Vladimir Putin had been named the "world's most powerful person" last year by Forbes magazine well before he annexed Crimea. The land grab added to the string of geopolitical victories credited to the Russian leader—including his rescue of Syria's Bashar Assad in the chemical-weapons standoff and the safe harbor he gave to the American secrets-spiller Edward Snowden. But Mr. Putin's real power base, the economy, is crumbling.

Russia's economic growth rate has plummeted from the 7% average annual pace of the last decade...