Question

In general, uncompensated cross-price effects are not equal. That is, \\[\frac{\partial x_{i}}{\partial p_{j}} \neq \frac{\partial x_{j}}{\partial p_{i}}.\\] regardless of relative prices. (This is a generalization of Problem \(6.1 .)\)

Short answer

Based on the above analysis and solution, the short answer would be: The assertion that uncompensated cross-price effects are not equal is generally true, as the cross-price effects depend on the responsiveness of one good's demand to the price change of another good, and this responsiveness may vary between goods. However, it is essential to consider the specific context and functional forms of the demand functions when analyzing these relationships, as the inequality may not hold true in every situation.

Step-by-step solution

1. Write down the demand functions for two goods.

Let's consider the demand functions for two goods, \(x_i\) and \(x_j\). These demand functions represent the quantity demanded of goods \(i\) and \(j\) respectively, and depend on their respective prices \(p_i\) and \(p_j\). The demand functions can be written as: $$ x_i = x_i(p_i, p_j) $$ $$ x_j = x_j(p_i, p_j) $$

2. Calculate the partial derivatives of the demand functions with respect to prices.

Next, we need to find the cross-price effects by calculating the partial derivatives of the demand functions with respect to the other good's price. This will give us: $$ \frac{\partial x_i}{\partial p_j} = \frac{\partial x_i(p_i, p_j)}{\partial p_j} $$ and $$ \frac{\partial x_j}{\partial p_i} = \frac{\partial x_j(p_i, p_j)}{\partial p_i} $$

3. Compare the cross-price effects derivatives.

Now we need to compare the cross-price effects by examining if the inequality holds true: $$ \frac{\partial x_i}{\partial p_j} \neq \frac{\partial x_j}{\partial p_i} $$ We can't determine if the inequality holds true for every possible pair of demand functions, as it depends on the specific functional forms and the price changes involved. However, typically in economics, it is reasonable to assume that this inequality holds true in general, since the cross-price effects depend on the responsiveness of one good's demand to the price change of another good and this responsiveness may vary between goods. In conclusion, the given assertion is generally correct for most cases, but it may not hold true in every specific situation. It is important for students to understand the concept of cross-price effects and consider the specific context when analyzing these relationships.

Key concepts

Uncompensated Cross-Price Elasticity

When discussing the concept of uncompensated cross-price elasticity, we are looking at how the demand for one good changes in response to a change in the price of another good. This is a vital concept in understanding how goods are interrelated in a market. Its calculation is done using partial derivatives, reflecting the rate of change of one variable when another one changes, holding everything else constant.

Crucially, this form of elasticity is called uncompensated because it doesn't account for changes in consumer income or any other adjustments in the market, focusing purely on the price change and demand reaction aspects. When you calculate these elasticities, you might notice that they are not necessarily equal for two goods, indicating an asymmetric relationship in how the demand for each good reacts to the other's price changes.

• Uncompensated: Ignores income changes.
• Cross-Price: Involves two different goods.
• Elasticity: Measures responsiveness of demand.

Demand Functions

Demand functions are mathematical representations that calculate the quantity of a good that consumers will purchase at various prices and under specific conditions. Two goods have their own distinct demand functions reflecting consumers' preferences and the market structure.

Consider two goods, where the demand function of good 'i' is shown as \(x_i = x_i(p_i, p_j)\). This means that the quantity demanded of good 'i' depends on its own price \(p_i\) as well as the price of good 'j', \(p_j\). Similarly, the demand for good 'j' can be expressed as \(x_j = x_j(p_i, p_j)\).

• Demonstrates consumer choices.
• Includes prices of both goods.
• Utilizes distinct functions for each good.

Partial Derivatives

Partial derivatives are a mathematical tool used to understand how a function changes in relation to one of its variables while keeping other variables constant. This is crucial in analyzing cross-price effects as it helps determine the responsiveness of the demand for a good with respect to the price of another.

In terms of demand functions, we compute the partial derivative like \(\frac{\partial x_i}{\partial p_j}\), which indicates how the demand for good 'i' will change with a unit increase in the price of good 'j'. It's a crucial step in economic analysis that clarifies relationships between products in a market.

• Helps isolate effects of one variable.
• Crucial for understanding interaction between goods.
• Used extensively in economic analysis.

Price Responsiveness

Price responsiveness refers to how sensitive the quantity demanded or supplied of a good is to changes in its price or the price of another good. In the context of cross-price elasticity, it highlights how the demand for one good responds to price changes in another, which can be critical for setting pricing strategies and understanding market dynamics.

For instance, in a competitive market, if the price of a complementary good increases, the demand for the related good might decrease due to reduced consumer interest. Conversely, if one good is a substitute for another, a price increase in one may lead to a rise in demand for the other. Understanding these dynamics helps businesses and consumers make informed decisions.

• Reflects demand sensitivity to price changes.
• Varies with substitutes or complements.
• Key in setting prices and market strategies.