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# Appendix: Further details of the elasticity computations

## Expenditure elasticities

Expenditure elasticities were obtained using the published summary table of average expenditures over a range of income groups in the 2001 NZ Household Economic Survey (HES), obtained from http://www.stats.govt.nz/. This table divides all households in the sample into income groups. Within each group the budget shares for each of commodity groups were obtained (by dividing average expenditure in each category by average total expenditure).[15] Denote the arithmetic mean total expenditure of the th group by and the budget share of the th commodity group and th total expenditure group by.

The raw values of these budget shares cannot be used to obtain elasticities because sampling variations (particularly for low and high income groups) give rise to negative elasticities. Regressions were carried out of the form:

(21)

for each commodity group, . In addition to generally providing a good fit, this specification has the advantage that weights based on the estimated parameters add to unity.[16] The smoothed budget shares were then used to calculate the total expenditure elasticities.

Differentiating (21), and dropping the subscript, gives:

(22)

so that unequivocally falls as rises if and or if so long as Alternatively, the share rises as income rises (that is, the income elasticity exceeds 1) if and or if so long as

The coefficient estimates are reported in Appendix Tables 1 and 2. The required expenditure elasticities were obtained using:

(23)

with taken from differentiation of (21).

Appendix Table 1 – Budget share regressions
a b c R-squared
Fruit -0.00829 0.002749 2.087083 0.26022
Vegetables 0.054475 -0.00579 -1.37903 0.794889
Meat 0.225018 -0.02806 -13.3021 0.732829
Poultry 0.052294 -0.00617 -3.62406 0.302579
Fish 0.001745 0.000182 0.429412 0.195002
Farm products, fats, oils 0.126825 -0.0152 -4.4229 0.869153
Cereals, cereal products 0.135981 -0.01589 -5.7208 0.887933
Sweet products, beverages 0.139186 -0.01627 -7.28867 0.713447
Other foodstuffs 0.152218 -0.01774 -8.35408 0.381223
Meals away from home -0.16374 0.03029 4.718327 0.971466
Rent 0.622274 -0.08312 -1.83007 0.928584
Payments to local authorities -0.0882 0.013836 15.20095 0.950052
Property maintenance goods 0.081228 -0.0069 -8.63439 0.718421
Property maintenance -0.68323 0.100973 43.35151 0.787053
Housing expenses n.e.c. -0.03517 0.004634 3.377435 0.741892
Domestic fuel and power 0.196174 -0.02413 -0.94521 0.976625
Home appliances -0.00112 0.003079 2.134009 0.024083
Household equipment -0.00563 0.001527 0.727346 0.041477
Furniture -0.10488 0.017014 6.453542 0.741345
Furnishings -0.00076 0.000831 -0.64299 0.451658
Floor coverings -0.06074 0.008517 4.952152 0.461902
Household textiles -0.01107 0.002411 0.852601 0.124718
Household supplies 0.078421 -0.00923 -4.4682 0.502538
Household services -0.08467 0.015257 16.66589 0.919618
Men's clothing -0.0772 0.01134 5.588882 0.523455
Women's clothing -0.17633 0.025524 13.84376 0.742386
Children's clothing -0.00054 0.000862 0.267848 0.027241
Other Clothing 0.029442 -0.00308 -3.3534 0.867552
Clothing supplies & services -0.00295 0.000601 0.434927 0.015415
Men's footwear -0.02062 0.003132 1.23944 0.597497
Women's footwear -0.06346 0.008629 5.976196 0.878226
Children's footwear 0.002005 -8.35E-05 -0.35364 0.250285
Other Footwear 0.010164 -1.05E-03 -1.32814 0.780407
Footwear supplies & services 0.003267 -3.81E-04 -0.3558 0.337931
Public transport within NZ -0.06172 0.009624 5.100902 0.503206
a b c R-squared -0.47099 0.07091 28.55522 0.787953 0.291782 -0.02883 -31.0849 0.652508 0.625611 -0.0743 -42.9299 0.569541 -0.01354 0.002442 1.589636 0.058705 0.166991 -0.0214 -8.79842 0.620878 -0.12924 0.021882 7.49086 0.629366 -0.02832 0.004974 3.036628 0.105698 0.00956 0.000191 -0.92042 0.395319 0.032463 -0.00257 -3.42718 0.58933 0.120291 -0.01498 -7.75913 0.470165 0.070482 -0.00593 -5.10796 0.308231 0.019047 0.000662 -4.02311 0.757771 0.011469 -0.00076 -2.11383 0.635397 -0.0119 0.002321 0.9711 0.140835 -0.12371 0.020146 11.10986 0.545345 -0.10878 0.015507 9.209476 0.352852 0.026106 -0.00101 1.068944 0.17518 -0.07752 0.011855 4.266511 0.679455 0.097855 -0.00857 -2.83561 0.185043 -0.00239 0.000818 -0.75566 0.896757 -0.03519 0.009307 0.15564 0.86498 0.061889 -0.00639 -7.72831 0.46135 0.207629 -0.02421 -17.3681 0.078868

#### Notes

• [15]Several commodity groups were excluded on the grounds that they more closely represented savings rather than expenditure. The ratio of averages is of course not the same as the average budget share (though earlier experiments using data for individual households showed that the differences were minor).
• [16]However, it does not guarantee that the predicted weights always lie in the range , though in practice this was not a serious problem; a few negative values at low total expenditure levels for some goods were set to zero and the other shares were adjusted accordingly to maintain the adding-up requirement.
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