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Why are Economies Unstable? Research Project

But Why are Economies Stable?

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Principal Investigator: Professor Robert MacKay, FRS

Robert Sinclair MacKay FRS FInstP FIMA is a British mathematician and professor at the University of Warwick. Robert’s research focuses on the theory and application of nonlinear dynamics. Highlights include his discovery and renormalisation explanation of how invariant tori break for Hamiltonian systems, and a proof of the existence of spatially localised time-periodic movements in networks of oscillators with an analysis of their stability, interaction and mobility.

He is also responsible for the construction and proof of a mechanical example of a uniformly chaotic system, and the construction of indecomposable spatially extended deterministic dynamical systems exhibiting more than one space–time phase.

Robert is currently a Professor of Mathematics, Director of Mathematical Interdisciplinary Research and Director of the Centre for Complexity Science at the University of Warwick. He has published around 180 papers and articles in his field, as well as being the recipient of over 100 research grants. He served as President of the Institute of Mathematics and its Applications from 2012–2013.

Co-Investigators: Nicholas Beale (Sciteb), Richard Gunton (Winchester), Samuel Johnson (Brimingham), and Marcus Miller (Warwick)

Project Summary

The title of our project was a question, “But why are economies stable?”, which merits an answer.  At one level, we suspect the answer is natural selection: economies evolve their structures and regulations towards states that exclude extremes. The level at which we propose an answer is structural: that economies have low trophic incoherence. 


Trophic incoherence is a concept that was introduced by Sam Johnson and co-uthors in 2014, to explain why ecological food webs are relatively stable despite arguments that such large complex systems should be unstable.  It quantifies the extent to which the nodes of a directed network can not be assigned levels so that the change of level along each edge is +1.


Our main results are: (i) an improved version of “trophic analysis”, quantifying upstreamness in production, financial and other networks; and quantifying the directedness of such networks (called “trophic coherence”), (ii) observation of correlations between stability of such networks and their trophic coherence.  In addition, (iii) we have analysed model of the leverage cycle, and (iv) we have written a paper on the importance of value-trackers to market stability.


To investigate the possible role of trophic incoherence in economics, we first had to overcome an obstacle, namely that trophic analysis requires basal nodes (nodes with no incoming edges); or, alternatively, top nodes (with no outgoing edges).  Although such nodes are natural in production networks, they are uncommon in input/output networks (indicating supply between sectors) and in financial networks.  We proposed and developed an improved version of trophic analysis that does not require basal or top nodes, and a correspondingly improved notion of trophic incoherence. 


We then carried out investigations of possible relations between trophic coherence and various notions of stability for economic and financial networks.  The most recent one is on stress-tests of simulated financial networks.  We found that for a range of networks with the same set of nodes, number of links, leverage ratio and total exposure, subject to the same shock on assets, but with different trophic incoherence, the total loss increased with incoherence.  This work resulted in a suggested policy measure: to incentivise banks to borrow and lend in such a way as to increase the trophic coherence of the network.


Another investigation is on the strength of co-movement of different sectors in an economy and its relation to trophic incoherence.  We computed the correlation between employment in different sectors over the period 2005-15 for 24 countries.  We compared with the trophic incoherence of their input-output networks) and found a significant positive correlation.


A third subject was volatility of share price of companies compared to the trophic incoherence of their neighbourhood in Bloomberg’s supply network.  We found a positive correlation again.


These and other investigations all suggest that higher trophic incoherence is associated with more instability. 


In summary, we have made innovative developments in network analysis that have led to evidence of relevance of trophic structure to stability of financial and economic systems, and advanced two other investigations on financial stability.

Interdisciplinarity: Our team brings together a variety of expertise, and consists of:

  • Nicholas Beale, Director of Sciteb, a consultancy with offices in London, Cambridge MA and Beijing, and Director of the Global Collaboration on Financial Systems Stability (in which Gunton, Miller and MacKay are active) with strong connections to top policy-makers in economics and finance;

  • Richard Gunton, ecologist, lecturer in mathematics at Winchester and postdoctoral research fellow in CECAN, an ESRC network on evaluation of policy in complex systems;

  • Samuel Johnson, Lecturer in Mathematics, University of Birmingham, and Turing Fellow, with expertise in ecology and seminal papers (e.g. PNAS 111 (2014): 17923-17928) on trophic coherence.

  • Marcus Miller, Professor of Economics and Research Associate of the ESRC Centre for Competitive Advantage in the Global Economy, both at the University of Warwick;

  • Robert MacKay, Professor of Mathematics and Director of Mathematical Interdisciplinary Research at Warwick, with particular expertise in the application of dynamical systems theory, and involvement in the Turing Institute’s Financial and Economic Data Science programme, and CECAN.



Working Paper I

How Directed is a Directed Network?

R.S. MacKay, S. Johnson, B. Sansom | 28 January, 2020

WP 1

Working Paper II

Trophic Incoherence Drives Systemic Risk in Financial Exposure Networks

R.S. MacKay, S. Johnson, B. Sansom | 28 January, 2021


Both leverage and interconnectedness are widely recognized as key factors for systemic risk and may interact. The magnitude of network-based amplification of distress depends on financial exposure network structure, and may be crucially influenced for example by the presence of destabilising feedback loops in an exposure network. It has been shown that the number of feedback loops in a network, as well as the eigenvalues of associated matrices, are related to a structural property called trophic coherence. In this paper we investigate the impact of trophic coherence on systemic risk - measured using DebtRank - and its interaction with leverage in simulated networks of banks connected to each other by direct exposures. The mechanism is simple: when a bank suffers a loss, distress propagates to its creditors, who in turn suffer losses, and so on. We show that trophic coherence has a crucial influence on contagion dynamics: shock amplification is moderated even at high leverage in more coherent networks; and high even where leverage is low in incoherent networks. This result not only suggests that it may be worthwhile to monitor the trophic coherence of financial networks; but also implies that in principle systemic risk could be significantly reduced simply by “rewiring” the interbank network (without any increase in capital requirements or reduction in interbank loans). We propose a simple strategy to incentivise the self-organised formation of more coherent network structures without impairing market functionality.

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