Housing demand in the developing metropolis : estimates from Bogota and Cali, Colombia

Urban and Regional Report No. 81-11 I;ii HOUSING DEMAND IN THE DEVELOPING METROPOLIS: EST2LMATES FROM BOGOTA AND CALI, COLOMBIA By Gregory K. Ingram June, 1981 This report was prepared under the auspices of the City Study Research Project (RPO 671-47) as City Study Project Paper No. 20. The views reporced here are those of the author, and they should not be interpreted as reflecting the views of the World Bank. This report is being circulated to stimulate discussion and comment. It was originally prepared for presentation at the Annual Meetings of the Eastern Economic Association, Phildelphia, Pa., April, 1981. Urban and Regional Economics Division Development Economics Department Developme.nt Policy Staff The World Bank Washington, D.C. 20433  PREFACE This paper forms part of a large program of research grouped under the rubric of the "City Study" of Bogota, Colombia, being conducted at the torld Bank in collaboration with Corporacion Centro Regional de Poblacion. The goal of the City Study is to increase our understanding of the workings of five major urban sectors -- housing, transport, employment location, labor markets, and the public sector -- in order that the impact of policies and projects can be assessed more accurately. The author has benefitted from comments and discussions with Michael Hartley, Steve Mayo, Janet Pack, Peter Schmidt, Joseph de Salvo and participants in seminars at Princeton, Michigan State, MIT, The World Bank, and Corporacion Centro Regional de Poblacion. He thanks Sungyong Kang for research assistance, Maria Elena Edwards for manuscript preparation, and the staff of Departamento Administrativo Nacional, Estadistica, Colombia, for aid with the data. Other City Study Papers dealing with housing and housing markets include: 1. Rafael Stevenson, "Housing Programs and Policies in Bogota: An Historical/Descriptive Analysis", Washington,D.C., The World Bank, Urban and Regional Report No. 79-8, June, 1978 (City Study Project Paper No. 3). 2. Alan Carroll, "Pirate Subdivisions and the Market for Residential Lots in Bogota", Washington D.C., World Bank Staff Working Paper No. 435, October, 1980. 3. Jose Fernando Pineda, "Residential Location Decisions of Multiple Worker Households in Bogota, Colombia," Washington, D.C., The World Bank, Urban and Regional Report iNo. 81 - 10, July, 1981 (City Study Project Paper No. 22).  ABSTRACT This paper presents estimates of housing demand equation parameters separately for owners.and renters in Bogota and Cali, Colombia in 1978, and for Bogota renters only in 1972. The demand estimation procedure uses a work place based stratification to introduce price variation in the equations. The demand equations estimated using this procedure give very significant results fdr the income elasticity of the demand for housing, with estimates of the income elasticity generally lying in the upper end of the range 0.2 to 0.8. Although the price term in the demand appears to be less than one. Other household characteristics involved in the demand equations have low demand elasticities, typically less than 0.5 in absolute magnitude. The age of the head has a positive elasticity over most of its range while family size usually has a positive elasticity for renters and a negative elasticity for owners. The demand equations suggest that female headed households consume more housing than male headed households, but this result is rarely statistically significant. Distance from home to work is entered into the demand equations as an adjustment to income, but it is undoubtedly also representing price variation within the workplace strata that are used as the main representation of price variation. The distance elasticity is small, less than -0.2, and is almost always negative. Comparisons of elasticity estimates with those obtained from U.S. data sets indicate that the range of the Colombian estimates generally overlaps the range of the U.S. estimates. Simple experiments involving the aggregation of the household survey data used to obtain micro data estimates suggest that income elasticity estimates based on correctly aggregated data can be good proxies for estimates based on fully specified models using household observations. Atthe same time, estimates based on micro data that are incorrectly aggregated can produce estimates of the income elasticity of demand that are badly biased. I.e P.ts..A.. 6 - ..:S-f ,,.. .z ,w . .UAYe.,s* M..ov.aS .............s.sfil=to)S>Uhl*<nul<~7*AS asvsd)  I. INTRODUCTION This paper reports three sets of results related to the estimation of housing demand equations. First, it presents estimates of housing demand parameters based on household interview data from Bogota and Cali, Colombia. A comparison of parameter values to those obtained from North American data sets shows that the Colombia demand elasticities are generally comparable in magnitude to those from the United States. Second, the approach employed to represent housing price variation in the demand equations uses a theoretically attractive and computationally straight forward procedure that is based on residential location a.heory. Finally, a simple exercise illustrates the magnitude of bias of the income elasticity of demand that can result from incorrect data aggregation techniques. Moreover, correctly aggregated data produce' income elasticity estimates that are similar to those obtained from disaggregate or micro data. II. THE PRICE TERM IN HOUSING DEMAND EQUATIONS Estimating demand equations for housing from cross sectional data presents many challenges, but measuring the variation in the unit price of housing is probably one of the greatest difficulties. Data sets typically report the total expenditure oTn housing rather than a unit price and quantity of housing. Hence the unit price must be inferred by relating variations in expenditure to variations in quantity. Moreover, housing is inherently multidimensional, including attributes of size, dwelling quality, location, public services, and neighborhood  amenities that are obtained in a single tied purchase. Since there is no widespread agreement as to how we should measure the quantity of housing,one analyst's price variation may be another analyst's quantity variation. Finally, even if we can agree that housing prices may vary, it is not obvious that all price variation is relevant for inclusion in a housing demand equation. For example, if a metropolitan area's housing prices vary with the quantity of housing but households can locate anywhere, we cannot simply put the price actually paid by the household into the demand equation because the household faces the whole schedule of prices. Simple inclusion of price indices in a demand equation requires that households be in different market segments. Numerous approaches have been employed to deal with one or more of these difficulties. Some examples include: (i) Assume intra-metropolitan price variation does not exist so that all variation in expenditures reflects variations in quantities; use expenditures in demand analysis as an index number to measure quantities. 