Microeconomics: Preferences &

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### Consumer Preferences Consumer theory models how individuals make choices to maximize their well-being. A **decision-maker (DM)** is assumed to: - Compare and rank commodity bundles. - Identify the most preferred alternative. - Choose this alternative from available options. For any two bundles (x, y), a DM can state: - x is at least as good as y ($x \succsim y$) - y is at least as good as x ($y \succsim x$) - Both (indifference) ### Rational Weak Preferences A weak preference relation ($\succsim$) is considered **rational** if it satisfies three properties: #### 1. Reflexivity (P1) For any commodity bundle $a \in X$, it must be that $a \succsim a$. - A bundle is at least as good as itself. This is a "sanity check" requirement. #### 2. Completeness (P2) For any two distinct bundles $a, b \in X$, at least one of the following must be true: - $a \succsim b$ (a is at least as good as b) - $b \succsim a$ (b is at least as good as a) - Both $a \succsim b$ and $b \succsim a$ (indifference) - This means any two bundles can be compared. #### 3. Transitivity (P3) For any three bundles $a, b, c \in X$: - If $a \succsim b$ and $b \succsim c$, then it must be that $a \succsim c$. - This ensures consistency in preferences. ### Strict and Indifference Preferences Weak preferences ($\succsim$) can be decomposed into: #### Strict Preference ($>$) $x > y$ (x is strictly preferred to y) if and only if $x \succsim y$ and $\neg (y \succsim x)$. - x is at least as good as y, and y is NOT at least as good as x. #### Indifference ($\sim$) $x \sim y$ (x is indifferent to y) if and only if $x \succsim y$ and $y \succsim x$. - x is at least as good as y, and y is at least as good as x. ### Most Preferred Bundle **Lemma:** Let $X$ be a non-empty finite set and let $\succsim$ be a rational weak preference relation on $X$. There exists at least one member $x_0$ of $X$ such that for all $y \in X$, $x_0 \succsim y$. - In a finite set, a rational consumer can always find a most preferred bundle. ### Utility Representation Preferences as binary relations can be abstract. A **utility function** ($u: X \to \mathbb{R}$) provides a concrete way to represent preferences. - A utility function $u$ represents a consumer's preferences if $a \succsim b \iff u(a) \ge u(b)$. - It makes comparing choices easier (e.g., $u(a)=5, u(b)=1 \implies a$ is preferred to $b$). - It simplifies thinking about ordering and finding optimal choices. #### Utility Representation Theorem Suppose an agent's weak preferences $\succsim$ are rational and $X$ is finite. Then there exists a utility function $u: X \to \mathbb{R}$ which represents $\succsim$. - Moreover, if $g: \mathbb{R} \to \mathbb{R}$ is an increasing function, then the composition $g \circ u$ also represents the same preference relation. #### Debreu's Theorem (for continuous commodity spaces) Suppose the agent's weak preferences $\succsim$ are rational, continuous, and $X \subset \mathbb{R}^k$. Then there exists a continuous utility function $u: X \to \mathbb{R}$ which represents $\succsim$. - Without continuity, a utility representation may not exist (e.g., Lexicographic Preferences). ### Lexicographic Preferences Lexicographic preferences on $\mathbb{R}^2_+$ are defined for $x = (x_1, x_2)$ and $y = (y_1, y_2)$ as $x \succsim y$ if: - $x_1 > y_1$, OR - $x_1 = y_1$ AND $x_2 \ge y_2$. - The corresponding strict preference $x > y$ is if $x_1 > y_1$ OR ($x_1 = y_1$ AND $x_2 > y_2$). - **Crucially, there exists no utility representation for this rational preference.