This dissertation consists of two essays in economics of industrial organization on private Medicare insurance markets. In the first chapter, together with Naoki Aizawa, we study the incentives for private health insurers to use advertising to attract healthy individuals in the market for private Medicare plans called Medicare Advantage (MA). Using data on the advertising expenditures of MA plans, individual-level and county-level MA enrollment, we document a large difference in an insurer's potential profits from healthy vs. unhealthy individuals. We then develop and estimate an equilibrium model of the MA market, which incorporates strategic advertising by insurers. Parameter estimates show that advertising has much larger effects on the demand of the healthy. We find that advertising accounts for 15% of the selection of healthier individuals into Medicare Advantage. In the second chapter, I study how consumer search frictions affect adverse selection and competition in the Medicare Supplement Insurance (Medigap) market. Using data on individual-level and market-level Medigap enrollment and claim costs, I estimate an equilibrium model of the Medigap market, which incorporates consumer search frictions and adverse selection. Parameter estimates show that search frictions are significant and that unhealthy consumers tend to face greater search costs. I find that the correlation between search costs and health risks reduces the extent of adverse selection by keeping unhealthy consumers with high search costs outside the market. In a counterfactual experiment where the government provides more information on available options, I find that the extent of adverse selection increases and that healthy individuals become worse off.