An agent consumer quantity (X1, X2) of good1 and 2, she has utility U(X½,X½).
a) Derive MRS
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 is calculated as the ratio of the marginal utilities of the two goods.
MRS = MU1/MU2
b) To show that the preferences are convex, we need to show that the MRS is decreasing as the consumer increases the quantity of one good and decreases the quantity of the other. This means that the consumer is willing to give up more of one good for a given amount of the other good as the quantity of the first good increases. This can be shown by taking the derivative of the MRS with respect to the quantity of the first good. If the derivative is negative, then the MRS is decreasing and the preferences are convex.