** This is because lexicographic preferences are not continuous. ### Continuity of Preferences The weak preference $\succsim$ is called **continuous** if for all sequences $a_n \to a$ and $b_n \to b$: - If $a_n \succsim b$ for all $n$, then $a \succsim b$. - If $a \succsim b_n$ for all $n$, then $a \succsim b$. - This means small changes in bundles do not cause abrupt changes in preferences. ### Indifference Curve The set of all bundles equally preferred to a bundle $a$ is the **indifference curve** containing $a$: $I(a) = \{b \in X : b \sim a\}$. #### Indifference Curve as subset of $\mathbb{R} \times \mathbb{R}$ - For continuous goods, indifference curves are typically smooth lines. - For discrete goods, indifference "curves" consist of discrete points. ### Upper and Lower Contour Sets Given a utility representation $u(x)$: - **Indifference Set:** $I(a) = \{b \in X : u(b) = u(a)\}$. - **Upper Contour Set:** $U(a) = \{b \in X : u(b) \ge u(a)\}$. (Bundles at least as good as $a$) - **Lower Contour Set:** $L(a) = \{b \in X : u(b) \le u(a)\}$. (Bundles no better than $a$) ### Marginal Rate of Substitution (MRS) If a utility function $u(x_1, x_2)$ exists and is differentiable, the MRS is defined as: $$MRS(x) = \frac{\partial u(x) / \partial x_1}{\partial u(x) / \partial x_2}$$ - The MRS measures the rate at which a consumer is willing to substitute $x_2$ for $x_1$ while remaining on the same indifference curve (maintaining the same utility level). - From $u(x_1, x_2) = \text{constant}$, differentiating gives: $$\frac{\partial u(x)}{\partial x_1} dx_1 + \frac{\partial u(x)}{\partial x_2} dx_2 = 0$$ $$\frac{dx_2}{dx_1} = - \frac{\partial u(x) / \partial x_1}{\partial u(x) / \partial x_2} = -MRS(x)$$ - Approximately, $\Delta x_2 \approx -MRS(x) \Delta x_1$. - It tells us how much $x_2$ a consumer must get in exchange for giving up one unit of $x_1$ to stay on the same indifference curve. ### Types of Utility Functions #### 1. Linear Preferences - **Utility Function:** $u(x_1, x_2) = \alpha x_1 + \beta x_2$. - **Marginal Rate of Substitution:** $MRS(x) = \frac{\alpha}{\beta}$. - To remain on the same indifference curve, for every 1 unit increase in $x_1$, $\frac{\alpha}{\beta}$ units of $x_2$ must be given up. This rate is constant and does not depend on amounts consumed. - **Perfect Substitutes:** Goods are perfectly substitutable for each other. - **Indifference Curve:** Straight lines with slope $-\alpha/\beta$. #### 2. Leontief Preferences - **Utility Function:** $u(x_1, x_2) = \min\{\alpha x_1, \beta x_2\}$. - **Perfect Complements:** Goods are consumed in fixed proportions. - No substitution is possible without changing utility, so the notion of MRS is meaningless. - **Indifference Curve:** L-shaped, with kinks at the fixed proportion (e.g., $\alpha x_1 = \beta x_2$). #### 3. Cobb-Douglas Preferences - **Utility Function:** $u(x_1, x_2) = A x_1^\alpha x_2^\beta$. - **Marginal Rate of Substitution:** $MRS(x) = \frac{\alpha x_2}{\beta x_1}$. - **Diminishing MRS:** As $x_1$ increases and $x_2$ decreases along an indifference curve, the MRS decreases (the curve becomes flatter). This means consumers are willing to give up less of $x_2$ for an additional unit of $x_1$ as $x_1$ increases. - **Indifference Curve:** Convex to the origin. #### 4. Quasi-linear Preferences - **Utility Function:** $u(x_1, x_2) = f(x_1) + x_2$. - **Marginal Rate of Substitution:** $MRS(x) = f'(x_1)$. - If $f$ is concave ($f''(x_1) ### Cardinal vs Ordinal Utility - A monotonic transformation $g: \mathbb{R} \to \mathbb{R}$ (increasing function) of a utility function creates another utility function that represents the same preference relation. - $x \succsim y \iff u(x) \ge u(y) \iff g(u(x)) \ge g(u(y))$. - **MRS is unchanged by this positive monotonic transformation.