Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Easter Island’s Collapse: a Tale of Population Race David de la Croix 1 and Davide Dottori 2 1 IRES, Dept. of economics & CORE, Univ. cath. Louvain 2 IRES, ept. of economics, Univ. cath. Louvain February 2007 1 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Easter Island and Tikopia: maps 2 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Chronology of the rise and fall • 400 CE. First settlers, likely Polynesians (Marquesas Islanders, less than 100 units) • 600 CE. Beginning of deforestation. • 1000-1500 CE. Moai building by competing clans. (more than 800 in total, each weighting up to 80 tons) • 1500 CE. Population’s peak at about 10000 (but other estimates come up to 20’000) • 1500-1700 CE. Appearance of new weapons, cannibalism, movements into fortified dwellings: conflicts . Deforestation completed. Population crash. • 1722 CE. Discovery by Dutch explorers. Estimated population: less than 3000. No tree. 3 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Easter island, Earth Island Monuments of Easter Island, 1775 CE, by William Hodges 4 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Easter Island: An Ecological Catastrophe 1 0.9 0.8 0.7 % forest pollen 0.6 0.5 0.4 0.3 0.2 0.1 0 0 500 1000 1500 2000 Year (CE) (data: John Flenley) 5 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion A counter-example: Tikopia • 900 BCE; First settlers from Eastern Polynesia; second wave from Eastern Polynesians tribes later. • Population achieved 1200 units by 1100 CE and then remained roughly steady over centuries • Control over population growth through several practices (cult of virginity, infanticides, celibacy, sea voyaging by young males, abortion, contraception, expulsion of segments of population in excess, fono (an annual address by the chief) 6 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Population in Easter and Tikopia 12000 2000 Easter (left scale) 1800 10000 Tikopia (right scale) 1600 1400 8000 1200 6000 1000 800 4000 600 400 2000 200 0 0 -1000 -500 0 500 1000 1500 2000 Year 7 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Literature • Brander & Taylor (AER, 1998): Population-resources interaction. Fertility determined by nutrition (Malthus). Extensions to account for abrupt decrease: Pezzey and Anderies (JDE, 2003), Erickson and Gowdy (LE, 2000), Reuveny and Decker (EE, 2000). • Conflicting groups: groups may conflict to encroach crop. A (static) problem of optimal allocation between working and fighting. Malthusian fertility. Myopic behaviors often implicitly postulated. Maxwell and Reuveny (JEBO, 2005; 2001), Lasserre and Souberyan (JEBO,2003), Prskawetz et al. (EMA,2003). 8 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion What we Do We are not satisfied with the way fertility is modeled. Also, assuming myopic behavior is unsatisfactory. We propose a first model with endogenous fertility decisions to include strategic complementarities between groups. 9 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Outline of the model • Utility Maximizing Fertility • Absence of strong property rights: crop distribution among clans follows a non cooperative bargaining • Bargaining power depends on the threat of fighting a war • Success in conflict depends on the relative size of groups • Incentive to increase clan’s population? ⇒ Population Race 10 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion The economy • OLG where agents live for 2 periods. Every agent belongs to a clan. 2 clans. • N it young individuals at time t in clan i . • The young work, rear children, support parents (and fight in case of conflict). • The old consume out of a portion of their sons’ income. • Child rearing has a disutility cost • Timing of clans’ decisions: 1. choice of own fertility rate as a social norm, considering the others’ as given, and having perfect foresight, 2. bargaining on crop sharing. 11 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion • Old age support: Share of income paid to old parents decreasing in the number of siblings : τ , 0 < τ < 1 . 