Here is where inequality comes in--if when Chen and co-authors look at children born to advantaged individuals (meaning married, college-educated and white) in the US, they survive at the same rates as their counterparts in Austria and Finland. But the trio find that children of disadvantaged parents in the US have much lower survival rates than children of disadvantaged parents in the other countries. This may well be because Europe's safety nets make the disadvantaged less disadvantaged.
The NBER paper appears to be carefully done. But, contrary to Green, it does not show that inequality is bad for babies. Here is the authors' policy conclusion:
Instead, the facts documented here suggest that, in general, if the goal is to reduce infant mortality, then policy attention should focus on either preventing preterm births or on reducing postneonatal mortality. Although the former has received a tremendous amount of policy focus (MacDorman and Mathews (2009); Wilcox et al. (1995)), the latter has - to the best of our knowledge - received very little attention. Our estimates suggest that decreasing postneonatal mortality in the US to the level in Austria would lower US death rates by around 1 death per 1000. Applying a standard value of a statistical life of US$7 million, this suggests on a standard cost-benefit test it would be worth spending up to $7000 per infant to achieve this gain. If policies were able to focus on individuals of lower socioeconomic status - given our estimates that advantaged groups do as well in the US as elsewhere - even higher levels of spending per mother targeted might be justified.
Identifying particular policies which could be effective is beyond the scope of this paper and is an area that deserves more research attention. One policy worth mentioning is home nurse visits. Both Finland and Austria, along with much of the rest of Europe, have policies which bring nurses or other health professionals to visit parents and infants at home. These visits combine well-baby checkups with caregiver advice and support. While such small scale programs exist in the US, they are far from universal, although provisions of the Affordable Care Act will expand them to some extent. Randomized evaluations of such programs in the US have shown evidence of mortality reductions, notably from causes of death we identify as important such as SIDS and accidents (Olds et al. 2007).
Whatever you think about the wisdom or ethics of such policies, you need to distinguish such policies from the issue of inequality. Imagine that the government reduced inequality with a higher tax on high-income people, but then spent none of the additional revenues in the way that the authors suggest above. Effect on inequality: a reduction. Effect on infant mortality: zero.
Or consider this policy: the government does not add any resources to help lower-income women but, instead, takes a lot of resources from higher-income white women in their child-bearing years. Effect on inequality: a reduction. Effect on infant mortality: possibly an increase.
Posing the policy issue in terms of inequality simply confuses the issue.