Question

As defined in Chapter \(3,\) a utility function is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope: The \(M R S\) depends on the ratio \(y / x\) a. Prove that, in this case, \(\partial x / \partial I\) is constant. b. Prove that if an individual's tastes can be represented by a homothetic indifference map then price and quantity must move in opposite directions; that is, prove that Giffen's paradox cannot occur.

Short answer

Question: Prove two properties of a homothetic utility function. (a) Show that the rate of change of x with respect to income (I) is constant. (b) Demonstrate that in the case of homothetic preferences, Giffen's paradox cannot occur. Answer: (a) For a homothetic utility function, the rate of change of x with respect to income (I) is constant because its Marginal Rate of Substitution (MRS) depends only on the ratio y / x. (b) Giffen's paradox cannot occur in the case of homothetic preferences because under such preferences, good x is always a normal good. The substitution effect prevails, leading the consumer to buy less of the more expensive good and more of the relatively cheaper good, thus contradicting the occurrence of Giffen's paradox.

Step-by-step solution

Define a homothetic utility function

A utility function u(x, y) is homothetic if the Marginal Rate of Substitution (MRS) only depends on the ratio y / x. The MRS is defined as the slope of the indifference curve and represents the amount of y the consumer is willing to give up to get one more unit of x.

Calculate the MRS

To find the MRS, take the partial derivative of the utility function u(x, y) with respect to x and y. The MRS is the negative ratio of these partial derivatives: MRS = -(\partial u / \partial x) / (\partial u / \partial y) Since the utility function is homothetic, the MRS depends only on the ratio y / x.

Prove that the rate of change of x with respect to I is constant

To show that ∂x / ∂I is constant, we will take the total differential of the MRS expression above: d(MRS) = -(\partial^2 u / \partial x^2) * (y/x) * dx - (\partial^2u / \partial x \partial y)*(y/x) * dy. Since MRS depends only on the ratio y / x, d(MRS) = 0. Rearranging to solve for dx/dy: dx/dy = (\partial^2u /\partial x \partial y) / (\partial^2 u / \partial x^2) * (y/x). Thus, the rate of change of x with respect to I is independent of the level of income, as it only depends on the ratio y / x. This completes part a of the exercise.

Explain normal and inferior goods

A good is called normal if its demand increases as income increases, while a good is said to be inferior if its demand decreases as income increases.

Define Giffen's paradox

Giffen's paradox describes a situation where the demand for a good increases even as its price increases. This is contrary to the Law of Demand, which states that the demand for a good should decrease as its price increases, holding all else constant.

Prove that Giffen's paradox cannot occur with homothetic preferences

Recall that ∂x / ∂I is constant for homothetic preferences as shown in Step 3. Since the rate of change of x with respect to income is constant, the demand for good x will always increase as income increases no matter the level of income. Therefore, good x is always a normal good under homothetic preferences. Now suppose that the price of good x increases. The substitution effect will always lead the consumer to buy less of the more expensive good and more of the relatively cheaper good. Since the substitution effect prevails and good x is always a normal good under homothetic preferences, the demand for good x must decrease as its price increases. This means that Giffen's paradox cannot occur with homothetic preferences, completing part b of the exercise.

Key concepts

Marginal Rate of Substitution (MRS)

Understanding the concept of the Marginal Rate of Substitution (MRS) is essential for grasping how consumers make choices between different goods. In the realm of economics, MRS indicates the rate at which a consumer is willing to exchange one good for another, maintaining the same level of utility or satisfaction. This rate reflects the consumer's personal preferences and is represented graphically as the slope of an indifference curve.

Specifically, if the utility function of goods x and y is homothetic, the MRS, which is given by the formula \( -\frac{\partial u}{\partial x} \div \frac{\partial u}{\partial y} \), will depend solely on the ratio of y to x. This means that the consumer's willingness to trade one good for the other does not change with the amounts of x or y they have, but only on their proportion to one another. This characteristic helps simplify the analysis of consumer choice under certain conditions.

Giffen's Paradox

Giffen's paradox is a fascinating anomaly within the study of economics. It occurs when a consumer increases the quantity demanded of a good as the price rises, violating the basic law of demand. This paradox is typically associated with 'Giffen goods,' which are typically inferior goods that take up a significant portion of a consumer's budget.

Within the context of a homothetic utility function, Giffen's paradox cannot manifest. The exercise illustrates that under homothetic preferences, the consumption pattern does not support the requirements for Giffen's paradox. Since the demand for a good increases with income consistently and the substitution effect always leads to the consumer choosing less of the good as its price increases, it prevents the emergence of the paradox. Essentially, the model of homothetic preferences precludes the possibility of a Giffen good scenario, where the demand curve appears to slope upward.

Substitution Effect

The substitution effect is a critical concept in consumer choice theory, describing how consumers respond to changes in the relative prices of goods. When the price of a good increases, consumers tend to substitute away from the more expensive good towards less expensive alternatives. This behavior occurs as they strive to maintain their level of utility within the constraints of their budget.

The homothetic utility function simplifies this analysis by ensuring that the preference for one good relative to another is consistent across different income levels, which implicitly makes the substitution effect predictable. As seen in the exercise, an increase in the price of good x naturally leads to a decrease in its quantity demanded due to the substitution effect, which is invariably present and unaffected by variations in income, reinforcing the consistency within homothetic preferences.

Normal and Inferior Goods

In economics, goods can be characterized as either normal or inferior, based on how their demand changes in relation to income. Normal goods are ones where demand increases as the consumer's income rises. In contrast, inferior goods have a negative relationship with income; as income increases, the demand for inferior goods declines.

Within the structure of a homothetic utility function, as cited in the exercise, goods are akin to normal goods. Since the rate of change of x with income \( \partial x / \partial I \) is constant and independent from the level of income, increases in income will consistently lead to an increased demand for good x. This interpretation aligns well with the notion of normal goods and illustrates the innate characteristic of homothetic utility functions to only represent scenarios where all goods are normal by their nature.