💰Intro to Mathematical Economics Unit 4 – Economic Modeling with Differential Equations

Key Concepts and Definitions

• Economic models simplify complex economic phenomena into mathematical representations to understand and predict economic behavior
• Endogenous variables determined within the model itself (price, quantity)
• Exogenous variables determined outside the model and taken as given (government policies, technological changes)
• Parameters numerical values that represent the characteristics of the economic agents or the system (preferences, technology)
  • Assumed to be constant in the model but can change in comparative static analysis
• Variables quantities that can change within the model (price, quantity, income)
• Functions mathematical equations that describe the relationship between variables in the model (demand function, production function)
• Differential equations describe the rate of change of a variable with respect to another variable, often with respect to time

Mathematical Foundations

• Calculus fundamental mathematical tool in economic modeling, used to analyze marginal changes and optimize economic decisions
  • Derivatives measure the rate of change of one variable with respect to another (marginal cost, marginal revenue)
  • Integrals used to calculate total quantities from marginal quantities (total cost, total revenue)
• Linear algebra used to represent economic models with multiple variables and equations
  • Matrices and vectors represent the coefficients and variables in a system of linear equations
  • Matrix operations (addition, multiplication, inversion) used to solve systems of linear equations
• Optimization techniques used to find the best solution to an economic problem given certain constraints
  • Lagrange multipliers method to find the maximum or minimum of a function subject to one or more constraints
  • Kuhn-Tucker conditions generalization of Lagrange multipliers for inequality constraints
• Probability theory used to model economic situations involving uncertainty and risk
  • Expected value weighted average of all possible outcomes, used to make decisions under uncertainty
  • Variance and standard deviation measure the dispersion of outcomes around the expected value

Types of Economic Models

• Microeconomic models focus on the behavior of individual economic agents (consumers, firms) and their interactions in markets
  • Consumer theory models how consumers make decisions to maximize their utility subject to budget constraints
  • Producer theory models how firms make decisions to maximize their profits subject to technological constraints
• Macroeconomic models focus on the behavior of the economy as a whole, including variables such as GDP, inflation, and unemployment
  • Keynesian models emphasize the role of aggregate demand in determining economic output and employment
  • Neoclassical models emphasize the role of supply-side factors (capital accumulation, technological progress) in long-run economic growth
• Growth models analyze the factors that contribute to long-run economic growth, such as capital accumulation, technological progress, and population growth
  • Solow model neoclassical growth model that emphasizes the role of capital accumulation and technological progress
  • Endogenous growth models incorporate the role of human capital, research and development, and innovation in driving long-run growth
• Trade models analyze the patterns and effects of international trade between countries
  • Ricardian model comparative advantage based on differences in labor productivity across countries
  • Heckscher-Ohlin model comparative advantage based on differences in factor endowments (capital, labor) across countries

Differential Equations in Economics

• Differential equations describe the rate of change of a variable with respect to another variable, often with respect to time
• First-order differential equations involve only the first derivative of the dependent variable
  • Exponential growth model $\frac{dP}{dt} = rP$, where $P$ is population, $t$ is time, and $r$ is the growth rate
  • Logistic growth model $\frac{dP}{dt} = rP(1-\frac{P}{K})$, where $K$ is the carrying capacity
• Second-order differential equations involve the second derivative of the dependent variable
  • Harrod-Domar model $\frac{d^2Y}{dt^2} = s\frac{dY}{dt} - \delta Y$, where $Y$ is output, $s$ is the savings rate, and $\delta$ is the depreciation rate
• Partial differential equations involve derivatives with respect to multiple variables
  • Heat equation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$, where $u$ is temperature, $t$ is time, $x$ is position, and $\alpha$ is thermal diffusivity
• Solving differential equations involves finding a function that satisfies the equation and any initial or boundary conditions
  • Separation of variables method to solve first-order differential equations
  • Laplace transforms method to solve linear differential equations with initial conditions

