Kathryn R. Johnson and Claudia Persico have a new NBER working paper on exactly that topic:

Between 2000 and 2008, access to high-speed, broadband internet grew significantly in the United States, but there is debate on whether access to high-speed internet improves or harms wellbeing. We find that a ten percent increase in the proportion of county residents with access to broadband internet leads to a 1.01 percent reduction in the number of suicides in a county, as well as improvements in self-reported mental and physical health. We further find that this reduction in suicide deaths is likely due to economic improvements in counties that have access to broadband internet. Counties with increased access to broadband internet see reductions in poverty rate and unemployment rate. In addition, zip codes that gain access to broadband internet see increases in the numbers of employees and establishments. In addition, heterogeneity analysis indicates that the positive effects are concentrated in the working age population, those between 25 and 64 years old. This pattern is precisely what is predicted by the literature linking economic conditions to suicide risk.

It seems broadband is indeed (was indeed?) good for you.

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