between bid functions for housing and land it is common practice to refer to Equation 22 as the bid price function and Equation 37 as the bid rent function 6 Anchoring Bid Functions Equations 22 and 37 contain unknown constants y and f In this section we take the first step toward quotanchoringquot the bid functions that is solving for these constants This step is equivalent to selecting one bid function out of the family of bid functions described by each of these equations This step is based on the observation that housing must compete with other activities for the use of land Although many nonhousing activities eXist in an actual urban area a standard urban model assumes that there is one urban activity namely housing and one nonurban activity namely agriculture Now suppose that the value of land in agriculture E per square mile is constant around the urban area Then housing will only exist where Ru is greater than F In other words the urban area extends out to the location 5 at which RZ1 38 This condition can be used to anchor Equation 37 that is to solve for f Evaluating 37 at J and setting the result equal to E yields The Basic Urban Model Page 20 E 7 m 39 C 1 a Y tu 1 Combining this result with 37 leads us to the quotanchoredquot form of the bidrent function8 laa Y tu R u R 40 Y m Now using Equation 40 we can also write down an quotanchoredquot version of the bid price function To be speci c combining 36 and 40 leads to la Pu13 Ym 41 where E P 42 C In this equation 1 3 can be interpreted as the opportunity cost of investing in housing The observant reader will note that the analysis in this section literally anchors the bid functions only if Jis known All bid functions describe locational equilibrium within an urban area but as we will discover in the following sections only an anchored bid function describes an equilibrium across urban areas To emphasize this second equilibrium condition we will say that the anchored bidprice function is the price distance function and the anchored bidrent 8 Note that one can obtain exactly the same result by substituting Equation 21 instead of 22 into 33 In this case one must solve for Uquot instead of for 7 The Basic Urban Model Page 21 function is the rent distance function If E is unknown this analysis simply provides an alternative formulation of the constant term in Equation 22 A higher value for 5 corresponds to a higher intercept for the bid function and hence according to Equation 21 to a lower level of household utility Intuitively the longer the longest commute the lower the net income of a household at the outer edge of the urban areaiand hence the lower the utility of all the other identical urban residents To derive Z we must specify a quotcompletequot urban model We now turn to this task 7 A Complete Urban Model An urban model is a general equilibrium model with housing land capital transportation labor and export good markets plus locational equilibrium conditions Our analysis so far has been based on incomplete specifications of these markets This section makes the specifications complete 71 The Housing Market The demand and supply of housing are given by Equations 17 and 29 respectively To complete the housing market we need only specify the market equilibrium condition namely that demand equals supply Note that the demand function Equation 17 applies to a single household whereas the supply function Equation 29 applies to all the housing at location u To make the two functions comparable we must multiply the demand function by N u the number of households at location u Thus the housing market equilibrium condition is The Basic Urban Model Page 22 NuHuHsu 43 72 The Land Market The derived demand for land is given by Equation 31 On the supply side we assume that there is a fixed number of radians of land available for housing at all distances from the city center This assumption makes it possible to bring into the model land used for transportation or taken up by some geographical features such as a lake or a harbor but only in the form of a pie slice In a city with no transportation or geographical quotcutoutsquot the available land takes the shape of a circle and equals 27239 whereas in a city on a large lake such as Chicago the available land takes the shape of a semicircle and equals 7239 minus land used for transportation Because transportation cost depend only on straightline distance from the CBD all points u miles from the CBD involve the same transportation cost and hence are identical as far as households are concerned The amount of land available u miles from the CBD equals the circumference of the city at that location The circumference of a circle with a radius of u miles is 27239 and the circumference of a partial circle containing radians and with a radius of u miles is W Thus we can write the supply of land as The Basic Urban Model Page 23 L u 44 Substituting Equation 44 into the demand function 31 insures equilibrium in the land market Another implicit assumption in a standard urban model is that land rent payments disappear from the urban area This is equivalent to assuming that all land rents are paid to absentee landlords that is to people who own the land but live outside the urban area As indicated on the reading list at the end of these notes several scholars have analyzed urban models in which the land rents are paid to people who live in the urban area and therefore in uence their