Hi, I have to answer the following question: A person with the utility function U(x, y) = y + x^2 has convex preferences, true or false?
I don't know how to do it, do I put U(x,y) = to a K
and make up some bundles? Can someone please explains the step to solve this?
To determine whether a person's preferences are convex, we need to examine the utility function and analyze its properties. In this case, the utility function is given as U(x, y) = y + x^2.
Convex preferences imply that individuals prefer a mixture of goods over extreme cases. To determine convexity, we need to check if the utility function exhibits diminishing marginal rates of substitution (MRS) and increasing marginal rates of substitution (MRS).
To calculate the MRS, we take the first-order partial derivatives ∂U/∂x and ∂U/∂y:
∂U/∂x = 2x
∂U/∂y = 1
Now, we can calculate the ratio of the MRS:
MRS = (∂U/∂x) / (∂U/∂y) = (2x) / 1 = 2x
Since the MRS is a linear function of x, it does not vary based on the level of consumption of x and is, therefore, independent of y. This implies that the MRS does not change as we move along the indifference curve, indicating constant marginal rates of substitution.
As a result, the utility function U(x, y) = y + x^2 does not satisfy diminishing marginal rates of substitution (MRS). Hence, it does not exhibit convex preferences.
In conclusion, the claim that a person with the utility function U(x, y) = y + x^2 has convex preferences is false.