There are at least two major problems with direct democracy. One is logistical/practical and the other has to do with the ability of the people to rule.
First, a direct democracy is very unwieldy and is likely to work only in a very small polity. It is very difficult to get anything done quickly if all the people are able to be involved in the decision-making process.
Second, in a direct democracy, the people are asked to make decisions on all sorts of issues. The issues that arise, especially in a modern country, can be very complex. The people as a whole will not typically have the time or the training to truly understand most of the issues involved. This means that a direct democracy might end up creating a number of policies that are ill-informed or short-sighted. This is not saying that the people are too dumb to rule themselves, simply that they are too busy and they lack specialized knowledge.
How did the cost play a role as a con of direct democracy?