The University of Chicago has 2 current professors as Nobel Prize winners in Economics. LSE has a great academic reputation. You need to ask yourself what kind of a person you want to be after 3-4 years. If you would like a liberal arts education and then major in one of the truly outstanding subject at UChicago, then go to Chicago. If you want to experience the delights of London and have a more focused degree with one subject done rigerously for 3 years, LSE is for you. Moreover, I have a few friends at LSE doing Economics. The first year is not very strenous in terms of workload so you certainly get to enjoy yourself in arguably the most exciting city in the world. I have a friend at Chicago and he keeps on speaking about the workload and the fact that they make you work immensely hard there.
Granted, Chicago is not the most appealing and safe city, especially in the south side (i think) of chicago which is the less prosperous area, except for the presence of the university itself. The university campus itself is stunning and similar to the architecture seen at Oxford. If I were you, I'd pick Chicago. Considering the fact that you're an international student and you'd be away from your parents, homesickness may occur at both places. So, it really comes down to the structure of the course and the type of intellectual development you wish to undertake in.