Problem Set 2 Econ 203 Intermediate Microeconomics ✓ Solved

Problem Set 2 Econ 203 Intermediate Microeconomicsplease Show All T

This assignment involves analyzing consumer behavior through budget constraints, indifference curves, demand functions, and utility maximization. You are asked to illustrate and decompose changes in consumer choice due to price variations, examine demand elasticities, analyze utility maximization under different utility functions, and graphically represent various market scenarios for normal, inferior, and Giffen goods. Carefully show all work, including graphs and calculations, to demonstrate understanding of core microeconomic principles.

Sample Paper For Above instruction

Introduction

Consumer choice theory forms a cornerstone of microeconomics, providing insights into how individuals allocate their limited income across various goods to maximize utility. This paper explores the theoretical and graphical analysis of consumer behavior in response to price changes, demand functions, and utility optimization, with particular emphasis on the concepts of substitution and income effects, elasticity, and the classification of goods as normal, inferior, or Giffen. To elucidate these concepts, detailed calculations, decompositions, and graph constructions are presented, illustrating the intricate relationship between prices, income, and consumer preferences.

Question 1: Effect of a Price Increase on Consumer Choice

The first problem examines how an increase in the price of steak influences a consumer's choice, represented through budget constraints and indifference curves. Initially, the consumer faces a budget line A, with a utility-maximizing bundle at point a. When the price of steak rises, the budget line shifts to B, prompting a new utility-maximizing point b. To understand this, we must analyze how the consumer's optimal bundle shifts and decompose this movement into substitution and income effects.

Graphical depiction

The initial budget line A touches the indifference curve at point a, representing the original optimal bundle with consumer choice (x, y). When the price of steak increases, the budget line B pivots inward, reflecting reduced purchasing power. The new tangency point at b indicates the new optimal bundle. The movement from a to b can be split into two segments: first, a substitution effect where the consumer adjusts their consumption to maintain utility relative to changing prices, and second, an income effect reflecting the change in purchasing power resulting from the price increase.

Decomposition into substitution and income effects

To decompose, draw an income-compensated budget line parallel to B that is tangent to the original indifference curve. The point where this line touches the original indifference curve is the 'halfway point' m. The move from a to m represents the substitution effect (keeping utility constant), while the shift from m to b captures the income effect. Analyzing these segments reveals whether steak is a normal or inferior good, based on whether the quantity demanded increases or decreases with income.

Assessment of steak as a normal or inferior good

If the quantity of steak decreases after the price increase (from a to b), it suggests that steak is a normal good—demand falls as the real purchasing power declines. Conversely, if demand increases, steak may be classified as an inferior good. The shape of the indifference curves and the direction of the movement aid in this determination.

Could an increase in steak’s price increase demand?

In standard demand theory, an increase in price generally reduces demand for a normal good. However, for inferior or Giffen goods, an unexpected increase in demand could occur. Giffen goods are inferior goods for which the income effect outweighs the substitution effect, leading to a paradoxical increase in demand with rising prices. Therefore, under certain conditions—specifically, in Giffen scenarios—an increase in the price of steak could indeed result in higher demand.

Question 2: Market Demand and Price Elasticity

Next, the demand functions for Travis and Maya are provided, and the task is to derive market demand, compute elasticity, and analyze revenue effects.

Market demand function

Travis’s demand: QT = 12 – 3P; Maya’s demand: QM = 6 – P. The market demand is the sum of individual demands at each price point, with attention to demand cessation at zero quantities.

Calculating combined demand:

  • At prices where demand remains positive, sum the quantities: QT + QM = (12 – 3P) + (6 – P) = 18 – 4P.
  • Demand is zero for individual demands when the quantity would be negative, so the combined demand is valid for P ≤ 3.

Price elasticity at P = 2

Market demand at P=2 is (18 – 4*2) = 10 units. Price elasticity of demand (E) is calculated as:

  • E = (dQ/dP) * (P/Q).
  • dQ/dP = -4.
  • At P=2, Q=10, so E = -4 * (2/10) = -0.8.

Demand nature at P=2

The magnitude of elasticity |E|=0.8, which indicates demand is inelastic at this price because |E|

Total revenue implications

When demand is inelastic, a price increase leads to a rise in total revenue because the percentage decrease in quantity demanded is less than the percentage increase in price.

Question 3: Utility Maximization with Logarithmic Utility

Bob’s utility function: U(x,y) = ln y + ε * ln x. With given prices and income, we analyze optimal consumption, effect of price changes, and the utility function’s properties.

Marginal rate of substitution (MRS)

The MRS is the rate at which Bob is willing to substitute y for x, holding utility constant, derived from partial derivatives:

MRS = (∂U/∂x) / (∂U/∂y) = (ε / x) / (1 / y) = ε * y / x.

Equating MRS to price ratio to find optimal bundles

At optimality: MRS = price ratio = p_x / p_y.

Consumption bundle calculations

Given initial prices and income, solving the consumer’s problem yields initial units of x and y. Increasing p_y to 2 and 3 impacts the optimal choice, altering the consumption bundle accordingly.

Effect of changing utility functions

If the utility function is scaled or modified, the MRS changes proportionally. The qualitative nature of the demand—whether increasing or decreasing with price—remains consistent, provided the form remains logarithmic, because adjustments in utility scale do not alter the optimization conditions qualitatively.

Question 4: Graphical Illustrations for Different Good Types

Construct diagrams demonstrating cases where x is a normal good, an inferior good, and a Giffen good. For each, depict budget lines, indifference curves, and the shape of the income- and price-consumption curves. These visualizations illustrate theoretical distinctions, showing how the consumer's choice responds to shifts in income and price, and how demand curves are shaped accordingly.

Conclusion

This analysis underscores the complexity underlying consumer decision-making. Understanding the mechanics of substitution and income effects, elasticity, and the classification of goods helps predict behavioral responses to market changes. Graphical representations reinforce these principles, providing visual clarity that complements quantitative analysis. Recognizing conditions for Giffen goods is particularly important, illustrating exceptions within demand theory that challenge naïve assumptions about price and demand relationships.

References

  • Varian, H. R. (2014). Intermediate Microeconomics: A Modern Approach. W.W. Norton & Company.

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