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Urban housing with discrete structures
JOURNAL
OF URBAN
ECONOMICS
Urban
11,
13 I- 147 (I 982)
Housing
with Discrete
Structures’
C. DUNCAN MACRAE Department o/the Treasuq: Washington, D. C. 20210
Received January 3, 1980; revised June 18, 1981 Competitive equilibrium in an urban housing model with structures that cannot be readily aggregated or subdivided is analyzed in this paper. The price and quantity of services forthcoming from each dwelling occupied by a household is determined by the equality of competitive supply and demand. A stack algorithm is then developed to ensure households are assigned to dwellings so that they would not prefer to live in any other dwelling with an equilibrium price less than the one they would pay if they lived there. Using the Urban Institute Housing Model as an example, the results of this algorithm are compared with those of the algorithm developed by de Leeuw and Struyk (D-S). The stack algorithm yields greater price discounting of existing dwellings below the price of a new dwelling than does the D-S algorithm and, thus, a greater potential for housing prices to be increased by a demand subsidy such as a housing allowance.
The standard model of urban residential land use* assumesthat housing structures can readily be converted from single-family to multi-family use and back again. Moreover, the quantity of structures can be perfectly subdivided or aggregated so that the available quantity can be allocated across households according to their demand for housing services. Since there is assumed to be no problem of conversion, the stock of housing can be viewed as a continuous variable. One implication of assuming that housing is a continuous variable is that in competitive equilibrium a household’s location is determined so that it is indifferent between living where it does and its next best alternative. A second implication is that the household’s consumption of housing services at a location is determined by the intersection of the competitive supply of services at the location with the household’s demand, which is the result of budget constrained utility maximization taking the price of servicesas given. ‘The work forming the basis for this paper was conducted pursuant to a contract with the Office of Policy Development and Research, Department of Housing and Urban Development while the author was with the Urban Institute. Opinions expressed are those of the author and do not necessarily represent the views of the Department of the Treasury, of the Department of Housing and Urban Development or of the Urban Institute. I wish to thank Margery Turner for her programming assistance, David Roscnbaum for his research assistance, and Michael Andreassi for his helpful comments. ‘See. for example, Muth (1969). 131 0094- I 190/82/020 I3 I - 17SO2.C13/0 Copyright E. 1982 by Academic Prcsr. Inc. All rights of reproduction in any form resewed.
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C. DUNCAN MACRAE
An alternative assumption and one employed in both the NBER Urban Simulation Model3 and the Urban Institute (U.I.) Housing Model4 is that there is no possibility of conversion. Structures neither can be subdivided so that the stock previously occupied by one family is instead occupied by two, nor can it be combined so that a multi-family dwelling is converted to single-family use. Housing services are no longer continuous but rather come in discrete bundles. While it is possible either to augment a household’s bundle through additional inputs or to allow the bundle to diminish, it is not possible to divide or combine dwellings. Once dwellings are viewed as discrete it is no longer possible to satisfy both conditions for competitive equilibrium in the housing market in the short run. Since structure cannot be transferred from one dwelling unit to another, it is in general not possible both for a household’s equilibrium demand for housing services to be equal to a dwelling’s competitively determined supply and for the household to be indifferent between the dwelling in which it lives and the next best alternative. The concept of equilibrium must, therefore, be modified. The approach taken by de Leeuw and Struyk to the specification of equilibrium in the U.I. Housing Model is to assumethe supplier of housing services takes the price of services as given but producer’s surplus is maximized by setting the price at the maximum level so that a household will not leave the dwelling in which it is living and move to the next best alternative.’ This means that the household is indifferent between its location and the next best one with the quantity of housing services being determined not by demand but by supply. But there is an inconsistency in this supply behavior. On the one hand, the supply relation is competitively determined; on the other hand, the price is set as if the supplier were a monopolist. However, a monopolist would not take the price as given; rather he would take into account the effect of price on marginal revenue. In specifying housing market equilibrium with durable structures, Muth also assumesthat a household is indifferent between the dwelling it occupies and the next best altemative.6 To allocate households to dwellings efficiently he employs rent-offer functions based on the condition that alI households with the same income and preferences must have the same utility. His allocation algorithm does not allow, however, for supplier response in either new or existing dwellings to the price of housing services; dwelling supply is assumedto be perfectly price inelastic. The purpose of this paper is to present and analyze an alternative concept of urban housing equilibrium with discrete structures. First, the price and 3Ingram et al. (1972). 4de Leeuw and Struyk (1975). ‘de Leeuw and Struyk (1975, Chap. 3). 6Muth (1978).
HOUSING WITH DISCRETE STRUCTURES
133
quantity of services forthcoming from each dwelling occupied are determined by the equality of demand and supply. Competitive behavior is assumed both because the majority of households are owner-occupants, whose housing consumption and production behavior can be decomposed into competitive demand and supply behavior, and because there is no evidence that the remaining households, renter-occupants, are confronted by landlords with significant monopoly power. Then an algorithm is developed to ensure households are assigned to dwellings so that they would not prefer to live in any other dwelling whose equilibrium price is less than the one they would pay if they lived there. In this way the two conditions for spatial equilibrium are maintained. Moreover, as we will see, the computational burden associated with calculating market equilibrium is reduced by an order of magnitude. The paper begins with a specification of housing market equilibrium with discrete structures using the demand and supply relations of the U.I. Housing Model as an illustrative example. An algorithm is developed to solve for market equilibrium, where the search is restricted only to valid intersections of households’ demands with dwellings’ supplies. To analyze the effect of the alternative concept of equilibrium, the U.I. Model is then recalibrated and the resulting structural parameters and predicted variables are compared with those generated by the algorithm developed by de Leeuw and Struyk. The paper concludes with a summary of the results of this comparative analysis. I. MARKET EQUILIBRIUM This section begins with the specification of demand for housing services by households and supply of these servicesby dwellings. The conditions for both microequilibrium and market equilibrium are then set forth. Housing Demand
The urban area is imagined to be composed of a finite number, I, of households living in dwellings located in a fixed number, n, of mutually exclusive and completely exhaustive zones or sectors. The i th household is then assumedto base its consumption of housing serviceson utility maximization subject to a budget constraint. The utility function for the household can be written as:
where Qi is the flow of housing services, X, is the quantity of other goods consumed by the household, and Z,, is a vector of neighborhood characteristics such as transportation accessibility, relative wealth, and racial composition for the k th sector in which the household chooses to live. The
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C. DUNCAN
MACRAE
P;Qi+X,=
r,-
budget equation is:
T,;,,
where Pi is the price of housing services in terms of other goods, which are measured in nominal terms, Y is household income, usually expressed in permanent form, and q, is the cost of transportation. The household demand for housing services resulting from constrained maximization is:
If (1) is separable in externalities (Z), then demand (3) is not a function of Z although the utility a household receives from housing will be. Moreover, if the accessibility of a zone is given exogenously then demand may be taken as fixed for a given zone. In the case of the U.I. Housing Model, the households utility function is separable in Z with accessibility treated as an externality rather than as the cost of transportation in the budget equation (2.2). Thus, Q is a function solely of P and Y. In particular, demand is of the linear expenditure form
Q; =[Pi/f’n - ~t(l - a;)(f’Jf’i
- lj]Qrt
(4)
where the parameter (Y~varies by household type, y, is constant across all household types, and P,, is the price of new housing services, which is the sum of the given operating, PO,and capital costs, PC.per unit of output,’ and Q: = qq/P,, the demand for servicesat the new house price. Housing services (Q) are viewed as forthcoming from a discrete number of dwellings, m, located in the n zones. The level of housing services produced by the jth dwelling is based on the maximization of the present discounted value of the rental stream-imputed or actual-from the dwelling, after operating costs, property taxes and depreciation, and subject to a production function relating Q to operating and capital inputs. In general, the supply of services in any period is a function of the expected path of prices into the indefinite future. However, supply can be expressed simply as a function of current prices: Qj = S,{p/. P,. PC}, if price expectations are static. ‘de Leeuw and Struyk (1975, footnote 3, p. 85).
