Nicholson Snyder Microeconomics

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1. Economic Models Economic Model: Simplified representation of reality to capture essentials. Focus on key variables and relationships. Assumes "ceteris paribus" (all else equal) for analysis. Verification Methods: Direct: Establish validity of assumptions. Often difficult due to complexity of human behavior. Indirect: Confirm validity by correct predictions of real-world events. Most common approach in economics. Key Features of Economic Models: Ceteris Paribus: "Other things being equal" assumption, isolates the effect of one variable. Optimization Assumptions: Economic actors pursue goals rationally. Firms: Profit maximization. Individuals: Utility maximization. Government: Social welfare maximization. Positive-Normative Distinction: Positive Economics: Explains and predicts economic phenomena. Deals with "what is" or "what will be." (e.g., "A tax on cigarettes will reduce smoking.") Normative Economics: Makes recommendations about what should be done. Deals with "what ought to be." (e.g., "The government should tax cigarettes to reduce smoking.") Exogenous Variables: Inputs into models, determined outside the model. (e.g., prices in consumer theory, technology in production theory). Endogenous Variables: Outputs of models, determined within the model. (e.g., quantity demanded/supplied, utility level, profit). Example 1.1: Profit Maximization Illustrates how a simple model can predict behavior. Profit Function: $\pi = Pq - C(q)$, where $P$ is price, $q$ is quantity, $C(q)$ is cost. First-Order Condition (FOC): $\frac{d\pi}{dq} = P - C'(q) = 0 \Rightarrow P = C'(q)$ (Price equals Marginal Cost). Second-Order Condition (SOC): $\frac{d^2\pi}{dq^2} = -C''(q) 0$ (Marginal Cost must be increasing). Prediction: How does $q^*$ change with $P$? Differentiate FOC with respect to $P$: $1 - C''(q^*) \frac{dq^*}{dP} = 0 \Rightarrow \frac{dq^*}{dP} = \frac{1}{C''(q^*)} > 0$. This implies the firm's supply curve is upward sloping. Marshallian Supply-Demand Synthesis: Price determined by interaction of supply and demand. Focus on single markets. Production Possibility Frontier (PPF): Shows alternative combinations of two goods that an economy can produce given its resources and technology. Opportunity Cost: The value of the next best alternative forgone. Reflected in the slope of the PPF. Rate of Product Transformation (RPT): The absolute value of the slope of the PPF, indicating how much of one good must be given up to produce more of another. General Equilibrium: Model of the entire economy, considering interconnections among all markets. Contrasts with partial equilibrium (single market). Econometrics: Statistical methods used to estimate relationships in economic models and test hypotheses. 2. Mathematics for Microeconomics Unconstrained Optimization Maximization of a Function of One Variable: $y = f(x)$ First-Order Condition (FOC): $\frac{dy}{dx} = f'(x) = 0$ (necessary for a local max/min). Second-Order Condition (SOC): $\frac{d^2y}{dx^2} = f''(x) $\frac{d^2y}{dx^2} = f''(x) > 0$ (for a local minimum). Rules for Differentiation: Constant: $\frac{da}{dx} = 0$ Constant multiple: $\frac{d[af(x)]}{dx} = af'(x)$ Power rule: $\frac{dx^a}{dx} = ax^{a-1}$ Logarithm: $\frac{d \ln x}{dx} = \frac{1}{x}$ Exponential: $\frac{da^x}{dx} = a^x \ln a$ (special case $e^x$: $\frac{de^x}{dx} = e^x$) Sum rule: $\frac{d[f(x) + g(x)]}{dx} = f'(x) + g'(x)$ Product rule: $\frac{d[f(x)g(x)]}{dx} = f(x)g'(x) + f'(x)g(x)$ Quotient rule: $\frac{d[f(x)/g(x)]}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$ Chain rule: $\frac{dy}{dz} = \frac{dy}{dx} \cdot \frac{dx}{dz}$ Functions of Several Variables: $y = f(x_1, x_2, \dots, x_n)$ Partial Derivatives: $\frac{\partial y}{\partial x_i} = f_i$ (holding all other $x_j$ constant). Elasticity: $e_{y,x} = \frac{\% \Delta y}{\% \Delta x} = \frac{\partial y}{\partial x} \cdot \frac{x}{y} = \frac{d \ln y}{d \ln x}$. Second-Order Partial Derivatives: $f_{ij} = \frac{\partial^2 f}{\partial x_j \partial x_i}$. Young's Theorem: $f_{ij} = f_{ji}$ (order of differentiation doesn't matter). Chain Rule (many variables): If $y = f(x_1(a), \dots, x_n(a), a)$, then $\frac{dy}{da} = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\frac{dx_i}{da} + \frac{\partial f}{\partial a}$. Implicit Functions: For $f(x_1, x_2) = k$, $\frac{dx_2}{dx_1} = -\frac{f_1}{f_2}$. Maximization of Functions of Several Variables: FOC: $f_1 = f_2 = \dots = f_n = 0$. (All partial derivatives must be zero). SOC (for 2 variables): $f_{11} 0$ (for a strict local maximum). This implies the Hessian matrix of second derivatives is negative definite. Concave Function: A function where the Hessian matrix is negative semidefinite (or negative definite for strict concavity). For two variables, this means $f_{11} Example 2.1: Profit Maximization with Two Inputs Illustrates FOC for multi-variable optimization. Profit $\pi(k,l) = Pq(k,l) - vk - wl$. FOC: $\frac{\partial \pi}{\partial k} = P \frac{\partial q}{\partial k} - v = 0$ and $\frac{\partial \pi}{\partial l} = P \frac{\partial q}{\partial l} - w = 0$. This means $P \cdot MP_k = v$ and $P \cdot MP_l = w$. The Envelope Theorem: For an optimized function $y^*(a) = f(x_1^*(a), \dots, x_n^*(a), a)$, the effect of a change in parameter $a$ on the optimal value $y^*$ is given by the partial derivative of $f$ with respect to $a$, holding $x_i$ at their optimal values: $\frac{dy^*}{da} = \frac{\partial f}{\partial a} |_{x_i=x_i^*(a)}$. Simplifies comparative statics by ignoring the indirect effects of $a$ on $x_i^*$. Example 2.2: Envelope Theorem Application Revisits Example 1.1, showing how to find $\frac{dq^*}{dP}$ using the envelope theorem on the profit function $\pi^*(P) = Pq^*(P) - C(q^*(P))$. $\frac{d\pi^*}{dP} = q^*(P)$. This is Hotelling's Lemma (Chapter 11). Constrained Optimization Constrained Maximization: Maximize $f(x_1, \dots, x_n)$ subject to $g(x_1, \dots, x_n) = 0$. Lagrangian: $\mathcal{L} = f(x_1, \dots, x_n) + \lambda g(x_1, \dots, x_n)$. FOCs: $\frac{\partial \mathcal{L}}{\partial x_i} = f_i + \lambda g_i = 0$ for all $i=1, \dots, n$. $\frac{\partial \mathcal{L}}{\partial \lambda} = g(x_1, \dots, x_n) = 0$ (the constraint itself). Interpretation of $\lambda$: $\lambda = -\frac{f_i}{g_i}$. Represents the marginal change in the objective function ($f$) for a marginal relaxation of the constraint ($g$). Often called the "shadow price" of the constraint. Second-Order Conditions: Involves the bordered Hessian matrix. For two variables and one constraint, it ensures the solution is a constrained maximum. Duality: Every constrained maximization problem has a dual constrained minimization problem (e.g., utility maximization vs. expenditure minimization). Example 2.3: Constrained Utility Maximization Maximize $U(x,y)$ subject to $P_x x + P_y y = I$. FOCs lead to $\frac{U_x}{U_y} = \frac{P_x}{P_y}$ (MRS equals price ratio). Interpretation of $\lambda$: $\lambda = \frac{U_x}{P_x} = \frac{U_y}{P_y}$, the marginal utility of income. Inequality Constraints (Kuhn-Tucker Conditions): For maximizing $f(x_1, \dots, x_n)$ subject to $g(x_1, \dots, x_n) \ge 0$. Lagrangian: $\mathcal{L} = f(x_1, \dots, x_n) + \lambda g(x_1, \dots, x_n)$. FOCs (for $x_i$): $f_i + \lambda g_i = 0$. Additional conditions: $\lambda \ge 0$ $\lambda g(x_1, \dots, x_n) = 0$ (complementary slackness: if constraint is not binding, $\lambda=0$; if $\lambda>0$, constraint must be binding). Quasi-Concave Functions: Functions whose upper contour sets are convex. Indifference curves or isoquants are convex to the origin. For $f(x_1, x_2)$, quasi-concavity requires $f_{11}f_2^2 - 2f_{12}f_1f_2 + f_{22}f_1^2 Ensures that a tangency point in constrained optimization is a maximum. Special Function Types Homogeneous Functions: $f(tx_1, \dots, tx_n) = t^k f(x_1, \dots, x_n)$. Degree $k$. Derivatives are homogeneous of degree $k-1$. Euler's Theorem: If $f$ is homogeneous of degree $k$, then $x_1f_1 + \dots + x_nf_n = kf$. Homothetic Functions: Monotonic transformation of a homogeneous function. The slopes of their level curves (e.g., MRS, RTS) depend only on the ratios of the variables, not their absolute levels. Example 2.4: Homothetic Functions Illustrates that $f(x,y) = x^a y^b$ (Cobb-Douglas) is homogeneous of degree $a+b$, and thus homothetic. $f(x,y) = \ln(x^a y^b) = a \ln x + b \ln y$ is not homogeneous, but it is homothetic because its MRS ($y/x$) is the same as Cobb-Douglas. Integration and Dynamic Optimization Integration: Antiderivative: $F(x) = \int f(x)dx$ where $F'(x) = f(x)$. Definite Integral: $\int_a^b f(x)dx = F(b) - F(a)$ (area under curve). Differentiating Definite Integral: $\frac{d}{dx} \int_a^{g(x)} f(t)dt = f(g(x))g'(x)$. Dynamic Optimization (Optimal Control): Maximizing an integral (objective function over time) subject to differential equations (state variable evolution). Hamiltonian: $H = f[x(t), c(t), t] + \lambda(t)g[x(t), c(t), t]$, where $f$ is the instantaneous payoff, $x$ is the state variable, $c$ is the control variable, and $g = \dot{x}$. Maximum Principle: $\frac{\partial H}{\partial c} = 0$ (optimality condition for control variable). $\frac{\partial H}{\partial x} = -\dot{\lambda}(t)$ (costate equation, evolution of shadow price). $\dot{x}(t) = g[x(t), c(t), t]$ (state equation). Mathematical Statistics: Random Variable: Numerical outcome of a random event. Probability Density Function (PDF): $f(x)$ showing probabilities of outcomes. $\int_{-\infty}^{\infty} f(x)dx = 1$. Expected Value: $E(x) = \sum x_i f(x_i)$ (discrete) or $\int xf(x)dx$ (continuous). Variance: $Var(x) = \sigma_x^2 = E[(x - E(x))^2]$. Covariance: $Cov(x,y) = E[(x - E(x))(y - E(y))]$. $Var(x+y) = Var(x) + Var(y) + 2Cov(x,y)$. 3. Preferences and Utility Basic Axioms and Utility Functions Axioms of Rational Choice: Underlie the existence of a utility function. Completeness: For any two bundles A and B, an individual can always state one of the following: A is preferred to B (A > B), B is preferred to A (B > A), or A and B are equally preferred (A ~ B). Transitivity: If A > B and B > C, then A > C. Ensures consistency of preferences. Continuity: If A > B, then situations "close to" A are also preferred to B. Rules out "knife-edge" preferences. Utility Function: A mathematical representation of preferences, $U(x_1, x_2, \dots, x_n)$. Assigns a numerical value to each consumption bundle. Ordinal Utility: Only the ranking of bundles matters, not the magnitude of utility differences. Utility functions are unique up to any positive monotonic (order-preserving) transformation. Ceteris Paribus: Other factors affecting utility are held constant (e.g., psychological, social, environmental). Indifference Curve: A set of consumption bundles among which an individual is indifferent (yield the same level of utility). Cannot intersect. Negatively sloped (assuming "more is better"). Convex to the origin (due to diminishing MRS). Higher indifference curves represent higher utility. Marginal Utility: The additional utility gained from consuming one more unit of a good, holding other goods constant. $MU_x = \frac{\partial U}{\partial x}$. Diminishing Marginal Utility: The marginal utility of a good decreases as more of the good is consumed. Not strictly required for convex indifference curves. Marginal Rate of Substitution (MRS): The rate at which an individual is willing to trade one good for another while maintaining the same level of utility. $MRS_{x \text{ for } y} = -\frac{dy}{dx} |_{U=\text{constant}} = \frac{MU_x}{MU_y}$. Diminishing MRS: The MRS decreases as more of good $x$ is consumed (and less of good $y$). This implies indifference curves are convex to the origin. Convexity of Indifference Curves: Implies individuals prefer balanced bundles to extreme ones. Mathematically, it is linked to the quasi-concavity of the utility function. Specific Utility Function Forms Cobb-Douglas Utility: $U(x,y) = x^a y^b$ (or $x^\alpha y^{1-\alpha}$). $MRS = \frac{a}{b} \frac{y}{x}$. Homothetic, as MRS depends only on the ratio $y/x$. Example 3.1: Cobb-Douglas Utility Calculates $MU_x, MU_y$ and $MRS$ for $U(x,y) = x y$. Shows that for $U(x,y) = \ln x + \ln y$, the MRS is identical, illustrating monotonic transformation. Perfect Substitutes: $U(x,y) = ax + by$. $MRS = \frac{a}{b}$ (constant). Indifference curves are straight lines. Perfect Complements (Leontief): $U(x,y) = \min(ax, by)$. Indifference curves are L-shaped. No unique MRS at the corner. CES (Constant Elasticity of Substitution) Utility: $U(x,y) = (\alpha x^\rho + \beta y^\rho)^{1/\rho}$. $MRS = \frac{\alpha}{\beta} (\frac{x}{y})^{\rho-1}$. Elasticity of substitution ($\sigma$) between $x$ and $y$ is $\sigma = \frac{1}{1-\rho}$. As $\rho \to 1$, $\sigma \to \infty$ (perfect substitutes). As $\rho \to 0$, $\sigma \to 1$ (Cobb-Douglas). As $\rho \to -\infty$, $\sigma \to 0$ (perfect complements). Example 3.2: CES Utility Calculates MRS for $U(x,y) = x^{0.5} + y^{0.5}$ (where $\rho=0.5$). Demonstrates how to test for diminishing MRS ($MRS$ decreases as $x$ increases). Quasi-linear Utility: $U(x,y) = f(x) + y$. MRS depends only on $x$, not $y$. (e.g., $U(x,y) = \ln x + y \Rightarrow MRS = 1/x$). Not homothetic. 