Ambrose's brother Patrick has a utility function U(x₁, x₂) = 16x ₁ + x₂. His income is $82, the price of good 1 (nuts) is $2, and the price of good 2 (berries) is $1. How many units of nuts will Patrick demand? A) 26 B) 12 C) 14 D) 16 E) 30

Question

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To solve for Patrick's optimal bundle, we need to find his marginal rate of substitution (MRS) and set it equal to the ratio of prices (Pnuts / Pberries). MRS = MU1 / MU2 Taking partial derivatives of Patrick's utility function with respect to each good: MU1 = 16x₁ MU2 = x₂ Now we can set up the equation: MU1 / MU2 = Pnuts / Pberries 16x₁ / x₂ = 2 / 1 Simplifying: 16x₁ = 2x₂ 8x₁ = x₂ 8 = x₂ / x₁ So Patrick is willing to trade 8 units of berries for one unit of nuts. We can now use Patrick's budget constraint to find how many nuts he can afford: 82 = 2x₁ + x₂ Rearranging: x₂ = 82 - 2x₁ Plugging this into the MRS equation: 8 = (82 - 2x₁) / x₁ Simplifying: 8x₁ = 82 - 2x₁ 10x₁ = 82 x₁ = 8.2 Therefore, Patrick will demand approximately 16 units of nuts (rounded to the nearest whole number). The answer is (D) 16.

Ambrose's Brother Patrick Has a Utility Function U(x1, X2) =

Ambrose's Brother Patrick Has a Utility Function U(x₁, X₂) =