Microeconomics Exam Study Guide: Consumer Preferences, Utility, Subsidies, Market Equilibrium, and Oligopoly Models

Q1. How does a rational and insatiable consumer with strictly convex preferences compare baskets x3=(0,2), x4=(6,2), x5=(1,4), x6=(4,2), x7=(2,2) to basket x1=(1,3)?

Background

Topic: Consumer Preferences and Indifference Curves

This question tests your understanding of rational, insatiable, and strictly convex preferences, and how a consumer ranks different consumption bundles relative to a reference bundle.

Key Terms:

• Rational preferences: Complete and transitive.

• Insatiable: More is always preferred to less.

• Strictly convex: Consumers prefer averages to extremes.

• Indifference curve: Set of bundles equally preferred.

Step-by-Step Guidance

1. List the coordinates of each basket and compare them to x1=(1,3).

2. Recall that insatiability means any basket with more of both goods is preferred to x1.

3. Strict convexity implies that mixtures of baskets are preferred to extremes.

4. Check if each basket has more, less, or equal amounts of goods compared to x1.

5. Consider if any basket is a convex combination of x1 and another basket, and what that implies for preference ranking.

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Q2. Given the utility function $u(x_1, x_2) = x_1^{1/2} x_2^{1/2}$, find the Hicksian demand functions and the expenditure function.

Background

Topic: Hicksian (Compensated) Demand and Expenditure Function

This question tests your ability to derive the Hicksian demand (minimizing expenditure for a given utility) and the expenditure function for a Cobb-Douglas utility function.

Key Terms and Formulas:

• Hicksian demand: $h_i(p_1, p_2, u)$

• Expenditure function: $e(p_1, p_2, u)$

• Cobb-Douglas utility: $u(x_1, x_2) = x_1^{\frac{1}{2}} x_2^{\frac{1}{2}}$

Step-by-Step Guidance

1. Set up the expenditure minimization problem: Minimize $p_1 x_1 + p_2 x_2$ subject to $x_1^{\frac{1}{2}} x_2^{\frac{1}{2}} = u$.

2. Write the Lagrangian: $\mathcal{L} = p_1 x_1 + p_2 x_2 + \lambda (u - x_1^{\frac{1}{2}} x_2^{\frac{1}{2}})$.

3. Take partial derivatives with respect to $x_1$, $x_2$, and $\lambda$, and set them equal to zero.

4. Solve the first-order conditions to express $x_1$ and $x_2$ in terms of $p_1$, $p_2$, and $u$ (Hicksian demand functions).

5. Plug the Hicksian demands back into the expenditure function $e(p_1, p_2, u) = p_1 x_1^* + p_2 x_2^*$.

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Q3. For utility $u(x_1, x_2) = x_1 x_2$, prices $p_1 = 8$, $p_2 = 2$, income $M = 20$, and a unit tax $t = 6$ on good 2, what is the minimum lump-sum subsidy needed so the consumer is not worse off?

Background

Topic: Taxation, Subsidies, and Consumer Welfare

This question tests your ability to analyze the impact of a unit tax and calculate the compensating subsidy required to maintain utility.

Key Terms and Formulas:

• Utility function: $u(x_1, x_2) = x_1 x_2$

• Budget constraint before tax: $8x_1 + 2x_2 = 20$

• Budget constraint after tax: $8x_1 + (2+6)x_2 = 20 + S$ (where $S$ is the subsidy)

• Compensating variation: Minimum subsidy to keep utility unchanged.

Step-by-Step Guidance

1. Find the consumer's optimal bundle before the tax using the original budget constraint.

2. Calculate the utility at this bundle.

3. Set up the new budget constraint with the tax and unknown subsidy $S$.

4. Find the new optimal bundle under the taxed prices and subsidy.

5. Set the utility at the new bundle equal to the original utility and solve for $S$.

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Q4. The market demand for a perfectly competitive industry with constant costs is $Y_d(p) = 4400 - 200p$. The short-run average cost curves are $SAC(y, K) = \frac{2}{3}y + \frac{23}{4}y + 40 - K + K^2 y$. Find the equilibrium price and number of firms in the long run.

Background

Topic: Perfect Competition, Long-Run Equilibrium, and Cost Functions

This question tests your ability to analyze market equilibrium in a perfectly competitive industry, using demand and cost functions to find equilibrium price and firm count.

Key Terms and Formulas:

• Market demand: $Y_d(p) = 4400 - 200p$

• Short-run average cost: $SAC(y, K)$

• Long-run equilibrium: $p = min AC$ and firms earn zero profit.

Step-by-Step Guidance

1. Find the minimum of the average cost function to determine the long-run equilibrium price.

2. Set $p = min AC$ and equate to the market demand to solve for total output $Y$.

3. Determine the output per firm at the minimum average cost.

4. Calculate the number of firms by dividing total output by output per firm.

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Q5. The inverse market demand is $p = 100 - Y$, and both firms have cost functions $c(y) = 28y$. Find and sketch the Cournot, Stackelberg, collusive, and perfect competition equilibria in the long run.

Background

Topic: Oligopoly Models (Cournot, Stackelberg, Collusion, Perfect Competition)

This question tests your understanding of different oligopoly models and how to derive equilibrium outputs and prices for each.

Key Terms and Formulas:

• Inverse demand: $p = 100 - Y$

• Cost function: $c(y) = 28y$

• Cournot equilibrium: Firms choose quantities simultaneously.

• Stackelberg equilibrium: One firm is leader, the other follower.

• Collusive equilibrium: Firms maximize joint profit.

• Perfect competition: Price equals marginal cost.

Step-by-Step Guidance

1. For Cournot: Set up each firm's profit function and solve for Nash equilibrium quantities.

2. For Stackelberg: Solve for the leader's optimal output, anticipating the follower's response.

3. For collusion: Maximize joint profit as a single monopolist.

4. For perfect competition: Set $p = MC$ and solve for total output.

5. Sketch the equilibria in output space, showing differences between models.

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