(5)
HOUSING WITH DISCRETE STRUCTURES
135
In the case of the U.I. Housing Model, there are m - 1existing dwellings, located in the first n - 1 zones. The supply from these dwellings is given by
where 1 - p, is the period depreciation rate, & is a price sensitivity parameter and Q, is the level of housing services produced by the dwelling in the previous period.* In the new construction zone, n, there is available to every household a new dwelling. For these 1 dwellings, the price of housing services is fixed at P, with the only constraint on supply being that Qj must not be less than the building code minimum, Q,,,. Microequilibrium
For the housing market to be in equilibrium every household must be in equilibrium with the dwelling which it occupies. This means that for every dwelling that is occupied the supply of services forthcoming from the dwelling must be equal to the level of services demanded by the household living in the dwelling:
where Pi, is the price the ith household would pay if it were to live in the ith dwelling. Thus, the first step in determining market equilibrium is to ascertain what are valid intersections of households’ demands with the dwellings’ supplies. Since supply from existing dwellings in the U.I. Housing Model (6) is linear in P and demand is quadratic in P, microequilibrium Pij is the positive root of the quadratic aP2 + bP + c = 0,
(8)
where a = v~I~~Q,P,, b
=P,Q,- 2/3w,~,/p~-
4
- &wP,~
c = (y,(l - CX)- l)aY, and subject to the supply constraint that Pj 1 P, and the demand constraints that for the implied level of housing service Qii 2 y,QT and Xi 1 y, X,?. For new houses, Qij is given simply by substituting P,, into (4) to obtain Qi” = aiq:/Pn, (9) *de Leeuw and Struyk (1975, p. 21).
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C. DUNCANMACRAE
subject to the supply constraint that Qi, 2 Q, and the same demand constraints as for existing dwellings. Equilibrium Assignment While dwellings are assumedimmobile, households are not. Therefore, for the housing market to be in equilibrium, households must be assigned to dwellings so that they have no motivation to move. An equilibrium assignment is then an assignment satisfying two conditions. The first condition is that given the vector of prices associated with the assignment, one Pi for each dwelling, no household prefers to live in a dwelling for which the equilibrium price is less than what the household would be willing to pay, i.e., for allj such that P, < Pij it is also true that Ujj 5 y, where Uij is the utility the household would receive if it were to live in thejth dwelling. This does not mean that the household is indifferent between the dwelling in which it lives and the next best alternative; but it does mean that it has no motivation to move there. While the intersections of household demands with dwelling supplies may not be a function of the level of externalities, household preferences regarding dwellings surely are. The utility a household associates with a dwelling is a function of the externalities in the dwelling’s zone. Thus, the second condition for an assignment to be an equilibrium one is that the externalities generated by the assignment be equal to the externalities on which the assignment was based. In the U.I. Model the household’s utility function is of the form Q = (Qi - y,Qrp(
Xi - y,X;)‘-lZi:,Z,‘Zi’,,
(10)
where Zil,, Zfk, and Zi’k are measures of the transportation accessibility, relative averagewealth, and racial composition, respectively, of the k th zone as perceived by the i th household. Since transportation accessibility (Z,‘,) for a zone is given in the U.I. Model, equilibration of assumed and actual externalities in this model amounts to ensuring that the actual relative wealth (Z,‘,) and racial composition (Z;‘,) for each zone are equal to their corresponding assumedvalues. II. STACK ALGORITHM In this section an algorithm for determining an equilibrium assignment of households to dwellings is presented. First, the data structures underlying the algorithm are described and then the procedure itself is designed. Data Structures Associated with the assignment algorithm are three data structures, which describe the status of the assignment at any point in the algorithm. These
HOUSING
WITH
DISCRETE
STRUCTURES
137
i Dwelllnp ” I FIG. I.
Assignment algorithm data structures.
structures are portrayed in Fig. 1. The first structure is the dwelling occupancy array indicating which household, if any, occupies each dwelling and what is the price that corresponds to the intersection of that household’s demand curve with the dwelling’s supply curve. Initially all dwellings are unoccupied, including new dwellings-potentially one for every household -and the price of services from each dwelling is set, somewhat arbitrarily, at zero. The second structure is a set of ordered dwelling stacks, one stack for each household, with dwellings arranged in order of their utility to the household, the most preferred at the top of the stack and the least preferred at the bottom. Initially each stack contains all dwellings that yield valid intersections with the corresponding household’s demand curve. The third structure is the stack of unassigned households which at the beginning of the algorithm contains all households to be assigned. Algorithmic
Procedure
The basic objective of the algorithm is to assign households one at a time to their preferred dwelling and reassign them only when another household is willing to pay a higher price for the dwelling. Starting with the household at the top of the unallocated household stack, the first step is to find the preferred dwelling for that household. This can be ascertained by examining the top of that household’s ordered dwelling stack. Having ascertained the preferred dwelling, the next step is to examine the dwelling occupancy array to seeif that dwelling is currently occupied. If the dwelling is not occupied, as initially we know it will not be, the row of the occupancy array corresponding to that dwelling is altered to reflect the household occupying it and the microequilibrium price the household would pay for it. The household at the top of the unassigned household stack is then pulled from the stack so that the next unassigned household can be considered. This procedure is continued until either the unassigned household stack is
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MACRAE
emptied, or, more likely, the household’s preferred dwelling is already occupied by a previously assigned household. If the preferred dwelling is occupied, a comparison is made between the current price paid for the dwelling and the price that the unassigned household is willing to pay in microequilibrium. If the unassigned household is not willing to pay more than the current occupant is, the current tenant remains in the dwelling and the unassigned household must look for another dwelling. To do this the dwelling at the top of the dwelling stack for the unassigned household is pulled bringing up the next dwelling in order of preference. This procedure is continued until either the unassigned household’s dwelling stack is empty or the household is willing to pay a higher price than the current occupant. If the dwelling stack is emptied, then the unassigned household is competed out of the housing market and has no viable assignment. Once the dwelling stack is emptied, that household is pulled from the unassigned household stack so that another household can be considered. If the unassigned household is willing, however, to pay a higher price than the current occupant, the dwelling is reassigned to that household. The dwelling at the top of the dwelling stack of the displaced household is pulled and the dwelling occupancy array is altered to reflect the new tenant and price. Also the household at the top of the unassigned household stack is pulled, since that household has now been assigned, and then the displaced household is pushed onto this stack so that it can now be considered for assignment. Again this procedure is continued until the unassigned household stack is emptied at which point the algorithm is terminated. Since every household considers dwellings in its preferred order and only loses out when another household is willing to pay a higher price, we can be sure that no assigned household would prefer to live in another dwelling for which another household is paying a lower price than the assigned household would have; otherwise, the household would be living there. Moreover, since the assignment choice is restricted to valid supply-demand microequilibria, we know that in the final assignment supply equals demand for every household-dwelling pair. We cannot know in advance, however, whether a household ultimately gets assigned to a dwelling since the household could be totally outcompeted in the housing market. Also, we cannot be sure that the externalities generated by the assignment are the same as the extemalities upon which the assignment was based. The most straightforward procedure for equilibrating assumedand generated externalities if they are not equal in the initial assignment is to substitute the generated for the assumed externalities and then carry out another assignment. Indeed, this is the procedure used by de Leeuw and Struyk to equilibrate assumed and generated zonal racial compositions but not the zonal wealth externality, which is equilibrated as part of their algorithm.
