Econ 102 Intermediate Microeconomics Assignment 3 Due April
Econ 102 Intermediate Microeconomics Assignment 3 Dueapril 19 In T
Econ 102 Intermediate Microeconomics Assignment # 3 (DueApril 19, in the class) Part 1 Utility function from consuming a bundle of goods (X, Y) is given as U=X1/2Y1/2 and Prices are given as P= (PX, PY) = ($2, $4), and income M= $. Derive the optimum consumption bundle. 2. If Prices now change to P’= (PX’, PY’) = ($4, $4). Derive the new optimum consumption bundle, and the substitution effects on the consumption of each goods. 3. Under the new prices, what are the income effects on the consumption of each goods? 4. Graphically show the price effect, substitution effect and income effect on the consumption of goods X. Part 2 Employ the idea of Slustky Equation to answer the following questions: 1. Define normal goods, inferior goods and Giffen goods. 2. Using Slutsky Equation to explain why the price effect on normal goods has to be negative? 3. Why the demand for Giffen goods increases when its price goes up?
Paper For Above instruction
Introduction
Microeconomics seeks to understand how individuals and households make choices about the allocation of scarce resources among competing goods and services. Central to this analysis is the concept of utility maximization, wherein consumers select bundles of goods that maximize their satisfaction subject to their budget constraints. This paper explores a series of microeconomic concepts—including optimal consumption bundles, substitution and income effects, and the classification of goods—using a specific utility function and price changes as illustrative examples. Additionally, the Slutsky equation is employed to explain the nuanced responses of demand to price variations, particularly concerning normal, inferior, and Giffen goods.
Part 1: Utility Maximization and Price Effects
Initial Consumption and Budget Constraints
The utility function provided is \( U = X^{1/2}Y^{1/2} \), reflecting a Cobb-Douglas form where \( X \) and \( Y \) are two goods, and the parameters suggest consumer preferences where both goods are essential and consumed in a balanced proportion. The initial prices are \( P_X = 2 \) dollars for good X and \( P_Y = 4 \) dollars for good Y, with income \( M \) unspecified but assumed to be sufficient for optimization.
Optimal Bundle with Initial Prices
The consumer’s problem is:
\[
\text{Maximize } U = X^{1/2}Y^{1/2}
\]
\[
\text{subject to } 2X + 4Y = M
\]
Using the method of Lagrange multipliers, the first-order conditions lead to:
\[
\frac{\partial U}{\partial X} = \lambda P_X \quad \Rightarrow \quad \frac{1}{2}X^{-1/2}Y^{1/2} = \lambda \times 2
\]
\[
\frac{\partial U}{\partial Y} = \lambda P_Y \quad \Rightarrow \quad \frac{1}{2}X^{1/2}Y^{-1/2} = \lambda \times 4
\]
Dividing the two equations yields:
\[
\frac{Y}{X} = \frac{P_X}{P_Y} = \frac{2}{4} = \frac{1}{2}
\]
which implies:
\[
Y = \frac{1}{2}X
\]
Substituting into the budget constraint, the optimal bundle is:
\[
2X + 4 \times \frac{1}{2}X = M \quad \Rightarrow \quad 2X + 2X = M \quad \Rightarrow \quad 4X = M
\]
\[
X^* = \frac{M}{4}
\]
\[
Y^ = \frac{1}{2}X^ = \frac{M}{8}
\]
This optimal bundle balances both goods in proportion to their relative prices and the consumer's income.
Price Change and New Optimal Bundle
When prices change to \( P' = (4, 4) \), the consumer's problem becomes:
\[
\text{Maximize } U = X^{1/2}Y^{1/2}
\]
\[
\text{subject to } 4X + 4Y = M
\]
Similarly, the ratio of marginal utilities is:
\[
\frac{Y}{X} = \frac{P'_X}{P'_Y} = 1
\]
which implies:
\[
Y = X
\]
Substituting into the budget constraint:
\[
4X + 4X = M \quad \Rightarrow \quad 8X = M \quad \Rightarrow \quad X' = \frac{M}{8}
\]
and
\[
Y' = X' = \frac{M}{8}
\]
The consumer adjusts consumption to this new bundle where both goods are consumed equally due to identical prices.
Substitution and Income Effects
The substitution effect isolates the change in consumption due solely to the change in relative prices, holding utility constant. The income effect reflects the change in consumption resulting from the change in purchasing power due to price variation.
- Substitution Effect: Moving from the initial bundle \( (X^, Y^) \) to a point on the new budget line that maintains the original utility level, the consumer substitutes toward the relatively cheaper good, resulting in increased consumption of the less expensive good.
- Income Effect: As prices rise to \( P' \), the consumer’s real income or purchasing power decreases, leading to a reduction in consumption of both goods if they are normal.
Graphically, these effects are represented by the Slutsky decomposition, where the total change in demand can be split into substitution and income effects, illustrating the consumer's response to price variation.
Part 2: Goods Classification and Slutsky Equation
Definitions of Goods
- Normal Goods: Goods for which demand increases as income rises.
- Inferior Goods: Goods for which demand decreases as income rises.
- Giffen Goods: Inferior goods for which demand increases when the price increases, contrary to normal demand behavior.
Slutsky Equation and Normal Goods
The Slutsky equation mathematically decomposes the total effect of a price change into the substitution and income effects:
\[
\Delta X = \underbrace{\text{Substitution effect}}_{negative \; for normal \; goods} + \underbrace{\text{Income effect}}_{positive \; for \; normal \; goods}
\]
For normal goods, because demand tends to decrease with an increase in price, the substitution effect is normally negative (consumers tend to buy less as the price rises). The income effect further reduces demand because the higher price effectively reduces the consumer's real income, causing a decrease in quantity demanded.
Giffen Goods and Price Increase
Giffen goods exhibit an unusual demand response: when their price rises, demand also increases. This counterintuitive behavior occurs because the income effect dominates the substitution effect; the increase in price reduces real income so significantly that consumers are compelled to buy more of the inferior, Giffen good, often because they cannot afford more desirable substitutes.
Conclusion
Understanding the interplay of utility maximization, substitution, income effects, and goods classification provides valuable insights into consumer behavior. The Slutsky equation plays a crucial role in decomposing the total demand response to price changes, especially in explaining the peculiar demand patterns of Giffen goods. Accurate economic models incorporating these concepts are essential for predicting market behavior and informing policy decisions.
References
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- Microeconomic Theory: Basic Principles and Extensions. Cengage Learning.
- Microeconomic Theory. Oxford University Press.
- Microeconomics. W.W. Norton & Company.
- Microeconomics. Pearson.
- A Course in Microeconomic Theory. Princeton University Press.
- Microeconomic Analysis. W.W. Norton & Company.
- Intermediate Microeconomics. McGraw-Hill Education.
- Demand Theory and Its Applications. Routledge.
- Advanced Microeconomic Theory. Pearson.