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## Equivalent and Compensating Variation

Equivalent and Compensating Variation:If the utility function is not of the quasi-linear type, the usual consumers’ surplus calculations are not valid measures of the real welfare changes (i.e., changes in utility) due to, for example, a government project financed by increased taxation. John Hicks suggested a different approach to measure welfare changes where we may contemplate asking the typical consumer what change in income is equivalent for her, to the proposed project-tax package. The consumer would when give the answer in terms of a sum of money which she would either want to receive (if she think that the project stinks), or which she would be willing to pay if she thinks the project is good. This question is natural to ask before the project has been implemented, and before any prices have been affected by the project and the accompanying taxes. If the project has been undertaken, a different question could be asked: How much would you like to receive in order to be compensated for the impact of the project on your utility? This will also produce an answer in terms of a sum of money, but now the prices have been affected by the project/tax, and it is therefore not certain (and will in general not be the case, unless the utility function is actually of the quasi-linear type) that the answer to these (ex ante and ex post) questions, will give the same money sum. Hicks called these sums of money “variations” for some reason, and proved that they are correct measures of welfare changes, expressed in terms of money, but for two different set of prices. The ex ante question will produce a measure, which Hicks called the equivalent variation, and the ex post question produces a measure he called the compensating variation.

Let us ignore the benefits of the project and concentrate on the taxation issue. Only good 1 is taxed so its price will increase, as will the relative price ratio. The equivalent- and compensating variations are illustrated in Figure shown. There are three points on each diagram which are common to both; the first point is the intercept on the vertical axis, Y

_{0}, which is, as usual, the initial money income (we’ve set p_{2}= 1, and it is constant all along); the second point is A which is the initial optimal bundle, located at the initial indifference curve, U_{0}; and the third is point B, which shows the final optimal bundle, located at the final indifference curve, U1, after the increase in price of good 1. If we use the equivalent variation measure (EV), we use the original relative price ratio to push the original budget set (budget set 0) inwards until it touches the final indifference curve (budget set 2). EV is here the difference between the intercepts of the budget sets 0 and 2, and is thus the amount of money which should be taken away from the consumer in order for him to just reach utility level U_{1}. The new money income is:Y

^{EV}_{1 }= Y_{0}+ EV,or,

EV = Y

^{EV}_{1}− Y_{0},which is negative in this case, since the price of good 1 has increased.

The compensating variation (CV) is shown in the right diagram in figure shown. Here we ask what income addition the consumer needs to attain the original utility level (U

_{0}), at the new relative prices. Hence, push budget set 1 until it touches the original indifference curve. CV is again the difference between the intercepts of the budget sets 0 and 2:Y

^{CV}_{1}= Y_{0 }+ CV,or,

CV = Y

^{CV}_{1}− Y_{0},which is positive in this case, since the price of good 1 has increased, and the consumer must get a positive sum of money to be compensated for this.

What should we conclude from all this? Well, we had a simple measure of the welfare changes of a price increase, the change in the consumers’ surplus, which is also observable, but it turned out to be inadequate in most cases. The theoretically correct measure could be defined in two separate ways, and these measures will not in general coincide. So the situation seems depressing. However, we may go a little bit further and use the Slutsky equation to put some bounds on the EV and CV measures. The simplest case is then good 1 is a normal good, which we’ll assume for now on. The CV measure basically assumes that the consumer is compensated for the price increase so that she reaches here original utility level. I’ve talked about this sort of compensation before and called it the Hicksian compensation then. I also mentioned that we can define a compensated demand function, which shows the pure substitution effect (it is cleansed from the income effect). In the case of a normal good we’ve seen that the Slutsky equation implies that the compensated demand curve has a steeper slope than the ordinary demand curve (since we said that in this case the income effect reinforces the substitution effect). In Figure shown I’ve drawn an ordinary demand curve together with two income-compensated demand curves. The income-compensated demand curves are functions, not of prices and money income, as the ordinary demand curve, but of prices and a given level of utility. That is why we have two of these sorts of demand curves to deal with. The one corresponding to EV is a function of U

_{1}, the final utility level, and the one corresponding to CV is a function of U_{0}(look back at Figure to check this out). Since U_{0}> U_{1}in our case (because we considered a price increase), the CV–curve must lie to the right of the EV -curve, because good 1 is a normal good and a higher utility level requires a higher income, and this will shift the demand curve outwards. The CV measure is now the area under its income-compensated demand curve between the prices, i.e., A + B + C; the EV measure is equal to A. Note that the change in the (ordinary) consumers’ surplus (ΔCS) is equal to A+B, hence this measure lies between (is bounded by) the two “correct” measures of the welfare change of the price increase. This is in a way favorable to the usual consumer surplus measure, since it is sort of an average of the two correct measures, furthermore, in most cases the difference between, e.g. the CV-measure (the one most likely to be used in practice) and the change in the ordinary consumers’ surplus is quite small, for small price changes. That supports the use of the latter in practice.For a price increase the change in the consumers’ surplus is negative, but CV is positive (since the consumers’ need to be compensated by extra income) and EV is negative (since welfare has decreased). In terms of figure all the mentioned areas are of course positive. It is therefore quite confusing to keep track of which measure is positive or negative and the crucial questions are the welfare effect for the consumer. In this case we should calculate the absolute amount, so that we always get a positive number. For a normal good we therefore have the following relationships between the three welfare measures:

|CV | > |ΔCS| > |EV |,

while for an inferior good:

|EV | > |ΔCS| > |CV |.

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