Question

Consider the following utility functions: a. \(U(X, Y)=X Y\) b. \(U(X, Y)=X^{2} Y^{2}\) \(c \quad U(X, Y)=\ln X+\operatorname{In} Y\) 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, all three utility functions exhibit diminishing Marginal Rate of Substitution. However, they show different patterns in terms of marginal utility: constant for function a, increasing for function b, and decreasing for function c.

Step-by-step solution

Calculate the Marginal Rate of Substitution (MRS)

The MRS is the rate at which a consumer is willing to trade off one good for another. It is given by the ratio of marginal utilities, i.e., $$MRS = -\frac{MU_x}{MU_y}.$$ We will calculate the marginal utilities of each utility function, then find the MRS.

Utility Function a: \(U(X, Y)=XY\)

First, calculate the marginal utilities for this utility function: $$MU_x = \frac{\partial U}{\partial X} = Y$$ $$MU_y = \frac{\partial U}{\partial Y} = X$$ Next, find the MRS: $$MRS = -\frac{MU_x}{MU_y} = -\frac{Y}{X}$$ Since \(X > 0\) and \(Y > 0\), the MRS is negative but diminishing as X and Y increase, as required. The marginal utility for goods X and Y is constant because they do not depend on the consumption of the other good.

Utility Function b: \(U(X, Y)=X^2 Y^2\)

First, calculate the marginal utilities for this utility function: $$MU_x = \frac{\partial U}{\partial X} = 2X Y^2$$ $$MU_y = \frac{\partial U}{\partial Y} = 2X^2 Y$$ Next, find the MRS: $$MRS = -\frac{MU_x}{MU_y} = -\frac{2XY^2}{2X^2Y} = -\frac{Y}{X}$$ The MRS is once again negative and diminishing for this utility function. However, since marginal utilities are increasing in both X and Y this time, we have increasing marginal utility.

Utility Function c: \(U(X, Y)=\ln X+\ln Y\)

First, calculate the marginal utilities for this utility function: $$MU_x = \frac{\partial U}{\partial X} = \frac{1}{X}$$ $$MU_y = \frac{\partial U}{\partial Y} = \frac{1}{Y}$$ Next, find the MRS: $$MRS = -\frac{MU_x}{MU_y} = -\frac{\frac{1}{X}}{\frac{1}{Y}} = -\frac{Y}{X}$$ The MRS is once again negative and diminishing for this utility function. However, since marginal utilities are decreasing in both X and Y, we have decreasing marginal utility.

Conclusion

Based on the analysis above, we can conclude that all three utility functions exhibit diminishing MRS. However, they exhibit different patterns in terms of marginal utility: constant for function a, increasing for function b, and decreasing for function c. This result shows that the relationship between MRS and marginal utility can be different for different types of utility functions, with varying degrees of responsiveness to changes in the consumption of goods.

Key concepts

Utility Functions

In the realm of economics, utility functions provide a mathematical representation of consumers' preferences.
These functions are pivotal in determining how individuals prioritize and choose among various bundles of goods.
For example, utility functions can take on different forms to capture varying behaviors, such as linear, quadratic, or logarithmic. Each form corresponds to specific assumptions about consumers' preferences.

• Linear Utility Function: This exhibits constant Marginal Utility (MU), reflected in utility function (a) in the exercise, where the utility increases at a constant rate as consumption of either good increases.
• Quadratic Utility Function: In function (b), the utility scales quadratically with consumption, indicating a higher level of satisfaction with more consumption, translating into increasing MU.
• Logarithmic Utility Function: Function (c), on the other hand, is characterized by decreasing MU, meaning additional units of consumption bring progressively less satisfaction.

Understanding the nature of utility functions is fundamental to grasp higher-level concepts such as Marginal Rate of Substitution (MRS).

Marginal Utility

Marginal Utility (MU) is a concept that refers to the additional satisfaction, or utility, a consumer gains from consuming one more unit of a good or service. It's an incremental measure that reflects the benefit derived from a small change in consumption.

Calculated Marginality

Regarding the exercise, MU is calculated by taking partial derivatives of the utility function with respect to goods X and Y. For instance, function (a) has a constant MU for both goods, while function (b) demonstrates increasing MU, and function (c) shows decreasing MU.

Changing Satisfaction

The concept of MU is crucial because it embodies the principle of diminishing returns—additional units of a good typically enhance utility at a decreasing rate. This decreasing MU aligns with everyday experience; the first slice of pizza can be incredibly satisfying, but the satisfaction from each additional slice tends to be smaller. Understanding MU is essential for comprehending how consumers make trade-offs between goods.

Diminishing MRS

The Marginal Rate of Substitution (MRS) reflects a consumer's willingness to trade one good for another while maintaining the same level of overall satisfaction.
It's important to note, as shown in the exercise solutions, that MRS diminishes as the consumer substitues more of one good for the other. This typical behavior follows the law of diminishing MRS, which is a cornerstone of consumer choice theory.

Understanding Trade-Offs

If you consider drinking water to quench your thirst, the first few sips have a high substitution rate for any other drink because they provide significant relief. As you continue to drink, your willingness to substitute those sips for something else decreases since your thirst is largely quenched. This is the manifestation of the diminishing MRS in real life. Each of the provided utility function examples in the exercise corroborates that, despite the type of marginal utility involved—constant, increasing, or decreasing—the diminishing nature of the MRS remains a consistent property.