Consider the following utility functions: a \(U(x, y)=x y\) \(U(x, y)=x^{2} y^{2}\) \(c(x, y)=\ln x+\ln y\) Show that each of these has a diminishing \(M R S\) but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?

To calculate the MRS, we will find the partial derivatives of each utility function with respect to x and y. Then, we will find the ratio of these partial derivatives. a) For the first utility function (\(U(x, y)=xy\)), we have: \(MU_x = \frac{\partial U}{\partial x} = y \text{ and } MU_y = \frac{\partial U}{\partial y} = x\) \(MRS = -\frac{MU_x}{MU_y} = -\frac{y}{x}\) b) For the second utility function (\(U(x, y)=x^2y^2\)), we have: \(MU_x = \frac{\partial U}{\partial x} = 2x y^2 \text{ and } MU_y = \frac{\partial U}{\partial y} = 2x^2 y\) \(MRS = -\frac{MU_x}{MU_y} = -\frac{2 x y^2}{2 x^2 y} = -\frac{y}{x}\) c) For the third utility function (\(U(x, y)=ln(x) + ln(y)\)), we have: \(MU_x = \frac{\partial U}{\partial x} = \frac{1}{x} \text{ and } MU_y = \frac{\partial U}{\partial y} = \frac{1}{y}\) \(MRS = -\frac{MU_x}{MU_y} = -\frac{\frac{1}{x}}{\frac{1}{y}} = -\frac{y}{x}\)

For all three utility functions, we see that MRS has the same form, \(MRS = -\frac{y}{x}\). Since the MRS is decreasing in x and increasing in y, we can conclude that all three utility functions have a diminishing MRS.

We have already found the marginal utility of each good (x and y) for each utility function in Step 1.

a) For the first utility function (\(U(x, y)=xy\)): The marginal utility of Good x is constant with respect to x (\(MU_x = y\)), and the marginal utility of Good y is constant with respect to y (\(MU_y = x\)). Therefore, this utility function exhibits constant marginal utility. b) For the second utility function (\(U(x, y)=x^2y^2\)): The marginal utility of Good x is increasing with respect to x (\(MU_x = 2x y^2\)), and the marginal utility of Good y is increasing with respect to y (\(MU_y = 2x^2 y\)). Therefore, this utility function exhibits increasing marginal utility. c) For the third utility function (\(U(x, y)=ln(x) + ln(y)\)): The marginal utility of Good x is decreasing with respect to x (\(MU_x = \frac{1}{x}\)), and the marginal utility of Good y is decreasing with respect to y (\(MU_y = \frac{1}{y}\)). Therefore, this utility function exhibits decreasing marginal utility.

In conclusion, all three utility functions have diminishing MRS. However, they exhibit different types of marginal utility: the first utility function has constant marginal utility, the second utility function has increasing marginal utility, and the third utility function has decreasing marginal utility. This shows that utility functions can have a diminishing MRS while still exhibiting different types of marginal utility.