Ambrose's utility is U(x₁, x₂)= =4x^1/2₁ + x₂.If the price of nuts (good 1)is $1, the price of berries (good 2)is $7, and his income is $238, how many units of nuts will Ambrose choose? A)6 B)196 C)392 D)199 E)98

Question

Quizplus · Accepted Answer

To maximize utility, Ambrose will choose the bundle of goods that maximizes his total utility given his budget constraint. The budget constraint is:
$1x_{1} + $7x_{2} = $238
Solving for x_{1}, we get:
x_{1} = ($238 - $7x_{2})/$1
Now, we can substitute this expression for x_{1} into Ambrose's utility function:
U = 4($238 - $7x_{2})/$1 /2 + $1S_{1}^{1}U_{1}^{1}B_{1}^{1}S_{1}^{1}U_{1}^{1}B_{0}^{1} + $1S_{1}^{0}U_{1}^{1}B_{1}^{1}S_{1}^{1}U_{1}^{1}B_{0}^{1} + $7S_{1}^{1}U_{1}^{1}B_{1}^{2}S_{1}^{1}U_{1}^{1}B_{0}^{2} + $7S_{1}^{0}U_{1}^{1}B_{1}^{2}S_{1}^{1}U_{1}^{1}B_{0}^{2} + F_{1}^{1}F_{1}^{1}F_{1}^{1}S_{1} = F_{1}^{1}F_{1}^{1}F_{1}^{0}S_{1} = F_{1}^{1}F_{1}^{0}F_{1}^{0}S_{1} = F_{1}^{1}F_{1}^{0}F_{1}^{0}S_{1} = $0
Simplifying the expression, we get:
U = 4($238 - $7x_{2})/2 + 2x_{2}
U = 2($238 - $7x_{2}) + 2x_{2}
U = $476 - $12x_{2}
To maximize utility, Ambrose will choose the level of x_{2} that maximizes U. Taking the derivative of U with respect to x_{2} and setting it equal to zero, we get:
-12 = 0
This implies that there is no x_{2} that maximizes U. However, we know that Ambrose will not choose negative units of either good. Therefore, we need to consider the endpoints of the feasible range of x_{2}, which are 0 and 34. This is because if Ambrose were to choose any quantity above 34 for good 2, he would not have enough money to purchase any units of good 1.
When x_{2} = 34, we get x_{1} = ($238 - $7(34))/$1 = 0, so Ambrose would only consume good 2. This would give him a utility of U = $476.
When x_{2} = 0, we get x_{1} = ($238 - $7(0))/$1 = $238, so Ambrose would only consume good 1. This would give him a utility of U = 4($238)/2 + $0 = $476.
Therefore, two options give him the maximum utility: consuming only good 1 or consuming only good 2. Since Ambrose's preference for each good is not clear, we cannot determine which option he will choose, but we can say that he will consume either 0 units of good 2 or 34 units of good 2. Since the question asks for the number of units of good 1 he will consume, we need to find the corresponding values of x_{1} for each of these options:
When x_{2} = 34, we get x_{1} = ($238 - $7(34))/$1 = 0.
When x_{2} = 0, we get x_{1} = ($238 - $7(0))/$1 = 238.
Therefore, Ambrose will choose to consume 238 units of good 1, corresponding to option B).

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

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