1. Household Heterogeneity Supplementary Slides
ECN741: Urban Economics
Household Heterogeneity
Supplementary Slides
Professor John Yinger, The Maxwell School, Syracuse University, 2018

2. Household Heterogeneity
Contents
1. Steps in deriving a bid-function envelope
2. Recent research on the economics of density

3. Household Heterogeneity
Contents
1. Steps in deriving a bid-function envelope
2. Recent research on the economics of density

4. Household Heterogeneity
Steps in Deriving a Bid-function Envelope
1. Derive a bid function
2. Find the constant term that preserves the intersection where 2 bid functions cross.
3. Assume a general form for the equilibrium relationship between bid function slopes and access to worksites.
4. Integrate the equation defined by steps 1 to 3 to find the bid-function envelope.

5. Household Heterogeneity
Step 1: Deriving a Bid Function
A. Assume that the time cost of commuting as a fraction, λ, of the wage is constant across households.
B. Assume that operating costs of commuting affect location choice but do not affect the demand for housing.
C. Define m as a measure of access to jobs, which can be in miles or minutes.

6. Household Heterogeneity
The Bid Function
A. Assume that the demand for housing has a constant elasticity form, with a unitary price elasticity.
B. Use the differential-equations approach for finding a bid-function with these assumptions.

7. Household Heterogeneity
The Bid Function, 2
The resulting bid function is:
where

8. Household Heterogeneity
Step 2: Deriving the Constant
As written, the above bid function does not have an envelope.
The following slide shows bid functions using this formula with the same constant term and different values of ψ .

9. Household Heterogeneity
Bid Functions with the Same Constant

10. Household Heterogeneity
Step 2: Deriving the Constant, Cont.
So the math problem is to find the constant that preserves an intersection between two bid functions when the slope (ψ) changes.

11. Household Heterogeneity
Finding the Bid-Function Constant

12. Household Heterogeneity
Step 3: Characterizing the Equilibrium
The equilibrium is a relationship between the slope of the bid function (ψ)and the measure of access (m).
The most general form that is tractable is:
where the σs are parameters to be estimated.

13. Household Heterogeneity

14. Household Heterogeneity
Step 4: Solving for the Envelope
Plugging this equilibrium equation into the equation for the constant term yields a differential equation that can be solved for the constant term as a function of ψ.
The solution depends on the value of σ3 and of γ, the income elasticity of demand for housing.
Then the equilibrium equation can be use to replace ψ with m, yielding the envelope.

15. Household Heterogeneity

16. Household Heterogeneity
Estimating the Envelope
The case of a linear equilibrium and a unitary income elasticity reduces to:
This looks like the semi-log form in most studies, but the estimated coefficient contains the parameters from the equilibrium function and cannot be interpreted as a measure of transportation costs.
If one uses outside information to determine γ and tmY, these envelopes can all be estimated with OLS.
That is what I am doing now.

17. Household Heterogeneity
Contents
1. Steps in deriving a bid-function envelope
2. Recent research on the economics of density

18. Household Heterogeneity
A New Approach to Testing Urban Models
A recent, prize-winning paper in Econometrica develops a totally different approach to empirical work in urban economics.
Gabriel M. Ahlfeldt, Stephen J. Redding, Daniel M. Sturm, and Nikolaus Wolf “The Economics of Density: Evidence from the Berlin Wall.” Econometrica 83 (6) (November):
This is a technically difficult paper and we will not cover it in detail, but it is instructive to compare its basic assumptions with the ones we have been discussing.

19. Household Heterogeneity
The Basic Approach
As the authors put it:
“we develop a model in which the internal structure of the city is driven by a tension between agglomeration forces (in the form of production and residential externalities) and dispersion forces (in the form of commuting costs and an inelastic supply of land).”
The biggest different from what we have done is that, as in Lucas and Rossi-Hansberg, the production sector, including its location, is endogenous.

20. Household Heterogeneity
Households
Households have a form of Cobb-Douglas utility.
“Workers are risk neutral and have preferences that are linear in a consumption index: Uijo = Cijo , where Cijo denotes the consumption index for worker o residing in block i and working in block j. This consumption index depends on consumption of the single final good (cijo); consumption of residential floor space (ℓijo); residential amenities (Bi) …; the disutility from commuting from residential block i to workplace block j (dij ≥1); and an idiosyncratic shock that is specific to individual workers and varies with the worker's blocks of employment and residence (zijo). This idiosyncratic shock captures the idea can have idiosyncratic reasons for living and working in different parts of the city.”

