The _LV_ and the _PV_ derive from index number theory initially proposed by Fisher (1922). As discussed by Fleurbaey (2009), index number theory has developed in three directions. The first direction aims at defining desirable properties of an index and find indexes that satisfy these properties. For example, Diewert (1992) shows that the original indexes proposed by Fisher satisfy a set of 21 desirable axioms which make these indexes superior to others. The second direction is a tradition that seeks indexes that depend only on prices and quantities and that are good approximations of welfare changes. Diewert (1992) has shown for example that it is possible to find functional forms of the expenditure function that are both simple and flexible and that result in indexes such as the geometric mean of the Laspeyers and Paasche indexes. The third direction initiated by Samuelson and Swamy (1974) seeks indexes that depend on individual preferences such as the money-metric utility function. In this paper, we will not discuss further these different developments of index number theory. What is important to stress here is that the _LV_ and _PV_ indexes are routinely used to measure welfare changes under price changes and to adjust longitudinally and spatially welfare measures such as the poverty headcount index.

As already clear from these first formulations, while the _CS_ , _CV_ and _EV_ methods require knowledge of utility functions and demand functions while the _LV_ and _PV_ as defined above require information on the demand function only. The difference between measures becomes clearer when we illustrate these methods geometrically in the next section while we will see that it is possible to estimate these measures also in the absence of direct knowledge on the Marshallian demand function. This is particularly important for countries where it is not possible to measure this demand function directly because of data constraints.

# **3.2 Geometric interpretation**

Figure 1 illustrates the five estimation methods discussed in a classic geometric setting. In the top panel, the initial budget line is the continue blue line and the initial equilibrium is at _A_ . The slope of this curve is _−p_<sup>_a_</sup> . After an increase in price from _p_<sup>_a_</sup> to _p_<sup>_b_</sup> , the budget constraint curve rotates as shown by the red line adjusting to a slope _−p_<sup>_b_</sup> and the final state is reached in _B_ .

The _LV_ measurement evaluates the change in welfare with the initial bundle of goods. Thus _LV_ = _−x_<sup>_a_</sup> ( _p_<sup>_b_</sup> _−p_<sup>_a_</sup> ) which is (minus) the distance between _A_ and _A_ 2. Similarly, we can evaluate the potential change in expenditure with the final quantities such that _PV_ = _−x_<sup>_b_</sup> ( _p_<sup>_b_</sup> _−p_<sup>_a_</sup> ), which is (minus) the distance between _B_ and _B_ 2. The _CV_ is the required budget to offset the loss in wellbeing with the new prices. This amount is equal to _−BB_ 1, which leads to _D_ . The _EV_ is the price equivalent loss in wellbeing. It equals _−AA_ 1, which leads to _C_ .

The bottom panel of Figure 1 shows the geometric interpretation of the five methods in the case of a price increase from _p_<sup>_a_</sup> to _p_<sup>_b_</sup> , a change in quantity from _x_<sup>_a_</sup> to _x_<sup>_b_</sup> , linear demand schedules derived from known utility functions and changes in the price of only one product (assumptions derived from the top panel). Following from the definitions provided in the previous section, the _CS_ is the area below the Marshallian demand curve and between initial and final prices, which is the area delimited by points _p_<sup>_a_</sup> _EAp_<sup>_b_</sup> . Consequently, the _EV_ is equal to the area _p_<sup>_a_</sup> _CAp_<sup>_b_</sup> , the _CV_ is equal to the area _p_<sup>_a_</sup> _EDp_<sup>_b_</sup> , the _LV_ is the rectangle _p_<sup>_a_</sup> _EFp_<sup>_b_</sup> , which is equal to _−x_<sup>_a_</sup> ∆ _p_ and the _PV_ is equal to the area of the rectangle _p_<sup>_a_</sup> _BAp_<sup>_b_</sup> , which is equal to _−x_<sup>_b_</sup> ∆ _p_ .

From Figure 1, which is inspired from the Hicks (1942), we can derive a first set of results: 10