Microeconomics Problem Set Guidance: Marginal Analysis, Elasticity, and Consumer Theory

Q1. Marginal Analysis in Advertising Allocation

Background

Topic: Marginal Analysis and Optimal Allocation

This question tests your understanding of how a maximizing agent (like a firm) should allocate resources (here, advertising budget) across different options to maximize returns, using the concept of marginal benefit versus marginal cost.

Key Terms and Formulas

• Marginal Benefit (MB): The additional benefit (e.g., sales) from spending one more unit of resource (e.g., $100K on advertising).

• Marginal Cost (MC): The additional cost incurred from increasing the resource by one unit.

• Optimal Allocation Rule: Allocate the next unit of resource to the option with the highest marginal benefit, as long as MB >= MC.

Step-by-Step Guidance

1. List the marginal benefits (additional sales) for each advertising channel (radio, TV, internet) for the first $100K spent.

2. Identify which channel offers the highest marginal benefit for the first $100K and allocate accordingly.

3. For the second $100K, recalculate the marginal benefits for each channel, considering diminishing returns (if provided), and decide where to allocate the next increment.

4. Continue this process, always comparing the marginal benefits for each channel at each increment, and allocate the budget to the channel with the highest remaining marginal benefit.

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Q2. Demand Curve, Elasticity, and Tax Incidence

Background

Topic: Demand Curve Estimation, Elasticity, and Tax Incidence

This question reviews how to calculate the price elasticity of demand using two points, derive both linear and constant elasticity demand curves, and analyze the effect of a per-unit tax on equilibrium and tax revenue.

Key Terms and Formulas

• Price Elasticity of Demand: $\varepsilon = \frac{\%\ \text{change in } Q}{\%\ \text{change in } P}$

• Linear Demand Curve: $Q = a - bP$

• Constant Elasticity Demand Curve: $Q = AP^{\varepsilon}$

• Tax Revenue: $\text{Tax Revenue} = \text{Tax per unit} \times \text{Quantity sold after tax}$

Step-by-Step Guidance

1. Use the two given points on the demand curve to calculate the slope and intercept for the linear demand curve.

2. Calculate the point elasticity of demand at one of the points using the elasticity formula.

3. Set up the constant elasticity demand curve using the elasticity value and solve for the constant $A$ using one of the points.

4. For the tax incidence part, use the given tax revenue and after-tax quantity to solve for the per-unit tax.

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Q3. Log-Linear Demand and Supply, Elasticity, and Comparative Statics

Background

Topic: Log-Linear (Constant Elasticity) Demand and Supply, Elasticity Calculation, and Comparative Statics

This question focuses on recognizing and working with log-linear (constant elasticity) demand and supply functions, calculating elasticities directly from coefficients, and performing comparative statics to analyze equilibrium changes.

Key Terms and Formulas

• Log-Linear Demand: $\ln Q^D = a + b \ln P + c \ln Y$

• Elasticity: The coefficient on $\ln P$ or $\ln Y$ gives the price or income elasticity, respectively.

• Comparative Statics: Analyzing how equilibrium price and quantity change when an exogenous variable (like income) shifts.

Step-by-Step Guidance

1. Write the demand and supply equations in log-linear form and identify the coefficients representing elasticities.

2. Calculate the price and income elasticities directly from the demand equation.

3. Set demand equal to supply to solve for equilibrium price and quantity before the income change.

4. Repeat the equilibrium calculation after the income change, and compare the results to analyze the effect of the shift.

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Q4. Indifference Maps and Preference Structures

Background

Topic: Consumer Preferences, Indifference Curves, and Utility Representation

This question asks you to interpret and sketch indifference maps for different types of preferences (goods, bads, perfect complements, perfect substitutes), and to understand how these preferences affect the shape of indifference curves.

Key Terms and Concepts

• Indifference Curve: A curve showing all combinations of two goods that provide the same utility to the consumer.

• Perfect Complements: Goods that are always consumed in fixed proportions; indifference curves are L-shaped.

• Perfect Substitutes: Goods that can be substituted at a constant rate; indifference curves are straight lines.

