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UPSC MainsECONOMICS-PAPER-I201410 Marks150 Words
Q1.

Examine the relationship between own and cross price elasticities for a compensated demand function.

How to Approach

This question requires a nuanced understanding of demand theory, specifically focusing on compensated demand and its implications for price elasticities. The answer should begin by defining compensated demand and its distinction from ordinary demand. It should then explain how own and cross-price elasticities are calculated within the context of a compensated demand function. Crucially, the answer must highlight the relationship – specifically, that in a compensated demand function, cross-price elasticities represent the substitution effect only, while ordinary demand includes both substitution and income effects. A clear explanation of Slutsky equation will be beneficial.

Model Answer

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Introduction

In economics, understanding consumer behavior is central to analyzing market dynamics. Demand functions describe the relationship between the price of a good and the quantity demanded. However, changes in price also affect a consumer’s purchasing power, leading to both substitution and income effects. To isolate the pure effect of price changes, the concept of *compensated demand* is introduced. A compensated demand function represents the quantities demanded when a consumer is compensated for any change in purchasing power resulting from a price change, maintaining their initial utility level. This question asks us to examine the relationship between own and cross-price elasticities specifically within this framework of compensated demand.

Compensated Demand and Price Elasticities

A compensated demand function differs from an ordinary (uncompensated) demand function in that it accounts for the change in purchasing power. When the price of a good changes, the consumer’s real income changes, affecting demand. Compensated demand eliminates this income effect, allowing us to focus solely on the substitution effect.

Own Price Elasticity

The own-price elasticity of demand measures the responsiveness of the quantity demanded of a good to a change in its own price. For an ordinary demand function, it captures both substitution and income effects. However, for a compensated demand function, the own-price elasticity represents only the substitution effect. This is because the consumer is compensated to maintain the same level of utility, eliminating the income effect. Mathematically, it is represented as:

εxxc = (∂qx/∂px) * (px/qx)

Where εxxc is the compensated own-price elasticity, qx is the quantity demanded of good x, and px is the price of good x.

Cross Price Elasticity

The cross-price elasticity of demand measures the responsiveness of the quantity demanded of one good to a change in the price of another good. Similar to the own-price elasticity, the cross-price elasticity in an ordinary demand function includes both substitution and income effects. However, in a compensated demand function, the cross-price elasticity only reflects the substitution effect. This is a crucial distinction.

Mathematically, it is represented as:

εxyc = (∂qx/∂py) * (py/qx)

Where εxyc is the compensated cross-price elasticity, qx is the quantity demanded of good x, and py is the price of good y.

The Relationship: Slutsky Equation

The relationship between compensated and uncompensated elasticities is formally captured by the Slutsky equation. The Slutsky equation decomposes the total effect of a price change into substitution and income effects:

∂qx/∂py = (∂qx/∂py)c - (∂qx/∂I) * (py/I)

Where:

  • ∂qx/∂py is the uncompensated (ordinary) cross-price elasticity
  • (∂qx/∂py)c is the compensated cross-price elasticity
  • ∂qx/∂I is the income elasticity of demand for good x
  • py is the price of good y
  • I is the consumer’s income

This equation demonstrates that the uncompensated cross-price elasticity is the sum of the compensated cross-price elasticity (the pure substitution effect) and a term that reflects the income effect. Therefore, the compensated cross-price elasticity is always less in absolute value than the uncompensated cross-price elasticity (except for inferior goods where the income effect can reinforce the substitution effect).

Implications

Understanding the distinction between compensated and uncompensated elasticities is vital for policy analysis. For example, when evaluating the impact of a tax on a good, policymakers need to consider both the substitution effect (captured by compensated elasticities) and the income effect. The compensated elasticities provide a more accurate measure of the direct impact of the price change on consumer behavior, independent of changes in purchasing power.

Conclusion

In conclusion, the relationship between own and cross-price elasticities for a compensated demand function is fundamentally different from that of an ordinary demand function. Compensated elasticities isolate the substitution effect, providing a clearer understanding of how consumers respond to price changes when their utility level is held constant. The Slutsky equation formalizes this relationship, demonstrating how the income effect influences the overall price elasticity. This distinction is crucial for accurate economic analysis and effective policy formulation.

Answer Length

This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.

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Additional Resources

Key Definitions

Substitution Effect
The tendency of consumers to substitute cheaper goods for more expensive ones when relative prices change, holding nominal income constant.
Income Effect
The change in consumption resulting from a change in purchasing power, caused by a change in price.

Key Statistics

According to a 2023 study by the National Bureau of Economic Research (NBER), the income elasticity of demand for luxury goods is significantly higher than that for necessities.

Source: NBER Working Paper Series (2023)

The price elasticity of demand for gasoline in the United States is estimated to be around -0.2 in the short run and -0.5 in the long run (Energy Information Administration, 2022).

Source: U.S. Energy Information Administration (EIA), 2022

Examples

Coffee and Tea

If the price of coffee increases, consumers may substitute tea for coffee. The compensated demand elasticity would measure this substitution effect alone, while the uncompensated elasticity would also consider the impact of the higher coffee price on consumers’ overall purchasing power.

Normal vs. Inferior Goods

For a normal good, both the substitution and income effects work in the same direction, leading to a positive uncompensated elasticity. For an inferior good, the income effect is negative, potentially offsetting the substitution effect and resulting in a lower or even negative uncompensated elasticity.

Frequently Asked Questions

▶Why is it important to consider compensated demand?

Compensated demand provides a more accurate measure of the pure effect of a price change on consumer behavior, isolating the substitution effect from the confounding influence of changes in purchasing power. This is crucial for policy analysis and understanding market responses.

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