Am I correct that cyclical unemployment can be negative?

Question

Let's start with stating and emphasizing some truths that we will need later. Overall unemployment rate=Frictional unemployment + Structural unemployment + Cyclical unemployment. Also, Natural rate of unemployment = Frictional unemployment + Structural unemployment. From this we can see that Cyclical unemployment would need to be equal to zero in order to our economy to have natural rate of unemployment.

But suppose we have overall level of unemployment that in LOWER than the natural rate of unemployment. Now things get interesting. We can't get overall level unemployment become lower than the natural rate of unemployment by decreasing frictional and/or stuctural unemployment. Fortunately, there is the third component of overall unemployment, namely cyclical unemployment. If we agree that cyclical unemployment can be negative, then this way we can allow overall unemployment to be under the natural rate of unemployment.

It looks reasonable mathematically, but I have some doubts because I have never heard that unemployment can be negative. Is my reasoning correct, can we have negative cyclical unemployment?

And if I'm NOT correct, if cyclical unemployment can NOT be negative, then HOW do we get overall unemployment below the natural rate of unemployment?

Answer 1

Cyclical unemployment $U^C$ is by definition difference between actual observed rate of employment $U$ and the natural rate of employment $U^N$ so we have: $U^C_t \equiv U_t-U_t^N$, see for example this paper.

Hence cyclical unemployment can be negative if actual observed rate of unemployment is lower than natural rate, because it is defined as a difference between the two.