'Muth). (ii) Allow intra-metropolitan prices to vary across neigh- borhoods; estimate neighborhood based price indices; then estimate demand equations assuming that residents of each neighborhood face only the prices in their own neighborhood. (King) (iii) Allow intra-metropolitan prices to vary by individual dwelling units; estimate a dwelling unit price index using a production function for housing and varying input prices; estimate demand equations assuming that occupants of each d-welling unit face only the price of their own dwelling unit. (Polinsky and Elwood)  3- (iv) Allow the marginal cost of attributes to differ within a metropolitan area; estimate a non-linear hedonic price index and use the first derivative of the index with respect to specific attributes as the price term in a demand equation for the attribute. (Witte, et al) These approaches each have potential shortcomings. Omitting price variation, as in (i), can bias other demand equation parameters if the omitted price term is correlated with included variables. Assuming that households face only their neighborhood or dwelling unit prices, as in (ii) and (iii), may fundamentally mis-state the price variation in the sample if households are not limited in their choices to specific neighborhoods or dwelling units. If all purchasers face all prices, the price "chosen" may reflect the impact of other household characteristics. Neighborhood-based or dwelling unit-based price variation requires a justification for market segmentation based on those dimensions. Estimating demand equations for specific attributes of housing, as in (iv), may not be relevant if we are really interested in the demand for housing as a composite good. A relatively simple application of residential location theory suggests an alternative way of incorporating price variation into a demand equation for housing as a composite good. Simple models of residential location theory are essentially based on the precepts of cost minimization. A worker surveys the housing market from his workplace, j, and he typically observes that housing prices , R, decline with distance, d, from his work- place in at least one direction. However, travel costs, t, increase with distance from his workplace. For any given amount of housing, H, he faces a total expenditure on housing, Z, composed of a housing expenditure plus a transport expenditure,  Z. = R.(d) .H + t.(d). (1) For quantity H the worker can solve for the least cost distance by 0 taking derivatives Z. R. (d) .H + t. (d) = O (2) and solving the expression for d., the optimal distance or location f-r quantity H and workplace j. This least cost distance can be substituted back into equation 1 to calculate the minimum total expenditure for quantity H , as * * * Z. = R.(d. ).H + t. (d ). (3) Consider carrying out this exercise for different work- places in a metropolitan area. The decline of housing prices with distance9 Rj (d), will differ systematically across workplaces, very likely showing steep rates of decline with distance for centrally located workplaces and gradual rates of decline for peripheral work- places. Travel costs per unit distance may also differ by workplace but in ways that may be difficult to generalize. For example, transit speeds may be higher but transit headways longer at peripheral locations as compared to central locations. As the workplace varies, however, there will be variation in the optimal housing and travel expenditure required for housing quantity Ho. This variation in expense by workplace for a given quantity of housing will be used as a measure of price variation in the housing demand equations estimated here. A price index will be estimated for each workplace zone. Households whose heads work  -5 at a partiTular workplace zone will face the same housing price index. Households with heads at another workplace will face the price index at their workplace, and so forth. Price variation will be across work- places. If housing prices vary by workplace, it is worth asking why workers all do not try to obtain jobs at the workplace that has the lowest housing price index. Urban economists have long argued that a metropolitan area with multiple workplaces and a price gradient for housing will have to have differential wage levels across workplaces for households to be in equilibrium (Moses). Accordingly, workplaces can have different housing prices, but they then must also have : 1 compensating differentials in wages to keep households in equilibrium. The existence of wage gradients across workplaces thus becomes a necessary condition for the workplace based housing price variation approach taken here. III. HOUSING DEMAND AND WORK PLACE-BASED PRICE VARIATION In developing a workplace based price index for housing, we have two possible formulations for the demand system that vary with the definition of the price of housing used. Different definitions will alter the specification of the demand equations that we estimate. In one formulation the price of housing will be based only on the housing ex- penditure and will not include the travel expenditure. In this case the budget constraint will be written Y P H-+ P V + t (d) (4) H' v *1 Preliminary empirical work indicates that a wage gradient with a peak in the Central Business District does exist in Bogota. .... L =..<E-.' .iW.t.- ..'Oe.'i. E..... : 1. 42p. ..bUsb.1U r.ee ........1&..a;s.o .t^-.so.'y liXAE  where Y is income; P., the price of housing; and Pv, the price of a composite commodity V. In this formulation, the travel expenditure, t (d), is included in the income constraint, and the derived demand equation will be of the form H = f [P,, (td] (5) That is, travel costs will have to be subtracted from income in the demand equation. If travel costs are an unknown function of distance, d, thenl d will be included in the demand equation as a separate variable. In the second possible formulation the price of housing will be the so-called gross price and will be based on the housing expenditure plus the travel expenditure. In this case the budget constraint will be written. Y = ZH .I+ P . V, (6) where ZH is tI. .i gross price term. In this case the travel cost does not enter separately into the budget constraint and the distance term will not appear in the demand equations. However, to implement this second approach one must be able to specify a priori the travel cost function which will be a combination of out of pocket cost and the opportunity cost of travel time. Since not enough information is avail- able for Bogota and Cali to allow us to specify the travel cost function with confidence, the first approach has been implemented here. Therefore, the estimated demand equations will have distance to the workplace in them as in Equation 5, and the workplace based price term will be based on housing expenditure only.  