Destabilization of an urban subsystem by a virtual collapse in its space

Abstract

It is shown that a virtual collapse in the space–time structure of a geographically distributed system may induce changes in the information flow field of the system and lead to instability. A case study of the commercial banking system of New York City demonstrates how a virtual collapse in the space–time structure of its check clearing process led to a transformed information flow environment in which banks had, for the first time, accurate and timely information concerning the decision-making processes of other banks. A dynamical systems model representing the behavior of the banks, both before and after the transformation, demonstrates how their competitive use of this information led to decision-making based on the decisions of others. This process of internal referencing resulted in a structurally unstable state in which perturbations, arising from asymmetric responses to seasonally changing deposit flows, accumulated with little or no decay. This caused loans to diverge from deposits and made the system susceptible to random financial perturbations that led to a banking panic. While the behavior of the aggregate system was structurally unstable, the behavior of the individual banks was stable, with the banks behaving as a coherent group. The internal referencing structural instability is of general significance for the aggregate behavior of geographically distributed systems that undergo a virtual collapse in their space–time structures and a transformation of their information flow fields.

Keywords

Growth and change analysis multiagent systems socioeconomic analysis urban dynamics self-organization

Introduction

The space–time structure of a geographical space evolves continuously as organizational and technological innovations influence human activities within the space. An important class of transformations that characterize this evolution are those that change relevant measures of separation between events in space–time, and especially measures of cost associated with flows of information.

A significant body of literature characterizes the manner in which such transformations change, and typically diminish, the costs of interaction in terms of the concepts of space–time compression (Harvey, 2001; Wiggins, 2008) and space–time convergence (Janelle, 2014). Common to both concepts is the idea that decreasing the costs of interaction effectively results in a virtual contraction of the space.

An especially interesting class of transformations are those by which the effective costs are so reduced that one may say that the space has virtually collapsed. Under such conditions, the decision-making agents in the space are no longer separated by an effective distance and one may expect them to possess a more complete knowledge of each other’s decisions. Their behavior may then change as they incorporate such information into their own decision-making.

Examples of virtual collapse are of great interest because the resulting behavioral changes may lead to significant changes in the qualitative dynamics and stability of the system. It is clearly of interest to understand the processes by which such changes unfold.

This paper investigates the response of the commercial banking system of New York City (NYC) to a virtual collapse in the space–time structure of its check clearing process. It is shown that the resulting transformation of its information flow fields led to a structural instability (see Mattuck et al., 2011; Wiggins, 2008) in which decision-makers increasingly based their decisions on the decisions of others.

As a result, the banks behaved increasingly as a coherent group in which previously stabilizing mechanisms were neutralized. The aggregate behavior of the system was then driven by asymmetrical perturbations that accumulated in the system instead of dissipating, leading to an inevitable financial crisis.

The structural instability underlying this behavior, which is termed the internal referencing structural instability (IRSI), has implications for many social systems in which decision-makers base their decisions on the decisions of others.

The basic assumptions and methodology of paper

A reasonable strategy for understanding the consequences of a virtual collapse of some part of the space–time structure of a system is to identify and investigate specific cases of collapse for which: (1) the transformation of the geographical space and its associated information flows are well-defined and well-documented and (2) it is possible to model mathematically the effects of the transformation on the dynamics and stability of the system.

It is clear that this strategy typically entails investigations of past events. As such, it is based on an assumption that human behavior exhibits fundamental invariances over differing organizations of space–time. This is analogous to the invariance of physical laws over different frames of reference according to the general theory of relativity (see, for example Stenger, 2007). In terms of the present case, for example, it is assumed that the fundamental profit-seeking behavior within a banking system is invariant over different organizations of space–time.

Such an assumption is supported by Riddiough and Thompson (2012) who note strong analogies between the behavior of the financial institutions that led to the financial crisis of 2007–8 and the behavior of the commercial banks examined in this paper that led to the crisis of 1857. While the behavior leading to the latter crisis occurred a 150 years earlier and under a significantly different organization of space–time, it provides remarkable insight into the behavior underlying the crisis of 2007–8.

As noted in the “A virtual collapse of the space–time structure of the banks” section, the virtual collapse of the space–time structure in which the banks cleared their checks and the resulting response of the banks to this contraction are both well-defined and well-documented. The virtual collapse of the space–time structure of check clearing process occurred essentially instantaneously with the establishment of the Clearing House Association (CHA) in October 1853. This allows one to characterize the behavior of the banks as taking place in two distinct space–time structures, with one occurring before and one occurring after this event.

The analysis is further facilitated by the fact that an understanding of the behavior of the banks in each of the space–time structures does not require a consideration of the locations of the banks in geographical space. The essential spatial component of the phenomena is encapsulated in the virtual collapse of the space–time structure of the system. The focus of the analysis is on the consequences of the relatively sudden change in space–time structure and the resulting changes in information flows.

Finally, the banking system of the 1850s in NYC was sufficiently simple and well-defined that one may construct, with some confidence, time-dependent mathematical models of the behavior of the system.

A virtual collapse of the space–time structure of the banks

In October 1853, the space–time structure in which the commercial banks of NYC operated underwent a major transformation. In particular, the banks organized themselves into a CHA (see Cannon, 1911; Myers, 1923) by which 51 of 57 banks settled their balances every day in a room at a central location. This innovation induced a virtual collapse in the space–time structure of the clearing process of the banks, from an area the size of NYC to an area the size of a room, and from exchange times of many hours to exchange times of minutes.