** - **Ordinal Utility:** Utility only has relative meaning (rankings) rather than its absolute value. Most of consumer theory uses ordinal utility. - **Cardinal Utility:** Assigns a specific value of utility to different options, implying that utility differences are meaningful. This is not typically assumed in standard consumer theory. ### Household Consumption - Households allocate spending across various goods and services (food, utilities, health, education, transport, rent, leisure). - **Budget Share Regularity:** As income rises, the budget share of food tends to fall (even if food quality rises). - **Marketing and Product Design:** Focus on how demand responds to income and prices. - Consumer theory provides tools for **demand functions**, **indirect utility**, and **comparative statics**. ### Indian Households: Heterogeneity - Households differ by income, location, family size, housing constraints, and access to markets. - Preferences are not identical; constraints differ; prices can differ by neighborhood and time. - **Consumer theory** offers a disciplined language to analyze choice given preferences and constraints. - **Low Income:** Tighter constraints, smaller choice set. - **High Income:** Broader choice set. ### Rich and Poor: Consumption Habits - **Poorer households:** Necessities dominate; many margins are "corner-like" (can't afford everything). - **Richer households:** More discretionary categories; stronger responses to variety/quality, brands, time costs. - Models need to handle **interior solutions**, **corners**, and **kinks**. ### Lean and Super Spenders - Two people with the same income can choose very different baskets. - This can be rationalized by different preferences or different constraints. - In this context, given prices and income, how does a rational agent choose? ### Budget Shares: Groceries vs Utilities/Medical - **Empirical Observation:** Food share falls with income; shares of manufactured goods and services rise. - This is a micro-foundation for **structural transformation**: agriculture share in GDP falls, manufacturing/services rise as incomes grow. ### Marketing and Household Products - Firms focus on how demand responds to income and prices. - Consumer theory provides **demand functions**, **indirect utility**, and **comparative statics**. - Key concepts: **substitution vs income effects**, Slutsky equation, welfare measures. ### Utility Maximization **Problem:** Maximize $u(x_1, x_2)$ subject to $p_1 x_1 + p_2 x_2 \le m$. **Budget Set:** $B(p, m) = \{(x_1, x_2) \in \mathbb{R}^2_+ : p_1 x_1 + p_2 x_2 \le m\}$. - **Geometry:** Pick the most north-east indifference curve that still touches the budget set. - For well-behaved preferences, the maximum is at a tangency point. #### Compactness and Existence - $B(p, m)$ is closed and bounded in $\mathbb{R}^2$, so it is compact. - If $u$ is continuous, then a maximum exists on the compact set $B(p, m)$. - This is a fundamental step before computing solutions. #### Rational Constrained Choice: The Budget Line - **Budget line:** $p_1 x_1 + p_2 x_2 = m$. - **Intercepts:** $(m/p_1, 0)$ and $(0, m/p_2)$. - **Slope:** $\frac{dx_2}{dx_1} = -\frac{p_1}{p_2}$. #### Rational Constrained Choice: Indifference