Question

A formal definition of what we have been calling the substitution elasticity is $$_{a}-d \frac{\operatorname{din} Y / X}{d(\ln M R S)}-\frac{/ \operatorname{din} M R S}{\sim \operatorname{din} Y / X} |-\mathrm{i}$$ a. Interpret this as an elasticity-what variables are being changed and how do these changes (in proportional terms) reflect the curvature of indifference curves. (See also the discussion in Chapter 11 of the elasticity of substitution in the context of a produc tion function.) b. Apply the definition of \(a\) given above to the CES utility function $$X^{s} \quad Y^{s}$$ Show that \(a=j^{\wedge}\) and that this value is constant for all values of \(X\) and \(Y\), thereby justifying the CES function's name.

Short answer

In summary, the substitution elasticity measures how easily a consumer can substitute between two goods in consumption as their relative prices change. The curvature of indifference curves represents the sensitivity of the consumer's preferences to changes in the consumption of the goods, with a higher value of elasticity implying flatter indifference curves and easier substitution between the two goods. Applying the definition of substitution elasticity to the CES utility function, we derived the expression \(\sigma = \frac{1}{1-\delta}\), which demonstrates that the substitution elasticity is constant for all values of X and Y, depending only on the \(\delta\) parameter. This justifies the name "Constant Elasticity of Substitution" for the CES utility function.

Step-by-step solution

Substitution Elasticity: Definition

The substitution elasticity (\(\sigma\)) measures how easily a consumer can substitute between two goods, X and Y in consumption, as their relative prices change. It is defined as the percentage change in the ratio of goods consumed, given a percentage change in the marginal rate of substitution (MRS).

Curvature of Indifference Curves

The curvature of indifference curves represents how sensitive the consumer's preferences are to changes in the consumption of goods X and Y. If the indifference curves are very curved (or convex), the consumer has strong preferences for consuming both goods in specific proportions, making it more difficult to substitute between them. On the other hand, if the indifference curves are less curved (or flatter), the consumer has less strict preferences for consuming the goods and can more easily substitute between them. This curvature is reflected in the value of the substitution elasticity.

Relation of Elasticity to Indifference Curves

In general, a higher value of substitution elasticity (\(\sigma\)) indicates a flatter indifference curve, allowing for easier substitution between goods X and Y. Conversely, a lower value of substitution elasticity implies a more curved indifference curve and less substitutability between the two goods. #b. Applying the Definition of σ to the CES Utility Function#

CES Utility Function: Definition

The CES Utility function has the following form: \\[ U(X,Y) = (AX^{\delta} + BY^{\delta})^{\frac{1}{\delta}} \\] where A and B are positive constants, and \(\delta\) is a parameter that governs the curvature of the utility function.

Marginal Utilities and MRS

First, we'll derive the marginal utilities for goods X and Y, and then the MRS. \\[ M_U^X = \frac{\partial U(X,Y)}{\partial X} = A\delta X^{\delta-1}\left(AX^{\delta} + BY^{\delta}\right)^{\frac{1}{\delta}-1} \\] \\[ M_U^Y = \frac{\partial U(X,Y)}{\partial Y} = B \delta Y^{\delta-1}(AX^{\delta} + BY^{\delta})^{\frac{1}{\delta}-1} \\] The Marginal Rate of Substitution (MRS) is defined as the ratio of the marginal utilities of the two goods: \\[ MRS = \frac{M_U^X}{M_U^Y} =\frac{A\delta X^{\delta-1}\left(AX^{\delta} + BY^{\delta}\right)^{\frac{1}{\delta}-1}}{B \delta Y^{\delta-1}(AX^{\delta} + BY^{\delta})^{\frac{1}{\delta}-1}} \\] We can simplify this expression a bit: \\[ MRS = \frac{A X^{\delta-1}}{B Y^{\delta-1}} \\]

