Moral hazard can be defined as the risk that someone will do something that is not wise because they will not bear the costs if they fail. Moral hazard is likely to be present when incentives are set up so that the costs of mistakes will not be felt. Insurance is a classic example of something that can, theoretically, lead to moral hazard.
In the case of the FDIC, we can argue that moral hazard exists because the government will guarantee a bank’s deposits in the event that the bank fails. What this means is that the bank and its depositors face less risk. Because this is so, the bank might be inclined to make riskier loans and investments. The depositors might be willing to allow the bank to do this. They do so because they know that, if the bank fails, the federal government will pay the depositors and no harm will be done.
In this way, we can say that the FDIC gives rise to moral hazard because it protects the banks and the depositors from the consequences of a failure on the part of the bank.