The application "Indifference Curves Between Food and Clothing" postulates that there are minimum levels of food and clothing necessary to support life. Suppose that the amount of food one has is F, the minimum level to sustain life is F, the amount of clothing one has is C, and the minimum necessary is C. We can then modify the Cobb-Douglas utility function to reflect these minimum levels: U(C, F) = (C - C)a(F - F)1- a, where C ( C and F ( F. Using the approach similar to that in Solved Problem 3.6, derive the optimal amounts of food and clothing as a function of prices and a person's income. To do so, introduce the idea of extra income, Y*, which is the income remaining after paying for the minimum levels of food and clothing:
Y* = Y - pCC - pFF. Show that the optimal quantity of clothing is C = C + aY*/pC and that the optimal quantity of food is F = F + (1 - a)Y*/pF.
Derive formulas for the share of income devoted to each good.