The relevant housing expenditure that will be used to define a price index for a given workplace will be the "efficient" or optimal expenditure implicit in the solution of equations 2 and 3 above. Corresponding to each quantity of housing, H, will be an optimal location or optimal distance, d, and an optimal expenditure, R (d ).H. If households are employing the kind of locational calculus embodied in residential location theory, the choices made by households with a head employed at a particular workplace will be at or near the optimal location for that workplace, and their housing expenditure will ap- proximate the optimal expenditure for their workplace and housing * quantity. The relation between housing expenditures and housing quantity for a given workplace can be captured by regressing the observed housing expenditure on measures of housing quantity for households whose heads work at the same work zone. The relation between housing expenditure and housing quantity can then be used to formulate a price index for the given workplace. This procedure can be repeated for each workplace so that price indices can be calculated for each workplace. These work- place-specific price indices then can be used as a price term in demand equations for housing as a composite good. The specific procedure that has been implemented in this paper can be summarized as follows. We have a sample of M households whose household heads have jobs located at one of J workzones, and there are N *1 Equation 3 can be solved for the expansion path of expenditures as the quantity of housing increases,as shown in Annex 1.  -8- households associated with workplace j. We know for household i (i = 1 to N.) at workplace j the monthly expenditure on housing (or the dwelling unit value), Ri., and a set of K dwelling unit characteristics, X... For each of the J workzones we estimate the equation K Pi k l = 1 k ij'k (7) by regressing housing expenditure on the measure of dwelli,ag characteristics, and we obtain J sets of parameters which indicate how the cost of housing attributes varies by workplace. We then define a representative dwelling unit in the housing market as the uinit that has the sample wide average amount of each dwelling unit characteristic,, where the average quantity is N. J J Xk = 1 i Xk. (8) M j=1 i=l The dwelling with attributes X then becomes the standard unit or the k equivalent of the standardized market basket for housing. For each workplace we use the estimated parameters in equation 7 to calculate the cost of the standard unit as R. = kBi X. (.9) 3 k jk k This cost of a standardized unit is used to formulate a workplace price index by choosing workplace 1 as a numeraire and calculating a price index R. (10) The households in the sample also have C hQusehold characteristics, Ho associated with them that affect household demand for housing. c These charac.teristics of the households and the distance from home to  -9- work, dii' are used in a demand equation whose dependent variable is housing expenditure divided by the price index in equation 10, or a quantity index of housing. The demand equation is of the form R.. i f (, H.i dHC) (11) .___ ijc 1J and is estimated over the sample of all M households as a single pooled demand function. In this paper both linear and double log specifications are used for equation 11. IV. THE SETTING AND THE DATA The household interview data used to implement the housing demand procedure just outlined are from Bog9ta and Cali, Colombia. The major data set used was collected in 1978 and covers both owners and renters, for whom equations are estimated separately, in both Bogota and Cali. A second data set is available for Bogota in 1972 but data for only renters can be used to ->stimate the demand for housing in 1972. In 1978 Bogota had a population of roughly 3.5 million and Cali, a population of roughly 1.1 million. Both cities have experienced rapid rates of population growth in the past, e.g. Bogota's population in 1972 was 2.8 million, but current population growth rates are moderating in both cities. Per capita income in 1978 was about $800 per annum in the two cities. The cities differ significantly in climate because of their differences in altitude. Bogota is 8000 feet above sea level arid has temperate weather with cool nights. Cali, at 3000 feet above sea level, is semi-tropical and warmer than Bogota. Differences in size and climate may well explain some of the differences in housing demand ir. the two cities.  - 10 - To implement the workplace-derived price indices it was necessary to divide the two cities into a number of workzones. The work zones that resulted are arbitrary but are based on considerations including compactness, respect for significant internal boundaries, and a requirement that there be an adequate number of observations in each work zone. The same work zone system was used in Bogota in 1972 and 1978 and the same work zorne system was used for renters and owners in each city. Tabulations of residence and workplace by annular ring and radial sector indicate a high degree of association between place of work of the household head and place of residence of the household. Empirical analyses indicate that the workplace of secondary workers may have a slight influence on a households' residential location, but the workplace of the household head is clearly a dominant determinant of residential location (Pineda). Average commute lengths in kilometers for each work zone and tenure type are shown in Exhibit 1 for Bogota and Exhibit 2 for Cali. In both cities these averages differ by up to a factor of 3. In both cities commute lengths are long for centrally located work- places and also for workplaces located along the mountains. V. THE HEDONIC PRICE EQUATIONS Separate hedonic equations were estimated for each work zone and tenure type in Bogota and Cali in 1978. For 1972 in Bogota a hedonic equation was estimated for renters only because no data were available about the value of owner occupied units in thel972 sample. For renters the dependent variable is the monthly rent and for owners the dependent variable is the value of the dwelling uuit in thousands of pesos. The  EXHIBIT 1 Bogota - 13 Work Zone Average Distances from Home to Work Place by Work Zone / ~5880 1 ~7820l < L 7256 8560 /5830 i 6710 6079 8036v / ENTRIES ARE: <<1 57 Distance for Renters, 1972- 420fF Distance for Renters, 1978 Distance for Owfners, 1978 (distance are in meters) Circled nlumber in zones.>X 3767 4454 310 3  - 12 - EXHIBIT 2 CALI 8 Work Zones Home to Workplace Average Distance by Workzone 3350 3698 1526 ENTRIES ARE: Distance for Renters, 1978 Distance for Owners, 1978 (distance in meters) Circled numbers are zones  - 13 - 1978 data were all collected in the same survey with the same questionaire so it is possible to use the same specification for the four sets of 1978 equations. In the 1978 equations the independent variables used included the dwelling unit area in square meters, DUAREA; the lot 2 area in m , LOTAREA; the number of blocks to the nearest bus line, BLKTOBUS; a dummy variable equal to 1 if the residence had a private or public phorie, DPHONACSS; a dummy variable equal to 1 if the dwelling unit had its own non-shared kitchen and bathroom facilities, DEXCLUSE; and a dummy variable equal to 1 if the dwelling unit had its garbage picked up by municipal authorities, DGARBCOL. The average values for the dependent and independent variables for the 1978 data are shown in Exhibits 3 through 6. It is interesting to note the similarities and differences between tenure classes and cities in these exhibits. Renters in Bogota and Cali, for example, have similar sized units on similar sized lots but Bogota renters have more phones while Cali renters have better garbage collection. Bogota owners have larger, more expensive