Suppose financial intermediaries hold assets that will each pay either zero or (in expected value) 1 next year. The proportion of bad assets is 10%, and each intermediary holds the same proportion of good and bad assets. The intermediaries know the exact quality of each asset they hold, but are liquidity constraint and value a payoff of $1 a year from now as .8. Deep pocket investors are willing to buy the asset at its expected value.
1. What is the equilibrium price if buyers know that sellers put for sale a fraction μ > 10% of the assets.
2. Show that there exist at least two equilibrium prices pl and ph > pl When equilibrium price is pi there is no sales of good assets, when it is ph all assets are sold to long term investors.
3. Extra credit and harder: Show that there are in fact exactly three equilibria and argue that the middle one is not of interest because it is "unstable".
4. Suppose now the intermediaries are forced to sell at least 70% of their asset holdings. Show that in any equilibrium intermediaries sell more than the minimum required. Calculate an equilibrium
5. Suppose intermediaries are not required to sell any of the asset, but the Central Bank announces that is ready to buy the asset at $.81 in the open market. What is the equilibrium price and how much would the government effectively buy?
6. Suppose instead the Central Bank is willing to lend money to the intermediaries with the guarantee of the asset. The CB would lend
$.81 at zero interest rate for each unit of the asset pledged. How much would be pledged to the CB?
Hint: Assume that an intermediary has pledged a certain number of good and bad assets and is deciding whether to pledge an extra good asset or to sell that asset in the market. If she pledges the extra good asset she gets 81 cents today and 19 cents tomorrow which she values at .8 x .19.
7. In less than 5 lines explain what this exercise tells you about the CB as a lender of lost resort or as a buyer of last resort.