The Possibility of Undercompensating Price Changes

In a recent issue of this Journal,' the author stated that it is not always possible to ' undercompensate ' arbitrary price changes, whereas in fact it is always possible under the conditions employed in that paper.2 The point is of some interest because of the use of undercompensated price changes in elementary taxation theory, in tracinlg out 'inferior Houthakker sequences '3 in revealed preference theory, and in defining lower limits to ' true' price index numbers. Let an individual's demands be q7,...,q, when the corresponding prices are p',...,p? and his total expenditure (income) is E?=0p qq Similarly, let his demands be qi,...ql at prices p'...p,t an-d inicome El =Zpql. Givein the prices, demands depend upon income, thus determining the associated income-consumption curve and Engel curves. The question is whether or not there always exist values of the ql such that