homes on larger lots than Cali owners. Between renters and owners the most striking differences are in the average area of the unit and the proportion of units having exclusive bath and kitchen facilities; owners are better housed than renters. Finally, there is more variability in the average dependent variable across work zones than there seems to be in the average independent variables. The independent variables used in the 1972 equations differ from those used in 1978 because the questionaire was quite different. The definition of the 1972 variables and their mean value by work zone  EXHIBIT 3 REDONIC PRICE ESTIMATION - MEAN VALUES BOGOTA RENTERS 1978 Household Survey All Variables Work Zones Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 Zone 9 Zone 10 Zone 11 Zone 12 Zone 13 DUAREA 67.75 72.76 70.91 62.79 87.68 71.28 69.74 74.37 71.26 42.47 54.99 64.86 68.50 61.49 LOTAREA 125.20 104.20 141.85 140.00 104.34 111.75 114.67 116.83 117.05 136.47 135.16 144.85 133.15 143.68 BLKTOBUS 1.84 1.68 1.60 1.77 2.05 1.93 1.64 1.57 1.56 1.88 1.74 1.98 2.38 2.31 DPHONACSS 0.58 0.66 0.56 0.56 0.65 0.59 0.59 0.67 0.66 0.41 0.47 0.59 0.54 0.48 DEXCLUSE 0.43 0.51 0.37 0.41 0.60 0.38 0.42 0.46 0.38 0.37 0.38 0.39 0.49 0.32 DGARBCOL 0.54 0.49 0.51 0.56 0.68 0.62 0.62 0.49 0.60 0.37 0.35 0.60 0.58 0.55 MEAN RENT 2104.36 2606.86 2218.67 1928.57 2948.00 1758.52 1873.26 2050.00 2284.07 1277.55 1618.88 1986.02 2307.36 1768.56 HEDONIC PRICE 2104 2358 2147 1964 2485 1735 1763 1906 2301 1972 1818 2139 2333 1973 INDEX  EXHIBIT 4, HEDONIC PRICE ESTIMATIONS - MEAN VALUES BOGOTA OWNERS VARIABLES ALL WORK ZONES ZMne 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 Zone 9 Zone 10 Zone 11 Zone 12 Zone 13 DUAREA 172.84 212.69 239.75 182.02 183.57 160.70 160.84 172.75 165.07 129.26 147.21 140.26 169.32 145.06 LOTAREA 150.19 168.08 116.18 130.76 153.12 153.70 140.38 137.87 160.16 158.63 134.96 143.42 176.62 153.29 BLKTOBUS 1.90 1.70 1.61 1.80 1.87 1.86 2.27 2.04 1.81 1.76 2.25 2.03 1.75 1.98 DPHONACSS 0:65 0.83 0.51 0.78 0.75 0.57 0.59 0.79 0.65 0.46 0.62 0.57 0.59 0.55 DEXCLUSE 0.84 0.91 0.84 0.88 0.94 0.75 0.83 0.79 0.82 0.80 0.74 0.83 0.84 0.84 DGARBCOL 0.52 0.64 0.35 0.51 0.58 0.57 0.52 0.43 0.59 0.43 0.38 0.54 0.57 0.47 MEAN VALUE 626.96 892.76 393.92 638.55 851.34 484.64 574.84 577.92 693.65 301.48 380.55 484.03 918.41 511.76 HEDONIC PRICES 627 699.2 436.8 632.6 744.7 533.1 591.8 562.4 705.3 428.0 450.8 552.8 950.1 607.5 INDEX Ln  EXHIBIT 5 HEDONIC PRICE ESTIMATIONS - MEAN VALUES CALL RENTERS VARIABLE ALt WORK ZONE!S zone I Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 DUAREA 65.35 60.41 56.47 54.00 77.32 73.07 61.58 63.74 76.08 LOTAREA 126.97 129.34 95.50 99.69 163.57 123.52 188.50 107.84 127.00 BLKTOBUS 1.34 1.39 0.85 1.92 1.00 1.41 1.21 1.50 1.42 DPHONACSS 0.20 0.20 0.12 0.19 0.25 0.17 0.25 0.24 0.16 DEXCLUSE 0.41 0.48 0.32 0.42 0.50 0.38 0.33 0.45 0.39 DGARBCOL 0.83 0.89 0.88 0.88 0.93 0.76 0.75 0.76 0.76 MEAN RENT 1805.11 1949.09 1835.29 1585.58 2167.86 2044.83 1781.25 1407.24 1724.34 HEDONI1C PRICES INDEX 1805 1890 2167 1703 1974 1944 1858 1412 1631 . . .. . -  EXHIBIT 6 HEDONIC PRICE ESTIMATI>ONS - MEAN VALUES CALI OWNER VARIABLES ALL WORK ZONES Zone I Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 DUAREA 124.87 158.32 137.60 140.11 105.00 112.78 149.80 98.74 93.46 LOTAREA 129.08 135.51 144.08 135.34 125.30 105.56 147.90 119.59 119.40 BLKTOBUS 1.58 1.59 1.08 1.54 1.12 1.74 1.60 1.41 2.43 DAHONACSS 0.24 0.32 0.20 0.34 0.15 0.26 0.33 0.06 0.26 DEXCLUSE 0.80 0.95 0.84 094 0.82 0.56 0.87 0.47 0.86 DGARBCOL 0.74 0.83 0.68 0.83 0.82 0.59 0.83 0.74 0.54 VALUE 361.49 558.17 467.40 511.63 270.00 243.52 398.83 186.47 220.57 HEDONIC PRICES 361.5 408.1 488.2 402.2 270.1 310.4 259.3 276.9 262.0 '-  - 18 - are shown in Exhibit 7. These variables are difficult to compare with those used in 1978, but there are some obviouis similarities in the spatial distribution of rents and services. Current prices are used in both time periods, and the consumer price index approximately tripled from 47 in 1972 to 150 in 1978. The coefficients from the hedonic price equations are shown in Exhibits 8 through 12. Again, the 1978 results are the most comparable. In 1978 there are equations for 13 Bogota and 8 Cali work zones and for 2 tenure types, or a total of 42 equations. The only variable that always has the correct sign in all 42 equations is dwelling unit area. Access to a phone, exclusive bath and kitchen facilities, and garbage collection also perform well, having the expected sign 36, 37, and 32 times respectively. The number of blocks to a bus is only positive half of the time, but it is possible that there is some disamenity associated with being too close to the nearest bus route. Lot area does not perform well in the hedonic equations, and it does very poorly in Cali where owners in particular do not seem to value additional lot size. The hedonic equations for the 1972 Bogota renters, shown in Exhibit 12, are similar to those for 1978 in that the measure of interior space, the number of rooms, has a positive effect on rent. A measure of the explanatory power of the hedonic price equations is shown in Exhibit 13 which summarizes the explanatory power of the regression equations and the workplace stratification in an analysis of variance framework. Overall the analysis explains from 45 to 69 percent  EXHIBIT 7 HEDONIC PRICE ESTIMATION - MEAN VALUES BOGOTA RENTERS 1972 Dousehold Survey VARIABLES All Work Zone 1 Zone 2 Zonie 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 Zone 9 Zone 10 Zone 11 Zone 12 Zone 13 Zones BLDGAGE 16.6 16.88 20.97 1.8.88 18.58 15.94 15.11 12.51 18.98 13.63 13.00 14.57 14.97 13.79 ROOM 2.48 2.68 2.52 2.17 2.85 2.40 2.27 2.34 2.78 2.00 2.38 2.30 2.46 2.39 SQROOM 8.65 10.05 8.20 6.25 11.81 8.12 7.18 7.61 10.69 5.69 7.64 6.73 9.54 7.61 GARBAGE 1.08 1.04 1.05 1.08 1.05 1.07 1.11 1.07 1.07 1.19 1.09 1.11 1.11 1.06 DISTBUS . 144.93 144.59 131.50 135.71 132.72 110.08 145.79 142.92 137.80 165.25 186.63 168.24 141.20 158.89 DHOUSE 0.41 0.39 0.52 0.31 0.46 0.39 0.36 0.43 0.45 0.42 0.41 0.47 0.38 0.41 PAPT 0.24 0.28 0.21 0.30 0.27 0.23 0.26 0.19 0.27 0.18 0.26 0.22 0.14 0.18 DDILAP 0.08 0.09 0.08 0.08 0.02 0.06 0.05 0.09 0.07 0.10 0.09 0.04 0.09 0.06 DUETER 0.28 0.25 0.29 0.27 0.25 0.31 0.33 0.25 0.23 0.34 0.31 0.36 0.26 0.30 DOTHERLU 0.80 0.84 0.69 0.81 0.90 0.65 0.88 0.81 0.83 0.71 0.74 0.84 0.78 0.88 OPUBLI 0.04 0.03 0.04 0.01 0.00 0.04 0.,05 0.04 0.06 0.08 0.10 0.03 0.06 0.02 DPRIVA 0.42 0.48 0.51 0.51 0.48 0.42 C.35 0.42 0.50 0.20 0.30 0.24 0.40 0.41 MEAN RENT 862.61 1016.89 881.00 858.12 1067.59 735.28 616.84 689.38 1190.55 516.53 619.48 682.09 951.85 817.50 HEIDONIC PRICE 863 918 850 883 1056 696 669 659 982 676 526 -708 1126 755 INDEX NOTE: Variable definiclons shown on next page. F  - 20 - EXHIBIT 7 (continued) Variable Definitions for Exhibit 7 BLDGAGE: Building age in years. ROOM: Number of rooms. SQROOM: Number of rooms squared. GARBAGE': 1 = garbage collection, 2 = mo. garbage collection. DISTBUS: Distaflnce to nearest bus line in meters. DHOUSE: Dummy variable 1 = unit is house. DAPT: Dummy variable 1 = unit is apartment. DDILAP: Dummy variable 1 = unit is in dilapidated condition. DDETER: Dummy variable 1 = unit is in deteriorated condition. DOTHERU: Dummy variable 1 = building also has non-residential use. DPUBLI: Dummy variable 1 = unit has public phone. DPRIVA: Dummy variable 1 = unit has private phone.  