WASHINGTON—When House Speaker Paul Ryan leaves Congress, the Republican Party will lose its most influential advocate for changes to Social Security, Medicare and Medicaid.

As Budget committee chairman, a vice-presidential running mate to Mitt Romney in 2012, Ways and Means chairman and finally House speaker, Mr. Ryan had pressed for curbs on federal spending on the three programs. These retirement and health care programs are popular with voters, but their costs are rising faster than the current funding to pay for them.

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