Threshold Rule and Scaling Behavior in a Multi-Agent Supply Chain

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1 Threshold Rule and Scaling Behavior in a Multi-Agent Supply Chain Valerio Lacagnina, Davide Provenzano Dipartimento di scienze statistiche e matematiche Silvio Vianelli Faculty of Economics University of Palermo

2 Previous works Bak P, Chen K, Scheinkman J, Woodford M (1993), Aggregate fluctuations from independent sectoral shocks: self-organized criticality in a model of production and inventory dynamics. Ricerche Economiche 47:3 30 Delli Gatti D, Di Guilmi C, Gaffeo E, Giulioni G, Gallegati M, Palestrini A (2005), A new approach to business fluctuations: heterogeneous interacting agents, scaling laws and financial fragility. Journal of Economic Behavior and Organization 56(4): Battiston S, Delli Gatti D, Gallegati M, Greenwald B, Stiglitz J.E. (2007), Credit chains and bankruptcy propagation in production networks. Journal of Economic Dynamics and Control 31(6): Weisbuch G, Battiston S (2007), From production networks to geographical economics. Journal of Economic Behavior and Organization 64:

3 The structure of the network ARTIFICIAL ECONOMICS 2010 M firms K production levels L=M/K firms per level Firms have the same number of suppliers and clients No border effects Cylindrical lattice (R, nq) order strategy

4 The structure of the network ARTIFICIAL ECONOMICS 2010 Firms at the bottom level (retailers) sell only one type of final good For each production level output is qualitatively different from input The total number of firms M and the number of connections among firms are constant over time Time runs discretely in periods t = 1, 2,, T The retailers' level is buffeted by exogenous random shocks which can take the value zero with probability 1-p and the value one with probability p

5 Demand and supply of goods a ARTIFICIAL ECONOMICS 2010 where a

6 Production of goods a where a

7 Backlogs and the rationing of the inventory a a with

8 The inventory dynamics a ARTIFICIAL ECONOMICS 2010

9 The price of goods Industry price of output Individual selling price for each firm where u j (t) is uniformly distributed in [1-ξ, 1+ξ]

10 Firms profits +

11 Banckruptcy and re-birth of firms Bankruptcy occurs when firm s capital is not sufficient to cover fixed costs A m (t) < rr m After a latency period, a re-birth process occurs in the corresponding vertex for a new firm. The re-birth capital is assumed to be proportional to the average firm capital A re-birth (t) = α Ā(t)

12 Parameters choice M = firms K = 5 layers L = 2000 firms per layer A m (0) = A init [10, 11] I m (0) = I init [1, 4] R = 1 Q = 3 T = 5000 time steps (20 years) ρ = 0.2 per unit in excess in the inventory λ = 0.05 per unit of unfilled demand r = 0.15 per unit of good in the reorder quantity

13 Parameters choice c = 0.3 per produced unit ξ = 0.2 θ = 1 τ = 1

14 SIMULATIONS RESULTS

15 Firms size distribution

16 Zipf plot

17 Scaling property

18 Mean excess plot

19

20 Conclusions and future works The (R, nq) economy with finite agents qualitatively reproduces the non normal distribution of firms very common in industrial demography. Lead time and batch ordering are responsible for the sectorial dynamics which can generate avalances of bankruptcies when the fluctuations hamper the financial solvency of the firms. At the moment we are working on a more elastic structure of the economy in order to investigate which structures emerge spontaneously when the network is let to evolve in time. Caliration of the model to real data.

21 Thank you for your kind attention

22

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