EXHIBIT 8 ESTIMATIONS OF THE HIEDONIC PRICE EQUATIONS BOGOTA RENTERS 1973 VARIABLES ALL WORK ZONES Zone I Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 Zone 9 Zone 10 Zone 11 ZDne 12 Zone 13 CONSTANT 130.31 -40.83 -231.02 238.42 -1505.18 617.41 1377.30 872.59 376.82 323.67 57.44 833.00 -436.22 699.52 DUAREA 15.59 14.02 25.47 10.26 9.58 12.75 8.23 4.66 13.52 21.14 26.54 23.78 22.43 13.84 F-ratio (239.08) (19.00) (37.93) (9.17) (4.7) (57.89 (18.16) (4.2) (17.5) (21.51' (28.42) (30.15) (23.01' (28.70' LOTAREA 1.30 0.965 1.41 1.553 9.46 -0.15 -2.98 0.635 -4.41 -1.93 4.77 -2.27 5.26 -0.305 F-ratio (4.30) (0.30) (0.36) (0.84) (10.16) (0.01) (2.56) (0.15) (3.17) (2.21) (6.48) (0.92) (4.43) (0.05) BLKTOBUS 152.99 36.11 -38.36 -192.18 156.25 -43.86 -306.76 -165.46 310.66 17.09 -183.30 -56.83 -111.85 -94.42 F-ratio (2.18) (0.07) (0.05) (1,90) (0.84) (0.39) (10.07) (2.87) (4.56) (0.05) (2.12) (0.12) (0.93) (2.51) DPHONACSS 656.46 904.05 272.97 722.62 1908.01 -199.02 552.12 507.71 560.46 533.46 -38.91 171.73 1169.99 596.69 F-ratio (23,69) (3.59) (0.27) (3.47) (6.80) (0.79) (3.41) (2.06) (1.25) (4.94) (0.01) (0.11) (3.07) (3.60) DEXCLUSE 941.69 1595.70 442.79 1094.12 1846.87 774.51 1017.27 1236.86 1482.85 -134.78 -249.63 338.27 530.18 98.13 F-ratio (37.66) (10.25) (0.38) (4.88) (5.97) (8.92) (8.14) (12.62) (6.84) (0.19) (0.26) (0.36) (0.51) (0.05) DGARBOOL 127.43 103.35 367.79 562.12 -51.34 255.76 16.95 221.38 58.42 323.21 -314.66 -299.19 -204.71 295.77 F-ratio (1.00) (0.06) (0.48) (1.98) (0.01) (1.13) (0.00) (0.47) (0.02) (1.46) (0.69) (0.34) (0.09) (0.97) 2 ADJ R 0.4241 0.3471 0.6101 0.4752 0.5029 0.6748 0.5632 0.4458 0.4468 0.4265 0.4899 0.3713 0.3817 0.5017 No. OBS 1025 156 75 70 65 61 69 63 91 49 89 88 72 77  EXHIBIT 9 ESTIMATION OF THE HEDO6,IC PRICE EQUATIONS BOGOTA OWNERS 1978 VARIABLES ALL WORK ZONES Zone I Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 Zone 9 Zone 10 Zone 11 Zone 12 Zone 13 CONSTANT -370.84 -604.17 -48.76 -462.58 -287.52 65.25 -463.18 -177.40 -148.96 -123.39 -18.19 -486.19 -795.87 -179.80 DUAREA 1.097 1.08 0.150 0.288 3.266 1.505 0.869 0.926 2.148 3.585 1.605 0.704 2.071 3.522 F-ratio (45.52) (5.44) (0.86) (0.33) (8.32) (8.49) (1.73) (1.31) (8.01) (37.19) (6.51) (0.60) (2.50) (14.09) LOTAREA 1.822 2.73 0.947 1.643 -0.075 -0.060 0.864 1.387 0.907 -0.139 -0.063 2.815 1.506 0.859 F-ratio (109.21) (31.93) (4.46) (4.57) (0.00) (0.01) (0.99) (4.27) (3.69) (0.27) (0.01) (16.45) (2.77) (1.36 BLKTOBUS 18.66 1.89 -39.83 145.63 -46.93 -11.39 89.27 -2.900 38.62 2.292 -0.688 8.655 166.00 12.17 F-ratio (2.44) (0.00) (1.93) (6.87) (0.56) (0.09) (3.14) (0.01) (1.70) (0.01) (0.00) (0.05) (3.76) (0.14) DPiIONACSS 309.82 182.59 147.64 259.97 252.83 273.43 432.58 2601.77 121.02 -183173 157.70 333.47 642.27 154.44 F-r.tio (42.25) (1.11) (2.21) (1.46) (0.92) (4.10) (5.85) (2.05) (0.63) (3.99) (1.94) (5.49) (5.85) (1.07) DEXCLUSE 273.45 535.00 310.14 378.71 374.12 73.45 339.66 340.35 20i.88 f1.38 20.09 237.87 462.90 83.91 F-ratio (22.61) (5.27) (5.87) (1.75) (0.79) (0.28) (2.63) (4.14) (1.40) (0.43) (0.03) (2.58) (2.43) (0.17) DGARBCOL 127.22 252.76 67.78 65.06 168.13 65.25 70.79 -152.81 47.06 32.18 157.65 114.59 72.23 -277.40 (8.59) (3.68) (0.42) (0.12) (0.65) (0.00) (0.1$) (1.11) (0.14) (0.16) (2.51) (1.00) (0.09) (3.14) ADJ 0 0.3560 0.4290 0.1770 0.2696 0.2743 0.3126 0.2846 0.2221 0.3368 0.5252 0.2091 0.4937 0.4891 0.3812 No. OBS 838 129 51 51 67 44 64 53 74 46 73 72 63 51  EXHIBIT 10 ESTIMATION OF THE HEDONIC PRICE EQUATIONS, CALI RENTERS 1978 VARIABLES ALL WORK ZONES Zone I Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 CONSTANT 475.91 618.44 519.93 131.10 983.85 -469.57 1238.27 1619.87 1185.73 DUAREA 14.07 6.82 13.35 12.93 19.54 22.34 16.61 13.21 11.94 F-ratio (102.18) (5.90) (7.38) (3.47) (41.75) (14.70) (4.27) (5.03) (18.78) LOTAREA 0.4813 0.1635 3.42 1.7039 -0.5709 4.544 -2.556 -3.1275 -4.358 F-ratio (0.35) (0.01) (1.61) (0.17) (0.06) (1.01) (0.47) (1.38) (3.94) BLKTOBUS -114d19 -41.91 0.22 116.52 -71.94 -141.16 -461.04 -379.92 -67.31 F-ratio (4.89) (0.12) (0.00) (0.50) (0.16) (0.32) (3.58) (8.88) (0.43) DAHONACSS 457.67 514.53 1385.18 234.85 -323.58 -284.33 1363.55 1034.76 653.64 F-ratio (5.44) (1.37) (3.53) (0.09) (0.47) (0.13) (4.57) (5.01) (1.58) DEXCLUSE 288.56 1037.62 17.36 692.80 -545.80 663.77 -686.04 -805.03 758.29 F-ratio (2.22) (6.63) (0.001) (0.83) (1.55) (0.63) (0.74) (2.08) (2.70) DGARBCOL 353.23 400.65 75.08 27.16 206.80 418.82 595.65 -42.22 -161.55 F-ratio (3.24) (0.66) (0.009) (0.001) (0.07) (0.30) (0.73) (0.01) (0.13) 2 ADJ R 0.5254 0.4582 0.5440 0.4183 0.7311 0.7224 0.2576 0.2244 0.6325 No. OBS 261 44 34 26 28 29 24 38 38 ) .3  EXHIBIT 11 ESTIMATION OF HEDONIC PRICE EQUATIOUS CALI OWNER 1978 VARIABLES ALL WORK ZONES Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 CONSTANT -188.88 -244.16 -456.90 -292.86 247.68 -129.24 -514.36 -29.91 50.70 DUAREA 4.00 3.10 6.84 1.36 1.94 1.26 5.31 3.02 1.64 F-ratio (172.01) (16.28) (99.06) (0.70) (3.48) (5.46) (24.98) (30.53) (3.57) LOTAREA -1.06 -0.66 -2.52 1.25 -1.38 0.25 -1.27 -0.816 -2.01 F-ratLo (9.93) (0.42) (5.89) (0.62) (3.27) (0.12) (1.09) (3.19) (10.50) BLKTOBUS 21.24 23.01 120.09 21.63 -58.39 25.47 3).83 14.51 -4.89 F-ratio (2.39) (0.17) (4.f4) (0.09) (1.67) (1.01) (0.46) (0.63) (0.13) DPHONACSS 199.67 95.18 349.88 492.92 -11.57 108.96 166.62 84.82 342.20 F-ratio (15.48) (0.51) (2.67) (5.77) (0.01) (1.16) (1.11) (1.09) (20.53) DEXCLUSE 89.01 135.50 151.74 131.04 7.45 185.22 228.19 4.05 148.30 F-ratio (3.11) (0.22) (0.60) (0.15) (0.00) (5.39) (1.29) (0.01) (3.06) DGARBCOL 46.28 248.55 29.24 143.38 65.12 49.17 -9.25 -15.58 99.46 F-ratio (0.92) (2.06) (0.03) (0.44) (0.48) (0.28) (0.003) (0.12) (2.08) ADJ R2 0.56304 0.4175 0.8381 0.405;8 0.1183 0.5200 0.7107 0.5343 0.6651 No. OBS 260 41 25 35 33 27 30 34 35  CIXIH8T 12 ZSTiaTIOtl O TliE h4DOIC PAICE WQUAT1toas SOCOTI REIIERS - 1972 All V.r1ablm UIrk Zines Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zoie 7 Zon* 8 ZC'ne 9 Zone 10 Zone 11 Zone 12 Zone 13 Constant 335.45 380.47 496.30 1172.48 711.64 260.85 671.02 613.87 400.47 270.84 518.59 715.71 -145.41 336.58 cLDGAGE 02.00 01.18 -0.54 -15.09 -6.18 -1.51 3.50 -13.68 -3.65 2.31 -6.60 -0.61 -4.35 -3.82 (2.15) (0.17) (Q.02) (2-79) (0.65) (0.09) (0.92) (5.12) (0.47) (0 34) (1.79) (0.01) (0 34) (0.37) ROOM 234.85 202.63 252.19 44.89 716.26 141.58 72.68 214.52 144.22 126.00 -70.01 6.23 571.64 -22. 17 (33.71) '(5.87) (2.09) (0.03) (8.31) (1.07) (0.44) (6.05) (0.72) (1.01) (0.15) (0.00) (9.20) (0.01) SQUOUII 2.04 3.41 -13.66 -8.81 -54.46 9.74 20.29 4.98 19.09 6.90 47.86 42.66 -35.67 61.01 (0.15) (0.11) (0.25) (0.07) (3.46) (0.29) (1.79) (0.20) (0.83) (0.12) (2. 71) (1.88) (2.54) (2.30) CAAGE -144.06 -210.76 -320.60 -32.34 -803 85 -9.61 -tO0.31 -206.02 40.93- 133.94. -98.49 -243.25 -253.36 289.89 (4;.94) (1.32) (1.12) (0-01) (2.94) (0.00) (0.44) (1.73) (0.02) (1.49) 10.24) (1.22) (1.10) (0.76) DIST1US -0.11 -0.31 0.42 -1.35 -0.69 0.17 0.08 -0.11 -0.55 -0.18 -0.11 0.60 0.21 0.91 (0.77) (1.42) (0.71) (4.61) (0.56) (0.15) (0.06) (0.12) l1.08) (0.29) (0.11) (2.18). (0.10) (2.50) DIOUSE 205.