Universität Bonn

Frontiers in Economics and Mathematics

Selected Topics Lecture Course

October 20, 2023 - January 12, 2024

Venue: Lipschitz-Saal, Endenicher Allee 60, Bonn and Hörsaal F, Juridicum, Bonn

Organizers: Jürgen Dölz, Martin Rumpf

Description: This lecture course will present recent developments in economics linked to modern methods in mathematics.It is addressing in particular students and doctoral students both in economics and in mathematics with bundles of 90-minute lectures combining the economics and the mathematics perspective.

Schedule

    Statistics in Economics and Uncertainty Quantification

    20.10.23 14 c.t. Christoph Breunig Lipschitz Saal, Endenicher Allee 60
    27.10.23 14 c.t. Jürgen Dölz Hörsaal F, Juridicum

    The Hamilton-Jacobi-Bellman Equation and Viscosity Solutions in Microeconomic Applications

    10.11.23 14 c.t. Tim Laux Hörsaal F, Juridicum
    17.11.23 14 c.t. Sven Rady             Lipschitz Saal, Endenicher Allee 60

    The Optimal Transport Problem, its Solutions, Numerical Approximation, and an Application to Mechanism Design

    24.11.23 14 c.t. Peter Gladbach    Hörsaal F, Juridicum
    01.12.23 14 c.t. Eva Kopfer Hörsaal F, Juridicum
    08.12.23 14 c.t. Andreas Kleiner Lipschitz Saal, Endenicher Allee 60
    15.12.23 14 c.t. Martin Rumpf Hörsaal F, Juridicum

    Extreme Points and Optimization under Majorization Constraints

    12.01.24 14 c.t. Benny Moldovanu Lipschitz Saal, Endenicher Allee 60

    Statistics in Economics and Uncertainty Quantification

    Uncertainty quantification refers to the characterization and computation of statistical quantities of interest that depend on random parameters. In the presence of infinite-dimensional parameters, we make the following contributions: In the first lecture we propose a Bayesian procedure to conduct inference on the average treatment effect. Our novel procedure enhances the robustness of the traditional Bayesian approach while imposing reduced regularity restrictions. In the second lecture we discuss how the prohibitive computational cost of the classical sample covariance estimator can be accelerated through the use of matrix compression techniques and efficient multilevel sampling strategies. Numerical examples which estimate covariance matrices with tens of billions of entries are presented.

    The Hamilton-Jacobi-Bellman Equation and Viscosity Solutions in Microeconomic Applications

    The value function in many optimization problems solves a nonlinear degenerate elliptic partial differential equation, called the Hamilton-Jacobi-Bellman (HJB) equation. Often, this equation has many solutions, but the theory of viscosity solutions allows us to single out the relevant one for applications. In the first lecture, we will show how the HJB equation arises from optimal control problems, derive a comparison principle, define viscosity solutions, and show their well-posedness. We will first work with a simple first-order example and build our way up to the degenerately elliptic case. In the second lecture, we will consider selected applications of the viscosity solution approach in continuous-time decision problems and games. Leading examples are Bayesian learning and (strategic) information acquisition.

    The Optimal Transport Problem, its Solutions, Numerical Approximation, and an Application to Mechanism Design

    In the first lecture, we introduce the discrete optimal transport problem and its dual. We show how to find a solution using the transportation simplex method.Finally, we briefly touch on the history of the optimal transport problem. The second lecture will provide the optimal transport problem between general probability measures. We characterize minimizers and establish the dual problem. In the third lecture, we apply these tools to mechanism design and study a multi-product monopoly problem. The fourth lecture will present the entropy regularization of optimal transport and derive from that effective numerical schemes.

    Extreme Points and Optimization under Majorization Constraints

    Majorization was introduced by Hardy, Littlewood and Polya and is related to the convex stochastic order from probability theory and to Choquet's theory from convex analysis. We characterize the set of extreme points of monotonic functions that are either majorized by a given function f or themselves majorize f. These extreme points play a crucial role in many economic design problems because these problems can be reduced to the optimization of linear functionals under majorization constraints. Our main results show that each extreme point is uniquely characterized by a countable collection of intervals. Outside these intervals the extreme point equals the original function f and inside the function is constant. Further consistency conditions need to be satisfied pinning down the value of an extreme point in each interval where it is constant. We apply these insights to a varied set of economic problems: equivalence and optimality of mechanisms for auctions and (matching) contests, Bayesian persuasion, optimal delegation, and decision making under uncertainty. In several dimensions we show that extreme points are characterized by regular polyhedral subdivisions or by power (Laguerre) diagrams.