1. Economics 331b Population dynamics in economics 1

2. Course logistics The enrollment in the course is full. You can participate if you received an email from me. There are always appeals for special cases. There is no waiting list. 2

3. Schedule Wednesday (today): Malthus and Cohen Friday: No class. Nordhaus lecture on Economics of Climate Change, Yale Climate Institute, 12:00 – 1:15, Kroon Hall Monday: no class Wednesday: Solow model with deomgraphy; tipping points Friday: Kremer model Presentations: any volunteers for Kremer, demography, or Cohen would be welcome (Wed and Fri of next week) 3

4. Importance of population Obviously important part of social sciences In environmental economics, part of the stress on natural systems. Can see in the “Kaya identity”: Pollution ≡ Pop * (GDP/Pop) * (Pollution/GDP), This equation is often used for energy, CO2, and other magnitudes. Warning: It is an identity, not a behavioral equation. It doesn’t explain anything. 4

5. Different world views on population 1.Malthus-Cohen: population bumping against resources. 2.Solow-Demographic transition: Need to make the big push to get out of the low-level Malthusian trap. 3.Kremer: people are bottled up and just waiting to be the next Mozart or Einstein. 5

6. Malthusian economics Basic propositions: 1. It may safely be pronounced, therefore, that population, when unchecked, goes on doubling itself every twenty-five years, or increases in a geometrical ratio. 2. It may be fairly pronounced, therefore, that, considering the present average state of the earth, the means of subsistence, under circumstances the most favourable to human industry, could not possibly be made to increase faster than in an arithmetical ratio. 3. Taking the whole earth … and, supposing the present population equal to a thousand millions, the human species would increase as the numbers, 1, 2, 4, 8, 16, 32, 64, 128, 256, and subsistence as 1, 2, 3, 4, 5, 6, 7, 8, 9. In two centuries the population would be to the means of subsistence as 256 to 9 ; in three centuries as 4096 to 13, and in two thousand years the difference would be almost incalculable. 4. In this supposition no limits whatever are placed to the produce of the earth. It may increase for ever and be greater than any assignable quantity; yet still the power of population being in every period so much superior, the increase of the human species can only be kept down to the level of the means of subsistence by the constant operation of the strong law of necessity, acting as a check upon the greater power. [This theory led to Darwin, social Darwinism, poorhouses, and many other social ideas.] 6

7. 7 Review of basic production theory Classical production model. Aggregate production function (for real GDP, Y) (1)Y = F( K, L) Standard assumptions: positive marginal product (PMP), diminishing returns (DR), constant returns to scale (CRTS): CRTS: mY = F( mK, mL) PMP: ∂Y/∂K>0; ∂Y/∂L>0 DR: ∂ 2 Y/∂K 2 <0; ∂ 2 Y/∂L 2 <0

8. The simplest Malthusian model Production function: (1) Y t = F(L t ) (1M) Y t = F(L t ) = 1 + ln 2 (L t ) Where L = population, B = births, D = deaths, w t = wage rate. Income: Population dynamics (3) and subsistence assumption (4): 8

9. n (population growth) Wage rate (w) 0 w* (Malthusian subsistence wage) n=n[w] 9

10. Demographic transition G.T. Miller, Environmental Science 10

11. Dynamics 1. Long-run equilibrium when population is constant: (5) P = P* → w = w* → wages at long run subsistence wages. 2. What happens if productivity increases? -If productivity takes a jump, then simply increase P (next slide) -More complicated if have continuous population growth, then can have a growth equilibrium. -Even more complicated if have demographic transition: 11

12. 12 L MPL, Real wage (w) MPL Neoclassical distribution of output/income w* S short-run S longrun MPL’

13. Malthus with continuous growth Assume Cobb-Douglas production function: This is the major anti-Malthus theorem: Rapid technological change can outstrip population growth even in the subsistence version. 13

14. Modern Malthusians Left-wing neo-Malthusians : This school that believes we are heading to low consumption because we are exhausting our limited resources (alt., climate change, …). See Limits to Growth, P Ehrlich, The Population Bomb Right-wing neo-Malthusians : This school believe that the “underclass” is breeding us into misery due to overly generous welfare programs. See Charles Murray, Losing Ground: American Social Policy 1950–1980. 14

15. Carrying Capacity Basic idea from ecology: the maximum number of individuals that the environmental resources of a given region can support. Demographers have sometimes assumed this applies to the upper limit on human populations that the earth can support. (maximum supportable human population). 15 Source: J. Cohen, “Population Growth…,” Science, 1995.

