Keeping all the setups unchanged, except the setting on uncertainty: instead of assuming crises are low probability events, from now on, assume that in \(t=1\), the probability for the normal state \(\pi_{H}\), can take any value between 0 and 1.

(a) Show that banks coordinate to choose \(\alpha_{H}\), as long as \(\pi_{H}\) is above a threshold. Compute the threshold, call it \(\bar{\pi}_{1}\). Are there bankruptcies in either of the two states?

(b) Show that banks coordinate to choose \(\alpha_{L}\), as long as \(\pi_{H}\) is below a threshold. Compute the threshold, call it \(\bar{\pi}_{2}\). Are there bankruptcies in either of the two states?

(c) Show that, for any \(\pi\) between \(\bar{\pi}_{1}\) and \(\bar{\pi}_{2}\), banks can coordinate neither on \(\alpha_{H}\) nor on \(\alpha_{L}\), and there are always bankruptcies despite the state of the world.