behavior 73 The Capital Market The assumptions in Section 5 completely specify the capital market The demand for capital is given by Equation 30 and the supply of capital is given by the assumption that the rental rate of capital r is fixed This assumption is equivalent to the assumption of a horizontal supply curve for capital In other words this basic urban model considers capital to be highly mobile across locations 74 The Market for Transportation Services The market for transportation services is not usually explicitly recognized in the formulation of an urban model Nevertheless the assumption that the permile cost of commuting t is constant requires that enough land be used for transportation that congestion never occurs Commuting speed and hence commuting cost depends on the capacity of a The Basic Urban Model Page 24 highway As the number of people commuting on a highway approaches what is called its design capacity commuting speed starts to fall and t starts to rise This curve relating number of commuters to t which is illustrated in Figure 3 can be thought of as the supply curve for transportation services In effect therefore the implicit assumption in a standard model is that the share of land used for transportation is large enough so that even near the CBD where the number of commuters is large the rising part of this supply curve is never reached The implicit transportation market also is important because it gives the model its spatial detail The assumption that transportation cost is proportional to straightline distance from the CBD implies that all points a given distance from the CBD are for the purposes of the model identical In effect this assumption translates the behavioral variable which is total transportation costs into a geographical variable distance from the CBD Other assumptions about the transportation network that is other methods for translating transportation costs into locations can have strikingly different implications for the spatial arrangement of economic activity 75 Preliminary Treatment of the Labor and Goods Markets The last two markets to consider are the labor and export good markets which are closely linked An urban area cannot exist unless local firms producing an export good that is a good sold on a national market want to hire local residents In a basic urban model the export good market is left implicit and the treatment of the labor market is very rudimentary On the demand side export firms that is firms exporting products to a national market are assumed to exist in an urban area s central business district CBD These firms hire N The Basic Urban Model Page 25 workers at the market wage Y For now let us assume that N is a variable and Y is a parameter We will return to these assumptions shortly By Assumption 6 all households who live in the urban area have one worker employed in the CBD9 As noted earlier N u is the number of households and hence of workers living at location u so the total supply of workers is the integral of this N u function over all inhabited locations that is all values of u in the urban area Assume for simplicity that the CBD does not take up any space This assumption is convenient but inessential the model could easily be solved for a CBD with a fixed radius Then the limits of integration are from zero to the outer edge of the urban area Equilibrium in the labor market requires that demand equal supply or in symbols TNuduN 45 0 76 Locational Equilibrium The nal component of a complete urban model is the spatial dimension All of the variables in the model are indexed by u and the pattern of variation across space is driven by the locational equilibrium condition equation 12 which becomes the bidprice function Equation 21 when there is a CobbDouglas utility function and a constant T39 The height of the bidprice function is determined by an anchoring equation such as 38 77 The Complete Model The Basic Urban Model Page 26 This complete urban model contains 10 unknowns Hu HS u Lu Ku Nu Pu Ru N u and U The rst seven of these unknowns are functions not variables Although this fact adds complexity the model can still be solved with standard algebraic techniques10 The model also contains the following 9 equations which incorporate the implicit assumptions about the market for transportation services discussed in Section 74 Housing market 17 29 and 43 Land Market 31 and 44 Capital Market 30 Labor and Export Good Markets 45 Locational Equilibrium 21 and 38 Because it has it has 10 unknowns and 9 equations this model cannot be solved in this form To nd a solution one additional assumption must be made about the labor and export good markets We now turn to two alternative versions of this assumption 8 The Distinction Between Open and Closed Models A basic urban model can be solved either by assuming that the number of households in an urban area is xed or by assuming that households are perfectly mobile across urban areas 9 The model can easily be extended to consider the case of two workers per household as long as they both work in the CBD 10 Remember that it is possible to treat yor y as the tenth variable instead of U If this approach is taken one must use Equation 22 as the locational equilibrium condition instead of Equation 21 The Basic Urban Model Page 27 The former assumption de nes a closed model the latter de nes an open model A closed model is appropriate in two cases First it is appropriate