Today we will go over the problem: When the price of X
is 10 and the price of Y is 30, a consumer purchases 100 units of X and 50
Units of Y. Because 100 Units of X and 50 Units of Y are purchased, the
consumer must be willing to substitute 2 units of X for 1 unit of Y to remain
indifferent. Given the prices, 3 units of X can be
substituted for each unit of Y along the budget constraint. Therefore,
the consumer is not maximizing utility. Explain why you agree or disagree with
this statement.

Ok, so the relevant information is:

The Price of X is 10

The price of Y is 30

The consumer purchases 100 units of X therefore she spends
1,000 on X.

The consumer purchases 50 units of Y, therefore she spends
1,500 on Y.

This means that the total budget is 2,500 which means:

Up to 250 units of X can be purchased, and up to 83.33 units
of Y can be purchased.

Look at the graph below and check out the three different
possible shapes of the utility functions.
Note that BC = budget constraint, IC1 = Indifference curve for perfect
substitutes, IC2 = “normal” indifference curve, and IC3 = indifference curve
for perfect complements.

The only indifference curve here that COULD make the above
statement true is IC1 (indifference curve 1), the perfect substitute indifference
curve. The reason for this is that with
perfect substitutes, we either buy one good or the other (a corner solution),
or can buy any combination of the goods because the budget line and the
indifference curve overlap. Here I show
this by making IC1 equal to the budget curve.
However, the question says that the person is willing to trade 1 Y for 2
X which violates what we see with IC1.
With IC1 the person is willing to trade 1 Y for 3 X so the above
paragraph has contradicted itself.

The next indifference curve is IC2, which is a normal
indifference curve. We can see that IC2
is tangent to the budget constraint at Y = 50 and X = 100, which shows that
this person IS maximizing their utility.
The reason for this is because at this point the MUx/MUy = Px/Py, which
means that the ratio of marginal utilities is equal to the ratio of
prices. Since the price of Y is 3 times
the price of X, we know that the marginal utility of Y is also three times the
marginal utility of X WHEN X = 100 and Y =50.
Remember that for normal indifference curves, as we consume more of a
good, the less utility we get from it (diminishing marginal utility). This is why we have a curved indifference
curve. This is also why the above
paragraph is wrong.

Finally we have IC3 which shows the indifference curve for
perfect complements. Here, it might be
possible that for every Y consumed, the consumer needs 2 X. Even though the price of Y is three times as
high as X, the consumer needs 2 X to get
utility from Y. Imagine that X is bread
and Y is a pack of salami. In order to
get a salami sandwich (this is what makes him happy), he must have 2 bread (X)
and one Y (a pack of salami). So even if
the prices are in a different ratio, he is not willing to substitute them based
on prices.

Ultimately, you always have to degree with this
statement. Remember that prices don’t
dictate what someone is willing to trade something for (although that would be
easy). What ultimately matters is the
combination of utility AND prices, which is why we try to equate marginal
utility per dollar spent when we maximize utility. Also, remember the equation: MUx/MUy = Px/Py
always occurs at the maximum and you should always get these types of questions
right.