Harvard ECON1010A - Intermediate Microeconomics: Game Theory & Consumer Choice Flashcards

Question 1

Axioms of Consumer Preference

Accepted Answer

To be represented by a continuous utility function, consumer preferences must be:
1. **Complete**: For any goods $A$ and $B$, $A \succ B$, $B \succ A$, or $A \sim B$.
2. **Transitive**: If $A \succ B$ and $B \succ C$, then $A \succ C$.
3. **Monotonic** (More is better): If $A$ contains strictly more of at least one good and no less of any other, $A \succ B$.

Question 2

Marginal Rate of Substitution (MRS)

Accepted Answer

The rate at which a consumer is willing to substitute good $y$ for good $x$ while maintaining the same utility level. Mathematically defined as:
$$MRS_{x,y} = -\frac{MU_x}{MU_y} = -\frac{\partial U / \partial x}{\partial U / \partial y}$$
Graphically, it represents the slope of the indifference curve.

Question 3

Consumer Utility Maximization Condition

Accepted Answer

For an interior solution with well-behaved convex preferences, optimal consumption occurs where the indifference curve is tangent to the budget constraint $p_x x + p_y y = I$:
$$MRS_{x,y} = -\frac{p_x}{p_y} \implies \frac{MU_x}{p_x} = \frac{MU_y}{p_y}$$

Question 4

Lagrangian Method for Utility Maximization

Accepted Answer

To maximize utility $U(x,y)$ subject to budget constraint $p_x x + p_y y \le I$, formulate the Lagrangian:
$$\mathcal{L}(x, y, \lambda) = U(x,y) + \lambda (I - p_x x - p_y y)$$
Set first-order conditions (FOCs) to zero:
1. $\frac{\partial \mathcal{L}}{\partial x} = MU_x - \lambda p_x = 0$
2. $\frac{\partial \mathcal{L}}{\partial y} = MU_y - \lambda p_y = 0$
3. $\frac{\partial \mathcal{L}}{\partial \lambda} = I - p_x x - p_y y = 0$

Question 5

Cobb-Douglas Utility Function & Demand

Accepted Answer

For a Cobb-Douglas utility function $U(x,y) = x^a y^b$:
- Expenditure shares are constant: a fraction $\frac{a}{a+b}$ of income $I$ is spent on $x$.
- **Marshallian Demand for $x$**:
$$x^*(p_x, p_y, I) = \left(\frac{a}{a+b}\right) \frac{I}{p_x}$$
- **Marshallian Demand for $y$**:
$$y^*(p_x, p_y, I) = \left(\frac{b}{a+b}\right) \frac{I}{p_y}$$

Question 6

Indirect Utility Function $V(p, I)$ & Roy's Identity

Accepted Answer

The indirect utility function $V(p_x, p_y, I)$ gives maximum utility achievable at prices $p$ and income $I$.

**Roy's Identity** recovers Marshallian demand from indirect utility:
$$x^*(p_x, p_y, I) = -\frac{\partial V / \partial p_x}{\partial V / \partial I}$$

Question 7

Expenditure Minimization Problem (EMP) & Hicksian Demand

Accepted Answer

Dual problem to utility maximization: $\min_{x,y} p_x x + p_y y$ subject to $U(x,y) \ge u$.
- Yields **Hicksian (compensated) demand functions** $h_x(p_x, p_y, u)$.
- Expenditure function: $E(p_x, p_y, u) = p_x h_x + p_y h_y$.
- **Shephard's Lemma**: $h_x(p_x, p_y, u) = \frac{\partial E}{\partial p_x}$.

Question 8

Slutsky Equation

Accepted Answer

Decomposes the total effect of a price change on demand into Substitution Effect (SE) and Income Effect (IE):
$$\frac{\partial x_i}{\partial p_i} = \underbrace{\frac{\partial h_i}{\partial p_i}}_{SE \ (\le 0)} - \underbrace{x_i \frac{\partial x_i}{\partial I}}_{IE}$$
- For normal goods: SE and IE reinforce each other.
- For inferior goods: SE and IE work in opposite directions.

Question 9

Expected Utility & Risk Preferences

Accepted Answer

For uncertain outcomes with payoffs $w_i$ and probabilities $p_i$, expected utility is $EU = \sum p_i U(w_i)$.
- **Risk Averse**: $U''(w) < 0$ (Concave), prefers $E[w]$ for sure over the gamble.
- **Risk Neutral**: $U''(w) = 0$ (Linear).
- **Risk Loving**: $U''(w) > 0$ (Convex).

