Question

A utility function is called separable if it can be written as \\[U(x, y)=U_{1}(x)+U_{2}(y),\\] where \(U_{i}^{\prime} > 0, U_{i}^{\prime \prime} < 0,\) and \(U_{1}, U_{2}\) need not be the same function. a. What does separability assume about the cross-partial derivative \(U_{x y}\) ? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to conclude definitively whether \(x\) and \(y\) are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.

Short answer

#Answer# a. The cross-partial derivative \(U_{xy}\) is equal to zero for a separable utility function. This means that the marginal utilities of goods \(x\) and \(y\) are independent of each other. This is plausible in situations where the consumption of one good does not influence the marginal utility of the other good, such as when consuming completely unrelated goods from different categories, like apples and televisions. b. If utility is separable, then neither good can be inferior. This can be shown by contradiction: assume that one good is inferior and its marginal utility is negative; however, this contradicts the given condition that marginal utilities are positive. c. Separability does not provide any information regarding how the quantity demanded for either good changes with the price of the other good. Since gross substitutes and gross complements are defined by the relationships between the quantities demanded with respect to the prices of goods, separability does not allow us to determine if \(x\) and \(y\) are gross substitutes or gross complements. d. Consider a Cobb-Douglas utility function, and apply the natural logarithm as a monotonic transformation. The transformed function is no longer separable, as the constants \(\alpha\) and \(\beta\) affect the utility functions of \(x\) and \(y\). Thus, separability is not invariant with respect to monotonic transformations.

Step-by-step solution

Find cross-partial derivative of separable utility function.

To find the cross-partial derivative \(U_{xy}\), we first differentiate the separable utility function with respect to \(x\), and then differentiate the resulting expression with respect to \(y\): \\[U_x = \frac{\partial (U_1(x) + U_2(y))}{\partial x} = U_1'(x).\\] Next, we differentiate the expression for \(U_x\) with respect to \(y\): \\[U_{xy} = \frac{\partial U_1'(x)}{\partial y} = 0.\\]

Discuss the Intuitive Meaning of \(U_{xy}\)

The cross-partial derivative \(U_{xy}\) measures the effect of a change in \(x\) on the marginal utility of \(y\) and vice versa. In this case, since \(U_{xy} = 0\), changes in one good (\(x\) or \(y\)) do not affect the marginal utility of the other good. This means that the goods \(x\) and \(y\) are independent of each other in terms of their marginal utilities. The separability condition is plausible in situations where the consumption of one good does not influence the marginal utility of the other good, such as when consuming two completely unrelated goods, like apples and televisions. b. Inferior Goods

Prove no inferior goods by contradiction.

Assume that good \(x\) is inferior. If \(x\) is inferior, then as income increases, the quantity demanded for good \(x\) decreases, and so \(U_1'(x) < 0\). However, this contradicts the given condition that \(U_1'(x) > 0\). Therefore, good \(x\) cannot be an inferior good. Similarly, we can show the same for good \(y\), and hence, neither good can be inferior. c. Gross Substitutes or Gross Complements

Explain why separability cannot definitively conclude if \(x\) and \(y\) are gross substitutes or gross complements.

The assumption of separability indicates that the marginal utilities of \(x\) and \(y\) are independent of each other; it does not provide any information regarding how the quantity demanded for either good changes with the price of the other good. Since gross substitutes and gross complements are defined by the relationships between the quantities demanded with respect to the prices of goods, separability does not allow us to determine if \(x\) and \(y\) are gross substitutes or gross complements. d. Separability and Monotonic Transformations

Write down the Cobb-Douglas utility function.

The Cobb-Douglas utility function can be written as: \\[U(x, y) = x^\alpha y^\beta,\\] where \(\alpha, \beta > 0\).

Perform a monotonic transformation and show the function is not separable anymore.

