Equilibrium analysis is a cornerstone of mathematical economics. It explores how economic systems reach stability, with supply meeting demand and prices settling at optimal levels. This concept helps us understand market behavior and predict outcomes in various scenarios.

From simple supply-demand models to complex general equilibrium theory, this topic covers diverse applications. We'll examine how changes in one variable affect others, analyze market stability, and explore equilibrium in game theory and macroeconomics.

Definition of equilibrium

• Equilibrium represents a state of balance or stability in economic systems where opposing forces offset each other
• In mathematical economics, equilibrium occurs when key variables remain constant over time, allowing for analysis of market behavior and outcomes

Types of equilibrium

• Static equilibrium maintains constant values for economic variables over time
• Dynamic equilibrium involves changing variables that maintain a stable relationship
• Partial equilibrium focuses on a single market or sector, assuming other factors remain constant
• General equilibrium considers interactions between multiple markets and sectors simultaneously

Equilibrium conditions

• Supply equals demand in market equilibrium, with no excess supply or demand
• Marginal cost equals marginal revenue in firm equilibrium, maximizing profit
• No incentive for economic agents to change their behavior in Nash equilibrium
• Equality of savings and investment in macroeconomic equilibrium

Supply and demand equilibrium

• Supply and demand equilibrium forms the foundation of microeconomic analysis in mathematical economics
• Understanding market equilibrium helps predict price and quantity outcomes in competitive markets

Market clearing price

• Price at which quantity supplied equals quantity demanded
• Graphically represented by the intersection of supply and demand curves
• Eliminates shortages and surpluses in the market
• Mathematically expressed as P* = f(Q_d) = g(Q_s)$ where $P* is the equilibrium price

Excess supply vs excess demand

• Excess supply (surplus) occurs when quantity supplied exceeds quantity demanded at a given price
• Excess demand (shortage) arises when quantity demanded exceeds quantity supplied at a given price
• Market forces drive prices towards equilibrium to eliminate excess supply or demand
  • Surplus leads to price decreases
  • Shortage results in price increases

Comparative statics

• Comparative statics analyzes how changes in exogenous variables affect equilibrium outcomes
• Essential tool in mathematical economics for predicting market responses to external shocks or policy changes

Shift in supply curve

• Caused by changes in production costs, technology, or number of suppliers
• Rightward shift (increase in supply) leads to lower equilibrium price and higher quantity
• Leftward shift (decrease in supply) results in higher equilibrium price and lower quantity
• Magnitude of price and quantity changes depends on elasticity of demand

Shift in demand curve

• Triggered by changes in income, preferences, or number of buyers
• Rightward shift (increase in demand) raises equilibrium price and quantity
• Leftward shift (decrease in demand) lowers equilibrium price and quantity
• Impact on equilibrium outcomes influenced by elasticity of supply

General equilibrium theory

• General equilibrium theory examines the simultaneous equilibrium of all markets in an economy
• Provides a comprehensive framework for analyzing complex economic systems and their interdependencies

Walrasian equilibrium

• Named after economist Léon Walras
• Describes simultaneous equilibrium in all markets of an economy
• Achieved when supply equals demand for all goods and services
• Prices adjust to clear all markets simultaneously
• Represented mathematically as a system of equations $\sum_{i=1}^n x_i(p) = \sum_{i=1}^n w_i$ for all goods $i$

Pareto efficiency

• State where no individual can be made better off without making someone else worse off
• Characteristic of general equilibrium under perfect competition
• Achieved through optimal allocation of resources
• Mathematically expressed using utility functions and budget constraints

Partial equilibrium analysis

• Partial equilibrium analysis focuses on a single market or sector while holding other factors constant
• Simplifies complex economic systems for targeted analysis of specific markets or policies

Single market analysis

• Examines equilibrium in one market assuming other markets remain unchanged
• Useful for analyzing impact of specific policies or shocks on a particular industry
• Ignores potential spillover effects or feedback from other markets
• Applicable when the market under study has minimal impact on the broader economy

Ceteris paribus assumption

• Latin phrase meaning "all other things being equal"
• Assumes all variables except those under consideration remain constant
• Allows isolation of effects of specific variables on equilibrium outcomes
• Simplifies analysis but may overlook important interactions between markets

