Ambrose's brother Bartholomew has an income of $109 and a utility function U(x₁, x₂) = . The price of good 1 (nuts) is $4 and the price of good 2 (berries) is $1. How many units of nuts will Bartholomew demand?

Question

Quizplus · Accepted Answer

To maximize utility within Bartholomew's budget constraint, we need to find the optimal combination of nuts and berries that he can purchase.

We can start by setting up Bartholomew's budget constraint as:

4x₁ + x₂ = 109

where x₁ is the amount of nuts and x₂ is the amount of berries.

To solve for the optimal amount of nuts, we can use the marginal rate of substitution (MRS):

MRS = MU₁ / MU₂ = 4 / 1

where MU₁ is the marginal utility of nuts and MU₂ is the marginal utility of berries.

Using Bartholomew's utility function, we can find the marginal utility of nuts as:

MU₁ = ∂U / ∂x₁ = 3x₁⁻²/₃x₂⁰⁰⁰⁰⁰⋯

We can simplify this to:

MU₁ = 2x₁⁻²

Using the MRS equation and plugging in the values, we get:

4 / 1 = 2x₁⁻²

Simplifying, we get:

x₁ = 16

Therefore, Bartholomew will demand 16 units of nuts.

Ambrose's Brother Bartholomew Has an Income of $109 and a Utility