Question 1
(a). Write down the problem of an agent that maximizes ex-ante utility in autarky. Find the conditions that characterise the allocation in autarky. Explain how the allocation changes with β.
Autarky is a situation where no trading takes place between agents. Each agent needs to provide for his own needs in an autarky, ie he independently chooses the amount of I that he wants to invest in the long run technology. The issue of liquidity insurance arises here.
Every agent wants to maximise his ex-ante utility but the problem is that at time t=0 he does not know about his type whether he wants to consume early at t=1 or late at t=2 resulting in asymmetric information. Hence, there is a risk that more than is optimal may be invested.
The conditions that characterise the allocation in autarky are bounded by the constraints of C1 and C2. If agent decides to consume early, he will get savings (1-I) and liquidated investment (ÉI).
C1 = 1 – I + ÉI = 1 – I (1-É)
If agent decides to consume late, he will obtain savings (1-I) and returns from investment (RI).
C2 = 1 – I + RI = 1 + I(R-1)
Agent will choose his consumer profile (C1, C2) that will maximise his ex-ante utility U based on the above constraints.
However, the allocation is not efficient in autarky as shown in the next part of the question.
Max U(C1,C2) = u(C1) + βu(C2) = [1 – I + ÉI]+ [1 – I + RI]= 2+ ÉI + RI
We set up the lagrangian method to explain the allocation changes in β where the constraint in the below equation is the maximum utility.
L = πu(C1) + (1-π)βu(C2) + λ[2+ ÉI + βRI]
= π + λÉI = 0
= (1- π)β + λRI = 0
= 2+ ÉI + βRI = 0
Complementary Slackness Condition: λ*[2+ ÉI + βRI] = 0
If values were given for the variables, we could even have solved and get the value of β. If a value close to zero is obtained for β, it means agent is impatient anda value close to one indicates that agent is patient.
This argument is further supported by the marginal rate of substitution concept where 
= R. If β=0, no returns obtained as the agent wants to consume immediately. If β=1, returns will result for the patient agent. Hence, it shows that the discount factor β will not change the basic results of the model.
(b) Write down the conditions that characterise the Pareto-optimal allocation. Show that autarky is not efficient. Explain how the allocation changes with β.
The conditions that characterise the allocation in autarky are bounded by the constraints of C1 and C2.
π1C1 = 1 – I => C1 = 
(1-π)C2= RI => C2 = 
The constraints can be combined in a single one.
π1C1 + (1-π)
= 1
The key result is that allocation is inefficient in autarky as shown below:
Recall in autarky: C1 = 1 – I + ÉI = 1 – I (1-É)
C2 = 1 – I + RI = 1 + I(R-1)
If C1 < 1 (unless I = 0) and C2 < R (unless I = 1), then combining these two facts we obtain
π1C1 + π2 < 1 which states that efficiency is not reached. It is true as less money and fewer resources exist in an autarky than in Pareto optimal allocation as no trade occurs. Therefore, consumption level is lower in autarky.
Max U(C1,C2) = u(C1) + βu(C2) = + β 
We set up the lagrangian method to explain the allocation changes in β where the constraint in the below equation is the maximum utility.
L = πu(C1) + (1-π)βu(C2) + λ[ + β ]
= π + λ
= 0
= (1- π)β + λ β = 0
= + β = 0
Complementary Slackness Condition: λ*[ + β ] = 0
If values were given for the variables, we could even have solved and get the value of β. If a value close to zero is obtained for β, it means agent is impatient anda value close to one indicates that agent is patient.
The argument of marginal rate of substitution is also applicable here where = R. If β=0, no returns obtained as the agent wants to consume immediately. If β=1, returns will result for the patient agent. Hence, it shows that the discount factor β will not change the basic results of the model.
(c) Assume the agents are now infinitely risk-averse. That is U(c1,c2) = min{c1,c2}. What is the Pareto-optimal allocation?
Pareto optimal is an allocation of resources where it is impossible to distribute resources without making at least one consumer worst off. Pareto optimal is the best outcome that could result in an economy with trade taking place and thus there is higher consumption level. It is like a desired state where assets are increased for patient people and consumption is increased for impatient people.
The Pareto optimal allocation for risk neutral agents satisfies the following first order condition:
Uʹ(C1) / Uʹ(C2) = R
which means that agents would like to equate the marginal rate of substitution between consumption levels at t=1 and t=2 with the returns on the long run technology.
