Setups stay the same, except that there are multiple asset classes in banks' balance sheets. In particular, let's assume that \(P_{1}\) is exogenous, and a fraction \(\beta\) of each bank's assets is composed of financial securities that can be sold in \(t=0\) at \(P_{0}\) or in \(t=1\) at \(P_{1}\). The rest of the assets can be liquidated. Each asset has liquidation value denoted by \(l\), distributed uniformly between 0 and \(Z\). Both the securities and the other assets have face value \(Z\) in period 2. The value of deposits is \(D\), a shock hits in period 1 with probability \(q\), and in case of shock depositors withdraw an amount \(f D\) of deposits in period 1. Assume that \(P_{1}\) is sufficiently large that the bank expects to be solvent in period \(t=1\) even if it does not sell any security in period \(t=0\).

(a) What condition should \(P_{0}\) and \(P_{1}\) satisfy to ensure that investors are indifferent to buy securities in either \(t=0\) or \(t=1\) ?

(b) Let's say that the bank plans to sell a fraction \(\eta_{1}\) of its securities at \(t=1\) (if the shock hits), and not to sell any security in \(t=0\). What fraction of the assets will be liquidated in case of shock? What is the average \(l\) of the liquidated assets? Can you find an expression for \(\eta_{1}\) ?

(c) Let's say that the bank plans to sell a fraction \(\eta_{0}\) of its security at date 0 and not to sell any security in \(t=1\). Moreover, assume that \(P_{0}\) and \(P_{1}\) are such that the bank is indifferent to sell its securities in \(t=0\) or in \(t=1\). What fraction of the assets will be liquidated in case of shock? What is the average \(l\) of the liquidated assets?
Can you find an expression for \(\eta_{0}\) ?

(d) Is it true that, under \(P_{1}\) and \(P_{0}\) that satisfy the condition you found in question (a), the bank is indifferent between selling securities in \(t=0\) and in \(t=1\) ?