Question 10

Monopoly Profit Maximization & Lerner Index

Accepted Answer

A monopolist maximizes profit where $MR = MC$. Since $MR = p\left(1 + \frac{1}{\epsilon}\right)$ (where $\epsilon < 0$ is price elasticity of demand):
$$p\left(1 + \frac{1}{\epsilon}\right) = MC$$
**Lerner Index of Market Power**:
$$L = \frac{p - MC}{p} = -\frac{1}{\epsilon}$$

Question 11

Degrees of Price Discrimination

Accepted Answer

- **First-Degree (Perfect)**: Charges each customer their exact willingness to pay. Consumer surplus = 0; allocation is Pareto efficient.
- **Second-Degree**: Charges different prices based on quantity/quality tiers (self-selection, e.g., bulk discounts).
- **Third-Degree**: Separates markets into distinct demographic groups based on price elasticity $\epsilon_i$. Charges higher price to less elastic groups: $p_1\left(1 + \frac{1}{\epsilon_1}\right) = p_2\left(1 + \frac{1}{\epsilon_2}\right) = MC$.

Question 12

Cournot Duopoly Model

Accepted Answer

Firms simultaneously choose quantities $q_1, q_2$. Aggregate price is $P(Q)$ where $Q = q_1 + q_2$.
- Each firm maximizes $\pi_i(q_i, q_j) = P(q_i + q_j)q_i - C_i(q_i)$.
- Solving $\frac{\partial \pi_i}{\partial q_i} = 0$ yields **Best-Response Functions** $r_1(q_2)$ and $r_2(q_1)$.
- Cournot-Nash Equilibrium is the intersection $(q_1^*, q_2^*)$ of both reaction curves.

Question 13

Bertrand Duopoly Model

Accepted Answer

Firms simultaneously set prices $p_1, p_2$ for homogenous products with identical marginal cost $MC$.
- Consumers buy exclusively from the firm with the lower price.
- Unique Nash Equilibrium: $p_1^* = p_2^* = MC$.
- Result: Zero economic profit, matching perfectly competitive market outcome despite having only two firms (Bertrand Paradox).

Question 14

Stackelberg Oligopoly Model

Accepted Answer

Sequential quantity competition where Firm 1 (Leader) chooses $q_1$ first, and Firm 2 (Follower) observes $q_1$ and chooses $q_2$.
- Solved via **backward induction**:
  1. Find Follower's reaction function $q_2 = r_2(q_1)$.
  2. Substitute $r_2(q_1)$ directly into Leader's profit function $\pi_1(q_1, r_2(q_1))$.
  3. Maximize $\pi_1$ with respect to $q_1$.
- The leader gains a first-mover advantage and produces higher output than in Cournot.

Question 15

Pure Strategy Nash Equilibrium (PSNE)

Accepted Answer

A strategy profile $(s_1^*, s_2^*, \dots, s_n^*)$ where no player can unilaterally deviate to achieve a strictly higher payoff:
$$u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*) \quad \forall s_i \in S_i, \forall i$$

Question 16

Mixed Strategy Nash Equilibrium (MSNE)

Accepted Answer

An equilibrium where players randomize over pure strategies according to probability distributions.

**Indifference Principle**: Player $A$ chooses probabilities for their actions such that Player $B$ is indifferent between all pure strategies played with positive probability in equilibrium.

Question 17

Subgame Perfect Nash Equilibrium (SPNE)

Accepted Answer

A refinement of Nash Equilibrium for dynamic/extensive-form games. A strategy profile is an SPNE if it induces a Nash Equilibrium in **every subgame** of the original game.

Solved using **backward induction**, which eliminates non-credible threats.

Question 18

Payoff Matrix & Dominant Strategies

Accepted Answer

Consider a standard normal form game:
| | Strategy L | Strategy R |
|---|---|---|
| **Strategy U** | $(a_1, a_2)$ | $(b_1, b_2)$ |
| **Strategy D** | $(c_1, c_2)$ | $(d_1, d_2)$ |

- **Strictly Dominant Strategy**: A strategy that yields a strictly higher payoff regardless of what opponent chooses.
- If $a_1 > c_1$ and $b_1 > d_1$, Strategy U strictly dominates Strategy D for Player 1.

Question 19

Edgeworth Box & Pareto Efficiency

Accepted Answer

A graphical representation of a 2-consumer, 2-good exchange economy ($A, B$ and $x, y$).
- **Pareto Efficiency**: An allocation where no person can be made better off without making someone else worse off.
- Tangency condition for interior Pareto efficient allocations:
$$MRS_{x,y}^A = MRS_{x,y}^B$$

Question 20

First Fundamental Theorem of Welfare Economics

Accepted Answer

If preferences are locally non-satiated and markets are complete with no transaction costs or externalities, any competitive (Walrasian) equilibrium allocation is **Pareto efficient**.

Question 21

Second Fundamental Theorem of Welfare Economics

Accepted Answer

Given convex preferences and technology, any Pareto efficient allocation can be supported as a **competitive equilibrium** through a suitable initial redistribution of endowments via lump-sum transfers.

Question 22

Walras' Law

Accepted Answer

In a general equilibrium framework, the total value of aggregate excess demand across all markets is identically zero for any price vector $p$:
$$\sum_{i=1}^k p_i z_i(p) = 0$$
*Implication*: If $k-1$ markets are in equilibrium, the $k$-th market must also be in equilibrium.

Harvard ECON1010A - Intermediate Microeconomics: Game Theory & Consumer Choice

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