Curves - **Indifference curve:** Set of bundles yielding same utility, $u(x_1, x_2) = \bar{u}$. - If $u$ is increasing, higher indifference curves are north-east. - If preferences are convex, indifference curves are convex to the origin. #### Rational Constrained Choice: Interior Optimum Idea - If the optimum $(x_1^*, x_2^*)$ is interior ($x_1^* > 0, x_2^* > 0$), then the optimal indifference curve is tangent to the budget line. - **Tangency condition:** $Slope(IC) = Slope(Budget Line)$. #### Rational Constrained Choice: The "Budget is Exhausted" Idea - If $u$ is increasing, any interior optimum must satisfy the budget with equality: $p_1 x_1^* + p_2 x_2^* = m$. - **Intuition:** If income is left unspent, you could buy more goods and be better off. #### What Can Go Wrong? - **Corner solutions:** Optimum on an axis ($x_i^* = 0$ for some $i$). - **Kinks:** Tangency fails (e.g., perfect complements). - **Non-convexities:** Multiple local optima can appear. ### Rational Constrained Choice (Notation) - Partial derivatives $u_1 = \frac{\partial u}{\partial x_1}$ and $u_2 = \frac{\partial u}{\partial x_2}$ are often used. - **Marginal utility of good $i$ ($u_i$):** Incremental utility from more of $x_i$, holding other goods fixed. ### Tangency at an Interior Optimum - At the utility maximum (interior), the slope of the budget line equals the slope of the indifference curve. - $\frac{p_1}{p_2} = \frac{u_1(x_1^*, x_2^*)}{u_2(x_1^*, x_2^*)} \equiv MRS$. - This means $\frac{u_1}{p_1} = \frac{u_2}{p_2}$ (marginal utility per dollar is equal across goods). #### MRS via Implicit Differentiation - On an indifference curve, $u(x_1, x_2) = \bar{u}$. - Differentiating: $u_1 dx_1 + u_2 dx_2 = 0 \implies \frac{dx_2}{dx_1} = -\frac{u_1}{u_2}$. - This is the rate at which $x_2$ must rise to compensate a small fall in $x_1$ to keep utility constant. #### MRS via First-Order Approximation - Moving from $(x_1, x_2)$ to $(x_1 + \Delta x_1, x_2 + \Delta x_2)$ on the same indifference curve: $\Delta u \approx u_1 \Delta x_1 + u_2 \Delta x_2 = 0 \implies \frac{\Delta x_2}{\Delta x_1} \approx -\frac{u_1}{u_2}$. - In the limit, $\frac{dx_2}{dx_1} = -\frac{u_1}{u_2}$. - The condition $\frac{p_1}{p_2} = MRS$ equates the market trade-off with the preference trade-off. #### If $MRS \ne p_1/p_2$, You Can Improve Utility - If MRS is not equal to the price ratio, a consumer can reallocate spending to achieve a strictly higher utility level. - This implies that at an interior optimum, $MRS = p_1/p_2$. This is the **tangency/no-arbitrage condition**. ### No Arbitrage Derivation of Interior FOC - **First-order Taylor's approximation:** $u(x_1 + h_1, x_2 + h_2) - u(x_1, x_2) = u_1 h_1 + u_2 h_2$. - If DM is at utility maximum, moving 1 rupee of consumption from good 2 to good 1 ($h_1 = 1/p_1, h_2 = -1/p_2$) should not increase utility: $u_1(1/p_1) + u_2(-1/p_2) \le 0$. - Similarly, moving 1 rupee from good 1 to good 2 should not increase utility: $u_1(-1/p_1) + u_2(1/p_2) \le 0$. - Combining these implies $\frac{u_1}{p_1} = \frac{u_2}{p_2}$, or $MRS = \frac{p_1}{p_2}$. ### Lagrangian: The Canonical Formulation - Maximize $u(x_1, x_2)$ subject to $m - p_1 x_1 - p_2 x_2 = 0$. - **Lagrangian:** $\mathcal{L}(x_1, x_2, \lambda) = u(x_1, x_2) + \lambda(m - p_1 x_1 - p_2 x_2)$. - **First-order conditions (FOCs):** - $\frac{\partial \mathcal{L}}{\partial x_1} = u_1 - \lambda p_1 = 0 \implies u_1 = \lambda p_1$ - $\frac{\partial \mathcal{L}}{\partial x_2} = u_2 - \lambda p_2 = 0 \implies u_2 = \lambda p_2$ - $\frac{\partial \mathcal{L}}{\partial \lambda} = m - p_1 x_1 - p_2 x_2 = 0 \implies p_1 x_1 + p_2 x_2 = m$ - From the first two FOCs, $\frac{u_1}{p_1} = \frac{u_2}{p_2} = \lambda$, and this implies $MRS = \frac{p_1}{p_2}$. ### Interpreting $\lambda$: Marginal Utility of Income - $\lambda$ is the **marginal utility of income** (or a "shadow price"). - If income increases by one rupee: - Spent on good 1: utility gain $\approx u_1(1/p_1) = u_1/p_1$. - Spent on good 2: utility gain $\approx u_2(1/p_2) = u_2/p_2$. - At the optimum, these gains must be equal: $\frac{u_1}{p_1} = \frac{u_2}{p_2} = \lambda$. - So, $\lambda$ represents the incremental utility from one extra unit of income. ### Lagrange Theorem (General n-Good Statement) - Maximize $u: \mathbb{R}^n_+ \to \mathbb{R}$ over $B(p, m) = \{x \in \mathbb{R}^n_+ : \sum_{j=1}^n p_j x_j \le m \}$, with $p \gg 0, m > 0$. - If an interior maximum occurs at $x^* \in \mathbb{R}^n_{++}$ and $u$ is differentiable at $x^*$, then there exists $\lambda \in \mathbb{R}$ such that: $\nabla u(x^*) = \lambda p$. - In row-derivative notation: $Du(x^*) = \lambda p$. #### Orthogonality Intuition - Feasible reallocations along the budget line satisfy $p \cdot h = 0$. - At an interior optimum, a small feasible move does not increase utility: $0 = \Delta u \approx \nabla u(x) \cdot h$. - Thus, $\nabla u(x)$ is orthogonal to the same set of feasible directions $h$ as $p$. This means $\nabla u(x^*)$ and $p$ must be parallel: $\nabla u(x^*) = \lambda p$. ### Several Equality Constraints (Setup) - **General problem:** Maximize $f: \mathbb{R}^n \to \mathbb{R}$ subject to $m$ linear constraints ($m ### Indirect Utility and Marshallian Demand - **Indirect Utility Function:** $v(p_1, p_2, m) = \max\{u(x_1, x_2) : (x_1, x_2) \in B(p, m)\}$. - This is the maximum utility attainable given prices $p$ and income $m$. - **Marshallian (Ordinary) Demand Functions:** The utility-maximizing bundle depends on $(p_1, p_2, m)$: - $x_1^*(p_1, p_2, m)$ - $x_2^*(p_1, p_2, m)$ ### Cobb-Douglas Preferences Example **Problem:** Maximize $u(x_1, x_2) = x_1^a x_2^{1-a}$ (where $0 ### Why the Method Works: A Checklist - **Existence:** Continuity of utility + compact budget set. - **Interior vs Corner:** Increasing preferences typically make the budget bind; strict quasiconcavity often yields uniqueness. - **First-order conditions:** Solve $u_i = \lambda p_i$ plus the budget constraint. - **Verify Solution:** Ensure the solution is feasible and respects non-negativity. ### Corner Solutions - Occur when the optimal bundle is on an axis (e.g., $x_i^* = 0$). - Tangency conditions may not hold. - **Causes:** - Perfect substitutes / linear utility (if prices are very different). - Very strong relative prices (one good is disproportionately expensive). - Non-convex preferences (consumer prefers extremes, not averages). - We need **inequality versions of the FOCs** at corners. #### Perfect Substitutes: An Example - **Utility:** $u(x_1, x_2) = x_1 + x_2$. - **Indifference curves:** Straight lines with slope $-1$. - **Rule of Thumb:** - If $p_1/p_2 > 1$ (good 1 is relatively expensive), buy only good 2 ($x_1^*=0$). - If $p_1/p_2 ### Kinky Solutions: Perfect Complements - **Utility:** $u(x_1, x_2) = \min\{x_1, x_2\}$. - **Indifference curves:** L-shaped; "balanced bundles" matter. - Tangency is not the right tool; the optimum occurs at the **kink point**. #### Perfect Complements: How to Compute the Optimum - For $u(x_1, x_2) = \min\{x_1, x_2\}$, optimal choices equalize the goods: $x_1^* = x_2^* = t^*$. - Substitute into