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