Do producer prices eventually feed into consumer prices? If so, what's the lead or lag time?

The Wall Street Journal article Why the Inflation Picture Looks Starkly Different for Businesses and Consumers got me thinking about these questions, and I do not believe they came up with the correct answer.

This month consumers said they expected a 2.7% rise in inflation over the next year, a level unchanged since December, according to the University of Michigan's latest sentiment survey.

Other survey data indicate businesses are feeling inflationary pressures. Take, for instance, the rising percentage of executives in the Institute for Supply Management's manufacturing survey who say they're paying higher prices for materials: In January, 46.6% reported higher prices, up from 42% a year earlier.