4. Utility Maximization and Choice The Budget Constraint and Utility Maximization Budget Constraint: $I = p_1x_1 + p_2x_2 + \dots + p_nx_n$. Shows all affordable consumption bundles given income ($I$) and prices ($p_i$). For two goods, $x_2 = \frac{I}{p_2} - \frac{p_1}{p_2}x_1$. Slope is $-\frac{p_1}{p_2}$. Changes in Income: Parallel shift of budget constraint. Changes in Price: Rotation of budget constraint around the axis of the unchanged good. Utility Maximization Principle: Individuals choose the consumption bundle that maximizes their utility subject to their budget constraint. Graphical Solution (2 goods): Tangency between the highest attainable indifference curve and the budget constraint. At this point, $MRS_{x,y} = \frac{p_x}{p_y}$. First-Order Conditions (Lagrangian Method): Maximize $\mathcal{L} = U(x_1, \dots, x_n) + \lambda(I - p_1x_1 - \dots - p_nx_n)$. $\frac{\partial \mathcal{L}}{\partial x_i} = U_i - \lambda p_i = 0 \Rightarrow \frac{U_i}{p_i} = \lambda$ for all $i$. This implies $\frac{U_i}{p_i} = \frac{U_j}{p_j}$ for any two goods $i, j$, or $\frac{U_i}{U_j} = \frac{p_i}{p_j}$. $\frac{\partial \mathcal{L}}{\partial \lambda} = I - \sum p_ix_i = 0$. Lagrange Multiplier ($\lambda$): Represents the marginal utility of income. How much utility increases if income increases by one unit. Second-Order Conditions: Diminishing MRS (strictly quasi-concave utility function) ensures that the tangency point is a maximum. Corner Solutions: If the indifference curve is not tangent to the budget constraint, the consumer may choose to consume zero of a good. This occurs if $MRS_{x,y} > \frac{p_x}{p_y}$ at $x=0$ (or vice versa). Kuhn-Tucker conditions apply here. Example 4.1: Cobb-Douglas Utility Maximization Maximizes $U(x,y) = x^\alpha y^\beta$ subject to $p_x x + p_y y = I$. Derives demand functions: $x = \frac{\alpha}{\alpha+\beta} \frac{I}{p_x}$ and $y = \frac{\beta}{\alpha+\beta} \frac{I}{p_y}$. Shows that the share of income spent on each good is constant. Example 4.2: CES Utility Maximization Maximizes $U(x,y) = x^\rho + y^\rho$ subject to $p_x x + p_y y = I$. Derives demand functions: $x = \frac{I}{p_x[1+(p_x/p_y)^{\rho/(1-\rho)}]}$ and $y = \frac{I}{p_y[1+(p_y/p_x)^{\rho/(1-\rho)}]}$. Illustrates how the demand for each good depends on its own price and the relative price ratio. Indirect Utility and Expenditure Functions Indirect Utility Function: $V(p_1, \dots, p_n, I) = U(x_1^*(p,I), \dots, x_n^*(p,I))$. Shows the maximum utility an individual can achieve given prices and income. Properties: Homogeneous of degree 0 in all prices and income (doubling all prices and income leaves utility unchanged). Nondecreasing in income, nonincreasing in prices. Quasi-convex in prices. Roy's Identity: Derives Marshallian demand functions from the indirect utility function. $x_i(p,I) = -\frac{\partial V/\partial p_i}{\partial V/\partial I}$. Example 4.3: Indirect Utility and Roy's Identity Uses the Cobb-Douglas demand functions from Example 4.1 to construct the indirect utility function. Applies Roy's Identity to this indirect utility function to show it correctly yields the Cobb-Douglas demand functions. Expenditure Minimization (Dual Problem): Minimize the expenditure required to achieve a given level of utility ($U_0$). Minimize $E = p_1x_1 + \dots + p_nx_n$ subject to $U(x_1, \dots, x_n) = U_0$. Lagrangian: $\mathcal{L}^E = p_1x_1 + \dots + p_nx_n + \mu(U_0 - U(x_1, \dots, x_n))$. First-Order Conditions: Same as for utility maximization: $\frac{U_i}{U_j} = \frac{p_i}{p_j}$. Expenditure Function: $E(p_1, \dots, p_n, U) = \min E$. Shows the minimum expenditure needed to achieve a given utility level $U_0$ at given prices. Properties: Homogeneous of degree 1 in prices. Nondecreasing in utility and prices. Concave in prices. Shephard's Lemma: Derives compensated (Hicksian) demand functions from the expenditure function. $x_i^c(p,U) = \frac{\partial E}{\partial p_i}$. Example 4.4: Expenditure Functions Derives the expenditure function for Cobb-Douglas utility. Applies Shephard's Lemma to derive the compensated demand functions for Cobb-Douglas. The Lump Sum Principle: Taxes on general purchasing power (income taxes) are more efficient than taxes on specific goods, because they do not distort relative prices and thus avoid welfare losses from substitution effects. 5. Income and Substitution Effects Demand Functions and Their Properties Demand Functions (Marshallian): $x_i^* = x_i(p_1, \dots, p_n, I)$. Show the quantity demanded of good $i$ as a function of all prices and income. Homogeneity: Demand functions are homogeneous of degree 0 in all prices and income. (If all prices and income double, quantity demanded remains unchanged). Changes in Income: Normal Good: $\frac{\partial x_i}{\partial I} > 0$. Quantity demanded increases with income. Inferior Good: $\frac{\partial x_i}{\partial I} Engel Curve: Relationship between quantity demanded of a good and income, holding prices constant. Individual Demand Curve: Shows the relationship between the quantity demanded of a good and its own price, holding other prices, income, and preferences constant. Compensated Demand and the Slutsky Equation Compensated (Hicksian) Demand Curves: $x_i^c = x_i^c(p_1, \dots, p_n, U)$. Show the quantity demanded of good $i$ as a function of all prices and a fixed utility level. Reflect only the substitution effect of a price change. Always downward sloping ($\frac{\partial x_i^c}{\partial p_i} \le 0$). Derived from the expenditure function via Shephard's Lemma: $x_i^c(p,U) = \frac{\partial E(p,U)}{\partial p_i}$. Slutsky Equation: Decomposes the total effect of a price change on quantity demanded into a substitution effect and an income effect. Total Effect: $\frac{\partial x}{\partial p_x}$. Substitution Effect: $\frac{\partial x^c}{\partial p_x}$. (Movement along an indifference curve). Always non-positive. Income Effect: $-x \frac{\partial x}{\partial I}$. (Movement to a new indifference curve). Negative for normal goods (price increase reduces purchasing power, reducing demand). Positive for inferior goods (price increase reduces purchasing power, increasing demand). Slutsky Equation: $\frac{\partial x}{\partial p_x} = \frac{\partial x^c}{\partial p_x} - x \frac{\partial x}{\partial I}$. Giffen's Paradox: For an inferior good, if the positive income effect outweighs the negative substitution effect, the demand curve can be upward sloping ($\frac{\partial x}{\partial p_x} > 0$). This is rare in practice. Example 5.1: Slutsky Equation for a Cobb-Douglas Case Calculates the own-price elasticity of demand for a Cobb-Douglas utility function. Decomposes this elasticity into substitution and income effects using the Slutsky equation. Shows that for Cobb-Douglas, income elasticity is 1, and the share of income spent on the good is constant. Demand Elasticities and Consumer Surplus Demand Elasticities: Measure the responsiveness of quantity demanded to changes in prices or income. Price Elasticity of Demand: $e_{x,p_x} = \frac{\% \Delta x}{\% \Delta p_x} = \frac{\partial x}{\partial p_x} \cdot \frac{p_x}{x}$. Elastic: $|e_{x,p_x}| > 1$. Unit Elastic: $|e_{x,p_x}| = 1$. Inelastic: $|e_{x,p_x}| Income Elasticity of Demand: $e_{x,I} = \frac{\% \Delta x}{\% \Delta I} = \frac{\partial x}{\partial I} \cdot \frac{I}{x}$. Normal good: $e_{x,I} > 0$. Inferior good: $e_{x,I} Luxury good: $e_{x,I} > 1$. Necessity good: $0 Cross-Price Elasticity of Demand: $e_{x,p_y} = \frac{\% \Delta x}{\% \Delta p_y} = \frac{\partial x}{\partial p_y} \cdot \frac{p_y}{x}$. Substitutes: $e_{x,p_y} > 0$. Complements: $e_{x,p_y} Relationships Among Elasticities: Homogeneity: For any good $x_i$, $\sum_{j=1}^n e_{x_i, p_j} + e_{x_i, I} = 0$. (Sum of all elasticities equals zero). Engel Aggregation: $\sum_i s_i e_{x_i, I} = 1$, where $s_i$ is the share of income spent on good $i$. (Weighted average of income elasticities is 1). Cournot Aggregation: $\sum_i s_i e_{x_i, p_j} = -s_j$. (Weighted sum of cross-price elasticities with respect to $p_j$ equals negative of income share of good $j$). Slutsky Equation (Elasticity Form): $e_{x,p_x} = e_{x,p_x}^c - s_x e_{x,I}$, where $s_x = \frac{p_x x}{I}$. Consumer Surplus: The difference between what consumers are willing to pay for a good and what they actually pay. Area below the demand curve and above the market price. Approximation for changes in welfare due to price changes. Compensating Variation (CV): The amount of money that would have to be given to a consumer to offset the harm from a price increase (or taken away for a price decrease), returning them to their original utility level. $CV = E(p_x^1, p_y, U_0) - E(p_x^0, p_y, U_0)$. Equivalent Variation (EV): The amount of money that would have to be given to a consumer (or taken away) to achieve the new utility level at the original prices. $EV = E(p_x^0, p_y, U_1) - E(p_x^0, p_y, U_0)$. CV and EV are generally not equal but become closer for small price changes. They can be visualized using compensated demand curves. Revealed Preference: If a consumer chooses bundle A when bundle B is also affordable, then A is "revealed preferred" to B. Allows inference of preferences without explicit utility functions. Implies the negativity of the substitution effect. Weak Axiom of Revealed Preference (WARP): If A is revealed preferred to B, then B cannot be revealed preferred to A. Strong Axiom of Revealed Preference (SARP): If A is directly or indirectly revealed preferred to B, then B cannot be directly or indirectly revealed preferred to A. (SARP implies WARP and guarantees the existence of a utility function). 