139
HOUSING WITH DISCRETE STRUCTURES P
p* ----------
D
PC--------
0, Y, Q*
Qc
Q" Q
Q
FIG. 2. One household, one existing dwelling, and one new dwelling.
III. COMPARATIVE
ANALYSIS
This section begins with a simplified theoretical comparison for the U.I. Housing Model of the results of the de Leeuw-Struyk algorithm and the stack algorithm for assigning households to dwellings. The results of the two algorithms are then compared empirically for six metropolitan areas to which the U.I. Model has been calibrated. Theoretical Analysis For a given set of household demand and dwelling supply parameters, the de Leeuw-Struyk algorithm, in general, yields higher prices and quantities of housing service than does the stack assignment algorithm. The reason can be seen most easily in a market where there are two dwellings, one new and one existing, and only one household, as illustrated in Fig. 2. Associated with the household’s demand curve, D, is a map of pricequantity indifference curves.9 Each curve represents price-quantity combinations that yield the same utility to the household while satisfying its budget constraints. For a given quantity of housing services (Q) the lower the price paid (P), the higher the amount of income available for expenditure on other goods (X) and hence the higher the household’s utility. 9de Leeuw and Struyk (1975, pp. 28-31).
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C. DUNCAN MACRAE
FIG. 3. One household and two existing dwellings.
Therefore, in Fig. 2, the lower is the indifference curve the higher is the household’s utility (U” 1 v’). The demand curve then describes the Q chosen by the household so as to maximize utility given P. It is determined by the tangency of the price line with the price-quantity indifference curve. For example, at the price P” utility is maximized at the level U” by consuming Q” of housing services. Alternatively, the demand curve can be viewed as the maximum price a household would pay to achieve a given level of utility. Whichever way the curve is seenthe associatedprice-quantity indifference map is useful in comparing the results of the two assignment algorithms. For the stack algorithm, supply-demand equilibrium for the household in a new dwelling with supply S, occurs at the point (Q,, PC) and for the existing dwelling equilibrium occurs at (Q”, P”). Since the level of utility is higher for the household in an existing house, the household chooses this house achieving a utility level of U”. In the de Leeuw-Struyk algorithm, however, the household is not allowed to pay the competitive price. Rather producer’s surplus is maximized for the landlord so that the price is increased to P’ where the household is indifferent between living in the existing house as opposed to the new house. The price is still lower than the new price (PC), but the utility received by the household, u’, is the same as if it were living in the new house.
HOUSING WITH DISCRETE STRUCTURES
141
While the de Leeuw-Struyk algorithm, in general, yields higher prices and quantities than does the stack algorithm, it is possible for the opposite to occur. In Fig. 3, we see an example of a market again with only one household and two dwellings but this time both dwellings are existing with supply curves given by S, and S,, respectively. Under the stack algorithm, the only choice open to the household is the first dwelling at a price P” since the second dwelling withdraws from the market before the price on that dwelling drops low enough to equilibrate its supply with the household’s demand. Under the de Leeuw-Struyk algorithm, however, households are not constrained to be on their demand curve. Therefore, the household choosesthe second dwelling at the price P’, which yields a level of utility of u’ instead of u”. Empirical Analysis
Having seen that there is a potential for the de Leeuw-Struyk algorithm to generate higher prices and quantities than the stack algorithm, let us now ascertain the degree to which this potential is realized. The U.I. Housing Model was resolved with the stack algorithm, for each of the six metropolitan areas first using the parameters that best explain housing market behavior in the 1960s under the de Leeuw-Struyk algorithm. Rather than report the results for all areas, I examine only two areas in 1970: Washington, which had little discounting of existing house prices below the new house price under the de Leeuw-Struyk algorithm, and Chicago, which had significant price discounting. For Washington, the resulting price structure curve for existing dwellings under the stack algorithm for 1970 is portrayed in Fig. 4. For comparison, the price curve under the de Leeuw-Struyk algorithm is also given.” As can readily be seen, except for quite large and quite small dwellings, there is little difference in the price structures under the alternative algorithms. In this area, the demand vs. the supply for mid-range dwellings is sufficiently large to yield existing house prices that are not significantly different from the price of new houses (P,,). Only in the case of a few large dwellings, which cannot be absorbed except at a lower price, and the small dwellings, which cannot meet most household’s demand except at a price significantly in excessof P,, is the potential for the de Leeuw-Struyk algorithm to yield higher prices than the stack algorithm realized. Using the parameters that “best fit” under the de Leeuw-Struyk algorithm, it would appear that the only slack in the Washington housing market is truly at the very upper and lower ends of the market. Using the de Leeuw-Struyk parameters for Chicago in 1970, we observe comparative results somewhat different from those observed in Washington. “de Leeuw and Stmyk (1975, p. 108).
Po= .47
20
I 40
III 60 80 om=
IIll I IIll III III Ii 100 120 140 160 180 200 220 240 2M) 280 300 320 340 360 380 4W 420
72
FIG. 4.
D
1970Washington existing dwelling price structures.
I.”
P P” = 1.37
1.5
1.4 1.3 1.2 1.1
1.0
1 .9 .8
; : :
.7
Po= 56
.6
:
I
I
I
20
40
60
Q,= ----de
Leeuw-Struyk
-stack
algorithm
I! “p
I
I
I
I
I
100
120
140
160
180
I ZM)
I 220
I
I
I
I
I
240
260
280
300
320
(DS) algorithm using
:
cl
81.2
DS parameters
FIG. 5. 1970 Chicago existing dwelling price structures.
