I have recently discussed Banking intermediation, FRB and bank-runs in the context of the Diamond 1997 model in line with my effort to understand the banking/financial sector and record some of the knowledge I acquired over the years.
When, in 1981, Stiglitz and Weiss published their credit rationing article, it was part of a wave of information economics articles generally presumed to have started with Akerloff’s “A Market for Lemons” article. The issue of asymmetric information permeates any analysis of the financial sector, be it of hedge funds or of commercial or investment banking.
I now propose to spend some time peering over the quality of the quantity of money being lent in the economy. This matters because if banks are lending too little, then the economy will perform beneath capacity, while if they are lending too much it will perform above capacity (for the specific meaning of “capacity” in this context, you might be interested in reading about the Output gap and how it is calculated in macro). In the next few lines I consider the main concepts and models available to understand these issues.
My understanding, is that the equilibrium in the lending market is one characterized by overinvestment, as proposed by De Meza and Webb 1987. During the period of adjustment to some shock, the commercial and financial banking sectors will experience what Modigliani and Jaffee coined as Dynamic Credit rationing, while once the shock has faded, the economy turns to overlending (or overinvestment). I have an unsubstantiated hunch that these extremes are more pronounced in the commercial banking sector than in the financial markets. This might be due to the fact that asymmetries of information are less pronounced in commercial banking and that creditors in that market are less able to delay their investment, to scale down the project or to reduce their consumption. For now, however, that is a tangent, rather than the focus of this post.
Equilibrium and Dynamic Credit Rationing:
Jaffee and Modigliani (1969:851) provide the first study of credit rationing that I could find. They separate it into two different forms. “Equilibrium rationing is defined as credit rationing which occurs when the loan rate is set at its long-run equilibrium level. Dynamic rationing is defined as credit rationing which may occur in the short run when the loan rate has not been fully adjusted to the long-run optimal level”. In their model (which I found well summarised in this study of credit rationing in Japan by Rimbara and Santomero 1976:6), equilibrium rationing occurs because the banker, unable to distinguish between low and high risk investors, charges a common “pooling solution” interest rate to both investors, which leads to a rationing of the credit available to the riskier investor. According to the same article,
“dynamic rationing is defined as the transitory difference in the quantity of excess demand for funds that arises due to the sluggish movement of R to its new equilibtium position, after a disturbance”.
This model was dismissed on account of the argument that, in banking as in insurance, a pooling solution could not be sustainable due to the fact that any competitor would offer a lower rate and capture the least risky investor, leaving the “pooler” to serve only the riskiest investor and thus exposed to too much risk. It is my understanding that the fallacy with this argument is that in the absence of a separating equilibrium, the pooling equilibrium actually leads to overinvestment, as I will argue later on, as market operators continue to lower prices (interest rates or insurance premia) , thus more or less competing towards a pooling equilibrium. This is a generally decent assumption for the financial sector in light of the fact that the asymmetries of information are probably exacerbated by a more or less perfectly competitive market for borrowing, where any single lender does not have the power to demand more information from lenders in order to avoid adverse selection. The same point is however less likely to be relevant in the private banking sector where the banks have access to an enormous volume of information about their debtors.
Over-Lending: The De Meza and Webb 1987 model
The De Meza and Webb 1987 model responded to the original, Stiglitz and Weiss 1981 article ( a quick description of the Stiglitz and Weiss 1981 model here). What separates the models are the assumptions made, “in Stiglitz and Weiss all projects have the same expected return but the dispersion of returns is different, whereas in our model the expected returns differ between projects.” This is the main cause of the differences between the models. Because De Meza and Webb 1987 considers both models, it is the one I discuss here, but I do not discuss the Stiglitz and Weiss 1981 model here. A simplified and selected version of the model is described in 3 parts, below, discussiong the entrepreneurs, the banks, and how overinvestment is a feature of this market.
Entrepreneurs
Assume that there is a continuum of entrepreneurs, all of whom have a project they want to invest in. The investment they need to carry out their project costs “K”. Each project can be either successful or a failure, which is surmised by whether it yields returns “Rs” or “Rf”, respectively. Moreover, Rs occurs with probability “p”, while Rf occurs with probability “1-p”. Thus we have
Rs > Rf > 0
and
1 > p > 1-p > 0
Although the project requires an initial investment of K, the entrepreneurs only own “W” amounts of wealth, such that W < K. This means that in order to finance their project, entrepreneurs will have to contract debt, B, in the case of this model, with a bank, such that B = K – W.
The debt contract, “D”, is such that if the project is successful the entrepreneurs will repay 
where “ri” is the interest rate on the bank loan entrepreneurs contracted with the bank.
