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ECN741: Urban Economics More General Treatment of Housing Demand Housing Demand Theory Class Outline Alternative Utility Functions An Exponential Density Function? The Envelope Theorem Maximizing a Bid Function Introduction to Comparative Statics with General Functional Forms Housing Demand Theory Stone-Geary Utility Function One simple extension to the Cobb-Douglas utility function is to a Stone-Geary utility function: U Z SZ 1 H SH where SZ and SH are “survival quantities.” In this case, the demand functions reflect the “required” spending on Z and H: H SH Y tu S z P {u } S H P {u } Z S Z (1 ) Y tu S z P {u } S H Housing Demand Theory Stone-Geary Utility Function, 2 Thus, the indirect utility function is: Y tu S z P {u } S H U* k P { u } What a disappointment! This expression cannot be solved for P{u}! So this utility function does not help. Housing Demand Theory CES Utility Function Another well-known utility function is the “constant elasticity of substitution” or CES function, which is U (1 ) Z H 1/ Because demand functions are unaffected by a monotonic transformation of utility, one can save a lot of algebraic mess by simplifying this to U (1 ) Z H Housing Demand Theory CES Utility Function, 2 The demand functions for this utility function can be derived and so can the indirect utility function. This indirect utility function can be solved for P{u}. I leave this derivation as an exercise. A CES utility function would be a good choice for a simulation model or urban spatial structure, since it includes the Cobb-Douglas utility function as a special case. Housing Demand Theory An Exponential Density Function? Some scholars (Mills, Muth) have argued that an exponential density function can be derived from an urban model. This was a pretty amazing claim because the exponential density function had been used in many empirical studies—and it worked very well! Let’s see what assumptions are needed for this to be true. Housing Demand Theory An Exponential Density Function, 2? Start with the standard locational equilibrium condition, and assume that the demand for housing takes a constant elasticity form. A constant elasticity form, by the way, can be derived from an “incomplete demand system,” which has a composite commodity essentially as a residual. See LaFrance, American Journal of Agricultural Economics, August 1986. Housing Demand Theory An Exponential Density Function, 3? So we have P {u } t H H B (Y tu ) P {u } Putting these together leads to the differential equation P {u } P {u } t B (Y tu ) Housing Demand Theory An Exponential Density Function, 4? The general solution to this differential equation is: 1 P {u } 1 Q (Y tu ) 1 B (1 ) where Q is a constant of integration. This obviously yields a pretty messy nonlinear form for P{u}, and hence for R{u} and D{u}. It does not lead to an exponential form! Housing Demand Theory An Exponential Density Function, 5? We can simplify this result by assuming that the price elasticity of demand for H, μ, is -1. This leads to P {u } P {u } t B (Y tu ) The solution to this differential equation is ln{ P {u }} Q (Y tu ) 1 A (1 ) This won’t lead to an exponential density, either! Housing Demand Theory An Exponential Density Function, 6? Now let’s change the demand function to: H BY P {u } This changes the differential equation to P {u } P {u } t BY and the solution to t ln{ P {u }} Q u BY or Q P {u } e e (t / BY )u Housing Demand Theory An Exponential Density Function, 6? Now remember from a basic urban model that R {u } 1/ a R {u } C P {u } and D {u } a (Y tu ) In addition, the net income expression in the denominator of D{u} comes from the housing demand equation. With the same simplification (=no tu), D {u } C P {u } 1/ a e Q / a a Y An exponential form at last! e ( t / aAY ) u a Y e 0e c c1u Housing Demand Theory An Exponential Density Function, 6? In short, to obtain an exponential density function, one has to assume that the demand for housing depends on Y, not on (Y - tu), and that the demand elasticity is -1. But if tu is not in the budget constraint, bid functions do not arise in the first place! Consistency with bid functions requires the assumption that θ = 0 (Kim and McDonald, Journal of Regional Science, 1987). The empirical evidence does not support this assumption. Housing Demand Theory The Envelope Theorem The Envelope Theorem is an important tool in economics and several scholars have employed it to good effect in the case of urban models. We will go over what the Envelope Theorem is, including a proof, and then show how it can be used to simplify urban model comparative statics. This is not