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Modern Theory of Banking and Finance

Autumn Semester 2020/21


1. An entrepreneur has a project that requires an initial investment 1. He/She has initial cash A; if A < 1, then the entrepreneur needs outside financing to fund the project. Banks lend with an interest rate i = 0. If the project is funded, the entrepreneur can choose to work or to shirk. If he/she works, the project succeeds for sure and yields an amount equal to R .  If he/she shirks, the entrepreneur obtains private benefits B < R but the project always fails and yields nothing.

(a)  Show how credit rationing could be caused by moral hazard.

(b) How could the entrepreneur boost his/her borrowing capacity?

(c) How would your answer in Part (a) change if we were to assume i > 0?

(d)  Show what happens to the minimum level of capital required to access financial markets if there is a monetary policy tightening.


2. Using the baseline model of adverse selection, assume that there is one bank and it is not able to distinguish between the two types of entrepreneurs. It extends loans and can choose freely the (risky) interest rate r at which it lends to the entrepreneurs. Because the bank cannot distinguish between the two types of entrepreneurs, it has to offer the same interest rate to all of them. Moreover, entrepreneurs have initially no cash in hand so they need to raise 1 from the bank. The repayment to the bank is therefore the minimum of R = 1 + r and of the payoff of the project. It is assumed that an entrepreneur accepts the offer of the bank if and only if implementing the project brings him a strictly positive profit.

(a)  Show that under asymmetric information good entrepreneurs are always worse

off and bad entrepreneurs are always better off.

(b)  Calculate the implicit amount of subsidy that goes from the Good entrepreneurs

to the Bad entrepreneurs.

(c)  Show when it is optimal for the good borrower to signal herself by posting the optimal level of collateral C* and when it is optimal for the good borrower to pool with the bad borrower.


3.  Consider a three-period economy with a continuum of agents. Each agent is endowed with one unit of a good in period 0, which she must use to finance consumption cl , c2  in periods 1 and 2.  Agents have access to two technologies.  Technology α is a one-period technology that returns A for every unit invested in the previous pe- riod. Technology β is a two-period technology: each unit invested in this technology returns B after two periods, while premature liquidation after one period returns nothing.  The agents are ex-ante identical but subject to consumption uncertainty: with probability π an agent will want to consume only in period 1, and with proba- bility 1 − π she will want to consume only in period 2. The ex-ante utility function of all agents is

U = π ln(cl ) + δ(1 − π) ln(c2 ),

where δ < 1 is a discount factor. Assume that δB > 1.

Hint: this is a slightly altered version of the standard model we covered in class.  The classroom version had different notation, assumed A = 1, and premature liquidation yielded L > 0 .

(a) What is the outcome in a situation of financial autarky. Identify any assumptions

you make and explain the outcome.

(b)  Specialise the analysis to the case where A = 1. Obtain the optimal symmetric

allocation in this economy.

(c) Explain how a system of fractional reserve banking can implement the optimal outcome. Identify any assumptions.

(d) What is a bank run? Why is fractional reserve banking vulnerable to bank runs?