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(1)

* Throughout this discussion, we will treat all income and price elasticities as constant over the relevant ranges. Of course, budget constraints must invalidate this assumption when the changes are large enough. This in turn means that our conclusions will not be able to be taken completely seriously for certain parameter values. As equation (1) might suggest, the suspect cases are those in which u is close to zero. 1

In particular, since c is always negative, an increase in income yields a positive net benefit whenever u is negative, and a negative net benefit when u is positive, which is the Giffen case. In view of the Slutsky equation u = c − k · η

(2)

(where k is the fraction of income spent on housing and η is the income elasticity of demand for housing), formula (1) can be rewritten as c . c − k · η Holding c and k fixed and letting η vary, we get the following graph depicting the relationship between η and the income elasticity of net benefit:

The region to the left of the vertical asymptote is precisely the range of η that makes housing a Giffen good. If η falls between the asymptote and the vertical axis, then housing is inferior but not Giffen, and if it falls to the right of the axis then housing is a normal good. Except in the Giffen case, exogenous income is always good thing—though it becomes less of a good thing as the income elasticity for housing increases. If housing is an inferior 2

good, things are even better, provided we don’t enter the Giffen range. A bit surprisingly, even though the Giffen case is the bad case, it is still true that if housing must be a Giffen good, then “the more Giffen the better”. That is, it is bad for η to be less than c /k, but better for it to be far less than just a little less. The case where η is very close to c /k is the hairtrigger one: If η is just slightly above this value, then exogenous income yields enormous gains; if it is just slightly below, then exogenous income yields enormous losses. (But the arbitrarily large gains and losses suggested by the asymptotic behavior should be taken with a grain of salt in light of footnote 1.) Now we will give a proof of formula (1). Suppose that income increases by 1% and that this leads to a t% increase in housing prices. The quantity of housing demanded will increase by η% on account of the income increase and t · u % (which is negative in the normal case) on account of the price increase, for a total increase of (η + t · u )%. But in equilibrium the quantity of housing demanded cannot change, so we get η + t · u = 0 or t=−

η . u

(3)

The percentage change in income spent on housing is equal to the percentage change in housing prices times the fraction of income spent on housing, or t · k. So the percentage increase in income available to spend on other things is equal to the original 1% increase in income, minus k · t. In view of equation (3), this is η u u + k · η = , c

1−t·k =1+k·

which, in view of equation (2), is equal to expression (1). There is also a nice geometric proof, using indifference curves, which the reader might prefer to discover for himself.

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