I did a quick search and the library service in England costs 840 million per year for an entire country.

Or roughly $12/resident/year. Admittedly, England has also been cutting deeply into their social services budgets, with library spending falling by over a quarter over the last decade. The UK has lost over a fifth of its public libraries during this time and continues to cut deeper and deeper into these budgets. So its per-capita cheap, but also a popular target for enormous budget cuts.

Compare that to the city's police budgets and you'll find a yawning gap that's only growing larger with time.

It's about a 13% cut in funding. Between the aprox. 207 libraries it cuts almost 300k from each leaving about 1.9 million per branch. These numbers are inflated in straight division like that because some of the libraries are not normal libraries, but research libraries which would have higher budgets so maybe around 1 - 1.2 million per regular branch. If you think that sounds like a lot because 1.2 million sounds like a big number, you've obviously never run a business.

I'm not sure this is the big gotcha you think it is. To serve a large population in a big relatively small area, you can easily achieve your goals with fewer, larger locations. This will allow a greater selection per location, which reduces the odds you will have to wait for the desired product to be shipped in. Moreover, land isn't cheap in New York, whereas it may well be in smaller locales in England. Either way, a reasonable metric is cost per capita to provide services for a region, and England is only about 2/3 as expensive as New York. I suspect the cost of living is higher in New York than England, and it will certainly have an impact of the relative budgets of the two organizations. And we haven't talked about the climate control requirements in England vs. New York because, frankly, I'm not too clear about the relative climate differences to say whether that's a significant issue in the first place.