Private sector business activity across the eurozone shrank at its fastest rate in three-and-a-half years in October, a key survey showed on Tuesday. The Purchasing Managers Index (PMI), a leading indicator compiled by the Markit research firm, produced a combined manufacturing and services score of 45.7 points, down from 46.1 in September and below an earlier flash estimate of 45.8. This was “a level historically consistent with the region’s economy contracting at a quarterly rate of around 0.5 percent,” said Markit senior economist Rob Dobson. Citing yet more evidence of a widespread downturn, with all of the big-four economies seeing output decline, Dobson said there were signs the contraction in Germany was quickening, with Ireland the only real “brighter spot.” Ireland logged a score of 55.5 — any result above 50 indicates growth — as it continues its painful recovery from a banking and property crash. But Germany fell to 47.7, Italy to 45.6 and France 43.5 — with Spain down at 41.5 points. The big four eurozone economies all saw fresh job losses. A breakdown of the data showed that new business fell for the 15th month running, and that average input costs had continued their three-year rise.