Charlie has the utility function u (xa, xb) = xaxb. If Charlie's income is $40, the price of apples is $4, and the price of bananas is $2, how many apples are there in the best bundle Charlie can afford?

There are 5 apples in the best bundle Charlie can afford

Step-by-step explanation:

If the utility function is u (xa, xb), where xa represent the quantity of apples and xb is the quantity of bananas then we want to choose the quantity of bananas and apples that maximises the utility of Charlie for the same budget restriction (get the most benefit for the same money).

The budget restriction is

$4 * xa + $2 * xb = $40

then

u (xa, xb) = xa*xb

4*xa + 2*xb = 40 → xb = (40 - 4*xa) / 2 = 20 - 2*xa

replacing in the utility function

u (xa, xb) = xa * (20 - 2*xa) = 20*xa - 2*xa²

the maximum of this function is obtained when the derivative of the utility function with respect to xa is 0. Thus

du/dxa = 20 - 4*xa = 0 → xa = 20/4 = 5

then for

xa=5 apples

xb=20 - 2*xa = 20 - 2*5 = 10 bananas

Charlie maximises his utility. Therefore there are 5 apples in the best bundle Charlie can afford