Housing costs swell, hampering home buyers and pushing up rents.

Housing costs jumped last month, as higher prices continued to constrain aspiring home buyers and push up the demand for apartment and home leases.

Rent costs rose 0.4 percent in December, according to government data released on Wednesday, helping to drive the Consumer Price Index up 7 percent in the year through December.

Rent, along with “owners’ equivalent rent” — a measure that tries to put a price on how much homeowners would pay for housing if they hadn’t purchased their home — make up about a third of the index, so...