In "Paul's Case," why does Paul leave home and what he is hoping to find?
First, let's look at some background. In order to tolerate his respectable but drab life as a typical, middle class boy, Paul escapes from his real life though his work at Carnegie Hall, where he works as an usher. He also has a very strong relationship with the acting troop, particularly with Charley Edwards, a young actor whom Paul admires tremendously.
The more involved Paul becomes with the theater, the more he falls behind in school, getting to a point where he is bold enough to tell the teachers that he has no business with academics, and that his job at the Carnegie is more important.
Matters went steadily worse with Paul at school. In the itch to let his instructors know how heartily he despised them and their homilies, and how thoroughly he was appreciated elsewhere, he mentioned once or twice that he had no time to fool with theorems...
This is when the problem begins. When the headmaster suggests to Paul's father that Paul be taken out of school and put to work, Paul's father asks that Paul be removed from his position as an usher, too. He asks Charley to cease his acquaintance with Paul as well.
Paul is sent to work as a clerk at Denny & Carson's, and he clearly does not like this. In a desperate act, Paul steals money from the company, which he was asked to deposit, and executes a plan that he and Charley Edwards talked about many times at the Carnegie: To go to New York and live like a dandy.
Paul has everything planned perfectly. He takes the right train, orders new clothes, and lands at the Waldorf, using the money that he took from the firm to pay for all of these luxuries.
Ultimately, what Paul aims to achieve is the feeling of "being there," among people who are beautifully dressed and aesthetically perfect in his opinion. Only someone with Paul's extreme sensitivity to beauty and detail would understand the need that he has to be a part of that world. He needs to experience it firsthand, thus breaking away entirely from the life in Cordelia Street which he detests so much. It is a way for him to find meaning in his life. Sadly, this will be his very last act, and he will die once he is threatened to be taken away from what he feels is his "element."