Show that the following utility functions have decreasing MRS and explain their marginal utility • U (x, y) = xy • U (x, y) = x2y2 • U (x, y) = ln x + ln y

To show that a utility function has a decreasing Marginal Rate of Substitution (MRS), we need to calculate the MRS and show that it decreases as consumption increases.

1. For the utility function U(x, y) = xy:
The marginal utility of x is \frac{\partial U}{\partial x} = y and the marginal utility of y is \frac{\partial U}{\partial y} = x.

The Marginal Rate of Substitution (MRS) is given by:
MRS = \frac{MU_x}{MU_y} = \frac{y}{x}

As consumption increases, the MRS = \frac{y}{x} will decrease.

2. For the utility function U(x, y) = x^2y^2:
The marginal utility of x is \frac{\partial U}{\partial x} = 2xy^2 and the marginal utility of y is \frac{\partial U}{\partial y} = 2x^2y.

The MRS is:
MRS = \frac{MU_x}{MU_y} = \frac{2xy^2}{2x^2y} = \frac{y}{x}

Just like in the previous case, as consumption increases, the MRS = \frac{y}{x} will decrease.

3. For the utility function U(x, y) = \ln x + \ln y:
The marginal utility of x is \frac{\partial U}{\partial x} = \frac{1}{x} and the marginal utility of y is \frac{\partial U}{\partial y} = \frac{1}{y}.

The MRS is:
MRS = \frac{MU_x}{MU_y} = \frac{\frac{1}{x}}{\frac{1}{y}} = \frac{y}{x}

Just like in the previous cases, as consumption increases, the MRS = \frac{y}{x} will decrease.

\boxed{\text{Answer: The utility functions provided have decreasing MRS as consumption increases.}}