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# Appendix B: Welfare Changes and Demand Elasticities

This appendix describes the computation of the welfare measures and the method used to compute the required parameters for each demographic group and total expenditure level. Only the main results are stated, as their derivations are available elsewhere.[18] The basis of the approach is the use of the linear expenditure system to model households’ behaviour. The total expenditure of each household is assumed to remain fixed when prices of goods and services change. Thus, possible changes in production (associated with the changing structure of demands) and factor prices, along with the distribution of income, are ignored.

The direct utility function for the linear expenditure system is:

(B1)

with and Here, and are respectively the total and the committed consumption of good If is the price of good and is total household expenditure, the budget constraint is In the present context, the parameters of the utility function differ according to both household type and total expenditure, as discussed further below. The next two subsections define equivalent variations and money metric utility, which are used in the distributional analyses.

## Equivalent Variations

The equivalent variation, , is defined in terms of the expenditure function as where is the minimum expenditure required to reach utility level at prices Defining the terms and respectively as and , the indirect utility function, , is:

(B2)

The expenditure function is found by inverting this and substituting for to get:

(B3)

Suppose that the vector of prices changes from to . Substituting for using (B3) and assuming that total expenditure remains constant at gives:

(B4)

Substituting for using equation (B2) into (B4) gives:

(B5)

The term is a Laspeyres type of price index, using s as weights. The term simplifies to which is a weighted geometric mean of price relatives.[19] A convenient feature of the present approach is that the expression for the equivalent variation requires only the percentage changes in prices to be specified.

## Money Metric Utility

For distributional analyses of tax reforms, it is necessary to have a money metric measure of each household’s utility. A suitable money metric is defined as the value of total expenditure, , which, at some reference set of prices, , would give the same utility as the actual total expenditure.[20] A feature of this metric is that it ensures that alternative situations are evaluated using a common set of reference prices. It is, importantly, invariant with respect to monotonic transformations of utility. Using the expenditure function gives:

(B6)

For the linear expenditure system, this is found to be:

(B7)

The effect on welfare can be measured in terms of a change in from to , where, as before, the indices and refer to pre- and post-change values respectively. If pre-change prices are used as reference prices, so that for all is simply the value of actual total expenditure after the change less the value of the equivalent variation; that is, . Hence the proportionate change, , is conveniently the ratio of to .

#### Notes

• [18]For example, see Powell (1974), Allen (1975), Creedy (1998a,b).
• [19]The corresponding result for the compensating variation follows by substituting into .
• [20]In terms of the indirect utility function,  is defined by  This metric was called ‘equivalent income’ by King (1983), but this term can lead to confusion when used in conjunction with adult equivalent scales.
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