collateral and adverse selection and investment

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1. A Model of collateral, investment and adverse selection Alberto Martin. Presented by: Amit Kumar Shreyash Nipun

2. Objective To establish a relationship between entrepreneurial wealth and aggregate investment under adverse selection. The prevailing literature: The prevailing view that emerges from the existing model is that whenever lenders are constrained because of information friction, investment and credit must be increasing in entrepreneurial wealth. Contributions of this paper: Pooling equilibria in which investment is independent of entrepreneurial wealth. Separating equilibria in which investment is increasing in entrepreneurial wealth. For a given rate of interest an increasing entrepreneurial wealth may generate a discontinuous fall in investment.

3. Assumptions: Banks and entrepreneurs, both are risk neutral.

4. Assumption 2 : (Loan contracts). Entrepreneurs and banks sign a contract of the form (I, R, c), where I is the amount borrowed and invested, R is the interest factor on the loan and c is the percentage of the loan that entrepreneurs must collateralize by using their own wealth. In the event of a successful state, entrepreneurs pay back the amount borrowed adjusted by the interest factor: otherwise, they default and the bank keeps the goods put up as collateral, the interest borne by them, and the residual value of the project. Expected profit of entrepreneurs can be written as :

5. Bank’s expected profit of accepting an application to contract from a j type entrepreneurs is given by: Assumption regarding bank competition : No cross subsidization : Banks are not allowed to offer contracts that loose money in expectation. Exclusivity : Entrepreneurs can apply to at most one of the contacts only.

6. Case of Symmetric Information The economy under full information displays many equilibria. Banks are indifferent between making entrepreneurs pay only in the event of success (i.e., setting cj = 0) and making them pay partially in the event of success and partially in the event of failure (i.e., setting cj > 0). All of these equilibria result the same level of investment and are equivalent in terms of efficiency. The expected profit of any j type entrepreneurs is : Zero profit condition of banks is :

7. In case of complete information the equilibrium values of I, R, c will satisfy: In the case of success the entrepreneur will pay

8. Asymmetric Information With the assumption of Exclusivity and no cross subsidization, a separating equilibrium can be defined as: for a given set of (r, W) a separating equilibrium satisfies the following conditions: 1. Feasibility: A contract must respect collateralization constraint 2. Incentive compatibility constraint: Each entrepreneur applies to the contract designed for his type.

9. 3. Zero profit condition for banks: Each contract yields banks zero profit in expectation. 4. No bank can profit by offering alternative contracts. Given a set of r and W the separating equilibrium is characterised by a pair of contracts satisfying: For B type entrepreneurs:

10. For G Type entrepreneurs : (Maximizing the profit function of G type) In order to provide a simple graphical representation of separating equilibrium, we define a NMC (No mimicry constraint) condition which is defined by combining ICC and the zero profit condition for the bank for G type agents.

11. Now the optimization problem reduces to maximizing the profit function of G type agents subject to the NMC and the collateralization constraint, by which And manipulating the above to get C in terms of I for G type agents, we arrive at:

12. Pooling Equilibria For a given interest rate-entrepreneurial wealth pair ( r,W ), a pooling equilibrium is a contract which satisfies the following conditions : Feasibility : This contract must adhere to the collateralization constraint. Zero profit condition of banks : The contract must give bank zero profits in expectation. No bank can profit by offering any alternative contracts. . The following proposition shows that pooling equilibrium will bring about a binding collateralization constraint, though the amount if investment is independent of entrepreneurial wealth (W).

13. Given ( r,W ), a pooling equilibrium is a contract , Which satisfies: Here, denotes the average probability of success of both B and G type projects in the economy.

