Question

a. Show that the CES function \\[ \alpha \frac{X^{8}}{8}+\beta \frac{Y^{8}}{8} \\] is homothetic. How does the \(M R S\) depend on the ratio \(Y / X ?\) b. Show that your results from part (a) agree with Example 3.3 for the case \(\delta=1\) (perfect substitutes \()\) and \(\delta=0\) (Cobb-Douglas). c. Show that the \(M R S\) is strictly diminishing for all values of \(\delta<1\) d. Show that if \(X=Y\), the \(M R S\) for this function depends only on the relative sizes of \(\alpha\) and \(\beta\) e. Calculate the \(M R S\) for this function when \(Y / X=.9\) and \(Y / X=1.1\) for the two cases \(\delta=.5\) and \(\delta=-1 .\) What do you conclude about the extent to which the \(M R S\) changes in the vicinity of \(X=Y\) ? How would you interpret this geometrically?

Short answer

In conclusion, the given CES production function is homothetic, and its MRS depends on the ratio of Y/X. As the ratio of Y/X approaches 1, the MRS becomes more stable, showing less curvature in the indifference curve. The calculated MRS for the different cases of Y/X ratios provides insight into the sensitivity of MRS to changes, with higher ratios deviating more significantly from 1, resulting in higher MRS compared to cases when Y/X is close to 1. The interpretation of the results indicates that the MRS and the shape of indifference curves can be affected by the relative proportions of X and Y, particularly when these proportions are further apart.

Step-by-step solution

a. Show that CES function is homothetic and find MRS

We are given the CES function: \[ \alpha \frac{X^{8}}{8}+\beta \frac{Y^{8}}{8} \] To show that the function is homothetic, we need to show that for any positive scalar t, \[ \alpha \frac{(tX)^{8}}{8}+\beta \frac{(tY)^{8}}{8} = t(\alpha \frac{X^{8}}{8}+\beta \frac{Y^{8}}{8}). \] Now, let's calculate: \( \alpha \frac{(tX)^{8}}{8}+\beta \frac{(tY)^{8}}{8} \) \( = \alpha \frac{t^{8}X^{8}}{8}+\beta \frac{t^{8}Y^{8}}{8} \) \( = t^{8}(\alpha \frac{X^{8}}{8}+\beta \frac{Y^{8}}{8}) \) Since t is a positive scalar, the function is indeed homothetic. Now, find the MRS which is the ratio of marginal utilities of X and Y. First, find the marginal utilities. Marginal utility of X: \[ \frac{\partial}{\partial X}\left(\alpha \frac{X^{8}}{8}+\beta \frac{Y^{8}}{8}\right) = \alpha X^{7} \] Marginal utility of Y: \[ \frac{\partial}{\partial Y}\left(\alpha \frac{X^{8}}{8}+\beta \frac{Y^{8}}{8}\right) = \beta Y^{7} \] Now, we find the MRS as the ratio of the marginal utilities: \[ MRS = \frac{\alpha X^{7}}{\beta Y^{7}} \] Notice the \(MRS\) depend on the ratio \(\frac{Y}{X}\) as follows: \[ MRS = \frac{\alpha }{\beta}\left(\frac{X}{Y}\right)^{7} \]

b. Show agreement with Example 3.3

In Example 3.3, the general CES function is given as: \[ (\alpha X^{\delta} + \beta Y^{\delta})^{\frac{1}{\delta}} \] When \(\delta = 1\), the function represents perfect substitutes: \[ (\alpha X^1 + \beta Y^1) = \alpha X + \beta Y \] When \(\delta = 0\), the function represents Cobb-Douglas: \[ (\alpha X^0 + \beta Y^0)^\infty = X^\alpha Y^\beta \] Our given CES function can be written as: \[ (\alpha \frac{(X^8)^1}{8} + \beta \frac{(Y^8)^1}{8}) \] Comparing with the general CES function, we can set \(\delta = 1\) \[ \alpha \frac{X^8}{8} + \beta \frac{Y^8}{8} = \alpha X^{1*8} + \beta Y^{1*8} \] We can see that our given CES function agrees with the CES function for the case perfect substitutes (\(\delta = 1\)). However, this specific CES function doesn't match the Cobb-Douglas case (\(\delta = 0\)).

c. Show that MRS is strictly diminishing for all values of δ < 1

To determine the elasticity of substitution (ES), we derive the logarithmic derivative of the MRS with respect to the ratio \(\frac{Y}{X}\): Elasticity of substitution: \[ ES = \frac{d(\ln MRS)}{d(\ln\frac{Y}{X})} = \frac{d(\ln (\frac{\alpha }{\beta}\left(\frac{X}{Y}\right)^{7}))}{d(\ln\frac{Y}{X})} \] Calculate the partial derivative : \[ ES = \frac{d(7 \ln \frac{X}{Y})}{d(\ln\frac{Y}{X})} = -7 \] Since \(ES = -7 < 1\), the MRS is strictly diminishing for all values of δ < 1.

d. Show that if X = Y, the MRS depends only on α and β

We already found the MRS as: \[ MRS = \frac{\alpha X^{7}}{\beta Y^{7}} \] Now let's set \(X = Y\) and substitute into the MRS expression: \[ MRS = \frac{\alpha X^{7}}{\beta X^{7}} = \frac{\alpha}{\beta} \] When \(X = Y\), we can see that the MRS depends only on the relative sizes of \(\alpha\) and \(\beta\).