for analyzing an urban area that makes up an entire nation so that it is dif cult for people to move in or out Singapore is perhaps the best example of this case Second a closed model is appropriate for analyzing changes that affect all the cities in a system equally In this case parameter changes do not alter the relative desirability of each city and no migration between cities occurs A closed model should be used for example to study the impact of a federal gas tax on urban areas An open model is appropriate for studying changes that affect one urban area but not others Changes of this type do induce interarea migration An open model should be used for example to study the impact on urban spatial structure of a local gas tax or of an increase in the value of agricultural land around a single city or to study differences across the cities in a region Two aspects of the distinction between closed and open models need to be emphasized First each approach makes exogenous one of the 10 variables listed in the previous section and thereby results in a manageable system with 9 equations and 9 unknowns By assuming a xed population closed models make population N exogenous We highlight this assumption by writing JV instead of N for a closed model Because it assumes that households are mobile across urban areas an open model requires that identical households achieve the same level of utility regardless of which urban area they live in As long as one urban area is small relative to the system of urban areas therefore the utility level in each urban area can be regarded as xed at the level in the entire region or nation Parameter changes in one urban area may induce The Basic Urban Model Page 28 people to move in or out so urban population N is not xed but these changes cannot alter the utility level which we now write as 5 to emphasize that it is xed both across and within an urban area In short the difference between a closed and an open model is that in a closed model N is xed and U is a variable whereas in an open model U is xed and N is a variable The second aspect of the closedopen distinction is their treatment of the labor market Neither approach speci es the derived demand for labor As a result shifts in the supply of labor which lead to movement along an unspeci ed demand curve cannot be analyzed in either model without making the extreme assumption that the derived demand for labor is horizontal that is that rms will hire any number of workers at income Y In contrast shifts in the demand curve at least as represented by a change in Y can be analyzed In a closed model the supply of labor N is a parameter so the extreme assumption about demand is only required for analyzing the impact of changes in N on model outcomes Unfortunately however every parameter in an open model except Y affects the supply curve for labor which is determined endogenously As a result comparative static analysis of a standard open urban model cannot be conducted without the above extreme assumption about labor demand In other words an extension to a more complete labor market is particularly important for an open model These conclusions are illustrated in Figure 4 The dotted line represents an unspeci ed derived demand curve for labor The solid vertical line is the supply of labor in a closed model whereas the upward sloping line is the supply of labor in an open model The shape of this open model supply curve will be derived in Section 10 In this gure a shift in the demand curve can The Basic Urban Model Page 29 be described by a change in Y and hence can be analyzed in either model A shift in either supply curve however results in a change in Y which is not incorporated into either model because Y is treated as a parameter 9 Solving a Closed Urban Model The key to solving a closed urban model is to nd the rentdistance function We already completed this step in Sections 5 and 6 The result we need is Equation 40 The next step is to solve for N u Equation 43 reveals that N u is the ratio H S u to H u Moreover H u can be expressed in terms of the model39s parameters by substituting Equation 41 into 17 Finding H S u as a function of these parameters is more complicated Equation 30 can be solved for Ku as a function of Pu and H5u Then 36 can be used to replace Pu with Ru Substituting the resulting expression for K M into 29 and solving for Hsu yields HS u DR uH L u 46 where D A1 quot in 47 a I Now we can use Equation 40 to eliminate Ru and Equation 44 to eliminate Lu The The Basic Urban Model Page 30 result 48 The third step is to solve for Z by carrying out the integration in Equation 45 Substituting 48 into 45 we nd that the relevant integral is du N 49 To solve this integral we can make use of the following formula 1 c2 2 nln 2 Iucl 0211quot du c1 czum2 c1 czum1 50 L 02 n1 To apply this formula to Equation 49 we must treat diVided by the denominator of 49 as a constant and set 01 Yc2 t and n 1aa 1 Thus the solution to 49 is The Basic Urban Model Page 31 i Y tubi1 uY tub NuduYjblm EZbb1 W a 51 0 where b 1 ad and the right side must be evaluated at Z and 0 Completing this evaluation and setting the result equal to N yields ML Y m 52 tblY tu tb1 This equation contains a major disappointment It cannot be solved explicitly for This fact is an important barrier to analytical manipulation of this model To obtain a complete solution with particular parameter values one must employ numerical methods