Stability of equilibrium

• Stability analysis examines whether an economic system returns to equilibrium after a disturbance
• Critical for understanding long-term behavior of markets and effectiveness of policies

Stable vs unstable equilibrium

• Stable equilibrium returns to its original state after small perturbations
• Unstable equilibrium moves away from its original state when disturbed
• Stability determined by slopes of supply and demand curves at equilibrium point
• Mathematically analyzed using differential equations and phase diagrams

Cobweb model

• Dynamic model of price fluctuations in agricultural markets
• Illustrates how production decisions based on previous prices can lead to cyclical behavior
• Stable cobweb converges to equilibrium over time
• Unstable cobweb leads to expanding price oscillations
• Mathematically represented as $Q_t = a + bP_{t-1}$ and $P_t = c - dQ_t$

Nash equilibrium

• Nash equilibrium represents a stable state in strategic interactions where no player can unilaterally improve their outcome
• Fundamental concept in game theory with applications in various economic scenarios

Game theory applications

• Oligopoly pricing strategies (Bertrand, Cournot models)
• Public goods provision and free-rider problem
• Labor market negotiations between firms and unions
• International trade agreements and tariff setting

Best response strategies

• Strategy that produces the most favorable outcome for a player, given the strategies of other players
• Determined by maximizing payoff functions for each player
• Nash equilibrium occurs when all players are using their best response strategies
• Mathematically found by solving simultaneous equations of best response functions

Dynamic equilibrium models

• Dynamic equilibrium models incorporate time-dependent processes and adjustments in economic systems
• Essential for analyzing long-term economic growth, business cycles, and policy impacts

Adjustment processes

• Price adjustment mechanisms (tâtonnement process)
• Quantity adjustment in disequilibrium situations
• Expectations formation and learning in dynamic markets
• Represented mathematically using differential equations or difference equations

Disequilibrium dynamics

• Transitional behavior of economic systems moving towards equilibrium
• Analysis of short-run fluctuations and convergence paths
• Stability conditions for dynamic equilibrium models
• Applications in business cycle theory and growth models

Equilibrium in macroeconomics

• Macroeconomic equilibrium involves balance in aggregate economic variables at the national or global level
• Crucial for understanding overall economic performance and policy effectiveness

IS-LM model

• Integrates goods market (IS curve) and money market (LM curve) equilibrium
• Determines equilibrium interest rate and output level
• IS curve equation: $Y = C(Y-T) + I(r) + G$
• LM curve equation: $M/P = L(r,Y)$
• Equilibrium occurs at the intersection of IS and LM curves

AD-AS model

• Combines aggregate demand (AD) and aggregate supply (AS) to determine equilibrium price level and output
• Short-run AS curve reflects sticky prices and wages
• Long-run AS curve represents potential output level
• AD curve derived from IS-LM model
• Equilibrium adjusts to shocks through price and output changes

Mathematical techniques

• Mathematical techniques provide powerful tools for analyzing and solving equilibrium problems in economics
• Essential skills for advanced study and research in mathematical economics

Systems of equations

• Linear and nonlinear systems used to model multiple market equilibria
• Matrix algebra for solving large systems of equations
• Cramer's rule and Gaussian elimination methods
• Applications in general equilibrium models and input-output analysis

Optimization methods

• Constrained optimization for finding equilibrium points (Lagrange multipliers)
• Numerical methods for solving complex equilibrium problems (Newton-Raphson method)
• Dynamic programming for intertemporal equilibrium models
• Applications in consumer and producer theory, welfare economics

Applications in policy analysis

• Equilibrium analysis provides insights for evaluating and designing economic policies
• Helps policymakers understand potential impacts and unintended consequences of interventions

Price controls

• Analysis of price ceilings (rent control) and price floors (minimum wage)
• Deadweight loss calculation in non-equilibrium situations
• Long-term effects on supply and demand in controlled markets
• Spillover effects on related markets and overall economic efficiency

Tax incidence

• Distribution of tax burden between buyers and sellers in equilibrium
• Impact of taxes on equilibrium price and quantity
• Elasticity of supply and demand determines tax incidence
• Analysis of efficiency and equity implications of different tax policies