Not really, it's more like private lending. The hospital has no ability to compete in a competative market, it's still controlled by the gov't and the revenues and staff are essentially paid for by the gov't. The private sector is providing the facility. So it's more of a really kind of odd lease program for the hospital. That does generate a LITTLE economic activity but very little in comparison to the actual money being spent.There are several PPP hospitals around so that sort of makes the government a partner in a for profit enterprise.
I mean ALL gov't activity has SOME impact on the private sector - they buy pens made by private companies, paper and computers made by private companies etc. But as a percent of the actual expenditures it's negligible.
Here's a good way to measure it - if the gov't money was taken away and nothing else changed, would the 'business' survive? in the case of PPP hospitals no, they would die. They wouldn't be allowed to charge for services directly under our laws and with no gov't money they'd be shut down. So it's not really a 'free enterprise' kind of business. It's not regulated by the free market. It requires gov't to drain money from the economic system in order to provide services.
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A capitalist money system is based on the idea that all income is a result of a fair exchange of services and goods for other services or goods (including currency). That tends to level inflation because the more money being made means the more goods and services that were produced for people to buy. Gov't provided services step outside of that model but rely on that model for their money.
And that's not necessarily automatically 'evil' or anything. But - if the private sector is shrinking or slowing down in comparison to the gov't then that's a bad thing, and the gov't tends to be a bit of an anchor for the private sector as it draws off money and resources from that system without replacing it with equal value.