Question

Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether they obey the assumption of a diminishing \(M R S\) ): a. \(U=3 X+Y\) b. \(U=\sqrt{X \cdot Y}\) \(\mathbf{c}, \quad U=\sqrt{X^{2}+Y^{2}}\) \(\mathrm{d} . \quad U=\sqrt{X^{2}-Y^{2}}\) \(\mathbf{e}, \quad U=X^{2 / 3} Y^{1 / 3}\) \(f_{.} \quad U=\log X+\log Y\).

Short answer

a. \(U=3X+Y\) b. \(U=\sqrt{X\cdot Y}\) c. \(U=\sqrt{X^2+Y^2}\) d. \(U=\sqrt{X^2-Y^2}\) e. \(U=X^{2/3}Y^{1/3}\) f. \(U=\log X+\log Y\) Answer: The utility functions that generate convex indifference curves are b, c, e, and f. Utility function a does not generate convex indifference curves because the MRS is constant. Utility function d does not generate convex indifference curves due to the utility function being undefined in certain regions (e.g., when \(X < Y\)).

Step-by-step solution

Find the equation of the indifference curve

To find the equation of the indifference curve, we need to set the utility function equal to a constant level of utility (\(U_0\)): \(U = U_0 = 3X + Y\)

Compute the MRS

The MRS is given by the negative ratio of marginal utilities. The marginal utility of X (\(MU_X\)) and Y (\(MU_Y\)) can be found by taking the partial derivatives with respect to X and Y. \(MU_X = \frac{\partial U}{\partial X} = 3\) \(MU_Y = \frac{\partial U}{\partial Y} = 1\) \(MRS = -\frac{MU_X}{MU_Y} = -\frac{3}{1} = -3\)

Analyze the MRS

Since the MRS is a constant, it does not show diminishing MRS. Therefore, the indifference curve for this utility function is not convex. #b. \(U=\sqrt{X\cdot Y}\)#

Find the equation of the indifference curve

Set the utility function equal to a constant level of utility (\(U_0\)): \(U = U_0 = \sqrt{X\cdot Y}\)

Compute the MRS

Find the marginal utilities of X and Y: \(MU_X = \frac{\partial U}{\partial X} = \frac{Y}{2\sqrt{X\cdot Y}}\) \(MU_Y = \frac{\partial U}{\partial Y} = \frac{X}{2\sqrt{X\cdot Y}}\) \(MRS = -\frac{MU_X}{MU_Y} = -\frac{Y}{X}\)

Analyze the MRS

The MRS depends on the ratio of Y to X, which means that it is diminishing as more units of X are consumed. Therefore, the indifference curve for this utility function is convex. Keep following the same steps for the remaining utility functions to determine if their indifference curves are convex: #c. \(U=\sqrt{X^2+Y^2}\)# Indifference curve: \(U = U_0 = \sqrt{X^2 + Y^2}\) MRS: \(MRS = -\frac{X}{Y}\) Conclusion: Convex. #d. \(U=\sqrt{X^2-Y^2}\)# Indifference curve: \(U = U_0 = \sqrt{X^2 - Y^2}\) MRS: Undefined, as there are regions where the utility function is not valid (e.g., when \(X < Y\)). Conclusion: This utility function does not generate convex indifference curves. #e. \(U=X^{2/3}Y^{1/3}\)# Indifference curve: \(U = U_0 = X^{2/3}Y^{1/3}\) MRS: \(MRS = -\frac{2Y}{3X}\) Conclusion: Convex. #f. \(U=\log X+\log Y\)# Indifference curve: \(U = U_0 = \log X + \log Y\) MRS: \(MRS = -\frac{Y}{X}\) Conclusion: Convex.