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A Dynamic Analysis of Schelling’s Binary Corruption Model: A Competitive Equilibrium Approach

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Abstract

Schelling (in Micromotives and Macrobehavior, Norton, New York, 1978) suggested a simple binary choice model to explain the variation of corruption levels across societies. His basic idea was that the expected profitability of engaging in corruption depends on its prevalence. The key result of the so-called Schelling diagram is the existence of multiple equilibria and a tipping point. The present paper puts Schelling’s essentially static approach into an intertemporal setting. We show how the existence of an unstable interior steady state leads to thresholds such that history alone or history in addition to expectations (or coordination) is necessary to determine the long-run outcome. In contrast to the related literature, which classifies these two cases according to whether the unstable equilibrium is a node or a focus, the actual differentiation is more subtle because even a node can lead to an overlap of solution paths such that the initial conditions alone are insufficient to uniquely determine the competitive equilibrium. Another insight is that a (transiently) cycling competitive equilibrium can dominate the direct and monotonic route to a steady state, even if the direct route is feasible.

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Notes

  1. In particular, we omit the discussion of the shape of the marginal utilities U X (i,X) for i=0,1; see, however, Andvig [15].

  2. Subscript B refers to the corresponding point in the Schelling diagram.

  3. For a detailed presentation of the Lagrangian technique taking into consideration inequality constraints, see Grass et al. [27, Sect. 3.6].

  4. For this reason, the payoff comparisons are irrelevant. To stress this point, note that the pure externality terms, here −δX 2, affect the payoffs but not the dynamics. Hence, by choosing a proper δ, we can choose the Pareto-dominant outcome at will.

  5. More precisely, Krugman [1] also shows a large overlap but only due to an error; the true overlap is small. In Fukao and Benabou [24] and Krugman [1], both long-run outcomes are attainable for almost all admissible initial conditions, x 0∈(0.05,0.99).

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Acknowledgements

The authors like to thank Marjorie Carlson, two referees, and the editor for their helpful comments. This research was supported by the Austrian Science Fund (FWF) under Grant P21410-G16 and P23084-N13.

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Authors and Affiliations

  1. H. John Heinz III College, Carnegie Mellon University, Pittsburgh, USA

    Jonathan P. Caulkins

  2. Department for Operations Research and Control Systems, Institute for Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria

    Gustav Feichtinger, Dieter Grass & Andrea Seidl

  3. Wittgenstein Centre for Demography and Global Human Capital (IIASA,VID/OEAW,WU), Vienna Institute of Demography/Austrian Academy of Sciences, Vienna, Austria

    Gustav Feichtinger & Andrea Seidl

  4. Department of Business Administration, University of Vienna, Vienna, Austria

    Richard F. Hartl, Andreas J. Novak & Franz Wirl

  5. Department of Econometrics and Operations Research & CentER, Tilburg University, Tilburg, The Netherlands

    Peter M. Kort

  6. Department of Economics, University of Antwerp, Antwerp, Belgium

    Peter M. Kort

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  1. Jonathan P. Caulkins
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  2. Gustav Feichtinger
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  3. Dieter Grass
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  7. Andrea Seidl
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  8. Franz Wirl
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Correspondence to Gustav Feichtinger.

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Caulkins, J.P., Feichtinger, G., Grass, D. et al. A Dynamic Analysis of Schelling’s Binary Corruption Model: A Competitive Equilibrium Approach. J Optim Theory Appl 161, 608–625 (2014). https://doi.org/10.1007/s10957-013-0420-7

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  • DOI: https://doi.org/10.1007/s10957-013-0420-7

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