Let's consider a logarithmic transformation, which is a monotonic transformation. Apply the natural logarithm to the Cobb-Douglas utility function: \\[\ln U(x, y) = \ln(x^\alpha y^\beta) = \alpha \ln x + \beta \ln y.\\] Now, try to express the transformed function as the sum of two functions, where one depends only on \(x\) and the other depends only on \(y\): \\[\ln U(x, y) = \alpha \ln x + \beta \ln y \ne U_1(x) + U_2(y),\\] since \(\alpha \ln x\) and \(\beta \ln y\) are not "pure" utility functions of \(x\) and \(y\), as they are affected by the constants \(\alpha\) and \(\beta\). Therefore, the transformed function is not separable, and separability is not invariant with respect to monotonic transformations.

Key concepts

Cross-Partial Derivative

In the study of separable utility functions, a key concept is the cross-partial derivative, represented as \(U_{xy}\). It is obtained by first taking the partial derivative of the utility function with respect to one good, say \(x\), and then taking the partial derivative of the result with respect to the other good, \(y\). What is pivotal here is that in a separable utility function, these cross-partial derivatives are equal to zero. This zeroes outcome tells us that the increment in satisfaction (utility) gained from consuming a bit more of one good is not affected by the amount we consume of the other good. Essentially, this signals the absence of interaction effects between goods in the consumer's utility evaluation.

Consider the separable utility function \(U(x, y) = U_1(x) + U_2(y)\). Differentiating with respect to \(x\) returns something that is independent of \(y\), and vice versa. Hence, the cross-partial derivative \(U_{xy}\) is zero, showing that the two goods are non-interactive in the context of consumption satisfaction. This simplicity is useful in economic models as it allows for easier analysis of consumer behavior, albeit at the loss of complexity in describing preferences.

Inferior Good

Grasping the concept of inferior goods helps to understand consumer behavior in response to changes in income. An inferior good is one where demand decreases as consumer income rises, invoking a unique demand curve shape when plotted. In the realm of separable utility functions, however, neither good can be classified as inferior. This is derived from the assumption that the marginal utility, \(U'\), for each good is positive. In line with the presumption that an increase in consumption leads to an increase in utility, it contradicts the notion of an inferior good, where higher income should prompt less consumption of the good.

The implication here is that separation of utility dispels the possibility of goods being inferior; they are either normal or luxury goods. Both see either a proportionate or more than proportionate increase in consumption as incomes rise. Thoughtful consideration of these goods is vital when evaluating policies or economic states that influence income.

Gross Substitutes and Complements

In consumer theory, a nuanced understanding of how goods interact with each other is crucial. Gross substitutes and gross complements are terms that depict the nature of this interaction. When the price of one good goes up and consumers react by buying more of another good, those two goods are considered gross substitutes. On the flip side, if the price increase of one good causes a decrease in the purchase of another, those goods are termed gross complements.

The twist with separable utility functions is that they don't convey enough information to decisively categorize goods into these buckets. Since separability entails independent marginal utilities of the goods, it remains silent on how quantity demanded is affected by price changes of the counterpart good. This illustrates the importance of not solely focusing on utility but also considering price and income effects when examining consumer choices and market dynamics.

Cobb-Douglas Utility Function

Among the various representations of consumer preferences, the Cobb-Douglas utility function stands out for its mathematical elegance and real-world applicability. It is expressed as \(U(x, y) = x^\alpha y^\beta\), where \(\alpha\) and \(\beta\) are constants that represent the relative importance, or elasticity, of goods \(x\) and \(y\) respectively. It's a particular form that implies specific, constant relative rates of substitution, showcasing how consumers are willing to trade off one good for another.

However, performing a monotonic transformation, such as taking the logarithm, on this utility function disrupts its separable nature. Post transformation, it no longer maintains the individual utility representations for each good, rather they become entwined with the constants \(\alpha\) and \(\beta\). This facet illuminates the delicate balance in utility functions; they can morph with transformations, altering their analytical properties. The Cobb-Douglas function is a cornerstone in understanding the proportionality of goods' consumption—vital for analyzing the elasticity of demand and the effects of price changes.