budget constraint: $p_1 t^* + p_2 t^* = m \implies (p_1 + p_2)t^* = m$. - $t^* = \frac{m}{p_1 + p_2}$. - So, $x_1^* = x_2^* = \frac{m}{p_1 + p_2}$. - (More general: $\min\{\alpha x_1, \beta x_2\}$ gives a fixed proportion line $\alpha x_1 = \beta x_2$). ### Quasi-linear: Solving the Problem - Maximize $v(x_1) + x_2$ subject to $p_1 x_1 + p_2 x_2 = m$. - Substitute $x_2 = (m - p_1 x_1)/p_2$: Maximize $v(x_1) + \frac{m - p_1 x_1}{p_2}$ - **FOC (interior in $x_1$):** $v'(x_1^*) - \frac{p_1}{p_2} = 0 \implies v'(x_1^*) = \frac{p_1}{p_2}$. - Then $x_2^* = (m - p_1 x_1^*)/p_2$. - **What is special?** $x_1^*$ depends on the price ratio $p_1/p_2$ but not directly on $m$ (when interior). Income changes mostly show up in $x_2^*$. - $\frac{\partial x_1^*}{\partial m} = 0$ (interior) - $\frac{\partial x_2^*}{\partial m} = \frac{1}{p_2}$ - This property is why quasi-linearity is convenient for welfare and policy calculations. #### Quasi-linear Example (Corner Check) - If $m$ is very small, the computed $x_2^*$ might be negative. - Since $x_2 \ge 0$ is required, the true optimum hits the corner $x_2^*=0$ when income is too low. - In that case, maximize $v(x_1)$ subject to $p_1 x_1 \le m$. So $x_1^* = m/p_1$, $x_2^* = 0$. - Always check non-negativity constraints after solving interior FOCs. ### Summary So Far - Utility maximization subject to a budget constraint is the core consumer problem. - **Interior optimum:** $MRS = p_1/p_2$ and $u_i = \lambda p_i$ with the budget binding. - $\lambda$ is the marginal utility of income. - **Marshallian demands** ($x_1^*(p, m), x_2^*(p, m)$) summarize behavior as a function of $(p, m)$. ### Monopoly: From Price-Taking to Price-Setting - **Price-taking (competitive firm):** Maximize $pq - C(q)$ where $p = MC(q)$. Price is given. - **Price-setting (monopoly/market power):** Faces a downward-sloping demand curve, $q = D(p)$ or $p = P(q)$. Price is not given; it depends on how much is sold. ### Why Study Monopoly? 1. **Efficiency:** Monopolies typically produce too little ($q^M ### Where Does Monopoly Come From? - **Natural monopoly:** Huge fixed costs + low marginal cost (one network is cheaper than many). - **Economies of scale/scope:** Learning curves, logistics, data, distribution. - **Network effects & platforms:** Value rises with number of users; feedback loops create dominance. - **Switching costs & lock-in:** Compatibility, ecosystems, habits. - **Legal barriers:** Licensing, patents, exclusive rights. - **Scarcity/capacity:** Limited slots, spectrum, gates, parking. *Note: "Monopoly" is not a binary label. Most real-world market power is partial: firms face a downward-sloping residual demand.* ### Residual Demand - Even with multiple sellers, a firm often faces **residual demand**. - Example: selling chai in a corridor with two other chai points; raising price makes some customers go elsewhere. - **Key idea:** Residual demand summarizes all constraints imposed by "the rest of the market." A monopolist chooses a point on $P(q)$, not taking $p$ as given. ### Primitives and Definitions - **Inverse demand:** $p = P(q)$, typically $P'(q) 0$, we typically have $MR(q) ### MR is the Slope of TR - $TR(q)$ is a curve in $(q, TR)$-space. $MR(q)$ is its slope. - **Profit:** $\pi(q) = TR(q) - C(q)$. - **First-order condition (interior optimum):** $\pi'(q) = MR(q) - MC(q) = 0$. - This means $MR(q^M) = MC(q^M)$ at the monopoly output $q^M$. - Normally $\pi''(q) \le 0$ (second-order condition). - **One-line meaning:** Increase output as long as the next unit adds more revenue