The Order of Acquisition of Durable Goods and The Multidimensional Measurement of Poverty Joseph Deutsch and Jacques Silber August 2005 Department of Economics, Bar-Ilan University, Ramat-Gan, Israel.

Incidence of poverty - ownership of durables - order of acquisition of durables Rational individual, maximizes utility given tastes and limited resources Income increase – a change along the acquisition path of durables Order of acquisition –permanent income Order of acquisition- ordered logit regression- latent deprivation variable Paroush, J., 1965, "the Order of Acquisition of Consumer Durables," Econometrica (33(1): Guttman, L. 1950, “Relation of Scalogram Analysis to Other Techniques, Measurement and Prediction”, Studies in Social Psychology in World War II, vol. 4. Guttman, L. 1959, “Simple Statistical Methods for Scalogram Analysis”, Psychometrika.

Empirical analysis Israeli Census of population – N= 204,098 households. Durables Analyzed:

Table 1: Ownership of Durable Goods by Gender of Head of Household Note that quite important differences are observed (with male headed households being evidently better endowed than female headed households)

Table 2: Ownership of Durable Goods by Household Size An inverted-U type of relationship is observed.

Table 3: Ownership of Durable Goods by Age of Head of Household With the exeption of ownership of dwelling, we observe again an inverted-U relationship.

The highest degree of ownership is found among households whose head is married. With the exception of car ownership, the lowest levels of ownership are observed either among singles or among widow(er)s. Table 4: Ownership of Durable Goods by Marital Status of Head of Household

In many cases the degree of ownership decreases monotonically with the year of immigration. Table 5: Ownership of Durable Goods by Year of Immigration of Head of Household

Table 6: Ownership of Durable Goods by Schooling Level (Years of Schooling) of Head of Household In most cases the degree of ownership increases monotonically with the schooling level

In most cases the greater the number of months the head of the household worked during the last twelve months, the higher the degree of ownership of the various durable goods. Table 7: Ownership of Durable Goods by Number of Months Worked by the Head of the Household During the Last 12 Months

Table 8: Ownership of Durable Goods by Status at Work of Head of Household The degree of ownership is highest among self-employed individuals. Note also that in most cases the degree of ownership is smallest when the head of the household did not work during the last twelve months.

In many cases this degree of ownership is highest among Jewish heads of household and lowest for Muslims. Table 10: Ownership of Durable Goods by Religion of Head of Household

Order of acquisition of durable goods - Paroush (1965): For the case with 3 durables: A,B and C. Sample space =8 possible outcomes Table 11: List of possible orders of acquisition when there are 3 goods

Distribution of Profiles Ranked by Number of Households With k=11 durables the number of possible profiles is 2 11 =2048. In fact we found just 695 different profiles for the 204,000 households. 22 profiles account for more than 50% of the households.

Assume that the order of acquisition is A,B and C, then all the consumers will be distributed along the path of acquisition with profiles, 1-4 and there will be no consumers with profiles 5-8. In this case we say that there is a perfect scale. When comparing actual figures, some consumers may deviate from the path of acquisition. For practical purpose, we will conclude that there is an order of acquisition if 90% of the profiles are reproducible. Table 11: List of possible orders of acquisition when there are 3 goods

List of possible profiles with acquisition order A,B,C Consumer’s Profile, X * * Guttman developed the index of reproducibility as: We then calculate de minimum deviation for each consumer in the sample, N i is the number of consumers with deviation S i and k is the number of commodities. Where S i is the minimum distance of the profile of individual i to the closest profile p j in the acquisition path. That is, suppose a consumer with the profile 0,1,0. If the order of acquisition is A,B,C then the closest profile in the path of acquisition to the consumer’s profile are profiles 1 or 3 with a deviation S=1.

Household’s Profile Closest Profile =  x i -p i  =k/2 - S=  X-P  Maximal Deviation When there is a perfect scale S i =0 for all consumers and R=1. Guttmans showed that the coefficient R is bounded between 0.5 and 1. When the profile is randomly determined,the maximum deviation will be obtained when the consumer’s profile is a series of consecutives 0,1. In this case the value of S i =k/2. If all the consumers have the same profile then R=0.5.

The calculation of the index of reproducibility assumes a given order of acquisition. Paroush suggested to find the coefficient of reproducibility for all the possible orders of acquisition and estimate the population order of acquisition as the order of acquisition with the highest coefficient R provided that it is greater than 0.9. Estimating the order of acquisition requires a very high number of computations. For a given order of acquisition with k commodities, the path of acquisition has k+1 possible profiles. Therefore, for each individual household i in the sample, the determination of the minimum distance S i from his profile to one of the possible profiles in the path of acquisition is based on 12 comparisons. As our sample is based on 204,098 household, 2,449,176 comparisons are needed in order to determine the reproducibility index R for a given order of acquisition. This procedure has to be repeated 11! =39,916,800 times which is the total number of possible order of acquisition resulting from 11 durable goods. As a result, the total number of iterations needed to find the order of acquisition with the highest index of reproducibility R is 2,449,176  39,916,800 = 9.77 

Table 12: Order of acquisition with highest proximity coefficient R (R = 0.92) The order of acquisition is similar but does not completely coincide with the rank of the durables ordered by the percentage of ownership. Also, the proportion of households with a profile of acquisition of durable goods corresponding to the different stages of the order of acquisition is 32% (65,333 households).

Let D i denote the continuous level of deprivation of household i such that a higher value of D i corresponds to higher degrees of deprivation. The deprivation score is assumed to be a function of H factors whose value for household i are X ih, h = 1 to H. We may therefore express this latent variable D i as where  is assumed to be a random logistic variable. We assume that this deprivation level is related to the stage of acquisition of durable goods where the household is located. We define the observed variable Y i as the number of durables not owned by household i. That is: if the household owns all the 11 durable goods (the lowest level of deprivation) the household owns only the first 10 durables in the acquisition path the household owns only the first j-1 durables in the acquisition path the household does not own any of the durable goods (the highest level of deprivation) if

Number of observations: Pseudo R-square: Log-Likelihood: Turning point=6 Turning point=57 Table 13: Results of Ordered Logit Regression (Dependent Variable = Latent deprivation index)

Table 14: Estimated Boundaries of the Deprivation Levels

We calculated for each household i in the sample the value of its latent deprivation variable D i and defined as “poor” the top 25%. Table 15: Incidence of Poverty by Gender of Head of Household by Household Size

by Age of Head of Household by Marital Status of Head of Household by Year of Immigration of Head of Household by Years of Schooling of Head of Household

by Status at Work of Head of Household by Place of Residence of Head of Household by Religion of Head of Household by Number of Months Worked by the Head of the Household During the Last 12 Months