As a social science, economics analyzes the production, distribution, and consumption of goods and services. The study of economics requires the use of mathematics in order to analyze and synthesize complex information.
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Using mathematics allows economists to form meaningful, testable propositions about complex subjects that would be hard to express informally. Math enables economists to make specific and positive claims that are supported through formulas, models, and graphs. Mathematical disciplines, such as algebra and calculus, allow economists to study complex information and clarify assumptions.
Algebra is the study of operations and their application to solving equations. It provides structure and a definite direction for economists when they are analyzing complex data. Math deals with specified numbers, while algebra introduces quantities without fixed numbers (known as variables). Using variables to denote quantities allows general relationships between quantities to be expressed concisely. Quantitative results in science, economics included, are expressed using algebraic equations.
Concepts in algebra that are used in economics include variables and algebraic expressions. Variables are letters that represent general, non-specified numbers. Variables are useful because they can represent numbers whose values are not yet known, they allow for the description of general problems without giving quantities, they allow for the description of relationships between quantities that may vary, and they allow for the description of mathematical properties. Algebraic expressions can be simplified using basic math operations including addition, subtraction, multiplication, division, and exponentiation.
In economics, theories need the flexibility to formulate and use general structures. By using algebra, economists are able to develop theories and structures that can be used with different scenarios regardless of specific quantities.
Calculus is the mathematical study of change. Economists use calculus in order to study economic change whether it involves the world or human behavior.
Calculus has two main branches:
- Differential calculus is the study of the definition, properties, and applications of the derivative of a function (rates of change and slopes of curves) . By finding the derivative of a function, you can find the rate of change of the original function.
- Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite and definite integral (accumulation of quantities and the areas under curves) .
Calculus is widely used in economics and has the ability to solve many problems that algebra cannot. In economics, calculus is used to study and record complex information - commonly on graphs and curves. Calculus allows for the determination of a maximal profit by providing an easy way to calculate marginal cost and marginal revenue. It can also be used to study supply and demand curves.
Common Mathematical Terms
Economics utilizes a number of mathematical concepts on a regular basis such as:
- Dependent Variable: The output or the effect variable. Typically represented as
$y$, the dependent variable is graphed on the $y$-axis. It is the variable whose change you are interested in seeing when you change other variables.
- Independent or Explanatory Variable: The inputs or causes. Typically represented as
$x_1$, $x_2$, $x_3$, etc., the independent variables are graphed on the $x$-axis. These are the variables that are changed in order to see how they affect the dependent variable.
- Slope: The direction and steepness of the line on a graph. It is calculated by dividing the amount the line increases on the
$y$-axis (vertically) by the amount it changes on the $x$-axis (horizontally). A positive slope means the line is going up toward the right on a graph, and a negative slope means the line is going down toward the right. A horizontal line has a slope of zero, while a vertical line has an undefined slope. The slope is important because it represents a rate of change.
- Tangent: The single point at which two curves touch. The derivative of a curve, for example, gives the equation of a line tangent to the curve at a given point.