4. A consumer has the additive utility function U(x1, x2, x) = u(xi) i=1 where u; is C2, with u(x) > 0 and u(x) < 0 for all x > 0. (a) Explain why, whenever a consumer is maximizing a monotone utility function (and not necessarily the one above) then, at any (p, w) > 0, there must be some good k (which may depend on (p, w)) for which dik dw (p, w) > 0. In other words, there must be some good k whose demand increases with income. (b) Show that for the additive utility function, the marginal utility of income di- minishes with income, i.e., d'v dw2 (p, w) < 0 for all (p, w) > 0. [Hint: Recall that the marginal utility of income equals the Lagrange multiplier.] (c) Show that, for the additive utility function, we may strengthen the conclusion in (a) to the following: di dw (p, w) > 0 for all (p, w) 0 and for every good i.