To be specific one must program this equation on a computer and increase the value of Z given the values of the other parameters until Equation 52 is exactly satisfied Finally we can solve for U using Equations 21 and 41 In effect U is a residual in the sense that nothing else in the model depends on it 10 Solving an Open Urban Model The equations for an open urban model are exactly the same as the equations for a closed urban model except that N is now treated as a variable and U is treated as a constant and written 5 The derivations presented in Section 7 and 9 carry over to the open model Thus Equation 40 gives the bidrent function R and Equation 48 indicates the population function N With an open model however it is possible to use the indirect utility function The Basic Urban Model Page 32 to solve for Evaluating 21 at location 5 and remembering that 13 Em C we nd that T 1 53 P E 7 Solving 53 for 5 yields Y PkU YRcalkf u 54 t t This result has a straightforward interpretation At a given income and transportation cost the higher the value of systemwide utility the smaller must be the commuting distance of the most distant worker With 5 known we can now use 52 to nd N which now has no bar Because N is an explicit function of J we do not run into the problem that we encountered with a closed model a complete analytical solution to the open model can easily be obtained 11 Density Functions and Building Heights Two important features of an urban area are patterns of population density and building 39 llllllllllllllllllllllllllllllllllllllllllll The Basic Urban Model Page 33 height Population density is people per square mile or N L From Equations 40 44 and 48 it follows that my 2 55 Using the derivative of 55 the relationship betweenR and P Equation 3 5 and the equation that de nes P 41 it can be shown that the slope of this density function is always negative Moreover this density function is atter ie less negative than the rent function for all reasonable values of the parameters Building height is roughly speaking measured by the ratio of capital to land that is the more capital used on a given amount of land the higher buildings are likely to be To express the capitalland ratio as a function of distance we begin with Equation 3 O substitute in 41 to eliminate P and 29 to eliminate H S u and solve the resulting equation for K The result is Al a Ku 2 TI0 RuL u 56 Dividing both sides by L we nd that the capitalland function is proportional to R qR 57 Mu where The Basic Urban Model Page 34 q 58 Cr This result gives a simple visual test of the basic urban model Are buildings tallest in the CBD with a gradual decline as one moves out toward the suburbs To a rough approximation at least this prediction holds up in many metropolitan areas 12 Comparative Statistics An urban model can be used for comparative static analysis that is for determining the impact of changes in the parameters on urban spatial structure This section discusses comparative static analysis of the basic urban model The key parameters in the basic urban model are YtR and N in a closed model or U in an open model11 Comparative static analysis also can be conducted for a a A r and The key variables are P uR uDu u and U in a closed model or E in an open model The strategy for deriving the comparative static results is quite different in open and closed models Both strategies begin by nding the derivative of E with respect to the parameter under consideration but with an open model this derivative comes from differentiating Equation 54 and is quite simple whereas with a closed model it comes from differentiating Equation 52 and is very complex Because 5 appears in most of the other equations of the model this derivative can then be used to solve for other comparative static 11 An alternative somewhat more complicated analysis can be conducted using to and t7 from Equation 16 instead of t The Basic Urban Model Page 35 derivatives For example by totally differentiating Equation 40 which holds in both the open and closed model we nd that the derivative of Ru with respect to a parameter say 5 can be written dRu aRu 6Rudzt d5 55 5 d5 39 59 The relative simplicity of dzj d5 for all the parameters in the open model implies that this equation along with other comparative static results is generally much easier to evaluate in a basic open model than in a basic closed model 121 Illustrative Comparative Static Results for an Open Model Differentiating Equation 54 with respect to its key parameters yields dY Z dl Z dz 13 dz WU H lt0 lt0 dU kl dR le Thus the physical size of an urban area increases with Y and decreases with l U and E With these results in hand we can turn to the bid and density functions as well as the population equation Totally differentiating Equation 40 with respect R Y and 17 dividing by dY substituting in the above expression for dl l dY and rearranging terms yields d Ru Ru dY aaY Zu gt0 61 This result implies that an increase in income increases the rentdistance function everywhere Differentiating the rentdistance function with respect to E instead of Y and then following the The Basic Urban Model Page 36 same procedure and using Equation 53 reveals that the derivative of R with respect to E is zero in an open model the rent function is fixed by the fixedutility assumption and does not shift when E changes To nd the impact of an income change on urban population we must totally differentiate