Measuring the efficient unemployment rate using the Beveridge curve
In this paper Michaillat and Saez suggest that the best marker of full employment is the efficient unemployment rate, u*, defined as the unemployment rate that minimizes the nonproductive use of labor—both jobseeking and recruiting. The nonproductive use of labor is well measured by the number of jobseekers and vacancies, u + v. Note that the Beveridge curve implies that unemployment cannot be reduced to zero because people constantly move in and out of jobs.
Through the Beveridge curve, the number of vacancies is inversely related to the number of jobseekers. With such symmetry, the labor market is efficient when there are as many jobseekers as vacancies (u = v), too tight when there are more vacancies than jobseekers (v > u), and too slack when there are more jobseekers than vacancies (u > v). Accordingly, the efficient unemployment rate is the geometric average of the unemployment and vacancy rates: u*=√uv. We compute u* for the United States between 1930 and 2022. We find for instance that the US labor market has been over full employment (u < u*) since May 2021.
[…]Empirically the Beveridge curve is very stable over the business cycle. It does not seem to respond to stabilization policies (either monetary or fiscal policy) in any systematic way. Given this stability, it makes sense that the Fed takes the Beveridge curve as given.

U* = √Uv
Authors: Pascal Michaillat, Emmanuel Saez
From: Brown University, University of California, Berkeley
Policy implications from the US Beveridge curve
The labor market in the US needs to cool off to decrease the pressure on inflation. In this paper, Blanchard et al. question whether the Fed can cool off the labor market through a decrease in vacancies, leaving unemployment unaffected. They argue that the U.S. Beveridge curve implies that the very high vacancy-to-unemployment ratio is due to not only an overheating labor market but also to a substantially increased natural unemployment. Therefore, to quiet the labor market will require substantial unemployment pain:
[I]t is highly unlikely that the decrease in the vacancy rate can be achieved without a substantial increase in the unemployment rate. Theory suggests that this would require a strong improvement in matching, through either a slowdown in reallocation or an improvement in matching efficiency, on which the Fed has no control and which we see no reason to expect—the empirical evidence indicates that it has never happened before.

Bad News for the Fed from the Beveridge Space
By: Olivier Blanchard, Alex Domash, Lawrence Summers – Peterson Institute, Harvard University