-f7 331:19 -61.88 593.77 101.90 179.40 75.52 92.53 367.14 88.94 323.96 142.10 241.61 230.93 (20.53) (11.01) (0.14) (3.99) (0.08) (1.64) (0.39) (0.78) (2.44) (0.82) (4.44) (0.83) (1.14) (.135) DAf 212.32 394.26 47.43 544.,O -259.12 144.29 185.73 -133.99 181.27 162.48 246.79 264.10 322.70 77.61 (17.62) (14.04) (0.06) (7.86) (0.48) (10.2) (2.24) (1.11) (0.56) (1.78) (2.36) (.193) (1.22) (0.11) DDILAp -z31.50 -396.94 -106.75 -24.13 -652.93 -26.79 47.27 -334.71 -39.83 -22.09 14.57 -795.89 -54.99 -532.55 (13.21) (8.99) (0.18) (0.01) (1.00) (0-02) (0.06) (5.91) (0.02) (0.03) (0.01) (5.64) (0.04) (2.69) DDEIT -152.15 -189.29 -93.29 293.90 3.88 58.06 -80.73 -34.58 -217.98 -127.78 -0.58 -218.43 -111.92 -306.23 (14.84) (4.52) (0.35) (1.88) (0.00) (0.27) (0.71) (0.14) (1.34) (1.89) (0.00) (2.81) (0.38) (3.65) OOTIIE9D -177.57 -96.94 -105.87 -518.42 -204.07 -279.10 -461.96 -162.71 -280.07 19.83 -209.20 -300.10 -44.17 -564.03 (17.91) '(0.97) (0.52) (6.05) (0.34) (6.29) (12.65) (2.25) (1.86) (0.04) (2.54) (3.04) (0.06) (5.67) DF0BLI 189.15 77.98 514.21, 198.50 - 239.50 -65.79 220.69 327.45 164.77 147.20 202.77 453.75 -83.59 (5.17) (0.14) (2.53) (0.07) (0.90) (0.12) (1.10) (0.94) (1.06) (0.58) (0.29) (1.95) (0.03) DP81VA 451.03 555.24 519.24 583.71 475.49 299.15 117.04 217.93 674.57 276.40 198.53 314.74 580.43 442.08 (145.01) (50.61) (12.83) (9.47) (3.98) (6.79) (1.53) (5.76) (17.41) (6.22). (1.88) (4.71) ('f-'.l- - (7.36) 2 ADO b 0.4633 0.4623 0.2924 0.3603 0.4345 0.4321 0.5259 0.5361 0.5303 0.3498 0.4044 0.4283 0.5818 0.5191 No. of Gb.. 1637 444 100 77 81 124 95 113 127 118 86 - 74 I08 90 Number in parenthe.e sre F-ratio.. "3  -26- .EXHIBIT 13: ANALYSIS OF VARIANCE: HEDONIC PRICE EQUATIONS PERCENT OF VARIATION EXPLAINED BY WORK ZONE DATA STRATIFICATION EQUATIONS TOTAL 1972 Bogota Renters 4.7 49.3 54.0 1978 Bogota Renters 2.5 47.6 50.1 1978 Bogota Owners 8.7 36.4 45.1 1978 Cali Renters 1.9 64.3 66.2 1978 Cali Owners 8.0 60.9 68.9 I .  27 - of the variation in housing prices with the equations having much more explanatory power than the workplace stratification. Interestingly, the workplace stratification has much more explanatory power for owner occupied units than for renter occupied units. This is consistent with the empirical regularity that owner occupied units have steeper price gradients in urban areas than do renter occupied units. Hence, work- place location matters more in the owner market than in the renter market. The "standardized" rents and values obtained by plugging the average renter and owner unit characteristics for Bogota and Cali into their respective workplace hedonic equations are shown in the last row of Exhibits 3 through 7 above. For-use in the demand equations these rents and values are transformed into spatial price indices by dividing through by the relevatit rent or value for workzone 1, the central business district. The resulting normalized price indices are displayed for Bogota in Exhibit 14 and for Cali in Exhibit 15. There obviously is variation in these price indices across workzones. In both Bogota and Cali there is more variation in the price index for owners (the range covers a factor of 2) than for renters. VI. THE HOUSING DEMAND EQUATIONS The dependent variable in the demand equations is the monthly rent or value divided by the workplace-specific price index as shown in equation 11. The independent variables are monthly household income (a measure of current income) in pesos, the price index described above, and the airline distance from home to work in meters. Three additional  13 WOI(vZONES 1972 and 1978 Workplace Price Indices BOGOTA /223 .849 12>S\/ a 0 f tA6 ' 1 ,. 1.871. ENTRIES ARE <V| 6 1972 Rent index 1978 Rent index J{( 1978 Value index X. Zone 1 = 1.0 (nu6eraire)v Circled numbers denote zones. | H H. h  - 29 - EXHI.BIT 15 CALI 8 WORKzoTES 1978 Workplace Price Indices. .75 < X /Circled numbers denote zones; / fi Entries are: 1978 Rent index / 1978 Value index i Zone 1 = 1.0 (numeraire) I.9 \.6 a-, tZ C8 ..86 , . .......... ;0  -30 household characteristics are included in the demand equations: a dummy variable for the sex of the household head (1= male); family size measured by the number of persons in the household; and the age of the household head in years. These three characteristics are hypothesized to capture differences in taste (sex of head), differences in the need for housing (family size), and differences in assets or wealth (age of the head). Two functional forms are estimated, double log and linear. In the linear specifications squaredterms for family size and the age of the head are entered to capture non-linearity in the effects of those variables. Five sets of equations are estimated for each year, tenure choice, and city combination. The ten fully specified equations are displayed in column 1 (linear specificationsy and column 3 (log-log specification) in the tables in Annex II. Column 5 shows the mean value of each variable in the demand equations. A comparison of these mean values across the five samples shows that renters have younger heads, smaller families, and lower incomes than owners. Differences between Bogota and Cali are slight- xcept for income: Bogota owners have much higher average incomes than Cali owners whereas Bogota renters have average incomes similar to Cali renters. In comparing Bogota renters over time, 1978 Bogota renters had smaller families and younger heads than did 1972 Bogota renters. 2 The demand equations in Annex II perform well with R statistics ranging from 0.25 to 0.6. Income is by f^ar the most important explanatory  variable. Age of the head and family size are usually significant while sex of the head is usually not significant, although it always has a negative sign. The housing price index is significant in two of the five samples, and it always has the correct sign. Distance from home to work is significant in four of the five samples and has the correct sign in 9 of the 10 equations. A summary of the fully specified demand equation results are displayed in Exhibits 16 and 17 for renters and owners in the form of of elasticities f6r each independent variable. These elasticities are calculated in the linear equations using the mean value of each independent variable except income. The linear elasticities are shown for approximately the first, second, and third quartiles of each sample's income distribution. In each case, the sample mean and the 75th percentile of the income disttibution are essentially identical. The magnitude of the various elasticities obviously vary across the samples shown, but they also display a consistent and stable pattern for most of the variables. All income elasticities are less than one, and at the sample mean they lie in a narrow range of 0.6 to 0.8 except for the Cali renter equations. The elasticity of the sex of the head is always negative and small being absolutely less than -0.2. Family size elasticities show an interesting pattern, being negative for owners and usually positive for renters. Since renter occupied units are usually smaller than owner occupied units, it appears that space is a binding constraint for renters, and larger renter families obtain more housing. Owner occupants, on the other hand, seem to be able to reduce the quantity of housing demanded as family size increases because they have, larger units  EXHIBIT 16 A -2- DEMAND ELASTICITIES AT VARIOUS INCOME LEVELS RENTERS Income Income | Head Family Age of Home to Work Pctile Level Income Sex Size Head Price Distance 1972 - LINEAR - BOGOTA 25 1000 0.32 -0.16 0.30 0.23 -0.91 -0.05 50 1700 0.45 -0.13 0.25 0.19 -0.75 -0.04 75 3079* 0.59 -0.09 O.18 0.14 -0.55 -0.04 1972 - LOG/LOG - BOGOTA All 0.77 -0.14 0.14 0.12 -0.70 -0.06 1978 - LINEAR- BOGOTA 25 3500 0.55 -003 -0.24 0.95 -0.17 -0.23 50 7100 0.71 -0.02 -0.16 0.61 -0.11 -0.15 75 11260* 0.80 -0.003 -0.11 0.43 -0.08 -0.10 1978 - LOG/LOG - BOGOTA All 0.72 -0.07 0.10 0.07 -O.28 -0.06 1978 - LINEAR - CALI 25 3500 0.05 -0.01 0.48 0.62 -0.34 -0.16 50 7300 0.10 -0.01 0.46 0.59 -0.32 -0.15 75 12829* 0.16 -0.01 0.42 0.55 -0.30 -0.14 1978 - LOG/LOG - CALI All 0.47 -0.20 0.36 0.43 -0.48 -0.03 *Samiple Mean.  -33- EXHIBIT 17 DEMAND ELASTICITIES AT VARIOUS INCOME LEVELS OWNERS Income Income Head Family Age of Home to Work Pctile Level Income Sex Size Head Price Distance 1978 - LINEAR - BOGOTA 25 6000 0.33 -0.03 -0.34 0.66 -0.31 0.02 50 10900 0.47 -0.02 -0.27 0.52 -0.24 0.01 75 17942* 0.60 -0.02 -0.21 0.40 -0.19 0.01 1978 - LOG/LOG - BOGOTA All 0.78 -0.09 -0.25 0.25 -0.44 -0.02 1978 - LINEAR - CALI 25 5000 0.39 -0.06 -0.57 0.53 -0.27 -0.06 50 8800 0.53 -0.05 -0.44 0.18 -0.21 -0.05 75 13841 0.64 -0.04 -0.34 0.13 -0.16 -0.04 1978 - LOG/LOG - CALI All 0.76 -0.06 -0.30 0.08 -0.33 -0.02 *Sample Mean.  - 34 - on the average, and the quantity of housing is not constrained by family size. Age of the household head has a consistently positive demand elasticity when evaluated at the sample mean. Using the linear demand equation with the squared term for head's age, it is possible to calculate the age at which housing demand is a maximum. This is consistently within the range 50 to 57 except for Bogota owners, for which it is 112. The price elasticity of demand is consistently less than one and becomes absolutely quite small for some of the linear specifications. Finally, the distance elasticity is almost consistently negative and quite small. Exhibit 18 summarizes the range of demand elasticities obtained from Cali and Bogota and compares them with estimates obtained from household surveysfrom the Ufnited States and Korea. The general pattern of results is quite similar between the U.S. and Colombid, with both countries differing somewhat from Korea. The Colombian income elasticities are somewhat higher than those obtained in the U.S. while the Colombian price elasticities miay be lower than those from the U.S. The elasticities of housing demand with respect to family size and age of the head cannot be compared with numbers from the U.S. but are somewhat similar to the Korean estimates. Finally, the effect of the sex of the household head, although usually statistically insignificant in Colombia, is also always negative as in the U.S. There are three possible explanations for this result. First, female headed households may have stronger preferences for housing than male headed households. Second, female headed households may be discriminated against and face higher prices  -35- EXHIBIT 18 RANGE OF HOUSIING DE'!ND ELASTICITIES FROM VARIOUS COUNTRIES (Based on Household Observations) Elasticitv of Housinz Demand with Resnect to Current Family Age of Sex of Country Income Price Size Read Head (1 = Uale) __________RENTERS Colombia .2 to .8 -.1 to -.7 -.1 to .4 .1 to .6 -.01 to -.2 consistently USA .1 to .4 -.2 to -.7 ? ? negative Korea .12 -.06 to .03 .15 to .25 - ._____ OWNERS Colombia .6 to .8 -.15 to -.40 -.2 to -.35 .1 to .4 -.02 to -.1 USA1 .2 to .5 -.5 to -.6 ? ? negative Korea2 .21 -.05 to .07 -.02 to .15 - - 1 From Stephen K. Mayo, "Theory and Estimation in the Economics of Housing Demand?" Journal of Urban Economics. 2 From J. Follain, G.C. Lim, and B. Renaud, "The Demand for Housing in Developing Countries: The Case of Korea," Journal of Urban Economics.  - 36 - which could produce larger expenditures on housing. Those larger ex- penditures could show up as a preference for larger quantities in the demand equations for renters, but the discrimination hypothesis is unconvincing for owner occupants. Third, female household heads have shorter commute distances than male household heads and may therefore systematically pay higher prices for housing because they commute less far down the rent gradient. The demand equations used should account for this, however, because distance is included. Accordingly, the fiirst explanation, based on preference differences, may be the most plausible. In order to investigate the effect of distance on the sex of head coefficient, and to see how sensitive the other parameters were to both the price and distance terms, the housing demand equations were estimated without the price and distance terms. The results are shown in columns two and four in the tables in Annex II. Examination of these tables indicates that omitting the price and distance terms tends to reduce the income coefficient very sligthtly, often only in the third significant digit. The family size effects are also minimally affected by the omission of these two terms. The sex of head and age of head coefficients do change quite a bit in percentage terms, however. This seems to be largely due to the omission of the distance term. Female headed households live closer to the headrs workplace than do male headed households, as do households with older heads as compared to households with younger heads. In general, however, the parameter estimates for the included variables are very stable with respect to the omission of the price and distance terms. I.  -37- These exercises suggest that neither the housing prices as specified in these demand equations nor the distance from home to work are collinear with household incom.e. Indeed, in Bogota and Cali, as in many other cities, the use of micro data dramatically reduces problems of multi-collinearity in the estimation of housing demand equations. VII. AGGREGATE ESTIMATES OF INCOME ELASTICITIES All of the parameter estimates that have been presented so far have been obtained from computer based multivariate regressions using individual households as observations. In many situations it may not be possible to gain access to individual household records because of confidentia2.ity restrictions while in other situations sufficient time or adequate computer facilities may make parameter estimation with micro data impossible. In this section we briefly investigate the adequacy of parameter estimates that could be estimated from published aggregated data. We focus on the estimation of the income elasticity of the demand for housing because that parameter is often of interest in both the design and evaluation of housing programs,policies, and projects. Each of the five samples we have analyzed was summarized in a matrix dimensioned by rent or value and income. Eight income categories were used for the 1978 data and nine for the 1972 data. The average rent or value was caloulated for each income category; this average was then regressed on the mid-points of the income  -38- categories in a log-log specification using a hand held calculator. The equations resulting from this exercise are shown in Exhibit 19, and the resulting income elasticities are compared to those from the disaggregate, fully specified equations in Exhibit 20. The aggregate