Consider a risk-neutral firm, protected by limited liability, that wants to finance a project at a cost \(I=1\). The project takes one period to complete. The firm has no initial wealth; hence to undertake the project the firm has to borrow from somewhere. There are many households with enough resources ("deep pockets") to finance the project. If the firm and a household sign a contract (claiming that the household lends \(x\), and \(R\) is to be repaid per unit lent if the project succeeds), the household grants a loan and the firm takes an action that affects the riskiness of the project. The firm has two options. The first option is to take a safer project, as given by the lottery \(\{(p, G) ;(1-p, 0)\}\). This project returns \(G\) with probability \(p\), and returns 0 with probability \(1-p\). The other option is a riskier project, as given by the lottery \(\{(p q, B) ;(1-p q, 0)\}\). The household cannot observe the action taken. The risk-free rate of return available to the household is normalized to 1.

We assume that \(p G>1>p q B\), and \(B>G\). The two inequalities can be satisfied at the same time only if \(1>q>0\). Hence, the risky project has a higher gross return in case of success than the safe project \((B>G)\), but succeeds less often \((q<1)\). Suppose that the household observes whether the project succeeds or fails (but in case of success the household does not observe whether the return is \(G\) or \(B\) ).

(a) Provide a graphical illustration of how the payoffs of the two projects to the firm vary with \(R\), and derive a critical value of \(R\), denoted by \(\hat{R}\), below which the firm chooses the safe project and above which the firm chooses the risky project.

(b) Which conditions have to be satisfied in a (competitive) credit market equilibrium?
Now let's introduce banks in the economy. Banks have a monitoring technology. By incurring a monitoring cost, \(c\), a bank is able to prevent the firm from undertaking the risky project, and to induce the firm to choose the safe project. A bank monitors whenever it finances a project. Suppose that the banking sector is competitive.

(c) Derive the conditions for a competitive equilibrium if banks are the only lenders available.

(d) For which values of \(p\) will we have only direct finance, only bank lending, and no lending, respectively?