142
0
143
HOUSING WITH DISCRETE STRUCTURES TABLE I U.I. Housing Model Calibration Results Metropolitan area Durham Austin Portland Pittsburgh Washington Chicago
Parameters
Mean error
Assignment &o~~m’
yI
y2
YS
81
DS Stack DS Stack DS Stack DS2 Stack DS Stack DS Stack
0.8 0.5 0.8 0.5 0.9 0.7 0.9 0.0 0.9 0.9 0.7 0.4
0.4 1.0 0.2 10.0 0.6 10.0 0.3 4.0 0.3 0.3 0.4 0.7
0.7 1.0 0.9 1.0 0.9 10.0 0.7 0.0 0.6 4.0 0.1 0.7
0.7 0.9 0.4 0.9 0.4 0.9 0.5 0.7 0.6 0.7 0.7 0.7
82
0.4 0.3 0.9 0.4 0.9 0.7 0.7 0.3 0.9 0.9 0.4 0.4
1960
I970
0.083 0.085 0.116 0.062 0.092 0.066 0.093 0.080 0.047 0.056 0.102 0.093
0.142 0.05I 0.171 0.202 0.100 0.111 0.088 0.099 0.089 0.08I 0.165 0.171
‘The de Leeuw-Struyk (DS) results are obtained from de Leeuw and Struyk (1975, Tables II and 12). ‘The model was recalibrated for 1970 since the results in de Leeuw and Struyk (1975, Table I I), are based on a reversal of the housing expense to income ratios (a,) for black nonelderly families and white elderly/single households,
As shown in Fig. 5, for mid-range dwellings the results of the algorithms are still quite similar but the discounting for large dwellings is more substantial in Chicago than in Washington. Under the stack algorithm almost the entire upper third of the existing housing stock has prices significantly below P,,. There does not appear to be the household demand to support housing prices for larger dwellings that there is in Washington. At the bottom of the market, we observe the exception to the rule that the stack algorithm results in higher prices than does the de Leeuw-Struyk algorithm. Calibration
Analysis
To complete the comparison with the de Leeuw-Struyk algorithm, the U.I. Housing Model was recalibrated for the 1960’s using the stack algorithm. The results of both algorithms are given in Table 1. Looking first at the demand parameters we see that, in general, the wealth (y2) and racial externalities ( y3) parameters are higher under the stack algorithm than with the de Leeuw-Struyk algorithm, while the minimum consumption parameter (y,) is nowhere greater. Indeed, in all but one of the six metropolitan areas the minimum consumption levels are lower under the stack algorithm. Moreover, they are more in keeping with independent estimates of minimum consumption with linear expenditure systems.” The only exception to ’ ’ Ring ( 1979)
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C. DUNCAN
MACR4E
the greater importance of externalities in determining household behavior under the stack algorithm is Pittsburgh for ys. This is the one area where households’ behavior is best explained by assuming no racial preferenceson the part of blacks or whites. Turning to the supply parameters we see that the decennial depreciation rate, 1 - &, is in general, lower under the stack algorithm than under the de Leeuw-Struyk algorithm with the exception of Chicago, where the parameter is the same under the two algorithms. Moreover, the implied annual rates which range from 1.1 to 3.5% under the stack algorithm also correspond more closely to other estimates of structure depreciation.‘* In general, the elasticity parameter, &, is also lower with the exception of Washington and Chicago, where the parameter is the same. Comparing the algorithms in terms of explanatory power as measured by mean error, it is apparent that in 1960 for four of the metropolitan areas the stack algorithm does a superior job, and for one of the areas there is not a significant difference between the results of the two algorithms. In 1970, the de Leeuw-Struyk algorithm, however, has the edge doing a better job in four of the areas with the stack algorithm doing a better job in only two of the areas. While no one algorithm dominates in terms of explanatory power, the stack algorithm is vastly more efficient computationally than the de LeeuwStruyk algorithm. Since both algorithms are programmed to run on the same computer (CDC 6600) we can make a direct cost comparison. On average the cost of solving the U.I. Housing Model with the stack algorithm is 15%of the cost of solving the model with the de Leeuw-Struyk algorithm. Price Structures
I conclude the comparative analysis of the two algorithms by examining the price structure curves that result with the recalibration under the stack algorithm. Again, only the curves for Washington and Chicago, are reported as shown in Figs. 6 and 7. The greater degree of price discounting under the stack algorithm than under the de Leeuw-Struyk algorithm is striking. With the parameters that best fit under the de Leeuw-Struyk algorithm, the difference between the two algorithms was noticeable primarily at the lower and upper ends of the housing market. However, after recalibration the difference in the results of the two procedures is more apparent. It is particularly striking in Chicago, where prices for many dwellings are at least 20% lower under the stack algorithm. Even in Washington, where there continues to be little price discounting, there is more than under the de Leeuw-Struyk algorithm. ‘*Muth
(1975, p. 314).
FIG. 6.
1970Washington existing dwelling price structures.
1.6
P
1.5 P”= 1.37
1.4 1.3 1.2 1.1 1.0 .9 -
: : :
.8 -
:
.J -
:
PO’
56
:
:
:: .6-
:
III 20 40
L III 60
80
a,= ***. -
de LeeuwStruyk stack algorithm
FIG. 7.
100
Ill 120
140
160
III 180 200
220
Ill 240
260
280
300
320
0
81.2 (DS) algorithm
1970Chicago existing dwelling price structures. 145
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C. DUNCAN
MACRAE
The result that there is more price discounting with the stack algorithm has implications for policy analysis with the U.I. Model. In comparing demand subsidies, such as housing allowances, with supply subsidies such as new construction subsidies, de Leeuw and Struyk focused attention on the importance of price discounting in the absence of the subsidy.13It is only where there is appreciable discounting that a demand subsidy can significantly raise housing prices. While the housing allowance supply experiments have found relatively little evidence of price inflationI the de Leeuw-Struyk results indicate the potential for inflation from a housing allowance in a number of metropolitan areas. The results of the stack algorithm indicate the potential is even greater. Since producer’s surplus is not maximized, there is still the opportunity for increases in the surplus through increased demand. IV. CONCLUSIONS In this paper we have developed and analyzed a stack algorithm for determining equilibrium in a model of urban housing with discrete structures. The algorithm ensures both that the competitive supply of services forthcoming from a dwelling is equal to the demand for those services by the resident household and that given the structure of prices in the housing market, the household would not prefer to live in any other dwelling than the one in which it resides. Using the Urban Institute Housing Model as an example, the stack algorithm was employed to calibrate the model for the six metropolitan areas for which de Leeuw and Struyk had already calibrated the model with their algorithm. On the demand side the stack algorithm implies lower minimum levels of consumption, which are more in accord with other research using the linear expenditure system, and a more important role for neighborhood externalities than does the de Leeuw-Struyk algorithm. On the supply side, the stack algorithm yields a lower structure depreciation rate, again more in keeping with independent estimates, and a lower elasticity of supply. Whether using the original parameters for the U.I. Model or the recalibrated parameters, the stack algorithm indicates that there was greater discounting of the price of existing dwellings below the new dwelling price in 1970 than does the de Leeuw-Struyk algorithm. Therefore, if demand subsidies such as housing allowances had been introduced, there would have been greater inflation in the price of housing. Finally, while neither algorithm obviously dominates in terms of explanatory power as measured by mean square error-one algorithm does a better job in some metropolitan “de Leeuw and Struyk (1975. pp. 115- 149) 14Bamett and Lowry (1979).
HOUSING WITH DISCRETE STRUCTURES
147
areas, the other does a superior job in other areas-the stack algorithm is an order of magnitude more efficient in obtaining a solution than is the de Leeuw-Struyk algorithm. Hence, with the stack algorithm a model of discrete housing such as the U.I. Model can be enhanced in ways, such as the addition of more households, dwellings, or zones, that were not previously computationally feasible. REFERENCES C. Lance Barnett and Ira S. Lowry, “How Housing Allowances Affect Housing Prices,” Rand Paper R-2452-HUD, September 1979. Frank de Leeuw and Raymond Struyk, “The Web of Urban Housing,” The Urban Institute, Washington, D.C. 1975. Gregory K. Ingram, John F. Kain, and J. Royce Ginn, “The Detroit Prototype of the NBER Urban Simulation Model,” Nat. Bur. Econ. Res., New York, 1972. A. Thomas King, Estimation of a linear expenditure system for the United States in 1973.J. Econ. Business, 31, No. 3 (September 1979). Richard F. Muth, “Cities and Housing,” Univ. of Chicago Press,Chicago, 1969. Richard F. Muth. Numerical solution of urban residential land-use models, J. Urbun Econ., 2, No. 4 (October 1975). Richard F. Muth, The allocation of households to dwellings, J. Reg. SC.. 18, No. 2 (August 1978).