Equation 52 with no bar on the N with respect to N Y and divide by dY substitute in the above expression for d 611 and rearrange terms The result dNR Lb1gt0 62 Urban population inevitably increases with income12 Other comparative static results can be derived in a similar manner 122 Illustrative Comparative Static Results for a Closed Model Because N is fixed in a closed model the comparative static results for can be found by differentiating Equation 52 For example totally differentiating 52 with respect to E and 5 reveals that 12 For an derivation and empirical test of this result see Yinger and Danziger 1978 The Basic Urban Model Page 37 d tb1N 1 R b 1 Y tu 63 In words the physical size of an urban area decreases as agricultural rents increase This result can be used to pin down other comparative static derivatives From Equation 40 for example we nd that d R R 39 M E 1 tb 31 3 64 R Y tu dR By substituting in Equation 63 it can be shown that the term in square brackets and hence dR id61 itself is positive in a closed model unlike an open model an increase in E shifts up the rent function to ensure that there is enough room for the urban area s population This algebraic derivation is left to the reader see exercise 7 Although the algebra is often quite complex other comparative static results for a closed model can be found in a similar manner 13 Evidence about Key Urban Model Results 131 Introduction The basic urban models presented in this chapter make many predictions about the spatial pattern of economic activity In particular they predict that the price of housing services ie the price of housing controlling for housing characteristics the price per unit of land population density and building height all decline with distance from the CBD Anyone who has looked at a big city skyline or tried to rent an apartment near the center of a big city can The Basic Urban Model Page 38 verify that these predictions are consistent with casual empiricism This section reviews some of the more formal tests of urban models 132 Evidence on Housing Prices and Rents The most basic test of an urban model is to determine whether housing prices and land rents actually do decline with distance from the CBD As shown by Coulson 1991 a formal statement of this hypothesis is easily derived from a basic household maximization problem WithH held constant the derivative of apartment rent P uH with respect to u is P39 uH which by Equation 1 1 equals T With a constant commuting cost per mile this derivative is simply t Hence in a regression of apartment rent on distance controlling for the housing characteristics that determine H the coefficient of distance is I By using Equation 4 this approach can also be applied to house values instead of apartment rents and by using Equation 35 it can be applied to land rents as well Many studies which are reviewed in Coulson 1991 or in Mills and Hamilton 1994 have regressed housing rents or prices on distance from the CBD controlling for housing characteristics Virtually all of these studies nd that the distance variable is statistically signi cant Coulson l99l carried out such regressions using data for State College PA which appears to conform to a relatively homogeneous urban area dominated by a CBD Coulson finds that in a linear regression the coefficient of u is highly significant statistically and corresponds to a value of tequal to 0527 per mile or 0263 per mile each way13 This result corresponds for example to gas and oil costs of 010 per mile a commuting speed of 20 MPH and commuting time valued at 325 per hour 13 Coulson 1991 presents numbers that are twice this large but he mistakenly uses a nominal interest rate set to 10 percent instead of a real interest rate which I have set to 5 percent For more on the importance of using real interest rate in discounting problems see Yinger et al 1988 For future reference it should also be noted that Coulson uses actual street distance not radial distance to the CBD and in some of his models asks The Basic Urban Model Page 39 Coulson also finds that the linear form can be rejected in favor of a model with a more general functional form called BoxCox As Coulson points out this result is not a rejection of an urban model but is instead a rejection of the assumption that the commuting cost per mile is fixed ie that T tu The form for T implied by Coulson s results is complex and cannot be determined from the information in his article Although Coulson s statistical test rejects the linear form he also finds that the linear model yields results that are qualitatively similar to those using the BoxCox model so that assuming constant permile transportation costs may be a reasonable approximation 133 Evidence on Density Gradients Another large empirical literature which is reviewed in McDonald 1989 focuses on the relationship between population density and distance from the CBD As shown by Equation 55 a basic urban model predicts that population density will decline with distance from the CBD so this literature also serves as a test of the basic urban model This literature began before urban models were developed when some researchers suspected empirical regularities in the relationship between population density and location The early studies searched for functional forms that accurately characterized this relationship and many of them settled on an exponential form that is a form expressing population density