Question

Comment on the validity of the statement: "If the derivative \(\left(\partial x^{*} / \partial P_{x}\right)\) is negative, then \(x\) cannot possibly represent an inferior good."

Short answer

The statement is not valid; a negative derivative indicates a normal response to price changes but doesn't rule out the good being inferior.

Step-by-step solution

Understand the Statement

The statement involves a derivative \(\left(\partial x^{*} / \partial P_{x}\right)\), where \(x^{*}\) is the quantity demanded of good \(x\) and \(P_{x}\) is its price. The claim is that if this derivative is negative, \(x\) cannot be an inferior good.

Define Inferior Goods

An inferior good is one for which the demand decreases as consumer income increases. However, the statement involves the price of the good, \(P_{x}\), not income.

Examine Price Effect

The derivative \(\partial x^{*} / \partial P_{x} < 0\) implies that as the price of good \(x\) increases, the quantity demanded of \(x\) decreases, which is a characteristic of a normal good rather than an inferior good. This is because, generally, for normal goods, demand decreases when price increases.

Correlation with Normal Goods

Normal goods exhibit a negative relationship between price and demand, i.e., demand decreases as price increases. Inferior goods can also exhibit this behavior depending on their elasticity, so \(\partial x^{*} / \partial P_{x} < 0\) does not directly prove that \(x\) is not an inferior good.

Conclude the Analysis

The initial statement is not valid because a negative derivative \(\partial x^{*} / \partial P_{x}\) does not preclude a good from being inferior. It only suggests that \(x\) behaves like a normal good with respect to price changes, but does not provide information about how demand changes with income.

Key concepts

Derivative

A derivative \(rac{\partial x^*}{\partial P_{x}}\) can be understood as the rate at which the quantity demanded of a good, denoted as \(x^*\), changes with respect to its price, \(P_{x}\). When we see this partial derivative, it's telling us how sensitive the quantity demanded is to changes in price. For example, if this derivative is negative, it indicates that as the price increases, the quantity demanded decreases. This behavior generally aligns with the laws of demand for most goods.

In simple terms:

• A negative derivative suggests an inverse relationship between price and quantity demanded.
• This is typical for many goods, especially those classified as normal goods.

Understanding the derivative helps us analyze consumer behavior in response to price changes, a key component in economic models of demand.

Demand Elasticity

Demand elasticity refers to how much the quantity demanded of a good responds to a change in its price. If a small change in price leads to a significant change in quantity demanded, the demand is said to be elastic. On the other hand, if the quantity demanded doesn't change much with a price change, the demand is inelastic.

In relation to the derivative \(\partial x^* / \partial P_{x} < 0\), it gives us insight into the price elasticity of demand:

• If demand is elastic, consumers are highly responsive to price changes, leading to larger swings in quantity demanded.
• If demand is inelastic, consumers are less responsive, and quantity demanded changes minimally.

This concept is critical because it affects revenue and pricing strategies for businesses. Knowing how sensitive consumers are to price changes allows companies to adjust prices strategically.

Normal Goods

Normal goods are products whose demand increases as consumer income increases. This is because consumers tend to buy more of these goods when they have more money.

• They exhibit a positive elasticity concerning income.
• Their demand rises with an increase in consumers’ purchasing power.

However, with regards to price, normal goods typically show a negative relationship. That means, for most normal goods, as the price goes up, the demand tends to fall, and vice versa. This is what is generally understood as a rational consumer behavior. It's important to note that a negative price derivative \(\partial x^* / \partial P_{x} < 0\) indicates a behavior aligned with normal goods, but doesn't necessarily exclude the possibility of the good being inferior under certain conditions of income changes.

Price Effect

The price effect describes the influence that changes in a product's price have on the quantity demanded.

• When the price of a product increases, the quantity demanded typically decreases (negative price effect).
• Conversely, when the price decreases, the quantity demanded usually increases.

This effect is a fundamental principle of the law of demand. However, the type of good—be it normal or inferior—can influence how this change manifests.For normal goods, the price effect generally leads to a straightforward decrease in demand as prices rise, consistent with the derivative \( \partial x^* / \partial P_{x} < 0 \).
Inferior goods, despite their peculiar income-demand relationship, can also experience a negative price effect. This means they can follow the general demand pattern when examined purely from the perspective of price changes, thus indicating that the classification as an inferior good is primarily influenced by income changes rather than price alone.