Intro to Mathematical Economics Unit 11 Review: General Equilibrium in Economics

Key Concepts and Definitions

• General equilibrium analyzes the behavior of supply, demand, and prices in a whole economy with multiple interacting markets
• Partial equilibrium focuses on a single market in isolation, assuming that other markets remain unchanged
• Walrasian equilibrium named after Léon Walras, is a state where all markets simultaneously clear and demand equals supply for all commodities
• Pareto efficiency occurs when no individual can be made better off without making at least one other individual worse off
• Edgeworth box is a graphical tool used to analyze the trading of two goods between two people, representing the preferences and endowments of both parties
• Competitive equilibrium is a situation in which the market price is such that the quantity supplied equals the quantity demanded, and no market participant has an incentive to alter their behavior
• Excess demand is the situation where the quantity demanded exceeds the quantity supplied at a given price level
• Excess supply occurs when the quantity supplied exceeds the quantity demanded at a given price level

Mathematical Foundations

• Set theory is used to define the basic elements of the economy, such as the set of consumers, producers, and commodities
  • Example: Let $I = \{1, 2, ..., m\}$ be the set of consumers and $J = \{1, 2, ..., n\}$ be the set of producers
• Vector spaces are employed to represent the consumption bundles and production plans
  • Example: A consumption bundle for a consumer $i$ is represented as $x_i = (x_{i1}, x_{i2}, ..., x_{il}) \in \mathbb{R}^l_+$
• Convex analysis is crucial for studying the properties of preference relations and production sets
  • A set $X \subseteq \mathbb{R}^n$ is convex if for any $x, y \in X$ and $\lambda \in [0, 1]$, $\lambda x + (1 - \lambda) y \in X$
• Optimization techniques are used to characterize the behavior of consumers and producers
  • Consumers maximize utility subject to budget constraints, while producers maximize profits subject to technological constraints
• Fixed point theorems, such as Brouwer's and Kakutani's, are employed to prove the existence of equilibrium
  • Brouwer's fixed point theorem states that any continuous function from a compact and convex set to itself has a fixed point

General Equilibrium Model Setup

• The economy consists of a finite number of consumers $I = \{1, 2, ..., m\}$ and producers $J = \{1, 2, ..., n\}$
• There are a finite number of commodities $L = \{1, 2, ..., l\}$ in the economy
• Each consumer $i$ has a preference relation $\succsim_i$ over the consumption set $X_i \subseteq \mathbb{R}^l_+$ and an initial endowment $\omega_i \in \mathbb{R}^l_+$
  • The preference relation is typically assumed to be complete, transitive, and locally non-satiated
• Each producer $j$ has a production set $Y_j \subseteq \mathbb{R}^l$ representing the feasible production plans
  • The production set is usually assumed to be closed, convex, and satisfying the no free lunch condition (i.e., $Y_j \cap \mathbb{R}^l_+ = \{0\}$)
• The total endowment of the economy is given by $\omega = \sum_{i=1}^m \omega_i$
• A price vector is an element of the price space $P = \mathbb{R}^l_+$, where $p_k$ represents the price of commodity $k$

Walrasian Equilibrium

• A Walrasian equilibrium is a price vector $p^* \in P$ and an allocation $(\mathbf{x}^*, \mathbf{y}^*)$ such that:
  1. For each consumer $i$, $x_i^*$ maximizes $\succsim_i$ subject to the budget constraint $p^* \cdot x_i \leq p^* \cdot \omega_i + \sum_{j=1}^n \theta_{ij} p^* \cdot y_j^*$
  2. For each producer $j$, $y_j^*$ maximizes profits $p^* \cdot y_j$ over the production set $Y_j$
  3. Markets clear: $\sum_{i=1}^m x_i^* = \omega + \sum_{j=1}^n y_j^*$
• In equilibrium, consumers maximize their utility given prices and budget constraints, producers maximize profits given prices and production sets, and markets clear
• The budget constraint for each consumer includes the value of their initial endowment and their share of profits from producers, where $\theta_{ij}$ represents consumer $i$'s ownership share in producer $j$
• The market clearing condition ensures that the total consumption equals the total endowment plus the total production

