1 9.734 (5.489 1 ) - . 105 1 (.00432) - 1 .998 1 (.59973) -.6307 (. 17977) .20 1 8 (. 1 1063 ) 2.4270 (.68468) .0450 (.00 1 63) 7.9826 ( 1 . 8 1 94 1 ) -.0379 .09 1 5 (.10908) .2284 (. 1 1553) .3 108 (.20042) -.0075 -2.9206 ( .99240) .0 147 (.00203) -. 1499 (.1059 1 ) -.2803 (. 1 1 887 ) .005 1 (.00 1 1 4) -23.796 (6.3989) . 1 283 (.0078 1 ) -2.4575 (.79063) -.0426 (.00270) 1 34 1 .844 .8 139 .3956 .6688
a 8; are the demographic parameters
b Parameters {33 and 83 computed as residuals
c LL is the value of the Log-Likelihood function d R square between observed and predicted.
Italy
-224.66 (94.082) 36.373 ( 14.6 1 30) - . 1 560 (.00057) -5.353 1 (2.4284) -.5920 (.29378) 1 .7540 (.78837) 4. 1 9 10 ( 1 .7340) .0445 (.00 1 75) 1 .6920 ( 1 .3668) -.0087 -.2 158 (. 10 197) .208 1 (.07550) .5997 (.24252) -.0326 - 1 1 .584 (4.7929) .05 1 1 (.00 143) -.4098 (.26057) - 1 .5522 (.58534) -.0085 (.00066) -25.48 1 ( 10.384 1 ) . 1 136 (.00205) -3.2385 ( 1 .3490 1 ) -.0034 (.00 104) 2290.260 .8638 .8074 .8733Figures in brackets are standard errors. There are no equation 2 parameters as the Housing
commodity is not considered. See chapter 1, on data.
to identify. We tried to set � to some a-priori value, as suggested in the literature, but, as a result of this, estimation became even more difficult.
The f3.s, and 8.s, the total expenditure, household size, and the time trend9 parameters were quite robust, and their values did not change much among estimation runs, whatever the starting points, and the convergence criterion. However, all the time trend parameters, except the one in the equation for apparel, were insignificant.
3.2.3 The Estimation Results for Italy
For the Italian data, the estimation procedure yvas marginally better, and we are confident to have achieved a maximum of the LL function, but once again the price parameter estimates proved to be the most difficult to estimate. As in the New Zealand case, the total expenditure, household size, and time trend parameters were quite robust, and had very small standard errors of estimate; with the exception of the time trend parameters for Food and Housing Operations which were insignificant.
The parameter estimates corresponding to the highest value of the LL function attained over a large number of trials are shown in Table 3 . 1 for both countries. We must remember, however, that the estimates of the price parameters for both countries, especially for New Zealand, must be treated with great caution. The "chaotic" characteristics of the estimation procedure are an indication of the possible inadequacy of the AIDS model to describe household consumption behaviour.
3.3
The Elasticities
3.3.1 The Elasticity Computations
From the parameter estimates shown in Table 3 . 1 we have computed, for both Italy and New Zealand, the elasticity of consumption with respect to its own price, the prices of the other commodities (the cross-price elasticities), the household size, and total expenditure. All the elasticities were obtained by numerical differentiation of the
9 As we did for the LES model we included in all equations a trend variable with values from I to 9 for New Zealand and 1 to 1 3 for Italy. The inclusion of a time trend is suggested to eliminate possible bias
functional relationships defining the point elasticities
10
of consumption with respect to the above explanatory variables.The elasticity of the share expenditure on the ith commodity wi versus the price of the jth commodity Pp for example, can be obtained from :
e .. = [8 (log w.) I 8 (log p.)]
l) .. - • l J (3.3 . 1)
where the derivative are calculated numerically at the variables' sample means, and other selected values.
From the share expenditure elasticities (3.3. 1 ), it is possible to obtain the familiar own-price elasticities, which are defined as the change in the quantity demanded of a commodity which follows a marginal change in its price, other things being the same. A simple relationship exists between share and quantity elasticities. In symbolic terms if we define the quantity elasticity as
E;; = [8 (log
q;)
I 8 (log p;)] (3.3.2)then it is: E;; = e;; - 1 .