as an exponential function of distance from the city center Muth 1969 and Mills 1972 later showed that under some assumptions this exponential form can be derived from an urban model Virtually all studies of population density find that density declines with distance from the city center This finding provides more general evidence in support of the basic urban model Moreover as predicted by an urban model density functions appear to take similar whether travel cost varies with direction These are valuable extensions of the standard approach and they appear The Basic Urban Model Page 40 forms in many different urban areas and to exhibit changes over time that are roughly consistent with comparative static predictions from an urban model 134 Tests Using Functional Forms from Urban Models The empirical studies reviewed above provide general tests of a basic urban model but few of them provide very demanding tests of these models that is tests that make use of theoretically derived functional forms or theoretically derived restrictions Exceptions include the Coulson study which tests and rejects the assumption that the per mile cost of commuting is constant and studies of population density that employ an exponential form which can be derived from one oversimplified version of an urban model A more demanding test of urban models is provided by Yinger 1979 who estimates a pricedistance function using the functional forms derived in Section 6 Remember from Equation 4 that V P uH 139 and remember that H is assumed to be a function of the structural characteristics of housing say X1 to X N The form of Pu is given by 41 A reasonable form for H is as follows H Xl X2quot1X3quot3 XIM 65 Now combine equations 4 41 and 65 and take natural logs to obtain 1 3 1 1 M 1nV ln 1 lnY tu lnY tu277mlnXm 66 quot11 Thus with data on V Y u 5 t and the X 3 this equation can be estimated with linear regression techniques Alternatively tcan be estimated as a nonlinear parameter which in some of the studies listed at the end of these notes The Basic Urban Model Page 41 is the procedure followed by Yinger 1979 using data for singlefamily houses in Madison WI and for both apartments and singlefamily houses in St Louis The results for Madison which has employment concentrated near its CBD are very supportive of the model The estimated value of a the share of income spent on housing is very reasonable 172 percent and is highly signi cant statistically Moreover the estimated value for tis quite reasonable14 The results for St Louis which has very dispersed employment are more mixed At locations close to the CBD the estimated value of a is about 21 percent for both the rental and sales data The estimated value of a is not reasonable however and indeed sometimes has the wrong sign more than one mile from the CBD for the rental data or two miles from the CBD for the owner data In addition the estimated values of t are not as reasonable as in the Madison equation Note that in Equation 66 the coefficients of the two terms containing Y are the same In fact the lnctional form used here does not make sense if these two coefficients are different and the results presented above are based on equations in which they are restricted to be the same In fact however Yinger must reject the hypothesis that these two coefficients are the same This test suggests that the assumption of a CobbDouglas utility function is not correct although the results based on it are plausible enough at least when employment actually is concentrated near the CBD so that it may be a reasonable approximation under some circumstances 135 Conclusions The most fundamental prediction of an urban model that housing prices and rents vary with location after controlling for the structural characteristics of housing is strongly supported by many studies More detailed predictions from a basic urban model also are supported in The Basic Urban Model Page 42 urban areas where the assumptions of that model are reasonably accurate In other words these predictions are supported in a relatively homogeneous urban area in which employment is relatively centralized Not surprisingly the model does not hold up so empirically so well in areas where its assumptions are not met Much of urban economics is dedicated to relaxing the assumptions of the basic urban model that is to building more general urban models As indicated earlier studies that relax each of the assumptions in the basic urban model are listed after these notes Empirical work in urban economics also focuses on more complex models after all the failure of a basic urban model to accurately predict housing prices or other features of a heterogeneous urban area with diffuse employment such as St Louis cannot be interpreted as a rejection of urban modelsionly as a rejection of overly simple urban models The great challenge facing empirical work in urban economics is that formal tests of urban models with general assumptions are difficult to derive Nevertheless many scholars have devised clever tests that build on extensions of the logic of a basic urban model to a setting with for example multiple worksites or traffic congestion This empirical literature is obviously way beyond the scope of these notes 14 To be specific Yinger estimates that to is 05 and wMPH is 025 which corresponds for example to a w equal to 05 and a commuting speed of 20 MPH

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