Microeconomics Study Guidance: Elasticity, Demand, Utility, and Consumer Choice

Q1. How should you allocate your $1 million advertising budget to maximize beer sales? Do TV, Radio, and Internet advertising exhibit diminishing returns?

Background

Topic: Production and Diminishing Returns

This question tests your understanding of how to allocate resources to maximize output and whether the production function exhibits diminishing returns.

Key Terms and Formulas:

• Diminishing Returns: The principle that as more of a variable input is added to a fixed input, the marginal product of the variable input eventually decreases.

• Marginal Product: The additional output generated by spending an additional $100K on advertising.

Step-by-Step Guidance

1. Examine the table to determine the marginal increase in beer sales for each $100K spent on TV, Radio, and Internet advertising.

2. Compare the marginal product for each advertising type at each spending level to identify which provides the greatest increase in sales per dollar spent.

3. Allocate your budget sequentially, starting with the advertising type that gives the highest marginal product for the first $100K, then the next highest for the second $100K, and so on, until the $1 million is spent.

4. To determine if diminishing returns are present, observe whether the marginal increase in sales decreases as more money is spent on each advertising type.

Try solving on your own before revealing the answer!

Final Answer:

Allocate the first $100K to Internet advertising, as it yields the highest marginal increase in sales. Continue allocating funds based on the highest marginal product at each increment. All advertising types exhibit diminishing returns, as the marginal increase in sales decreases with additional spending.

Q2. Given data on price and quantity for gasoline before and after a tax, calculate the price elasticity of demand and fit a demand function.

Background

Topic: Price Elasticity of Demand

This question tests your ability to calculate price elasticity and use it to derive a demand function.

Key Terms and Formulas:

• Price Elasticity of Demand: Measures the responsiveness of quantity demanded to a change in price.

• $\text{Elasticity} = \frac{\%\text{ change in quantity}}{\%\text{ change in price}}$

• Point elasticity formula: $\varepsilon = \frac{\partial Q}{\partial P} \cdot \frac{P}{Q}$

Step-by-Step Guidance

1. Identify the initial and new price and quantity values from the table (before and after the tax).

2. Calculate the percentage change in price and quantity using the midpoint method.

3. Apply the elasticity formula to find the price elasticity of demand.

4. Use the calculated elasticity to fit the data into a demand function, such as a constant elasticity or linear form.

Try solving on your own before revealing the answer!

Final Answer:

The price elasticity of demand is calculated using the midpoint method. The demand function can be fitted using the elasticity value and the observed data points.

Q3. The demand and supply for automobiles is given by $Q^d = 0.1P^{-2.1}$ and $Q^s = 6400P^{0.9}$. Calculate equilibrium price and quantity, and analyze elasticity values.

Background

Topic: Market Equilibrium and Elasticity

This question tests your ability to solve for equilibrium in a market and interpret elasticity values.

Key Terms and Formulas:

• Market Equilibrium: Occurs where quantity demanded equals quantity supplied.

• Set $Q^d = Q^s$ and solve for $P$.

• Elasticity: Measures responsiveness of demand or supply to changes in price, income, or other factors.

Step-by-Step Guidance

1. Set the demand and supply equations equal to each other: $0.1P^{-2.1} = 6400P^{0.9}$.

2. Combine like terms and solve for $P$ algebraically.

3. Once you have $P$, substitute back into either equation to solve for $Q$.

4. Interpret the elasticity values given for price, income, and wage, and relate them to the market context.

Try solving on your own before revealing the answer!

Final Answer:

The equilibrium price and quantity are found by solving the system of equations. Elasticity values indicate how sensitive demand and supply are to changes in price, income, and wage.

Q4. Billy Joe's preferences for two goods (x and y) are represented by different utility functions. Analyze monotonicity, convexity, and marginal utility.

Background

Topic: Consumer Preferences and Utility

This question tests your understanding of utility functions, monotonicity, convexity, and marginal utility.

Key Terms and Formulas:

• Monotonicity: Preferences are monotonic if more of a good is always preferred.

• Convexity: Preferences are convex if mixtures of bundles are preferred to extremes.

• Marginal Utility: The additional utility from consuming one more unit of a good.

• Utility function example: $U(X, Y) = X^{1/8}Y^{1/10}$

Step-by-Step Guidance

1. Examine the utility function to determine if it is monotonic (does utility increase as X or Y increases?).

2. Check for convexity by considering whether combinations of X and Y yield higher utility than extremes.

3. Calculate marginal utility for X and Y by taking partial derivatives of the utility function.

4. Analyze whether marginal utility diminishes as X or Y increases.

Try solving on your own before revealing the answer!

Final Answer:

The utility function is monotonic and convex if both exponents are positive and less than one. Marginal utility diminishes as X or Y increases.

Q5. Using the Bureau of Labor Statistics table, characterize U.S. consumer preferences for food and clothing. Are they Cobb-Douglas? What about income elasticity?

Background

Topic: Consumer Choice and Income Elasticity

This question tests your ability to interpret real-world data and relate it to theoretical models like Cobb-Douglas preferences.

Key Terms and Formulas:

• Cobb-Douglas Preferences: Utility functions of the form $U(X, Y) = X^aY^b$.

• Income Elasticity of Demand: Measures how quantity demanded changes as income changes.

Step-by-Step Guidance

1. Compare the percentage change in expenditures for food and clothing across income groups.

2. Determine if the income elasticity of demand for each good is equal to one (a characteristic of Cobb-Douglas preferences).

3. Analyze whether the data supports the Cobb-Douglas model for U.S. consumer preferences.

Try solving on your own before revealing the answer!

Final Answer:

Consumer preferences for food and clothing are Cobb-Douglas if income elasticity is equal to one. The data suggests whether this is the case based on expenditure changes.