Economics….truly the dismal science
For intellectual masochists economics takes some beating. Many mathematicians have said the stuff i do is harder maths than anything they have done in their degree. And applied maths is how i would describe the degree.
Basically economics is a series of assumptions (supposedly axioms—–in accordance with the geometric method…but far from verifiably true) which leads to some logical implications and predictions about consumer and producer behaviour, the impact of certain policies etc.
Despite the unrealism of many of the assumptions, as they supposedly predict well that is all that matters….or so Friedman would have you believe.
For example very few of the population would be able to solve the devilish algebreic functions economists use to model decisions. But because real life behaviour approximates the behaviour one would take if one solved these equations then economic models are thus valid.
In addition a considerable amount of theory is based on revealed preference: if a consumer chooses good X when faced with a choice between X and Y then he prefers X to Y and we can construct a preference ordering X>Y and from it an ordinal (rank) utility function. Of course we are assuming stability of consumers preferences and a whole host of other unrealistic assumptions such as transitivity. So basically we are explaining consumer behaviour using er consumer behaviour…a circular argument.
We also have that most especial creature "home economus" rational economic man who is presumed to be both maximising his utility AND making consistent choices. Sadly this mythical beast has never been sighted in real life and just remains a figment of economic imagination.
Moreover any challenges to the idea of a self–interested man trying to maximise his utility can be easily met by incorporating amongst other things "altruism" into his utility function. Amartya Sen challenges this line of reasoning by introducing the idea of commitment: doing things that harm your wellbeing because it is the right thing to do, rather than because you get some sick pleasure from saving drowning puppies.
Other economists have introduced notions such as bounded rationality, use of rules of thumb and even herd type behaviour where we do what every one else does as we lack the information to make the best decision. A fascinating application of this by Paul Ormerod looks at network economics with a certain number of "celebrity types" who are able to influence the behaviour of others resulting in ripple like effects through the network
Another fascinating application of derived utility functions are attempts to aggregate preferences to get a social welfare function and use this to inform notions of how to maximise happiness or justice etc. The latter represented by a Rawlsian SWF where we maximise the utility of the least well off individuals in society. The problem of course this literature faces is the requirement to make interpersonal comparisons to get a complete social welfare function, something which economists are loath to do. One fun logical absurbity is the famous Paretian Impossibility theorem which says the only consistent social choice mechanism is one where you have a dictator (albeit a benevolent one)
Something which has been confined mainly to the periphery of economics is institutional debates over which is best: central planning or free markets. Fascinating stuff.
Oh and of course we have good old Keynes and his brand of socialism which involved pumping loads of money into the economy via public works programmes (a strategy used by Hitler which is kinda ironic) with resulting multiplier effects.
Keynes is pretty much mud for most economists as his theories lack mathematical (micro) foundations..despite being more logical to the impartial observer.
Preferred to this is a theory built up on dynamic programming (basically deriving a time path of optimal decisions) and using this to see the impact of changes from steady state. As you may guess…maths maths and more maths and none of the beloved graphs of good old economics 101.
Moreover a brand of statistics called econometrics is used to empirically test the mathematical models economists so enjoy making. This is basically dressed up curve fitting and the relationships they derive break down invariably when predicting into the future. It is absolute hell to learn.
Second to this for sadism is microeconomic theory especially general equilibrium theory, a series of mathematical proofs invoking numerous fixed point theorems and separating hyperplane theorems, involving my beloved Walrasian auctioneer and the "core". Of course like every other theory relies on loads of ludicrous assumptions which theory can only accomodate for to an extent.
Thirdly in this triumvarate of incestruos marriage with mathematics is game theory. Yeah it is a cute idea and logically appealing. But once you introduce mathematical notation, bayesian updating and the bloody Folk theorem it is a task fit for the underworld.