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[match_transport.md] Fix typos #302

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20 changes: 10 additions & 10 deletions lectures/match_transport.md
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Original file line number Diff line number Diff line change
Expand Up @@ -15,19 +15,19 @@ kernelspec:

## Overview

Optimal transport theory studies how a marginal probabilty measure can be related to another marginal probability measure in an ideal way.
Optimal transport theory studies how a marginal probability measure can be related to another marginal probability measure in an ideal way.

* here ideal means to minimize some cost criterion.

The output of such a theory is a **coupling** of the two probability measures, i.e., a joint probabilty
The output of such a theory is a **coupling** of the two probability measures, i.e., a joint probability
measure having those two marginal probability measures.

This lecture describes how Job Boerma, Aleh Tsyvinski, Ruodo Wang,
This lecture describes how Job Boerma, Aleh Tsyvinski, Ruodu Wang,
and Zhenyuan Zhang {cite}`boerma2023composite` used optimal transport theory to formulate and compute an equilibrium of a model in which wages and allocations of workers across jobs adjust to match measures of different types with measures of different types of occupations.

Production technologies allow firms to reshape costs of mismatch so that they become concave.

It is then possible that in equilibrium there is neither **positive assortive** nor **negative assorting** matching, an outcome that {cite}`boerma2023composite` call **composite assortive** matching.
It is then possible that in equilibrium there is neither **positive assortative** nor **negative assortative** matching, an outcome that {cite}`boerma2023composite` call **composite assortative** matching.

For example, with composite matching in an equilibrium model with workers of different types, ex ante identical *workers* can sort into different *occupations*, some positively and some negatively.

Expand Down Expand Up @@ -665,7 +665,7 @@ def find_layers(self):
(H_z[None, 1:] <= layers_height[:-1, None])
* (layers_height[1:, None] <= H_z[None, :-1]))

# each layer is reshaped as a list of indices correponding to types
# each layer is reshaped as a list of indices corresponding to types
layers = [self.type_z[layers_01[ell]]
for ell in range(len(layers_height)-1)]

Expand Down Expand Up @@ -1194,7 +1194,7 @@ We then return the full matching, the off-diagonal matching, and the off-diagona

```{code-cell} ipython3
def solve_primal_pb(self):
# Compute on-diagonal matching, create new instance with resitual types
# Compute on-diagonal matching, create new instance with residual types
off_diagoff_diagonal, match_tuple = self.generate_offD_onD_matching()
nonzero_id_x, nonzero_id_y, matching_diag = match_tuple

Expand Down Expand Up @@ -1440,7 +1440,7 @@ There are two feasible matchings, one corresponding to PAM, the other to NAM.

Evidently,

* PAM corresponds to the matching with two medium side displacement because the correponding cost is strictly convex and increasing in the the displacement.
* PAM corresponds to the matching with two medium side displacement because the correponding cost is strictly convex and increasing in the displacement.

* NAM corresponds to the matching with a small displacement and a large displacement because the gain is strictly convex and increasing in the displacement.

Expand Down Expand Up @@ -1485,7 +1485,7 @@ To explore the coincidental resemblence to a NAM outcome, let's shift left typ

PAM and NAM are invariant to any such shift.

However, for a large enough shift, composite sorting now coindices with PAM.
However, for a large enough shift, composite sorting now coincides with PAM.

```{code-cell} ipython3
N = 2
Expand Down Expand Up @@ -2138,7 +2138,7 @@ if len(exam_assign_OD.n_x) * len(exam_assign_OD.m_y) < 1000:

+++ {"user_expressions": []}

Having computed the dual variables of the off-diagonal types, we compute the dual variables for perfecly matched pairs by setting
Having computed the dual variables of the off-diagonal types, we compute the dual variables for perfectly matched pairs by setting

$$
\begin{aligned}
Expand Down Expand Up @@ -2402,7 +2402,7 @@ ConcaveCostOT.plot_marginals_pdf = plot_marginals_pdf

+++ {"user_expressions": []}

We plot the hystograms and the measure of underqualification for the worker types and job types. We then compute the primal solution and plot the matching.
We plot the histograms and the measure of underqualification for the worker types and job types. We then compute the primal solution and plot the matching.

```{code-cell} ipython3
# Plot pdf
Expand Down
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