OF URBAN
ECONOMICS
Urban
11,
13 I- 147 (I 982)
Housing
with Discrete
Structures’
C. DUNCAN MACRAE Department o/the Treasuq: Washington, D. C. 20210
Received January 3, 1980; revised June 18, 1981 Competitive equilibrium in an urban housing model with structures that cannot be readily aggregated or subdivided is analyzed in this paper. The price and quantity of services forthcoming from each dwelling occupied by a household is determined by the equality of competitive supply and demand. A stack algorithm is then developed to ensure households are assigned to dwellings so that they would not prefer to live in any other dwelling with an equilibrium price less than the one they would pay if they lived there. Using the Urban Institute Housing Model as an example, the results of this algorithm are compared with those of the algorithm developed by de Leeuw and Struyk (D-S). The stack algorithm yields greater price discounting of existing dwellings below the price of a new dwelling than does the D-S algorithm and, thus, a greater potential for housing prices to be increased by a demand subsidy such as a housing allowance.
The standard model of urban residential land use* assumesthat housing structures can readily be converted from single-family to multi-family use and back again. Moreover, the quantity of structures can be perfectly subdivided or aggregated so that the available quantity can be allocated across households according to their demand for housing services. Since there is assumed to be no problem of conversion, the stock of housing can be viewed as a continuous variable. One implication of assuming that housing is a continuous variable is that in competitive equilibrium a household’s location is determined so that it is indifferent between living where it does and its next best alternative. A second implication is that the household’s consumption of housing services at a location is determined by the intersection of the competitive supply of services at the location with the household’s demand, which is the result of budget constrained utility maximization taking the price of servicesas given. ‘The work forming the basis for this paper was conducted pursuant to a contract with the Office of Policy Development and Research, Department of Housing and Urban Development while the author was with the Urban Institute. Opinions expressed are those of the author and do not necessarily represent the views of the Department of the Treasury, of the Department of Housing and Urban Development or of the Urban Institute. I wish to thank Margery Turner for her programming assistance, David Roscnbaum for his research assistance, and Michael Andreassi for his helpful comments. ‘See. for example, Muth (1969). 131 0094- I 190/82/020 I3 I - 17SO2.C13/0 Copyright E. 1982 by Academic Prcsr. Inc. All rights of reproduction in any form resewed.
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An alternative assumption and one employed in both the NBER Urban Simulation Model3 and the Urban Institute (U.I.) Housing Model4 is that there is no possibility of conversion. Structures neither can be subdivided so that the stock previously occupied by one family is instead occupied by two, nor can it be combined so that a multi-family dwelling is converted to single-family use. Housing services are no longer continuous but rather come in discrete bundles. While it is possible either to augment a household’s bundle through additional inputs or to allow the bundle to diminish, it is not possible to divide or combine dwellings. Once dwellings are viewed as discrete it is no longer possible to satisfy both conditions for competitive equilibrium in the housing market in the short run. Since structure cannot be transferred from one dwelling unit to another, it is in general not possible both for a household’s equilibrium demand for housing services to be equal to a dwelling’s competitively determined supply and for the household to be indifferent between the dwelling in which it lives and the next best alternative. The concept of equilibrium must, therefore, be modified. The approach taken by de Leeuw and Struyk to the specification of equilibrium in the U.I. Housing Model is to assumethe supplier of housing services takes the price of services as given but producer’s surplus is maximized by setting the price at the maximum level so that a household will not leave the dwelling in which it is living and move to the next best alternative.’ This means that the household is indifferent between its location and the next best one with the quantity of housing services being determined not by demand but by supply. But there is an inconsistency in this supply behavior. On the one hand, the supply relation is competitively determined; on the other hand, the price is set as if the supplier were a monopolist. However, a monopolist would not take the price as given; rather he would take into account the effect of price on marginal revenue. In specifying housing market equilibrium with durable structures, Muth also assumesthat a household is indifferent between the dwelling it occupies and the next best altemative.6 To allocate households to dwellings efficiently he employs rent-offer functions based on the condition that alI households with the same income and preferences must have the same utility. His allocation algorithm does not allow, however, for supplier response in either new or existing dwellings to the price of housing services; dwelling supply is assumedto be perfectly price inelastic. The purpose of this paper is to present and analyze an alternative concept of urban housing equilibrium with discrete structures. First, the price and 3Ingram et al. (1972). 4de Leeuw and Struyk (1975). ‘de Leeuw and Struyk (1975, Chap. 3). 6Muth (1978).
HOUSING WITH DISCRETE STRUCTURES
133
quantity of services forthcoming from each dwelling occupied are determined by the equality of demand and supply. Competitive behavior is assumed both because the majority of households are owner-occupants, whose housing consumption and production behavior can be decomposed into competitive demand and supply behavior, and because there is no evidence that the remaining households, renter-occupants, are confronted by landlords with significant monopoly power. Then an algorithm is developed to ensure households are assigned to dwellings so that they would not prefer to live in any other dwelling whose equilibrium price is less than the one they would pay if they lived there. In this way the two conditions for spatial equilibrium are maintained. Moreover, as we will see, the computational burden associated with calculating market equilibrium is reduced by an order of magnitude. The paper begins with a specification of housing market equilibrium with discrete structures using the demand and supply relations of the U.I. Housing Model as an illustrative example. An algorithm is developed to solve for market equilibrium, where the search is restricted only to valid intersections of households’ demands with dwellings’ supplies. To analyze the effect of the alternative concept of equilibrium, the U.I. Model is then recalibrated and the resulting structural parameters and predicted variables are compared with those generated by the algorithm developed by de Leeuw and Struyk. The paper concludes with a summary of the results of this comparative analysis. I. MARKET EQUILIBRIUM This section begins with the specification of demand for housing services by households and supply of these servicesby dwellings. The conditions for both microequilibrium and market equilibrium are then set forth. Housing Demand
The urban area is imagined to be composed of a finite number, I, of households living in dwellings located in a fixed number, n, of mutually exclusive and completely exhaustive zones or sectors. The i th household is then assumedto base its consumption of housing serviceson utility maximization subject to a budget constraint. The utility function for the household can be written as:
where Qi is the flow of housing services, X, is the quantity of other goods consumed by the household, and Z,, is a vector of neighborhood characteristics such as transportation accessibility, relative wealth, and racial composition for the k th sector in which the household chooses to live. The
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C. DUNCAN
MACRAE
P;Qi+X,=
r,-
budget equation is:
T,;,,
where Pi is the price of housing services in terms of other goods, which are measured in nominal terms, Y is household income, usually expressed in permanent form, and q, is the cost of transportation. The household demand for housing services resulting from constrained maximization is:
If (1) is separable in externalities (Z), then demand (3) is not a function of Z although the utility a household receives from housing will be. Moreover, if the accessibility of a zone is given exogenously then demand may be taken as fixed for a given zone. In the case of the U.I. Housing Model, the households utility function is separable in Z with accessibility treated as an externality rather than as the cost of transportation in the budget equation (2.2). Thus, Q is a function solely of P and Y. In particular, demand is of the linear expenditure form
Q; =[Pi/f’n - ~t(l - a;)(f’Jf’i
- lj]Qrt
(4)
where the parameter (Y~varies by household type, y, is constant across all household types, and P,, is the price of new housing services, which is the sum of the given operating, PO,and capital costs, PC.per unit of output,’ and Q: = qq/P,, the demand for servicesat the new house price. Housing services (Q) are viewed as forthcoming from a discrete number of dwellings, m, located in the n zones. The level of housing services produced by the jth dwelling is based on the maximization of the present discounted value of the rental stream-imputed or actual-from the dwelling, after operating costs, property taxes and depreciation, and subject to a production function relating Q to operating and capital inputs. In general, the supply of services in any period is a function of the expected path of prices into the indefinite future. However, supply can be expressed simply as a function of current prices: Qj = S,{p/. P,. PC}, if price expectations are static. ‘de Leeuw and Struyk (1975, footnote 3, p. 85).
