Tom's Dec 8 tweaks to composite matching lecture

lectures/_static/quant-econ.bib

@@ -3,6 +3,20 @@
 Note: Extended Information (like abstracts, doi, url's etc.) can be found in quant-econ-extendedinfo.bib file in _static/
 ###
 
+@techreport{boerma2023composite,
+  title={Composite sorting},
+  author={Boerma, Job and Tsyvinski, Aleh and Wang, Ruodu and Zhang, Zhenyuan},
+  year={2023},
+  institution={National Bureau of Economic Research}
+}
+
+@article{delon2011minimum,
+  title={Minimum-weight perfect matching for non-intrinsic distances on the line},
+  author={Delon, Julie and Salomon, Julien and Sobolevski, Andrei},
+  journal={arXiv preprint arXiv:1102.1558},
+  year={2011}
+}
+
 @article{sargent1973stability,
   title={The stability of models of money and growth with perfect foresight},
   author={Sargent, Thomas J and Wallace, Neil},

lectures/match_transport.md

@@ -31,8 +31,8 @@ import pandas as pd
 ## Introduction
 
 This notebook presents  Python code for solving **composite sorting** problems of the kind
-studied in the August 2023 paper *Composite Sorting* by Job Boerma, Aleh Tsyvinski, Ruodo Wang,
-and Zhenyuan Zhang.  
+studied in  *Composite Sorting* by Job Boerma, Aleh Tsyvinski, Ruodo Wang,
+and Zhenyuan Zhang  {cite}`boerma2023composite`.  
 
 +++ {"user_expressions": []}
 
@@ -487,7 +487,11 @@ Suppose that agents $i$ of type $z_i$ and $j$ of type $z_j$, with $z_i < z_j,$ a
 Then there is an equal number of agents from each side in $\{i+1, \dots, j-1\},$ if this set is not empty. 
 
 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
-$$z_i < z_k < z_j < z_\ell$$
+
+$$
+z_i < z_k < z_j < z_\ell
+$$
+
 with matches $(z_i,z_j)$ and $(z_k, z_\ell),$ violating the no intersecting pairs property.
 
 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). 
@@ -937,8 +941,7 @@ example_off_diag.plot_layer_matching(layer_example, matching_layer)
 
 We will now present two key results in the context of OT with concave type costs.
 
-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))
-XXXX  for proofs. 
+We refer {cite}`boerma2023composite` and {\cite}`delon2011minimum` for proofs. 
 
 
 Consider the problem faced within a layer, i.e., types from $Y \sqcup X$
@@ -1603,7 +1606,7 @@ The dual solutions of $V_D$ and $W_D$ are related by $u_x = \alpha_x - \phi_x$ a
 
 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.
 
-The authors propose an efficient method to compute the dual variables from the optimal matching (primal solution) in the case of composite sorting.
+{cite}`boerma2023composite` propose an efficient method to compute the dual variables from the optimal matching (primal solution) in the case of composite sorting.
 
 Let's generate an instance and compute the optimal matching.
 
@@ -2047,7 +2050,7 @@ print('Value of primal solution: ', (assignment * example_assignment.cost_x_y).s
 
 +++ {"user_expressions": []}
 
-We now replicate the empirical analysis carried out by the authors.
+We now replicate the empirical analysis carried out by {cite}`boerma2023composite`.
 
 The dataset is obtained from the American Community Survey and contains individual level data on income, age and occupation.