Mature Economy
  
See: Mature Industry. Same idea. Just with a country.
Like...the U.S. is a pretty mature economy. We still grow, albeit not very fast. Engliand's economy is an even slower grower. And Italy's economy is shrinking. Think: beyond mature...like, one-foot-in-the-grave mature.
The opposite would be something like an emerging market, like China was 20 years ago, or like Indonesia probably is today, if their government's corruption doesn't spoil the party for its people.