Question

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?

Short answer

Answer: Yes, utility functions can exhibit diminishing Marginal Rate of Substitution while still displaying different types of marginal utility. As demonstrated in the examples, 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. All three utility functions have diminishing MRS.

Step-by-step solution

Calculate the MRS for each utility function

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: \(MU_x = \frac{\partial U}{\partial x} = \frac{1}{x} \text{ and } MU_y = \frac{\partial U}{\partial y} = \frac{1}{y}\) \(MRS = -\frac{MU_x}{MU_y} = -\frac{\frac{1}{x}}{\frac{1}{y}} = -\frac{y}{x}\)

Describe the MRS behavior

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.

Calculate the marginal utility

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

Describe the behavior of marginal utility

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.

State the findings and conclusion

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.

Key concepts

Marginal Rate of Substitution (MRS)

The Marginal Rate of Substitution (MRS) is a fundamental concept in economics that expresses the trade-off rate between two goods for a consumer who maintains the same level of utility. Simply put, it tells us how much of one good a person is willing to give up to gain more of another good, while remaining equally satisfied.

The calculation of MRS involves finding the ratio of the marginal utilities of the two goods. In mathematical terms, if we denote the utility function as \( U(x, y) \), the MRS is given by the ratio \( -\frac{MU_x}{MU_y} \), where \( MU_x \) and \( MU_y \) are the marginal utilities of goods \( x \) and \( y \), respectively. It's important to note that the MRS is often negative, indicating that as we substitute one good for another, there is a loss of one to gain the other.

• For the utility function \( U(x, y) = xy \), the MRS is \( -\frac{y}{x} \)
• For \( U(x, y) = x^2y^2 \), the MRS remains \( -\frac{y}{x} \)
• And for \( U(x, y) = \ln(x) + \ln(y) \), the MRS is also \( -\frac{y}{x} \)

This consistency in form across these functions illustrates that despite different functional forms, a decreasing MRS can be present, indicating that with more of good \( x \), a consumer is willing to give up less of good \( y \).

Marginal Utility

Marginal utility reflects the additional satisfaction a consumer gains from consuming one additional unit of a good. In economics, understanding marginal utility helps explain consumer choices and demand curves, as it reveals how values change when consumption alters.

Calculating marginal utility involves taking the derivative of the utility function with respect to the good in question. For instance, if the utility function is \( U(x, y) = xy \), the marginal utility of \( x \) is expressed as \( MU_x = \frac{\partial U}{\partial x} = y \), indicating that it's constant in this context. Each type of utility function demonstrates differing behaviors of marginal utility:

• For \( U(x, y) = xy \), both \( MU_x = y \) and \( MU_y = x \) are constant.
• For \( U(x, y) = x^2y^2 \), \( MU_x = 2xy^2 \) and \( MU_y = 2x^2y \) increase with larger \( x \) and \( y \) values.
• For \( U(x, y) = \ln(x) + \ln(y) \), both \( MU_x = \frac{1}{x} \) and \( MU_y = \frac{1}{y} \) decrease as \( x \) and \( y \) grow.

These variations illustrate how utility functions can deliver a constant, increasing, or decreasing marginal utility, impacting decision-making processes.

Economic Theory

Economic theory often makes use of utility functions and concepts like MRS and marginal utility to analyze and predict consumer behavior. By examining how individuals make choices given their preferences and constraints, economists can derive conclusions about market dynamics and individual decision-making.

Utility functions like \( U(x, y) = xy \), \( x^2y^2 \), and \( \ln(x) + \ln(y) \) embody varied forms of preference expressions. They offer insights into how changes in quantity consumed influence perceived satisfaction levels. In economics, understanding diminishing MRS and different types of marginal utility sheds light on how consumers shift consumption patterns and optimize satisfaction.

• Diminishing MRS across different utility functions suggests a common consumer behavior trait - willingness to trade less of one good for another as more of it is consumed.
• Different marginal utility behaviors highlight the importance of recognizing how satisfaction changes with consumption, affecting demand.

Together, these components of economic theory enhance our understanding of consumer choice and market behavior, forming a backbone for further exploration of economic models and real-world applications.