sou · GTB357d11 · 2006-01-01 00:00:00 — Consumer Demand and Labor Supply in Box 513, SE-751 20 This paper analyses the demand for consumer goods and leisure in for the period 1980-2003. A dynamic version of the AIDS demand system is used. An error correction model is used where a form of habit formation is applied. The results show reasonable estimates of price and income elasticities for the consumer goods. Also, the labor supply elasticity is in line with previous results using micro data. The estimation of demand elasticities has a long tradition in applied economics. During the 1970ies and 1980ies demand systems based on specic utility or cost functions were estimated and e¤ort was put on deriving attractive functional forms of these systems. The most successful demand systems were those with a exible functional form, such as the Translog1 or the AIDS ()2 . Demand systems were often estimated with aggregate time series data, though the theory from which the systems were derived consid- ered the behavior of a single consumer or household. Initially, systems like the Translog and AIDS were static. Later, dynamic versions of these models were used and t the data better. The purpose of this paper is to estimate demand elasticities for the most recent Swedish aggregate consumption data. The elasticities can be used for example for estimating deadweight losses for di¤erent tax schedules or for com- puting optimal tax rates according to the Ramsey or inverse rule3 , i.e. tax rates being set inversely to the compensated price elasticity of demand4 . These elas- ticities must also include the price elasticities (own and cross price elasticities) of leisure time. If leisure time is excluded, the implicit assumption is that the cross-price elasticities between leisure and other goods is zero, in which case a uniform tax is optimal. Therefore, the estimated demand systems include Swedish consumption goods and leisure time. The paper is organized as follows. The next section gives a brief introduction to the theory of consumer demand and some of the problems that occur in this type of exercise. The third section discusses the problems of specifying the demand system, which in this case is a dynamic version of the AIDS model. The fourth section gives a detailed description of the data and how the treatment of leisure time is done. The fth section gives the main empirical results, while detailed tables can be found in the Data appendix. The last section gives the conclusions. 2Theory of consumer demand A general description of the consumer decision problem is given in the life cycle hypothesis. There, the consumer maximizes expected discounted utility from a bundle of consumption goods and services as well as leisure time - given an intertemporal budget constraint. With many goods and services and demand dependent on many expected future discounted prices this problem becomes very complex and di¢ cult to handle in empirical applications. Various assumptions have to be done in order to keep the problem manageable. Rather than starting from the most general and compley theory, here the most simple static model is used but extended with some dynamics. This is one of the ways in which the 1 See , Jorgensen and (1979). 2 See Deaton and Muellbauer (1980). 3 See and Stiglitz (1980), p. 4 The compensated price elasticity of demand for good i with respect to a change in the price of good j is dened as "ij = belongs to a bundle of goods, gives the same utility as before the price change, i.e. u(q ) = u(q development has proceeded in the literature5 . Generally, the consumer maximises the expected utility max (1 + r) (1+ )Et fu(qt+ )g subject to an intertemporal budget constraint At+1 = At + !tht ptqt where the horizon is from t to T, r is the real rate of interest, is the rate of time preference, E is the expectations operator, u is the utitilty function, At is wealth at time t, !t is the wage rate, ht is the number of hours worked, pt = p1t; :::; pnt is a vector of prices and qt = q1t; :::; qnt a vector of quantities. Assuming intertemporal separability, and for convenience dropping the time subscript, the problem can be stated max u(q) subject to x = where x is total expenditure. The solution to this problem gives the Marshal- lian demand functions qi = gi(x; p). If these demand functions are substituted into the direct utility function u(q) the indirect utility function u = '(x; p) is obtained. The indirect utility function shows the highest possible utility that can be attained for alternative prices and total expenditure. The dual problem is to minimize the total expenditure required to obtain a given utility level, which can be stated as min x = pq u = v(q) The solution to this problem gives the Hicksian demand