## Working papers

### Regulating vertically structured platforms (with Ellen Muir) | [PDF]

Novermber 2021

We study a platform that sells productive inputs (such as e-commerce and distribution services) to a fringe of producers in an upstream market, while also selling its own output in the corresponding downstream market. The platform faces a tradeoff: any output that it sells downstream increases competition with the fringe of producers and lowers the downstream price, which in turn reduces demand for the platform's productive inputs and decreases upstream revenue. Adopting a mechanism design approach, we characterize the optimal menu of contracts the platform offers in the upstream market. These contracts involve price discrimination in the form of nonlinear pricing and quantity discounts. If the platform is a monopoly in the upstream market, then we show that the tradeoff always resolves in favor of consumers and at the expense of producers. However, if the platform faces competition in the upstream market, then it has an incentive to undermine this competition by engaging in activities, such as "killer" acquisitions and exclusive dealing, that harm both consumers and producers.

### Robust bounds for welfare analysis (with Shosh Vasserman) | [PDF]

September 2021

Economists routinely make functional form assumptions about consumer demand to obtain welfare
estimates—often for convenience, tractability, or both. How sensitive are welfare estimates to these
assumptions? In this paper, we answer this question by providing bounds on welfare that hold for
families of demand curves commonly considered in different literatures. We show that typical functional
forms—such as linear, exponential and CES demand—are extremal in different families: they yield either
the highest or lowest welfare estimate among all demand curves in those families. To illustrate the
flexibility of our approach, we apply our results to the welfare analysis of trade tariffs, income
taxation, and surge pricing.

### Optimal redistribution through public provision of private goods | [PDF]

August 2021

**Accepted for presentation at EC'21 **

How does a private market influence the optimal design of a public program? In this paper, I study a
designer who has preferences over how a public option and a private good are allocated. However, she
can design only the public option. Her design affects the distribution of consumers who purchase the
private good—and hence equilibrium outcomes. I characterize the optimal mechanism and show how it can be
computed explicitly. I derive comparative statics on the value of the public option and show that the
optimal mechanism generally rations the public option. Finally, I examine implications on the optimal
design when the designer can intervene in the private market or introduce an individual mandate.

### Fixed-price approximations in bilateral trade (with Francisco Pernice and Jan Vondrák) | [PDF]

August 2021

**Accepted for presentation at SODA'22 **

We consider the bilateral trade problem, in which two agents trade a single indivisible item. It is known that the only dominant-strategy truthful mechanism is the fixed-price mechanism: given commonly known distributions of the buyer's value $B$ and the seller's value $S$, a price $p$ is offered to both agents and trade occurs if $S \leq p \leq B$. The objective is to maximize either expected welfare, $\mathbb{E}\!\left[S + (B-S) \mathbf{1}_{S \leq p \leq B}\right]$, or expected gains from trade, $\mathbb{E}\!\left[(B-S) \mathbf{1}_{S \leq p \leq B}\right]$.

We improve the approximation ratios for several welfare maximization variants of this problem. When the agents' distributions are identical, we show that the optimal approximation ratio for welfare is $\left(2+\sqrt{2}\right)/4$. With just one prior sample from the common distribution, we show that a $3/4$-approximation to welfare is achievable. When agents' distributions are not required to be identical, we show that a previously best-known $(1-1/e)$-approximation can be strictly improved, but $1-1/e$ is optimal if only the seller's distribution is known.

### Markets for goods with externalities | [PDF]

April 2020

I consider the welfare and profit maximization problems in markets with externalities. I show that when
externalities depend generally on allocation, a Pigouvian tax is often suboptimal. Instead, the optimal
mechanism has a simple form: a finite menu of rationing options with corresponding prices. I derive
sufficient conditions for a single price to be optimal. I show that a monopolist may ration *less*
relative to a social planner when externalities are present, in contrast to the standard intuition that
non-competitive pricing is indicative of market power. My characterization of optimal mechanisms uses a
new methodological tool—the *constrained maximum principle*—which leverages the combined mathematical
theorems of Bauer (1958) and Szapiel (1975). This tool generalizes the concavification technique of Aumann
and Maschler (1995) and Kamenica and Gentzkow (2011), and has broad applications in economics.

## Older working papers

### Fixed-price approximations to optimal efficiency in bilateral trade (with Jan Vondrák) | [PDF]

September 2019

* Partially superseded by "Fixed-price approximations in bilateral trade" (with Francisco Pernice and Jan Vondrák) *

This paper studies fixed-price mechanisms in bilateral trade with ex ante symmetric agents. We show that the optimal price is particularly simple: it is exactly equal to the mean of the agents’ distribution. The optimal price guarantees a worst-case performance of at least 1/2 of the first-best gains from trade, regardless of the agents’ distribution. We also show that the worst-case performance improves as the number of agents increases, and is robust to various extensions. Our results offer an explanation for the widespread use of fixed-price mechanisms for size discovery, such as in workup mechanisms and dark pools.