John’s utility function is U(x,y) = (x+2)(y+1). His marginal rate of substitution is −6 and he consumes x∗ = 8 units, how many units of good y is he consuming?
To find out how many units of good y John is consuming, we need to use the concept of the marginal rate of substitution (MRS).
The MRS represents the rate at which an individual is willing to substitute one good for another while maintaining the same level of utility. It is given by the ratio of the marginal utility of x to the marginal utility of y:
MRS = MU(x) / MU(y)
Since John's utility function is U(x,y) = (x+2)(y+1), we can find the marginal utilities of x and y by taking partial derivatives with respect to each variable:
MU(x) = ∂U/∂x = y + 1
MU(y) = ∂U/∂y = x + 2
Now we can calculate the MRS using the given value:
-6 = (y + 1) / (x + 2)
Plugging in x* = 8:
-6 = (y + 1) / (8 + 2)
Simplifying the equation:
-6 = (y + 1) / 10
Multiplying both sides by 10:
-60 = y + 1
Subtracting 1 from both sides:
y = -61
Therefore, John is consuming -61 units of good y, which is not a realistic quantity. It is possible that there was a mistake in the calculations or in the given information.