Fundamental Methods of Mathematical Economics

1. Mathematical Economics: A Powerful Analytical Approach

The major difference between "mathematical economics" and "literary economics" lies principally in the fact that, in the former, the assumptions and conclusions are stated in mathematical symbols rather than words and in equations rather than sentences; moreover, in place of literary logic, use is made of mathematical theorems—of which there exists an abundance to draw upon—in the reasoning process.

Symbols and Logic. Mathematical economics is not a separate branch but a method using mathematical symbols and theorems to analyze economic problems. It offers a concise and precise language, leveraging a wealth of mathematical theorems for deductive reasoning. This approach contrasts with "literary economics," which relies on verbal arguments and less formal logic.

Advantages of the Mathematical Approach:

• Conciseness and precision in expressing assumptions and conclusions.
• Access to a vast library of mathematical theorems.
• Explicitly stating assumptions, preventing unintentional adoption of implicit ones.
• Ability to handle general n-variable cases, overcoming the limitations of geometric methods.

A Mode of Transportation. The mathematical approach is a tool that accelerates the journey from postulates to conclusions. While geometric methods offer visual insights, they are limited by dimensionality. Mathematical techniques, such as calculus and algebra, allow for the analysis of complex, multi-variable relationships that are impossible to visualize geometrically.

2. Economic Models: Simplified Frameworks for Understanding

Such a deliberately simplified analytical framework is called an economic model, since it is only a skeletal and rough representation of the actual economy.

Abstraction and Essential Factors. Economic models are simplified representations of the real world, designed to isolate and analyze key factors and relationships. These models, often mathematical, consist of equations that describe the structure and assumptions of the system. By focusing on the primary elements, models allow economists to study complex phenomena without being overwhelmed by real-world complexities.

Ingredients of a Mathematical Model:

• Variables: Endogenous (determined within the model) and exogenous (determined outside the model).
• Constants and Parameters: Fixed magnitudes that influence variable relationships.
• Equations: Definitional, behavioral, and equilibrium conditions that link variables.

Solving for Endogenous Variables. The goal of a mathematical model is to solve for the values of endogenous variables, such as market-clearing prices or profit-maximizing output levels. These solutions are expressed in terms of parameters and exogenous variables, providing insights into how changes in external factors affect the system's equilibrium.

3. Equilibrium Analysis: Finding the Balance

According to one definition, an equilibrium is "a constellation of selected interrelated variables so adjusted to one another that no inherent tendency to change prevails in the model which they constitute."

A State of Rest. Equilibrium in economics refers to a state where opposing forces are balanced, and there is no inherent tendency for change within the model. This balance is achieved when all variables are simultaneously at rest, and their states are compatible with each other. External factors, such as parameters and exogenous variables, are assumed to be fixed when defining an equilibrium.

Types of Equilibrium:

• Market Equilibrium: Quantity demanded equals quantity supplied.
• National-Income Equilibrium: Aggregate demand equals national income.
• Goal Equilibrium: An optimal state achieved through conscious effort (e.g., profit maximization).

Statics and Limitations. Equilibrium analysis, or statics, focuses on the characteristics of the equilibrium state itself, rather than the process of reaching it. This approach disregards the time element and the potential for instability, which are addressed in dynamic analysis.

4. Linear Models and Matrix Algebra: Organizing Complexity

With the model thus constructed, the next step is to solve it. i.e., to obtain the solution values of the three endogenous variables, Qd. Qs, and P.

Compact Notation. Matrix algebra provides a powerful tool for representing and solving systems of linear equations, which are common in economic models. It allows for the concise expression of complex relationships and facilitates the analysis of multi-variable systems.

Matrices and Vectors:

• Matrices: Rectangular arrays of numbers, parameters, or variables.
• Vectors: Special matrices with only one column (column vectors) or one row (row vectors).
• Matrix Operations: Addition, subtraction, scalar multiplication, and matrix multiplication, each with specific rules and conformability conditions.

Solving Linear Systems. Matrix algebra enables the expression of an equation system as Ax = d, where A is the coefficient matrix, x is the vector of variables, and d is the vector of constants. The solution, if it exists, can be found using the inverse of the coefficient matrix: x = A⁻¹d.

5. Testing Nonsingularity: Determinants as Gatekeepers

In general, in order to apply that process, make sure that A) the satisfaction of any one equation in the model will not preclude the satisfaction of another and B) no equation is redundant.

Squareness and Linear Independence. For a matrix to have an inverse (be nonsingular), it must be square (number of rows equals the number of columns) and its rows (or columns) must be linearly independent. Linear independence means that no row can be expressed as a linear combination of the other rows.

Determinants as a Test. The determinant of a square matrix is a scalar value that provides a test for nonsingularity. A nonzero determinant indicates that the matrix is nonsingular and has an inverse, while a zero determinant implies singularity and linear dependence.

Calculating Determinants:

• 2x2 Matrix: |A| = ad - bc
• Higher-Order Matrices: Laplace expansion using minors and cofactors.