(5)
HOUSING WITH DISCRETE STRUCTURES
135
In the case of the U.I. Housing Model, there are m - 1existing dwellings, located in the first n - 1 zones. The supply from these dwellings is given by
where 1 - p, is the period depreciation rate, & is a price sensitivity parameter and Q, is the level of housing services produced by the dwelling in the previous period.* In the new construction zone, n, there is available to every household a new dwelling. For these 1 dwellings, the price of housing services is fixed at P, with the only constraint on supply being that Qj must not be less than the building code minimum, Q,,,. Microequilibrium
For the housing market to be in equilibrium every household must be in equilibrium with the dwelling which it occupies. This means that for every dwelling that is occupied the supply of services forthcoming from the dwelling must be equal to the level of services demanded by the household living in the dwelling:
where Pi, is the price the ith household would pay if it were to live in the ith dwelling. Thus, the first step in determining market equilibrium is to ascertain what are valid intersections of households’ demands with the dwellings’ supplies. Since supply from existing dwellings in the U.I. Housing Model (6) is linear in P and demand is quadratic in P, microequilibrium Pij is the positive root of the quadratic aP2 + bP + c = 0,
(8)
where a = v~I~~Q,P,, b
=P,Q,- 2/3w,~,/p~-
4
- &wP,~
c = (y,(l - CX)- l)aY, and subject to the supply constraint that Pj 1 P, and the demand constraints that for the implied level of housing service Qii 2 y,QT and Xi 1 y, X,?. For new houses, Qij is given simply by substituting P,, into (4) to obtain Qi” = aiq:/Pn, (9) *de Leeuw and Struyk (1975, p. 21).
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C. DUNCANMACRAE
subject to the supply constraint that Qi, 2 Q, and the same demand constraints as for existing dwellings. Equilibrium Assignment While dwellings are assumedimmobile, households are not. Therefore, for the housing market to be in equilibrium, households must be assigned to dwellings so that they have no motivation to move. An equilibrium assignment is then an assignment satisfying two conditions. The first condition is that given the vector of prices associated with the assignment, one Pi for each dwelling, no household prefers to live in a dwelling for which the equilibrium price is less than what the household would be willing to pay, i.e., for allj such that P, < Pij it is also true that Ujj 5 y, where Uij is the utility the household would receive if it were to live in thejth dwelling. This does not mean that the household is indifferent between the dwelling in which it lives and the next best alternative; but it does mean that it has no motivation to move there. While the intersections of household demands with dwelling supplies may not be a function of the level of externalities, household preferences regarding dwellings surely are. The utility a household associates with a dwelling is a function of the externalities in the dwelling’s zone. Thus, the second condition for an assignment to be an equilibrium one is that the externalities generated by the assignment be equal to the externalities on which the assignment was based. In the U.I. Model the household’s utility function is of the form Q = (Qi - y,Qrp(
Xi - y,X;)‘-lZi:,Z,‘Zi’,,
(10)
where Zil,, Zfk, and Zi’k are measures of the transportation accessibility, relative averagewealth, and racial composition, respectively, of the k th zone as perceived by the i th household. Since transportation accessibility (Z,‘,) for a zone is given in the U.I. Model, equilibration of assumed and actual externalities in this model amounts to ensuring that the actual relative wealth (Z,‘,) and racial composition (Z;‘,) for each zone are equal to their corresponding assumedvalues. II. STACK ALGORITHM In this section an algorithm for determining an equilibrium assignment of households to dwellings is presented. First, the data structures underlying the algorithm are described and then the procedure itself is designed. Data Structures Associated with the assignment algorithm are three data structures, which describe the status of the assignment at any point in the algorithm. These
HOUSING
WITH
DISCRETE
STRUCTURES
137
i Dwelllnp ” I FIG. I.
Assignment algorithm data structures.
structures are portrayed in Fig. 1. The first structure is the dwelling occupancy array indicating which household, if any, occupies each dwelling and what is the price that corresponds to the intersection of that household’s demand curve with the dwelling’s supply curve. Initially all dwellings are unoccupied, including new dwellings-potentially one for every household -and the price of services from each dwelling is set, somewhat arbitrarily, at zero. The second structure is a set of ordered dwelling stacks, one stack for each household, with dwellings arranged in order of their utility to the household, the most preferred at the top of the stack and the least preferred at the bottom. Initially each stack contains all dwellings that yield valid intersections with the corresponding household’s demand curve. The third structure is the stack of unassigned households which at the beginning of the algorithm contains all households to be assigned. Algorithmic
Procedure
The basic objective of the algorithm is to assign households one at a time to their preferred dwelling and reassign them only when another household is willing to pay a higher price for the dwelling. Starting with the household at the top of the unallocated household stack, the first step is to find the preferred dwelling for that household. This can be ascertained by examining the top of that household’s ordered dwelling stack. Having ascertained the preferred dwelling, the next step is to examine the dwelling occupancy array to seeif that dwelling is currently occupied. If the dwelling is not occupied, as initially we know it will not be, the row of the occupancy array corresponding to that dwelling is altered to reflect the household occupying it and the microequilibrium price the household would pay for it. The household at the top of the unassigned household stack is then pulled from the stack so that the next unassigned household can be considered. This procedure is continued until either the unassigned household stack is
138
C. DUNCAN
MACRAE
emptied, or, more likely, the household’s preferred dwelling is already occupied by a previously assigned household. If the preferred dwelling is occupied, a comparison is made between the current price paid for the dwelling and the price that the unassigned household is willing to pay in microequilibrium. If the unassigned household is not willing to pay more than the current occupant is, the current tenant remains in the dwelling and the unassigned household must look for another dwelling. To do this the dwelling at the top of the dwelling stack for the unassigned household is pulled bringing up the next dwelling in order of preference. This procedure is continued until either the unassigned household’s dwelling stack is empty or the household is willing to pay a higher price than the current occupant. If the dwelling stack is emptied, then the unassigned household is competed out of the housing market and has no viable assignment. Once the dwelling stack is emptied, that household is pulled from the unassigned household stack so that another household can be considered. If the unassigned household is willing, however, to pay a higher price than the current occupant, the dwelling is reassigned to that household. The dwelling at the top of the dwelling stack of the displaced household is pulled and the dwelling occupancy array is altered to reflect the new tenant and price. Also the household at the top of the unassigned household stack is pulled, since that household has now been assigned, and then the displaced household is pushed onto this stack so that it can now be considered for assignment. Again this procedure is continued until the unassigned household stack is emptied at which point the algorithm is terminated. Since every household considers dwellings in its preferred order and only loses out when another household is willing to pay a higher price, we can be sure that no assigned household would prefer to live in another dwelling for which another household is paying a lower price than the assigned household would have; otherwise, the household would be living there. Moreover, since the assignment choice is restricted to valid supply-demand microequilibria, we know that in the final assignment supply equals demand for every household-dwelling pair. We cannot know in advance, however, whether a household ultimately gets assigned to a dwelling since the household could be totally outcompeted in the housing market. Also, we cannot be sure that the externalities generated by the assignment are the same as the extemalities upon which the assignment was based. The most straightforward procedure for equilibrating assumedand generated externalities if they are not equal in the initial assignment is to substitute the generated for the assumed externalities and then carry out another assignment. Indeed, this is the procedure used by de Leeuw and Struyk to equilibrate assumed and generated zonal racial compositions but not the zonal wealth externality, which is equilibrated as part of their algorithm.
