Question

utility function u(x,y) = x3 ·y2

I am going to walk you through the process of deriving the optimal quantity of apples and bananas that will make you the happiest. To do this, we are going to apply what we learnt about derivatives.

a) First, you have a budget. You cannot just buy an infinite amount since you would not be able to afford it. Suppose the price of a single apple is Px = 2 while the price of a single banana is Py = 4. If you buy x apples and y bananas, how much would that cost you (your answer should depend on x and y)? Make that equal to 100, meaning that you have a budget of $100 that you are trying to allocate between apples and bananas. The equation that you got is what we call a budget constraint.

a) General form of budget constraint is

P1* X + P2 * Y =m

where P1 = price of good 1 and P2 = price of good 2 and m = income

So, here

Budget constraint is 2X + 4Y = 100

b) Take your budget constraint and solve it for x in terms of y.

2X + 4Y = 100

2X = 100-4Y

X = 50 - 2Y

c) Substitute the expression that you got in part b) into the utility function. You should now have and expression for your utility that depends only of y, that is u(y).

d) Find the value of y that maximizes your utility. You should get more two solutions but explain which one(s) solve(s) your problem and why the other solution is obviously not a maximum.

e) Take that value of y you found in part d), plug it back into your budget constraint and solve for x

Please answer C D E

Answer 1