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 6.28 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

In conclusion, we have determined that X and Y, as well as X and Z, are gross complements based on the negative partial derivatives of the demand function for X with respect to the prices of Y and Z. To further analyze if these goods are net complements or net substitutes, one would need to compute the Hicksian demand functions for X and find the partial derivatives with respect to the prices of Y and Z. A positive sign for these derivatives would indicate net substitutes, while a negative sign would indicate net complements. However, calculating the Hicksian demand functions is beyond the scope of this solution.

Step-by-step solution

Compute the demand function for X

To find out the demand function for X, we need to write the marginal utility of X and apply the equation of the demand function. The given utility function is: $$ U(X, Y, Z) = -\frac{1}{X} - \frac{1}{Y} - \frac{1}{Z} $$ Now, let's find the marginal utility of X with respect to X: $$ MU_X = \frac{\partial U}{\partial X} = \frac{1}{X^2} $$ The demand function for X in Equation 6.28 is: $$ X(p_X, p_Y, p_Z, M) = \frac{M*p_X}{p_X^2 + p_Y^2 + p_Z^2} $$

Analyze the demand function for X

Now that we have the demand function for X, let's analyze how the demand for X varies with the change in the price of Y and the price of Z. Take the partial derivative of the demand function with respect to the price of Y and the price of Z. Partial derivative of the demand function with respect to the price of Y: $$ \frac{\partial X}{\partial p_Y} = \frac{-2M*p_X*p_Y}{(p_X^2 + p_Y^2 + p_Z^2)^2} $$ Partial derivative of the demand function with respect to the price of Z: $$ \frac{\partial X}{\partial p_Z} = \frac{-2M*p_X*p_Z}{(p_X^2 + p_Y^2 + p_Z^2)^2} $$

Determine if X and Y or X and Z are gross substitutes or gross complements

Now, let's analyze the sign of the above derivatives. A positive sign indicates that the goods are gross substitutes, while a negative sign indicates that the goods are gross complements. $$ \frac{\partial X}{\partial p_Y} < 0 $$ $$ \frac{\partial X}{\partial p_Z} < 0 $$ Since both partial derivatives are negative, it indicates that X and Y, as well as X and Z, are gross complements.

Discuss how to determine if X and Y or X and Z are net substitutes or net complements

To determine if X and Y or X and Z are net substitutes or net complements, we should calculate the compensated demand functions for X. The compensated demand functions take into consideration the income effect, represented by the substitution effect. For this, we need to calculate the Hicksian demand function for X and find the partial derivatives with respect to the prices of Y and Z. If the sign of these derivatives is positive, it would indicate that X and Y or X and Z are net substitutes. If the sign is negative, then the goods will be net complements. To compute the Hicksian demand function, we can apply the method of integrating the marginal rate of substitution (MRS) with respect to the price ratio. Unfortunately, calculating the Hicksian demand functions is beyond the scope of this explanation.

Key concepts

Gross Substitutes

When we talk about gross substitutes, we mean goods that, when the price of one increases, the demand for the other also rises. In the context of the CES utility function given in the exercise, we determine this relationship by examining the sign of the partial derivatives of the demand functions. Specifically, we look at relations like \( \frac{\partial X}{\partial p_Y} \). If this is positive, then \(X\) and \(Y\) are gross substitutes.

In our example, however, these derivatives were negative. Therefore, \(X\) and \(Y\) (and similarly \(X\) and \(Z\)) are not gross substitutes. Remember that examining the price and quantity changes can illustrate how consumers substitute one good for another when relative prices change.

• If demand for both goods changes in the same direction, they are substitutes.
• If demand for a good increases as the price of another good rises, they too are typically substitutes.

Understanding these relationships helps us grasp consumer behavior under varying market conditions.

Gross Complements

Gross complements are goods that, when the price of one increases, the demand for the other decreases. This is a common concept in consumer choice theory. For the CES utility function in the exercise, this relationship is confirmed by negative partial derivatives \( \frac{\partial X}{\partial p_Y} \) and \( \frac{\partial X}{\partial p_Z} \).

Both derivatives indicated negative values, implying that goods \(X\) and \(Y\) as well as \(X\) and \(Z\) are indeed gross complements. This means that if the price of \(Y\) or \(Z\) increases, the demand for \(X\) decreases, suggesting a dependency where one good enhances the use of the other.

• Negative cross-price elasticity showcases complementarity.
• These relationships are essential in predicting how changes in one market will affect consumption in another.

This understanding allows economists to predict product bundling strategies and consumer purchasing trends.

Hicksian Demand Function

The Hicksian demand function, also known as compensated demand, plays a key role in economic theory by isolating the substitution effect from the income effect. Unlike the usual demand functions, Hicksian demand keeps utility constant to show how consumption changes when prices change, given the consumer's satisfaction level.

To determine if goods are net substitutes or complements using Hicksian demand, we assess how demand changes relative to price changes, again looking at partial derivatives. For net substitutes, \( \frac{\partial h_X}{\partial p_Y} \) being positive implies \(X\) and \(Y\) are substitutes. Conversely, if it's negative, they are net complements.

Here's what you need to know:

• The Hicksian demand is derived from the cost minimization problem, rather than utility maximization.
• It removes the income effect, focusing strictly on how substitution impacts demand.
• While more complex to calculate, it offers insights into consumer reactions purely to price changes.

This function provides valuable insights for economists to understand true substitution and complementarity beyond income effects.