functions as qi = hi(u; p) which may simply be derived as the partial derivative of the cost function x = c(u; p) w.r.t. The cost function shows the minimum cost to obtain utility level u at prices p. Substituting the indirect utility function (which is the inverse of the cost function) into the Hicksian demand functions gives the Marshallian demand functions. In this paper the dual approach to obtain the demand functions is used. The total expenditure elasticity is dened as Ei = @qi x and shows the percentage change of the demand for good i following a one percentage change in total expenditure (often also referred to as the income elasticity). wi = denote the budget share of the i-th good we have the general property that wiEi = 1: The uncompensated6 price elasticity of demand is dened as eij = @qi pj and shows the percentage change of the demand for good i following a one percentage change in the price of good j. For i = j we refer to the own-price elasticity and for i 6= j to the cross-price elasticity. 5 See Deaton and Muellbauer (1980b) and Edgerton et al (1996) where various dynamic versions of the AIDS model were developed. 6 Uncompensated implies that the consumer is not compensated for the loss in income induced by a price increase. property is that j eij + Ei = 0: Finally, the compensated7 price elasticity is dened as eij = eij + wj Ei which shows the percentage change of the demand for good i following a one percentage change in the price of good j - at utility. 2.1Static and dynamic model The consumer decision problem with respect to both consumption goods and leisure is dynamic and is probably best analysed within the life-cycle model framework. The literature using aggregate data however tend to divide the problem into a dynamic optimization problem about aggregate consumption and a (more or less) static optimization problem about the allocation between di¤erent consumer goods. Once leisure - or labor supply - is considered the problem is clearly a dynamic one. Typical for the aggregate consumption prob- lem is the marginal utility condition governing the Euler equations in this area. This condition can also be used for the allocation of the individual consumption items, i.e. the demand functions could be stated in terms of mar- ginal utility as qi = hi ( @u@c ; p) , as in et al (1994). However, here we follow the simpler route and introduce simple dynamics in an otherwise static framework. Particular problems also arise in the context of durable goods, i.e. goods which are not consumed within the data period one quarter of a year. The consumption of Housing is for rental apartments estimated as rents in nom- inal and xed prices, while for owner-occupied housing a rental equivalent is computed. However, capital gains are not included. For other durables like vehicles, household equipment, etc. consumption is estimated as purchases in current and xed prices. Some studies have chosen to exclude durables and some to include them as here. In this study durables measured as purchases account for approximately 8-9 percent of total expenditure including Leisure. 2.2Utility tree and multi-stage budgeting The number of goods available to the consumer and the number of observations present in the data obviously poses a problem, particularly to the researcher occupied with time-series data. At the most there are about 90 observations available in this study (75 for the data up to 1998). This put a restriction on the number of goods that can be included. A system with 10 goods to be esti- mated with FIML has 10 10+20=120 free parameters to estimate without any dynamics. Various +restrictions therefore must be imposed on the systems in order to make estimation possible. Restrictions like Slutsky symmetry clearly helps to some extent. A more radical approach is the idea of multi-stage budget- ing, a procedure previously dealt with in the literature8 . Consider a two-stage budgeting process in which the consumer is supposed to decide in two di¤erent stages. The rst stage considers n groups of goods, r=1,...n and in the second stage the rth group consists of i=1,...,m goods. Then the consumer preferences 7 Compensated implies that the demand is calculated at the point where the consumer is compensated for the loss in income induced by the price change so that the utility is the same as before the price change. The compensated demand hence measures the pure price e¤ect. 8 See Gorman (1959), Pollak (1971), Deaton and Muellbauer (1980a, Ch. 5.2), Pudney (1981), (1982), Laisney (1991) or Edgerton et al. (1996). are said to be weakly separable if they can be represented by a utility function of the form u = f [v1(q1); :::; vn(qn)] where qr represents the vector of quantities in the rth group. Utility max- imization