Ambrose's utility function is U(x₁, x₂)=4x^1/2₁ + x₂.If the price of nuts (good 1)is $1, the price of berries (good 2)is $6, and his income is $264, how many units of nuts will Ambrose choose? A)20 B)144 C)288 D)147 E)72

Question

Quizplus · Accepted Answer

To maximize utility, Ambrose must allocate his entire budget between nuts (good 1) and berries (good 2), subject to their prices and his income. This can be represented as:

1x1 + 6x2 = 264

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

Now, using Ambrose's utility function, we can form his marginal rate of substitution (MRS) as the ratio of the marginal utilities of the two goods:

MRS = MU1/MU2 = (2x1^-0.5)/(x2^-1)

To maximize utility, Ambrose must equate the MRS with the price ratio of the two goods (P1/P2 = 1/6) in order to achieve the highest possible satisfaction per dollar spent. Setting up this equation and simplifying, we get:

2x1^-0.5 / x2^-1 = 1/6
x1/x2 = (1/6)^2/3
x1 = (1/6)^2/3 * x2

Substituting this into the budget constraint, we get:

1((1/6)^2/3 * x2) + 6x2 = 264
(1/6)^2/3 * x2 = 44
x2 = 144

Substituting back into the equation for x1, we get:

x1 = (1/6)^2/3 * 144
x1 ≈ 48

Therefore, Ambrose will choose 48 units of nuts (good 1).

Ambrose's Utility Function Is U(x1, X2)=4x1/21 + X2

Ambrose's Utility Function Is U(x₁, X₂)=4x^1/2₁ + X₂