Must-Read: : Adding More Periods to Diamond-Dybvig: Fear of Illiquidity, Not Insolvency: “We simply add an extra time period…. It’s a friendly amendment…
…Agents are ex ante identical. Each agent has an endowment of apples. There is a costless storage technology for apples. There is also an investment technology (planting apples in the ground) which gives a strictly positive rate of return at maturity, but a negative rate of return if you cancel the investment before maturity. Each agent has a 10% probability of becoming impatient (getting the munchies) and wanting to eat all his apples this period. Those probabilities are independent across agents, and there is a large number of agents, so exactly 10% of agents will become impatient each period. Getting the munchies is private information….
Standard Diamond-Dybvig… has… an initial period where agents lend their apples to the bank; a second period where 10% of agents get the munchies and ask for their apples back; and a third period when the investment matures. Banks exist to provide insurance against risk of munchies by pooling assets; normal insurance won’t work because the information is private…. Make it a 4 period model:
- An initial period where agents lend their apples to the bank;
- A second period where 10% of agents get the munchies and ask for their apples back;
- A third period where another 10% of agents get the munchies and ask for their apples back; and
- A fourth period when the investment matures….
The bank credibly commits that it will never cancel an investment before maturity, and stores 20% of apples in reserve. In the good equilibrium… only agents who get the munchies ask for their apples back. Now suppose there is a… run on the bank in the second period…. An agent who does not have the munchies in the second period will rationally join that run on the bank, falsely claiming that he does have the munchies… [because] he might get the munchies in the third period, and if the bank suspends redemptions he will be unable to satisfy his future cravings, so he wants to join the line before the bank runs out of stored apples, so he can store apples at home…. Even if people are 100% confident that the bank is solvent, there can still be bank runs if people cannot predict their own future needs for liquidity, and fear that the bank might become illiquid…. Having a deposit in an illiquid bank is functionally not the same as having a deposit in a liquid bank, even if both are solvent…