AEB 6106 Problem Set 3 Due October 10 At Class Start
Aeb 6106 Problem Set 3due Oct 10 at the start of class1 In problem
Derive the own-price Slutsky equation for various utility functions and analyze the terms and implications for different scenarios, including whether goods are normal or inferior, and whether they act as substitutes or complements.
Evaluate the economic relationships such as elasticities for given demand functions and determine the nature of goods in terms of normality, inferiority, substitution, and complementarity based on the provided utility and demand functions.
Paper For Above instruction
The analysis of demand responses to price and income changes is central to understanding consumer behavior in microeconomics. The Slutsky equation is pivotal in decomposing the total change in demand due to a price change into substitution and income effects. This paper derives the own-price Slutsky equations for various specified utility functions and discusses the economic interpretations of their terms and implications for the nature of goods involved.
Introduction
The Slutsky equation provides a framework for analyzing how the quantity demanded of a good responds to changes in its own price, considering both substitution effects (how consumers reallocate consumption when relative prices change) and income effects (how changes in real purchasing power influence demand). It is expressed as:
\( \frac{\partial x}{\partial p_x} = \frac{\partial x^c}{\partial p_x} - x \frac{\partial x^s}{\partial I} \)
where \(x^c\) is the compensated (Hicksian) demand, and \(x^s\) is the Marshallian (uncompensated) demand. Deriving the specific form of this equation depends on the utility function, as demand functions are linked to preferences and constraints.
Derivation for Utility Function 1
The first utility function considered is \(U = (x^{4/5} y^{1/5})\), with derived demand functions, indirect utility, and expenditure functions. The Marshallian demands are: \(x = \frac{4}{5} \frac{I}{p_x}\) and \(y = \frac{1}{5} \frac{I}{p_y}\). To derive the own-price Slutsky equation for \(x\), we note that in the case of Cobb-Douglas preferences, the demand functions are linear in income and inversely proportional to the price, and demand responses are well-behaved.
The compensated demand for \(x\) is \(x^c = \frac{4}{5} \frac{U^}{p_x}\), assuming a fixed utility level \(U^\). The demand derivative with respect to \(p_x\) at constant utility is then:
\( \frac{\partial x^c}{\partial p_x} = - \frac{4}{5} \frac{U^*}{p_x^2} \)
Given the demand and the substitution effect, the Slutsky equation describes the demand response as:
\( \frac{\partial x}{\partial p_x} = - \frac{4}{5} \frac{U^*}{p_x^2} - x \frac{\partial x}{\partial I} \)
where \(x = \frac{4}{5} \frac{I}{p_x}\), and \(\frac{\partial x}{\partial I} = \frac{4}{5} \frac{1}{p_x}\). The negative derivative indicates that an increase in the price reduces demand, consistent with the law of demand.
Interpretation of Terms in the Slutsky Equation
The terms in the equation include the substitution effect, represented by the derivative of compensated demand, which captures how demand changes purely due to relative price changes, holding utility constant. The income effect term, \(x \times \frac{\partial x}{\partial I}\), captures the change in demand due to the change in purchasing power resulting from the price change. In the given example, the demand functions are smooth and strictly decreasing in prices, reflecting normal goods.
Scenario Analysis and Goods Nature
For the specified utility functions, goods generally turn out to be normal, as demand increases with income. In the case of Cobb-Douglas preferences, goods are always favorable, and demand is always proportional to income, thus indicating normality.
Conclusion
Deriving the own-price Slutsky equation for various utility functions reveals how substitution and income effects shape demand. For Cobb-Douglas utilities, the demand functions are linear in income, and substitutability hinges on the relative prices, with demand responding predictably to price changes. Understanding this decomposition enhances our comprehension of consumer choices, especially in determining the impact of price policies and shifts in market conditions.
References
- Varian, H. R. (2014). Intermediate Microeconomics: A Modern Approach. W. W. Norton & Company.