What are the welfare effects of a policy that facilitates for insurance customers to privately and covertly learn about their accident risks? We endogenize the information structure in Stiglitz's classic monopoly insurance model. We first show that his results are robust: For a small information acquisition cost c, the consumer gathers information and the optimal contracts are close to the ones in the Stiglitz model. If c is so low that the consumer already gathers information (c < c), both insurer and consumer benefit from a policy that reduces c further. For
c > c, marginally reducing c hurts the insurer and weakly benefits the consumer.
Paradoxically, a reduction in c that is "successful," meaning that the consumer
gathers information after the reduction but not before it, can hurt both parties.
The reasons for this are that, after the reduction, (i) the cost is actually incurred
and (ii) the contracts can be more distorted.