6. Demand Relationships among Goods Substitutes and Complements Two-Good Case: The Slutsky equation for cross-price effects: $\frac{\partial x}{\partial p_y} = \frac{\partial x^c}{\partial p_y} - y \frac{\partial x}{\partial I}$. The sign of the total effect is ambiguous because the substitution effect and income effect can work in opposite directions. The cross-price substitution effect ($\frac{\partial x^c}{\partial p_y}$) is usually positive. Gross Substitutes: $\frac{\partial x_i}{\partial p_j} > 0$. An increase in the price of good $j$ leads to an increase in the demand for good $i$. Gross Complements: $\frac{\partial x_i}{\partial p_j} Gross definitions are generally asymmetric ($\frac{\partial x_i}{\partial p_j}$ may not equal $\frac{\partial x_j}{\partial p_i}$). Net Substitutes (Hicksian Substitutes): $\frac{\partial x_i^c}{\partial p_j} > 0$. An increase in the price of good $j$ leads to an increase in the compensated demand for good $i$. Reflects only the substitution effect, holding utility constant. Net Complements (Hicksian Complements): $\frac{\partial x_i^c}{\partial p_j} Net definitions are symmetric: $\frac{\partial x_i^c}{\partial p_j} = \frac{\partial x_j^c}{\partial p_i}$. (This is because the expenditure function is concave, making its Hessian matrix symmetric). Hicks' Second Law of Demand: Most goods must be net substitutes. It is impossible for all goods to be net complements. ($\sum_{j \ne i} \frac{\partial x_i^c}{\partial p_j} \ge 0$). Example 6.1: Net Substitutes for Cobb-Douglas Uses compensated demand functions for Cobb-Douglas utility to show that goods are net substitutes. Demonstrates that $\frac{\partial x^c}{\partial p_y} > 0$. Composite Commodities and Household Production Composite Commodities: A group of goods whose relative prices move together can be treated as a single commodity. Allows simplification of demand analysis by reducing the number of variables. Example 6.2: Composite Commodity Illustrates how to treat a group of goods (e.g., "all other goods") as a single composite commodity with a "price" equal to a weighted average of individual prices. Household Production Model (Becker): Individuals derive utility from "attributes" or "basic commodities" that they produce using market goods and their own time. The household acts as a small firm, combining inputs to produce outputs that directly provide utility. Implies implicit (shadow) prices for these basic commodities. Almost Ideal Demand System (AIDS): A system of demand equations that is consistent with utility maximization. Allows for flexible specification of price and income elasticities. Share of budget spent on good $i$ is $s_i = \alpha_i + \sum_j \gamma_{ij} \ln p_j + \beta_i \ln(I/P^*)$, where $P^*$ is a price index. 7. Uncertainty Expected Utility and Risk Aversion Random Variable: A variable whose outcome is subject to chance. Probability Density Function (PDF): $f(x)$ for outcomes of a random variable. $P(x_i)$ for discrete outcomes. Expected Value: $E(x) = \sum x_i P(x_i)$ (discrete) or $\int xf(x)dx$ (continuous). Fair Gamble: A gamble where the expected monetary value is zero. St. Petersburg Paradox: Illustrates that individuals do not necessarily base decisions on expected monetary value if that value is infinite, pointing to the concept of expected utility. Expected Utility Hypothesis (von Neumann-Morgenstern): Individuals choose among risky alternatives (gambles) based on the expected value of the utility derived from the outcomes. $E(U) = \sum P_i U(x_i)$, where $P_i$ is probability of outcome $x_i$. Utility function $U$ is defined over wealth or consumption. Risk Aversion: An individual who prefers a certain amount of wealth over a risky gamble with the same expected monetary value. Implies diminishing marginal utility of wealth ($U''(W) Willing to pay a risk premium to avoid risk. Certainty Equivalent: The amount of certain wealth that yields the same utility as a risky gamble. Example 7.1: Coin Flip and Risk Aversion Calculates expected value of a gamble and expected utility for a risk-averse individual ($U(W) = \sqrt{W}$). Shows that expected utility is less than the utility of the expected wealth. Determines the risk premium. Risk Neutrality: $U''(W)=0$ (linear utility). Indifferent between a fair gamble and its expected value. Risk Loving: $U''(W)>0$ (convex utility). Prefers a fair gamble to its expected value. Measures of Risk Aversion Absolute Risk Aversion (Pratt): $r(W) = -\frac{U''(W)}{U'(W)}$. Measures risk aversion at a given wealth level. A higher $r(W)$ means greater risk aversion. Risk premium ($P$) for a small gamble with expected value 0 and variance $\sigma_h^2$ is approximately $P \approx \frac{1}{2} r(W) \sigma_h^2$. Relative Risk Aversion: $rr(W) = W r(W) = -W \frac{U''(W)}{U'(W)}$. Measures risk aversion relative to wealth. Often assumed to be constant (CRRA). Types of Utility Functions: CARA (Constant Absolute Risk Aversion): $U(W) = -e^{-AW}$ (for $A>0$). $r(W) = A$. CRRA (Constant Relative Risk Aversion): $U(W) = \frac{W^R}{R}$ (for $R \ne 0$) or $U(W) = \ln W$ (for $R=0$). $rr(W) = 1-R$. If $R=0$, $U(W)=\ln W$, $rr(W)=1$. If $R=1$, $U(W)=W$, $rr(W)=0$ (risk neutral). Example 7.2: Constant Relative Risk Aversion Calculates $r(W)$ and $rr(W)$ for $U(W)=\ln W$ and $U(W)=\sqrt{W}$. Shows that for $\ln W$, $rr(W)=1$. For $\sqrt{W}$, $rr(W)=1/2$. Reducing Uncertainty and State-Preference Approach Methods for Reducing Uncertainty/Risk: Insurance: Risk-averse individuals pay a premium to transfer risk to an insurer. Fair insurance: Premium equals expected loss. Risk-averse individuals will fully insure. Diversification: Spreading investments across multiple independent assets to reduce overall risk without sacrificing expected return. Flexibility (Real Options): Maintaining the ability to adjust decisions as new information becomes available, which has value in uncertain environments. (e.g., delaying investment). Information: Acquiring knowledge to make better decisions in uncertain situations. State-Preference Approach: Models choice under uncertainty as choices among contingent commodities (wealth in different "states of the world"). Let $W_1$ be wealth in state 1 (e.g., "good times") and $W_2$ in state 2 ("bad times"). Budget constraint: $p_1 W_1 + p_2 W_2 = I$, where $p_1, p_2$ are prices of contingent claims. If markets are "fair" (no transaction costs or risk premiums), then $p_1 = \pi_1$ and $p_2 = \pi_2$ (actual probabilities of states). Risk aversion implies preference for smoothing consumption across states, so $W_1=W_2$ if prices are fair. Example 7.3: State-Preference Model Uses a Cobb-Douglas utility function for wealth in two states. Demonstrates how to find optimal wealth allocation in each state given probabilities and prices of contingent claims. Shows that if prices are fair, optimal allocation is to equalize wealth in both states (full insurance). Asymmetric Information: One party has more information than another. Leads to: Moral Hazard: Hidden actions (e.g., insured person takes less care). Adverse Selection: Hidden characteristics (e.g., high-risk individuals are more likely to buy insurance). 8. Game Theory Basic Concepts and Simultaneous Games Game: A formal representation of a strategic situation involving multiple decision-makers. Players: The decision-makers in the game. Strategies: A complete plan of action for a player in every possible situation that might arise in the game. $s_i \in S_i$. Payoffs: The utility or profit that players receive for each possible combination of strategies. $u_i(s_i, s_{-i})$ where $s_{-i}$ are strategies of other players. Normal Form Game: A representation of a game using a payoff matrix, which shows players, their strategies, and the payoffs for each strategy combination. Prisoners' Dilemma: A classic example where individual rationality leads to a collectively suboptimal outcome. Each player has a dominant strategy to 'confess' (defect), even though both would be better off if they both 'remain silent' (cooperate). Best Response: A strategy $s_i^*$ for player $i$ is a best response to the other players' strategies $s_{-i}$ if $u_i(s_i^*, s_{-i}) \ge u_i(s_i', s_{-i})$ for all other strategies $s_i'$ in $S_i$. Dominant Strategy: A strategy that is a best response to all possible strategies of other players. If a player has a dominant strategy, they will always play it. Nash Equilibrium: A strategy profile $(s_1^*, \dots, s_n^*)$ such that each player's strategy $s_i^*$ is a best response to the other players' equilibrium strategies $s_{-i}^*$. No player can unilaterally improve their payoff by changing their strategy. Example 8.1: The Battle of the Sexes Illustrates a game with multiple Nash equilibria in pure strategies. Highlights the coordination problem when there are multiple equilibria. Example 8.2: A Cournot Equilibrium Applies Nash equilibrium concept to a quantity-setting duopoly. Derives best-response functions and finds the intersection as the Nash equilibrium. Mixed Strategy: A probability distribution over a player's pure strategies. Players choose probabilities for playing each pure strategy. Nash Equilibrium in Mixed Strategies: Each player's mixed strategy makes the other players indifferent between their pure strategies. Existence of Nash Equilibrium (Nash's Theorem): Every finite game has at least one Nash equilibrium, possibly in mixed strategies. Example 8.3: Mixed Strategies in the Battle of the Sexes Calculates the mixed strategy Nash equilibrium for the Battle of the Sexes game. Shows how each player's mixed strategy makes the other player indifferent between their pure strategies. Continuum of Actions: Games where players choose from a continuous range of actions (e.g., quantities in Cournot, prices in Bertrand). Nash equilibrium is found by setting partial derivatives of payoff functions to zero. Tragedy of the Commons: Game illustrating the overuse or degradation of a common resource when individuals act in their own self-interest, leading to an outcome that is worse for all. Sequential and Repeated Games Sequential Games: Players move in a specific order, and later movers observe the actions of prior movers. Extensive Form Game: A representation using a game tree, showing decision nodes, branches (actions), information sets, and payoffs. Information Set: A set of decision nodes where a player cannot distinguish between the nodes. (If a player is uncertain about a previous move, those nodes are in the same information set). Subgame: A part of a game tree that starts from a single decision node (or an information set of a single node) and includes all subsequent branches. Proper Subgame: A subgame that does not contain any information sets that cross its boundary. Subgame-Perfect Equilibrium (SPE): A strategy profile that is a Nash equilibrium in every proper subgame of the original game. Rules out non-credible threats. Backward Induction: The standard method for finding SPE in finite sequential games. Start from the end of the game and work backward. Example 8.4: A Sequential Game Illustrates backward induction in a sequential game (e.g., entry deterrence). Identifies the SPE and rules out non-credible threats. Repeated Games: The same "stage game" is played multiple times. Finitely Repeated Games: If the stage game has a unique Nash equilibrium, the only SPE is to repeat that Nash equilibrium in every period (due to backward induction). Infinitely Repeated Games: Cooperation can be sustained if players are sufficiently patient. Discount Factor ($\delta$): Measures how much players value future payoffs relative to current payoffs. ($\delta = 1/(1+r)$). Trigger Strategies: Strategies where players cooperate as long as others cooperate, but "punish" any deviation by reverting to a non-cooperative (e.g., Nash) strategy forever (grim trigger) or for a finite period. Folk Theorem: In infinitely repeated games, any individually rational payoff (better than minmax payoff) can be sustained as an SPE if players are sufficiently patient (i.e., $\delta$ is high enough). Games with Incomplete Information Incomplete Information (Bayesian Games): Some players have private information about certain parameters of the game (e.g., costs, valuations, "types"). Player Types ($t_i$): Represents a player's private information. Each player knows their own type, but not necessarily others'. Beliefs: Players form beliefs (probability distributions) about the types of other players. Bayesian Nash Equilibrium: A strategy for each player (a function mapping their type to an action) such that, given their beliefs about other players' types and strategies, no player can improve their expected payoff by unilaterally changing their strategy. Bayes' Rule: Used to update beliefs about types based on observed signals or actions. $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$. Signaling Games: An informed player moves first, sending a signal to an uninformed player. The uninformed player then takes an action based on the signal. Perfect Bayesian Equilibrium: A refinement of Bayesian Nash equilibrium for sequential games with incomplete information. Requires: Strategies are optimal given beliefs. Beliefs are updated using Bayes' rule whenever possible. Separating Equilibrium: Different types of the informed player choose different signals. The signal perfectly reveals the type. Pooling Equilibrium: Different types of the informed player choose the same signal. The signal provides no information about the type. Hybrid Equilibrium: One type plays a mixed strategy. Job-Market Signaling (Spence): Education can serve as a costly signal of high ability, even if it doesn't directly increase productivity. 