Question

Two goods have independent marginal utilities if \\[ \frac{d t U}{d Y d X} \quad \frac{d^{2} U}{d X d Y}=0 \\] Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing \(M R S\). Provide an example to show that the converse of this statement is not true. 3.10 a. Show that the CES function \\[ \begin{array}{c} x^{3} \\ 0 \end{array}+* \frac{Y}{T} \\] is homothetic. How does the \(M R S\) depend on the ratio \(\mathrm{F} / \mathrm{X}\) ? b. Show that your results from part (a) agree with Example 3.3 for the case \(6=1\) (perfect substitutes \()\) and \(5=0\) (Cobb-Douglas). c. Show that the \(M R S\) is strictly diminishing for all values of \(8<1\) d. Show that if \(X=Y\), the \(M R S\) for this function depends only on the relative sizes of a and j8. e. Calculate the \(M R S\) for this function when \(Y / X=.9\) and \(Y / X=1.1\) for the two cases \(8=5\) and \(8=-1 .\) What do you conclude about the extent to which the \(M R S\) changes in the vicinity of \(\mathrm{X}=\mathrm{F} ?\) How would you interpret this geometrically?

Short answer

Question: Explain the relationship between diminishing marginal utility and diminishing MRS, and provide an example showing the converse of the statement is not always true. Answer: Diminishing marginal utility implies that the additional satisfaction derived from consuming more of a good decreases as consumption increases. When the marginal utilities of two goods are independent, diminishing marginal utility results in a diminishing MRS - the rate at which a consumer is willing to substitute one good for another. Specifically, if the second derivative of MRS is positive, this implies diminishing MRS. However, the converse of the statement is not always true. An example can be seen in the utility function U(X, Y) = X^2 + Y^2, where the MRS is diminishing, but the marginal utilities are not independent.

Step-by-step solution

Diminishing marginal utility relationship to MRS

Starting from the given condition of independent marginal utilities: \[ \frac{d t U}{d Y d X} \frac{d^{2} U}{d X d Y}=0 \] Let's denote marginal utilities as: \[ MU_X = \frac{dU}{dX}, MU_Y = \frac{dU}{dY} \] Then, according to the condition above, we can rewrite it as: \[ \frac{dMU_Y}{dX} MU_X = 0 \] Diminishing marginal utility implies \( \frac{dMU_X}{dX} < 0 \) and \( \frac{dMU_Y}{dY} < 0 \) We can find the MRS as follows: \[ MRS = \frac{MU_X}{MU_Y} \] Now, we need to find the second derivative of MRS with respect to X and show that it is positive, which implies diminishing MRS: \[ \frac{d MRS}{dX} = \frac{\frac{dMU_X}{dX} \cdot MU_Y - MU_X\frac{dMU_Y}{dX}}{(MU_Y)^2} \] Given that independent marginal utilities condition: \(\frac{dMU_Y}{dX} = MU_X = 0\), we have: \[ \frac{d MRS}{dX} = \frac{\frac{dMU_X}{dX} \cdot MU_Y}{(MU_Y)^2} \] Now, we find the second derivative of MRS with respect to X: \[ \frac{d^2 MRS}{dX^2} = \frac{(\frac{d^2MU_X}{dX^2} \cdot MU_Y) \cdot (MU_Y)^2 - (\frac{dMU_X}{dX} \cdot MU_Y)^2}{(MU_Y)^4} \] Since diminishing marginal utility for X implies that \(\frac{dMU_X}{dX} < 0\) and \(\frac{d^2MU_X}{dX^2} \geq 0\), we can conclude that \(\frac{d^2 MRS}{dX^2} \geq 0\). Therefore, the MRS will be diminishing, proving the initial claim. 2. Converse of the statement not true (example)