To obtain the above relationship, between the quantity and share price elasticities, consider the definitions of the share variables appearing in (3.2. 1 ):
w = (pq)ly (3.3.3)
take the logarithms of both sides of (3.3.3), and differentiate with respect to log p to obtain the expression:
[8(log w) I 8(log p)] = 1 + 8(log q) I 8(log p) -8(log y) I 8(log p) (3.3.4)
where the left hand side term is e, the second term on the right hand side is E, and that 8(log y) I 8(log p) = 0 because income is independent of prices ' ' ·
10 Given a function y = f(x), the point elasticity of y with respect to X is : E = o log y I 8 log X , Chiang,
. �
1 984, p.305.
For the cross price, household size and income elasticities it is E = e because all these
v ariables, as they appear in model (3. 1 .8), and in its demographically extended v ersion (3.2. 1 ), are independent among themselves and therefore their cross derivatives are zero. Only the quantity derivatives remain12 •
One of the advantages of compu
��
g. the elasticities by numerical differentiation is that, as a by-product of the elasticity computations, we can obtain the Slutsky matrix- with the generic element sij =
( aqi 1 Jpj)-
from the matrix of the first derivativeso f the share expenditures sij =
(Jw;j(Jzj)
because _these two matrices are linked by thesimple relation: 13
S;; = (y s;; -
q; )
I p; and Su = ( su y)
I p; (3.3.5)For all the variable values at which we computed the elasticities we also computed the S lutsky matrices but, neither for Italy nor for New Zealand were the Slutsky matrices negative semi-definite, 14 thus pointing to a violation, by the empirical demand systems, of the customary assumption of concavity of the utility function.
In Part A of Table 3.2, and Table 3.3, we show the price elasticities - for New Zealand
arid Italy - computed at the means, for the four commodities considered.
In
Part B ofT able 3.2, and Table 3.3, we report the household size, and total expenditure
elasticities computed first at the sample means, and then for two other sets of c onsumption values: one taken midway between the lowest observation and the mean, the other midway between the mean and the highest observation. The first set of such v alues is defined as:
Q1 = lowest observation + 0.25 ( Range) (3.3.6)
1 1 I am grateful to Ranjan Ray for discussing with me the relationship between share and quantity
elasticities.
1 2 For household size for example it would be: [8 (log w) I 8 (log h)] = o (log p) I 8 (log h) -8 (log y) I
o (log h) + 8 (log q) I 8 (log h) = o (log q) I 8 (log h) because household size enters the demand
equations as an additive and linear element, independent of income and prices.
1 3 If we substitute into the relationship E = ( e -1 ) the expressions E = ( 8 q I 8 p) (p I q), and e = ( 8 w I
8 p) (p lw) we obtain : S (p I q) = [ s (p lw) -1 ] . Substituting into this last expression the share variable
where Range = (highest observation - lowest observation). The second set is defined
as:
Q3 = lowest observation + 0.75 ( Range). (3.3.7)
Finally, in Part C, and D of Table 3:2; and Table 3 .3, we show the household size, and total expenditure elasticities separately for all the different household sizes considered in our data. These elasticities have been computed at the means of the sub-samples obtained by grouping together the households of equal size.
3.3.2 The New Zealand Elasticities
There are several interesting features of the estimated elasticities. To consider the price elasticities first. For Food and Housing Operations the own-price elasticities have the "right" (negative) sign, and they are all less than unity. The consumption of these two commodities is price inelastic: a marginal increase (decrease) in their prices will generate a less than proportional fall (rise) in their consumption. When the prices of these two commodities increase (decrease) the consumers will decrease (increase) the amounts purchased less than proportionately and therefore will spend a larger (smaller) share of their budget on them. This is a logical conclusion as the amounts of food and housing needed by a household cannot be easily and rapidly changed.
Transport has negative a own-price elasticity which is very close to unity. This implies that marginal changes in the cost of transport will induce almost proportional changes in the demand for transport. Because Transport is an almost unit elastic commodity its total expenditure will remain more or less constant when its price changes.
For Apparel both the own-price, and all the cross-price elasticities, are negative, and larger than unity. The expenditure on Apparel is highly elastic, i.e. it is very sensitive to price changes, both with respect to its own price and to the prices of the other three
commodities considered here - New Zealanders give a very low priority to clothing expenditure.