9. Production Functions Basic Concepts and Marginal Productivity Production Function: $q = f(k, l)$ (or $f(x_1, \dots, x_n)$). Shows the maximum quantity of output ($q$) that can be produced from any given combination of inputs (e.g., capital $k$, labor $l$). Marginal Physical Product (MP): The additional output produced by employing one more unit of a specific input, holding all other inputs constant. $MP_k = \frac{\partial q}{\partial k} = f_k$. $MP_l = \frac{\partial q}{\partial l} = f_l$. Diminishing Marginal Productivity: The MP of an input eventually decreases as more of that input is used, holding other inputs constant. $f_{kk} Average Physical Product (AP): Total output divided by the quantity of an input. $AP_l = \frac{q}{l}$. Example 9.1: A Cobb-Douglas Production Function Calculates $MP_l, MP_k, AP_l, AP_k$ for $q = 10k^{0.5}l^{0.5}$. Shows that both marginal products are diminishing. Isoquants and Returns to Scale Isoquant: A curve that shows all technically efficient combinations of two inputs (e.g., $k$ and $l$) that can produce a given level of output ($q_0$). Analogous to indifference curves in consumer theory. Negatively sloped (assuming inputs are productive). Convex to the origin (due to diminishing RTS). Marginal Rate of Technical Substitution (RTS): The rate at which one input can be substituted for another while keeping output constant. $RTS(l \text{ for } k) = -\frac{dk}{dl} |_{q=q_0} = \frac{MP_l}{MP_k}$. Diminishing RTS: The RTS decreases as more labor is substituted for capital (and vice versa). This implies isoquants are convex. Mathematically, it requires the production function to be quasi-concave. Cross-productivity effects $f_{kl}$ are usually positive (e.g., more capital makes labor more productive). Example 9.2: RTS for Cobb-Douglas Calculates RTS for $q = 10k^{0.5}l^{0.5}$. Shows that RTS is $\frac{k}{l}$, demonstrating its diminishing nature as $l$ increases and $k$ decreases. Returns to Scale: How output responds to a proportional increase in all inputs. If $f(tk, tl) = t^j f(k,l)$: Constant Returns to Scale (CRS): $j=1$. Output increases proportionally with inputs. (e.g., doubling inputs doubles output). Increasing Returns to Scale (IRS): $j>1$. Output increases more than proportionally. Decreasing Returns to Scale (DRS): $j For CRS, the production function is homogeneous of degree 1. $MP_k, MP_l$ are homogeneous of degree 0. Homothetic Production Function: The RTS depends only on the ratio of inputs ($k/l$), not on the absolute levels. All homogeneous production functions are homothetic. Elasticity of Substitution ($\sigma$): Measures the responsiveness of the capital-labor ratio ($k/l$) to changes in the RTS. $\sigma = \frac{\% \Delta (k/l)}{\% \Delta RTS} = \frac{d \ln(k/l)}{d \ln RTS}$. Always positive. High $\sigma$ means inputs are easily substitutable. Low $\sigma$ means inputs are poor substitutes. Specific Production Function Forms Linear Production Function: $q = ak + bl$. Inputs are perfect substitutes. $\sigma = \infty$. RTS is constant. Fixed Proportions (Leontief) Production Function: $q = \min(ak, bl)$. Inputs are perfect complements. Isoquants are L-shaped. $\sigma = 0$. Cobb-Douglas Production Function: $q = Ak^\alpha l^\beta$. (Often $A$ is technology, $\alpha+\beta$ determines returns to scale). $\sigma = 1$. If $\alpha+\beta=1$, CRS. If $\alpha+\beta>1$, IRS. If $\alpha+\beta In $\ln q = \ln A + \alpha \ln k + \beta \ln l$, $\alpha$ and $\beta$ are output elasticities of capital and labor, respectively. Example 9.3: Returns to Scale for Cobb-Douglas Uses $q = k^\alpha l^\beta$ to show how $f(tk, tl) = t^{\alpha+\beta} f(k,l)$ determines returns to scale. CES (Constant Elasticity of Substitution) Production Function: $q = (\alpha k^\rho + \beta l^\rho)^{\gamma/\rho}$. Elasticity of substitution $\sigma = \frac{1}{1-\rho}$. $\gamma$ determines returns to scale (if $\gamma=1$, CRS). As $\rho \to 1$, CES approaches linear. As $\rho \to 0$, CES approaches Cobb-Douglas. As $\rho \to -\infty$, CES approaches Leontief. Example 9.4: CES Production Function Calculates RTS for a CES function. Determines the elasticity of substitution. Technical Progress Technical Progress: Shifts the production function, allowing more output to be produced with the same amount of inputs, or the same output with fewer inputs. Can be represented as $q = A(t)f(k,l)$, where $A(t)$ increases over time. Neutral Technical Progress: Does not change the RTS for a given $k/l$ ratio. Biased Technical Progress: Changes the RTS for a given $k/l$ ratio, favoring one input over another. Growth Accounting: Decomposes output growth into contributions from input growth and technical progress (total factor productivity, TFP). $G_q = G_A + e_{q,k} G_k + e_{q,l} G_l$, where $G$ denotes growth rate and $e$ are output elasticities. 10. Cost Functions Definitions of Costs and Cost Minimization Definitions of Costs: Accounting Cost: Explicit, out-of-pocket expenses, historical costs, depreciation according to accounting rules. Economic Cost (Opportunity Cost): The value of the next best alternative use of a resource. Includes both explicit and implicit costs (e.g., value of owner's time, forgone interest on invested capital). Sunk Costs: Costs that have already been incurred and cannot be recovered. Should not influence future decisions. Labor costs: $w$ (wage rate). Capital costs: $v$ (rental rate of capital). Entrepreneurial costs: Opportunity cost of the owner's time and financial capital. Total Cost (TC): The sum of all economic costs of production. For two inputs, $C = wl + vk$. Cost-Minimizing Input Choices: Firms choose input combinations to produce a given output level ($q_0$) at the lowest possible cost. Graphical Solution: Tangency between an isoquant ($q_0$) and the lowest possible isocost line. Isocost line: All combinations of inputs that cost the same amount: $C_0 = wl + vk$. Slope is $-\frac{w}{v}$. First-Order Conditions (Lagrangian Method): Minimize $\mathcal{L} = wl + vk + \lambda[q_0 - f(k,l)]$. $\frac{\partial \mathcal{L}}{\partial l} = w - \lambda f_l = 0 \Rightarrow \frac{w}{f_l} = \lambda$. $\frac{\partial \mathcal{L}}{\partial k} = v - \lambda f_k = 0 \Rightarrow \frac{v}{f_k} = \lambda$. Therefore, $\frac{w}{f_l} = \frac{v}{f_k} \Rightarrow \frac{f_l}{f_k} = \frac{w}{v}$. (RTS equals the input price ratio). Also, $\frac{MP_l}{w} = \frac{MP_k}{v}$ (marginal product per dollar spent is equal for all inputs). Lagrange Multiplier ($\lambda$): Represents the marginal cost of producing an additional unit of output. $\lambda = MC$. Example 10.1: Cost Minimization for a Cobb-Douglas Production Function Minimizes $wl+vk$ subject to $q_0 = k^\alpha l^\beta$. Derives the cost-minimizing input demand functions $k(w,v,q_0)$ and $l(w,v,q_0)$. Shows that the ratio $k/l = \frac{\alpha w}{\beta v}$. Expansion Path: The locus of cost-minimizing input combinations for varying output levels, holding input prices constant. It shows how input usage changes as output increases. Inferior Input: An input whose usage decreases as output expands. (Rare in production). The Cost Function and Input Demands Total Cost Function: $C = C(v, w, q)$. Shows the minimum cost of producing a given output level $q$ for given input prices $v$ and $w$. Derived by substituting the cost-minimizing input demand functions ($k^*(v,w,q), l^*(v,w,q)$) into the total cost equation ($C = vk + wl$). Example 10.2: Cost Function for Cobb-Douglas Uses the optimal input demands from Example 10.1 to derive the total cost function for $q = k^\alpha l^\beta$. $C(w,v,q) = q^{1/(\alpha+\beta)} B w^{\beta/(\alpha+\beta)} v^{\alpha/(\alpha+\beta)}$, where $B$ is a constant. Average Cost (AC): $AC(v,w,q) = C(v,w,q)/q$. Cost per unit of output. Marginal Cost (MC): $MC(v,w,q) = \frac{\partial C}{\partial q}$. Additional cost of producing one more unit of output. Properties of Cost Functions: Homogeneous of degree 1 in input prices (doubling $w,v$ doubles cost for given $q$). Nondecreasing in $q, v, w$. Concave in input prices (firms can substitute away from more expensive inputs). AC and MC are homogeneous of degree 1 in input prices. Elasticity of Substitution ($\sigma$): Can also be expressed in terms of input prices: $\sigma = \frac{\partial \ln(k/l)}{\partial \ln(w/v)}$. (Easier to observe than RTS). Technical Change: Shifts cost functions downward. If $q=A(t)f(k,l)$, then $C_t(v,w,q) = C_0(v,w,q)/A(t)$. Conditional Input Demand Functions: $k^c(v,w,q)$ and $l^c(v,w,q)$. Show the cost-minimizing quantity of inputs needed to produce $q$ at given prices. Shephard's Lemma: Derives conditional input demand functions from the cost function. $k^c(v,w,q) = \frac{\partial C(v,w,q)}{\partial v}$. $l^c(v,w,q) = \frac{\partial C(v,w,q)}{\partial w}$. Example 10.3: Shephard's Lemma Applies Shephard's Lemma to the Cobb-Douglas cost function (from Example 10.2) to re-derive the conditional input demand functions. Short-Run vs. Long-Run Costs Short Run: A period during which at least one input (usually capital) is fixed. Fixed Cost (FC): Cost of fixed inputs (e.g., $vk_1$). Variable Cost (VC): Cost of variable inputs (e.g., $wl$). Short-Run Total Cost (STC): $STC = FC + VC = vk_1 + wl$. Short-Run Average Total Cost (SATC): $SATC = STC/q$. Short-Run Average Variable Cost (SAVC): $SAVC = VC/q$. Short-Run Marginal Cost (SMC): $SMC = \frac{\partial STC}{\partial q} = \frac{\partial VC}{\partial q}$. Relationship: SMC intersects SAVC and SATC at their minimum points. Long Run: A period during which all inputs are variable. Long-Run Total Cost (LTC): $LTC = C(v,w,q)$. All costs are variable. Long-Run Average Cost (LAC): $LAC = LTC/q$. Long-Run Marginal Cost (LMC): $LMC = \frac{\partial LTC}{\partial q}$. Relationship: LMC intersects LAC at its minimum. Relationship between Short-Run and Long-Run Costs: Short-run costs are generally higher than long-run costs (except at the optimal fixed input level for a given output). The Long-Run Average Cost (LAC) curve is the envelope of all possible Short-Run Average Total Cost (SATC) curves. At the minimum point of LAC, $LAC = LMC = SATC = SMC$. Example 10.4: Short-Run and Long-Run Costs Illustrates the derivation of short-run cost curves when capital is fixed. Compares short-run costs to long-run costs for a specific output level. 