Appendix E The linear expenditure system and equivalent variation

E.1 Derivation of the equivalent variation metric

The direct utility function for the Linear Expenditure System is:

With 0 < β < 1, and . β_i is marginal expenditure on good i out of total expenditure; given β_i is positive it rules out inferior goods. x_i and γ_i are respectively the total expenditure on good i and the amount of committed expenditure on good i. Committed expenditure is expenditure that is considered the basic need and is consumed no matter what the income. If p_i is the price of good i and γ is total expenditure the budget constraint is:

Following Creedy and Sleeman (2006) we define the two terms A and B respectively as:

The indirect utility function, V (p, y), can be derived as:

inverting (E.5) and substituting the expenditure function E(p, U) for y you get:

Given

When prices change from p₀ to p₁, we can write:

assuming that total expenditure remains constant at y, this gives:

Substituting for U₁ into (E.9) using equation (E.6) and rearranging slightly gives:

The term A₁/A₀ is a Laspeyres type of price index, using committed expenditure of good i (γ_i) as weights:

Where

The term B₁/B₀ can be simplified to

which is a weighted geometric mean of the relative prices of each good in time 0 and time 1.

Given we have estimated budget shares wi (see Section E.2) and have expenditure levels for each household type as well as observed price changes,[39] to calculate the equivalent variation all we need is p_iγ_i and β_i.

Creedy and Sleeman (2006) show that:

Where e_i is the elasticity of the budget share of good i to expenditure. We are able to estimate this parameter, e_i for each household type; the next section describes the estimation procedure. Once e_i is estimated, and given we have values for w_i, we are able to calculate β_i, one of our unknowns.

Given w_i, β_i, ξ and e_i we are able to calculate η_ii - the price elasticity for each household type of good i to its own price using (E.15). ξ is the Frisch parameter and denotes the elasticity of marginal utility of total expenditure with respect to total expenditure. As Creedy and Sleeman (2006) do we impose the Frisch parameter. Following Creedy and Sleeman (2006) we assume it takes a fixed value of -1.9, in the next section we test the sensitivity of our results to different values of the Frisch parameter.

Given η_ii and total expenditure γ we can find p_iγ_i using:

We can now calculate the equivalent variation using (E.10) with the expenditure γ.

E.2 Estimating the budget share - expenditure elasticity

As we outlined above we need an estimate of the budget share of good i, w_i, and the elasticity of the budget share of good i with respect to expenditure, e_i. Again following Creedy and Sleeman (2006) we estimate the budget share of good i for each household type using the following functional form (omitting the i subscript for each good):

As Creedy and Sleeman (2006) note this form has the convenient property that if parameters are estimated using ordinary least squares, the adding-up condition that the budget shares must sum to 1 across all goods holds for predicted shares, at all total expenditure levels, γ.

As there are 36 commodity groups (see Table A.3) and 34 household types (12 cluster and 22 categories in the hard dimension analysis), a total of 1224 (36 times 34) budget share regressions were performed. Hence the estimated budget shares for each good and each group and cluster cannot be reported here.

Turning to estimating the elasticity of the budget share of good i with respect to expenditure, at any given level of γ, the expenditure elasticity (for a required commodity group and household type) can be expressed as:

Which we are able to estimate using ordinary least squared - again owing to the large number of results these are not reported.

E.2.1 Sensitivity of results to Frisch parameter

Following Creedy and Sleeman (2006) we set the Frisch parameter at -1.9. The Frisch parameter is the marginal utility of total expenditure with respect to total expenditure hence its plausible values are negative. As stated above it is used to calculate equivalent variation using the Linear Expenditure System. Figure E.1 and Figure E.2 show that while the values of our calculated equivalent variation measures differ for different various plausible values of the Frisch parameter. The relative positions of the clusters in terms of who was affected most and least remains the same until very high values (ie less negative) of the Frisch parameter are selected.

Figure E.1: Movements in equivalent variation

Figure E.2: Movements in equivalent variation normalised by expenditure