Affluence and Food: A Simple Way to Infer Incomes - The University ...

n(1) qˆ= η Mˆ+ ∑ η ′ p ˆ ,i i ij jj=1where ηiis the income elasticity of demand for i and η ′ijis the ( i, j ) thuncompensated price elasticity. If wedefine the change in the cost of living index as a budget-share weighted-average of the n price changes,Pˆ= w pˆ, then the change in real income is the excess of the change in money income change over thisn∑ j=1jjindex, Q ˆ = M ˆ − P ˆ , and the Slutsky demand equation takes the formi i ij jj=1( ˆ ˆ)n(2) qˆ= η Qˆ+ ∑ η p − P ,whereη is the ( i, j ) thcompensated price elasticity. In deriving this equation (2) from (1), we have used (i)ijnthe Slutsky decomposition η ′ij= ηij − wjη i, and (ii) demand homogeneity, according to which ∑ j = 1 ηij= 0.ŵ = pˆ + qˆ − M = pˆ − P + qˆ− Qˆ. Combining this with equation (2) then yieldsNext, we note that ˆ(ˆ)i i i i i( ) ˆ ∑n( ˆ ˆ) ( ˆ ˆ)(3) wˆ= η − 1 Q + η p − P + p − P .i i ij j ij=1If the n commodities are broad aggregates, it would be likely that there would only be limitedsubstitutability between them. We thus take the utility function to be of the preference independentnform, u ( q ,…,q ) = ∑ i = 1u ( q ) , with u ( )1 n i iii the sub-utility function for good i, so that the marginal utility ofi depends only on own consumption. This form of tastes implies that as an approximation, own-priceelasticties are proportional to income elasticities and cross-price elasticities are zero,(4) η ≈ φη , i = 1, …, n, η ≈ 0, i, j = 1, …, n,i ≠ j,ii i ijwhere φ is the reciprocal of the income elasticity of the marginal utility of income (the “income flexibility”for short). Equations (3) and (4) then imply( ) ˆ ( )( ˆ ˆ)(5) wˆ≈ η − 1 Q + φη + 1 p − P ,i i i iwhich shows the dependence of the budget share on income and the relative price of the good. The twoparameters in equation (5) are the income elasticity and the income flexibility.3. Income and FoodOn the basis of the budget shares, in most countries food is the most important single commodity.Thus, in what follows we concentrate on this commodity. In this section, we analyse the relationship3