Question

Consider the following utility functions: a. \(\quad U(X, Y)=X Y\) b. \(U(X, Y)=X^{2} Y^{2}\) c. \(\quad U(X, Y)=\ln X+\ln Y\) 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

Based on the analysis conducted, each utility function exhibits the following characteristics: 1. Utility function a: Diminishing MRS and constant marginal utility. 2. Utility function b: Diminishing MRS and increasing marginal utility. 3. Utility function c: Diminishing MRS and decreasing marginal utility.

Step-by-step solution

Calculate the Marginal Utilities of Good X and Good Y

To find the marginal utilities of each good, we must take the partial derivatives of the utility functions (U) with respect to goods X and Y. For utility function a, \(U(X, Y)=XY\): \(\frac{\partial U}{\partial X}=Y\) \(\frac{\partial U}{\partial Y}=X\) For utility function b, \(U(X, Y)=X^{2}Y^{2}\): \(\frac{\partial U}{\partial X}=2X^{1}Y^{2}\) \(\frac{\partial U}{\partial Y}=X^{2}2Y^{1}\) For utility function c, \(U(X, Y)=\ln X+\ln Y\): \(\frac{\partial U}{\partial X}=\frac{1}{X}\) \(\frac{\partial U}{\partial Y}=\frac{1}{Y}\)

Calculate the Marginal Rate of Substitution (MRS)

The MRS is the negative ratio of the two marginal utilities. For each utility function, we will find the MRS: For utility function a: \(MRS = -\frac{\frac{\partial U}{\partial X}}{\frac{\partial U}{\partial Y}} = -\frac{Y}{X}\) For utility function b: \(MRS = -\frac{\frac{\partial U}{\partial X}}{\frac{\partial U}{\partial Y}} = -\frac{2X}{2Y}\) For utility function c: \(MRS = -\frac{\frac{\partial U}{\partial X}}{\frac{\partial U}{\partial Y}} = -\frac{1/X}{1/Y}\)

Determine if the Marginal Rate of Substitution is Diminishing

For each utility function, we will check if the MRS is diminishing: For utility function a: MRS a \(= -\frac{Y}{X}\) is diminishing since the ratio of Y to X decreases as X increases. For utility function b: MRS b \(= -\frac{2X}{2Y}\) is diminishing since the ratio of X to Y decreases as X increases. For utility function c: MRS c \(= -\frac{1/X}{1/Y}\) is diminishing since the ratio of 1/X to 1/Y decreases as X increases.

Determine if the Marginal Utilities are Constant, Increasing, or Decreasing

We will now analyse the marginal utilities to determine if they are constant, increasing, or decreasing for each utility function: For utility function a: \(\frac{\partial U}{\partial X}=Y\), This marginal utility is constant with respect to X. \(\frac{\partial U}{\partial Y}=X\), This marginal utility is constant with respect to Y. For utility function b: \(\frac{\partial U}{\partial X}=2X^{1}Y^{2}\), This marginal utility is increasing with respect to X. \(\frac{\partial U}{\partial Y}=X^{2}2Y^{1}\), This marginal utility is increasing with respect to Y. For utility function c: \(\frac{\partial U}{\partial X}=\frac{1}{X}\), This marginal utility is decreasing with respect to X. \(\frac{\partial U}{\partial Y}=\frac{1}{Y}\), This marginal utility is decreasing with respect to Y. #Conclusion# In conclusion, all three given utility functions exhibit a diminishing MRS. However, they have different types of marginal utility. Utility function a has constant marginal utility, utility function b has increasing marginal utility, and utility function c has decreasing marginal utility.

Key concepts

Marginal Rate of Substitution

The Marginal Rate of Substitution (MRS) is a key concept in understanding how consumers make choices between different goods. It represents the rate at which a consumer can give up some amount of one good in exchange for another good while keeping their overall utility constant. In mathematical terms, the MRS is the negative ratio of the marginal utility of one good to the marginal utility of another. Let's break it down further.

For each utility function given, we calculate the MRS by taking the negative ratio of the partial derivative of the utility function with respect to X and Y.

• For utility function a, \(MRS = -\frac{Y}{X}\).
• For utility function b, \(MRS = -\frac{2X}{2Y} = -\frac{X}{Y}\), showing a similar relationship.
• For utility function c, \(MRS = -\frac{1/X}{1/Y} = -\frac{Y}{X}\).

Each MRS tells us how willing a consumer would be to trade one good for another, illustrating their preferences and ensuring that the utility level remains unchanged.

Diminishing MRS

The idea of a diminishing MRS involves the principle that as a consumer continues to consume more of one good, they are willing to give up less and less of the other good to maintain the same level of utility. This reflects the idea of indifference curves, which become flatter as one moves down them.

Applying this to the utility functions, you can check that each has a diminishing MRS. As you consume more of good X, the quantity of Y you're willing to trade for additional units of X decreases.

• For utility function a, \(MRS = -\frac{Y}{X}\), it's clear that as X increases, the MRS decreases.
• Similarly, for utility function b, \(MRS = -\frac{X}{Y}\), increasing X will decrease MRS.
• Utility function c with \(MRS = -\frac{Y}{X}\) follows the same logic.

These findings show that all three functions align with the principle of diminishing MRS, supporting the idea that consumers prefer balanced combinations over extremes.

Utility Functions

Utility functions are mathematical representations of a consumer's preferences, showing the satisfaction or happiness derived from consuming a bundle of goods. They allow us to explore concepts like MRS and marginal utility, providing a framework to analyze consumer behavior.

Here's a closer look at the given utility functions:

• Function a: \(U(X, Y) = XY\), represents a direct relationship between X and Y, leading to constant marginal utility. The satisfaction derived from consuming either product does not change with quantity.
• Function b: \(U(X, Y) = X^2Y^2\), leading to increasing marginal utility for both goods. More consumption of either good yields proportionally more satisfaction.
• Function c: \(U(X, Y) = \ln X + \ln Y\), shows decreasing marginal utility. As you consume more of either good, the additional satisfaction gained decreases.

These different forms offer insights into how different functions can represent varying consumer preferences and experiences of satisfaction, highlighting the rich diversity of economic behavior.