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ECN741: Urban Economics The Basic Urban Model: Assumptions The Basic Urban Model Outline of the Class Origins of Urban Economics Key Assumptions of a Basic Urban Model The Basic Household Maximization Problem The Basic Urban Model von Thünen Urban economics was invented (sort of) by a German agricultural economist, Johann Heinrich von Thünen, who lived from 1783 to 1850. More specifically, von Thünen invented the concepts of bidding and sorting, which form the basis for urban economics. He also, by the way, invented general equilibrium analysis! The Basic Urban Model von Thünen’s Model von Thünen modeled the location of agricultural activities around a central market place. Each activity had a maximum amount it was willing to pay for land at each location—its land bid. The winning activity at a given location was the one that bid the most there. This leads to the sorting of activities across locations. The Basic Urban Model von Thünen’s Model, Continued The key to the model is transportation costs. Some firms produced heavy or perishable products, so they would not pay much for land far from the center; transportation costs would eat up all their profits at distant locations. But some other factors, based on von Thünen’s practical knowledge of agriculture, also came into the model. The Basic Urban Model Annual Rent per Acre von Thünen's Model of Rents and Locations 0 2 City Milk 4 Wood 6 Rent for Milk/Vegetables 8 Grain 10 Rent for Wood 12 14 Livestock Rent for Grain 16 18 20 Rent for Livestock Distance from Central Market (miles) The Basic Urban Model Alonso In 1964, William Alonso published Location and Land Use, based on his dissertation in regional planning. This amazing book applied the von Thünen logic to the location of households in an urban area. People all work in a central work site. Each type of household bids on land in every location. Household types sorted into different locations based on their bids, and types with high commuting costs locate nearer to the center. The Basic Urban Model Mills and Muth Edwin Mills (at Princeton and my thesis adviser) and Richard Muth (at Chicago) extended Alonso to consider housing at about the same time. Mills: A 1967 publication (May AER), which cites a Muth working paper; a 1972 book (Studies in the Structure of the Urban Economy). Muth: A 1969 book (Cities and Housing). The Basic Urban Model Urban Models Urban models are designed to explain why urban areas look the way they do, with a focus on housing. Where do people live? How far do they commute? How much housing do they consume? What is the price of housing? The Basic Urban Model Key Assumptions We now turn to the key assumptions of a basic urban model. These assumptions lead to a simple, complete urban model that describes urban residential structure After we explore a model with these assumptions, this class will investigate how urban residential structure changes when more general assumptions are introduced. The Basic Urban Model Key Assumptions 1. Housing Demand 2. Housing Supply 3. The Transportation Network 4. Why Location Matters 5. Types of Households 6. The Labor Market 7. Household Mobility 8. Local Governments The Basic Urban Model 1. Housing Demand Household utility functions depend on a composite consumption good, Z, and housing, H, and take the Cobb-Douglas form. U (1 )ln{Z } ln{H }, H is measured in units of housing services = quality adjusted square feet. The Basic Urban Model 2. Housing Supply Housing services are produced with land, L, and capital, K, according to a Cobb-Douglas production function with constant returns to scale. H K 1 a a L Housing is owned by absentee landlords. Maintenance, rehabilitation, and conversion activities are ignored. The Basic Urban Model 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, t, using a single transportation mode. The urban area is located on a featureless plain (=von Thünen!). There are no commuting arteries or street grids. There is only one transportation mode (=car?) The Basic Urban Model 4. Why Location Matters Distance to work is the only locational characteristic households care about. This assumption rules out neighborhood amenities, such as School quality Access to shopping or recreation Air quality The characteristics of their neighbors. The Basic Urban Model 5. Types of Households All households are alike. All households have the same income, family structure, job location, and utility function!!! We obviously need to weaken this assumption—and eventually we will. The Basic Urban Model 6. The Labor Market Income is fixed and all households have a single worker with a job in the central business district (CBD). This assumption greatly limits labor market analysis; there is no explicit export good and the implicit demand curve for labor is horizontal. The Basic Urban Model 7. Household Mobility Households are assumed to be perfectly mobile within an urban area. If a household has an opportunity to improve its utility, it will take it! This assumption implies that all (identical!) households will achieve the same level of utility and is crucial to the logic of urban models. The Basic Urban Model 8. No Local Governments The basic urban model ignores local governments. There are no local government services and no property taxes. This is obviously and important omissions, and we will spend the last third of the class trying to fix it! The Basic Urban Model Preliminaries Now to set up the basic household problem that is the core of an urban model, we consider The price of housing Owning versus renting The Basic Urban Model The Price of Housing The annual price per unit of housing services is P. This price depends on household location, measured in miles from the CBD, u; that is, P = P{u}. Apartment rent is P{u}H. The derivative of P with respect to u, is negative: Pʹ{u} = dP{u}/du < 0 The Basic Urban Model Owner-Occupied Housing This model can also be applied to owners. The value of a house, V, is the present value of the rental benefits from owning it. Let i be a household’s real discount rate and M be the expected lifetime of a house in years. Then M P u H y 1 1 i V where i * y P{u}H P{u}H * i i i 1 1 i M . The Basic Urban Model The Household Problem The household problem is to Maximize U{Z, H} Subject to Y = Z + P{u}H + tu The Lagrangian is U Z , H Y Z P u H tu , The Basic Urban Model The First-Order Conditions The key first-order conditions are: U 0, Z Z U P u 0, H H P u H t 0. u The Basic Urban Model Implications The first two conditions imply that U / H MBH P u . U / Z The third condition indicates that a household moves away from the CBD until it finds the u* at which the savings in housing costs from moving 1 mile further out equals the associated increased commuting cost: P u H t The Basic Urban Model Location Choice $ -Pʹ{u}H t u* u The Basic Urban Model A Puzzle Can you see a problem with this result? Every household is alike, so every household picks the same u*. How can we have a city in which everyone lives the same distance from the CBD? We can’t! The Basic Urban Model The Urban Model Twist The key to solving this puzzle is to recognize that the form of P{u} is determined by the market. Hence, the equation for a household’s location choice can be interpreted as a equilibrium condition for P{u}, that is, as a locational equilibrium condition. If P{u} meets this condition, then every household will be satisfied no matter where it lives and no household will have an incentive to move. The Basic Urban Model The Locational Equilibrium Condition The locational equilibrium condition is written: t P u . H Because t and H are positive, the slope of P{u} must be negative. Econ 101 tells us that H declines with P, so the slope must get flatter as u increases. The Basic Urban Model The Bid Function for Housing (Price per Unit of Housing Services) P(u) Slope = ΔP/Δu = -t/H ΔP Δu CBD u The Basic Urban Model Features of Equilibrium 1. Note that the model does not say where any given household will live. Location choices are now idiosyncratic and outside the model. 2. Note also, that the locational equilibrium condition defines a family of bid functions. Higher bid functions correspond to a higher cost of living—and a lower utility level. The Basic Urban Model A Family of Bid Functions P(u) Lower Utility Higher Utility CBD u The Basic Urban Model Looking Ahead Next time we will see how this household problem fits into equations for the other markets in an urban area. The result is a general equilibrium model of urban residential structure.