the last class in which the Envelope Theorem will appear! Housing Demand Theory The Envelope Theorem, 2 Many of you may be familiar with the Envelope Theorem, but in case you are not, I’m going to prove it for you. The proof also serves as an explanation. Then I am going to restate the bidding problem in an urban model and show you how you can do some comparative statics derivations by applying the Envelope Theorem to this re-formulation. Housing Demand Theory The Envelope Theorem, 3 Let’s start with a standard maximization problem, which can be written as M axim ize y f { x1 , ..., x n , } Subject to g { x1 , ..., x n , } 0 The Lagrangian expression for this problem is £ f g Housing Demand Theory The Envelope Theorem, 4 Differentiating this Lagrangian leads to the following first-order conditions £ i fi g i 0 £ g 0 which lead, in turn, to the following solutions: x x { } * i * i { } * * Housing Demand Theory The Envelope Theorem By substituting the solutions back into the expression for y, which is what we are trying to maximize, we find can write the solution to the problem as y f { x , ..., x , } { } * * 1 * n Note that ϕ{α} is sometimes called the indirect objective function; it is the maximum value of y for given α using x’s that meet the constraint. Housing Demand Theory The Envelope Theorem Now suppose we want to know how this maximum value changes when we change α, which is just an exercise in comparative statics. Differentiating the above expression for y* yields y * i xi * fi f This looks pretty complicated. It looks like we have to re-solve the problem with the new values of α to find the new values of the xi*s and λ* and then substitute those values back into y. The Envelope Theorem gives us a shortcut! Housing Demand Theory The Envelope Theorem To get to this shortcut, first write the constraint at the optimal values of the xis. g { x , ..., x , } 0 * i * n This constraint must still hold when α changes, which implies that g g i x i * i g 0 Housing Demand Theory The Envelope Theorem Now if we multiply this result by λ and add it to the messy comparative statics result derived earlier, we find that * * * y i xi fi f g i g i xi f i i gi xi * f g But, by the first-order conditions, the terms in parentheses in the second line have to equal zero, so y * f g £ Housing Demand Theory The Envelope Theorem This is an amazing shortcut. To find the comparative static derivative of y* with respect to α, all we have to do is differentiate y and g with respect to α. We do not have to worry about changes in the xi*s!! Of course this only applies to small changes, but, as we will see, it is a very helpful result. Housing Demand Theory Maximizing Bids In an urban model, the household budget constraint is Y Z P {u } H tu One way to think about the household problem is to say that they have to figure out the most they are willing to bid for housing at each location, given their budget constraint and a fixed utility level. In other words, they M axim ize P {u } Y Z tu H S ubject to U {Z , H } U * Housing Demand Theory Maximizing Bids The decision variables in this problem are Z and H, and the parameters are Y, t, U*, and u. We are treating this as an open model. So we can use the envelope theorem to find the derivative of P{u} with respect to each of these parameters. Housing Demand Theory Maximizing Bids Although Z and H are formally choice variables, this formulation cannot be used to find their demand functions; this problem is to determine P{u}, not to determine responses to P{u}. Nevertheless, one first-order condition is needed to pin down the CS derivatives. Housing Demand Theory Maximizing Bids The Lagrangian is £ Y Z tu H U { Z , H } U The relevant first-order condition is: £ 1 U Z 0 Z H and the result we need is: 1 UZH * Housing Demand Theory Maximizing Bids Now we can use the Envelope Theorem to find the impact of Y, t, U*, and u on P{u} using y * f g £ The CS results are: P {u } Y 1 H 0 P {u } t u H 0 P {u } U * 1 UZH 0 Housing Demand Theory Maximizing Bids In this setting, it is appropriate to treat u as a parameter, and we can ask how bids changes when a household moves to a more distant location: The envelope theorem reveals that: P {u } u P {u } t 0 H which is, of course, the result we have derived before. Housing Demand Theory General Comparative Statics One final point to make about this exercise is that these CS results apply to any utility function. Hence they are part of a general CS analysis, such as the one in Brueckner’s Handbook chapter. In fact, the CS results with general functional forms are the same (i.e. have the same signs) as the results in the table presented in the last class.