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Revert "Tom's Dec 8 tweaks to composite matching lecture"
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lectures/_static/quant-econ.bib

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Note: Extended Information (like abstracts, doi, url's etc.) can be found in quant-econ-extendedinfo.bib file in _static/
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###
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@techreport{boerma2023composite,
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title={Composite sorting},
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author={Boerma, Job and Tsyvinski, Aleh and Wang, Ruodu and Zhang, Zhenyuan},
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year={2023},
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institution={National Bureau of Economic Research}
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}
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@article{delon2011minimum,
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title={Minimum-weight perfect matching for non-intrinsic distances on the line},
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author={Delon, Julie and Salomon, Julien and Sobolevski, Andrei},
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journal={arXiv preprint arXiv:1102.1558},
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year={2011}
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}
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@article{sargent1973stability,
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title={The stability of models of money and growth with perfect foresight},
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author={Sargent, Thomas J and Wallace, Neil},

lectures/match_transport.md

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## Introduction
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This notebook presents Python code for solving **composite sorting** problems of the kind
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studied in *Composite Sorting* by Job Boerma, Aleh Tsyvinski, Ruodo Wang,
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and Zhenyuan Zhang {cite}`boerma2023composite`.
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studied in the August 2023 paper *Composite Sorting* by Job Boerma, Aleh Tsyvinski, Ruodo Wang,
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and Zhenyuan Zhang.
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+++ {"user_expressions": []}
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@@ -487,11 +487,7 @@ Suppose that agents $i$ of type $z_i$ and $j$ of type $z_j$, with $z_i < z_j,$ a
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Then there is an equal number of agents from each side in $\{i+1, \dots, j-1\},$ if this set is not empty.
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Indeed, if this were not the case, then some agent $k \in \{i+1,j-1\}$ would be matched with some agent $\ell$ with $\ell \notin \{i,\dots, j\},$ i.e., there would be types
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$$
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z_i < z_k < z_j < z_\ell
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$$
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$$z_i < z_k < z_j < z_\ell$$
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with matches $(z_i,z_j)$ and $(z_k, z_\ell),$ violating the no intersecting pairs property.
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We conclude that we can define a binary relation on $[N]$ such that $i \sim j$ if there is an equal number of agents of each side in $\{i,i+1,\dots, j\}$ (or if this set is empty).
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We will now present two key results in the context of OT with concave type costs.
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We refer {cite}`boerma2023composite` and {\cite}`delon2011minimum` for proofs.
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We refer to the original papers XXXX (can cite both Boerma et al (2023) and [Delon, Salomon, Sobolevski (2011)](https://link.springer.com/article/10.1007/s10958-012-0714-6))
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XXXX for proofs.
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Consider the problem faced within a layer, i.e., types from $Y \sqcup X$
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The dual solution $(u,v)$ of $W_D$ can be interpreted as equilibrium utilities of the agents, which include the individual specific amenities and equilibrium shadow costs.
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{cite}`boerma2023composite` propose an efficient method to compute the dual variables from the optimal matching (primal solution) in the case of composite sorting.
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The authors propose an efficient method to compute the dual variables from the optimal matching (primal solution) in the case of composite sorting.
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Let's generate an instance and compute the optimal matching.
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+++ {"user_expressions": []}
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We now replicate the empirical analysis carried out by {cite}`boerma2023composite`.
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We now replicate the empirical analysis carried out by the authors.
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The dataset is obtained from the American Community Survey and contains individual level data on income, age and occupation.
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