Question

Example 6.3 computes the demand functions implied by the three-good CES utility function \\[U(x, y, z)=-\frac{1}{x}-\frac{1}{y}-\frac{1}{z}.\\] a. Use the demand function for \(x\) in Equation 6.32 to determine whether \(x\) and \(y\) or \(x\) and \(z\) are gross substitutes or gross complements. b. How would you determine whether \(x\) and \(y\) or \(x\) and \(z\) are net substitutes or net complements?

Short answer

What about net substitutes or net complements? Answer: Neither \(x\) and \(y\) nor \(x\) and \(z\) are gross substitutes or gross complements, as their cross-derivatives of the demand functions are zero, indicating that the demand of \(x\) does not change if the price of \(y\) or \(z\) changes. Similarly, neither \(x\) and \(y\) nor \(x\) and \(z\) are net substitutes or net complements, since their compensated demand functions remain the same as their ordinary demand functions.

Step-by-step solution

Calculate the Demand Functions

To calculate the demand functions for the three-good CES, we first need to find the marginal rates of substitution. The given utility function is: $$ U(x, y, z) = -\frac{1}{x} -\frac{1}{y} -\frac{1}{z} $$ First, we find the partial derivatives of the utility function with respect to each good. $$ \begin{aligned} \frac{\partial U}{\partial x} &= \frac{1}{x^2}\\\\ \frac{\partial U}{\partial y} &= \frac{1}{y^2}\\\\ \frac{\partial U}{\partial z} &= \frac{1}{z^2} \end{aligned} $$ Next, we find the marginal rate of substitution between each pair of goods by taking the ratio of their respective partial derivatives: $$ \begin{aligned} MRS_{xy} &= \frac{\partial U / \partial x}{\partial U / \partial y} = \frac{x^2}{y^2}\\\\ MRS_{xz} &= \frac{\partial U / \partial x}{\partial U / \partial z} = \frac{x^2}{z^2} \end{aligned} $$ Now, we must find the demand functions for \(x, y\) using both \(MRS_{xy}\) and \(MRS_{xz}\).

Examine Cross-Derivatives

To determine whether \(x\) and \(y\) or \(x\) and \(z\) are gross substitutes or gross complements, we should examine the cross-derivatives of the demand functions for \(x\) with respect to \(y\) and with respect to \(z\): $$ \begin{aligned} \frac{\partial}{\partial y} D_x(P_x, P_y, P_z, I) &= -\frac{I}{P_x} \frac{\partial MRS_{xy}}{\partial y}\\\\ \frac{\partial}{\partial z} D_x(P_x, P_y, P_z, I) &= -\frac{I}{P_x} \frac{\partial MRS_{xz}}{\partial z} \end{aligned} $$ Taking the partial derivatives of \(MRS_{xy}\) and \(MRS_{xz}\): $$ \begin{aligned} \frac{\partial MRS_{xy}}{\partial y} &= 0\\\\ \frac{\partial MRS_{xz}}{\partial z} &= 0 \end{aligned} $$ Now we have: $$ \begin{aligned} \frac{\partial}{\partial y} D_x(P_x, P_y, P_z, I) &= 0\\\\ \frac{\partial}{\partial z} D_x(P_x, P_y, P_z, I) &= 0 \end{aligned} $$ These derivatives indicate that neither \(x\) and \(y\) nor \(x\) and \(z\) are gross substitutes or gross complements. Since the cross-derivatives are zero, the demand of \(x\) does not change if the price of \(y\) or \(z\) changes.

Determining Net Substitutes or Net Complements

To determine whether \(x\) and \(y\) or \(x\) and \(z\) are net substitutes or net complements, we would need to find the compensated demand function. This can be done by applying Hicksian demand approach, which requires that we know the change in income resulting from a price change. In this particular problem, since neither \(x\) and \(y\) nor \(x\) and \(z\) are gross substitutes or gross complements, their compensated demand functions will remain the same as their ordinary demand functions. As a result, \(x\) and \(y\) or \(x\) and \(z\) are not net substitutes or net complements either.

Key concepts

Marginal Rate of Substitution

Understanding the Marginal Rate of Substitution (MRS) is crucial when analyzing consumer choice and preference between two goods. The MRS represents the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility. It is calculated by taking the ratio of the marginal utilities of two goods.

For example, in a CES (Constant Elasticity of Substitution) utility function such as the one given by \(U(x, y, z) = -\frac{1}{x} -\frac{1}{y} -\frac{1}{z}\), we can find the MRS by taking the partial derivatives of the utility function with respect to each good and forming ratios, like so:

\[MRS_{xy} = \frac{\frac{\text{\text{d}}U}{\text{\text{d}}x}}{\frac{\text{\text{d}}U}{\text{\text{d}}y}} = \frac{x^2}{y^2}\]
\[MRS_{xz} = \frac{\frac{\text{\text{d}}U}{\text{\text{d}}x}}{\frac{\text{\text{d}}U}{\text{\text{d}}z}} = \frac{x^2}{z^2}\]

These formulas yield the rate at which x can be substituted for y or z without changing the utility level. A lower MRS implies that a consumer is willing to give up less of one good for the other. Conversely, a higher MRS means the consumer would need to give up more of one good to obtain another unit of the second good, signaling a preference for the latter.

Gross Substitutes and Complements

The concepts of gross substitutes and gross complements pertain to how demand for one good responds to price changes in another good in a consumer's basket. Two goods are considered gross substitutes if an increase in the price of one good leads to an increase in the demand for another good. Conversely, two goods are gross complements if an increase in the price of one good causes a decrease in the demand for another good.

For the CES utility function provided, to clarify whether goods are gross substitutes or complements, we would analyze the cross-partial derivatives of the demand functions. In this case, we found that:

\[\frac{\text{\text{d}}}{\text{\text{d}}y} D_x(P_x, P_y, P_z, I) = 0\]
\[\frac{\text{\text{d}}}{\text{\text{d}}z} D_x(P_x, P_y, P_z, I) = 0\]

As both derivatives are zero, this indicates that goods x and y, as well as x and z, do not react to the price changes of each other. Therefore, they are neither gross substitutes nor gross complements in this scenario. However, it's important to note that this is a specific case due to the constant marginal rates of substitution in the CES utility function.

Compensated Demand Function

The compensated demand function adjust a consumer's demand based on their income and price changes to keep their level of utility constant; it reflects the concept of consumer's 'real' choice unaffected by income effects. This is also known as the Hicksian demand function. To identify it, we generally apply the Hicksian demand approach and factor in how an income change affects demand, compensating to hold utility constant.

In the context of the CES utility function problem, since we've determined that neither x and y, nor x and z are gross substitutes or gross complements, the compensated demand function would match the ordinary demand function. This implies that there would be no change in the quantity demanded of x when the prices of y or z change, even after adjusting for income effects.

Interestingly, this outcome tells us that for this utility function, the goods in question do not exhibit the characteristics of net substitutes or net complements. Essentially, this simplification in the CES utility function underscores the importance of understanding the impact of both price and income changes on consumer demand through the lens of the compensated demand function.