instructive to discuss the potential error size of this formula before undertaking more complex approximations of the welfare effect that invariably lead to demand modelling of some sort.

The question of the error size is, of course, non-trivial. If utility and demand functions were known, then changes in quantities due to price changes could be calculated with precision and one could judge which of the estimation methods come closer to reality. But if utility and demand are latent functions largely unknown to the observer, then we don’t have a benchmark to measure the true error size. In this case - which is the case for most analysts - the only tests we can conduct are about the difference between estimation methods for different price increases. This is what we do in this section treating separately individual and social welfare. Recall that we treat social welfare as a simple non-weighted aggregation of individual welfare. Hence, the question of small and large price changes is treated under individual welfare only while we focus on the question of comparing distributions under the social welfare section.

# **5.1 Individual welfare**

## **5.1.1 Small price variations**

For small price variations, the optimal strategy is to use _LV_ for two good reasons. One is that this is the measure that requires the least information to be computed and the second is that estimations produced by other measures converge to the _LV_ measure for small price variations. The question we want to address here is: what is “small”? What price variations are small enough to make _LV_ a viable tool. Or, at what point _LV_ and other measures start to diverge significantly. We consider the case of a single price change or multiple price changes and the role of the Coob-Douglas parameter “alpha”.

_Single price change_ . We can start with an illustrative example comparing _LV_ estimates with _EV_ estimates derived from Cobb-Douglas preferences in the case of a price change of only one product. We assume that the “true” utility is defined with Cobb-Douglas preferences with _U_ ( _x_ 1 _, x_ 2) = _x_<sup>_α_</sup> 1<sup>_x_</sup> 2<sup>1</sup><sup>_−α_</sup> and _α_ = 0 _._ 3. Assume also that initially we have income _m_ = 100, and prices _p_ 1 = _p_ 2 = 1. Figures 5 and 6 show respectively the impact on welfare and the error size estimated for price increases between 0 and 50 percent. It is shown that _LV_ provides similar estimates to _EV_ up to a price increase of around 10 percent while estimates diverge by about 12 percent for price increases around 20 percent (Figure 5). We can also estimate the percentage error between the two measures - _|_ ( _LV − EV_ ) _/EV |_ - (Figure 6) and this shows that the error is about 6 percent for price increases of about 10 percent and doubles to around 12 percent for price increases around 20 percent. It is also visible that the error is a linear function of the price increase. As already discussed, the marginal approach _LV_ tends to overestimate the negative impact on welfare as compared to other measures. This is due to the fact that the marginal approach essentially captures the income effect but not the substitution effect. The bias illustrated in Figure 6 is also discussed by Banks, Blundell, and Lewbel (1996). Using European data, they find a bias between 5 and 10 percentage points with a price change ranging from 10 to 20 percent. They also confirm the negligible impact when price changes are below 10 percent.

Using the same setting based on Cobb-Douglas preferences, we can now extend comparisons to the other welfare measures considered. Note that the income effect is approximated by _LV_ while the substitution effect is given by: 30