What is the marginal rate of substitution for U(x,y) = �ãxy? ,�ã(x+y)?

Aargh font didn't come out right. I meant MRS of U(x,y) = (xy)^0.5 (ie sqaure root of xy) and (x+y)^0.5.

Use basic calculus. MUx = .5y^.5/x^.5

MUy = .5x^.5/y^.5
MRS = MUx/MUy = y/x

take it from here

To find the marginal rate of substitution for the utility function U(x,y), we need to take the partial derivatives of U with respect to x and y.

The utility function U(x,y) = xy implies that the satisfaction or utility obtained from consuming x units of the good x and y units of the good y is given by the product of x and y.

Let's find the partial derivatives of U(x,y) with respect to x and y:

∂U/∂x = y

∂U/∂y = x

These partial derivatives give us the rate at which the utility changes with respect to changes in x and y, respectively.

The marginal rate of substitution (MRS) is the rate at which a consumer is willing to trade one good for another while staying on the same level of utility. It represents the amount of one good that a consumer is willing to give up to obtain an additional unit of the other good, while maintaining the same level of satisfaction.

In this case, the MRS can be obtained by taking the ratio of the partial derivatives:

MRS = ∂U/∂x / ∂U/∂y = (y/x)

For the utility function U(x,y) = (x+y), the partial derivatives are:

∂U/∂x = 1

∂U/∂y = 1

Therefore, the MRS for this utility function is:

MRS = ∂U/∂x / ∂U/∂y = 1/1 = 1