Before October 1853, each bank cleared its balances with other banks in a bilateral manner. Employees from each bank travelled to every other bank in NYC, exchanging checks drawn on the deposits of the other bank for gold and, in turn, exchanging gold for checks drawn on its own deposits. It was frequently the case that daily exchanges occurred.

The information that any bank possessed concerning the other banks before 1853 was highly constrained. In particular, the banks lacked useful information concerning the proportion of deposits that were loaned out by other banks and the reserves of gold that they kept to safeguard against runs on their deposits. While each bank had partial information concerning the banks with which it settled balances, accurate and timely information concerning the state of most of the banking system was unavailable. Quarterly statements were required by New York State law, but were too inaccurate and infrequent to be of value in banking operations.

Information flows between the banks increased greatly following the virtual space–time collapse, since the CHA recorded all transactions and posted daily proof sheets of exchanges between the banks. From August 1853 onwards, each bank was required to publish weekly statements in a local newspaper, and the CHA’s proof sheets made their falsification difficult. The CHA also collected weekly data on loans, deposits, and reserves that were published, in aggregate form, in influential magazines. As a result, every bank had access, for the first time, to accurate, timely, and global information characterizing the lending and reserve decisions of other banks in NYC.

The behavior to be explained: Change and crisis

The virtual collapse in the space–time structure of the banking system and the resulting transformation of its information flow field led to major changes in the dynamics of the banks. The main problem to be explained is the divergence of aggregate loans from deposits and reserves illustrated in Figure 1 from mid-1854 onwards and the rapid contraction in loans and deposits during the banking panic of late 1857, which is represented by their sharp decline beginning in about the 190th week of the time series.

Figure 1.

The divergence of loans from deposits and reserves for the NYC clearing house banks 1854–1858 and the crisis of 1857, as deviations from the mean values of 1855. Seasonal variations in the deposits, and hence in the loans, are well-defined in this time series.

The divergence is surprising since it is clear from Gibbons (1858) that bankers knew the dangers of loaning out too high a proportion of their deposits. A financial disturbance in 1854 had, in fact, caused the banks to decrease the loan-to-deposit ratio (LDR) to 0.69 by early 1855.

A question of interest concerns the existence of causal connections between (1) the virtual collapse of the space–time structure; (2) the resulting transformation of the information flow field; (3) the seasonally driven increases in the LDR; and (4) the resulting financial instability, all of which are represented in Figure 1.

Previous analyses of the financial crisis and panic of 1857

There is a significant literature on the crisis of 1857 and its causes including, for example the analyses of Calomaris and Schweikart (1991), Gibbons (1858), Hutcheson (1936), Huston (1987), Kelly and O’Grada (2000), Mishkin (1990), Riddiough and Thompson (2012), and Van Vleck (1943). This literature is, however, largely focused on traditional economic and historical descriptions of the processes and events leading to the panic.

There appears to be little discussion of the effects of the technological and organizational innovations that were transforming the space–time and information flow structures in which the NYC banking system was embedded. One of the few researchers (Hutcheson, 1936) to comment on the role of technological innovation in the panic merely noted that it was the first American banking panic to spread rapidly throughout the US as a result of the telegraph.

The only comment concerning the relation between the organizational innovation of the CHA and the crisis of 1857 appears to have been made by Gibbons (1858), who suggested a causal relation between the two events. As noted in the “A model-based explanation for the behavior of the banks” section, his suggestion indicates an intuitive understanding of the model of banking behavior that is discussed in this paper as well as the concept of an IRSI.

The hypothesis that an IRSI underlay the behavior of the banks in the crisis was investigated by Smith (1977), although he did not suggest that it resulted from a virtual collapse in the space–time structure of the banking system. The current paper not only investigates a significant generalization of the model of Smith (1977), but also provides analytic solutions to the model that provide significant insight into the origin and nature of the IRSI.

A dynamical systems model of the NYC banks 1853–58

The description of banking operations described above together with the CHA data illustrated in Figure 1, suggests representing the behavior of the banks in terms of a dynamical systems model of the temporal changes in loans and deposits. The creation of deposits and the issue of loans in a system of many banks operating in continuous time are well-approximated by continuous functions of time (see, for example Sumarti and Hasmi, 2015; Sumarti et al., 2015).

The determinants of the LDR

A critical choice for any commercial bank was the proportion of its deposits that it wished to loan out, as represented by the relation

l_{i} = γ_{i} d_{i}, i = 1, n

(1)

in which l_i and d_i are the loans and deposits, respectively, of the ith bank; γ_i is the bank’s choice of the critical LDR; and n is the number of banks.

It is clear from Gibbons (1858) that bankers understood that the choice of a rational LDR depended on the cost of holding insufficient reserves and the opportunity cost of not lending out the bank’s reserves. While they may not have applied a formal theory in deciding upon an LDR, they possessed a large body of knowledge, derived from the experience of many bankers, that was applied heuristically. An important question concerns the sources of information about the state of the financial environment that were employed by the banks, together with their experiential knowledge, in choosing an LDR.

