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What are the differences between the approaches of Keynes and Real Business Cycle theorists? What are the assumptions made by both the theories? How do they differ in their methodology? Do they contradict each other??
I wish to get a basic idea of the theories with a slight bit ofrigour.
The distinction is between "nominal" and "real" recessions and booms.
Real business cycle theory was developed to point out the fact that variations in employment and hours could occur even in an economy where markets were working competitively and there were no pricing frictions. In the RBC world, recessions and booms are driven by "real" factors: variation in technology, variation in the supply of commodities used as input in the production process, etc. A consequence is that in the RBC world while increased government spending on consumption goods indeed raises output and employment (the fiscal multiplier in RBC models is between 0 and 1), following such a policy during a recession is socially suboptimal. (You force people to work more and consume less.) To justify government spending during a recession is no easier and no harder than to justify it during "normal times", and then the argument becomes one about market failure, public finance, etc.
In contrast, Keynesian theory (and its modern cousins) say that while "real recessions" in the above sense may indeed exist, there are also "nominal recessions": these are recessions caused by "less spending" (for instance, if you're working with MV = PY, they are recessions caused by a fall in V). The reason this creates a recession is because there are nominal rigidities in the economy - workers are unwilling to accept nominal wage cuts, producers face "menu costs" which make price changes infrequent, etc. Because of these rigidities, a fall in "aggregate spending" manifests itself not only in the form of lower prices, but also in the form of lower output.
So far, everything I said is also consistent with monetarism, for instance. However, this is not all there is to Keynesian theory: Keynesians go further and claim that the solution to a shortfall in spending is for the government to increase spending and "make up the lost demand". In other words, if you think of the AD-AS model, Keynesians think that government spending can raise AD. This idea has been discredited in academia for decades because it ignores budget constraints - there's only so much money available in the economy, if the government borrows money from you and spends it, that's money someone else did not spend. It's not clear how government spending boosts "money velocity", and in fact there are New Keynesian models where increased government spending is deflationary and makes a nominal recession worse by creating more demand for real money balances.
Keynesian ideas have made a resurgence in the past 10 years thanks to the zero lower bound. The argument above about budget constraints is no longer true when nominal interest rates are stuck at zero, because now people will actually start hoarding cash instead of spending it. Therefore, increasing government spending does not "crowd out" private spending, because households and firms are just sitting on the cash they have anyway. Monetary stimulus is ineffective because any additional money is just held by the public. This is the "liquidity trap" a nutshell. Critics point out that permanent monetary stimulus as opposed to a temporary one can still work in a liquidity trap (as pointed out in Krugman (1998)) and that monetary models (say, with cash-in-advance constraints) are indeterminate at the zero lower bound: without anchoring consumption and prices in the future to some level chosen arbitrarily, it's not possible to pin down a unique equilibrium of the economy today.
There's more to be said on this subject, but this should be enough for an answer.