When U(c1,c2) = min[c1,c2], it shows agents’ attitude to risk aversion.
The pareto optimal allocation for the risk averse agent is u(C1) + πβu(C2G) + (1-π)βu(C2B) where the superscripts G and B denote good and bad state respectively.
L = u(C1) + πβu(C2G) + (1-π)βu(C2B) + λ[u(C1)]
The concaved utility function states that agents prefer to consume more to less and shows how consumption is smoothed out over time and across states in the future. The agent is risk averse in the sense that he does not want consumption in the bad state at t=2 to be too much different from consumption at t=1.
Question 2
(a) Write down the incentive constraint of the bank. How does collateral affect the repayment R the bank can promise?
Banks, regarded as information sharing coalitions, can easily overcome the problem of asymmetric information of investors. It is assumed that banks will use the signaling tool to invest in high quality projects which will benefit the investors. Banks are expected to behave in such a way that will maximise investors’ interest.
The firm chooses the good project if
pH(y-Ru-Rm) > pL(y-Ru-Rm) + b => Ru + Rm < y- 
The bank must also be encouraged to monitor the project:
pHRm– C > pLRm => Rm > 
The bank will borrow only least possible amount from banks as bank finance is more expensive than direct finance.
Im = Im (β) ≡ 
= 
where β denotes expected rate of return.
The bank will collect get the remaining finance Iu = 
from uninformed investors. Hence, the bank’s incentive constraint binds.
Using the incentive constraints we have: Ru < y- which states: Iu < 
[y – 
] indicating that the project will only be financed if:
A + Iu + Im > 1 => A > 
(β,r) ≡ 1 – Im(β) – 
[y – ]
Other constraints would include a lack of monitoring from the bank giving rise to the probability of non-monitoring pL and the inability to dispose the collateral, ie if the collateral appreciates, the bank will not be able to sell it until loan to investors has been repaid.
The collateral, usually in the form of assets, plays the role of a guarantee that banks give to investors as a security in case of failure of the project. Collateral is also seen as an alternative to monitoring as it saves efforts and reduces the risk of the bank. ϵ ∈ (0,1) can be interpreted as if K is close to one, bank will be able to refund the money to investors whereas if K is close to zero, bank will be unable to repay back the loan.
A better collateral equals better chance of getting money back as the bank will prefer to behave or else it will lose the collateral.
If the project is successful with expected probability p, the bank will gain returns X which will be used to refund the loan to investors and claim back the collateral. The higher the returns from the project, the bank will be able to distribute partly between the investors and keep partly as its own profits.
In case of failure of the project, the bank will obtain zero returns and is then unable to repay R to the investors. The latter will seize the collateral and will liquidate it to gain maximum money from it as refund of their investment in the unsuccessful project.
(b) Suppose investors have all the bargaining power. Write down their objective, find the optimal contract and their equilibrium profits.
If investors have all the bargaining power, they will be able to influence the project financing process significantly and dictate their terms. The objective of investors is to obtain maximum returns X from the project. They will want to have full details about the project to ensure that the bank is choosing a high quality project (θ) rather than making an adverse selection. Investors delegate the monitoring of the project to the bank as the latter has comparative advantage in monitoring activities hence monitoring costs will be reduced. Investors will use monitoring and auditing as tools to be free from asymmetric information and to improve efficiency. They will expect close monitoring and continuous feedback on the project from the bank.
The optimal contract for investors will be where lending will be most profitable and the below equation is taken from the Diamond Model (1984):
E[y] > 1 + r + C = E[y] > 1 + 1 + C = E[y] > 2+ C
where E[y] = Returns from investment
r = risk free rate, equal to 1 in the question
C = monitoring costs
The optimal contract is bounded by the break-even constraint of uninformed investors implying an upper bound on Iu:
pHRu > (1 + r) => Iu < < [y – 
]
Equilibrium profits of the investors will be at a feasible break-even point, usually where demand equals to supply:
A + Iu + Im > 1 => A > (β,r) ≡ 1 – Im(β) – [y – ]
(c) For which value of K can the bank borrow and invest?
The value of the collateral must be either equal or slightly higher than the investment in project (I) and monitoring costs (C) to encourage investors to finance the project as a lower value of the collateral will not attract them.
K = I + C or K > I + C
Ideally if K > I + C, this will attract more investors to finance the project and in turn banks will be able to borrow from them and invest in the project.