11. Profit Maximization Profit Maximization by a Price-Taking Firm Economic Profit: $\pi = R(q) - C(q)$, where $R(q)$ is total revenue and $C(q)$ is total economic cost. Profit Maximization Principle: A firm chooses the output level ($q^*$) that maximizes its economic profit. First-Order Condition (FOC): $\frac{d\pi}{dq} = \frac{dR}{dq} - \frac{dC}{dq} = 0 \Rightarrow MR(q^*) = MC(q^*)$. (Marginal Revenue equals Marginal Cost). Second-Order Condition (SOC): $\frac{d^2\pi}{dq^2} = \frac{dMR}{dq} - \frac{dMC}{dq} Marginal Revenue (MR): The additional revenue from selling one more unit of output. For a price-taking firm (perfect competition), $MR = P$ (market price). Therefore, for a price-taker, profit maximization implies $P = MC(q^*)$. Example 11.1: Profit Maximization in the Short Run Uses a short-run cost function (e.g., $STC = q^3 - 10q^2 + 75q + 100$) and a given price $P$. Calculates SMC and sets $P=SMC$ to find the profit-maximizing output. Verifies SOC and calculates profit. The Firm's Supply and Input Demand Short-Run Supply Curve (Price-Taking Firm): The portion of the Short-Run Marginal Cost (SMC) curve that lies above the Short-Run Average Variable Cost (SAVC) curve. Shutdown Decision: In the short run, a firm will continue to produce as long as $P \ge SAVC_{\min}$. If $P Profit Function: $\Pi(P, v, w) = \max_{k,l} [P f(k,l) - vk - wl]$. Shows the maximum profit a firm can achieve given output price ($P$) and input prices ($v, w$). Properties: Homogeneous of degree 1 in all prices ($P, v, w$). Nondecreasing in output price $P$. Nonincreasing in input prices $v, w$. Convex in output price $P$. Envelope Results (Hotelling's Lemma): Derive supply and unconditional input demand functions from the profit function. $\frac{\partial \Pi}{\partial P} = Q(P,v,w)$ (firm's supply function). $\frac{\partial \Pi}{\partial v} = -k(P,v,w)$ (unconditional demand for capital). $\frac{\partial \Pi}{\partial w} = -l(P,v,w)$ (unconditional demand for labor). Example 11.2: The Profit Function Uses a Cobb-Douglas production function to derive the firm's profit function. Applies Hotelling's Lemma to derive the supply function and unconditional input demand functions. Producer Surplus (Short Run): The area above the firm's supply curve and below the market price. Measures the extra return to fixed inputs (quasi-rents). Numerically, it is equal to $\Pi(P, \dots) + \text{Fixed Costs}$. Profit Maximization and Input Demand: Firms demand inputs to produce the profit-maximizing level of output. Marginal Revenue Product (MRP): The additional revenue generated by employing one more unit of an input. $MRP_l = MR \cdot MP_l$. For a price-taker, $MRP_l = P \cdot MP_l$. First-Order Conditions: Hire inputs until the MRP of each input equals its input price. $P \cdot MP_k = v$ and $P \cdot MP_l = w$. Second-Order Conditions: Ensure that the input choices lead to profit maximization (e.g., diminishing MRP). Unconditional Input Demand Functions: $k(P,v,w)$ and $l(P,v,w)$. Show the profit-maximizing quantity of inputs as a function of output price and input prices. Substitution Effect (Input Demand): Change in input demand due to a change in relative input prices, holding output constant (movement along an isoquant). Output Effect (Input Demand): Change in input demand due to a change in the optimal output level caused by a change in input price. Input demand curves are unambiguously downward sloping ($\frac{\partial l}{\partial w} Example 11.3: Input Demand for a Cobb-Douglas Production Function Derives the unconditional input demand for labor and capital using the profit-maximization conditions $P \cdot MP_l = w$ and $P \cdot MP_k = v$. Shows how these demands depend on output price and input prices. 12. The Partial Equilibrium Competitive Model Market Demand and Supply Market Demand: The horizontal summation of all individual demand curves. $Q_D(P, \text{other prices}, \text{incomes}, \text{preferences}) = \sum_{j=1}^m x_j(P, \dots)$. Shifts in market demand: Due to changes in aggregate income, prices of related goods, population, or aggregate preferences. Market Price Elasticity of Demand: $e_{Q,P} = \frac{\partial Q_D}{\partial P} \cdot \frac{P}{Q_D}$. Market Income Elasticity of Demand: $e_{Q,I} = \frac{\partial Q_D}{\partial I} \cdot \frac{I}{Q_D}$. Market Cross-Price Elasticity of Demand: $e_{Q,P'} = \frac{\partial Q_D}{\partial P'} \cdot \frac{P'}{Q_D}$. Timing of Supply Response: Very Short Run (Market Period): Supply is perfectly inelastic (fixed quantity). Price rations demand. Short Run: Number of firms is fixed. Existing firms can adjust variable inputs (e.g., labor) to change output. Market supply is the horizontal sum of individual firms' SMC curves above SAVC. Long Run: All inputs are variable. Firms can enter or exit the industry. Short-Run Price Determination: Equilibrium price is determined by the intersection of market demand and short-run market supply. Market Supply Curve: $Q_S(P,v,w) = \sum_{i=1}^n q_i(P,v,w)$, where $q_i$ is firm $i$'s supply function. Short-Run Elasticity of Supply: $e_{S,P} = \frac{\partial Q_S}{\partial P} \cdot \frac{P}{Q_S}$. Equilibrium Price: $Q_D(P^*) = Q_S(P^*)$. Comparative Statics: Analyzing how changes in demand or supply conditions affect equilibrium price and quantity. If demand shifts right: $P$ and $Q$ increase. If supply shifts right: $P$ decreases, $Q$ increases. Magnitude of change depends on elasticities. For a demand shift parameter $a$: $\frac{dP}{da} = \frac{\partial Q_D/\partial a}{e_{S,P} - e_{Q,P}}$. Long-Run Competitive Equilibrium and Welfare Long-Run Competitive Equilibrium: For each firm, $P = MC = AC_{\min}$. Firms earn zero economic profits. Entry/exit ensures this condition. All firms produce at the minimum point of their Long-Run Average Cost (LAC) curve. Example 12.1: Long-Run Equilibrium Given a firm's long-run cost function, finds the minimum AC and the corresponding output level. Uses market demand to determine the number of firms in long-run equilibrium. Long-Run Industry Supply Curve: Constant Cost Industry: Entry/exit does not affect input prices. LR supply curve is horizontal (perfectly elastic). $e_{LS,P} = \infty$. Increasing Cost Industry: Entry increases input prices (e.g., due to specialized inputs). LR supply curve is upward sloping. $e_{LS,P} > 0$. Decreasing Cost Industry: Entry reduces input prices (e.g., due to agglomeration economies). LR supply curve is downward sloping. $e_{LS,P} Ricardian Rent: Long-run profits for low-cost firms or owners of unique resources are capitalized into the prices of those inputs, becoming economic rents. Economic Efficiency and Welfare Analysis: Competitive equilibrium maximizes total surplus (consumer surplus + producer surplus). Consumer Surplus: The area below the market demand curve and above the market price. Producer Surplus (Long Run): The area above the long-run supply curve and below the market price. Represents rents to scarce inputs. Deadweight Loss: The reduction in total surplus due to market inefficiencies (e.g., price controls, taxes, monopolies). Price Controls and Shortages: A price ceiling set below the equilibrium price creates a shortage, reduces quantity supplied, and leads to deadweight loss. Tax Incidence Analysis: Who bears the economic burden of a tax. The burden of a tax falls more heavily on the side of the market (consumers or producers) that is less elastic. For a specific tax $t$ on producers: Change in price paid by demanders: $\frac{dP_D}{dt} = \frac{e_S}{e_S - e_D}$. Change in price received by suppliers: $\frac{dP_S}{dt} = \frac{e_D}{e_S - e_D}$. Deadweight loss of tax: Proportional to the square of the tax rate and inversely proportional to the sum of elasticities. Example 12.2: Tax Incidence Analyzes the impact of a per-unit tax on equilibrium price and quantity. Calculates the portions of the tax borne by consumers and producers based on demand and supply elasticities. 13. General Equilibrium and Welfare General Equilibrium Model of Exchange General Equilibrium: Analysis of how prices and quantities are determined in all markets simultaneously, considering interdependencies. Perfectly Competitive Price System: Assumes many markets, each competitive, with price-taking agents and perfect information. Law of One Price: Identical goods must trade for the same price in all markets. Edgeworth Box (Exchange Economy): A graphical tool to analyze trade between two individuals with fixed endowments of two goods. Dimensions of the box represent total quantities of goods. Each point in the box is an allocation of goods between the two individuals. Contract Curve: The set of all Pareto efficient allocations. These are points where the individuals' indifference curves are tangent to each other ($MRS_1 = MRS_2$). Core: The portion of the contract curve that is individually rational (each person is at least as well off as with their initial endowment). Walrasian Equilibrium (Competitive Equilibrium in Exchange): A set of prices for the goods such that, given their initial endowments, each individual maximizes utility, and all markets clear (aggregate demand equals aggregate supply). At equilibrium, $MRS_1 = MRS_2 = p_x/p_y$. Walras' Law: In an economy with $n$ markets, if $n-1$ markets are in equilibrium, the $n$-th market must also be in equilibrium. (The value of aggregate excess demand is zero for any set of prices: $\sum_i p \cdot (x_i - \bar{x}_i) = 0$). Existence of Equilibrium: Guaranteed under certain conditions (e.g., continuous demand functions, convex preferences). Proved using fixed-point theorems (Brouwer, Kakutani). Example 13.1: Exchange with Two Goods Sets up an Edgeworth Box for two individuals with Cobb-Douglas utility and endowments. Derives the contract curve and the competitive equilibrium prices and allocations. General Equilibrium with Production Edgeworth Box (Production): Used to allocate fixed total amounts of two inputs (e.g., capital $K$, labor $L$) between the production of two goods ($X$ and $Y$). Production Contract Curve: The set of all technically efficient allocations of inputs. These are points where the isoquants for the two goods are tangent ($RTS_{K,L}^X = RTS_{K,L}^Y$). Production Possibility Frontier (PPF): Shows the maximum amount of one good that can be produced for any given amount of another, given the fixed total inputs and technology. Derived from the production contract curve. Shape: Concave to the origin due to diminishing returns, specialized inputs, or differing factor intensities. Rate of Product Transformation (RPT): The absolute value of the slope of the PPF. It represents the opportunity cost of producing one more unit of good $X$ in terms of good $Y$. $RPT_{X,Y} = -\frac{dY}{dX} = \frac{MC_X}{MC_Y}$. Equilibrium in Production and Exchange: For overall efficiency, three conditions must hold: Efficiency in exchange: $MRS_1 = MRS_2 = \dots = MRS_m$ for all individuals. Efficiency in production: $RTS_X = RTS_Y = \dots$ for all goods. Product mix efficiency: $RPT_{X,Y} = MRS_{X,Y}$ for all individuals. In a perfectly competitive economy, all these conditions are met. Prices guide both consumers and producers to the efficient outcome: $MRS = P_X/P_Y = RPT$. Example 13.2: Production Possibility Frontier Derives a PPF from two Cobb-Douglas production functions and fixed total inputs. Calculates the RPT and illustrates its relationship to relative marginal costs. Comparative Statics: Analyzing how changes in technology, preferences, or endowments affect the general equilibrium. Corn Laws Debate: Illustrated how trade policy (tariffs on grain) affects factor prices (Stolper-Samuelson Theorem). Welfare Economics First Theorem of Welfare Economics: Every Walrasian (competitive) equilibrium is Pareto efficient. Requires: Price-taking behavior, complete markets, no externalities, perfect information. Second Theorem of Welfare Economics: Any Pareto optimal allocation can be achieved as a Walrasian equilibrium by an appropriate lump-sum redistribution of initial endowments (wealth). Implies that efficiency and equity can be separated. Redistribution can achieve desired equity, and then markets can achieve efficiency. Social Welfare Functions: A function that combines the utility levels of all individuals in society into a single measure of societal well-being. $SW = SW(U_1, U_2, \dots, U_m)$. Utilitarian (Benthamite): $SW = \sum U_i$. Maximizes the sum of utilities. Maximin (Rawlsian): $SW = \min(U_i)$. Maximizes the utility of the least well-off individual. Computable General Equilibrium (CGE) Models: Numerical models that simulate the behavior of complex economies to analyze policy changes. 