Reverse-Engineering the Business Cycle with Petri Nets

1. Reverse-Engineering the Business Cycle with Petri Nets Johnnie B. Linn III Concord University Athens, WV

2. Special Thanks for • Online input-output tables for the U.S. economy, 1998-2010. • U.S. Bureau of Economic Analysis, Department of Commerce, http://www.bea.gov. • Online eigenvalue/eigenvector calculator for a 32 x 32 matrix. • Bluebit Software, http://www.bluebit.gr.

3. The Problem • “I do not think that the currently popular DSGE models pass the smell test.” • Robert Solow Professor Emeritus, MIT • Prepared Statement, House Committee on Science and Technology Subcommittee on Investigations and Oversight. 1 • So what is DSGE?

4. Dynamic Stochastic General Equilibrium (DSGE) • Macroeconomic models founded on microeconomic principles.2 • But DSGE failed to foresee the financial crisis. Could agent-based modeling do better? • The Economist, Jul 22nd 2010

5. Agent-Based Modeling • “[C]omputational models for simulating the actions and interactions of autonomous agents (both individual or collective entities such as organizations or groups) with a view to assessing their effects on the system as a whole.”3

6. So Where am I Going with This? • The aggregate data we have are the products of agents. Can we find all possible sets of actions of agents that could have produced our results? • Then we can link each set of agents and their actions with a particular set of behavioral assumptions.

7. Petri Nets • Bipartite Directed Graphs • Used to Model Interconnected Causal Systems

8. Petri Nets and Their Firings

9. Petri Nets and Their Firings

10. Petri Nets and Their Firings

11. Petri Nets and Their Firings

12. Here, Causal Relationships of Quantities Disposable Income Consumption

13. Matrix Representation for a Set of Firings

14. An Adjacency Matrix and Firings =+ =+

15. Decomposition of the Adjacency Matrix

16. Key Assumption • All transitions are enabled, and can be fired in any order. • Makes internal cycles possible. • Models independent actions by agents.

17. The Business Cycle, Starting Model Value Added Gross Domestic Product

18. NIPA Use Table, No Intermediate Detail

19. Creation/Destruction Cycle Destructive Processes Gross Domestic Product/ Value Added Intermediate Commodities Creative Processes

20. Adjacency Matrix for the Creation/Destruction Cycle Int. GDP/VA Total • Creation Process is derived from the transpose of the use table. • Destruction Process is derived from the untransposed use table.

21. Hard-Wiring the Technical Transformation Processes Int. GDP/VA Total • A token is denominated in dollars. • Arc values can be scaled to best fit the eigenvalue/eigenvector calculating algorithm.

22. Reverse-Engineering the Cycle • Each firing delivers at least one new token to the cumulative marking. It will be used to fill the element of U that represents that firing. • Working backwards…

23. where Ct is a vector of constants that ensure that no other elements of U are changed. Continuing.. … So

24. but so

25. or • In a cyclical process, U is a set of initial processes C that have replicated themselves.

26. So let then

27. The matrix BA’ is a linear transformation of the vector U with eigenvalues (1-νi). Eigenvalues are normally expressed as λ , so we have where λ=1-ν

28. If the Arcs are Hard-Wired to Data: • The number of tokens accounted for by each element of C will vary according to its process’ contribution to volume. • Likewise the magnitudes of the elements of U will vary according to their processes’ contributions to volume. • The components of U remain orthogonal.

29. The Data • NIPA Use Tables, 1998-2010 • Two sets • Before redefinitions • After redefinitions • Sector Level

30. Part of a Use Table

31. The Computations • 15 creative processes produce 15 commodities plus scrap and noncomparable imports. • 17 destructive processes reduce 15 commodities plus scrap and noncomparable imports to intermediate or final uses. • Scrap and noncomparable imports do not have final uses.

32. The Computations (2) • There are 32 processes for 18 quantities, so 14 processes are redundant. • (λ I – BA’) is a 32 x 32 matrix. • The matrix has 32 eigenvalues and 32 corresponding eigenvectors. • We seek complex eigenvalues.

33. Summary of Results • Complex eigenvalues were found. • Average annual angle of rotation (alibi) about 24 degrees. • Corresponds to cycle length of 15 years. • Velocity of rotation increased significantly after 2007. • Phase angles between creation process and corresponding destruction process for each commodity appear to have significance.

34. Alibi Results, 1998-2010

35. What the Algorithm Did • Algorithm had 14 degrees of freedom. • Aligned the phase angles of all 15 creative processes. • Also, rotated each eigenvector to guarantee one process to have a phase angle of zero degrees, or sometimes 180 degrees. • Not always the same process selected.

36. Raw Phase Angles, Before Redefinitions

37. Raw Phase Angles, After Redefinitions

38. Phase Angle from Creative Datum, Before Redefinitions

39. Phase Angle from Creative Datum, After Redefinitions

40. Agriculture, forestry, fishing, and hunting Before Redefinitions After Redefinitions

41. Mining Before Redefinitions After Redefinitions

42. Utilities Before Redefinitions After Redefinitions

43. Construction Before Redefinitions After Redefinitions

44. Manufacturing Before Redefinitions After Redefinitions

45. Wholesale Trade Before Redefinitions After Redefinitions

46. Retail Trade Before Redefinitions After Redefinitions

47. Transportation and Warehousing Before Redefinitions After Redefinitions

48. Information Before Redefinitions After Redefinitions

49. Finance, insurance, real estate, rental, and leasing Before Redefinitions After Redefinitions

50. Professional and business services Before Redefinitions After Redefinitions