order polynomial expansion.<sup>15</sup> As a benchmark, we also report results for _LV_ .

For this purpose, we use data from the 1987/88 _Nationwide Food Consumption Survey_ , which is conducted by the United States Department of Agriculture. Demands for four categories of food are estimated: meats, fruits and vegetables, breads and cereals, and miscellaneous. In Figure 10, we assess the impact of a potential increase in the price of meats. Note that for the Cobb Douglas model, the compensated own-price and cross-price elasticities for good _i_ are ( _αi −_ 1) and _αj_ respectively ( _αl_ is the expenditure share of good _l_ ).<sup>16</sup> Of course, this may - slightly - underestimate the welfare effect because of the overestimation of the substitution effect component.This is also confirmed by the low estimated compensated cross-elasticities with the QUAIDS model compared to those of the Cobb-Douglas model.

Figure 10 shows that the marginal approach (LV) overestimates the impact on welfare as compared to all _EV_ estimations irrespective of the demand model used. For instance, when the price of meat quadruples, the impact is -80 USD, which is even higher than the maximum reasonable impact not exceeding total expenditures (less than 50 USD on average), in the case of perfect complement goods and non substitutable goods. Overall, the demand models show a clear ranking with the Cobb-Douglas model providing smaller estimates as compared to the EASI, QUAIDS, AIDS and LES models. All models provide very close estimates diverging of a few percentage points only for very high price changes. QUAIDS, AIDS and LES are particularly close even at price increases of 400 percent.

We shall conclude that the choice of the demand system is not a major discriminatory factor in determining differences in welfare estimations and that the use of the simple Cobb-Douglas model is reasonable even for very high price increases with the caveat that it may underestimate the welfare effect as compared to other more sophisticated demand models. However, given the simplicity of the Cobb-Douglas model and its nice properties in relation to the share of goods consumed over total consumption, to its capacity to estimate easily all measures of welfare change and in relation to the Taylor approximations, this model remains, in our view, the model of choice.

_Other approximation methods_ . The next question we want to address is how other estimation methods including the elasticity method, Hicks or Taylor approximations of various degrees compare to the Cobb-Douglas demand method. We start by comparing the CobbDouglas method with the Elasticity and Taylor approximations (Figure 11). Consider the following example. Suppose that the true preferences are Cobb-Douglas and that we consider as “errors” the differences from these estimates. With initial prices normalized to one, we have _CS_ = _−mαlog_ (1 + _dp_ 1). Now assume that the price of the first good doubles and that the global elasticity of the linear demand curve is equal to -0.5. Based on the approximation of equation (23), we find that _CS_ = _−_ 22 _._ 5 whereas the true (Cobb-Douglas) value is equal to _CS_ = _−_ 20 _._ 79. This error is evidently explained by the assumption of the linear demand curve and it can become quite large as shown in Figure 11. We can also see that estimates made with the elasticity method are rather close to the Cobb-Douglas demand method up to 100 percent price increases. The error size can be better appreciated in Figure 12. The elasticity method results in a rather good approximation of the Taylor’s higher order estimate and converges to

> 15See Deaton and Muellbauer (1980a), Banks, Blundell, and Lewbel (1997), Lewbel and Pendakur (2009) and Appendix A.

> 16See for instance Ramskov and Munksgaard (2001).

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