Josh likes to eat Bananas (b), and Fish (f). His preferences are Cobb-Douglas, and can be presented by the utility function u(b, f ) = (1/5) (b − 2)(f + 3). Each banana costs 25 dollars and each fish costs 12 dollars. Josh has $314 to spend. Assume that bananas and fish are both perfectly divisible goods.
(a) Find Josh's optimal bundle.
(b) Suppose that Josh is now $75. What is his optimal bundle now?
(c) Now, consider the general case, where Josh's income is m, the price of bananas is pb, and the price of fish is given by pf . In other words, treat fish as the numeraire. Find Josh's demand for each good in terms of m, pb and pf .
(d) Find the good on which Josh will always spend more than half of his income.
(e) Find the own-price, cross-price, and income elasticities of good 2 (fish).