max{x1,x2} where P1not=p2

Question

I have seen min{x1,x2} functions representing perfect compliments but have never seen a max{x1,x2} function anywhere in my book or lectures, I also have never seen anything about p1 not equaling p2. can anybody give me some insight to this problem.

2. Consider a consumer with a utility function U = max{x1,x2} the consumer faces prices p1 and p2 for goods 1 and 2, respectively, and has an income of m dollars. (assume throughout p1 dne p2). His optimal consumption solves the standard problem.

a)Find the optimal levels of consumption for goods 1 and 2 as a function of income and prices.

b) is good 1 a normal good?

c) Are goods 1 and 2 complements or substitutes?

any help would be much appreciated

Answer 1

Here is my suggestion:

1. I would say the optimal level of consumption is $\frac{m}{p_1}$ if $p_1 < p_2$ and $\frac{m}{p_2}$ if $p_2 < p_1$. So the individual consumes only whichever good is cheaper. Since utility is just the maximum of whichever two amounts you could consume, you would maximize that by consuming the highest amount possible.
2. Good 1 is a normal good if it is cheaper than Good 2, then the amount consumed rises with income.
3. I would say these goods are perfect substitutes, since you consume as much of the cheaper good as possible with no further constraints. EDIT: This wrong, they are not perfect substitutes, since they are not substitutable - if you consume only Good 1, you wouldn't want to exchange any for Good 2, except if you then had more of Good 2 than of Good 1.