Research
Working Papers
Cointegration-Driven Scale for Multidimensional Diffusions: An Extension of Feller’s Theory to Multidimensional Diffusions
New
Abstract
We resolve the non-uniqueness problem preventing extension of one-dimensional scale functions to multidimensional diffusions by exploiting cointegration structure. For a diffusion $X_t \in \R^d$ with cointegrating matrix $\beta \in \R^{r \times d}$ such that $Y_t = \beta X_t$ is ergodic with invariant density $\pi_Y(y) \propto e^{-U(y)}$, we prove that $h(x) = e^{-U(\beta x)}$ is the unique (up to constants) positive function that: (i) depends only on $\beta x$, (ii) yields a symmetric $h$-transformed generator along cointegrated directions, and (iii) admits the invariant density interpretation. This cointegration-scale transform enables exact exit probability formulas, sharp mean exit time bounds via Dirichlet forms, and complete boundary classification through Foster-Lyapunov criteria. We prove that no canonical extension exists to trending directions, establishing the fundamental limits of scale theory in multiple dimensions.
Countervailing Vertical Contracting
with Joseph Basford and Daniele Condorelli
Abstract
We study a setting in which a privately informed supplier must secure a publicly observed license from a monopolist before selling to a private-value buyer. Licensing fees, tied to subsequent sale terms, optimally take the form of royalties which impose penalties at low prices and which may be negative at high prices to enforce price-maintenance and screen the buyer, yielding allocations equivalent to second-degree price discrimination. Licensing’s welfare impact is ambiguous — it can reduce buyer-driven deadweight loss but introduce supplier inefficiencies. Applied to exclusive-dealing, our model shows such contracts can enhance aggregate welfare, contrary to Aghion & Bolton (1987).
Slides
Hierarchies of Belief Types in Games with Many Players
with Peter Hammond
Abstract
Following Mertens & Zamir (1985) and Brandenburger & Dekel (1993), universal type spaces have been constructed for each player in a general n-player game of incomplete information, where types encode hierarchies of beliefs. We introduce an analogous construction for games with a continuum of players indexed by the Lebesgue unit interval. For each player, belief types correspond to joint probability measures over triples consisting of (i) the player’s label, (ii) a non-belief type given by a cardinal equivalence class of payoff functions, and (iii) a recursively constructed hierarchy of belief types. The marginal over player labels must equal Lebesgue measure.
Publications
Hierarchies of Beliefs for Many Player Games
Abstract
Mertens and Zamir (1985) first provided the universal type space construction for finite player games of incomplete information with a compact state space. Brandenburger and Dekel (1993) complemented it for a Polish state space. This paper extends the construction of Brandenburger and Dekel (1993) to games with infinitely many players for Harsanyi’s notion of a type. The extension is formulated by randomly drawing a countably infinite set of actual players from a continuum of potential players, represented by their labels in [0,1]. The random distribution of the countably infinite set of actual players almost surely converges to Lebesgue due to the Glivenko–Cantelli theorem. A coherent type is shown to induce beliefs over other player’s types and common knowledge of coherency closes the model of beliefs. Implications of dropping the Polish space assumption are discussed and an informal extension to measurable spaces is provided for future work. The formalisation provided here allows Harsanyi’s notion of Type.
Estimating How Much Children Work: Questionnaires vs Time Use Diaries
Abstract
Current estimates of child labour often rely on questions such as, “How many hours did you work last week?” While biases in adult self-reports are well-documented in high-income countries, there is limited evidence on the accuracy of children's responses in low- and middle-income countries (LMICs). Using data from nine LMICs, including China and India, this paper shows that time diaries report more than twice as many work hours as standard questionnaires. This discrepancy suggests that current estimates may significantly understate child labour. Moreover, certain forms of work—such as collecting water or firewood—appear to contribute to these measurement gaps.
Data
- Multinational Time Use Study Database, Version 11 — dataset (2023). Dataset
Conferences
- Countervailing Vertical Contracting* — Warwick Micro Theory Work in Progress, June 2025.
- Estimating How Much Children Work: Questionnaires vs Time Use Diaries* — International Association for Time Use Research Conference, Tokyo, Nov-2023.
- Evaluating the Twin Deficit Hypothesis in the United Kingdom
- International Conference of Undergraduate Research, September 2022,
- British Conference of Undergraduate Research, April 2023.
- International Conference of Undergraduate Research, September 2022,
- Exploring the Relation Between Children and Gender Pay gap in the United Kingdom — International Conference of Undergraduate Research, Sept. 27-29, 2021.
* presented by co-authors