One may distinguish between sources of information available before and after the formation of the CHA. The available pre-CHA sources were limited, and characterized by a lack of accurate and timely information concerning the condition of the banks, as noted in the “A virtual collapse of the space–time structure of the banks” section. The most accurate and complete information would be the bank’s own records of its deposits and loans, although some information would derive from the clearing process. It is therefore assumed that the ith bank’s choice for a value of its LDR, denoted by $\hat{γ_{i}}$ , was based largely on this source and on the experiential knowledge of its bankers.

The post-CHA sources of information were significantly expanded, as noted in the “A virtual collapse of the space–time structure of the banks” section, with information concerning loans, deposits, and reserves of all banks published on a weekly basis, as well as the information made available by the CHA, newspapers, and bankers magazines. One may assume that the LDRs of other banks were viewed as providing accurate and timely information on safe and profitable values of the ratio.

It would have been natural to combine such information into a weighted average of the LDRs of all banks in the Association, with weights indicating the relative size of a bank’s deposits. The resulting ratio, based on information concerning every bank, would provide a guide to setting a bank’s LDR.

Combining this calculation with the assumption of an equilibrium state in which banks have loaned out their target proportion of deposits $l_{i} = γ_{i} d_{i}$ leads to a simple relation

\bar{γ} = Σ_{i} \frac{d_{i}}{D} γ_{i} = Σ_{i} \frac{d_{i}}{D} \frac{l_{i}}{d_{i}} = \frac{Σ_{i} l_{i}}{D} = \frac{L}{D}

(2)

in which

D = Σ_{i} d_{i}

and

L = Σ_{i} l_{i}

are the aggregate loans and deposits in the system.

The aggregate LDR $\bar{γ} = L / D$ was an easily obtainable and tempting piece of information for bankers to apply when making decisions about loans since it was (1) a natural analogue of the pre-CHA individual ratio $\hat{γ_{i}} = l_{i} / d_{i}$ , (2) based on a great deal more information than ${\hat{γ}}_{i}$ , and (3) published on a weekly basis.

It is reasonable to assume that the banks integrated the pre- and post-CHA sources of information in deciding on the proportion of deposits to loan out and that the value of $γ_{i}$ employed at any given time was a weighted average of the values ${\hat{γ}}_{i}$ and $\bar{γ}$ , namely

γ_{i} (μ) = (1 - μ_{i}) \hat{γ_{i}} + μ_{i} \bar{γ} = (1 - μ_{i}) \hat{γ_{i}} + μ_{i} L / D, 0 \leq μ_{i} \leq 1

(3)

in which the parameter μ_i to be assumed to be a slowly varying function of time that reflected the bankers’ relative confidence in the two LDRs. The value of μ_i may also be interpreted as a measure of market contagion and hence as characterizing the banks as a group.

The dynamical systems model for individual banks

In modeling the weekly changes in the loans l_i of a given bank, it is reasonable to assume that (1) loans were increased or decreased in response to changes in deposits so that the desired LDR was maintained and (2) corrections to loans were made when the ratio of loans to deposits deviated from the desired LDR γ_i.

It is further reasonable to assume that these relations were linear. A linear response in changing the level of loans in response to changes in the level of deposits is appropriate in maintaining the linear relation (1) of loans to deposits. Furthermore, if one chooses the time scale of the model to be that at which loan decisions were made, which was typically less than a week, it is reasonable to assume that with a difference of N dollars between the actual and desired level of loans, a bank would typically adjust its loan portfolio by the same amount, if N were not too large. Such an adjustment leaves the LDR intact and no interest income is lost. While a non-linear response is possible for N sufficiently large, frequent adjustments to loan portfolios would typically lead to relatively small adjustments in the linear range.

These assumptions lead immediately to the continuous-time model

{\dot{l}}_{i} = γ_{i} \dot{d_{i}} + (γ_{i} d_{i} - l_{i}) + ϵ_{i}, i = 1, n

(4)

in which

{\dot{l}}_{i}

and

{\dot{d}}_{i}

represent, respectively, the time derivatives of loans and deposits; n is the number of banks; and the term ϵ_i represents small perturbations to the system. The first term on the LHS indicates that changes in loans occur with changes in deposits while the second term indicates that loans change when the desired LDR does not equal the actual LDR.

Combining this equation with the relation (3) for the LDR leads to a loan equation for the ith bank

{\dot{l}}_{i} = ((1 - μ_{i}) \hat{γ_{i}} + μ_{i} (L / D)) (d_{i} + {\dot{d}}_{i}) - l_{i} + ϵ_{i}, i = 1, n

(5)

that is applicable to the pre-CHA system for

μ_{i} = 0

and to the post-CHA system for

{0 < μ_{i} \leq 1, i = 1, n}

. One may, without significant loss of generality, simplify the analysis by assuming that

μ_{i} = μ, i = 1, n

, which accords with the viewpoint that μ is a universal parameter representing the contagious confidence of the banks in the post-CHA banking system.

