This is a sample exam question to which I don't have a model answer. I'd appreciate your feedback.

Question.

It is thought that a consumer measures the utility of posessing a quantity of apples and a quantity of oranges by the formula:

.

It is know that, when a consumer's budget for apples and oranges is $1, he will buy 2 apples and 1 orange when they are equally priced. Find .

The price of oranges falls to half that of apples with the price of apples unchanged. How many apples and oranges will the consumer buy for $10?

[Hint. First solve a utility maximisation problem with as a parameter.

Answer.

First part, finding alpha.

I will maximise by

first - find critical points by setting the first derivative to zero, then

second - evaluate the second derivative at this critical point(s)

Then I use the fact that for $1 consumer would buy 2 apples and 1 orange, so I substitute x=2 and y=1:

Now when I try to evaluate the second derivative, it is Hessian matrix and it's determinant is 0. Therefore, the test is inconclusive.

What can I do here to show that alpha=2/3 gives the maximum value of the function? (if it does...)

Given and the utility function is

I will post the second part as a separate post, as this one is growing long.