page_title: Mathematical economics
section_title: Application
hierarchical_section_title: Mathematical economics / Application
context_page_description: Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.
  Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.
  Broad applications include:
  optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker
context_section_description: Much of classical economics can be presented in simple geometric terms or elementary mathematical notation. Mathematical economics, however, conventionally makes use of calculus and matrix algebra in economic analysis in order to make powerful claims that would be more difficult without such mathematical tools. These tools are prerequisites for formal study, not only in mathematical economics but in contemporary economic theory in general. Economic problems often involve so many variables that mathematics is the only practical way of attacking and solving them.  Alfred Marshall argued that every economic problem which can be quantified, analytically expressed and solved, should be treated by means of mathematical work.
  Economics has become increasingly dependent upon mathematical methods and the mathematical tools it employs have become more sophisticated.  As a result, mathematics has become considerably more important to professionals in economics and finance. Graduate programs in both economics and finance require strong undergraduate preparation in mathematics for admission and, for this reason, attract an increasingly high number of mathematicians. Applied mathematicians apply mathematical principles to practical problems, such as economic analysis and other economics-related issues, and many economic problems are often defined as integrated into the scope of applied mathematics.
  This integration results from the formulation of economic problems as stylized models with clear assumptions and falsifiable predictions.  This modeling may be informal or prosaic, as it was in Adam Smith's The Wealth of Nations, or it may be formal, rigorous and mathematical.
  Broadly speaking, formal economic models may be classified as stochastic or deterministic and as discrete or continuous.  At a practical level, quantitative modeling is applied to many areas of economics and several methodologies have evolved more or less independently of each other.
  Stochastic models are formulated using stochastic processes. They model economically observable values over time. Most of econometrics is based on statistics to formulate and test hypotheses about these processes or estimate parameters for them. Between the World Wars, Herman Wold developed a representation of stationary stochastic processes in terms of autoregressive models and a determinist trend. Wold and Jan Tinbergen applied time-series analysis to economic data. Contemporary research on time series statistics consider additional formulations of stationary processes, such as  autoregressive moving average models. More general models include  autoregressive conditional heteroskedasticity (ARCH) models and generalized ARCH (GARCH) models.
  Non-stochastic mathematical models may be purely qualitative (for example, models involved in some aspect of social choice theory) or quantitative (involving rationalization of financial variables, for example with hyperbolic coordinates, and/or specific forms of functional relationships between variables).  In some cases economic predictions of a model merely assert the direction of movement of economic variables, and so the functional relationships are used only in a qualitative sense: for example, if the price of an item increases, then the demand for that item will decrease.  For such models, economists often use two-dimensional graphs instead of functions.
  Qualitative models are occasionally used. One example is qualitative scenario planning in which possible future events are played out. Another example is non-numerical decision tree analysis. Qualitative models often suffer from lack of precision.
caption: The IS/LM model is a Keynesian macroeconomic model designed to make predictions about the intersection of "real" economic activity (e.g. spending, income, savings rates) and decisions made in the financial markets (Money supply and Liquidity preference).  The model is no longer widely taught at the graduate level but is common in undergraduate macroeconomics courses.[124]