The dynamical systems model for aggregate banking system

A model of the aggregate banking system, whose behavior is illustrated in Figure 1, is obtained by summing equation (5) over each of the banks to obtain a representation of the aggregate loans as a function of the aggregate deposits. Applying the assumption that $μ_{i} = μ$ , summing over i, and re-arranging terms in equation (5) leads to

\dot{L} + (1 - μ - μ (\dot{D} / D)) L = (1 - μ) Σ_{i} {\hat{γ}}_{i} (d_{i} + {\dot{d}}_{i}) + ϵ

(6)

in which L represents the aggregate loans

L = Σ_{i} l_{i}

An analysis of the model: Its solutions and their stability

The behavior of the banks of NYC, illustrated in Figure 1, may be understood by solving and analyzing the models for both the aggregate system and the individual banks. The strategy is to understand the solutions as functions of increasing values of the parameter μ, representing the banks’ increasing use of the partially self-referential information made available following the spatial collapse.

The analysis is organized into the following subsections. In the “The solution for aggregate loans and their dynamic equilibrium” subsection, the solution to equation (6) for aggregate loans is derived and it is shown that, while the dynamical equilibrium component of the solution satisfies the banks LDRs, it is sensitive to perturbations. In the “Perturbations and the stability of the aggregate solution” subsection, the sensitivity of the solutions to perturbations, and the resulting instability of the LDRs, is shown to increase with increasing values of μ. In the “The solutions and stability for the loans of individual banks” subsection, solutions to equation (5) for the individual banks are shown to remain stable around the evolving values of their LDRs. These values are increasingly determined by random perturbations in the aggregate data that the individual banks increasingly employed in setting their LDRs. Finally, in the “The IRSI for μ = 1” subsection, the structurally unstable state μ = 1 is characterized as one in the which the aggregate system is driven entirely by such perturbations while the individual banks behave increasingly as a coherent group of stable entities.

The solution for aggregate loans and their dynamic equilibrium

Equation (6) for aggregate loans is a first order ordinary differential equation whose explicit solution may be represented in terms of an integrating factor (see, for example Tenenbaum and Pollard, 1985). On multiplying the equation through by the integrating factor $D^{- μ} e^{(1 - μ) t}$ and re-arranging terms one obtains the equation

\frac{d}{d t} [L D^{- μ} e^{(1 - μ) t}] = (1 - μ) e^{(1 - μ) t} D^{- μ} Σ_{i} {\hat{γ}}_{i} (d_{i} + {\dot{d}}_{i})

(7)

which may be integrated between the limits

[0, t]

, using integration by parts on the RHS, to obtain the solution

\begin{array}{l} L = Σ_{i} {\hat{γ}}_{i} d_{i} + e^{- (1 - μ) t} {(D / D_{0})}^{μ} Σ_{i} (l_{0 i} - {\hat{γ}}_{i} d_{0 i}) \\ + μ e^{- (1 - μ) t} D^{μ} \int_{0}^{t} e^{(1 - μ) t} D^{μ} Σ_{i} {\hat{γ}}_{i} d_{i} ((\dot{D} / D) - (Σ_{j} {\hat{γ}}_{j} {\dot{d}}_{j} / Σ_{j} {\hat{γ}}_{j} d_{j})) d t \end{array}

(8)

in which

l_{0 i}

and

d_{0 i}

are the loans and deposits of the ith bank at time t = 0 and

D = Σ_{i} d_{i}

The first term on the RHS of the solution (8) represents a necessary, but not sufficient, condition for each bank to achieve and maintain its LDR ${\hat{γ}}_{i} = l_{i} / d_{i}$ . If the second and third terms on the RHS are both zero, the first term is a solution

\bar{L} = Σ_{i} {\hat{γ}}_{i} d_{i} \Leftrightarrow Σ_{i} ({\hat{γ}}_{i} d_{i} - l_{i}) = 0

(9)

that represents a desired state of dynamic equilibrium for the aggregate system. It should be noted that the time rate of change of this dynamic equilibrium is generally not zero, because on setting its time derivative to zero

\dot{\bar{L}} = Σ_{i} {\hat{γ}}_{i} {\dot{d}}_{i} = 0 i = 1, n

(10)

it follows that the deposits are not changing over time, which is generally not the case.

In order that the dynamic equilibrium be a solution, the second and third terms of equation (8) must be zero. While the third term

μ e^{- (1 - μ) t} D^{μ} \int_{0}^{t} e^{(1 - μ) t} D^{μ} Σ_{i} {\hat{γ}}_{i} d_{i} ((\dot{D} / D) - (Σ_{j} {\hat{γ}}_{j} {\dot{d}}_{j} / Σ_{j} {\hat{γ}}_{j} d_{j})) d t

(11)

will be zero if the positive and negative contributions to the integral cancel each other out, there is no apparent reason as to why this should be so. One may instead ask whether there are reasonable conditions by which the integrand in equation (11) is zero.

A first possibility is that the banks have identical LDRs ${{\hat{γ}}_{i} = \hat{γ}, i = 1, n}$ . Although the banks of the post-CHA period had reason to track the aggregate LDR L/D and, as shown in the “The solutions and stability for the loans of individual banks” subsection, the LDRs of a set of banks converge to a common LDR as $μ \to 1$ , the condition is unreasonable for the pre-CHA period, when banks had little information about each other’s reserve ratios.

A second possibility is that each bank had a relatively steady, but not necessarily identical, share of the total deposits $d_{i} = k_{i} D$ with $Σ_{i} k_{i} = 1$ and ${\dot{k}}_{i} = 0, i = 1, n$ . This condition appears reasonable for a time scale at which the loan process occurred. Furthermore, if a shift occurs in the deposit proportions of the banks, $k_{i} \to k_{i} + k_{i}'$ , the integrand is still zero since $Σ_{i} {\hat{γ}}_{i} {\dot{d}}_{i} / Σ_{i} {\hat{γ}}_{i} d_{i} = Σ_{i} {\hat{γ}}_{i} (k_{i} + k_{i}') \dot{D} / Σ_{i} {\hat{γ}}_{i} (k_{i} + k_{i}') D = \dot{D} / D$ .

