The President does not really have any constitutional authority to directly do much to craft a policy to repair the economy. The President has the power to appoint various government officials in the executive branch (people like the Secretary of the Treasury) who can help shape economic policy. The President has the power to give an annual speech to Congress and to recommend laws to them. But this is all that the President is given authority to do under Article II of the Constitution. The President cannot simply make laws to fix the economy even if he (or someday she) knew how to do so.
Though the President has little in the way of constitutional authority, we still expect him to be responsible for the economy. This is largely because he is the most visible political figure in our system and is therefore the easiest to blame when things are bad or to give credit to when things are good.