## 'math+econ+code' masterclass

## on equilibrium transport

## and matching models in economics

## June 21-25, 2021

This intensive course, part of the 'math+econ+code' series, is focused on the computation of competitive equilibrium with substitutes, which is at the core of surge pricing engines and allocation mechanisms. It will investigate diverse applications such as surge pricing, matching platforms and network congestion. It provides a bridge between theory, empirics and computation and will introduce tools from economics, mathematical and computer science. Students will have the opportunity to write their own code (in Python), in the spirit of cooking lessons. Mathematical concepts (such as lattice programming, supermodularity, discrete convexity, Perron-Froebenius theory, etc.) will be taught on a needs basis while studying various economic models. The same is true of computational methods (such as tatonnement algorithms, asynchronous parallel computation, mathematical programming under equilibrium constraints, etc.). Hence there are no prerequisite other than the equivalent of a first-year graduate sequence in econ, applied mathematics or other quantitative disciplines.

This course is very demanding from students, but the learning rewards are high.

### Practical information

• The course will be taught online over five consecutive days, June 21-25, 2021, 8am - 12noon US Eastern time / 2pm-6pm Paris time.

• Additionally, five advanced lectures will be taught online on a monthly basis afterwards, July - November 2021. Precise days of the lectures will be announced later.

Applications are now closed

### Course outline

## Day 1: competitive equilibrium

## with gross substitutes

## Monday 5/21, 4h

Walrasian equilibrium and gross substitutes. Hedonic pricing. Jacobi algorithm.

## Day 2: matching models with transferable utility

## Tuesday 5/22, 4h

##

Optimization and equilibrium formulation. Duality. Computation by descent methods. Sinkhorn's algorithm.

## Day 3: matching models with imperfectly transferable utility

## Wednesday 5/23, 4h

Distance-to-frontier function, matching function equilibrium. Matching models with taxes. Collective models with public consumption.

## Day 4: matching models with non-transferable utility

## Thursday 5/24, 4h

##

Gale and Shapley’s deferred acceptance algorithm. Adachi’s algorithm and Tarski's fixed point theorem. Aggregate stable matchings.

## Day 5: equilibrium on networks

## Friday 5/25, 4h

##

Equilibrium pricing on networks. Scheduling, dynamic programming and Bellman-Ford’s algorithm.

## + 5 advanced lectures

## July to November, 2h each

Lectures on related advanced mathematical / computational topics to complement the masterclasses and solidify the theoretical knowledge.

### Advanced lectures

## Advanced lecture 1

Parallel computing

(with Loan Tricot)

Principles of parallelization; computing on the cloud; use of the numba library.

July 30 2021

2pm Paris time

## Advanced lecture 2

Special matrices

Z-, P- and M-matrices, diagonally dominant matrices, Stieltjes matrices, and Perron-Froebenius theory. M-maps and inverse isotonicy.

September 17, 2021

2pm Paris time

## Advanced lecture 3

Submodularity

Lattices, submodularity and Topkis’ theorem. Veinott's order. Monotone comparative statics. Strategic complementarities.

October 15 2021

2pm Paris time

## Advanced lecture 4

Gross substitutes

Polymatroids, exchangeability.

Lovasz extensions.

Discrete convex analysis, L / L# / M / M# convexity. Unified Gross Substitutes.

November 12 2021

2pm Paris time

## Advanced lecture 5

Matching with contracts

Kelso and Crawford’s deferred acceptance algorithm. Hatfield and Milgrom’s model of matching with contracts.

December 10 2021

2pm Paris time