It is henceforth assumed that the condition d_i = k_iD holds and that the third term on the RHS of equation (8) is zero. It follows from this assumption that, in the equilibrium state

\bar{L} = Σ_{i} {\hat{γ}}_{i} d_{i} = Σ_{i} k_{i} {\hat{γ}}_{i} D \equiv \bar{\hat{γ}} D

(12)

in which

\bar{\hat{γ}}

is a weighted average of the individual LDRs.

The second term on the RHS of equation (8)

e^{- (1 - μ) t} {(D / D_{0})}^{μ} (Σ_{i} (l_{0 i} - {\hat{γ}}_{i} d_{0 i}))

(13)

is zero if the loans and deposits at time t = 0 satisfy the desired LDRs of the banks. If this is not the case, because of the occurrence of perturbations

_{l}^{0 i}'

in the initial values of the loans represented as

Σ_{i} l_{0 i} \to Σ_{i} l_{0 i} + Σ_{i}_{l}^{0 i}'

, it follows that

e^{- (1 - μ) t} {(D / D_{0})}^{μ} Σ_{i} (l_{0 i} +_{l}^{0 i}' - {\hat{γ}}_{i} d_{0 i}) = e^{- (1 - μ) t} {(D / D_{0})}^{μ}_{L}^{0}'

(14)

in which

_{L}^{0}' = Σ_{i} l_{0 i}

, since

Σ_{i} (l_{0 i} - {\hat{γ}}_{i} d_{0 i}) = 0

The rate of exponential decay of the discrepancy $_{L}^{0}'$ in equation (14) depends on the value of μ. Since the perturbation decays slowly if the value of μ is close to 1 and will not decay at all if μ = 1, one cannot assume that the second term of the solution (8) is zero if perturbations to the system occur when μ is significantly greater than zero. Hence it is important to investigate solutions to equations (4) and (6) when they are subject to perturbations.

Perturbations and the stability of the aggregate solution

Perturbations would have arisen regularly and frequently as banks made decisions on a weekly basis concerning LDRs and loans. Perturbations include random effects generated by the financial environment of the banks, such as loan delinquencies, as well as systematic structural perturbations to the form of the RHS of equation (5), affecting the rate of change in the loans of an individual bank. The origins and nature of structural perturbations arising at the level of the individual banks are discussed in the “Modeling the perturbations as a relative income effect (RIE)” section.

A critical question is whether perturbations affect the stability of the dynamic equilibrium $\bar{L} = \bar{\hat{γ}} D$ . In relation to equation (6) for aggregate loans, the issue may be investigated by allowing the term ϵ to represent the occurrence of perturbations to the loans at some arbitrary set of times ${t = t_{i}, i = 1, n}$

ϵ = Σ_{i} L_{i}^{'} δ (t - t_{i})

(15)

in which

_{L}^{i}'

is the size of the perturbation at time t_i and

δ (t - t_{i})

is the Dirac delta¹ which indicates the time at which the perturbation occurs.

Although the perturbations $_{L}^{i}'$ are assumed to be small, they may be expected to change the value of the loans with $\bar{L} \to \bar{L} + L^{'}$ in which $L' < < \bar{L} = \bar{\hat{γ}} D$ . One may determine the stability of the unperturbed solution to equation (6) by examining whether the perturbed terms $_{L}^{i}'$ grow, in which case the system is unstable, or decay, in which case the system is stable (see Mattuck et al., 2011; Warf, 2003).

Substituting the perturbed solution $L = \bar{L} + L'$ and the form of the perturbations (15) into equation (6) leads to two equations, since $_{L}^{i}' < < \bar{L}$ , with the first describing the dynamic equilibrium $\bar{L} = \bar{\hat{γ}} D$ and the second

\frac{d}{d t} [L' D^{- μ} e^{(1 - μ) t}] = Σ_{j} L_{j}^{'} e^{(1 - μ) t} D^{- μ} δ (t - t_{j})

(16)

describing the evolution of the perturbations

L_{j}^{'}

. The form of this equation follows from the fact that the deposits are unperturbed.

The solution for the perturbations, when added to the dynamic equilibrium solution $\bar{L}$ for the unperturbed loan equation, is

\bar{L} (t) + L' (t) = \bar{\hat{γ}} D (t) + Σ_{j} L_{j}' e^{- (1 - μ) (t - t_{j})} {(D (t) / D (t_{j}))}^{μ}

(17)

in which

L_{j}^{'}

represents a perturbation that is applied at time t = t_i when the deposit level is

D (t_{j})

. This expression is a generalization of the expression (14) representing the effect of perturbations on the second term on the RHS of the solution (15) for aggregate loans.For the pre-CHA banks, characterized by μ = 0, the solution (17) becomes

\bar{L} (t) + L' (t) = \bar{\hat{γ}} D (t) + Σ_{j} L_{j}' e^{- (t - t j)}

(18)

and the perturbations decay and the system is stable.

