Q$3$: Consider an economy with two types of firms, Good and Bad. $$ is the fraction of firms that are of

Good quality, and $$ is common knowledge. Assume that only managers know their firm type. Good

firms have a probability $p_G$ of an $"$up$"$ move in valuation each period, and Bad firms have a probability $p_B$

of an up move, where $p_G>p_B>0.$ Every time that a firm issues a bond, it incurs a cost $C.$ Assume that $C$

$=$$$1,=0\mathrm{.}4,p{G}_G=0\mathrm{.}7,p_B=0\mathrm{.}3,$ the face value of debt $=$$$6,M_3=$$$24\mathrm{.}5M_4=$$$12,$ and $M_5=$$$6,$ where

$M_i$ represents firm value at node i$.$ If short$-$term debt is rolled over, the values of $M_3,M_4,$ and $M_5$ are each

reduced by $$2($due to bond issuing costs$).$ If a long$-$term bond is issued at $t=0,$ then $M_3,M_4,$ and $M_5$ are

each reduced by $$1.$ Assume that the risk$-$free rate of interest is $0,$ and that $M_2$ is large enough so that

there is no chance of default at $t=1.$ Firm value is given by the figure below:

a$)(10$ points$)$ If there is a pooling equilibrium with a long$-$term bond issued at $t=0,$ what is the promised

interest rate $($from $t=0$ to $t=2)$ on this bond? What is the expected value of equity at $t=2$ i$)$ for the Good

firms and ii$)$ for the Bad firms? Assume all investors are risk$-$neutral. $($Note: firm value equals bond

value plus equity value.$)$

b$)(10$ points$)$ If there is a pooling equilibrium with short$-$term debt being rolled over by both the Good

and Bad firms, what is the promised interest rate that must be paid in the second period by a firm that

finds itself at $M_2($the fact that in the first period this firm had a down movement is observable to

everyone$)?$ What is the expected value of equity at $t=2$ i$)$ for the Good firms and ii$)$ for the Bad firms?

$($Note: If the firm value for the next period is guaranteed to be above the face value of the debt, the

promised interest rate on the debt is $0,$ the risk$-$free rate.$)$

c$)(10$ points$)$ If there is a separating equilibrium in which only Bad firms issue long$-$term bonds, what

will be the promised interest rate on the long$-$term bond? What will be the promised interest rate for

the second period for a firm that finds itself at $M_2?$ What is the expected value of equity $($expectation

at $t=0)$ at $t=2i)$ for the Good firms and ii$)$ for the Bad firms?

d$)(10$ points$)$ For these parameter values, will there be a separating equilibrium, a short pooling

equilibrium, or a long pooling equilibrium?