How to Write about Bid Functions and Sorting
PAI 786 Urban Policy
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.
- “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.]
- “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.
- “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.
- “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.
- “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.”