As μ increases in value, it follows from equation (17) that perturbations decay at decreasing rates, implying a possible accumulation of slowly decaying perturbations. In the full internal referencing state μ = 1, the solution is represented by

L (t) \equiv \bar{L} (t) + L' (t) = \bar{\hat{γ}} D (t) + Σ_{j} (L' (t_{j}) / D (t_{j})) D (t)

(19)

and while the perturbations increase or decrease in magnitude as the deposits increase or decrease, they do not decay, but accumulate in the aggregate system. Although the rate of growth of the accumulated perturbations is not exponential in the internal referencing state, the system is unstable in the sense that perturbations may accumulate without bound over time.

It follows that the aggregate LDR $\bar{γ}$ observed by the banks at any time may differ significantly from the LDR $\bar{\hat{γ}}$ of the dynamic equilibrium. It takes the form

\bar{γ} = L (t) / D (t) = \bar{\hat{γ}} + Σ_{j} (L' (t_{j}) / D (t_{j}))

(20)

and increases monotonically if the perturbations are positive.

A state of neutral stability, in which perturbations do not decay but accumulate and affect the dynamics of the system, is said to be structurally unstable (see, for example Mattuck et al., 2011; Wiggins, 2008) because its dynamics and stability characteristics may be significantly affected by structural perturbations.

The solutions and stability for the loans of individual banks

The decreasing stability of the equilibrium solution (17) for aggregate loans for increasing values of μ raises the issue of the stability of the equilibrium solution for the loans of the individual banks. Employing solutions for the aggregate loans, one may solve equation (5) for the loans of an individual bank, first for the case of the unperturbed equation, with aggregate loans $L = \bar{\hat{γ}} D$ , and then for the case of the perturbed equation, which allows one to determine the stability of the unperturbed solution for the individual banks.

The equation for individual loans in which the LDR involves the unperturbed aggregate loans may be solved using the methods employed in solving the aggregate loan equation (6) and the assumptions $L = \bar{L} = \bar{\hat{γ}} D$ and d_i = k_iD to give the analogous solution

{\bar{l}}_{i} = ((1 - μ) {\hat{γ}}_{i} + μ \bar{\hat{γ}}) d_{i} + e^{- t} (l_{0 i} - ((1 - μ) {\hat{γ}}_{i} + μ \bar{\hat{γ}}) d_{i 0}), i = 1, n

(21)

in which the first term on the RHS represents the dynamic equilibrium of the ith bank with a LDR that is a weighted average, changing with μ, of the individual LDR

{\hat{γ}}_{i}

and the aggregate LDR

\bar{\hat{γ}} = Σ_{i} k_{i} {\hat{γ}}_{i}

. The second term is non-zero only in the case of a perturbation in the initial conditions, in which case it is exponentially decreasing and the solution is stable with respect to such perturbations.

When equation (5) is perturbed with the term $ϵ_{i} = Σ_{j} l_{i j}^{'} δ (t - t_{j})$ not only are there perturbations to the value of $l_{i} \to {\bar{l}}_{i} + l_{i}^{'}$ but also to the value of $L \to \bar{L} + L'$ . In this case, one may employ the result of equation (17) and set $L' = Σ_{j}_{L}^{j}' e^{- (1 - μ) (t - t_{j})} {(D (t) / D (t_{j}))}^{μ}$ . On applying these perturbed variables to equation (5) one obtains, after re-arrangement, an equation, analogous to equation (7), for the evolution of perturbations to the loans of an individual bank

\begin{array}{l} \frac{d}{d t} [_{l}^{i}' e^{t}] = (k_{i} μ Σ_{j} L_{j}^{'} e^{(1 - μ) t_{j}} D {(t_{j})}^{- μ}) e^{μ t} D^{μ} (t) (1 + (\dot{D (t)} / D (t))) \\ + Σ_{j} e^{t} δ (t - t_{j}) l_{i j}^{'}, i = 1, n \end{array}

(22)

in which

l_{i j}^{'}

are perturbations to

_{l}^{i}'

. This equation has the solution

\begin{array}{l} _{l}^{i}' = k_{i} μ Σ_{j} L_{j}^{'} e^{- (1 - μ) (t - t_{j})} D {(t_{j})}^{- μ} D^{μ} (t) + (l_{i 0} - k_{i} F D_{0}^{μ}) e^{- t} \\ + Σ_{j} e^{- (t - t_{j})} l_{i j}^{'} i = 1, n \end{array}

(23)

in which the three terms relate to three sources of perturbation.

The first term on the RHS of equation (23) represents an accumulated contribution of perturbations that result from the bank’s use of the aggregate loan information, together with its perturbations, in setting their LDR. These perturbations decay at an increasingly slow rate with increasing values of μ. This term is important since it represents the change in the LDR of the ith bank which, in the limit μ = 1 takes the form

l_{i} / d_{i} = \bar{\hat{γ}} + k_{i} Σ_{j} (L_{j}^{'} / D (t_{j})), i = 1, n

(24)

which represents an increasing LDR if the perturbations are positive.

The second term indicates that any perturbation in the initial condition of the perturbed value of $l_{i}'$ decays at an exponential rate, while the third term represents the exponential decay of all other perturbations to l_i. The exponential decay rates of these terms indicate that the individual loan equations are stable with respect to perturbations about the evolving and perturbation-driven LDR.

