Compensated Demand: Understanding Hicksian Demand

An in-depth exploration of compensated demand (Hicksian demand), its mathematical foundations, significance in economic theory, and practical applications.

Compensated demand, also known as Hicksian demand, was introduced by Sir John Hicks, a prominent British economist. His formulation provided a significant advancement in the theory of consumer choice by linking consumer behavior to utility maximization under a constraint of constant utility.

Types/Categories

1. Individual Compensated Demand: Relates to a single consumer’s response to changes in prices while keeping utility constant.
2. Market Compensated Demand: Aggregates individual compensated demands to analyze market behavior.
3. Short-run vs. Long-run Compensated Demand: Differentiates responses in immediate time frames versus those allowing for full adjustment to price changes.

Key Events

• 1939: Publication of Hicks’s book “Value and Capital,” introducing the concept.
• 1950s: Hicks’s model further integrated into mainstream economic theory through advanced mathematical treatment.
• 1970s: Development of duality theory, connecting expenditure and utility functions in a rigorous manner.

Detailed Explanations

Compensated demand functions, denoted \( x^c(p_1, p_2, U) \) for goods 1 and 2, represent the quantities of goods demanded at given prices \( p_1 \) and \( p_2 \), keeping utility \( U \) constant.

Mathematical Formulation

The compensated demand for two goods \( x_1 \) and \( x_2 \) is derived from the minimization of expenditure subject to achieving a predetermined utility level \( U \).

1. Utility Function: \( U(x_1, x_2) \)
2. Expenditure Function: \( E(p_1, p_2, U) \), which represents the minimum expenditure required to achieve utility \( U \) at prices \( p_1 \) and \( p_2 \).

The problem can be formulated as:

$$ \min_{x_1, x_2} \{ p_1x_1 + p_2x_2 \} $$

$$ \text{subject to} \quad U(x_1, x_2) = U $$

The solution yields the compensated demand functions:

$$ x_1^c = h_1(p_1, p_2, U) $$

$$ x_2^c = h_2(p_1, p_2, U) $$

Deriving Compensated Demand

Compensated demands can be derived from the Lagrangian function:

$$ \mathcal{L} = p_1x_1 + p_2x_2 + \lambda (U - U(x_1, x_2)) $$

Taking first-order conditions with respect to \( x_1 \), \( x_2 \), and \( \lambda \) gives the system of equations needed to solve for \( x_1 \) and \( x_2 \).

Charts and Diagrams

    graph LR
	    A[Expenditure Minimization] --> B(Utility Level)
	    B --> C(Compensated Demand Functions)
	    C --> D(Quantities of Goods)

Importance and Applicability

Understanding compensated demand is crucial in:

• Welfare Economics: Evaluating the impact of policies without changing consumer welfare.
• Price Index Theory: Adjusting inflation measures by considering constant utility.
• Consumer Behavior Analysis: Gaining insights into how consumers reallocate their spending in response to price changes while maintaining the same level of satisfaction.

Examples and Considerations

Example

Consider a consumer with the utility function \( U(x_1, x_2) = x_1^{0.5} x_2^{0.5} \). If the prices are \( p_1 = 2 \), \( p_2 = 1 \), and utility level \( U = 10 \):

$$ \min_{x_1, x_2} \{ 2x_1 + x_2 \} $$

$$ \text{subject to} \quad x_1^{0.5} x_2^{0.5} = 10 $$

Solving this problem will provide the compensated demand functions for \( x_1 \) and \( x_2 \).

Considerations

• Substitution Effect: How consumption shifts due to price changes, isolated from the income effect.
• Duality in Economics: The relationship between utility maximization and expenditure minimization is a cornerstone of modern economic analysis.

Related Terms

1. Marshallian Demand: Quantity of goods demanded given income and prices.
2. Substitution Effect: Change in consumption patterns due to relative price changes, holding utility constant.
3. Income Effect: Change in consumption due to a change in purchasing power.

Comparisons

• Compensated vs. Uncompensated Demand: The latter includes both substitution and income effects, whereas compensated demand isolates the substitution effect by holding utility constant.

Interesting Facts

• Roy’s Identity: It links the Marshallian demand function to the utility function, while the Shepard’s Lemma relates compensated demand to the expenditure function.

Famous Quotes

“Value is in the eye of the beholder, but it’s also subject to the constraints of the wallet.” - John Hicks

Expressions, Jargon, and Slang

• “Substitution is key”: Indicates the importance of understanding the substitution effect.
• “Utility ain’t free”: Reflects that achieving satisfaction comes at a cost.

FAQs

What is compensated demand?

Compensated demand refers to the demand for goods that takes into account changes in prices while keeping the consumer’s utility constant.

Why is it important?

It is crucial for understanding how consumers adjust their spending to maintain the same level of satisfaction, especially useful in welfare analysis.

How is it different from Marshallian demand?

Marshallian demand reflects actual market behavior incorporating both income and substitution effects, whereas compensated demand isolates the substitution effect.

References

• Hicks, J.R. (1939). Value and Capital. Oxford University Press.
• Mas-Colell, A., Whinston, M.D., & Green, J.R. (1995). Microeconomic Theory. Oxford University Press.
• Varian, H.R. (1992). Microeconomic Analysis. W.W. Norton & Company.

Summary

Compensated demand is a pivotal concept in economics, distinguishing itself by holding utility constant to isolate the substitution effect. This approach provides clarity in analyzing consumer behavior, policy impacts, and theoretical economic constructs, making it a valuable tool for both economists and policymakers.

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By following this structured approach, the article provides a comprehensive, in-depth examination of compensated demand that can be easily integrated into the Encyclopedia, offering valuable insights into economic theory and consumer behavior.