Existence and Uniqueness of Equilibrium

• The existence of a Walrasian equilibrium is a fundamental question in general equilibrium theory
• Arrow-Debreu existence theorem provides sufficient conditions for the existence of a Walrasian equilibrium
  • The theorem requires assumptions such as convex preferences, convex production sets, and a bounded economy
• Kakutani's fixed point theorem is a key tool in proving the existence of equilibrium
  • It states that a upper hemicontinuous, nonempty-, convex-, and compact-valued correspondence from a convex and compact set to itself has a fixed point
• The uniqueness of equilibrium is not guaranteed in general and depends on the specific properties of the economy
  • Gross substitutability of commodities is a sufficient condition for uniqueness, but it is a strong assumption
• Sonnenschein-Mantel-Debreu theorem highlights the difficulty of obtaining uniqueness, as it shows that any continuous and homogeneous excess demand function can be rationalized as the excess demand of an economy with well-behaved consumers and producers

Efficiency and Welfare Analysis

• Pareto efficiency is a central concept in welfare analysis of general equilibrium
  • An allocation is Pareto efficient if there is no other feasible allocation that makes at least one individual better off without making anyone else worse off
• First Welfare Theorem states that, under certain assumptions (no externalities, complete markets, perfect competition), any Walrasian equilibrium is Pareto efficient
  • This theorem provides a strong justification for the efficiency of competitive markets
• Second Welfare Theorem asserts that, under similar assumptions, any Pareto efficient allocation can be supported as a Walrasian equilibrium with appropriate redistribution of initial endowments
  • This theorem suggests that efficiency and equity can be separated, and any desired Pareto efficient allocation can be achieved through lump-sum transfers followed by competitive exchange
• Social welfare functions, such as the utilitarian or the Rawlsian, can be used to evaluate and compare different allocations based on the society's value judgments
  • The utilitarian social welfare function seeks to maximize the sum of individual utilities, while the Rawlsian focuses on maximizing the utility of the worst-off individual
• Compensating and equivalent variations are monetary measures of welfare changes that can be used to quantify the impact of policy interventions or economic shocks on individuals

Applications and Real-World Examples

• International trade theory uses general equilibrium models to analyze the patterns of trade and the welfare effects of trade policies
  • The Heckscher-Ohlin model explains trade patterns based on differences in factor endowments across countries (labor and capital)
• Public finance employs general equilibrium frameworks to study the efficiency and distributional impacts of tax policies
  • The Diamond-Mirrlees production efficiency theorem shows that, under certain conditions, optimal tax systems should not distort production decisions
• Environmental economics utilizes general equilibrium models to examine the interaction between the economy and the environment
  • Integrated assessment models (IAMs) combine economic and climate modules to analyze the costs and benefits of climate policies (carbon taxes or emission permits)
• Macroeconomic models, such as the dynamic stochastic general equilibrium (DSGE) models, are based on the general equilibrium framework and are used for policy analysis and forecasting
  • The Real Business Cycle (RBC) model explains economic fluctuations as the result of real shocks (technology shocks) propagating through a competitive economy
• Urban economics employs spatial general equilibrium models to study the location choices of households and firms and the formation of cities
  • The Alonso-Muth-Mills model analyzes the trade-off between housing prices and commuting costs in determining the urban spatial structure

Common Pitfalls and Misconceptions

• Assuming that partial equilibrium analysis always provides accurate insights, ignoring the potential spillover effects across markets
  • For example, analyzing the impact of a subsidy on a single market without considering the general equilibrium effects on related markets
• Interpreting the First Welfare Theorem as a blanket justification for laissez-faire policies, neglecting the importance of the underlying assumptions (no externalities, complete markets, perfect competition)
  • Market failures, such as externalities or public goods, can lead to inefficient outcomes even in competitive equilibrium
• Overemphasizing the normative implications of the Second Welfare Theorem without considering the practical difficulties of implementing lump-sum transfers
  • Lump-sum transfers are rarely feasible in practice due to information constraints and political economy considerations
• Treating the existence of a Walrasian equilibrium as a guarantee of its stability or convergence under tatonnement processes
  • The stability of equilibrium depends on the specific adjustment processes and the properties of excess demand functions
• Neglecting the potential multiplicity of equilibria and the resulting coordination problems
  • In the presence of increasing returns or externalities, there may be multiple equilibria, and the economy may get stuck in a suboptimal equilibrium
• Assuming that the competitive equilibrium always results in a socially desirable outcome, ignoring distributional concerns and the limitations of the Pareto criterion
  • The Pareto criterion does not take into account the initial distribution of resources or the fairness of the allocation
• Overreliance on representative agent models, which may obscure important heterogeneity and distributional issues
  • Representative agent models assume that all individuals have identical preferences and endowments, which may not capture the diversity of real-world economies
• Ignoring the bounded rationality and behavioral biases of economic agents, which may lead to deviations from the predictions of standard general equilibrium models
  • Behavioral economics has shown that individuals may exhibit loss aversion, hyperbolic discounting, or other biases that are not captured by the standard rational choice framework