What is the difference between applied economics and mathematical economics?
Mathematical economics is the application of mathematical methods to represent theories and analyze theoretical problems in economics. By convention, the applied mathematical methods refer to those beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods.
An advantage claimed for the approach is its allowing formulation of theoretical relationships with rigor, generality, and simplicity.
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.
Applied economics is the study of economics in relation to real world situations, as opposed to the theory of economics. It is the application of economic principles and theories to real situations, and trying to predict what the outcomes might be.
Put simply, applied economics is the study of observing how theories work in practice. Applied economics may be practiced at macroeconomic (the whole, aggregate economy) or microeconomic (analyzing individual consumers and companies) levels.
When wondering whether to study Economics or Applied Economics at university, bear in mind that applied economics students take fewer core theoretical modules and a larger number of applied modules than those studying for an economics degree.
According to the University of St. Andrews in Scotland:
"This (Applied Economics degree) provides the student with a broader perspective on the application of economic theory to real world issues but less knowledge of core economic theory."
On its website, Duke University says Applied Economics is for people seeking a broad understanding of economics. While having to study several aspects of economic theory, they will also have considerable freedom in their elective courses.
Simply put, mathematical economics involves the rigorous mathematical formulation and systematization of economic theory. It does not involve a direct application of these models and theories to the real world through empirical testing and estimation.
On the other hand, applied economics involves the rigorous application of economic theory (both mathematical/quantitative and qualitative) to the real world. It includes:
- estimation of model-specified parameters through statistical inference (e.g. estimating the output elasticities of input in a production function);
- validation of theoretical models through econometric and other forms of empirical analysis (e.g. seeking evidence for whether unemployment rises as minimum wage is raised, as specified theoretically; or whether countries with greater trade openness have higher rates of economic growth on average);
- as well as the application of theory to predict outcomes given a set of considerations (e.g. whether a particular public policy will have its intended effect as theoretically-predicted ... e.g. changes in GDP in response to changes in fiscal policy as specified in Keynesian theory): and
- forecasting of economic variables using relationships theoretically-specified.
Case studies are also part of applied economics.
In general, it is simply theoretical analysis vs empirical analysis
Theorization in any subject (and more so in economics) is explicitly or implicitly axiom-based. Since Adam Smith built economics to be a pursuit of seeking knowledge about the evolution and functioning of the economies (the system that is related to the affairs of the individuals in the society with regard to meeting their material needs), theorization began.
As Bertrand Russell explained, most of the complications and paradoxes in philosophy (read a closer view to understanding any system, real or conceptual) arose due to the fact that the language used for stating the problems and resolving them was imprecise (and carrying many possible meanings). So he pleaded that the language should be exact for which he suggested that mathematical language should be used (see his History of Western Philosophy).
This way of putting the questions and providing answer (or seeking for the answer) gave rise to mathematical economics. Thus mathematical economics is a system of (1) clearly stated axioms - A, (2) deduction of inferences from those axioms - I, (3) relationship of those inferences with the issues related to an economic system - E. Thus, it is a triplet (A, I, E).
Take a simple but very important case. Will a society whose economy is exchange based (market economy) ensure harmony, stability, efficiency, growth, justice and best possible application of resources (Smith's problem)? Adam Smith answered this in the affirmative and argued that an invisible hand (perhaps groping in the darkness singularly led by the price signal) will ensure all these (Smith's solution).
Now, one must mathematically define as to what one means by an economy, what is harmony (of interests), what is stability, what is efficiency, what is growth, what is justice, what is best application of resources. That those are ensured, what will be the necessary and the sufficient set of axioms under which exchange takes place?
This attempt started with Leon Walras and went on to Arrow-Debreu. Sonnenschein–Mantel–Debreu theorem showed that existence of many markets would have many equilibria and the price signals will not be unique.
Investigations showed that a market economy will ensure Smith's solution to the Smith's problem only under very strict and mostly unrealistic axiomatic structure.
Studying economics in this manner - axiomatically, is called mathematical economics. It is a top down approach to understanding economics.
[There could, however, be another approach to understanding economics. Create some agents (on computer), define the rules under which they would act and interact with each other (acting in the same manner) and see what emerges. Does such a simulation, repeated for a large number of times, often lead to an ensemble that could be identified with the market economy? This is making economics from below.]
Now as to applied economics. We have vague understanding of the functioning of an economy in the specific domains and in specific areas. Nothing ensures the results, but experience suggests an effect of the effort to be only likely. Thus, we have some thumb rules, validated mostly by experience. Using these thumb rules with an objective to expect desired results is applied economics. This applied economics could be in the realm of agriculture, industry, welfare and so on. It can address to the problems in certain specific areas and regional economies. It is less concerned with economics as a science and more concerned with economics as a tool of social engineering, many a time economic policies.
One may note, however, that applied economics has a provisional (and not scientifically secure) theoretical backing and mathematical economics (which is purely theoretical) does not have an empirical (real life) support.
This is the weak point of economics. Unlike many other sciences where theory and real life are not at two poles, economic theory and economic application are mostly poles apart. The one is blind (can run, but does not know whereto and what it would collide against while running or whether it would not fall in the well) and the other is lame (knows, can see, but cannot take an enterprise to move, walk or run).