1. Econ 331b Kremer’s model and Mathematics, economics, science of catastrophes

2. Basics 1.Production function with labor, land, and technology 2.Diminishing returns leads to Malthusian population equilibrium. 3.Higher population leads to more rapid technological change. Fundamental point about technologies: They are ultimate externality because of non-rivalry. [Jefferson: “He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me.”] This leads to explosive growth in technology.

3. Jefferson on ideas If nature has made any one thing less susceptible than all others of exclusive property, it is the action of an idea, which an individual may exclusively possess as long as he keeps it to himself; but the moment it is divulged, it forces itself into the possession of every one, and the receiver cannot dispossess himself of it. Its peculiar character, too, is that no one possesses the less, because every other possesses the whole of it. He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me.

4. The simplest Kremer model

5. Time series: 1,000,000 BP - present Why does this flatten out in recent years?

6. Cross section This is really interesting!

7. Some interesting conclusions 1.Most important point is to remember that technology is endogenous, not exogenous. This is a central issue in energy policy and climate-change policy. 2.How can policy induce more rapid “green” technological change? Think about this. 3.Kremer suggests that pro-natal population is progressive rather than regressive. Population is a boon not a bomb. 4.Model misses the historical fact that invention is not just randomly distributed among people. (Why was Vienna the greatest center of classical music in all time?) What is the reason for the agglomeration of invention?

8. Economics 331b Tipping points and catastrophes 8

9. k k*** k**k* 9 Original locally stable equilibrium

10. k k***k**k* 10 Forcing function tips function (demography, global warming, financial worries, …)

11. k k***k**k* 11 OOPS!!!!!!!

12. k k*** k** k* 12 Note: Have new and different locally stable equilibrium

13. Mathematics of dynamics systems 1.Standard linear systems (boring) 2.Unstable dynamics (nuclear reactions) x t = βx t-1 + ε t (β > 0) 3. Unstable dynamics with boundaries (speculation, epidemics) x t = βx t-1 + ε t (β > 0; x min < x < x max ) 4.Multiple locally stable equilibria (Solow-Malthus, bank panics) 5.Hysteresis loops (Phillips curve, Greenland Ice Sheet, business cycles, snowball earth) 6. Chaotic systems or butterfly effect (weather) 7. Catastrophic disintegration (World Trade Towers, Katrina) 13

14. Examples from climate system Source: Lenton et al., “Tipping Elements,” PNAS, Feb 2008, 1786. 14

15. Source: T. Lenton et al., “Tipping Elements,” PNAS, Feb. 2008, 1786.

16. Hysteresis Loops When you have tipping points, these often lead to “hysteresis loops.” These are situations of “path dependence” or where “history matters.” Examples: -In low level Malthusian trap, effect of saving rate will depend upon which equilibrium you are in. -When have natural monopoly, “first mover advantage.” - In macroeconomics, the expectational Phillips curve theory shows hysteresis loop in inflation. - In climate system, ice-sheet equilibrium will depend upon whether in warming or cooling globe. 16

17. 17 Hysteresis loops and Tipping Points for Ice Sheets 17 Frank Pattyn, “GRANTISM: Model of Greenland and Antrarctica,” Computers & Geosciences, April 2006, Pages 316-325 Source: GRANTISM model (to examine later).

18. Snowball earth (Budyko-Sellars model) Source: Paul Hoffman (Harvard) and Snowball Earth 18

19. Policy Implications 1.(Economic development) If you are in a low-level equilibrium, sometimes a “big push” can propel you to the good equilibrium. 2.(Finance) Government needs to find ways to ensure (or insure) deposits to prevent a “run on the banks.” 3.(Climate) Policy needs to ensure that system does not move down the hysteresis loop from which it may be very difficult to return. 19