estimates each differ by less than 20 percent from the disaggregate 16g-log estimates, and in 4 out of 5 cases the aggregate lg-log estimates lie between the linear and log-log disaggregate estimates. It is obvious that aggregate based estimates of income elasticities of the expenditure for housing could be a very good approximation for the income elasticity of demand for housing in the samples used here. It is important to note, however, that the aggregate estimates obtained are very sensitive to the way in which the underlying micro data are aggregated. Two experiments illustrating this were performed with the 1972 sample of renters. First, the sample was aggregated- to the level of 63 zones for the city of Bogota,and average rents and incomes were calculated for each zone. A hand held calculator was then used to calculate a log-log regression of average z6nal rent on average zonal income using all 63 observations. The resulting income elasticity, 0.95, was substantially higher than the 0.71 estimate obtained using nine observations from the correctly aggregated sample. A third experiment was then run on the 1972 Bogota data. Fot this experiment the data in the rent-income matrix were incorrectly aggregated by calculating the average income for each rent category and regressing the rent category midpoints on the mean incomes. This rent stratified approach yielded an income elasticity estimate of 1.36, nearly twice  -39- EXHIBIT 19 HOUSING DEWAND EQUATIONS FRO"K AGGREGATE DATA Sample B 1 1972 Phase II Renter 2.92 .71 .99 1978 Bogota Renter 1.54 .79 .99 1978 Cali Renter 12.38 .55 .97 1978 Bogota Owner 9.11 .67 .99 1978 Cali Owner 7.81 .66 .97 B Equation, of form Rent B Income 1 0 Income stratification  - 40 - EXHIBIT 20 CO' ARISON OF AGGREC-ATE T.7D DISAGGREGATE INCOINE ELASTICITIES OF HOUSING DEMAD Sample and SDecification Aggregate Disagaregate 1972 Bogota Renter Log-Log |71 *77 Linear 1 .9 1978 Bogota Renter | Log-Log .79 .72 Linear - .80 1978 Cali Renter Log-Log .55 .47 Linear .16 1978 Cali Owner Log-Log .66 .76 Linear 64 1978 Bogota Owner Log-Log .67 .78 Linear .60  -41- the 0.71 obtained using an income stratified aggregation procedure. It is obvious that the aggregatioii bias in estimates of income elasticities can be very large, but that correctly aggregated data can give useful results. VIII. CONCLUSION This paper has described and implemented a two step estimation procedure for incorporating price variation in the estimation of demand equations for housing using household survey data from Bogota and Cali, Colombia. The demand equations estimated using this procedure give very significant results for the income elasticity of the demand for housing, with estimates of the income elasticity generally lying in the upper end of the range 0.2 to 0.8. Although the price term in the demand equations gave less significant results, the price elasticity of demand appears to be less than one. There is, however, greater uncertainty about the magnitude of the price elasticity than about the magnitude of the income elasticity. Other household characteristics involved in the demand equations have low demand elasticities, typically less than 0.5 in absolute magnitude. The age of the head has a positive elasticity over most of its range while familyvsize usually has a positive elasticity for renters and a negative elasticity for owners. The demand equations suggest that female headed households consume more housing than male headed households, but this result is rarely statistically significant. Distance from home to work is entered into the demand equations as an  -42- adjustment to income, but it is undoubtedly also representing price variation within the workplace strata that are used as the main representation of price variation. The distance elasticity is small, less than -0.2, and is almost always negative. Comparisons of elasticity estimates with those obtained from U.S. data sets indicate that the range of the Colombian estimates generally overlaps the range of the U.S. estimates. This similarity of values may seem surprising at first, but is much less so on reflection. Housing is a non-traded good and its price is endogenous to the local economy, reflecting, among other things, local income levels. Perhaps we shouYld be more surprised at the similarities between Bogota and Cali, two cities whose climates differ markedly. Simple experiments involving the aggregation of the household survey data used to obtain micro data estimates suggest that income elasticity estimates based on correctly aggregated data can be good proxies for estimates based on fully specified models using household observations. At the same time, estimates based on micro data that are incorrectly aggregated can produce estimates of the income elasticity of demand that are badly biased.  -43- ANNEX I. Housing Expenditure and Housing Quantity The residential location model used has been formulated in the location rent/transport cost trade off mode as a surface of total expenditure as Z (H, d) = R(d).H + t (d), (Al-1) where R(d) is a rent gradient, H is the quantity of housing, t(d) is a travel cost function, and d is a measure of distance or location. The relevant set of Z(H,d)'s for a household to consider are those where for each H, Z(H,d) is a minimum. These minimum points constitute a locus of efficient expenditure points for a household on a graph whose axes were labelled Z and H. This total expenditure expansion path can obviously be disaggregated into an expenditure expansion path for each of its two-components, transport and housing. We can solve for the housing expenditure expansion path, using general notation, by taking the derivative of Equation Al-l- Z' (H,D) = R' (d).H,+ t'(d) = 0 (Al-2) and solving for the optimal location, d*, as d = g (R', t', H). (Al-3) This can be substituted back into the housing expenditure expression to form an expansion path of housing expenditures as R (d*).H = R -g(RI, tI HI)] .FL, (Al-4) which is a function of workplace-specific price gradients and travel costs,as well as the quantity of housing consumed. It is this expenditure expansion path that we are trying to summarize with our workplace- specific price index.  -44- Note that one could substitute a housing demand equation for H into equation Al-3 and get an expression for d* as a function of a', t', and income plus other household characteristics.-/ We have not employed this completely reduced from approach because the goal of this exercise is the estimation of housing demand equations. Hence, we deal only with cost minimization concerns in order to define an efficient consumption possibility locus for a household. *1 I owe this point to Joseph DeSalvo.  - 45 - ANNEX II. Housing Demand Equations SPECIFICATION Or DENAND ECUATIONS The demand equations summarized in the next pages use two different specifications defined as follows: LINEAR EXP/P Bo + B Y + B P +3 X+ h 1 2h 3 4 Demand elasticities vary with independent variables. LOG/LOG 31 32 .33 E.YP/Ph = Bo S PB g E. /'hBOY ph *x Demand elasticities are constant. NOTATION EXP = Housing Expenditure (rent or value) Ph = Housing price index; workplace-specific Y = Current household income X = Other hom.eRhold characteristics B. = Paramete>rs AP .txva-z9 $zLa. o rismvsoN; i 2,.t...':