6. Comparative Statics: Examining Equilibrium Shifts

The purpose of any theoretical analysis, regardless of the approach, is always to derive a set of conclusions or theorems from a given set of assumptions or postulates via a process of reasoning.

Analyzing Changes in Equilibrium. Comparative statics examines how changes in parameters or exogenous variables affect the equilibrium values of endogenous variables in a model. It involves comparing the initial equilibrium state with the new equilibrium state after the change, without considering the adjustment process.

Qualitative vs. Quantitative Analysis:

• Qualitative: Focuses on the direction of change (increase or decrease).
• Quantitative: Determines the magnitude of the change.

The Role of Derivatives. The concept of the derivative, representing the rate of change, is central to comparative statics. Partial derivatives are used to analyze the impact of changes in individual parameters on equilibrium values.

7. Optimization: Seeking the Best Outcome

The sensible procedure is, therefore, to pick out what appeal to our reason to be the primary factors and relationships relevant to our problem and to focus our attention on these alone.

Goal-Oriented Equilibrium. Optimization problems involve finding the best possible state for an economic unit, such as maximizing profit or utility. This contrasts with nongoal equilibrium, where the equilibrium state arises from the impersonal balancing of forces.

Objective Functions and Choice Variables:

• Objective Function: A mathematical expression that represents the goal to be maximized or minimized.
• Choice Variables: The variables whose values can be chosen to achieve the optimal outcome.

First-Order and Second-Order Conditions. Optimization problems are solved by finding the values of choice variables that satisfy necessary and sufficient conditions for an extremum. These conditions often involve derivatives or differentials of the objective function.

8. Exponential and Logarithmic Functions: Modeling Growth and Change

The "language" used is more concise and precise; B) there exists a wealth of mathematical theorems at our service; C) in forcing us to state explicitly all our assumptions as a prerequisite to the use of the mathematical theorems, it keeps us from the pitfall of an unintentional adoption of unwanted implicit assumptions; and D) it allows us to treat the general ^-variable case.

Variable Exponents. Exponential functions, where the independent variable appears in the exponent, are essential for modeling growth and decay processes. Logarithmic functions, the inverses of exponential functions, are useful for solving equations involving exponents and for simplifying complex expressions.

Key Concepts:

• Exponential Function: y = bx, where b is the base and x is the exponent.
• Logarithmic Function: x = logb y, the inverse of the exponential function.
• Natural Exponential Function: y = ex, where e is Euler's number (approximately 2.71828).
• Natural Logarithm: x = ln y, the logarithm to the base e.

Applications in Economics. Exponential and logarithmic functions are used to model compound interest, population growth, and other phenomena involving rates of change. They also simplify optimization problems and provide insights into economic relationships.

9. Dynamic Analysis: Time and Economic Evolution

Common sense would tell us that, if you intend to go to a place 2 miles away, you will very likely prefer driving to walking, unless you have time to kill or want to exercise your legs.

Tracing Time Paths. Dynamic analysis focuses on the evolution of economic variables over time, examining how they adjust and converge to equilibrium. This approach contrasts with static analysis, which only considers the equilibrium state itself.

Continuous vs. Discrete Time:

• Continuous Time: Variables change at every point in time, modeled using differential equations and integral calculus.
• Discrete Time: Variables change only at specific intervals, modeled using difference equations.

Differential and Difference Equations. Dynamic models are often expressed as differential equations (continuous time) or difference equations (discrete time), which describe the patterns of change in the variables. Solving these equations provides the time path of the variables.

10. Dynamic Systems: Interacting Equations of Change

As more commodities enter into a model, there will be more variables and more equations, and the equations will get longer and more complicated.

Simultaneous Dynamic Equations. Dynamic systems arise when multiple variables interact and influence each other's patterns of change. These systems are represented by sets of simultaneous differential or difference equations.

Walrasian General-Equilibrium Model. A Walrasian model includes all the commodities in an economy in a comprehensive market model.

Solving Dynamic Systems. Solving dynamic systems involves finding the time paths of all the variables simultaneously, taking into account their interdependencies. Techniques from matrix algebra and calculus are often used to analyze these systems.

11. Linear Programming: Optimizing Under Constraints

Common sense would tell us that, if you intend to go to a place 2 miles away, you will very likely prefer driving to walking, unless you have time to kill or want to exercise your legs.

Optimization with Inequalities. Linear programming is a mathematical technique for optimizing a linear objective function subject to linear inequality constraints. This approach is particularly useful for resource allocation problems, where resources are limited and choices must be made within those limits.

Key Concepts:

• Objective Function: A linear expression to be maximized or minimized.
• Constraints: Linear inequalities that restrict the values of the choice variables.
• Feasible Region: The set of all points that satisfy all the constraints.
• Extreme Points: The corner points of the feasible region.

Simplex Method. The simplex method is an algorithm for finding the optimal solution to a linear programming problem. It involves systematically examining the extreme points of the feasible region to identify the one that yields the best value of the objective function.

Last updated: April 4, 2025