Consolidate mccall and lake model code

[jlperla]

I want to simplify and consolidate a few things following some code changes in https://jlperla.github.io/undergrad_computational_macro/slides/markov_chains_applications.html#/title-slide and https://jlperla.github.io/undergrad_computational_macro/slides/search_dynamic_programming.html#/title-slide

On the "Job Search II: Search and Separation"

• See the different calculation of the bellman equation in https://jlperla.github.io/undergrad_computational_macro/slides/search_dynamic_programming.html#/bellman-operator
  • Why don't you modify the calculation in https://github.com/QuantEcon/lecture-julia.myst/blob/main/lectures/dynamic_programming/mccall_model_with_separation.md to show that additional formulation, inside the T(...) function, which uses less vectorization

In the Lake Model lecture, you will see that it does its own Markov chain stuff but nests the "search with separation" lecture. We will want to make things consistent. I also want to remove the QuantEcon dependency and just do the Markov chain stuff manually. In particular inside of https://github.com/QuantEcon/lecture-julia.myst/blob/main/lectures/multi_agent_models/lake_model.md

• Rename LakeModel to lake_model for consistent naming convention of functions.
• Instead of the MarkovChain object, instead have the function from https://jlperla.github.io/undergrad_computational_macro/slides/markov_chains_applications.html#/simulating-markov-chains
  • You may need to make a minor change to the code which plots the s_bar path because you can't just do the cumsum(s_path) because it won't have the [0;1] as created in MarkovChain(P: [0;1]). Probably just something like cumsum(s_path .= 2)
  • After this, get to the point that the QuantEcon include can be removed.
• Then go through the changes already made in https://github.com/QuantEcon/lecture-julia.myst/blob/main/lectures/dynamic_programming/mccall_model_with_separation.md and copy/paste them into this Lake Model lecture. It nests that old one, so we want to make sure it is all consistent
  • If there are any reasons they can't be fully consistent, let me know.
• More generally, go through https://jlperla.github.io/undergrad_computational_macro/slides/search_dynamic_programming.html#/title-slide and see if there are any other nice changes in those lecture notes we should put into the quantecon there.