now means maximizing the functions vr(qr) separately, using the standard tools of demand analysis but replacing total expenditure by group expenditure xr: The Marshallian demand functions now can be written qri = gri(p1; :::; pn; xr) where pr denotes the price vector of the rth group and xr = i priqri is the total expenditure of the group. The rst stage allocation of total expen- diture into group expenditure is a problem since the price vectors pr must be replaced by some price pr: Deaton and Muellbauer (1980a) show that an approximation exists such that the demand function above can be replaced by Qr = gr(P1; :::; Pn; x) where Qr is a quantity index - or expenditures expressed in prices - and Pr a price index, where the latter could be an approximation to a true cost of living index. Though the assumption of weak separability means that one can study the consumer optimization problem at separate stages it has implications for the e¤ect of a price change on a certain group belonging to group r on the demand for another good belonging to another group s. In addition, the expenditure elasticity depends both on the rst stage elasticity and the elasticity within the group at lower stages of the budgeting process. If a price of a particual good increases it will a¤ect the demand for all goods in the group to which the good belong. But the price index of the group - Pr - is also a¤ected and hence the demand for all other goods belonging to groups outside r. The relationships are uncovered in Edgerton (1992, 1993) and Edgerton et al (1996, p. If qri is expenditure in prices and pri the implicit price derived by dividing expenditure in current prices with expenditure in prices then the price index can be written c(u; pr) c(u; ) where is the unit vector, xr = PrQr, x = substitution we obtain the Marshallian demand and Qr = i Qri. qri = gri [pr; Prgr(P1; :::; Pn; x)] = gri(p1; :::; pn; x): Following Edgerton et al (1996) and using the denitions E(r)i for the within group expenditure elasticity, E(r) for the group expenditure elasticity, and Ei for the total expenditure elasticity and similarly for price elasticities - eij for the uncompensated and eij for the compensated price elasticity - and budget shares we obtain the following denition of elasticities: = E(r)E(r)i = rse(r)ij + E(r)iw(s)j ( rs + e(r)(s)) = rs eij + E(r)iw(s)j e(r)(s) = rs e(r)ij + E(r)iw(s)j e(r)(s) where rs = 1 for r=s and zero otherwise9 . How should these denitions be interpreted? The expenditure elasticities are straightforward. The price elasticity is composed of two parts which can be labelled the direct and indirect e¤ects. The direct e¤ect - rse(r)ij - is the within group price elasticity measured in the usual way. The indirect e¤ect measures how much the price change of a certain good a¤ects the allocation among groups. It depends on the three factors E(r)i - the within group expenditure elasticity w(s)j - the budget share of the good which price changes andrs + e(r)(s) - the price elasticity between groups r and s The rst factor measures the e¤ect of the change in group expenditure (due to the price change) on the expenditure on the ith good. The second factor measures the relative change of the group price index caused by the change of the price of the ith good, which is measured by the budget share of the price changing good. The third factor measures the e¤ect on the demand for group r - on Qr - of a change in the price index of group s - of Pr. Note that if the latter own price elasticity equals -1 the elasticity collapses to eij = e(r)ij . On the other hand, if e(r)(r) = 0 then the price change implies a proportional change in the expenditure of the group and eij = e(r)ij : Note also that the total price elasticity is well approximated by the within group price elasticity if the within group budget share of the price changing good is small or within group expenditure elasticities are small. 3Specication of demand system The estimation of elasticities requires a specication of the demand model. Sev- eral alternatives exist in the literature. In Edgerton et al (1996) di¤erent speci- cations are evaluated empirically. The nally chosen model is the AIDS () model which is commonly applied in the empirical liter- ature. 