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We hear more and more these days about growth, particularly in the Eurozone and Britain, being beneath its long-term trend due to deficient demand. I can think of a few things which support that idea - such as a drying up of the credit which fuelled pre-crash consumer growth, low inflation and perhaps even deflation, long-term stagnant wages, and all this despite ultra low interest rates and QE. In short, the deficient demand story is one which has an intuitive appeal.

What I don't get is simply this: can we prove that deficient demand is the cause of the problem, and if so how?

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This is one of the issues with Macroeconomics - you usually cannot prove these kind of things, you can only find indicators.

That is, because what we observe is always - in the neoclassical language - the general equilibrium, never the "off equilibrium adjustment" (as in the Walrasian Auctioneer). That includes Keynesian logic - these days, the majority of Economists does not think about these traps as off-equilibrium paths, but rather as multiple equilibria [citation needed].

As a comparison, if you only observe the equilibrium outcome, it is hard to say whether a decrease in prices comes from an increase in supply or a decrease in demand. Now, if you could randomize or had a natural experiment at hand (say, randomly shut down half of the firms in the economy), you could actually "prove" the underlying issue with empirical data.

But since this is not feasible within many subjects of macroeconomic interest, disproving ideas using empirical evidence is quite difficult.

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