• Bad: A good that decreases utility as its quantity increases; indifference curves slope upward for a bad and a good.

Step-by-Step Guidance

1. Identify the type of preference structure described (good, bad, perfect complement, perfect substitute).

2. Draw or visualize the corresponding indifference curves for each case, noting their shape and slope.

3. Explain how the consumer's utility changes as you move along or between indifference curves for each preference type.

4. Relate the shape of the indifference curves to the consumer's willingness to substitute between the two goods.

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Q5. Monotonicity, Convexity, and Utility Maximization

Background

Topic: Consumer Preferences, Monotonicity, Convexity, and Utility Maximization

This question explores the axioms of rational choice, specifically monotonicity ("more is better") and convexity ("averages are preferred to extremes"), and asks you to verify these properties for a given utility function. It also introduces the Lagrangian method for finding optimal demands.

Key Terms and Formulas

• Strict Monotonicity: More of a good always increases utility.

• Strict Convexity: Averages of bundles are strictly preferred to extremes.

• Marginal Rate of Substitution (MRS): $MRS = \frac{MU_X}{MU_Y}$

• Lagrangian Method: $\mathcal{L} = U(X, Y) + \lambda (I - P_X X - P_Y Y)$

Step-by-Step Guidance

1. Check the utility function for strict monotonicity by verifying that the marginal utilities with respect to both goods are positive.

2. Check for strict convexity by analyzing whether the MRS diminishes as you move along an indifference curve.

3. Set up the Lagrangian for the consumer's utility maximization problem, including the budget constraint.

4. Take the first-order conditions with respect to $X$, $Y$, and $\lambda$ to derive the system of equations for optimal demands.

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Q6. Cobb-Douglas Preferences and Expenditure Shares

Background

Topic: Cobb-Douglas Utility, Expenditure Shares, and Income Elasticity

This question examines the property of Cobb-Douglas preferences that leads to constant expenditure shares on each good, and asks you to relate this to real-world consumption patterns (e.g., food vs. entertainment) and income elasticity.

Key Terms and Formulas

• Cobb-Douglas Utility: $U(X, Y) = X^{\alpha} Y^{1-\alpha}$

• Expenditure Share: The fraction of income spent on each good, determined by the exponents in the utility function.

• Income Elasticity of Demand: $\varepsilon_I = \frac{\%\ \text{change in } Q}{\%\ \text{change in income}}$

Step-by-Step Guidance

1. Recall that with Cobb-Douglas preferences, the consumer spends a constant share of income on each good, regardless of income level.

2. Calculate the percentage change in income using the median values of the given income ranges.

3. Compare the percentage change in consumption of each good to the percentage change in income to assess income elasticity.

4. Discuss whether the constant expenditure share property holds for all goods in reality, especially for necessities like food.

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Q7. Utility Maximization with Lagrangian and Corner Solutions

Background

Topic: Utility Maximization, Lagrangian Method, and Corner Solutions

This question asks you to analyze a utility function for linearity, monotonicity, and convexity, and then use the Lagrangian method to find optimal demands. It also explores the possibility of corner solutions, where the consumer may choose to consume only one good.

Key Terms and Formulas

• Lagrangian for Utility Maximization: $\mathcal{L} = U(X, Y) + \lambda (I - P_X X - P_Y Y)$

• First-Order Conditions: Set the partial derivatives of the Lagrangian with respect to $X$, $Y$, and $\lambda$ equal to zero.

• Corner Solution: Occurs when the optimal bundle involves consuming only one good (either $X^* = 0$ or $Y^* = 0$).

• Marginal Utility of Income: The value of the Lagrange multiplier $\lambda$.

Step-by-Step Guidance

1. Analyze the utility function to determine if it is linear in $X$ or $Y$, and check for monotonicity and convexity.

2. Set up the Lagrangian for the consumer's problem, including the budget constraint.

3. Take the first-order conditions with respect to $X$, $Y$, and $\lambda$ to derive the necessary equations for optimal demands.

4. Consider the conditions under which a corner solution would arise, and how to check if the interior solution is feasible.

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