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.

Key concepts

Utility Functions

Utility functions are mathematical representations that help us understand consumer preferences. They help to quantify how different combinations of goods provide satisfaction, or utility, to a consumer. Typically, the goal is to maximize this utility given budget constraints. They are fundamental in forming models of consumer choices in economics.
For instance, let’s consider a utility function of the form \(U=3X+Y\). This function suggests that the utility or satisfaction of a consumer increases with additional units of goods \(X\) and \(Y\), with \(X\) being weighted three times more heavily than \(Y\). The function dictates a linear relationship, implying constant returns to the consumption of \(X\) and \(Y\).
However, not all utility functions are linear. Take, for example, \(U=\sqrt{X \cdot Y}\). This function depicts a Cobb-Douglas-type utility, representing a multiplicative interaction between \(X\) and \(Y\), showcasing diminishing returns as more of a single good is consumed. Different forms of utility functions can illustrate different consumer behaviors and preferences as highlighted by their respective shapes on a graph. Understanding these differences is key to analyzing consumer decision-making.

Marginal Rate of Substitution (MRS)

The Marginal Rate of Substitution (MRS) defines the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility. It is a crucial concept in understanding indifference curves, which graphically represent combinations of goods that provide the same satisfaction level.
The MRS is calculated as the negative ratio of marginal utilities, that is, \(-\frac{MU_X}{MU_Y}\). For instance, given a utility function where \(U=3X+Y\), the MRS is constant at \(-3\). This suggests a fixed trade-off between \(X\) and \(Y\), illustrating a linear indifference curve.
In contrast, a utility function like \(U=\sqrt{X \cdot Y}\) results in \(MRS=-\frac{Y}{X}\). This MRS changes depending on \(X\) and \(Y\); indicating that as \(Y\) decreases relative to \(X\), the willingness to substitute \(X\) for \(Y\) decreases. This change in MRS reflects more typical consumer behavior where the MRS is diminishing- a characteristic of convex indifference curves. Understanding MRS helps in predicting how changes in price of goods will affect consumer behavior and market demand.

Convexity in Economics

In economics, convexity usually refers to the shape of an indifference curve, which plays a significant role in demonstrating consistent consumer preferences. Convex indifference curves depict diminishing MRS, implying that as a consumer has more of one good, they require increasingly larger amounts of the other good to maintain the same level of utility.
For instance, a utility function like \(U=\sqrt{X \cdot Y}\) will generate convex indifference curves, meaning they bow inward. These curves suggest that the more you have of one good (say \(X\)), the smaller the amount of the other good (\(Y\)) you are willing to part with to maintain satisfaction. This behavior is quite typical in real life, as we often see diminishing marginal utility with increased consumption.
This contrasts with linear indifference curves, such as those derived from \(U=3X+Y\), which are not convex. Rather, these linear forms imply a constant MRS and do not demonstrate the typical diminishing returns usually observed in consumer behavior.
Convexity is a central assumption in economics, underscoring predictable, rational consumer behavior which is essential for constructing various economic models and market analyses.