21. Household Heterogeneity
Households, 2
The aggregate consumption index is:
Transportation costs take the “iceberg” form, which means that they represent a “melting” of utility or (because, as we will see, utility is proportional to wages) as a “melting of wages.
For reasons of tractability, the random shock, zijo, is assumed to be drawn from a Fréchet distribution.

22. Household Heterogeneity
Households, 3
“The indirect utility resulting from residing in block i and working in block j can be expressed in terms of the wage paid at this workplace (wj), commuting costs (dij), the residential floor price (Qi), the common component of amenities (Bi), and the idiosyncratic shock (zijo).”

23. Household Heterogeneity
Households, 4
This approach makes it impossible to derive a bid function. As in previous models, we can solve the indirect utility function for the residential floor price and we can interpret zijo as unobserved household traits.
However, wages and vary across worksites, and zijo varies across worksites and residential locations. So we cannot look at the change in bid across residential locations as dij varies.
In other words, we cannot study the observable determinants of sorting.

24. Household Heterogeneity
Households, 5
Finally, they model “residential externalities as depending on the travel-time weighted sum of residential employment density in surrounding blocks:
where HRr/Kr is residence employment density per unit of land area; residential externalities decline with travel time (τit) through the iceberg factor exp{-ρτit} ϵ (0, 1]; ρ determines their rate of spatial decay; and η controls their relative importance in overall residential amenities. The parameter η captures the net effect of residence employment density on amenities, including negative spillovers such as air pollution and crime, and positive externalities through the availability of urban amenities.”

25. Household Heterogeneity
Firms
“Production of the tradable final good occurs under conditions of perfect competition and constant returns to scale. For simplicity, we assume that the production technology takes the Cobb-Douglas form, so that output of the final good in block j (yi) is
where Ai is final goods productivity and LMi is the measure of floor space used commercially.”

26. Household Heterogeneity
Firms, 2
They also model production “externalities as depending on the travel-time weighted sum of workplace employment density in surrounding blocks:
where HMs/Ks is workplace employment density per unit of land area; production externalities decline with travel time (τjs) through the iceberg factor exp{-δτjs } ϵ(0, 1]; δ determines their rate of spatial decay; and λ controls their relative importance in determining overall productivity.

27. Household Heterogeneity
Empirical Work
This is an amazing (long!) article. Let me end with a brief comment on some of the empirical work.
They have data for Berlin. One part of the data gives commuting flows across districts.
They regress the log of the probability that a person living in district i works in district j as a function of travel time, residential district traits, and work district traits. This regression can be derived from their model.
The results are on the next slide.
I have tract-to-tract commuting data for Cleveland if any of you are interested in replicating this.

28. Household Heterogeneity
Empirical Work, 2

29. Household Heterogeneity
Empirical Work, 2
They also estimate their entire structural model.
They assume some parameters,
They use their gravity model to identify some others.
They have amazing data for West Berlin from 1936, 1986, and 2006: before the wall, with the wall, and after the wall. They use the large changes in Berlin associated with the wall to identify other parameters.
They estimate the model with GMM, which essentially finds the parameter values that give the model the best fit.

30. Household Heterogeneity
Empirical Work, 3
They find significant evidence of agglomeration economies, which decline rapidly with distance.
They find significant evidence of neighborhood externalities, which decline very rapidly with distance (decline by half every 1,000 feet).
These effects are in addition to the impact of neighborhood traits they can observe.

31. Household Heterogeneity
Empirical Work, 3

32. Household Heterogeneity
Conclusions
This article is an amazing technical achievement, especially by linking residential location and production.
Many of the results are not very transparent. The paper does not discuss, for example, the estimated land price patterns.
The price of this general-equilibrium approach is high:
Cobb-Douglas forms,
Identical households except for a random shock (and hence no insight into sorting),
Iceberg commuting costs (=no variation in speed, no operating costs, and a constant value of commuting time as a fraction of the wage).