e. Calculate MRS for different Y/X ratios and interpret the results

Now we will calculate the MRS for the following cases: 1) \(Y/X = 0.9\) and \(\delta = 0.5\) 2) \(Y/X = 1.1\) and \(\delta = 0.5\) 3) \(Y/X = 0.9\) and \(\delta = -1\) 4) \(Y/X = 1.1\) and \(\delta = -1\) Recall the MRS expression: \[ MRS = \frac{\alpha }{\beta}\left(\frac{X}{Y}\right)^{7} \] For case 1) (\(Y/X = 0.9\) and \(\delta = 0.5\)): \[ MRS = \frac{\alpha }{\beta}\left(\frac{1}{0.9}\right)^{7} = \frac{\alpha }{\beta}(1.111)^{7} \] For case 2) (\(Y/X = 1.1\) and \(\delta = 0.5\)): \[ MRS = \frac{\alpha }{\beta}\left(\frac{1}{1.1}\right)^{7} = \frac{\alpha }{\beta}(0.909)^{7} \] For case 3) (\(Y/X = 0.9\) and \(\delta = -1\)): \[ MRS = \frac{\alpha }{\beta}\left(\frac{1}{0.9}\right)^{7} = \frac{\alpha }{\beta}(1.111)^{7} \] For case 4) (\(Y/X = 1.1\) and \(\delta = -1\)): \[ MRS = \frac{\alpha }{\beta}\left(\frac{1}{1.1}\right)^{7} = \frac{\alpha }{\beta}(0.909)^{7} \] The MRS remains the same for the changes in the \(\delta\). However, the ratio of \(Y/X\) impacts the MRS. When \(Y/X\) is closer to 1, the MRS is less sensitive to changes in the ratio. In other words, when X and Y are close in magnitude, the MRS becomes more stable and less sensitive to changes. Geometrically, this means that the indifference curve is less curved when X and Y are close to each other.

Key concepts

Marginal Rate of Substitution (MRS)

Marginal Rate of Substitution (MRS) is a concept in economics that measures the willingness of a consumer to substitute one good for another while maintaining the same level of utility or satisfaction. In terms of mathematical expression, it is the ratio of the marginal utilities of two goods.

When looking at the CES function given in the problem, \( \alpha \frac{X^{8}}{8}+\beta \frac{Y^{8}}{8} \), we derive the MRS by taking the partial derivatives of the function with respect to each good, X and Y. The MRS is significant because it illustrates how a consumer trades off between different goods. A diminishing MRS, as shown in the exercise, indicates that the consumer values the first few units of a good more than the later units, reflecting the law of diminishing marginal utility.

Key Insight:

When solving for the MRS and examining how it changes with the ratio of \( Y / X \), students should note that the exponent in the CES function directly affects the steepness of the MRS curve. A higher exponent means a more rapid decline in the MRS as one moves along an indifference curve.

Elasticity of Substitution

Elasticity of substitution is a valuable measure of responsiveness in consumer preference. It quantifies how easily two goods can be substituted for one another in response to changes in relative price or utility. In other words, it tells us how a change in the ratio of one good to another impacts the MRS between those goods.

With the CES function from our exercise, we can compute the elasticity of substitution by taking the logarithmic derivative of the MRS with respect to the ratio of the two goods, \( \frac{Y}{X} \). The result of this calculation will be negative for the CES function with a power of less than one, showing that the MRS is strictly diminishing. This concept is particularly important in understanding consumer behavior and how it is influenced by changes in relative prices or utilities of goods.

Practical Application:

Understanding elasticity of substitution helps in predicting how a change in the price ratio of goods will affect a consumer's purchasing decisions. This can influence pricing strategies for businesses and policy decisions in economics.

Cobb-Douglas Function

The Cobb-Douglas production function is a widely used model that represents the relationship between the quantities of inputs used in production and the amount of output produced. It has the general form \( X^\alpha Y^\beta \), where \( X \) and \( Y \) represent input quantities, and \( \alpha \) and \( \beta \) are the output elasticities of each input.

In the context of the problem, we compare the CES function to the Cobb-Douglas scenario for \( \delta=0 \). Although the given CES formula doesn't match exactly with the Cobb-Douglas form when \( \delta=0 \), it is critical to understand the distinct characteristics of the Cobb-Douglas function which include constant returns to scale and a unitary elasticity of substitution. These characteristics inform us about the proportional increase in output given proportional increases in inputs and the direct, one-to-one tradeoff between inputs, respectively.

Real-World Relevance:

A strong grasp of the Cobb-Douglas function is key for students entering fields like economics, business, or any industry involving production analysis, as it greatly aids in understanding the nature and productivity of various input combinations.

Perfect Substitutes

Perfect substitutes are two goods that can be exchanged for one another at a constant rate of substitution without affecting a consumer's utility. This implies a linear relationship and an MRS that is constant, regardless of the quantities consumed.

Returning to the CES function in the exercise, we're asked to consider the case of perfect substitutes when \( \delta=1 \). The function simplifies to a linear form, \( \alpha X + \beta Y \), which illustrates that the goods are interchangeable at a fixed rate, here defined by the ratio of their coefficients \( \alpha \) and \( \beta \).

Understanding in Context:

When teaching concepts involving perfect substitutes, it is vital to highlight their direct practical ties, such as in market scenarios where consumers can switch between similar brands of a product. Emphasizing this theory enables students to better analyze consumer choices and market dynamics, particularly in highly competitive environments where product differentiation is minimal and prices often dictate consumer preferences.