Dynamic Systems

1. Dynamic Systems Modeling Complex Interconnections

2. I. Dynamic Systems • Problem: Many variables and each one affects the others. x=f(y,z); y=f(x,z); z=f(x,y) As number of associated variables increases, econometric and game-theoretic models become intractable.

3. B. Solution: Computer Simulation • Begin with connections between variables over time (obtained from theory or regression analysis) • Diagram the causal loops and connections. Arrows illustrate causation and +/- indicates coefficient. Example: + -

4. 3. Functional forms • Just how is independent variable X related to dependent variable Y? • Just how is Yt-1 related to Yt? • In both cases, options include linear, exponential, S curve, bell curve, threshold (see Fig 2.2 in Hughes for examples).

5. 4. Feedback loops a. Negative feedback loops (odd number of negative linkages): Analogous to thermostats  tend to generate steady-state or cyclical patterns of change. Examples: Time 

6. + - Examples of negative feedback loops • Tiredness-Sleep

7. + - Examples of negative feedback loops • Tiredness-Sleep • Disease-Population

8. + - Examples of negative feedback loops • Tiredness-Sleep • Disease-Population • Deterrence Theory

9. + - Examples of negative feedback loops • Tiredness-Sleep • Disease-Population • Deterrence Theory • Prices

10. Examples of negative feedback loops • Tiredness-Sleep • Disease-Population • Deterrence Theory • Prices • Clouds-Temperature -

11. b. Positive feedback loops (even number of negative linkages) • Analogous to a snowball rolling downhill • Tend to produce exponential growth or exponential decay: Time 

12. + + Examples of positive feedback loops • Arms Races

13. Examples of positive feedback loops • Arms Races • Permafrost Thaw + + +

14. Examples of positive feedback loops • Arms Races • Permafrost Thaw • Population Growth + +

15. Examples of positive feedback loops • Arms Races • Permafrost Thaw • Population Growth • RiskTM + +

16. Examples of positive feedback loops • Arms Races • Permafrost Thaw • Population Growth • RiskTM • Hegemonic Decline - -

17. Examples of positive feedback loops • Arms Races • Permafrost Thaw • Population Growth • RiskTM • Hegemonic Decline • “Liberal Peace” - - + +

18. c. Feedback Linkage • What happens when the same variable is part of two different feedback loops? • If dominant loop changes, tends to produce S-shaped curves • Example: Adoption of a new technology

19. Adoption of a new technology

20. Another example: Population What happens if death rate is reduced but birth rate is unchanged? Birth Rate Population Death Rate + - + +

21. 5. Modules • Modules connect all the variables in a given area together: • Inputs  Module (relationships and feedback loops between inputs)  Outputs • Figure 2.1 in Hughes lists IFs modules

22. II. Evolution of World Models: World2 and World3 • World2: Possibly the first global model 1. Attempted to model the connections between stocks and flows of five factors: • Population (dynamic model) • Agricultural Production (limited by land and productivity) • Industrial Production (limited by capital and productivity) • Natural Resources (assumed as 250 years’ worth at 1970 consumption levels) • Pollution (generated by production)

23. 2. Structure of World2 (Handout)

24. 3. Base Case Prediction Note the changes in pollution and population:

25. B. World3 and “The Limits to Growth” • Origins: criticism of the simplistic nature of World2 • Use: Model forecasts used to prepare assigned reading by Meadows et al, The Limits to Growth, a 1972 best-seller by the “Club of Rome”

26. 3. Features • Each main variable from World2 gets its own module. For example: • Service sector of economy is added • Population is now tracked by age group • Natural resources renew themselves • Technological changes are permitted Increase from 244 (World2) to 1583 (World3) equations

27. b. Overall model (see Handout)

28. 4. Results a. Adding complexity did not change the fundamental “limits to growth” forecasts from World2

30. World2 “Standard Run” (Base Case)

31. World3 “Standard Run” (Base Case)

32. b. What is limiting growth? Same factors in both World2 and World3: • Initial run: Resource depletion • Assume more resources: Pollution (even worse than depletion because triggers massive die-off) • Assume more resources and pollution control: agricultural production Put simply, multiple consequences of growth create a negative feedback loop that limits growth – since infinite growth impossible, cannot sustain exponential growth forever

33. c. Critical Insights • Exponential growth in anything is hard to sustain  best hope is an S-shaped curve • Delays between elements (i.e. pollution may kill over decades rather than immediately  we may not realize it when we are past the tolerable limit)

34. Example: The pond • Algae grows in a pond. If it covers the whole pond, fish will suffocate. • Assume exponential growth of algal population, doubling time = 1 day • When should we act?

35. When is the threat imminent? • Day 1: 1/10 of 1% covered

36. When is the threat imminent? • Day 2: One-fifth of one percent covered

37. When is the threat imminent? • Day 3: Nearly 1% covered

38. When is the threat imminent? • Day 8: About 1/8 covered

39. When is the threat imminent? • Day 9: About 1/4 covered

40. When is the threat imminent? • Day 10: About 1/2 covered

41. When is the threat imminent? • Day 11

42. Example: The pond • Algae grows in a pond. If it covers the whole pond, fish will suffocate. • Assume exponential growth of algal population, doubling time = 1 day • What if we make the pond bigger? Double its size and we buy one more day. Make it 100 times bigger and we have about one more week.

43. 5. Assessments: The Standard Run

47. 5. Assessments: The Standard Run

48. Note the 1985 peak – and 2000s uptick

49. Calories per person to 2004 (note that the scale magnifies changes)

50. 5. Assessments: The Standard Run