I think Krugman’s problem is that he doesn’t have a formal model that can help him understand what makes banks so important. Instead he refers to Diamond-Dybvig and Tobin-Brainard. Diamond-Dybvig is an excellent workhorse model, but it’s a model of why banks face runs and doesn’t really attempt to take a close look at the role banks play in the broader economy. Since I’m not very familiar with Tobin-Brainard, I’m going to discuss Krugman’s description of it:
if we ask, “Is the volume of bank lending determined by the amount the public chooses to deposit in banks, or is the amount deposited in banks determined by the amount banks choose to lend?”, the answer is once again “Yes”; financial prices adjust to make those choices consistent.
But this discussion abstracts from the fact that the funds the public is able to deposit in banks depend fundamentally on whether or not the bank is willing to lend those funds. A classic example is the difficulty of borrowing against one’s future wage income. For purposes of consumption smoothing members of the public would often like to hold as deposits cash representing future wage income that was posted to a checking accounts against a loan that has to be paid back to the bank. However, when the banks refuse to provide this form of lending, the public doesn’t get to hold these deposits in banks or take advantage of the consumption smoothing opportunities that such deposits would allow. In short, the criteria that banks use for making lending decisions are (rather obviously) crucial inputs determining the size of the deposit base. Because of these dynamics, it appears that “the answer is ‘Yes'” only as long as the banks’ lending criteria are held constant. This, of course, is a unexceptional modelling decision, but it is incumbent on Krugman to understand that Tobin-Brainard depends fundamentally on not modelling the details of bank lending. [Note that Steve Waldman gives a more general explanation of the problem with Tobin-Brainard: “it must be reasonable to model the nonbank private sector as if it were a unified actor with preferences independent of the behavior of the banking system.”]
For a better understanding of the central role that banks play in the monetary system, I suggest that Krugman take a look at a paper I wrote half a decade ago, An Idealized View of Financial Intermediation. This paper looks only at the role banks play in facilitating payments and abstracts entirely from portfolios and the banking system’s role in long-term investment. All banks have is creditworthiness, but by adding this to the economy banks are able to offer credit lines and make it possible for everybody’s debts to be accounted for and made enforceable.
(More specifically, I introduce a monetary friction into a general equilibrium model of a repeated endowment economy and demonstrate that credit lines underwritten by banks are a better solution to the monetary problem than cash: as long as loss of access to bank lending is a sufficient penalty to induce bank customers to repay debt, banks need only set a credit limit for everyone that’s high enough for the richest person to overcome the monetary friction and then everyone will choose to use the right amount of credit. The model also shows why bank customers will choose not to default: loss of access to bank services restricts your ability to participate in the broader economy, and leaves you with a very bad set of choices. Because there is no investment in this environment the only role played by the infinite horizon is to provide for a future punishment for bank customers who do not repay debt.)
This is a formal model, so I abstract from issues like investment, portfolios, and bank default. On the other hand, since Krugman seems to need a model to explain to him in what way banks add something both unique and extraordinarily important to the economy, this paper provides an answer: banks make credit and credit-based transactions possible, because they are trustworthy. In short, banks are the cornerstones of the economy’s circulatory system, which is, I think, also Cullen Roche’s point.