Applying σ to CES Utility Function

Now, let's apply the definition of substitution elasticity to the derived MRS expression: \\[ \sigma = \left(\frac{d \ln MRS}{d \ln \frac{Y}{X}}\right)^{-1} \\]

Derivative and Final Expression

Taking the derivatives with respect to \( \ln \frac{Y}{X}\), we will find the expression for σ as a function of δ: \\[ \sigma = \left(\frac{A X^{\delta-1}}{B Y^{\delta-1}}\right)^{-1} \times \left(-(\delta-1)\right) \\] Simplifying the expression, we obtain that the substitution elasticity is: \\[ \sigma = \frac{1}{1-\delta} \\]

Justification of CES Name

This result demonstrates that the substitution elasticity, \(\sigma\), is a constant value for all values of X and Y, as it depends only on the \(\delta\) parameter. This justifies the name "Constant Elasticity of Substitution" for the CES utility function.

Key concepts

Marginal Rate of Substitution

Understanding the Marginal Rate of Substitution (MRS) is crucial for students delving into consumer choice theory. It reflects the rate at which a consumer is willing to give up one good in exchange for another, while still maintaining the same level of utility or satisfaction. This concept is illustrated when we imagine a consumer choosing between two products, say apples and bananas. If they highly value apples over bananas, they will only give up a few apples for a significant number of bananas, indicating a low MRS. Mathematically, we determine MRS by taking the ratio of the marginal utilities of the two goods. With the formula,
\[ MRS = \frac{MU_X}{MU_Y} \],
where \( MU_X \) and \( MU_Y \) are the marginal utilities of good X and good Y, respectively, we can see that the MRS is directly connected to how much utility a consumer gets from consuming a little more of each good.

CES Utility Function

The CES (Constant Elasticity of Substitution) utility function represents a specific type of utility function in economics used to describe consumer preferences. The term 'constant elasticity of substitution' indicates that this form of utility function has a fixed elasticity of substitution between any two goods across all levels of consumption, which means that the degree of substitutability between the goods does not change with changes in their quantities.

In technical terms, the CES utility function is given by:
\[ U(X,Y) = (AX^{\delta} + BY^{\delta})^{\frac{1}{\delta}} \],
where A and B are parameters showing the relative importance of goods X and Y, and \( \delta \) (delta) is a parameter influencing the curvature of the utility function and consequently the substitution elasticity. As the exercise showcases, CES utility functions lead to constant substitution elasticity irrespective of the quantities of X and Y consumed, clarifying its naming.

Indifference Curves

Indifference curves depict a consumer's preference for different combinations of two goods that provide the same level of satisfaction or utility. They are a graphical representation of the MRS and are crucial in consumer choice theory. These curves slope downwards from left to right, indicating that a consumer must give up a certain amount of one good to gain more of another while remaining equally satisfied.

A steeper curve means a high MRS and shows that the consumer values one good much more than the other. A flatter curve indicates a lower MRS, suggesting that the consumer is more willing to substitute between the goods. The shape of these curves also illustrates the concept of diminishing marginal rate of substitution; as you move down an indifference curve, the rate at which you are willing to trade goods generally falls, reflecting how the value of one more unit of a good decreases as you have more of it.

Elasticity of Substitution

Elasticity of substitution is a measure used to gauge how easily one good can be substituted for another, based on changes in relative prices or the MRS. This elasticity plays a significant role in both consumer theory and production theory. When discussing consumer preferences, it describes how a consumer's consumption pattern changes when the relative price of goods changes.

For instance, if tea and coffee are substitutes, an increase in the price of tea should theoretically lead to increased consumption of coffee, if the elasticity of substitution is high. In a symbolical form:
\[ \sigma = -\frac{d \ln(MRS)}{d \ln(Y/X)} \],
Elasticity of substitution reflects the curvature of indifference curves; a higher value indicates flatter curves and thus, a greater ease of substituting one good for another. It's integral to understand that this concept helps to explain a consumer's responsiveness to price changes and significantly impacts demand analysis.