Rents soaring across the country as would-be homebuyers slide back into the overcrowded rental market.

Rents rose in the US in June at the fastest pace since 1986, helping to propel overall inflation to a fresh four-decade high. An index measuring rent of a primary residence was 0.8% higher in June than the month before, an acceleration from the 0.6% increase recorded in May, according to the Labor Department’s report on consumer prices. In the 12 months through June, rents were up 5.8%. Those costs are soaring across the country as would-be homebuyers slide back into the overcrowded rental market.

But rent growth may be peaking as affordability concerns mount, and a surge in construction of new units is poised to start adding to the available inventory. The Labor Department measure tends to lag behind other estimates, so it is likely that rent increases will contribute to rising inflation in the consumer price index through the rest of this year, according to Mark Zandi, chief economist of Moody’s Analytics. “The big increase in CPI rents is catch-up with the consistent double-digit growth in market rents,” Zandi said. “The good news is that market rents appear to be topping out, as renters are not able to afford the higher rents and are balking. More rental supply is also coming, although this will take a year or two to have a meaningful impact on market rents.”