14. Monopoly Monopoly Behavior and Welfare Implications Monopoly: A market structure where a single firm is the sole producer of a good or service for which there are no close substitutes. The monopolist faces the entire market demand curve. Barriers to Entry: Factors that prevent new firms from entering a market, protecting the monopolist's position. Technical Barriers: Natural Monopoly: Average cost (AC) is declining over the entire range of market demand, making it more efficient for one firm to serve the market. Unique knowledge or production techniques. Control over essential raw materials or inputs. Legal Barriers: Patents, copyrights, government licenses, exclusive franchises. Created Barriers: Strategic actions by the monopolist (e.g., heavy advertising, predatory pricing, lobbying). Profit Maximization for a Monopolist: The monopolist chooses the output level ($Q^*$) where Marginal Revenue (MR) equals Marginal Cost (MC). $MR(Q^*) = MC(Q^*)$. The price ($P^*$) is then determined by the demand curve at $Q^*$. Since the demand curve is downward sloping, $P > MR$ for a monopolist. Therefore, $P > MC$ at the profit-maximizing output. Marginal Revenue: $MR = P(Q) + Q \frac{dP}{dQ}$. $MR = P(1 + \frac{1}{e_{Q,P}})$. A monopolist will only operate on the elastic portion of the demand curve ($e_{Q,P} Lerner Index: A measure of monopoly power: $\frac{P - MC}{P} = -\frac{1}{e_{Q,P}} = \frac{1}{|e_{Q,P}|}$. Monopoly profits (economic rents) can persist in the long run due to barriers to entry. No Unique Supply Curve: A monopolist does not have a unique supply curve because its output decision depends not only on MC but also on the shape of the demand curve. Example 14.1: Profit Maximization for a Monopolist Given a demand curve and a total cost function, derives MR and MC. Sets $MR=MC$ to find the profit-maximizing quantity and price. Calculates profit and deadweight loss. Welfare Implications of Monopoly: Monopoly leads to an inefficient allocation of resources. Output is lower ($Q_m P_c$) than under perfect competition. Deadweight Loss: The loss of total surplus (consumer surplus + producer surplus) that results from the monopolist producing less than the socially optimal output. This is a pure welfare loss to society. Monopoly transfers consumer surplus to producer surplus (monopoly profit). Product Quality and Durability: A monopolist chooses not only quantity and price but also product characteristics like quality and durability. The optimal quality choice maximizes profit, considering how quality affects demand and cost. Coase's conjecture suggests that a monopolist selling a durable good may face competition from its past self, leading to a tendency to overproduce and reduce prices over time. Price Discrimination and Regulation Price Discrimination: Charging different prices for the same good or service. Requires firms to have market power, be able to identify different customer groups, and prevent resale (arbitrage). First-Degree (Perfect) Price Discrimination: The monopolist charges each consumer their maximum willingness to pay for each unit. Extracts all consumer surplus. Results in the socially efficient output level (where $P=MC$ for the last unit sold), but all surplus goes to the monopolist. Second-Degree Price Discrimination (Nonlinear Pricing): Charging different prices based on the quantity consumed (e.g., bulk discounts, two-part tariffs). Two-Part Tariffs: A fixed fee for the right to buy the good, plus a per-unit price. $T(q) = A + Pq$. Third-Degree Price Discrimination (Market Separation): Dividing consumers into different groups (e.g., students vs. non-students, domestic vs. international) and charging different prices to each group. The monopolist sets $MR_1 = MC$ for market 1 and $MR_2 = MC$ for market 2. Higher prices are charged in markets with less elastic demand: $\frac{P_1}{P_2} = \frac{(1 + 1/e_{Q_2,P_2})}{(1 + 1/e_{Q_1,P_1})}$. Welfare effects are ambiguous; it can increase total output and potentially benefit some consumers, but creates deadweight loss relative to perfect price discrimination. Example 14.2: Price Discrimination Analyzes profit maximization for a monopolist selling in two separate markets with different demand elasticities. Compares total profit with and without price discrimination. Regulation of Monopoly: Marginal Cost Pricing: Setting price equal to marginal cost ($P=MC$). This achieves allocative efficiency but can lead to losses for natural monopolies. Average Cost Pricing (Rate of Return Regulation): Setting price equal to average cost ($P=AC$). Allows the firm to cover costs and earn a "fair" rate of return, but usually results in higher prices and lower output than MC pricing. May lead to the Averch-Johnson effect (over-capitalization) due to incentives to inflate the rate base. Two-Tier Pricing: Charging some users a high price and others a low price to cover total costs (e.g., peak-load pricing, lifeline rates). Price Caps: Setting a maximum price the monopolist can charge. Can provide incentives for efficiency. Dynamic Views of Monopoly: While static models show inefficiency, some argue that temporary monopoly profits (e.g., from patents) are necessary to incentivize innovation and R&D (Schumpeterian view). 15. Imperfect Competition Oligopoly Models and Strategic Interaction Imperfect Competition: Market structures between perfect competition and monopoly, characterized by a few firms, differentiated products, or barriers to entry. Oligopoly: A market with a small number of firms, where the actions of one firm significantly affect others. Strategic interaction is key. Bertrand Model: Firms compete by setting prices simultaneously. Assumes homogeneous products, constant marginal cost ($c$), and no capacity constraints. Nash Equilibrium: Both firms set price equal to marginal cost ($P_1 = P_2 = c$). Each firm earns zero economic profit. Bertrand Paradox: Even with only two firms, the outcome is the same as perfect competition. Cournot Model: Firms compete by choosing quantities simultaneously. Assumes homogeneous products, and each firm chooses its output level, taking the output of rivals as given. Firm $i$'s profit: $\pi_i = P(Q)q_i - C_i(q_i)$, where $Q = \sum q_j$. Nash Equilibrium FOC: $P(Q) + P'(Q)q_i = C_i'(q_i)$ (Each firm sets its Marginal Revenue equal to its Marginal Cost, but MR depends on rivals' output). The Cournot outcome is between perfect competition and monopoly. Price is greater than MC. As the number of firms ($n$) increases, the Cournot outcome approaches the perfectly competitive outcome. Best-Response Functions (Reaction Functions): $q_i = BR_i(q_{-i})$. Shows firm $i$'s optimal output for every possible output of its rivals. The intersection of these functions is the Nash equilibrium. Example 15.1: Cournot Equilibrium Derives best-response functions for two firms in a Cournot duopoly with linear demand and constant MC. Calculates the Cournot Nash equilibrium quantities, price, and profits. Capacity Constraints: If firms have limited capacity, the Bertrand paradox may not hold. Firms may price above MC if they cannot meet all demand at MC. Product Differentiation: Products are not identical (e.g., different brands, features, locations). Demand for a firm's product depends on its own price, rivals' prices, and product characteristics. With differentiated products, Bertrand competition typically results in prices above marginal cost. Example 15.2: Product Differentiation Analyzes a Bertrand duopoly with differentiated products (demand for each firm depends on both prices). Derives the best-response functions in prices and finds the equilibrium prices. Tacit Collusion and Dynamic Competition Tacit Collusion: Firms coordinate their behavior without explicit agreement, often through repeated interaction. Finitely Repeated Games: If the stage game (e.g., Cournot or Bertrand) has a unique Nash equilibrium, the only Subgame-Perfect Equilibrium (SPE) of the finitely repeated game is to play the stage-game Nash equilibrium in every period (by backward induction). Infinitely Repeated Games: Cooperation (e.g., sustaining monopoly-like prices) can be an SPE if firms are sufficiently patient. Discount Factor ($\delta$): The weight given to future profits. Higher $\delta$ facilitates collusion. Trigger Strategies (e.g., Grim Trigger): Firms cooperate as long as others cooperate. If any firm deviates, all firms revert to the non-cooperative (Nash) strategy forever. The condition for a firm to not deviate is that the immediate gain from deviation is less than the present value of future losses from the breakdown of cooperation. Collusion is harder with more firms (as the gain from deviation is larger for each firm relative to the loss from punishment) or less patient firms. Longer-Run Decisions: Investment, Entry, Exit: Flexibility vs. Commitment: Firms make strategic choices that can be flexible or commit them to a particular course of action. Irreversible investments (sunk costs) create commitment. Sunk Costs: Non-recoverable costs that can act as a barrier to entry or exit. First-Mover Advantage (Stackelberg Model): One firm (the leader) chooses its output (or price) first, and the other firm(s) (the follower(s)) observe this choice and then make their own decisions. The leader anticipates the follower's reaction and incorporates it into its own decision. Typically, the leader produces more and earns higher profits than in Cournot. Example 15.3: Stackelberg Leadership Revises the Cournot example, assuming one firm is a Stackelberg leader. Calculates the leader's and follower's quantities, price, and profits, showing the leader's advantage. Price Leadership: One firm sets its price, and others follow. Can be dominant firm leadership or barometric leadership. Strategic Substitutes: An increase in one firm's action (e.g., output) leads to a decrease in its rival's optimal action. (e.g., quantities in Cournot). Strategic Complements: An increase in one firm's action (e.g., price) leads to an increase in its rival's optimal action. (e.g., prices with differentiated products). Strategic Entry Deterrence: Incumbent firms take actions to prevent potential rivals from entering the market. May involve building excess capacity, predatory pricing, or product proliferation. Incumbent overproduction: If the incumbent commits to a high output level, it can reduce the residual demand available to a potential entrant, making entry unprofitable. Signaling (Entry Deterrence): An incumbent firm may use its actions (e.g., a low price) to signal its type (e.g., low cost or aggressive) to a potential entrant, deterring entry. How Many Firms Enter? (Long-run equilibrium): The number of firms in an industry depends on the fixed costs of entry ($K$) and the market size. Equilibrium $n^*$ satisfies $g(n^*) \ge K$ and $g(n^*+1) There's a trade-off: The appropriability effect (firms don't appropriate all consumer surplus from their entry, leading to too little entry) and the business-stealing effect (firms don't internalize the reduction in rivals' profits from their entry, leading to too much entry). Innovation: Dissipation Effect: Competition can dissipate profits, reducing the incentive for firms to innovate. Replacement Effect: An incumbent firm has less incentive to innovate than a potential entrant if the innovation would largely replace its existing product line. Firms may engage in "patent races" to be the first to innovate. 