1 + n i , t • Utility: (for simplicity) Linear U i = c i , t + β d i , t +1 − λ n i , t Budget constraints: � τ � = 1 − (1) c i , t y i , t 1 + n i , t − 1 τ = (2) d i , t +1 n i , t y i , t +1 1 + n i , t where: c it : 1st period consump d it +1 : 2nd period consumption : income of a young : number of children y it n i , t β > 0 : discount factor λ ≥ 0 : child-rearing disutility 12 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion • Population Dynamics: N it +1 = n it N it (3) • Production: Y t = A t L α ( N 1 , t + N 2 , t ) 1 − α L : Land (fixed: ⇒ L = 1); A t : TFP depending on stock of resources R t : A t = A ( R t ) • Resources Dynamics (Matsumoto 2002): 1 + δ − δ R t � � R t +1 = K − b ( N 1 , t + N 2 , t ) (4) R t δ : natural regeneration growth rate where: K : carrying capacity b : coefficient on human impact 13 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion The Clan’s Problem • θ ≡ group 1’s crop-share y 1 t = θ t Y t / N 1 t = (1 − θ t ) Y t / N 2 t y 2 t • Step 2 : Given n i , t , θ results from bargaining: ( U 1 − ¯ U 1 ) γ ( U 2 − ¯ U 2 ) 1 − γ γ : group 1’s exogenous contractual force with ¯ : fall back utility U i • Step 1 : Clans maximize U i over n i , t , c i , t , d i , t +1 s.t. budget constraints 14 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Fall back utility ¯ U i : expected pay-off in case of war: ¯ π t ˆ U 1 , t + (1 − π t )ˇ = U 1 t U 1 , t ¯ (1 − π t )ˆ U 2 , t + π t ˇ U 2 t = U 2 , t π t = p ( N 1 , t , N 2 , t ) : group 1’s probability to win, where ˆ (ˇ U i , t U i , t ) : utility if war is won (lost). 15 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Assumptions about war • War involves destruction of a portion ω ∈ [0 , 1) of total product. • War entails no human loss. • The winner encroaches the whole available crop. • The bigger group has more chances to win: p ′ N 1 > 0 and p ′ N 2 < 0. • An explicit form satisfying desirable properties (Skaperdas, ET 1996): N µ 1 , t p = ∈ (0 , 1) N µ 1 , t + N µ 2 , t where µ : sensitivity to size of the clan 16 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Step 2: Bargaining Outcome Proposition (1) Nash bargaining solution: N µ 1 , t θ t = γω + (1 − ω ) N µ 1 , t + N µ 2 , t The obtained share is endogenous and depends on group size. 17 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion The role of endogenous fertility Channels through which fertility affects utility: • Costs: • Disutility cost from child-rearing • Next period crop to be divided among more persons • Benefits: • Greater old age support • Positive effect on bargaining power: ( θ t +1 ) children ↑ ⇒ threat point tomorrow ↑ (perfect foresight) ⇒ share of crop tomorrow ↑ ⇒ income when old ↑ 18 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Step 1: Fertility Choice - Reaction Functions • Fertility is chosen taking as given the other group’s one ⇒ fertility reaction functions. Mutual dependency? Positive slope? • No explicit general solution can be found analytically. Analytical study under Assumption 1. General case studied numerically. Assumption (1) Parameters satisfy: µ = 1 , ω = 0 , λ = 0 , α = 1 N 1 , t implying: θ t = N 1 , t + N 2 , t and “coconuts”-type crop. 19 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Proposition (2) • The fertility reaction functions have positive slopes � N j , t n i , t = n j , t N i , t • The Nash equilibrium is: � � N 2 , t N 1 , t n ∗ 3 , n ∗ 3 1 , t = 2 , t = (5) N 1 , t N 2 , t • The Nash Equilibrium is stable. 20 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Comparative Statics Corollary 2 Corollary 1 n 1 n 1 high β , τ r 1 high µ r 1 low α high γ low A t +1 high ω , λ r 2 r 2 n 2 n 2 21 / 27
Introduction The model Bargaining and Fertility Choice Dynamics Conclusion Dynamics Proposition (3) • If a strictly positive Resources Steady State exists, then it is stable 1 − b (¯ N i + ¯ � N j ) � ¯ R = K δ • Under Assumption 1 • A positive s.s. exists if and only if initial populations are not too high: � � 2 b N 2 , 0 < δ N 1 , 0 • If so, population converge to ¯ N i = ¯ � � N j = N 1 , 0 N 2 , 0 • θ and π converge to 1 / 2 . 22 / 27
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