Example of converse statement not being true

Consider the following utility function: \[ U(X, Y) = X^2 + Y^2 \] Marginal utilities are: \[ MU_X = \frac{dU}{dX} = 2X, \ MU_Y = \frac{dU}{dY} = 2Y \] And MRS is: \[ MRS = \frac{MU_X}{MU_Y} = \frac{2X}{2Y} = \frac{X}{Y} \] Now we'll show that it has diminishing MRS, but doesn't have independent marginal utilities. First, let's find the second derivative of MRS with respect to X: \[ \frac{d^2 MRS}{dX^2} = \frac{d}{dX} \left( \frac{Y}{X^2} \right) = -\frac{2Y}{X^3} < 0 \] Since \(\frac{d^2 MRS}{dX^2} < 0\), the MRS is diminishing. However, the marginal utilities are not independent: \[ \frac{dMU_Y}{dX} = \frac{d(2Y)}{dX} = 0, \ \frac{dMU_X}{dY} = \frac{d(2X)}{dY} = 0 \] \[ \frac{dMU_Y}{dX} \cdot \frac{dMU_X}{dY} = 0 \cdot 0 \neq 0 \] This counterexample shows that the converse of the statement is not true, as even if MRS is diminishing, the marginal utilities might not be independent.

Key concepts

Marginal Rate of Substitution

Understanding the marginal rate of substitution (MRS) is key to grasping consumer behavior. The MRS measures how much of one good a consumer is willing to give up to gain an additional unit of another good, while keeping their overall utility unchanged. This is mathematically represented as \( MRS = \frac{MU_X}{MU_Y} \), where \( MU_X \) and \( MU_Y \) are the marginal utilities of goods X and Y respectively.
In economic terms, the MRS is the slope of the indifference curve at any given point, reflecting the consumer's willingness to substitute between two goods. When dealing with utility functions, such as the CES function, understanding the MRS helps in determining how substitution preferences change with varying quantities of goods.
It is important to highlight that the MRS usually diminishes as we consume more of one good relative to another. This is because, typically, the marginal utility decreases as more of a good is consumed, which reflects the principle of diminishing returns.

CES Function

The Constant Elasticity of Substitution (CES) function is a key concept in economics used to represent utility or production with a specific feature: the elasticity of substitution remains constant. A CES function is generally expressed as \( U(X, Y) = (aX^{-\rho} + bY^{-\rho})^{-\frac{1}{\rho}} \), where parameters \( a \) and \( b \) are positive, and \( \rho \) affects the degree of substitutability between X and Y.
The CES function is popular for analyzing situations where goods can be substituted for each other at a constant rate, providing a flexibility that other functions do not have. For instance, when \( \rho = 0 \), the function represents perfect substitutes, resulting in a linear indifference curve. Conversely, when \( \rho < 0 \), it represents goods that are less easily substituted, often taking a Cobb-Douglas form in economic models.
Learning about the CES function aids in understanding how changes in relative prices can influence optimal consumption bundles, which is crucial in many economic analyses.

Homothetic Functions

Homothetic functions are a class of functions where utility or production levels are multiplied by a scalar factor when all inputs are increased by the same factor. Simply put, a function is homothetic if consumer preferences are consistent, maintaining the same rate of substitution between goods, regardless of the level of consumption.
Mathematically, for a utility function \( U(X, Y) \), if for any positive scalar \( \lambda \), \( U(\lambda X, \lambda Y) = \lambda^k U(X, Y) \), the function is homothetic when \( k = 1 \). This indicates that an increase in inputs leads to proportional increases in utility, maintaining identical indifference curves at all scales.
Homotheticity is vital in simplifying economic models, as it allows for the preservation of consumer preferences regardless of income changes. This makes it easier to predict consumer behavior, as variations in consumption patterns are systematically proportionate.

Diminishing MRS

Diminishing Marginal Rate of Substitution is an essential concept in understanding consumer preferences. The principle states that as a consumer substitutes one good for another, the amount of the first good they're willing to give up to acquire additional units of the second good decreases.
This diminishing MRS is closely tied to the concept of diminishing marginal utility, whereby the additional satisfaction (or utility) gained from consuming one more unit of a good decreases as consumption increases. It reflects the consumer's decreasing willingness to sacrifice units of one good in exchange for another as they accumulate more of a given good.
The concept of diminishing MRS is graphically depicted by the curvature of the indifference curve, which becomes flatter as one moves down the curve. This curvature indicates a decrease in the rate of substitution and is a fundamental concept in consumer theory that underscores rational consumer choices.