Hence the n banks act increasingly as a coherent group of individually stable entities as μ increases.

The IRSI for μ = 1

In the structurally unstable state μ = 1, equation (6) becomes

\dot{L} - (\dot{D} / D) L = 0

(25)

and L is a solution if and only if

k L

is a solution for any k > 0. Hence the aggregate loans are not uniquely determined by deposits but are determined, in part and at any time, by the accumulation of non-decaying and slowly decaying perturbations, as demonstrated by the solution (23).

This accumulation of perturbations results in a changing aggregate LDR L/D and, as shown in the preceding section, the behavior of the individual banks is stable about this LDR since the stabilizing mechanisms involving the absolute LDRs ${\hat{γ}}_{i}$ are no longer active. In this state, therefore, the individual banks behave stably with respect to the group.

The state of structural instability when μ = 1 has been termed the IRSI by Smith (1977) since it emerges when decision-makers in a group integrate information concerning the decisions of the other decision-makers into their own decisions.

Modeling the perturbations as a relative income effect (RIE)

The preceding analysis of the dynamical systems model indicates the importance of perturbations in driving the behavior of the NYC banking system in its structurally unstable state. Key issues, therefore, are to determine the origin and nature of the perturbations and to construct an appropriate model of them.

Analysis of perturbations from a RIE

An analysis by Smith (1977) of the data shown in Figure 1 indicates that the perturbations driving the banking system of NYC were related to RIEs (see Duesenberry, 1949). In a system focused on earning income interest from loaning out deposits, a RIE suggests that the banks would have a tendency to raise their LDRs in order to perform competitively with other banks and maintain income from loans during times of decreasing deposits.

The publication of the LDRs of the banks in the CHA would have indicated whether a bank was losing income relative to banks with higher LDRs, especially in times of the seasonally decreasing deposit flows observable in Figure 1. A RIE would tempt banks to maintain loans at previously higher levels as deposits decreased, in order to maintain the previously higher levels of income and remain competitive with other banks.

Smith (1977) tested the hypothesis of RIE-related perturbations by examining the data for aggregate loans and deposits illustrated in Figure 1 for asymmetric responses to increases and decreases of deposits at various temporal frequencies. It was hypothesized that, at relatively long frequencies, loan reductions caused by deposit reductions should be less than loan increases caused by deposit increases, with the effects being less marked or reversed at shorter wavelengths of activity.

After filtering the data according to the methods of Jenkins and Watts (1968) to make it stationary, Smith (1977) found that in the frequency band of strong seasonal variations in activity (0.0–0.06 cycles per week), increases in loans in response to increases in deposits exceeded the decrease in loans in response to decreases in deposits of similar magnitude by a considerable and statistically significant margin as shown by the gain functions represented in Figure 2. At the longest wavelengths, for example, the increment to loans for each additional dollar of deposits was approximately 0.85 (in line with the LDR), but the decrement for each dollar of decreasing loans was only about 0.50, significantly lower than the LDR. These effects were especially marked from the third quarter of 1856 onwards, and were weaker or reversed at the less important quarterly and monthly frequencies.

Figure 2.

Estimates for the gain functions of the positive and negative increments to loans against the positive and negative increments to deposits, from Smith (1977).

On the basis of these results, Smith (1977) concluded that a RIE involving asymmetric responses to increasing and decreasing deposit flows provides a plausible hypothesis for the origin of the structural perturbations that destabilized the banking system. The results rule out a hypothesis of a steady divergence of loans from deposits, while the high coherencies (correlation coefficients in the spectral domain) indicate the reasonableness of the linearity assumption.

A model of the perturbations

In conformity with the results of Smith (1977) and the representation (15) of perturbations employed in the “The dynamical systems model for individual banks” subsection, the simplest model of perturbations to the loan equation for the ith bank (equation (5)) takes the form of small, positive increments to loans

ϵ_{i} = S (\dot{D}) δ (t - t_{j})_{l}^{i 0}' > 0, S (\dot{D}) = {\begin{matrix} 0, & if (\dot{D}) > 0 \\ 1, & if (\dot{D}) \leq 0 \end{matrix}

(26)

that were added during times of seasonally decreasing deposits to reduce the decrements that would normally have occurred under the prevailing LDR. The perturbations may be viewed as arising at least as frequently as the banks’ loan committees made decisions about new and existing loans.

A model-based explanation for the behavior of the banks

The preceding analysis of the dynamical systems model (5) of the banking system of NYC, when driven by the RIE-based structural perturbations represented by equation (26), leads to the following plausible explanation for the aggregate behavior represented in Figure 1.

Before the formation of the CHA in 1853, decisions were made by the banks with little valid information concerning other banks. At that time, the contagion factor had the value μ = 0 and the system was stable, although the lack of information probably resulted in many poor decisions concerning LDRs and loans, leading to the chaotic state of the banking system described by Gibbons (1858).

The formation of the CHA in October 1853 greatly increased the availability of information concerning each of the banks. Such information informed banks as to whether they were earning less interest on deposits ( $γ_{i} < L / D$ ) or behaving in a riskier manner ( $γ_{i} > L / D$ ) than other banks. Although financial disturbances in 1854 led banks to temporarily decrease their LDRs, the financial stability and rising expectations of 1855–57, together with the smooth operation of the CHA, led banks to increasingly trust and use the aggregate ratio L/D in making loan decisions. Hence μ increased steadily in value and, from the evidence of Figure 1, it is reasonable to assume that $μ \approx 1$ by the first quarter of 1856, and remained at that value until the crisis of 1857.

