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Over the last several years, I have read dozens of
Gridcity assignments concerning bid functions and on sorting, and I have
learned that these topics are difficult to write about. This memo
identifies a few arguments that are not correct -- and a few that are.
The key to clear understanding on these issues is to remember that
sorting depends on the slopes
of bid functions. Remember that the slope of a bid function is -t/H,
where t is commuting cost per mile and H is housing
consumption (in units of housing services).
To live farther from a worksite, people must be compensated for
commuting costs, as measured by t, so a higher t
implies that more compensation (i.e. a larger drop in housing prices) is
required.
Compensation comes in the form of a drop in
P(u), the price of
housing per unit of H. To get the same absolute compensation,
people with larger houses (i.e. more H) require less
compensation per unit of H. As a result, a large H
implies a lower required drop in P(u), all else equal.
Sorting depends on the relative
slopes of bid functions. If two groups are competing for housing, the
group with the steeper bid function will win the competition close to
the worksite and the other group will win the competition far from the
worksite.
If two groups with the same worksite differ in both
t and H,
then sorting depends on how the ratio of t to H
differs between them. The group with the highest absolute value for -t/H will live closest to the worksite.
1.
"High-income people live farther from
worksites (than do low-income people) because they are richer and want
bigger houses."
This is only half right. It is true that housing is a normal good so
that H increases with income. It is also true that a higher H leads to a flatter bid function and hence, all else equal, to
relatively high bids at distant locations. However, t also
increases with income, largely because the opportunity cost of time
increases with income. Hence, t/H might actually increase with
income, in which case high-income people would live closer to worksites.
The key is to recognize that an increase in income usually leads to a
smaller proportional increase in t than in H, which
implies that t/H declines with income. [Note also, that it is
possible to set the parameters in Gridcity so that high-income people
live closer to worksites. This occurs when high-income people have
relatively high transportation costs and relatively low demand for
housing. This kind of outcome is not unusual in many other countries.]
2.
"High-income people want a lot of
housing, so they move out to where housing is cheapest, namely in the
suburbs."
This argument picks up a the same element of truth as the previous one,
namely that a high demand for housing pushes a group away from a
worksite. However, like the previous argument, it misses the role of
transportation costs.
3.
"High-income people live farther from worksites because the
neighborhoods are nicer there."
This argument has an element of truth to it, but it is putting the cart
before the horse. In the most basic sorting models, including Gridcity,
neighborhood amenities do not matter. Higher-income people live farther
from worksites because the have a lower value of t/H, not
because of neighborhood amenities. Once a sorting pattern is
established, competition for entry into desirable neighborhoods (and
into jurisdictions with good public services) may reinforce it, but the
original sorting process does not involve neighborhood characteristics.
4. "High-income
people live farther from worksites because they have a higher utility
level." [In the terms of Gridcity, an equivalent argument is to say that
"high-income people live farther from worksites because they have a
higher system
income, as opposed to actual income."]
This
argument is simply wrong. Utility has nothing to do with it. The utility
level simply indicates the quality of a group's opportunities in other
areas and says nothing about their relative bidding power within the
urban area mapped by Gridcity. The utility level has no impact on the
slope of a bid function and therefore cannot affect sorting.
5.
"High-income people live farther from worksites because they do not need
as large a drop in the price of housing services (as do low-income
people) to compensate them for living in a distant location."
This version
is technically correct, although a bit cryptic. Indeed, a clear
explanation is virtually impossible in just a sentence or two, because
it must convey four points: (1) People must be compensated for living
far from worksites in the form of lower housing prices (per square
foot). (2) The amount of the compensation depends on the ratio of per
mile commuting costs (t) to housing consumption, H. (3) Groups with a
higher ratio of t to H live closer to worksites. (4) Higher-income
groups tend to have lower ratios of t to H. Here is a slightly longer
version that tries to make all these points:
"People will
not live far from a worksite unless they are compensated in the form of
a lower housing price per square foot. The required drop in housing
prices for each additional mile from a worksite equals per mile
commuting costs, t (which indicates the total compensation needed),
divided by the number of square feet of housing consumed, H (since any
decline in price per square foot applies to a person's entire housing
unit). A group that does not require a large drop in housing prices to
be willing to live far from a worksite, as indicated by a relatively low
ratio of t to H, has an advantage in bidding for housing at more distant
locations. Because housing demand, H, tends to increase much faster as
income increases than do commuting costs, t, high-income people tend to
have a relatively low ratio of t to H and hence tend to win the
competition for housing far from worksites.
Finally, here is a more complete version, which is probably too long for
a short decision memorandum:
"A household will not live farther from its worksite than similar
households unless it is compensated in the form of a lower price per
unit of housing. The required compensation depends on its transportation
costs and on the amount of housing it consumes, say in square feet. If
one more mile of commuting costs $2.00 per day and a household lives in
a 1000 square foot apartment, then the price per square foot must drop
by $0.002 per day (or 2/1000) to compensate the household for living one
mile further away. In other words, the housing cost savings from moving
one mile farther out ($.002 per day multiplied by 1000 square feet =
$2.00) just equals the increased cost of commuting. Thus, the amount of
compensation depends on the ratio of per-mile commuting costs to housing
consumption. The price that exactly compensates a group for its
commuting cost is called its 'bid."
"Now consider two groups that are competing with each other for housing
around some worksite. Housing suppliers will want to sell or rent to the
group willing to bid the most per unit of housing. The group that needs
more compensation, in the form of a larger drop in the price of housing,
to live far from a worksite will not bid as much for housing at distant
locations and will lose out to the other group there -- but will win the
competition close to the worksite. As just explained, this group can be
identified as the one with a higher ratio of per mile commuting costs to
housing consumption."
"Finally,
note that the ratio of per mile commuting costs to housing consumption
tends to decline as income goes up. Higher-income people consume more
housing than lower-income people and also tend to have higher commuting
costs, largely in the form of a higher opportunity cost of their time.
However, the operating costs of commuting are not clearly liked to
income, so income tends to have a smaller proportional impact on total
commuting costs than it has on housing consumption. It follows that
high-income households tend to be the ones that win the competition for
housing at distant locations, whereas low-income households tend to win
the competition close to a worksite. Note that low-income households do
not win the competition near worksites by paying more than high-income
households for the same units; instead, they bid a higher price per
square foot while consuming few square feet, that is, while accepting
small housing units or doubling up."