Perfect Information Case
For the sake of this model, the unsuccessful project yields a Rf and pays interest rate on loan rf,such that
As a result the entrepreneur must be declared insolvent and default on its debt, only paying Rf. The successful project however, will yield Rs at interest rs, such that
With limited liability, it is assumed that in case of success, the entrepreneur makes a profit, “Πi”, such that
For each of the specific cases, we obtain the following insights:
while in the case of failure, the entrepreneur make a null profit because,
(The limited liability lies in the fact that the bank can only reach for the assets related with the project, not any other part of the entrepreneur’s own wealth).
Thus, with limited liability, the return to entrepreneur i’s project can be summarised as such:
As a result the expected profit is
The participation condition (PC), is the necessary requirement to satisfy an entrepreneur to the point where he carries his project, must be such that
where “ ” is the “safe” rate of interests on deposit or savings. In other words, investing in a project must be at least as good as not investing at all, and just “sitting on the money”.
Therefore, with perfect information, only the successful projects will apply for financing and so, the market clears.
Asymmetric Information: From Separating to Pooling Equilibrium through competition
Suppose now that the banks cannot tell the difference between the successful and the unsuccessful projects. In this incarnation of the model, the authors argue that “it is easily shown that a separating equilibrium does not exist”. This is supposedly “so easy” that they don’t bother to show it and only consider the pooling equilibrium. The normal (graduate book) explanation for this is an application of Adverse Selection to the banking sector. If the bank decides to offer two contracts, one for the Good and another for the Bad investor, there is no set of contracts that can be offered where the Bad investor does not have an incentive to mask as the Good investor. Assuming that the banks are only interested in offering loans to the Good investors, with successful projects, a separating equilibrium requires that a number of such incentive and participating conditions must hold.
In other words, the interest rate charged to the successful project must be such that it is still high enough to make the Bad investor unable to pay for that debt contract with his failure returns.
To the best of my understanding, however , in a competitive investment market, competing banks would enter the market bringing the interest rate charged to Good investors down to its limit were it is equal to that charged to the bad projects.
Thus, it is my understanding of this model that in a competitive banking market “the separating equilibrium does not exists” because it will converge to the pooling equilibrium:
Banks
In this competitive banking environment, banks who lend the money to the entrepreneurs are assumed to be identical in size and funds, competitive (deposit rate “s” takers, rather than setters) and risk neutral (so they’ll assume the risk in the presence of risk averse depositors) expected profit (“Π”) maximisers. Although they have no prior knowledge of each specific project, they know the distribution of characteristics of the population of entrepreneurs, i.e.: the distribution of success and failure probabilities, “F(p)” and its density function “f(p)”. As a result, the pooling equilibrium for the loan contract of quantity B at interest rates r yields the following bank profit:
Where p* “is the success probability of the marginal project (that is the project for which the participation condition (PC) holds with equality). Moreover, it must be, by definition that
Which implies that
Which implies that at the margin the perfect competition banking equilibrium (when there can be no more profits will be
The Logic of Overlending
So why is there over lending in the pooling equilibrium? Because in the pooling equilibrium the banks will be making a profit in the marginal loan they make. How? Here’s the logic(where before I had “rs” I will now replace by “r” as there is only one interest rate being charged, the pooling equilibrium one):
So,
Subtracting the original efficient pooling equilibrium condition by this later relationship, we get
This however violates the banking equilibrium. Because at this stage the banks are still making a profit, they continue to lend beyond the perfect competition point described above.
As a result, there it is possible to say that there is overlending, Q.E.D.
Recent insights and implications for SME financing
A more recent version of De Meza & Webb’s insights from 2003 uses a model of Moral Hazard cum adverse selection model to make a more sophisticated and complete version of this argument. Because my point was just to show that the banking/financial sector has a propensity to overlend, I have not included it in this discussion. However, for the most mathematical and theoretical readers it might be of interest. The authors basically prove the point above and then go on to show the “near impossibility of credit rationing”, if investors can: (i) delay their investment, (ii) scaling down the project or (iii) reduce consumption.
These provide a further logic for the differences in access to credit of large corporations and private persons, other than the extent of informational asymmetries. During financial crises, commercial banks are disproportionately unwilling to lend given that commercial creditors are likely to have less room for manoeuvre than large corporations that have the resources to access financial markets directly.
Conclusion
Overinvestment matters because it exposes the market to excess risk, which would probably be mitigated if the market was perfect. This imperfection matters because it implies that financial bubbles are intrinsic to an unregulated financial system. The problem is that bubbles inevitably burst. This imperfection matter because it increases the risk of bankruptcies and bank runs that threaten the stability of our economies, societies and livelihoods in general.
PS: Any errors are my own and are in no way, necessarily, a reflection of errors committed by the authors referred to above.
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