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#### Abstract:

Most governments are mandated to maintain their economies at full employment. We propose that the best marker of full employment is the efficient unemployment rate, $u^\ast$, which is the unemployment rate that minimizes the nonproductive use of labor—both unemployment and recruiting. The nonproductive use of labor is well measured by the number of jobseekers and vacancies, $u + v$. 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 ($u > v$). Moreover, the efficient unemployment rate is the geometric average of the unemployment and vacancy rates: $u^\ast = \sqrt{uv}$. We compute $u^\ast$ for the United States between 1930 and 2022. We find that $u^\ast$ averages 4.1% over the period, that $u^\ast$ is quite stable over time and, for instance, that the US labor market was inefficiently tight from the middle of 2021 to the end of 2022.

#### Figure 12C: Efficient unemployment rate $u^\ast = \sqrt{uv}$ in the United States, 1930–2022

#### Citation:

Michaillat, Pascal, and Emmanuel Saez. 2022. “$u^\ast = \sqrt{uv}$.” arXiv:2206.13012. https://doi.org/10.48550/arXiv.2206.13012 .