3.1 The AIDS model The AIDS uses the time period t cost function log c(ut; pt) = 0 + k log pkt + log pkt log pjt +u where u is the utility level, p is a price vector, pkt is the price of the k-th good. If prices are normalised to unity (12) collapses into 0 + 0. u then can be seen as an index of utility, 0 u 1 , where for u = 0, 0 can be seen as the cost of subsistence while for u = 1, 0 + 0 can be seen as the cost of bliss. As described in section 2, taking the partial derivative w.r.t. the k-th price gives the Hicksian demand for the k-th good, which upon substitution of the indirect utility function gives the Marshallian demand as 9 rs is the so called Kroneckers delta. wit = i + Xj ij log pjt + i (log xt log Pt ) k log pkt+ 1 log pk log pj : the budget share for the i-th good. A linear version of the AIDS model was suggested by Deaton and Muellbauer (1980) where the price index Pt was replaced by the index Pt = 0 + wkt log pkt This approximation has become very popular in the literature. Here, both linear and nonlinear versions have been used. Though the linear version of AIDS has proven accurate in a number of studies10 I have nally chosen the nonlinear version. The additional computational burden is small and the results not particularly sensitive to the a priori choice of the parameter 0.11 . The linear version was evaluated here but it was the nonlinear version that was nally chosen. The simplest static model is not likely to perform well with time series data. Di¤erent dynamic versions of the model have been used in the literature. dynamic demand system suggested by (1991), Alessie and Kapteyn (1991) and Kesavan et al (1993) can be derived from a dynamic form of the cost function. Using the principle of demographic translation, as suggested by Pollak and (1981), results in demand functions wit = i + ij wjt 1 + ij log pjt + i (log xt log Pt ) (15) This type of system was evaluated here and short and long run elasticities = 0 was used, where the rst et al (1994),(1994) or Edgerton et al (1996). See for example Chalfant (1987),P 11 The parameter 0 was set to 30 percent of the log of total expenditure in 1995, the base year in which prices was normalised to unity. sum implies adding-up and the second identication. The restriction kj = jk, for all k,j, was rejected and the further restriction that is diagonal, was also rejected in this particular specication. 3.4AIDS in error correction form Another alternative, which was nally chosen, is the error correction form, which has become very popular recently and seems to t the data well. It allows both short term and long term as well as feedback responses to be estimated and it can be derived from the basically static AIDS framework through the method of demographic translation. The long run equilibrium can be desribed by (13) above and the error correction form as Xj {ij log pjt + 'i log(x=P ) + 0 + k log pkt + kj log pk log pj wit 1 i Xj ij log pjt 1 i (log xt 1 log Pt 1)(20) The matrix should be restricted in order for the system to be theoretically In particular, it is su¢ cient if i ij = j ij = 0 for adding up and be coherent (and possible to estimate identication to hold. For the system to P e.g. with Full Information Maximum Likelihood) parameters have to be further restricted. Di¤erent specications were tested and the nally chosen system is a restricted version which allows six commodity groups to be included. The preferred specication use a single scalar as the error correction term. The model can be rewritten in level form as wit = i + Xj {ij log pjt + 'i log(x=P )t + (1 + )wit 1 + Xj ij log pjt 1 + i log(x=P )t 1 which can be derived from the AIDS expenditure function (12) by translating the parameter i into i = i + (1 + )wit 1 + Xj kj log pjt 1 + i log(x=P )t 1 and the long run parameters be derived as {ij + ij 'i + i This is the specication nally used for estimating the elasticities and can be viewed as a compromise which on the one hand saves parameters through various restrictions derived from economic theory but on the other hand is exible enough to allow for the most important dynamics. The long run elasticities in the model are then given by (22) - (26): k kj + jk = eijl + wj Eil and the short run elasticities by (27) - (29): k ({kj + {jk) log pk where ij is Kroneckers delta. Seasonal dummies are included in the estimations but for simplicity have been excluded in the formulas above. Note that the short run elasticities are not directly identied due to the loss of the term in the di¤erence form. The parameter i is interpreted as the budget share for a household at the subsistence level and is identied in the derivation of the long run elasticities. The short run elasticities are identied by assuming that i in the short run is equal to the long run value. The data used is quarterly national accounts running from 1980-2003. For the period 1980-1998 data are from the old SNA denitions and the base year is 1991. For the period 1993-2003 data are based on the latest SNA denitions and comply better with international accounting systems. The denitions in some commodity groups are quite di¤erent and it has not been possible to adjust the data for the earlier period so that it completely matches with data for the later period. However, the best possible revisions were done and the data for the earlier period linked with the later period to give a longer data series. The series were then used in the estimations and compared with estimations run for the period 1980-1998. It turned out that the estimates with the consistent data series for 1980-1998 were more reasonable, both in terms of t and theoretical consistency12 . 