1. More General Treatment of Housing Demand
ECN741: Urban Economics
More General Treatment of Housing Demand
Professor John Yinger, The Maxwell School, Syracuse University, 2016

2. Class Outline 1. Alternative Utility Functions
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function
5. Introduction to Comparative Statics with General Functional Forms

3. Class Outline 1. Alternative Utility Functions
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function

4. Stone-Geary Utility Function
Housing Demand Theory
Stone-Geary Utility Function
One simple extension to the Cobb-Douglas utility function is to a Stone-Geary utility function:
where SZ and SH are “survival quantities.”
In this case, the demand functions reflect the “required” spending on Z and H:

5. Stone-Geary Utility Function, 2
Housing Demand Theory
Stone-Geary Utility Function, 2
Thus, the indirect utility function is:
What a disappointment! This expression cannot be solved for P{u}!
So this utility function does not help.

6. Housing Demand Theory
CES Utility Function
Another well-known utility function is the “constant elasticity of substitution” or CES function, which is
Demand functions are unaffected by a positive monotonic transformation of utility, so if δ is positive (which it may not be), one can save a lot of algebraic mess by simplifying this to

7. 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} with the added assumption that β = 0.5.
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, but the β = 0.5 assumption is pretty strong.

8. Class Outline 1. Alternative Utility Functions
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function

9. An Exponential Density Function?
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 striking claim because the exponential density function had been used in many empirical studies—and it worked very well!
What assumptions are needed for this to be true?

10. An Exponential Density Function?, 2
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.

11. An Exponential Density Function?, 3
Housing Demand Theory
An Exponential Density Function?, 3
So we have
Putting these together leads to the differential equation

12. An Exponential Density Function?, 4
Housing Demand Theory
An Exponential Density Function?, 4
The general solution to this differential equation is:
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!

13. An Exponential Density Function?, 5
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
The solution to this differential equation is
This won’t lead to an exponential density, either!

14. An Exponential Density Function?, 6
Housing Demand Theory
An Exponential Density Function?, 6
Now let’s use gross income in the demand function:
This changes the differential equation to
and the solution to

15. An Exponential Density Function?, 7
Housing Demand Theory
An Exponential Density Function?, 7
Now combining the housing production function, the capital market equilibrium condition, the housing S = D condition, and the definition of density from a basic urban model yields
where Φ is a constant.
With the above expression for P{u} and no commuting costs in the denominator, this equation simplifies to
where D0 and δ are constants.
An exponential form at last!

16. An Exponential Density Function?, 8
Housing Demand Theory
An Exponential Density Function?, 8
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, μ, is -1.
In other words, households consider tu when they decide where to live, but the impact of tu on their net income only affects their consumption of Z, not of H.
An alternative approach that leads to an exponential density is to assume that θ = 0 and μ = -1 (Kim and McDonald, Journal of Regional Science, May 1987). The empirical evidence (covered in the next class) does not support the first part of this assumption.

17. An Exponential Density Function?, 9
Housing Demand Theory
An Exponential Density Function?, 9
As it turns out, this analysis leaves out a step, because it focuses on deriving a bid function, not a bid-function envelope, which is what we see in an actual urban area.
The only clean way to rescue the algebra is to assume that all households are alike, which implies that the bid function equals the bid-function envelope.
If households are not all alike, the above assumptions are not sufficient to generate an exponential bid-function envelope.
In general, the form of envelopes (to be derived in a later class) is more complex than the form of the underlying bid functions. Even if each household type has an exponential bid function, the bid- function envelope and the overall density function area unlikely to take an exponential form.

18. Class Outline 1. Alternative Utility Functions
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function

19. 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!

20. 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.
The history of this theorem is recounted by Silberberg (Journal of Economic Education, Winter 1999)
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.

21. Housing Demand Theory
The Envelope Theorem, 3
Let’s start with a standard maximization problem, which can be written as
The Lagrangian expression for this problem is

22. Housing Demand Theory
The Envelope Theorem, 4
Differentiating this Lagrangian leads to the following first-order conditions
which lead, in turn, to the following solutions:

23. Housing Demand Theory
The Envelope Theorem, 5
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
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.

24. Housing Demand Theory
The Envelope Theorem, 6
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
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!

25. Housing Demand Theory
The Envelope Theorem, 7
To get to this shortcut, first write the constraint at the optimal values of the xis.
This constraint must still hold when α changes, which implies that

26. Housing Demand Theory
The Envelope Theorem, 8
Now if we multiply this result by λ and add it to the messy comparative statics result derived earlier, we find that
But, by the first-order conditions, the terms in parentheses in the second line have to equal zero, so

27. The Envelope Theorem, 9 This is an amazing shortcut.
Housing Demand Theory
The Envelope Theorem, 9
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.

28. Class Outline 1. Alternative Utility Functions
Housing Demand Theory
Class Outline
1. Alternative Utility Functions
2. An Exponential Density Function?
3. The Envelope Theorem
4. Maximizing a Bid Function

29. Maximizing Bids In an urban model, the household budget constraint is
Housing Demand Theory
Maximizing Bids
In an urban model, the household budget constraint is
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

30. Housing Demand Theory
Maximizing Bids, 2
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.

31. Housing Demand Theory
Maximizing Bids, 3
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.

32. Maximizing Bids, 4 The Lagrangian is
Housing Demand Theory
Maximizing Bids, 4
The Lagrangian is
The relevant first-order condition is:
and the result we need is:

33. Housing Demand Theory
Maximizing Bids, 5
Now we can use the Envelope Theorem to find the impact of Y, t, U*, and u on P{u} using
The CS results are:

34. Housing Demand Theory
Maximizing Bids, 6
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:
which is, of course, the result we derived before.

35. General Comparative Statics
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 1987 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.