16. Labor Markets Labor Supply: The Work-Leisure Choice Allocation of Time: Individuals make a fundamental choice between allocating their time to market work (earning income) or leisure (non-market activities). Utility: $U(c,h)$ where $c$ is consumption (goods) and $h$ is leisure. Time constraint: $T = l + h$, where $T$ is total time endowment (e.g., 24 hours), $l$ is hours worked, and $h$ is hours of leisure. Income constraint: $c = wl + n$, where $w$ is the wage rate and $n$ is nonlabor income. Full Income Constraint: Substituting $l = T-h$ into the income constraint: $c = w(T-h) + n \Rightarrow c + wh = wT + n$. The opportunity cost of an hour of leisure is the wage rate $w$. Utility Maximization FOC: The individual maximizes utility by choosing $c$ and $h$ such that the Marginal Rate of Substitution between leisure and consumption ($MRS_{h,c}$) equals the wage rate: $MRS_{h,c} = \frac{MU_h}{MU_c} = w$. Income and Substitution Effects of a Wage Change: An increase in the wage rate ($w$) has two effects: Substitution Effect: Leisure becomes relatively more expensive (its opportunity cost increases). This encourages the individual to substitute away from leisure, meaning they work more (increase $l$) and consume less leisure (decrease $h$). Income Effect: A higher wage increases purchasing power (income). If leisure is a normal good (which it usually is), the individual will demand more leisure (decrease $l$) and consume more goods. These two effects work in opposite directions for labor supply. If the substitution effect dominates, the labor supply curve is upward sloping. If the income effect dominates, the labor supply curve is backward bending. Example 16.1: A Backward-Bending Labor Supply Curve Uses a specific utility function (e.g., $U(c,h) = c^\alpha h^\beta$) to derive the labor supply function. Demonstrates that for certain parameter values, the labor supply curve can become backward-bending at high wage levels. Mathematical Analysis of Labor Supply (Slutsky Equation): Labor supply function: $l(w,n)$. The Slutsky equation for labor supply: $\frac{\partial l}{\partial w} = \frac{\partial l^c}{\partial w} + l \frac{\partial l}{\partial n}$. $\frac{\partial l^c}{\partial w}$: The compensated labor supply effect (pure substitution effect). Always positive (compensated labor supply curve is upward sloping). $l \frac{\partial l}{\partial n}$: The income effect. Since leisure is usually a normal good ($ \frac{\partial h}{\partial n} > 0 \Rightarrow \frac{\partial l}{\partial n} Market Supply Curve for Labor: The horizontal summation of individual labor supply decisions. Can be upward sloping, backward bending, or positively sloped due to increased labor force participation. Labor Demand and Market Equilibrium Demand for Labor by a Firm: A firm's demand for labor is derived from its profit-maximization decision. Marginal Revenue Product of Labor (MRP$_l$): The additional revenue generated by hiring one more unit of labor. For a price-taking firm, $MRP_l = P \cdot MP_l$. Profit Maximization Condition: Hire labor until $MRP_l = w$ (wage rate). The firm's demand curve for labor is its $MRP_l$ curve, which is downward sloping due to diminishing marginal productivity of labor. Example 16.2: Deriving a Firm's Demand for Labor Given a production function and output price, derives the $MP_l$ and $MRP_l$. Sets $MRP_l = w$ to find the firm's labor demand function. Market Demand for Labor: The horizontal summation of individual firms' labor demand curves. Labor Market Equilibrium: Determined by the intersection of market labor demand and market labor supply. This determines the equilibrium wage rate and employment level. Wage Variation and Imperfections Wage Variation: Differences in wages across individuals and occupations. Human Capital: Investment in education, training, and experience increases productivity, leading to higher wages. General human capital: Skills transferable across firms. Specific human capital: Skills valuable only to a particular firm. Compensating Wage Differentials: Wage differences that arise to compensate workers for non-pecuniary aspects of jobs (e.g., risk, unpleasantness, location). Dangerous jobs tend to pay higher wages. Job Search: The process by which workers find jobs and firms find workers. Search costs and imperfect information can lead to wage dispersion. Monopsony in the Labor Market: A market with a single buyer of labor (e.g., a dominant employer in a remote town). The monopsonist faces the entire upward-sloping market labor supply curve. Marginal Expense of Labor (ME$_l$): The additional cost of hiring one more unit of labor. Since hiring more labor raises the wage for all existing workers, $ME_l > w$ (wage rate). If $w(l)$ is the supply curve, $ME_l = \frac{d(w(l)l)}{dl} = w(l) + l \frac{dw}{dl}$. Profit Maximization: The monopsonist hires labor until $MRP_l = ME_l$. Outcome: A monopsonist hires less labor ($L_m$) and pays a lower wage ($w_m$) than would occur in a perfectly competitive labor market ($L_c, w_c$). This creates deadweight loss. Example 16.3: Monopsonistic Hiring Given a labor supply curve and $MRP_l$, derives the $ME_l$. Sets $MRP_l = ME_l$ to find the monopsonistic employment and wage. Compares with the competitive outcome. Labor Unions: Organizations of workers that act as a collective bargaining agent. Can exert monopoly power in the labor market. Unions may aim to maximize the total wage bill, economic rent for members, or employment. Their actions typically lead to higher wages and lower employment than in a competitive market. 17. Capital and Time Intertemporal Choice and the Rate of Return Capital: Long-lived inputs to production that yield services over time. Physical Capital: Machines, buildings, inventories. Human Capital: Skills and knowledge embodied in labor. Rate of Return: The terms at which current consumption can be transformed into future consumption. Single-period Rate of Return ($r_1$): If $s$ units of current goods yield $x$ units of future goods, $r_1 = \frac{x-s}{s} = \frac{x}{s} - 1$. Perpetual Rate of Return ($r_p$): If $s$ units of current goods yield $y$ units of goods per period forever, $r_p = \frac{y}{s}$. Price of Future Goods ($p_1$): The amount of current goods that must be given up to obtain one unit of future goods. $p_1 = \frac{1}{1+r}$. Individual Intertemporal Choice: Individuals allocate consumption over time to maximize utility. Utility: $U(c_0, c_1)$, where $c_0$ is current consumption and $c_1$ is future consumption. Budget constraint: $c_0 + p_1 c_1 = W$, where $W$ is total lifetime wealth (present value of current and future income). First-Order Condition: Maximize $U(c_0, c_1)$ subject to $W = c_0 + \frac{c_1}{1+r}$. $MRS_{c_0,c_1} = \frac{MU_{c_0}}{MU_{c_1}} = 1+r$. Intertemporal Impatience (Time Preference): Individuals often prefer current consumption to future consumption. Rate of Time Preference ($\delta$): The rate at which an individual discounts future utility relative to current utility. $MRS_{c_0,c_1} = \frac{1+\delta}{1+r}$. Consumption Smoothing: Individuals generally prefer to smooth consumption over time. Equilibrium Rate of Return ($r^*$): Determined by the interaction of aggregate supply and demand for future goods (i.e., aggregate savings and investment). Real vs. Nominal Interest Rates: Nominal Interest Rate ($i$): The observed market rate. Real Interest Rate ($r$): The nominal rate adjusted for expected inflation ($\dot{P}^e$). $1+i = (1+r)(1+\dot{P}^e) \approx 1+r+\dot{P}^e \Rightarrow i \approx r+\dot{P}^e$. Firm's Demand for Capital and Investment Capital Stock vs. Capital Services: Capital Stock ($K$): The physical asset (e.g., a machine). Capital Services ($k$): The flow of productive services provided by the capital stock over time. Competitive Rental Rate of Capital ($v$): The market price for renting a unit of capital services. The rental rate is determined by the interest rate ($r$), the depreciation rate ($\delta$), and the price of the capital good ($P_K$): $v = P_K(r+\delta)$. Firm's Demand for Capital: Firms demand capital until the Marginal Revenue Product of Capital (MRP$_K$) equals the rental rate ($v$). $MRP_K = P \cdot MP_K = v$. Investment: The process of adding to the capital stock. The demand for investment is derived from the firm's demand for capital. It is inversely related to the interest rate. Present Discounted Value (PDV) Approach to Investment: A project is worth undertaking if its PDV of future returns exceeds its cost. $PDV = \sum_{t=1}^T \frac{R_t}{(1+r)^t}$, where $R_t$ is the revenue (or profit) in period $t$. For a durable good, the firm will purchase it if its market price ($P_K$) is less than or equal to the PDV of the stream of future net revenues it generates. Example 17.1: Present Value of a Machine Calculates the PDV of a machine that yields a stream of net revenues over its lifetime. Compares PDV to the machine's cost to determine investment profitability. Natural Resources and Mathematics of Interest Natural Resource Pricing (Hotelling's Rule): For an exhaustible resource with constant marginal extraction cost ($c$), the price of the resource ($p$) must rise at the rate of interest ($r$) for producers to be indifferent between extracting now and extracting later. $\dot{p} = rp$. If marginal extraction costs are not constant, the net price ($p-c$) must rise at the rate of interest: $\frac{\dot{p}-\dot{c}}{p-c} = r$. Example 17.2: Hotelling's Rule Illustrates how the price of an exhaustible resource must grow at the rate of interest over time. Shows how a constant marginal cost affects the path of the resource price. Mathematics of Compound Interest (Appendix): Future Value: $FV = P_0(1+i)^n$. Present Discounted Value: $PDV = \sum_{j=0}^n \frac{N_j}{(1+i)^j}$. Perpetuity: A stream of constant payments $N$ forever, $PDV = N/i$. Annuity: A stream of constant payments for a fixed number of periods. Continuous Compounding: $FV = P_0 e^{rn}$. The effective annual rate $i = e^r - 1$. PDV (continuous): $\int_0^T f(t)e^{-rt}dt$. Duration: A measure of the interest rate sensitivity of an asset's price, defined as the weighted average of the times until cash flows are received, with weights proportional to the present value of the cash flows. 