This paper studies optimal bundling of products with inter-dependent values. I show that, under some conditions, a firm optimally chooses to sell only the full bundle of a given set of products if and only if the optimal sales volume of the full bundle is larger than the optimal sales volume for any smaller bundle. I then provide an interpretation of this characterization based on (i) the magnitude of the variation across consumers in how complementary they find different products, and (ii) how this variation correlates with price sensitivity.
Reclassification risk is a major concern in health insurance where contracts are typically one year in length but health shocks often persist for much longer. We theoretically characterize optimal long-term insurance contracts with one-sided commitment, and use our characterization to provide a simple computation algorithm for computing optimal contracts from primitives. We apply this method to derive empirically-based optimal long-term health insurance contracts using all-payers claims data from Utah, and then evaluate the potential welfare performance of these contracts. We find that optimal long-term health insurance contracts that start at age 25 can eliminate over 94% of the welfare loss from reclassification risk for individuals who arrive on the market in good health, but are of little benefit to the worst age-25 health risks. As a result, their ex ante value depends significantly on whether pre-age-25 health risk is otherwise insured. Their value also depends on individuals’ expected income growth.
This paper develops a strategy with simple implementation and limited data requirements to identify spatial distortion of supply from demand -or, equivalently, unequal access to supply among regions- in transportation markets. We apply our method to ride-level, multi-platform data from New York City (NYC) and show that for smaller rideshare platforms, supply tends to be disproportionately concentrated in more densely populated areas. We also develop a theoretical model to argue that a smaller platform size, all else being equal, distorts the supply of drivers toward more densely populated areas due to network effects. Motivated by this, we estimate a minimum required platform size to avoid geographical supply distortions, which informs the current policy debate in NYC around whether ridesharing platforms should be downsized. We nd the minimum required size to be approximately 3.5M rides/month for NYC, implying that downsizing Lyft or Via-but not Uber{can increase geographical inequity.
Reclassification risk is a major concern in health insurance where contracts are typically one year in length but health shocks often persist for much longer. While most health systems with private insurers emphasize short-run contracts paired with substantial pricing regulations to reduce reclassification risk, long-term contracts with one-sided insurer commitment have significant potential to reduce reclassification risk without the negative side effects of price regulation, such as adverse selection. In this paper, we theoretically characterize optimal long-term insurance contracts with one-sided commitment, extending prior models of this form in several key directions that are important for studying health insurance markets. We leverage this characterization to provide a simple algorithm for computing optimal contracts from primitives. We estimate key market fundamentals using data on all under-65 privately insured consumers in Utah and pair these estimates with our model to study comparative statics related to contract design and welfare. We find that the welfare value of a system that effectively implements these long-term contracts depends crucially on (i) the degree of public insurance pre-system health risk (ii) the distribution of expected lifetime income gradients in the population (iii) the stochastic process governing life-cycle health shocks (iv) the extent of consumer switching costs and (v) the degree of consumer myopia.
This paper studies the e ects of economies of density in transportation markets, focusing on ridesharing. Our theoretical model predicts that (i) economies of density skew the supply of drivers away from less dense regions, (ii) the skew will be more pronounced for smaller platforms, and (iii) rideshare platforms do not nd this skew ecient and thus use prices and wages to mitigate (but not eliminate) it. We then develop a general empirical strategy with simple implementation and limited data requirements to test for spatial skew of supply from demand. Applying our method to ride-level, multi-platform data from New York City (NYC), we indeed nd evidence for a skew of supply toward busier areas, especially for smaller platforms. We discuss the implications of our analysis for business strategy (e.g., surge pricing) and public policy (e.g., consequences of breaking up or downsizing a rideshare platform).
Reclassification risk is a major concern in health insurance where contracts are typically one year in length but health shocks often persist for much longer. We use rich individual-level medical information from the Utah all-payer claims database to empirically study one possible solution: long-term insurance contracts. We characterize optimal long-term contracts with one-sided commitment theoretically, derive the contracts that are optimal for consumers in Utah, and assess the welfare level that a full implementation of these contracts could achieve relative to several key benchmarks. We find that dynamic contracts perform very well for the majority of the population, for example, eliminating over 94% of the welfare loss from reclassification risk for individuals who arrive on the market at age 25 in good health. However, dynamic contracts instead provide very little benefit to the worst pre-age-25 health risks. Their value is also substantially lower for consumers whose income growth with age is relatively high. With pre-age-25 insurance in place, consumers with flat net income prefer dynamic contracts to an ACA-like environment, but consumers with steeper income profiles prefer the ACA-like environment. Overall, we show that there are scenarios in which dynamic contracts can provide substantial welfare benefits, but that complementary policies are crucial for unlocking these benefits.