16. Agenda for today Cohen and the idea of carrying capacity The neoclassical growth model: - the basics from macro - adding endogenous population growth Nonlinear dynamics and tipping points Schedule presentations 16

17. Alternative methods for estimating carrying capacity 1.Just assume a maximum population density 2.Extrapolate population trends. 3.Single factor model (e.g., food supply) 4.Single factor as function of multiple inputs 5.Multiple factor constraints (P < β water; P < γ food; …) 6.Multiple dynamic and stochastic constraints (P(t) < β water(t) + ε(t) ; P(t) < γ food(t) +ς(t) ; …] [Source: As described in Cohen] 17

18. Carrying Capacity from Cohen Basic idea is that there is an upper limit on the population that the earth can support. This is a variant of Malthus as follows: Not clear how to interpret (9). One possibility is the maximum L at subsistence wages, which would be MPL(Z)=w*, or in C-D framework: Which means that carrying capacity grows at 18

19. Economic interpretation of carrying capacity theories Carrying capacity is a concept foreign to economic demography. Is it a normative concept? A descriptive concept? As descriptive, it seems related to Malthusian subsistence wage. Carrying capacity changes over time with technological change. Basic trends in U.S. and rest of world outside of Africa is that technological shifts have outweighed diminishing returns. I.e., clear evidence that because of technological change, carrying capacity has increased over time. As normative, it seems inferior to concept of optimum population. This would be some social welfare function as U(C, L), maximized over L However, introducing L gives serious difficulties to Pareto criterion, which is central normative criterion of economics 19

20. Growth dynamics in neoclassical model* Major assumptions of standard model 1. Full employment, flexible prices, perfect competition, closed economy 2. Production function: Y = F(K, L) = LF(K/L,1) =Lf(k) 3. Capital accumulation: 4. Labor supply: * For those who are rusty on the neoclassical model, see handout as well as chapters from Mankiw on the course web site. 20

21. 21 k y = Y/L y = f(k) (n+δ)k y* i* = (I/Y)* k* i = sf(k)

22. Demographic transition G.T. Miller, Environmental Science 22

23. Current demography 23

24. n (population growth) Per capita income (y) 0 y* = (Malthusian or subsistence wages) n=n[f(k)] 24 Unclear future trend of population in high-income countries

25. Growth dynamics with the demographic transition Major assumptions of standard model 1. Full employment, flexible prices, perfect competition, closed economy 2. Production function: Y = F(K, L) = LF(K/L,1) =Lf(k) 3. Capital accumulation: 4. Labor supply: Now add endogenous population: 4M. Population growth: n = n(y) = n[f(k)] ; demographic transition This leads to dynamic equation (set δ = 0 for expository simplicity) 25

26. k y = Y/L y = f(k) k*** i = sf(k) k**k* Low-level trap n[f(k)]k 26 High-level equilibrium “TIPPING POINT”

27. Examples of traps and tipping points In social systems (“good” and “bad” equilibria) Bank panics and the U.S. economy of 2007-2009 Steroid equilibrium in sports Cheating equilibrium (or corruption) Epidemics in public health In climate systems (see next slide) Very interesting policy implications of tipping/trap systems 27

28. k k*** k**k* 28 Original locally stable equilibrium

29. k k***k**k* 29 Forcing function tips function (demography, global warming, financial worries, …)

30. k k***k**k* 30 OOPS!!!!!!!

31. k k*** k** k* 31 Note: Have new and different locally stable equilibrium

32. 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) 32

33. Examples from climate system Source: Lenton et al., “Tipping Elements,” PNAS, Feb 2008, 1786. 33

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

35. 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. 35

36. 36 Hysteresis loops and Tipping Points for Ice Sheets 36 Frank Pattyn, “GRANTISM: Model of Greenland and Antrarctica,” Computers & Geosciences, April 2006, Pages 316-325 Source: GRANTISM model (to examine later).

37. Snowball earth (Budyko-Sellars model) Source: Paul Hoffman (Harvard) and Snowball Earth 37

38. 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. 38

39. k y = Y/L y = f(k) k*** i = sf(k) The Big Push in Economic Development {n[f(k)]+δ}k 39