18. Asymmetric Information Principal-Agent Model and Moral Hazard Asymmetric Information: A situation where one party in an economic transaction has more or better information than the other party. Leads to market inefficiencies and potential market failures. Principal-Agent Model: A framework for analyzing situations where one party (the agent) acts on behalf of another (the principal), but their interests may not be aligned, and the principal cannot perfectly observe the agent's actions or information. First Best: The outcome that would be achieved if there were no asymmetric information (i.e., perfect information and no transaction costs). Principal extracts all surplus. Second Best: The optimal outcome that can be achieved given the informational constraints. Hidden Actions (Moral Hazard): Occurs when one party's actions are unobservable to the other party after a contract is signed. Example: Owner-Manager Relationship. Owner (principal) wants manager (agent) to exert effort ($e$). Manager's effort is costly and unobservable. Manager is typically risk-averse. Output ($y$) depends on effort and random shock: $y = f(e) + \epsilon$. Manager's utility depends on income ($s$) and effort cost ($c(e)$): $U(s,e)$. Owner designs a compensation contract $s(y)$ to incentivize effort. First-Best (Observable Effort): Owner pays a fixed salary $s^*$ to cover manager's reservation utility and extracts all surplus. Manager chooses efficient effort $e^*$ where $f'(e^*) = c'(e^*)$. Second-Best (Unobservable Effort): Owner must link manager's pay to observable output $y$. Manager's incentive compatibility constraint: Manager chooses $e$ to maximize own utility given $s(y)$. Manager's participation constraint: Manager's expected utility from the contract must be at least their reservation utility. Optimal contract balances incentives (strong link between pay and output) with risk-sharing (risk-averse manager prefers less variable pay). Typically, the second-best effort level is less than the first-best effort. Moral Hazard in Insurance: Once insured, individuals may take less care to prevent the insured event from occurring. (e.g., less careful driving after buying car insurance). Insurers address this with deductibles, co-pays, or monitoring. Example 18.1: Moral Hazard in a Linear Contracting Model Analyzes optimal effort and compensation when output is a linear function of effort and a random shock ($y=e+\epsilon$). Manager has CARA utility. Derives the optimal incentive coefficient ($b$) in a linear contract ($s=a+by$). Shows that $b Adverse Selection and Market Signaling Hidden Types (Adverse Selection): Occurs when one party has private information about their own characteristics (type) before a contract is signed. Example: Used Car Market (Akerlof's "Lemons" Problem). Sellers know the quality of cars, buyers don't. Buyers assume average quality, leading to lower prices, driving good quality cars out of the market. Only "lemons" remain. Adverse Selection in Insurance: High-risk individuals are more likely to purchase insurance than low-risk individuals. If insurers cannot distinguish types, they must charge an average premium. This makes insurance unattractive to low-risk individuals, who then drop out, leading to a market dominated by high-risk types. Insurers respond by offering a menu of policies (e.g., with different deductibles) to induce self-selection. High-risk types choose full coverage; low-risk types choose partial coverage. Nonlinear Pricing (Screening): A monopolist offers a menu of price-quantity packages designed to induce different types of consumers to self-select, thereby revealing their type. Example: Charging different prices per unit depending on the total quantity purchased. The monopolist distorts the quantity offered to lower-value consumers (downward distortion) to prevent higher-value consumers from choosing that package. "No distortion at the top": The highest-value consumer type is typically offered the efficient quantity. Market Signaling: An informed party takes an observable action (a signal) to convey their private information to an uninformed party. The signal must be costly and differentially costly for different types for it to be credible. Job-Market Signaling (Spence): Education can be a signal of high ability, even if it doesn't directly increase productivity. High-ability individuals find it less costly to acquire education, so they choose to do so, distinguishing themselves from low-ability individuals. Example 18.2: Signaling in Education Analyzes conditions under which education can serve as a separating signal for high-ability workers. Compares the cost of education for high- and low-ability workers and the resulting payoffs. Auctions Auctions: Mechanisms for selling goods where potential buyers submit bids. Often used when item value is uncertain or there are many buyers. Private Value Auction: Each bidder knows their own valuation, and these valuations are independent. Common Value Auction: The item has the same value to everyone, but bidders have private, imprecise estimates of that value. Auction Formats: English Auction (Ascending Price): Bidders openly call out progressively higher bids until only one bidder remains. Generally efficient and yields high revenue. Dutch Auction (Descending Price): The auctioneer starts at a high price and lowers it until a bidder accepts. First-Price Sealed-Bid Auction: Bidders submit sealed bids, and the highest bidder wins and pays their bid. Second-Price Sealed-Bid Auction (Vickrey Auction): Bidders submit sealed bids, and the highest bidder wins but pays the second-highest bid. Strategy: Bidding your true valuation is a weakly dominant strategy in a Vickrey auction. Revenue Equivalence Theorem: In private value auctions, under certain conditions (risk neutrality, independent valuations), many auction formats (English, Dutch, First-Price Sealed-Bid, Second-Price Sealed-Bid) yield the same expected revenue for the seller. Winner's Curse: In common value auctions, the winner is often the bidder with the most optimistic (and perhaps erroneous) estimate of the item's value, potentially leading to overpaying. Bidders must adjust their bids downward to avoid this. Example 18.3: A Sealed-Bid Auction Compares bidding strategies for first-price and second-price sealed-bid auctions. Illustrates why bidding true value is dominant in a second-price auction. 19. Externalities and Public Goods Externalities and Efficiency Externality: An activity by one economic agent that affects the welfare of another agent in a way that is not transmitted through market prices. Negative Externality: Imposes a cost on others (e.g., pollution, noise). Positive Externality: Provides a benefit to others (e.g., vaccination, R&D). Allocative Inefficiency of Externalities: When a negative externality exists, the private marginal cost (PMC) of production is less than the social marginal cost (SMC). Firms only consider PMC, leading to overproduction relative to the socially optimal level. When a positive externality exists, the private marginal benefit (PMB) is less than the social marginal benefit (SMB). This leads to underproduction. Example 19.1: An Interfirm Externality Illustrates a negative externality where an upstream firm's output (pollution) negatively affects a downstream firm's production. Compares the private profit-maximizing output with the socially optimal output, showing overproduction by the upstream firm. Solutions to Externality Problem: Coasian Bargaining: If property rights are well-defined and transaction costs are low, private parties can bargain to an efficient solution, regardless of who initially holds the property rights. Coase Theorem: The efficient allocation of resources will result from private bargaining, even in the presence of externalities, provided that property rights are well-defined and transaction costs are sufficiently low. Pigovian Tax: A tax imposed on an activity that generates a negative externality, equal to the marginal external cost at the socially optimal output level. Internalizes the externality, making firms face the full social cost. Example 19.2: A Pigovian Tax Calculates the optimal Pigovian tax that leads the polluting firm to produce the socially efficient output level. Regulation (Command and Control): Directly limiting the amount of the externality (e.g., limits on pollution emissions). Can be inefficient if regulators lack full information. Marketable Pollution Permits (Cap-and-Trade): The government sets a total limit (cap) on the externality and issues permits that allow a certain amount of the externality. Firms can then buy and sell these permits, creating a market price for the externality. Achieves efficiency by allowing firms with lower abatement costs to reduce pollution more, and sell permits to firms with higher abatement costs. Public Goods Public Goods: Goods that are both nonexclusive and nonrival. Nonexclusivity: It is impossible or prohibitively costly to exclude individuals from consuming the good once it is produced (e.g., national defense, clean air). Nonrivalry: Consumption by one individual does not diminish the amount available for others to consume (e.g., one person listening to a radio broadcast does not prevent others from listening). The marginal cost of an additional consumer is zero. Typology of Goods: Private Goods: Rival and exclusive (e.g., an apple). Club Goods: Nonrival but exclusive (e.g., subscription TV, private park). Common Property Resources: Rival but nonexclusive (e.g., a common fishing ground, public roads). Pure Public Goods: Nonrival and nonexclusive (e.g., national defense). Public Goods and Market Failure: Due to nonexclusivity, public goods suffer from the free-rider problem . Individuals have an incentive to consume the good without paying for it, leading to underprovision by private markets. Due to nonrivalry, charging a price for a public good is inefficient, as it would exclude some who would benefit at no additional cost. Efficient Provision of Public Goods (Samuelson Condition): For a private good, efficiency requires $MRS_1 = MRS_2 = \dots = P_X/P_Y$. For a public good ($g$), efficiency requires summing individual marginal benefits (MRS) and setting this sum equal to the marginal cost (RPT) of the public good: $\sum_{i=1}^n MRS_i = RPT_{g,X}$. This means that the total willingness to pay for an additional unit of the public good must equal its marginal cost. Example 19.3: Optimal Provision of a Public Good Given individual demand curves for a public good, derives the aggregate demand (vertical summation). Finds the optimal quantity where aggregate demand intersects the marginal cost. Voting and Resource Allocation Voting and Public Goods: Since private markets fail to provide public goods efficiently, governments often provide them, funded by taxes. Voting mechanisms are used to decide on the level of public good provision. Majority Rule: A common voting system. Can lead to Condorcet's Paradox (Cycling) : If voter preferences are not "single-peaked," majority rule can lead to inconsistent outcomes, where no single option wins a majority against all others. Single-Peaked Preferences: Preferences where there is one most preferred option, and utility decreases as choices move away from that option in any direction. Median Voter Theorem: If preferences are single-peaked and the issue is one-dimensional, then majority rule will select the outcome most preferred by the median voter (the voter whose preferred outcome is in the middle of all voters' preferred outcomes). Implies that political parties will tend to converge to the center to appeal to the median voter. Voting Mechanisms to Reveal Preferences: Lindahl Pricing: A conceptual solution where each individual pays a "tax price" for the public good equal to their marginal valuation. If everyone truthfully reveals their preferences, the efficient level of public good is provided. (Impractical because individuals have an incentive to understate their true valuations). Groves Mechanism: A mechanism designed to induce truthful revelation of preferences for public goods by providing payments (or taxes) that incentivize honest reporting. Clarke Mechanism: A variant of the Groves mechanism where a "pivotal" voter (whose vote changes the outcome) pays a tax related to the externality their vote imposes on others.