The asterisk refers to my skepticism about happiness surveys. But other people cite and believe this data, so I'll play along and show the implications of the actual survey data.
Megan McArdle has a very nice article discussing the Danish model, cited by progressives like Bernie Sanders as something we should emulate. I agree with virtually everything in the article. Nonetheless, I'd like to add a few points that McArdle does not make in order to put this all into perspective.
One reason that progressives love Denmark is that it has a lot of social welfare spending, and its workers are highly productive. Even so, GDP/person in America is about 20% higher than in Denmark (adjusted for PPP.) The gap between America and Europe as a whole is much larger, as Denmark is one of Europe's richest countries.
Is there any evidence that this gap is due to Europe's welfare state? I'd say yes. AFAIK, Switzerland is the only western European country with less government spending than America (as a share of GDP.) It's also the only Western European country that is richer, apart from oil-rich Norway and a few tiny countries. High welfare spending leads to fewer hours worked, and this factor largely explains why Denmark is poorer than the US, and Europe as a whole is much poorer.
In my view it's a mistake for progressives to cite Denmark's high levels of happiness. If you compare the US to Europe as a whole, America is happier than more than 90% of Europe, weighted by population. Indeed the Netherlands is the largest European country that rates happier than America. And yet Europe as a whole has a much more extensive welfare state than America. And Europe as a whole is a far better comparison than Denmark---much more similar to America in terms of size and diversity of population. Unfortunately for progressives, Europe is much poorer and less happy than Denmark. Indeed the US is roughly 40% to 50% richer than the EU on a PPP basis. That's partly due to fewer hours worked and partly due to lower productivity.
Putting this all together there's really no evidence that social welfare spending makes Denmark a happy country. If they are happier (I'm agnostic on that point) it's more likely due to their culture---a high trust society. Big government probably makes countries poorer than otherwise, and does not seem to make them happier. That's the lesson I draw from looking at both Denmark and Europe as a whole.
PS. Ten years ago I wrote a long article on this topic, entitled "The Great Danes".