An agent consumer quantity (X1, X2) of good1 and 2, she has utility U(X½,X½).
a) Derive MRS by calculating clearly
b) Show the prefernce are convex
a) The Marginal Rate of Substitution (MRS) is the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility. It can be calculated by taking the partial derivative of the utility function with respect to one good and dividing it by the partial derivative of the utility function with respect to the other good.
For the given utility function, U(X1,X2), the MRS is:
MRS = (∂U/∂X1)/(∂U/∂X2)
b) To show that the preferences are convex, we need to show that the MRS is decreasing as the consumer moves along the indifference curve. This can be done by taking the second partial derivative of the utility function with respect to both goods and showing that it is negative.
For the given utility function, U(X1,X2), the second partial derivatives are:
∂2U/∂X12 = -1
∂2U/∂X22 = -1
Since both second partial derivatives are negative, the MRS is decreasing as the consumer moves along the indifference curve, and the preferences are convex.