Final answer:
To maximize Charlie's utility subject to his budget constraint, we can substitute the budget constraint equation and the utility function into the objective function U(x) and solve for x.
Step-by-step explanation:
To maximize Charlie's utility subject to his budget constraint, we need to find the combination of apples and bananas that will give him the highest possible utility given his income.
Let's assume that Charlie consumes x units of apples and y units of bananas.
Since the price of apples is $4, the amount spent on apples will be 4x. Similarly, since the price of bananas is $1, the amount spent on bananas will be y.
According to Charlie's utility function U(A,B)=A²B, his utility is given by U(x,y)=x²y.
Considering his budget constraint, where his total income is $120, we have the equation 4x + y = 120.
To maximize his utility, we can substitute y = 120 - 4x into the utility function and find the maximum value of U(x).
Taking the derivative of U(x) with respect to x and setting it equal to 0, we can solve for x.
By substituting the value of x back into the budget constraint equation, we can find the value of y.
The solution is x = 10 and y = 20. Therefore, Charlie would consume 20 units of bananas if he chose the bundle that maximizes his utility subject to his budget constraint.