Question 3
(a) If A ≥ A3, the firm issues high-quality public debt (public debt that has a high probability of being re-paid)
We will discuss circumstances when the entrepreneur can issue high quality public debt:
- Well-capitalised firms [A >
] can issue direct debt as they possess high capital. - Low credit risk – High quality public debt refers that the entrepreneur is likely to meet payment obligations. This type of public debt is an attractive investment vehicle as it has a low risk of default.
- High dilution costs
- Good reputed firms can issue direct debt only if πs >
where πs is the probability of repayment at t=2, conditionally on success at t=0 and given all firms are monitored at t=0. - It is assumed that monitoring cost c is small such that
in the credit market at equilibrium. The entrepreneur has incentive to issue high quality public debt at a rate of 
when 
as the latter equation means high probability of success. The economic interpretation is when project is successful, returns (R) are obtained. The entrepreneur cannot ask for more than R as the firm will also keep some profits for itself. Every party in the transaction is happy and is in equilibrium when a good project is undertaken.
(b) If A3 > A ≥ A2, the firm borrows from a monitor (and from uninformed investors)
We will analyse circumstances when the firm borrows from a monitor and uninformed investors:
- Firms with medium capital [
(β,r) < A < ] borrow from banks. - Firms borrow from banks when they suffer from high credit risk and high dilution costs because banks can provide efficient renegotiation in case of default and can limit dilution costs though there will be an intermediation cost involved.
- Uninformed investors are ready to invest Iu in exchange of return Ru upon successful project. Firms must be encouraged to choose good project pH(y – Ru) > pL (y- Ru) + B <==> Ru < y –
- When the firm falls short of capital to issue a direct debt, it can borrow Im from banks (with return Rm if project succeeds) and Iu from uninformed investors (with return Ru if project succeeds).
The firm chooses the good project if
pH(y-Ru-Rm) > pL(y-Ru-Rm) + b => Ru + Rm < y-
The bank must also be encouraged to monitor the project:
pHRm– C > pLRm => Rm >
The bank will borrow only least possible amount from banks as bank finance is more expensive than direct finance.
Im = Im (β) ≡ = where β denotes expected rate of return.
The bank will collect get the remaining finance Iu = from uninformed investors
Hence, the bank’s incentive constraint binds.
- Two conditions are necessary for bank lending to be in equilibrium in credit market:
(i) Monitoring cost must be less than the returns of the good project
pH G – 1 > c
(ii) Direct lending which is cheaper must be impossible.
pHRc < 1
Firm should borrow from a monitor (for example a bank) and from uninformed investors at intermediate probability of success when pH 
] at a rate of R = 
.
(c) If A2 > A ≥ A1, the firm issues junk bonds (public debt that has a low probability of success)
We will discuss circumstances when the firm issues junk bonds:
- It is possible that firms with medium capital [(β,r) < A < ] issue junk bonds.
- High credit risk- Junk bonds refer to bonds with low credit quality and high default risk. They are attractive to risk seeker investors due to their high yielding returns.
- Low dilution costs as it limits exposure to bad firms but involves inefficient bankruptcy costs for good firms.
- The zero profit condition for investors is:
1 = pR + (1- p) A
This nominal return R is feasible (R < y) if py + (1- p) A > 1 and the expected profit of good firms is then:
πB = p (y- R)+ py
By substituting R, we will obtain: πB = 2py – 1 + (1- p) A
- When the monitoring element c is added, the monitor can reduce the entrepreneur’s private benefit of misbehaving from B to b.
pH 
> c >(pH −pL) R−pH 
If R > Rc, the firm will issue junk bonds with low probability of success. This states that the firm is indebted and have too much risk associated with it. The economic interpretation out of it is that the entrepreneur will ask for higher returns but the firm will not afford to provide it. This will lead the entrepreneur to choose the bad project and disequilibrium occurs. Hence, such a combination is not feasible because the maximum repayment is K.
(d) If A1 > A, the firm does not invest
We will analyse circumstances when the firm cannot invest:
- Firms with low capital [A < (β,r)] can neither invest nor borrow. Venture capitalists are the only solution for such firms.
- When monitoring costs are added, if pH <
it means there is a small probability of success. The equilibrium consists of no trade taking place and the credit market collapses because good projects cannot be funded and bad projects have a negative net present value. Hence, the firm should not invest as there is no trade equilibrium.
References
Frexias X. and Rochet J-C., (2006) Microeconomics of Banking, 2nd Edition
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