Texts: Our adoption of the first three axioms was justified primarily by the fact that they guarantee that the preference that satisfies them can be characterized by a full ranking of bundles, which then allows us to represent it using an indifference map (with the properties of indifference curves that we described), and more conveniently, a non-unique utility function. We have also started solving the consumer's problem for a given utility function and have used it to derive an individual's demand for goods. But we haven't said much about how this utility function is chosen in the case where we cannot observe a person's preferences and ask them to rank all bundles.
It turns out that in real life, a commonly observed aspect of consumers' behavior is that they tend to spend fixed shares of their incomes on certain goods (say housing, utilities, transport, health, leisure, clothing, telecommunications, etc...). How much consumers spend on each category of goods is data that is much easier to collect than abstract data on preferences. It turns out, this type of data can be used to decide on which utility function to use to represent preferences.
As an illustration, consider the problem below.
Use the utility function U(X1, X2, X3) = X1^a * X2^b * X3^c where a, b, and c are positive numbers.
a. Solve this consumer's utility maximization problem by finding the optimal quantities of X1, X2, and X3 as functions of prices P1, P2, P3, and income M. [HINT: remember that ln is an increasing function, so that ln(U) is an increasing transformation of U]
b. Find P1*X1*, the total spending on X1 at the optimal choice for that good. Find P2*X2*, the total spending on X2 at the optimal choice for that good. Find P3*X3*, the total spending on X3 at the optimal choice for that good.
c. What is the share of M that is spent on good X1 at the optimal choice? What is the share of M that is spent on good X2 at the optimal choice? [Note that this finding can be extended to any utility function of this form, for more than just 3 goods.]