To understand this result, let’s consider an example. Imagine that Alison’s
utility from the last dollar spent on juice is three times her utility from the
last dollar spent on burritos. In this case, Alison can increase her total
utility by spending less on burritos and spending more on juice.
If Alison wants to maximize her utility, she will continue to switch her
expenditure from burritos to juice as long as her last dollar spent on juice
yields more utility than her last dollar spent on burritos. This switching,
however, reduces the quantity of burritos consumed and, given the law of
diminishing marginal utility, raises the marginal utility of burritos. At the
same time, switching increases the quantity of juice consumed and
thereby lowers the marginal utility of juice.
Eventually, the marginal utilities will have changed enough so that the
utility received from the last dollar spent on juice is just equal to the
utility received from the last dollar spent on burritos. At this point, Alison
gains nothing from further switches. (In fact, switching further would
reduce her total utility.)
So much for the specific example. What can we say more generally about
utility maximization? Suppose we denote the marginal utility of the last
unit of product X by and its price by Let and refer,
respectively, to the marginal utility of a second product Y and its price.
The marginal utility per dollar spent on X will be For example,
if the last unit of X increases utility by 30 and its price is $3, its marginal
utility per dollar is If the last unit of Y increases utility by 10
and its price is $1, its marginal utility per dollar is With these
MUX pX. MUY pYMUX/pX.30 / 3 =10.
10 / 1 =10.