There is a new NBER Working Paper by Mark Duggan, Amanda Starc, and Boris Vabson, here is the abstract, with the bold emphasis added by me:
Governments contract with private firms to provide a wide range of services. While a large body of previous work has estimated the effects of that contracting, surprisingly little has investigated how those effects vary with the generosity of the contract. In this paper we examine this issue in the Medicare Advantage (MA) program, through which the federal government contracts with private insurers to coordinate and finance health care for more than 15 million Medicare recipients. To do this, we exploit a substantial policy-induced increase in MA reimbursement in metropolitan areas with a population of 250 thousand or more relative to MSAs just below this threshold. Our results demonstrate that the additional reimbursement leads more private firms to enter this market and to an increase in the share of Medicare recipients enrolled in MA plans. Our findings also reveal that only about one-fifth of the additional reimbursement is passed through to consumers in the form of better coverage. A somewhat larger share accrues to private insurers in the form of higher profits and we find suggestive evidence of a large impact on advertising expenditures. Our results have implications for a key feature of the Affordable Care Act that will reduce reimbursement to MA plans by $156 billion from 2013 to 2022.
There is an ungated version here (pdf).