139
HOUSING WITH DISCRETE STRUCTURES P
p* ----------
D
PC--------
0, Y, Q*
Qc
Q" Q
Q
FIG. 2. One household, one existing dwelling, and one new dwelling.
III. COMPARATIVE
ANALYSIS
This section begins with a simplified theoretical comparison for the U.I. Housing Model of the results of the de Leeuw-Struyk algorithm and the stack algorithm for assigning households to dwellings. The results of the two algorithms are then compared empirically for six metropolitan areas to which the U.I. Model has been calibrated. Theoretical Analysis For a given set of household demand and dwelling supply parameters, the de Leeuw-Struyk algorithm, in general, yields higher prices and quantities of housing service than does the stack assignment algorithm. The reason can be seen most easily in a market where there are two dwellings, one new and one existing, and only one household, as illustrated in Fig. 2. Associated with the household’s demand curve, D, is a map of pricequantity indifference curves.9 Each curve represents price-quantity combinations that yield the same utility to the household while satisfying its budget constraints. For a given quantity of housing services (Q) the lower the price paid (P), the higher the amount of income available for expenditure on other goods (X) and hence the higher the household’s utility. 9de Leeuw and Struyk (1975, pp. 28-31).
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C. DUNCAN MACRAE
FIG. 3. One household and two existing dwellings.
Therefore, in Fig. 2, the lower is the indifference curve the higher is the household’s utility (U” 1 v’). The demand curve then describes the Q chosen by the household so as to maximize utility given P. It is determined by the tangency of the price line with the price-quantity indifference curve. For example, at the price P” utility is maximized at the level U” by consuming Q” of housing services. Alternatively, the demand curve can be viewed as the maximum price a household would pay to achieve a given level of utility. Whichever way the curve is seenthe associatedprice-quantity indifference map is useful in comparing the results of the two assignment algorithms. For the stack algorithm, supply-demand equilibrium for the household in a new dwelling with supply S, occurs at the point (Q,, PC) and for the existing dwelling equilibrium occurs at (Q”, P”). Since the level of utility is higher for the household in an existing house, the household chooses this house achieving a utility level of U”. In the de Leeuw-Struyk algorithm, however, the household is not allowed to pay the competitive price. Rather producer’s surplus is maximized for the landlord so that the price is increased to P’ where the household is indifferent between living in the existing house as opposed to the new house. The price is still lower than the new price (PC), but the utility received by the household, u’, is the same as if it were living in the new house.
HOUSING WITH DISCRETE STRUCTURES
141
While the de Leeuw-Struyk algorithm, in general, yields higher prices and quantities than does the stack algorithm, it is possible for the opposite to occur. In Fig. 3, we see an example of a market again with only one household and two dwellings but this time both dwellings are existing with supply curves given by S, and S,, respectively. Under the stack algorithm, the only choice open to the household is the first dwelling at a price P” since the second dwelling withdraws from the market before the price on that dwelling drops low enough to equilibrate its supply with the household’s demand. Under the de Leeuw-Struyk algorithm, however, households are not constrained to be on their demand curve. Therefore, the household choosesthe second dwelling at the price P’, which yields a level of utility of u’ instead of u”. Empirical Analysis
Having seen that there is a potential for the de Leeuw-Struyk algorithm to generate higher prices and quantities than the stack algorithm, let us now ascertain the degree to which this potential is realized. The U.I. Housing Model was resolved with the stack algorithm, for each of the six metropolitan areas first using the parameters that best explain housing market behavior in the 1960s under the de Leeuw-Struyk algorithm. Rather than report the results for all areas, I examine only two areas in 1970: Washington, which had little discounting of existing house prices below the new house price under the de Leeuw-Struyk algorithm, and Chicago, which had significant price discounting. For Washington, the resulting price structure curve for existing dwellings under the stack algorithm for 1970 is portrayed in Fig. 4. For comparison, the price curve under the de Leeuw-Struyk algorithm is also given.” As can readily be seen, except for quite large and quite small dwellings, there is little difference in the price structures under the alternative algorithms. In this area, the demand vs. the supply for mid-range dwellings is sufficiently large to yield existing house prices that are not significantly different from the price of new houses (P,,). Only in the case of a few large dwellings, which cannot be absorbed except at a lower price, and the small dwellings, which cannot meet most household’s demand except at a price significantly in excessof P,, is the potential for the de Leeuw-Struyk algorithm to yield higher prices than the stack algorithm realized. Using the parameters that “best fit” under the de Leeuw-Struyk algorithm, it would appear that the only slack in the Washington housing market is truly at the very upper and lower ends of the market. Using the de Leeuw-Struyk parameters for Chicago in 1970, we observe comparative results somewhat different from those observed in Washington. “de Leeuw and Stmyk (1975, p. 108).
Po= .47
20
I 40
III 60 80 om=
IIll I IIll III III Ii 100 120 140 160 180 200 220 240 2M) 280 300 320 340 360 380 4W 420
72
FIG. 4.
D
1970Washington existing dwelling price structures.
I.”
P P” = 1.37
1.5
1.4 1.3 1.2 1.1
1.0
1 .9 .8
; : :
.7
Po= 56
.6
:
I
I
I
20
40
60
Q,= ----de
Leeuw-Struyk
-stack
algorithm
I! “p
I
I
I
I
I
100
120
140
160
180
I ZM)
I 220
I
I
I
I
I
240
260
280
300
320
(DS) algorithm using
:
cl
81.2
DS parameters
FIG. 5. 1970 Chicago existing dwelling price structures.
