Clara's utility function is U(X, Y)=(X + 2)(Y + 1).If Clara's marginal rate of substitution is -6 and she is consuming 8 units of good X, how many units of good Y is she consuming?
A)14
B)60
C)59
D)6
E)13

Question

Quizplus · Accepted Answer

The marginal rate of substitution (MRS) is defined as the rate at which a consumer is willing to trade one good for another while maintaining the same level of utility. Mathematically, MRS = ΔY/ΔX.
In this case, we are given that Clara's MRS is 6. That is, she is willing to give up 6 units of good X for an additional unit of good Y while maintaining the same level of utility. Therefore, we can set up the following equation:
6 = ΔY/ΔX
Solving for ΔY, we get:
ΔY = 6ΔX
We also know that Clara is consuming 8 units of good X. Therefore, we can find her consumption of good Y as follows:
U(X,Y) = F1F1F1S1F1F1F10 (X F1F1F1S1F1F1F10 2)(Y F1F1F1S1F1F1F10 1)
Taking partial derivatives with respect to X and Y, we can find Clara's marginal utility of X (MUx) and marginal utility of Y (MUy) as follows:
MUx = F1F1F12(Y F1F1F1S1F1F1F10 1)/(X F1F1F1S1F1F1F10 2)
MUy = F1F1F1S1F1F1F10(X F1F1F1S1F1F1F10 2)/Y
Dividing MUx by MUy, we get Clara's MRS as:
MRS = (MUx/MUy) = (Y/X) * (F1F1F12/(F1F1F1S1F1F1F10))
We can now solve for Y by substituting the values we know:
6 = ΔY/ΔX = (Y-X/F1F1F12)/(X/F1F1F12) = (Y/X) - (F1F1F1S1F1F1F10/(X F1F1F1S1F1F1F10 2))
Simplifying, we get:
(Y/X) = (F1F1F1S1F1F1F10 + 6)/(F1F1F1S1F1F1F10/2) = 59
Therefore, Clara is consuming 59 units of good Y. Answer choice C is correct.

Clara's Utility Function Is U(X, Y)=(X + 2)(Y + 1)