Knowledge that Transforms

We investigate how formularies used by pharmacy benefit managers (PBMs) affect equilibrium manufacturer rebates for branded drugs through tiering and exclusion. We develop a theoretical model of multidimensional contracting in which a PBM negotiates with drug manufacturers over menus of formulary‐contingent rebates and chooses a formulary. We then estimate consumer demand responses to tier placement for statins using claims data from Princeton University, a large employer contracting with a single PBM to offer prescription drug coverage to its employees. Combining the theoretical model with demand estimates and observed list prices, we quantify how allowing for differential tier placement and exclusion affect equilibrium rebates. Our predictions are consistent with available aggregate rebate data, and we find that allowing a PBM to place branded drugs on preferred‐ and non‐preferred tiers can substantially increase negotiated rebate payments.

We develop an optimal policy assignment rule that integrates two distinctive approaches commonly used in economics—targeting by observables and targeting through self‐selection . Our method can be used with experimental or quasi‐experimental data to identify who should be treated, be untreated, and self‐select to achieve a policymaker's objective. Applying this method to a randomized controlled trial on a residential energy rebate program, we find that targeting that optimally exploits both observable data and self‐selection outperforms conventional targeting. We use the Local Average Treatment Effect (LATE) framework (Imbens and Angrist (1994)) to investigate the mechanism in our approach. By estimating several key LATEs based on the random variation created by our experiment, we demonstrate how our method allows policymakers to identify whose self‐selection would be valuable and harmful to social welfare.

This paper studies how the risk of hold‐up affects procurement. I use data on the universe of solar power auctions in India. The Indian context allows clean estimates of counterparty risk, because solar plants set up in the same states, by the same firms, are procured in auctions intermediated by either risky states themselves or the trusted central government. I find that the counterparty risk of an average state increases solar prices by 10%. This risk premium sharply reduces investment, because demand for green energy is elastic. Contract intermediation by the central government eliminates the counterparty risk premium.

A buyer and seller bargain in continuous time over a good. Bargainers can be rational or committed to some fixed price. A rational buyer has a private value and outside option. If the set of buyer values and commitment types is rich and the probability of commitment vanishes, outcomes are partially consistent with the Coase conjecture: the seller chooses a price below the maximum of the lowest outside option and half the lowest value; the buyer immediately accepts or takes his outside option.

We cast the problem of communicating scientific uncertainty as one of reporting a posterior distribution on an unknown parameter to an audience of Bayesian decision‐makers. We establish novel bounds on the audience's regret when the analyst reports an approximation to a posterior that the audience treats as exact. Under a palatable restriction on the audience's decision problems, the bounds take an especially convenient form. Under a further restriction on the audience's priors, a bootstrap distribution can be used as a stand‐in posterior. We propose a practical recipe for checking whether a conventional statistical report (say, a normal parameterized by a point estimate and standard error) is a good approximation, and for improving the report if it is not. We illustrate our proposals using the articles in the 2021 American Economic Review that use a bootstrap for inference.

Gaussian empirical Bayes methods usually maintain a precision independence assumption: The unknown parameters of interest are independent from the known standard errors of the estimates. This assumption is often theoretically questionable and empirically rejected. This paper proposes to model the conditional distribution of the parameter given the standard errors as a flexibly parameterized location‐scale family of distributions, leading to a family of methods that we call close . The close framework unifies and generalizes several proposals under precision dependence. We argue that the most flexible member of the close family is a minimalist and computationally efficient default for accounting for precision dependence. We analyze this method and show that it is competitive in terms of the regret of subsequent decision rules. Empirically, using close leads to sizable gains for selecting high‐mobility Census tracts.

Internal labor markets are increasingly important for matching workers to jobs within organizations. We present evidence from a randomized trial that compares matching workers to jobs using the deferred acceptance (DA) algorithm to the traditional manager‐directed matching process. Our setting is the U.S. Army's internal labor market, which matches over 14,000 officers to units annually. We find that DA reduces administrative burden and increases match quality as measured by reduced justified envy, increased truthful preference reporting, and officers' and units' preferences over their matches. The overall impact of DA on officer retention and performance in the two years after officers started their new jobs is limited by strategic preference coordination between officers and units. However, DA leads to significant improvements in officer retention and promotions in markets with inexperienced managers. Our findings suggest that cross‐market communication between agents in internal labor markets can attenuate the benefits of strategyproof matching algorithms.

Consider a repeated interaction where it is unknown which of various stage games will be played each period. This framework separates the basic logic of intertemporal incentives from the requirement that any given strategy profile yields a well‐defined payoff vector. A natural solution concept is ex post perfect equilibrium: strategies must form a subgame‐perfect equilibrium for any realization of the sequence of stage games. When there is one long‐run player and others are short‐run, and public randomization is available, we can adapt the standard recursive approach to determine the maximum feasible gap between reward and punishment for the long‐run player. This allows us to identify which actions can be played in equilibrium and, assuming perfect monitoring, to fully characterize what outcome paths can arise. With multiple long‐run players or no public randomization, the approach fails; a diagnostic of this failure is that optimal penal codes may no longer exist.

Shrinkage methods are frequently used to improve the precision of least squares estimators of fixed effects. However, widely used shrinkage estimators guarantee improved precision only under strong distributional assumptions. I develop an estimator for the fixed effects that obtains the best possible mean squared error within a class of shrinkage estimators. This class includes conventional shrinkage estimators and the optimality does not require distributional assumptions. The estimator has an intuitive form and is easy to implement. Moreover, the fixed effects are allowed to vary with time and to be serially correlated, in which case the shrinkage optimally incorporates the underlying correlation structure. I also provide a method to forecast fixed effects one period ahead in this setting.