142
0
143
HOUSING WITH DISCRETE STRUCTURES TABLE I U.I. Housing Model Calibration Results Metropolitan area Durham Austin Portland Pittsburgh Washington Chicago
Parameters
Mean error
Assignment &o~~m’
yI
y2
YS
81
DS Stack DS Stack DS Stack DS2 Stack DS Stack DS Stack
0.8 0.5 0.8 0.5 0.9 0.7 0.9 0.0 0.9 0.9 0.7 0.4
0.4 1.0 0.2 10.0 0.6 10.0 0.3 4.0 0.3 0.3 0.4 0.7
0.7 1.0 0.9 1.0 0.9 10.0 0.7 0.0 0.6 4.0 0.1 0.7
0.7 0.9 0.4 0.9 0.4 0.9 0.5 0.7 0.6 0.7 0.7 0.7
82
0.4 0.3 0.9 0.4 0.9 0.7 0.7 0.3 0.9 0.9 0.4 0.4
1960
I970
0.083 0.085 0.116 0.062 0.092 0.066 0.093 0.080 0.047 0.056 0.102 0.093
0.142 0.05I 0.171 0.202 0.100 0.111 0.088 0.099 0.089 0.08I 0.165 0.171
‘The de Leeuw-Struyk (DS) results are obtained from de Leeuw and Struyk (1975, Tables II and 12). ‘The model was recalibrated for 1970 since the results in de Leeuw and Struyk (1975, Table I I), are based on a reversal of the housing expense to income ratios (a,) for black nonelderly families and white elderly/single households,
As shown in Fig. 5, for mid-range dwellings the results of the algorithms are still quite similar but the discounting for large dwellings is more substantial in Chicago than in Washington. Under the stack algorithm almost the entire upper third of the existing housing stock has prices significantly below P,,. There does not appear to be the household demand to support housing prices for larger dwellings that there is in Washington. At the bottom of the market, we observe the exception to the rule that the stack algorithm results in higher prices than does the de Leeuw-Struyk algorithm. Calibration
Analysis
To complete the comparison with the de Leeuw-Struyk algorithm, the U.I. Housing Model was recalibrated for the 1960’s using the stack algorithm. The results of both algorithms are given in Table 1. Looking first at the demand parameters we see that, in general, the wealth (y2) and racial externalities ( y3) parameters are higher under the stack algorithm than with the de Leeuw-Struyk algorithm, while the minimum consumption parameter (y,) is nowhere greater. Indeed, in all but one of the six metropolitan areas the minimum consumption levels are lower under the stack algorithm. Moreover, they are more in keeping with independent estimates of minimum consumption with linear expenditure systems.” The only exception to ’ ’ Ring ( 1979)
144
C. DUNCAN
MACR4E
the greater importance of externalities in determining household behavior under the stack algorithm is Pittsburgh for ys. This is the one area where households’ behavior is best explained by assuming no racial preferenceson the part of blacks or whites. Turning to the supply parameters we see that the decennial depreciation rate, 1 - &, is in general, lower under the stack algorithm than under the de Leeuw-Struyk algorithm with the exception of Chicago, where the parameter is the same under the two algorithms. Moreover, the implied annual rates which range from 1.1 to 3.5% under the stack algorithm also correspond more closely to other estimates of structure depreciation.‘* In general, the elasticity parameter, &, is also lower with the exception of Washington and Chicago, where the parameter is the same. Comparing the algorithms in terms of explanatory power as measured by mean error, it is apparent that in 1960 for four of the metropolitan areas the stack algorithm does a superior job, and for one of the areas there is not a significant difference between the results of the two algorithms. In 1970, the de Leeuw-Struyk algorithm, however, has the edge doing a better job in four of the areas with the stack algorithm doing a better job in only two of the areas. While no one algorithm dominates in terms of explanatory power, the stack algorithm is vastly more efficient computationally than the de LeeuwStruyk algorithm. Since both algorithms are programmed to run on the same computer (CDC 6600) we can make a direct cost comparison. On average the cost of solving the U.I. Housing Model with the stack algorithm is 15%of the cost of solving the model with the de Leeuw-Struyk algorithm. Price Structures
I conclude the comparative analysis of the two algorithms by examining the price structure curves that result with the recalibration under the stack algorithm. Again, only the curves for Washington and Chicago, are reported as shown in Figs. 6 and 7. The greater degree of price discounting under the stack algorithm than under the de Leeuw-Struyk algorithm is striking. With the parameters that best fit under the de Leeuw-Struyk algorithm, the difference between the two algorithms was noticeable primarily at the lower and upper ends of the housing market. However, after recalibration the difference in the results of the two procedures is more apparent. It is particularly striking in Chicago, where prices for many dwellings are at least 20% lower under the stack algorithm. Even in Washington, where there continues to be little price discounting, there is more than under the de Leeuw-Struyk algorithm. ‘*Muth
(1975, p. 314).
FIG. 6.
1970Washington existing dwelling price structures.
1.6
P
1.5 P”= 1.37
1.4 1.3 1.2 1.1 1.0 .9 -
: : :
.8 -
:
.J -
:
PO’
56
:
:
:: .6-
:
III 20 40
L III 60
80
a,= ***. -
de LeeuwStruyk stack algorithm
FIG. 7.
100
Ill 120
140
160
III 180 200
220
Ill 240
260
280
300
320
0
81.2 (DS) algorithm
1970Chicago existing dwelling price structures. 145
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C. DUNCAN
MACRAE
The result that there is more price discounting with the stack algorithm has implications for policy analysis with the U.I. Model. In comparing demand subsidies, such as housing allowances, with supply subsidies such as new construction subsidies, de Leeuw and Struyk focused attention on the importance of price discounting in the absence of the subsidy.13It is only where there is appreciable discounting that a demand subsidy can significantly raise housing prices. While the housing allowance supply experiments have found relatively little evidence of price inflationI the de Leeuw-Struyk results indicate the potential for inflation from a housing allowance in a number of metropolitan areas. The results of the stack algorithm indicate the potential is even greater. Since producer’s surplus is not maximized, there is still the opportunity for increases in the surplus through increased demand. IV. CONCLUSIONS In this paper we have developed and analyzed a stack algorithm for determining equilibrium in a model of urban housing with discrete structures. The algorithm ensures both that the competitive supply of services forthcoming from a dwelling is equal to the demand for those services by the resident household and that given the structure of prices in the housing market, the household would not prefer to live in any other dwelling than the one in which it resides. Using the Urban Institute Housing Model as an example, the stack algorithm was employed to calibrate the model for the six metropolitan areas for which de Leeuw and Struyk had already calibrated the model with their algorithm. On the demand side the stack algorithm implies lower minimum levels of consumption, which are more in accord with other research using the linear expenditure system, and a more important role for neighborhood externalities than does the de Leeuw-Struyk algorithm. On the supply side, the stack algorithm yields a lower structure depreciation rate, again more in keeping with independent estimates, and a lower elasticity of supply. Whether using the original parameters for the U.I. Model or the recalibrated parameters, the stack algorithm indicates that there was greater discounting of the price of existing dwellings below the new dwelling price in 1970 than does the de Leeuw-Struyk algorithm. Therefore, if demand subsidies such as housing allowances had been introduced, there would have been greater inflation in the price of housing. Finally, while neither algorithm obviously dominates in terms of explanatory power as measured by mean square error-one algorithm does a better job in some metropolitan “de Leeuw and Struyk (1975. pp. 115- 149) 14Bamett and Lowry (1979).
HOUSING WITH DISCRETE STRUCTURES
147
areas, the other does a superior job in other areas-the stack algorithm is an order of magnitude more efficient in obtaining a solution than is the de Leeuw-Struyk algorithm. Hence, with the stack algorithm a model of discrete housing such as the U.I. Model can be enhanced in ways, such as the addition of more households, dwellings, or zones, that were not previously computationally feasible. REFERENCES C. Lance Barnett and Ira S. Lowry, “How Housing Allowances Affect Housing Prices,” Rand Paper R-2452-HUD, September 1979. Frank de Leeuw and Raymond Struyk, “The Web of Urban Housing,” The Urban Institute, Washington, D.C. 1975. Gregory K. Ingram, John F. Kain, and J. Royce Ginn, “The Detroit Prototype of the NBER Urban Simulation Model,” Nat. Bur. Econ. Res., New York, 1972. A. Thomas King, Estimation of a linear expenditure system for the United States in 1973.J. Econ. Business, 31, No. 3 (September 1979). Richard F. Muth, “Cities and Housing,” Univ. of Chicago Press,Chicago, 1969. Richard F. Muth. Numerical solution of urban residential land-use models, J. Urbun Econ., 2, No. 4 (October 1975). Richard F. Muth, The allocation of households to dwellings, J. Reg. SC.. 18, No. 2 (August 1978).