Equilibrium Analysis

• Equilibrium state in which economic variables remain constant over time, with no tendency to change
• Market equilibrium occurs when the quantity supplied equals the quantity demanded at a given price
  • Solve for equilibrium price and quantity by setting supply and demand functions equal to each other
• Competitive equilibrium occurs when all markets in an economy are in equilibrium simultaneously
  • Solve for equilibrium prices and quantities in all markets using a system of equations
• Nash equilibrium occurs in game theory when each player's strategy is optimal given the strategies of the other players
  • Solve for Nash equilibrium by finding the best response of each player to the strategies of the other players
• General equilibrium analysis studies the simultaneous equilibrium in all markets of an economy
  • Arrow-Debreu model general equilibrium model with multiple commodities, consumers, and producers
  • Computable general equilibrium (CGE) models numerical models that simulate the effects of policy changes on the economy

Dynamic Systems and Stability

• Dynamic systems models that describe the evolution of economic variables over time
  • Discrete-time models variables change at fixed intervals (annually, quarterly)
  • Continuous-time models variables change continuously over time
• Phase diagrams graphical tool to analyze the behavior of dynamic systems in the space of state variables
  • Nullclines lines or curves where the rate of change of a variable is zero
  • Steady states equilibrium points where all variables remain constant over time
• Stability analysis studies the behavior of a dynamic system near its steady states
  • Stable steady state returns to equilibrium after a small perturbation
  • Unstable steady state diverges from equilibrium after a small perturbation
• Bifurcation analysis studies how the stability of a dynamic system changes as a parameter varies
  • Saddle-node bifurcation creation or destruction of two steady states as a parameter crosses a critical value
  • Hopf bifurcation emergence of periodic oscillations as a parameter crosses a critical value

Applications to Real-World Problems

• Economic growth models used to analyze the factors that contribute to long-run economic growth and living standards
  • Solow model emphasizes the role of capital accumulation and technological progress (United States, Japan)
  • Endogenous growth models incorporate the role of human capital, research and development, and innovation (South Korea, Taiwan)
• Environmental economics models used to analyze the economic causes and consequences of environmental problems
  • Optimal control models determine the optimal level of pollution control over time (carbon taxes, emissions trading)
  • Resource extraction models analyze the optimal depletion of non-renewable resources (oil, minerals)
• Financial economics models used to analyze the behavior of financial markets and institutions
  • Portfolio selection models determine the optimal allocation of assets in a portfolio (mean-variance analysis)
  • Option pricing models determine the fair price of options and other derivatives (Black-Scholes model)
• Labor economics models used to analyze the determinants of wages, employment, and income distribution
  • Human capital model explains wage differentials based on differences in education and training
  • Search and matching models analyze the process of job search and the determinants of unemployment

Limitations and Criticisms

• Assumptions economic models rely on simplifying assumptions that may not hold in reality
  • Perfect rationality assumes that economic agents have complete information and make optimal decisions
  • Perfect competition assumes that markets have many buyers and sellers, homogeneous products, and no barriers to entry or exit
• Aggregation problem macroeconomic models aggregate the behavior of individual agents, which may lead to fallacies of composition
  • Paradox of thrift increasing individual savings may lead to lower aggregate demand and output
  • Paradox of deleveraging reducing individual debt may lead to lower aggregate demand and output
• Lucas critique policy changes may alter the behavior of economic agents, rendering the model's predictions invalid
  • Rational expectations agents may anticipate policy changes and adjust their behavior accordingly
• Complexity economics criticizes the reductionist approach of traditional economic models and emphasizes the role of emergent properties and non-linear dynamics
  • Agent-based models simulate the interactions of heterogeneous agents using computer algorithms
  • Network models analyze the structure and dynamics of economic networks (trade networks, financial networks)
• Behavioral economics incorporates insights from psychology and other social sciences to model the bounded rationality and biases of economic agents
  • Prospect theory models how people make decisions under risk and uncertainty, emphasizing loss aversion and reference dependence
  • Nudge theory uses choice architecture to influence people's decisions without restricting their freedom of choice