Consider the following utility functions: a. \(\quad U(X, Y)=X Y\) b. \(U(X, Y)=X^{2} Y^{2}\) c. \(\quad U(X, Y)=\ln X+\ln Y\) 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 find the marginal utilities of each good, we must take the partial derivatives of the utility functions (U) with respect to goods X and Y. For utility function a, \(U(X, Y)=XY\): \(\frac{\partial U}{\partial X}=Y\) \(\frac{\partial U}{\partial Y}=X\) For utility function b, \(U(X, Y)=X^{2}Y^{2}\): \(\frac{\partial U}{\partial X}=2X^{1}Y^{2}\) \(\frac{\partial U}{\partial Y}=X^{2}2Y^{1}\) For utility function c, \(U(X, Y)=\ln X+\ln Y\): \(\frac{\partial U}{\partial X}=\frac{1}{X}\) \(\frac{\partial U}{\partial Y}=\frac{1}{Y}\)

The MRS is the negative ratio of the two marginal utilities. For each utility function, we will find the MRS: For utility function a: \(MRS = -\frac{\frac{\partial U}{\partial X}}{\frac{\partial U}{\partial Y}} = -\frac{Y}{X}\) For utility function b: \(MRS = -\frac{\frac{\partial U}{\partial X}}{\frac{\partial U}{\partial Y}} = -\frac{2X}{2Y}\) For utility function c: \(MRS = -\frac{\frac{\partial U}{\partial X}}{\frac{\partial U}{\partial Y}} = -\frac{1/X}{1/Y}\)

For each utility function, we will check if the MRS is diminishing: For utility function a: MRS a \(= -\frac{Y}{X}\) is diminishing since the ratio of Y to X decreases as X increases. For utility function b: MRS b \(= -\frac{2X}{2Y}\) is diminishing since the ratio of X to Y decreases as X increases. For utility function c: MRS c \(= -\frac{1/X}{1/Y}\) is diminishing since the ratio of 1/X to 1/Y decreases as X increases.

We will now analyse the marginal utilities to determine if they are constant, increasing, or decreasing for each utility function: For utility function a: \(\frac{\partial U}{\partial X}=Y\), This marginal utility is constant with respect to X. \(\frac{\partial U}{\partial Y}=X\), This marginal utility is constant with respect to Y. For utility function b: \(\frac{\partial U}{\partial X}=2X^{1}Y^{2}\), This marginal utility is increasing with respect to X. \(\frac{\partial U}{\partial Y}=X^{2}2Y^{1}\), This marginal utility is increasing with respect to Y. For utility function c: \(\frac{\partial U}{\partial X}=\frac{1}{X}\), This marginal utility is decreasing with respect to X. \(\frac{\partial U}{\partial Y}=\frac{1}{Y}\), This marginal utility is decreasing with respect to Y. #Conclusion# In conclusion, all three given utility functions exhibit a diminishing MRS. However, they have different types of marginal utility. Utility function a has constant marginal utility, utility function b has increasing marginal utility, and utility function c has decreasing marginal utility.