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Earlier I studied macroeconomics from Blanchard. But now while reading Mankiw's Macroeconomics I found different definition for natural rate of unemployment. According to Blanchard,

"the rate of unemployment (and by implication the level of output) that prevails if the price level and the expected price level are equal." (pg. 135, sixth edition) and

"The natural rate of unemployment is the rate of unemployment required to keep the inflation rate constant." (pg. 170, sixth edition)

But according to Mankiw,

"The natural rate is the rate of unemployment toward which the economy gravitates in the long run, given all the labor-market imperfections that impede workers from instantly finding jobs."(pg. 177. eighth edition),i.e., where unemployment rate reaches steady-state, it neither increases or decreases, where rate of job finding is equal to rate of job separation.

What is the common ground between all these definitions? Are they different or essentially the same? If so, then how?

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Mankiw's definition alludes to a situation where the only source of unemployment is frictions and imperfections in the labor/human capital market (you could also add inherent rigidities like "specificity" of labor/human capital, which is the price to pay for specialization). But also, in "imperfections" Mankiw may also include "wage rigidities".
We could say that this is the "micro" approach, focused on the special characteristics of the labor/human capital market.

Blanchard's 1st definition on the other hand is more "macroeconomic": it is a perfect-foresight equilibrium state (or maybe a self-fulfilled one). It indirectly says that, whenever (i.e. always) expectations and forecasts on the nominal magnitudes are not perfect, the real magnitudes of the economy do not come out exactly as expected, creating variations in, among other things, the unemployment rate. But when we have such an ideal state of perfect foresight, even temporarily, the unemployment observed will be the "natural rate". Blanchard does not go into why this "natural rate" is not zero (presumably, he has in mind the "micro-" approach as an explanation).

So we could say that Mankiw describes the "natural rate" as the rate due to causes over and beyond expectational disequilibrium on nominal magnitudes. On the other hand, Blanchard identifies the natural rate as -...exactly, the rate that will prevail in the absence of expectational disequilibrium on nominal magnitudes (he just doesn't state what other causes create unemployment).

Blanchard second definition is the NAIRU concept clearly explained in @BKay's answer.

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One definition of the natural rate of unemployment is the NAIRU: non-accelerating-inflation rate of unemployment. The classic story of the money illusion and unemployment works like this:

Both Friedman and Phelps argued that the government could not permanently trade higher inflation for lower unemployment. Imagine that unemployment is at the natural rate. The real wage is constant: workers who expect a given rate of price inflation insist that their wages increase at the same rate to prevent the erosion of their purchasing power. Now, imagine that the government uses expansionary monetary or fiscal policy in an attempt to lower unemployment below its natural rate. The resulting increase in demand encourages firms to raise their prices faster than workers had anticipated. With higher revenues, firms are willing to employ more workers at the old wage rates and even to raise those rates somewhat. For a short time, workers suffer from what economists call money illusion: they see that their money wages have risen and willingly supply more labor. Thus, the unemployment rate falls. They do not realize right away that their purchasing power has fallen because prices have risen more rapidly than they expected. But, over time, as workers come to anticipate higher rates of price inflation, they supply less labor and insist on increases in wages that keep up with inflation. The real wage is restored to its old level, and the unemployment rate returns to the natural rate. But the price inflation and wage inflation brought on by expansionary policies continue at the new, higher rates.

The Concise Encyclopedia of Economic: Phillips Curve by Kevin D. Hoover, emphasis mine.

At first this seems like a Blanchard-type definition. But why it is that the NAIRU takes one value and not another? The answer typically given is like that provided by Mankiw: labor-market imperfections impeding or incentivizing workers from instantly finding jobs. A decline in unemployment insurance, a reduction in the regulatory costs of firing workers, or a technological improvement in the ability of workers and employers to find each other all could reduce unemployment without altering inflation or inflation expectations.

In my eye, both definitions contribute something important to the idea of a natural rate of unemployment. If labor market frictions are moving around a great deal it is hard to speak of a natural rate of unemployment. If the fiscal or monetary policy authorities are manipulating inflation in a way that is at odds with inflation expectations there similarly is no natural rate of unemployment either. By "gravitates in the long run" I assume that Mankiw doesn't mean convergence. Rather, he means something more like mean reversion. Only when the monetary authority is catering rational expectations of inflation and labor market frictions are broadly stable that a natural rate of unemployment makes sense. It's going to be something more like a long run average rate of unemployment in that economy.

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In the Blanchard book itself, when discussing the WS-PS model there is also a definition similar to that given in Mankiw. Relating to the first definition you gave, in the long run expectations adapt and expected and observed price levels match. That relates the first and third definition. The second definition goes through the definition of the Phillips curve if I am mot mistaken and again has to do with expected and seen prices matching.

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