Question

Consider the following utility functions: a. \(U(x, y)=x y\) b. \(U(x, y)=x^{2} y^{2}\) c. \(U(x, y)=\ln x+\ln y\) 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?

Short answer

Answer: All three utility functions have a diminishing MRS. However, they exhibit different types of marginal utility - constant for the first function, increasing for the second function, and decreasing for the third function.

Step-by-step solution

Utility Function 1 - \(U(x, y)=x y\)

Find the partial derivatives: $$\frac{\partial U}{\partial x} = y$$ $$\frac{\partial U}{\partial y} = x$$ Compute the MRS: $$MRS = \frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}} = \frac{y}{x}$$ The MRS decreases as the value of x increases, which shows that it has a diminishing MRS. The second-order partial derivatives are constant in this case, so the marginal utility is constant.

Utility Function 2 - \(U(x, y)=x^{2} y^{2}\)

Find the partial derivatives: $$\frac{\partial U}{\partial x} = 2xy^{2}$$ $$\frac{\partial U}{\partial y} = 2x^{2}y$$ Compute the MRS: $$MRS = \frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}} = \frac{2xy^{2}}{2x^{2}y} = \frac{y}{x}$$ The MRS decreases as the value of x increases, which shows that it has a diminishing MRS. The second-order partial derivatives become larger as the values of x and y increase, so the marginal utility is increasing.

Utility Function 3 - \(U(x, y)=\ln x+\ln y\)

Find the partial derivatives: $$\frac{\partial U}{\partial x} = \frac{1}{x}$$ $$\frac{\partial U}{\partial y} = \frac{1}{y}$$ Compute the MRS: $$MRS = \frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}} = \frac{\frac{1}{x}}{\frac{1}{y}} = \frac{y}{x}$$ The MRS decreases as the value of x increases, which shows that it has a diminishing MRS. The second-order partial derivatives become smaller as the values of x and y increase, so the marginal utility is decreasing. From the analysis above, we can conclude that all three utility functions have a diminishing MRS, but they exhibit different types of marginal utility - constant, increasing, and decreasing, respectively.