Selgin’s flawed criticism of endogenous money

George Selgin has recently been arguing against the now mainstream view that banks are not intermediaries between lenders and borrowers, and in fact create credit out of nothing. He has criticized this Bank of England working paper, calling the paper rubish.

It is somewhat difficult to parse exactly what George is arguing. He appears to be arguing that individual banks cannot create infinite amounts of money, because the money will then be deposited at other banks. However, no-one who believes money is endogenous would deny this. The point of endogenous money advocates is that the banking system as a whole can create infinite credit, not that individual banks can do so.

The subthread of the discussion that best got to the heart of the matter is this exchange between George and Eric Tymoigne. The root of the remaining disagreement is George’s claim that the banking system cannot evade the problems caused by a single bank issuing a loan if all banks expand credit at the same rate. George appeals to a paper he wrote to justify this view.

Just saw this(not frequently here these days:turns me into a phone-checking maniac).Your main grudge is with the funding structure,which may get lost on some. As the adage goes"banks must grow in step with each other",otherwise regulatory & profit constraints bit if grow too fast
— Eric Tymoigne (@tymoignee) April 17, 2022

Unfortunately for George,the paper is full of outright errors, and conceptual problems that mean its conclusion does not hold. I briefly go through several of these errors below.

The right hand side of equation 2 should read $p/r$ instead of $r/p$ I believe. The argument that $R_j$ must be positive when $p>2r$ only holds with $r/p$ in the above expression. This error is just a typo and doesn’t change the results of the paper. I mention it only to facilitate discussion of the next issue.

The most glaring error in the paper is when George derives equation 2 from equation 1. I assume George was doing something like the following to get from equation 1 to equation 2. Equation 1 reads

$R_j r + \int_{R_j}^{\infty} p(X-R_j) \phi (x)\,dx \$

To find the FOC we take the derivative with respect to $R_j$ and set it equal to zero. Taking this derivative correctly is difficult, and will lead to a very complicated expression since $R_j$ is in the bounds of integration. If we simply ignore the fact when taking the derivative we get

$r - \int_{R_j}^{\infty} p \phi (x)\,dx \ = 0$ which can be rearranged to give

$\frac{r}{p}= \int_{R_j}^{\infty} \phi (x)\,dx \$ ,

which is equation 2 (after correcting the typo mentioned above). Unfortunately, ignoring the $R_j$ in the bounds of integration is simply not correct. To see this consider a simple derivative/integral expression such as

$\frac{d}{dx} \int_{0}^{x} y\,dy \$

If we simple ignore the x that appears in the bounds of integration this is equal to zero, however if we actually compute the integral before differentiating we get a different answer.

Even if there wasn’t a math error in the paper, George’s conclusion implies that banks want to hold negative reserves if the rate of interest banks pay on overnight loans is less than double the rate they pay on loans. George argues that neglecting frictions is the reason for this result, but he fails to demonstrate that including frictions changes anything. Taken seriously, this means his result should be used to conclude that in the empirically relevant case banks want to hold no reserves.

Finding the optimal level of reserves can’t even theoretically be used to conclude anything about bank lending. Suppose the supply of reserves is exogenously fixed. Showing banks would want to hold more reserves if given the choice does not show that banks will not make loans if their reserves are below that amount. In order to conclude that bank lending is limited by reserves you need to actually show that the amount of loans a bank will make is limited by the amount of reserves.

On a more conceptual level, the argument that higher variance in the deposits that a bank holds prevents bank expansion of credit is flawed because George fails to consider an important element of a the banks optimization problem. Banks have higher costs as the variance of the amount of reserves held in them increases, but since banks are borrowing in the overnight market from other banks, banks also earn more from lending to other banks as the variance increases. In general, these two factors will cancel each other out. To treat this issue formally would be relatively complicated, and is not required given the deficiencies of the formal argument in George’s paper.

Each of these problems individually is enough to invalidate the claims of the paper. Since this paper appears to be a crucial element of the argument behind George’s worldview, it seems either he must address the issues with it or concede that his view of the banking system is incorrect. Given how loudly he has proclaimed views based on this flawed paper I would encourage him to do so as publicly as possible.

Selgin, G. (2001). In-Concert Overexpansion and the Precautionary Demand for Bank Reserves. Journal of Money, Credit and Banking, 33(2), 294. doi:10.2307/2673887

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