Question

Why is the change in the quantity index from Year 1 to Year 2 ,using as -weights the prices of Year 1 , not always equal to the change in the quantity index, using as weights the prices of Year \(2 ?\) Explain this with the help of the conventional price line - indifference curve diagram.

Short answer

In short, the change in the quantity index from Year 1 to Year 2, using the prices of Year 1 as weights, is not always equal to the change in the quantity index using the prices of Year 2 as weights due to the changes in relative prices between the goods and the effect this has on consumer preferences and choices. The conventional price line - indifference curve diagram illustrates this phenomenon, showing that different slopes of price lines and their relation to indifference curves lead to different outcomes when calculating the quantity index using different price weights.

Step-by-step solution

Understand the Quantity Index

A quantity index measures the changes in quantities of a set of goods and services over time. When calculating a quantity index, the quantities are usually weighted by their prices in a specific base period. In this exercise, we are comparing the changes in the quantity index using the prices of Year 1 and Year 2 as weights.

Price Line and Indifference Curve Definitions

A price line represents the possible combinations of two goods that a consumer can purchase given a fixed income and the prices of those goods. The slope of the price line is equal to the ratio of the prices of the two goods. An indifference curve represents all combinations of two goods that provide the same level of satisfaction to a consumer. The slope of the indifference curve, called the marginal rate of substitution, represents the rate at which a consumer is willing to exchange one good for another while maintaining the same level of satisfaction.

Analyzing the Change in Quantity Index Using Year 1 and Year 2 Prices

We will now use the price line-indifference curve diagram to explain the difference in the quantity index change when using different weights. 1. Draw two price lines, one representing Year 1 prices and another representing Year 2 prices. The slopes of these price lines will differ due to the change in prices between the two years. 2. Draw an indifference curve passing through the point representing the quantity of goods consumed in Year 1. 3. Now, draw a second indifference curve passing through the point representing the quantity of goods consumed in Year 2. Notice that this curve will intersect both price lines since the quantities consumed can be priced in both Year 1 and Year 2 prices. 4. Observe the price lines and the indifference curves: When using Year 1 prices as weights, the change in the quantity index is measured along the price line representing the Year 1 prices. Similarly, when using Year 2 prices as weights, the change in the quantity index is measured along the price line representing the Year 2 prices.

Explanation of the Difference in Change of the Quantity Index

Since the price lines have different slopes due to the change in prices between Year 1 and Year 2, the change in the quantity index when using Year 1 prices as weights is not equal to the change in the quantity index when using Year 2 prices as weights. The quantity index with different weights reflects the changes in relative importance between goods as their prices change over time, leading to different outcomes when the quantity index is calculated using different price weights. In conclusion, the change in the quantity index when using as weights the prices of Year 1 is not always equal to the change in the quantity index, using as weights the prices of Year 2, due to the changes in relative prices between the goods and the effect this has on the consumer's preferences and choices. The conventional price line - indifference curve diagram helps illustrate this phenomenon through the differing slopes of the price lines and their relation to the indifference curves.

Key concepts

Price Line

The price line in economics represents the combinations of two goods that a consumer can buy with a fixed amount of money, taking into account the prices of those goods. It is essentially a budget constraint which shows the trade-offs a consumer faces when choosing how to allocate their limited resources.

The slope of the price line is determined by the ratio of the prices of the two goods. If the price of one good increases while the income and the price of the other good remain the same, the price line will rotate inwards, indicating that the consumer can afford less of the more expensive good. Conversely, if the price decreases, the price line rotates outwards. Understanding the concept of a price line is crucial when analyzing consumer behavior, as it sets the stage for the choices available within a consumer’s budget.

Indifference Curve

An indifference curve is a tool used to represent a consumer's preferences graphically. It shows all combinations of two goods between which a consumer is indifferent because they provide the same level of utility or satisfaction. These curves are typically convex to the origin, indicating the principle of diminishing marginal rate of substitution: as a consumer has more of one good, they are willing to give up less of a second good to maintain the same level of satisfaction.

Indifference curves do not intersect because each curve represents a different level of utility. They are a vital concept for understanding how consumers might substitute between goods and the trade-offs they are willing to make, particularly when paired with price lines to analyze consumer equilibrium.

Price Weights

Price weights are used in the calculation of quantity indexes to reflect the relative importance of different goods or services. In essence, a good with a higher price weight will have a greater impact on the index value. When we say that a quantity index from Year 1 to Year 2 uses the prices of Year 1 as weights, we mean that the quantities of the goods and services are valued at Year 1 prices.

The choice of price weights can significantly affect the result of an index. For example, if the price of a good increases by a large percentage from Year 1 to Year 2, using Year 1 prices as weights could understate the cost increase in a cost of living index. This variability highlights why comparing two quantity indexes with different price weights can lead to different conclusions about relative changes.

Marginal Rate of Substitution

The marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of overall satisfaction. It is illustrated by the slope of the indifference curve. In economic terms, the MRS decreases as you move down along the curve because as a consumer obtains more of one good, they are willing to sacrifice fewer units of another good.

The concept of the MRS is closely tied to the notion of opportunity cost and is fundamental for understanding how individuals make decisions about consumption. As it changes, it reflects shifts in a person's willingness to trade off goods, owing to their preferences and the diminishing additional satisfaction that each unit of a good provides.