The US Beveridge Curve, December 2000 to August 2026: An Exact Slope and Convexity Formula, a Decomposition of the Post-2022 Fall in Job Openings, and Why Matching Efficiency Has Not Recovered

Between March 2022 and August 2026 the US job-openings rate fell from 7.49% to 4.17% of the labour force while unemployment rose only from 3.67% to 4.14%. I use JOLTS and Current Population Survey data from FRED for December 2000 to August 2026 (308 months) to explain how. With a Cobb–Douglas matching function and flow balance between hires and separations, I derive the Beveridge curve in closed form and prove that its elasticity is −(α+u/(1−u))/(1−α) and that its curvature is exactly vα/[(1−α)²u²(1−u)²] > 0. Estimated on 2001–2019, the vacancy elasticity of the job-finding proxy is 0.6934 (se 0.0368), so along the curve near 4% unemployment a 1% fall in openings should raise unemployment by about 2%. The observed ratio since March 2022 is −4.875, so most of the fall in openings was not a move along the curve. An exact log decomposition assigns −0.3263 of the −0.5855 log fall in openings to lower separations, −0.1346 to a partial recovery of matching efficiency, and only −0.0531 to higher unemployment. Matching efficiency remains 0.23 log points below its 2001–2019 mean. I also show that the unrestricted matching regression rejects constant returns, because JOLTS hires barely respond to the number of unemployed.