BERLIN—German producer prices rose in January at their fastest rate since modern records began, soaring 25 percent and extending a run of sharp increases likely to keep businesses under financial stress and consumer inflation high. Monday’s Federal Statistics Office showed most of the rise was due to spiralling energy costs, which the BDI industry association said were threatening to hamper an economy that the country’s central bank said was probably shrinking due to increased coronavirus-induced worker absences. The jump in factory gate costs, considered a leading indicator for consumer prices, was the biggest since 1949, when West and East Germany were founded and the country’s post-war economic data series began. Analysts polled by Reuters had expected a repeat of December’s figure of 24.2 percent. Sharp increases of 18.4 percent and 19.2 percent were logged in October and November, respectively. That succession of jumps in the PPI measure, taken before products …