A Characterization for Optimal Bundling of Products with Inter-dependent Values

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

Eliminating Latent Discrimination: Train Then Mask

How can we control for latent discrimination in predictive models? How can we provably remove it? Such questions are at the heart of algorithmic fairness and its impacts on society. In this paper, we define a new operational fairness criteria, inspired by the well-understood notion of omitted variable-bias in statistics and econometrics. Our notion of fairness effectively controls for sensitive features and provides diagnostics for deviations from fair decision making. We then establish analytical and algorithmic results about the existence of a fair classifier in the context of supervised learning. Our results readily imply a simple, but rather counter-intuitive, strategy for eliminating latent discrimination. In order to prevent other features proxying for sensitive features, we need to include sensitive features in the training phase, but exclude them in the test/evaluation phase while controlling for their effects. We evaluate the performance of our algorithm on several real-world datasets and show how fairness for these datasets can be improved with a very small loss in accuracy

An Empirical Analysis of Optimal Nonlinear Pricing

In continuous-choice settings, consumers decide not only on whether to purchase a product, but also on how much to purchase. Thus, firms optimize a full price schedule rather than a single price point. This paper provides a methodology to empirically estimate the optimal schedule under multi-dimensional consumer heterogeneity. We apply our method to novel data from an educational-services firm that contains purchase-size information not only for deals that materialized, but also for potential deals that eventually failed. We show that this data, combined with identifying assumptions, helps infer how price sensitivity varies with "customer size". Using our estimated model, we show that the optimal second-degree price discrimination (i.e., optimal nonlinear tariff) improves the firm's profit upon linear pricing by at least 5.5%. That said, this second-degree price discrimination scheme only recovers 5.1% of the gap between the profitability of linear pricing and that of infeasible first degree price discrimination. We also conduct several further counterfactual analyses (i) empirically quantifying the magnitude by which incentive-compatibility constraints impact the optimal pricing and profits, (ii) comparing the role of demand- v.s. cost-side factors in shaping the optimal price schedule, and (iii) studying the implications of fixed fees for the optimal contract and profitability

Spatial Distribution of Supply and the Role of Market Thickness: Theory and Evidence from Ride Sharing

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

Risk Aversion and Double Marginalization

In vertical markets, eliminating double marginalization with a two-part tariff may not be possible due to downstream firms\u27 risk aversion. When demand is uncertain, contracts with large fixed fees expose the downstream rm to more risk than contracts that are more reliant on variable fees. In equilibrium, contracts may thus rely on variable fees, giving rise to double marginalization. Counterintuitively, we show that increased demand risk or risk aversion can actually mitigate double marginalization. We also characterize several sufficient conditions under which increased risk or risk aversion does exacerbate double marginalization. We conclude with an application to merger analysis

Spatial Distribution of Supply and the Role of Market Thickness: Theory and Evidence from Ride Sharing

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)

Eliminating Latent Discrimination: Train Then Mask

How can we control for latent discrimination in predictive models? How can we provably remove it? Such questions are at the heart of algorithmic fairness and its impacts on society. In this paper, we define a new operational fairness criteria, inspired by the well-understood notion of omitted variable-bias in statistics and econometrics. Our notion of fairness effectively controls for sensitive features and provides diagnostics for deviations from fair decision making. We then establish analytical and algorithmic results about the existence of a fair classifier in the context of supervised learning. Our results readily imply a simple, but rather counter-intuitive, strategy for eliminating latent discrimination. In order to prevent other features proxying for sensitive features, we need to include sensitive features in the training phase, but exclude them in the test/evaluation phase while controlling for their effects. We evaluate the performance of our algorithm on several real-world datasets and show how fairness for these datasets can be improved with a very small loss in accuracy

Optimal Long-Term Health Insurance Contracts: Characterization, Computation, and Welfare Effects

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

The Welfare Effects of Long-Term Health Insurance Contracts

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