In this structurally unstable state, positive perturbations $l_{0 i}' > 0$ to loans, resulting from the tendency of individual banks to decrease loans less in seasons of decreasing deposits than to increase them in seasons of increasing deposits, did not decay, but accumulated in the system. This caused aggregate loans and the aggregate LDR to increase, resulting in a growing divergence of loans from deposits.

Simultaneously, the behavior of the individual banks was stable in the sense that deviations of their loans from their increasing and perturbation-driven LDRs decayed exponentially, closing a positive feedback loop for the divergence of loans from deposits. In this state, the banks acted as a coherent group that moved in unison, with the individuals supporting the aggregate and the aggregate supporting the individuals.

By mid-1857 the divergence of loans from deposits was sufficient to make the system unstable with respect to random shocks from the financial environment. These occurred in 1857 in the form of the failure of the Ohio Life Insurance and Trust Company and the loss of a freighter carrying gold.

The model additionally explains the rapid decrease in loans following the crisis. The shocks of 1857 led to the sudden transition $μ : 1 \to 0$ , with the banks reverting to their conservative LDRs $\hat{γ_{i}}$ . This led to large negative values $\hat{γ_{i}} d_{i} - l_{i} < < 0$ and strong contractions in loans, which was further exacerbated by the withdrawal of deposits by panicked depositors.

Finally, the model demonstrates how the banking structure could be stabilized, since a system modeled by

{\dot{l}}_{i} = (Γ - γ_{i}) {\dot{d}}_{i} + ((Γ - γ_{i}) d_{i} - l_{i}) + ϵ_{i}, i = 1, n

(27)

in which Γ is a fixed, global LDR and

γ_{i} d_{i}

represents the bank’s discretion in holding back even more deposits, cannot exhibit structurally unstable states. The Associated Banks agreed to such a mechanism in 1858, although it was not enforced until after 1865.

Concluding comments

In his analysis of the panic of 1857, published in the year following the panic, Gibbons (1858) presents a remarkably prescient and beautiful summary of the behavior described by the dynamical systems model of the banking system:

The evil of disproportionate expansion by the banks separately which was so grievous before the organization of the Clearing House was effectually cured; but afterwards came the evil of harmonious expansion which was something quite new and uncontemplated. The balance wheel was perfect so far as to prevent individual members going astray, but for want of a fixed scale, it did not prevent the whole from going astray together.

This observation clearly indicates that an IRSI underlay the crisis.

The principal conclusion of the paper is that this IRSI, which emerges as a property of the model analyzed above, provides a satisfying explanation of the cause of the panic. An even deeper understanding follows from the realization that the IRSI emerged as result of a virtual collapse in the space–time structure of the banks’ clearing process and an associated transformation of their information flow environment.

The current analysis also demonstrates explicitly how behavior in the structurally unstable state μ = 1 was driven by seasonally asymmetric perturbations. The destabilizing effects of the perturbations ironically arose because of the stability of the individual banks which led, as a result of the IRSI, to their behaving as a coherent group moving in unison about a randomly driven LDR.

The IRSI appears to have great relevance for many situations in which decision-making agents determine their actions on the basis of the actions of other agents. It is of interest to note, for example, that Kelly and O’Grada (2000) investigated a further, and related, IRSI in the social networks of panicked depositors during the crisis of 1857.

The analysis presented in this paper indicates the importance of discovering and investigating the effects of significant changes in the space–time structure of geographic systems. As demonstrated in this paper, such changes, together with associated transformations in the information flow fields of a system, may have profound effects on the dynamics and stability of geographically distributed systems.

Footnotes

Acknowledgements

This paper is dedicated to the memory of Elliott Montroll.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Terence R Smith

Note

Terence R Smith is Professor Emeritus of Geography and Professor Emeritus of Computer Science at the University of California at Santa Barbara. His undergraduate degree was in Geography (Cambridge University, 1962–65) and his PhD in Geography and Environmental Engineering (Johns Hopkins University, 1965–71). He held positions at UWO and University of Rochester before moving to UCSB in 1976. He has published over 130 refereed articles on his research interests, which include the theory of fluvial landscape systems, human geographic systems and spatial decision-making, the application of mathematical and computational techniques, and the development of geo-referenced digital libraries.

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Destabilization of an urban subsystem by a virtual collapse in its space–time structure

Abstract

Keywords

Introduction

The basic assumptions and methodology of paper

A virtual collapse of the space–time structure of the banks

The behavior to be explained: Change and crisis

Previous analyses of the financial crisis and panic of 1857

A dynamical systems model of the NYC banks 1853–58

The determinants of the LDR

The dynamical systems model for individual banks

The dynamical systems model for aggregate banking system

An analysis of the model: Its solutions and their stability

The solution for aggregate loans and their dynamic equilibrium

Perturbations and the stability of the aggregate solution

The solutions and stability for the loans of individual banks

The IRSI for μ = 1

Modeling the perturbations as a relative income effect (RIE)

Analysis of perturbations from a RIE

A model of the perturbations

A model-based explanation for the behavior of the banks

Concluding comments

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

Note

References