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This exercise develops an alternative proof of Theorem 16.3.
(a) Choose the appropriate topology on Z so that U is continuous on X × X × Z.
(b) Use Theorem A.12 in Appendix A to show that the objective function in Problem 16.1 is continuous in the product topology; Theorem A.13 and Lemma A.2 to show that the constraint set is compact; and Theorems A.9 and A.16 to show that V∗(x(0), z(0)) is well defined, continuous, and bounded over X × Z.
(c) Deduce the same results for V (x(0), z(0)) by applying Theorem 16.1.
Think about the statement above and relate it to what you have learned in Chapter 4. What caused prices to rise and then to fall? Are there other products that may have experienced changes in prices as a result of Hurricane Irene? Why?
Now assume that Home has started trading with the Foreign country, which is exactly the same: it has the same demand curve and there is only one firm there that also produces apples at the marginal cost of $4 per kilo.
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