This is the first time I come accross a Max function inside an integral. I have looked around online and did not find anything about it. I would like to know the rules of what can I do when I have an integral over a Max. I am trying to figure out how one can get from equation (1) to equation (2). I get the intuition behind it, but want to know the mathematics behind it: the rules and steps.
The particular example that I came accross is the following:
$$w^{\ast }=\left( 1-\beta \right) b+\beta \int ^{w_{m}}_{0}\max \left\{ w,w^{\ast }\right\} dF\left( w\right)\tag{1}$$
If we substract $\beta w^{\ast}$ we get:
$$\left( 1-\beta \right) w^{\ast }=\left( 1-\beta \right) b+\beta \int ^{w_{n}}_{w^{\ast }}\left( w-w^{\ast }\right) dF\left( w\right)\tag{2}$$
1) How does the substraction enter the Max function under the integral? 2) How does it change the lower bound of integration? 3) Any links on the rules of max functions under integrals?
This comes from the search and matching models in economics (McCall model). We try to solve for w*, a particular wage, which is just a constant. The support of the distribution of wages goes from [0,$w_{m}$].