An Incentive Problem in the Dynamic Theory of Banking

Author

Ernst-Ludwig VON THADDEN — DEEP, University of Lausanne and CEPR

Date

December 2000

This paper has now been published and is no longer available as a part of our Research Paper Series. The reference to this paper is:

Von Thadden, E.-L. (2002): An Incentive Problem in the Dynamic Theory of Banking, Journal of Mathematical Economics, Vol. 38, 2002, 271-292.

Abstract

This paper develops a continuous-time model of liquidity provision by banks, in which customers can deposit and withdraw their funds strategically. The strategic withdrawal option introduces an incentive-compatibility problem that turns the problem of designing deposit contracts into a non-standard, non-convex optimal control problem. The paper develops a solution method for this problem and shows that, in this more general frame-work, the insights obtained from the traditional banking models change considerably, up to the point of liquidity provision becoming impossible. The continuous-time framework allows to discuss the problem elegantly and may help to make this part of the banking literature more operational in the sense of modern asset pricing theory.

Executive Summary

This paper revisits a centuries-old question in banking theory, by inquiring into the role and scope of maturity transformation by commercial banks. In a seminal paper, Diamond and Dybvig (1983), building on earlier work by Bryant (1980), have explained maturity transformation arguing that banks can provide intertemporal risk sharing possibilities to investors by taking on an illiquid portfolio. In their model, efficient intertemporal consumption allocations in a simple two-period general equilibrium model can be achieved through standard demand deposit contracts, and despite its illiquidity the banking system is essentially stable. This insurance argument provides the modern theoretical underpinning for what traditional banking theory has called maturity transformation: the creation of short-term liabilities backed by long-term investments.

The present paper generalizes Diamond and Dybvig’s model to the case of continuous time and studies the scope and structure of liquidity provision in a fully dynamic framework. This is of interest, because the Diamond-Dybvig model is, as far as incentive problems are concerned, essentially static. In that model, investors choose bank-intermediated investment in an ex-ante sense, because it provides higher expected utility than direct investment. Once the uncertainty concerning future consumption is resolved, there is no more interaction in the model.

However, if one introduces the possibility of repeated investment and ongoing uncertainty, investor behavior will be more complex, and some of the more traditional concerns by bankers again become relevant. Precisely because bank deposits provide greater liquidity than the underlying direct investment opportunities, a depositor has an incentive to withdraw her deposit, even without liquidity needs, thus realizing the liquidity premium the deposit provides, and to re-invest it directly. This arbitrage behaviour has a potentially destabilizing effect, because liquidity provision implies a transformation of the return structure of the intermediary’s asset base that reflects the liquidity preferences of depositors. If the depositors have incentives to misrepresent their preferences, the intermediary itself risks illiquidity.

This is the basic incentive problem studied in this paper. The problem studied here does not arise in the original model by Diamond-Dybvig (1983), because reinvestment after one period there makes no sense. The problem is, however, a principal one: liquidity provision is by its very nature a dynamic (i.e. ongoing) phenomenon, and incentive issues are at the core of banking. Hence, a theory of liquidity provision that does not address dynamic incentive issues is at best incomplete.

The main incentive problem for an individual depositor is a tradeoff between sticking to intermediated investment and withdrawing the funds for direct investment. This tradeoff is present at all dates at which the investor has not yet consumed her deposits. If the investor leaves her funds with the bank she gets high liquidity at lower levels of over-all returns; if she withdraws and re-invests them she obtains extra returns which are less liquid. It is, therefore, natural to study liquidity in a model with infinitely many periods.

The continuous-time model developed here is a convenient and relatively elegant frame-work, with advantages similar to those of continuous-time formulations a la Merton in asset pricing, albeit much simpler. Yet, even this relatively simple continuous-time model allows for a relatively rich analysis of the instabilities inherent in the bank-depositor relationship, and of the interplay between technological factors, consumer preferences, and uncertainty influencing them.

However, taking the incentives for strategic deviation by depositors into account, turns the problem of designing deposit contracts from a standard, convex problem into a non-convex second-best control problem with a complex constraint set. Most of Section III of the present paper is devoted to solving this problem, for which the literature does not seem to provide a solution method.

Several features of the solution are of interest. First, there is little general structure.

In particular, the set of dates at which the incentive compatibility constraints bind may consist of arbitrarily many time-intervals, alternating with intervals in which first-best liquidity provision can be achieved locally. While the (unconstrained) first-best solution is largely unaffected by the distribution of investors’ liquidity needs, the second-best strongly depends on them.

Second, while there is little general structure, two alternative additional restrictions on the data of the problem (explored in Propositions 3 and 4) yield a remarkably simple structure. First, incentive compatibility is binding over the whole time-horizon if and only if the investors’ intertemporal risk aversion is sufficiently large and the degree of irreversibility of investment not too large. In particular, the question of whether incentive compatibility is always binding is determined solely by a simple condition on preferences and technology.

Somewhat orthogonally, Proposition 4 presents a condition on the distribution of consumption uncertainty — regardless of preferences and technology — that implies a simple temporal structure of incentive compatibility. If the distribution of investors’ consumption needs over time has a log-concave density (i.e. is essentially single-peaked), then there is only one regime switch: early on, incentive compatibility binds, later on, when investment has been in place for a sufficiently long time, incentive compatibility does not bind.

The main economic conclusion from this analysis is that the scope for liquidity transformation, as determined by the second-best, may be much very different from what the first-best suggests is desirable. From the first-best analysis (a la Diamond and Dybvig (1983)), liquidity transformation is, ceteris paribus, socially the more valuable the larger the investors’ intertemporal risk aversion. However, as the second-best analysis shows, the scope for liquidity transformation decreases with intertemporal risk-aversion and is, independently of investors’ preferences, restricted by the degree of irreversibility of investment. The lower the latter, the less liquidity provision is possible. In the extreme case in which the economy has only short-term investment possibilities (the productivity of long-term investment equals the productivity of short-term investment, which means that investment is fully reversible), liquidity transformation is completely impossible, and deposit arrangements can only replicate the autarky allocation.

These results cast doubts on the robustness of the dynamic features of the Diamond Dybvig (1983) model, and therefore, in the spirit of finance theory after Merton (1990), on the practical usefulness of the basic version of that model. On the other hand, the results are compatible with the empirical observation that deposit contracts are typically not front-loaded, as the notion of liquidity by Diamond and Dybvig (1983) would suggest. The model may therefore provide a useful starting point for making Diamond and Dybvig (1983) operational.