Understanding marginal utility and maximized consumption.

This post goes through an example of how to figure out if someone is maximizing their utility buy looking at their consumption and figuring out their marginal utility per dollar spent.  In particular it answers the following question:

Roy spends her income of $150 per week on two goods; movies (which cost $5 each) and books (which cost $10 each). At his level of consumption, the marginal utility from the last movie consumed is 20 and the marginal utility from the last book consumed is 30. Is Roy maximizing her utility?Why or why not? If not, what should Roy do to achieve a higher level of utility?

There are two rules to see if someone is maximizing their utility.  First, the marginal utilities per dollar have to be equal.  Second, the entire budget has to be exhausted (used up).

Lets check the second rule first, is the entire budget being used? 
Unfortunately, this problem doesn't say how many of each good he is purchasing, so we will have to assume that it is being satisfied.  Let's move on to the first rule.


The marginal utilities for the goods aren't equal, this means Roy isn't maximizing his utility right? WRONG!  What we need to figure out is if the marginal utilities per dollar spent are equal, so let's calculate that first:

Marginal utility per dollar of movies = marginal utility / price = 20/$5 = 4.
Marginal utility per dollar of books = marginal utility/ price = 30/$10 = 3.

We now see that the marginal utilities per dollar spent are not equal, because 4 doesn't equal 3.  This means that his marginal utility per dollar spent is higher for movies than for books, so he gets a better return (more happiness per dollar) by buying less books and more movies.  Since we don't know the exact breakdown of his consumption, meaning how many of each he is currently buying, we can only suggest that he buy less books and more movies.