14. The third equation implies that this equilibrium must imply a binding collateralization constraint. The first equation implicitly implies the level of investment in the pooling equilibrium condition. When wealth is zero, the condition in the first condition should be satisfied since it equates Marginal productivity of investment of G type agents to the marginal cost of funds incurred by them. The second equation simply gives us the value of R in the pooling case. We had arrived at the first equation by solving the following optimization problem:

15. We know that both the pooling loan size and the size of loans given to bad agents in the separating case are independent of entrepreneurial wealth. Actually, their relative magnitude depends only on the value of , which is given by: For any given interest rate r, Note- We will call the above equation as the threshold equation. Now, the condition to surpass this threshold (denoted by the right hand side) depends on the ratio of G type to B type agents. If this ratio is high enough, then it makes surpass this threshold. And because of that there will be overinvestment of B type in the pooling allocation relative to the separating allocation.

16. Regime Switches and Investment Say denotes the equilibrium contracts. Now whether these will be pooling or separating, will depend on the profits of the agents in each case. (and it can be proved that when G type prefer optimal pooling to the optimal separating contract, B type agents prefer to do so.) Lemma 1: This lemma gives us a threshold value of above which the equilibrium is always pooling, given W=0.

17. The threshold value of in lemma 1 guarantees a unique switching point for each level of r, stated by the following lemma 2 : . When , the threshold equation implies that And it can also be shown that : the switching point equals the B type size. This means that in the limiting case, there is no cross subsidization at the switching point as the loan is fully collateralized. At this, it must hold: Intuition : As the loan here is fully collateralized, G type will be indifferent between this and the separating contract only when both result in equal loan sizes :

18. . When , however, there is a discontinuity in aggregate investment when the regime switches from pooling to separating. In this case, , so that the switch from pooling to separating must mean a contraction in the amount invested by bad entrepreneurs. The same will be true of their G type agents, for whom it must be the case that Intuition : At the switching point G type are indifferent between either of the contracts. The pooling case allows some level of cross subsidization, but separating does not. So, to remain indifferent, separating must result in lower loan size. . So, when the economy switches from pooling to separating equilibrium in this condition, there is a contraction in the amount of investment of both types of agents. This has been summarized in the following lemma 3 :

19. . Now, how does the lemma 3 synchronise with the intuition regarding the choice of screening device in the case when entrepreneurial wealth is low. In this case, From the perspective of financial intermediaries, c is significantly low to distinguish between the G type and B type agents present in the market. Hence, there is a strong tendency to pool all projects and have good entrepreneurs cross-subsidize their bad counterparts. Whereas cross-subsidization implies that the pooling equilibrium is costly for good entrepreneurs, it also benefits them by allowing them to expand their investment.

20. And here, G type and B type, both know (from lemma 3) that their investment will shrink in the separating case. So, both will refrain from accepting the Separating case. And, in the pooling case, since the G type will have to accept cross subsidization (CS) condition, they will reveal the financial intermediaries that they are G types. As, CS condition will imply them to reduce their expected profit. But B types will not reveal themselves, as their I anyway shrinks in the separating case, and they are being compensated for profits through cross subsidization.

21. Conclusion: This paper improves upon the prevailing literature which said, because of information friction, investment and credit must be always be increasing in entrepreneurial wealth . Instead, it enlightens us with the following idea: 1. Pooling equilibria in which investment is independent of entrepreneurial wealth. (It is the case when entrepreneurial wealth is low). 2. Separating equilibria in which investment is increasing in entrepreneurial wealth (when the wealth of entrepreneurs are high). 3. For a given rate of interest an increasing entrepreneurial wealth may generate a discontinuous fall in investment, where the equilibria switch from pooling to separating and the amount of total investment also changes.

22. 1. The author has assumed that entrepreneurs are risk neutral. Risk aversion in the preferences of entrepreneurs would generate an additional cost of pledging their wealth as collateral, since doing so would increase the degree of variation of consumption. So, this effect could restrain the amount of collateral pledged and of investment undertaken in the economy under the separating regime. On the other hand, because of this reason, collateral could be more effective as a screening device. The net impact on the level of collateralization and investment would depend on the relative importance of these two effects. 2. He has characterized the contracts under the assumption of a fixed interest rate. It could be thought that, since investment may be discontinuous, the existence of an equilibrium is not guaranteed in a general equilibrium case in which the interest rate is determined endogenously. Critical Points :