12 Compensated own-price elasticities were negative and the goodness of t were better for most equations. It could not be ruled out that this result were due to the inability to consistently link the old with the new series. , the older but longer series was chosen. 4.1Utility tree and specication of categories With quarterly data for the period 1980-1998 there are 76 observations. The static AIDS model with seasonal dummies and no theoretical restrictions implies 40 parameters in a system with 5 goods and 135 parameters in a system with 10 goods. , it is clear that the dimension of the system must be kept small. With the dynamic specication, 6 goods appear to be the limit, but the theoretical homogeneity and symmetry restrictions were applied and the dynamics kept simple. The design of the two-stage budgeting system is governed by the demand from the public inquiry (). It takes account of the design of the present system of the VAT system in which some goods are exempted, and tax rates vary across di¤erent commodities and services. Some a priori considerations are also done and some aggregation as done by Statistics in the SNA is kept here. Some di¤erent aggregations were tried and in particular estimates where some groups tend to have positive own- price elasticities were avoided. The aggregation nally considered is Goods in two-stage budgeting system Food, beverages, and health care Food including light beer 1_2 Alchohol and tobacco Housing, fuel, and furniture Housing, fuel, and furniture Household and personal care 3_3 Post and telephone 4_3 Foreign and consumption 4_4 Recreation including cultural activities Miscellaneous goods and services 5_3 Books and magazines with budgeting in two stages. The rst-stage has 6 goods and the second stage betweem one and ve goods in each category. Leisure is obviously a single good and the good with largest budget share. We now turn to its measurement, which is an important issue. 4.2The measurement and treatment of leisure Suppose the n-th good in the system of demand equations above is leisure time, such that pn is the price of leisure and qn the number of leisure hours. budget share of leisure then is wn = Ppnqn : The price of leisure has been measured as the wage rate less the average marginal tax rate13 . The budget restriction then can be written piqi = pnh + A; where pnh is labor income, h is the total number of hours worked and A is non-labor income14 . The budget restriction can be rewritten by taking account of the time restriction T = ml + z + s + n + qn, where T is the total number of hours available, ml = h is total numbers of hours, where m is mean hours and l the number of employed. z is the number of hours used for home production, s is the number of hours used for labor search, n is the number of subsistence hours and qn is leisure time. The budget restriction then becomes i=1 piqi = pn(T ml z s n) + A M where M usually is referred to as full or potential income15 . The problem then is how to estimate leisure time. The daily budget for an individual consumer could simply be determined as (24-h-0-0-8), the number of total hours less the number of hours worked less the assumed 8 hours used for sleeping16 . The number of daily hours is then aggregated to the total number of quar- terly leisure hours for the adult population, b. The conversion factor from daily to quarterly hours is , the number of days each quarter. These have been es- timated as the number of ordinary working days, i.e. ve days for an ordinary week and also exclusive of all "red" days. The simplest measure of the total number of quarterly leisure hours is therefore (24 8) b ml. However, we also assume that some of the time of the unemployed is used for home production and labor search. If the the number of unemployed persons are labelled u, then the labor force is l + u and we assume that 1mu hours is used for home pro- duction and labor search. Similarly, we assume that the elderly - or the adult population outside the labor force - use some time for home production and labor search, which is 2m(b (l + u)). Finally, there is the number of subsis- tence hours, which are related to sleeping, eating and for some health care, etc. which are di¢ cult to determine. We denote this by bn. The total number of quarterly leisure hours then can be determined as 24 b ml 1mu 2m(b (l + u)) nb where the respective terms are