than cost; stop when they match. ### Linear Demand Geometry - For linear inverse demand $P(q) = a - bq$ ($a, b > 0$). - $TR(q) = (a - bq)q = aq - bq^2$. - $MR(q) = a - 2bq$. - **MR properties:** - Same vertical intercept ($a$) as demand. - Twice the slope (steeper). - The $q$-intercept is halved. ### Monopoly Optimum in a p-q Picture - **Profit maximization:** $MR(q^M) = MC(q^M)$. - The monopoly price is then $p^M = P(q^M)$. - *Note: Do not set $P = MC$ for monopoly. $P = MC$ characterizes the efficient/competitive benchmark.* ### Why Society Cares: Regulation, Antitrust, Platforms - Once market power exists, policymakers ask: What do we do about it? - **Regulation:** Price caps, rate-of-return, access obligations (e.g., for natural monopolies). - **Antitrust:** Address dominance/platforms, tying/bundling, self-preferencing, exclusionary contracts. - **Merger policy:** "Will this make residual demand less elastic?" - **Micro theory's use:** Predicts markups, output restriction, welfare loss, and how pricing instruments change outcomes. - **Geek notes:** Policy debates often hinge on how elastic the residual demand is. ### Empirical Angle: Measuring Market Power (Residual Demand Elasticity) - How much does price go up if the firm cuts quantity? This is the residual demand elasticity. - **Why it matters:** - Direct measure of "competitive pressure." - Links cleanly to markup via **Lerner Index**: $\frac{P - MC}{P} = \frac{1}{\epsilon}$. - Very high elasticity ($\epsilon$) implies price is close to marginal cost (efficient). - Helps evaluate mergers, entry barriers, and exclusionary conduct. - **Practical lesson:** A firm doesn't need 100% market share to have market power; it just needs customers who won't switch quickly. ### Elasticity Form of MR and the Lerner Rule - **Point elasticity of demand:** $\epsilon = \frac{dq}{dp} \frac{p}{q}$ ($\epsilon ### Efficiency Benchmark and Deadweight Loss - **Total surplus (TS):** $\int_0^q P(z) dz - C(q)$. - **Efficient benchmark ($q^*$):** Achieved when $P(q^*) = MC(q^*)$. (Maximizes total surplus). - Monopoly output $q^M MC$) that do not occur under monopoly. ### Why Monopoly Has No Supply Curve - **Competitive supply:** Firm maps price to quantity via $p = MC(q) \implies q = q(p)$. - **Monopoly:** Chooses a point on the demand curve where $MR(q) = MC(q)$. - $MR(q)$ depends on the full demand curve (level and slope). - There is no single-valued "q as a function of p" supply curve for monopoly. - **Memory line:** Supply curve summarizes price-taking. Monopoly chooses the price-quantity pair. ### Monopoly Power is About Instruments, Not Just One Price - Monopoly often uses more sophisticated pricing strategies than a single price. - **Richer instruments:** - **Two-part tariffs:** Entry fee + per-unit price. - **Price discrimination:** Different prices for different markets or types. - **Menus (second-degree discrimination):** Offer a menu of (quantity, total payment) options; consumers self-select. - **Auctions:** When values are private. - **Punchline:** With a better instrument set, a monopolist can sometimes: - Eliminate DWL (achieve efficient quantity). - Extract surplus (via a fixed fee). ### Two-Part Tariff - **Structure:** Entry fee $r$ + per-unit (ride) price $p$. - **Quasilinear utility:** $U(q, y) = u(q) + y$, where $y$ is money. - Consumer pays $r + pq$ for $q$ rides. Income $m$. - **Participation constraint:** Consumer enters if $u(q^*) - pq^* - r \ge 0$. - **Monopolist's problem:** Choose $q$ to maximize total surplus $u(q) - C(q)$. - **Solution:** Set $p = u'(q^*) = C'(q^*) = MC(q^*)$. (Per-unit price equals marginal cost). - **Entry fee:** Set $r = u(q^*) - pq^*$ (extracts all consumer surplus). - **Diagram:** Set per-unit price $p = c$ (efficient quantity $q^*$), charge entry fee $r = \text{consumer surplus}$. ### Equivalently: A General Tariff (q, T) - Instead of $(r, p)$, think of a contract $(q, T)$: pay total $T$, receive $q$. - **Monopolist chooses $(q, T)$ to maximize:** $T - C(q)$. - **Subject to participation:** $u(q) - T \ge 0$. - At optimum, monopolist sets $T = u(q)$ (extracts all surplus). - **Monopolist chooses $q$ to maximize:** $u(q) - C(q)$. - **FOC:** $u'(q^*) = C'(q^*) = c$. - **Why general tariffs matter:** When consumers differ, a single two-part tariff may be too rigid; menus of $(q, T)$ become natural. ### Price Discrimination More Generally - Monopolist has more instruments. - **Types:** - Fixed fee + per-unit price (two-part tariff). - Different prices for different markets (third-degree). - Menu of $(q, T)$ options (second-degree screening / Mussa-Rosen style). - **Why it matters:** - Can increase profit (extract more surplus). - Can change output (sometimes reducing DWL, sometimes increasing it). - Forces consideration of **information** (hidden types) and **incentives** (self-selection). ### A Necessary Condition: No Resale / No Arbitrage - Price discrimination is feasible only if low-price buyers cannot profitably resell to high-price buyers. - **Typical anti-resale mechanisms:** - ID checks / personalization (student tickets). - Time/location restrictions (surge pricing, off-peak electricity). - Bundling with non-transferable services. - Quality/versioning (economy vs business class). - **Note:** If resale is easy, uniform pricing re-emerges (arbitrage equalizes prices). ### Taxonomy: Three Canonical Forms of Price Discrimination 1. **First-degree (Perfect Price Discrimination):** Charge each marginal unit at the buyer's marginal willingness to pay. 2. **Second-degree (Screening / Menus):** Offer a menu of $(q, T)$ or (quality, price); consumers self-select. 3. **Third-degree (Market Separation):** Different per-unit prices across distinct markets. ### Perfect Price Discrimination (Idea) - Monopolist extracts *all* consumer surplus. - **Interpretation:** Charges each marginal unit at the buyer's marginal willingness to pay. - Produces until willingness to pay falls to marginal cost. - With inverse demand $p = P(q)$ and constant $MC = c$, output satisfies $P(q^{PPD}) = c$. - Perfect PD achieves the **efficient quantity** (like two-part tariff for identical consumers). ### Diagram: Monopoly vs Perfect PD (Constant MC) - Monopoly: $MR = MC$. - Perfect PD / Efficiency: $P = MC$. - Perfect PD produces more output than a single-price monopoly. ### Third-Degree Price Discrimination: Separated Markets - Markets separated by geography, time, or identity (no arbitrage). - Firm chooses quantities $q_1, q_2$ for two markets with inverse demands $P_1, P_2$ and common cost $C(q_1 + q_2)$. - **FOCs:** $MR_1 = MC$ and $MR_2 = MC$. - This implies $MR_1 = MR_2 = MC$. #### Elasticity Pricing Rule - Using $MR_i = P_i(1 + 1/\epsilon_i)$, the FOCs imply: $P_1(1 + 1/\epsilon_1) = P_2(1 + 1/\epsilon_2) = MC$. - The market with **lower elasticity** (more inelastic demand) gets a **higher price**. - $|\epsilon_1| P_2$. #### Quick Examples - Student discounts (elastic group). - Peak vs off-peak pricing (electricity). - Geography: same product priced differently across cities. - Early-bird vs last-minute fares.