Mean Variance Optimization of Non-Linear Systems and Worst-case Analysis



implications of exogenous mergers under imperfect competition with the financial intermedi-
ation literature characterizing banks as liquidity providers. As in Deneckere and Davidson
(1985) and Perry and Porter (1985), banks have incentives to merge to acquire market
power. Unlike these papers, however, in our model banks’ incentives to merge are also
driven by financing cost advantages related to size, and in particular, by the gains from the
optimal adjustment of reserve holdings due to the presence of an internal money market.
In this sense, our paper also links the industrial organization literature on mergers with
the contributions of Yanelle (1989, 1997) and Winton (1995, 1997) on the relation between
competition and diversification in finite economies.

The field of research studying the role of banks as liquidity providers started with Dia-
mond and Dybvig (1983). More recently Kashyap, Rajan and Stein (2002) describe the links
between banks’ liquidity provision to depositors and their liquidity provision to borrowers
through credit lines; and Diamond (1997) discusses the relationship between the activities
of Diamond-and-Dybvig-type banks and liquidity of financial markets. Concerning liquidity
provision by public authorities, Holmstrom and Tirole (1998) analyze the role of government
debt management in meeting the liquidity needs of the productive sector. However, this lit-
erature has not considered one of our main concerns here: The implications of imperfect
competition and financial consolidation for private and public provision of liquidity.

Several authors have studied the rationale for an interbank market and its effect on
reserve holdings. For example, Bhattacharya and Gale (1987) show that banks can optimally
cope with liquidity shocks by borrowing and lending reserves; but they also argue that
moral hazard and adverse selection lead to under-investment in reserves. Bhattacharya
and Fulghieri (1994) add that with some changed assumptions reserve holdings can also
become excessive. These authors argue that the central bank has a role in healing these
imperfections. Allen and Gale (2000) and Freixas et al. (2000) analyze how small unexpected
liquidity shocks can lead to liquidity shortages in the banking system and thus, in the absence
of a central bank, to contagious crises. We discuss how the likelihood and the extent of such
shortages vary with changes in market structure when a central bank stands ready to offset
private market liquidity fluctuations through monetary operations.

The paper is also related to the literature on internal capital markets. Gertner et al.
(1994) and Stein (1997) discuss the efficiency-enhancing role of these internal markets. While



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