Co-evolution and complexity in population growth and economic development

Population dynamics and economic growth. Co-evolution and complexity M. M. Baldi, C. Mammana, and E. Michetti Department of Economics and Law University of Macerata, Macerata, Italy This work has been funded by the European Union - NextGenerationEU under the Italian Ministry of University and Research (MUR) National Innovation Ecosystem grant ECS00000041 - VITALITY - CUP D83C22000710005. I punti di vista e le opinioni espresse sono tuttavia solo quelli degli autori e non riflettono necessariamente quelli dell’Unione europea o della Commissione europea. N´e l’Unione europea n´e la Commissione europea possono essere ritenute responsabili per essi. https://www.safina-vitality.it/ AMASES 2025 1/25 In honor of Maffeo Pantaleoni Maffeo Pantaleoni (1857–1924) has been a great Italian Economist In his 1909 paper Di alcuni fenomeni di dinamica economica (Of Certain Phenomena of Economic Dynamics; partial English tr. with title The Phenomena of Economic Dynamics, 1910), he writes: given other economic systems it is questionable whether there are equilibrium positions that are static, i.e. whether, once they are reached, a new dynamism does not begin, so that strictly speaking there would only exist a succession of dynamic positions of which they would be part, as centres in which the sense of movement would change, even the pseudo positions of equilibrium 2/25 Pantaleoni’s genius Such a statement is truly remarkable because: Pantaleoni was not a mathematician The theory of dynamical systems reached its greatest development in the second half of the 20th century Computers for simulations did not yet exist 3/25 Population dynamics in discrete time. A recap. Let: nt: number of individuals at time t∈N r: growth rate h: carrying capacity The most historical population growth models are: nt+1 =nt(1 + r) Malthus model (1798) nt+1 =rnt(1 −nt) Verhulst (logistic) model (1838) nt+1 =rh h+(r−1)ntntBeverton and Holt model (1957) 4/25 The novelty Non-constant population labor force growth rate nt+1 =g(nt) gis the modified κ-logistic function rooted in the expκ function introduced next Shifted Cobb-Douglas production function within a B¨ohm and Kaas model. 5/25 The expκfunction - I The expκfunction: It is a generalization of the exponential function It depends on a parameter κ When the parameter κapproaches zero, it tends to the exp function With κlarge enough, it tends to infinity like a power function Its popularity is due to the capability to take rare events into account Ordinary events follow an exponential law Rare events are characterized by a Pareto (or power-tail) law 6/25 The expκfunction - II The expκfunction is a function mapping Ronto R++ given by expκ(x) = p1 + κ2x2+κx1 κ 7/25 The logistic (or sigmoidal) function σ:R→(0,1), σ(x) = 1 1 + e−x It maps Rto the interval (0,1), thus transforming a number into a probability For this reason, this function is widely used in the field of machine learning Its first derivative is always positive, meaning that larger numbers correspond to probabilities closer to one and vice versa 8/25 The κ-logistic function σκ:R→(0,1), σκ(x) = 1 1 + expκ(−x) 9/25 The map Putting all together, we get the following two-dimensional map: S:=                xt+1 =     1−δ 1+ntxt0≤xt≤xc 1 1+nt(1 −δ)xt+Aαsrxt(xt−xc)α−1xc<xt≤xw 1 1+nth(1 −δ)xt+Asw(xt−xc)+α(sr−sw)xt (xt−xc)1−αixt>xw. nt+1 =Mσκ(nt−nc)−σκ(−nc)) 1−σκ(−nc). 16/25 On the number of equilibrium points of g a) The origin is an equilibrium point of g. b) If nc= 0 and M≤2 then the origin is the only equilibrium point of g. c) If nc= 0 and M>2 then gadmits a unique positive equilibrium point. d) If nc>0 and M≥√1+κ2n2 c σκ(−nc)then gadmits a unique positive equilibrium point. e) If M<4, there exists an nc>0 such that for all nc≥nc, the origin is the only equilibrium point. f) If nc≥M, the origin is the only equilibrium point. 17/25 A necessary condition A necessary condition for the existence of two positive equilibrium points for gis that: i) Mσκ(−nc)≤p1 + κ2n2 c ii) nc<M 18/25 On the stability of the origin a) If Mσκ(−nc)<p1 + κ2n2 cand nc= 0 then the origin is globally asymptotically stable. b) If Mσκ(−nc)<p1 + κ2n2 cand nc>0 then the origin is asymptotically stable. c) If Mσκ(−nc)>p1 + κ2n2 cthen the origin is unstable and the only positive equilibrium point is asymptotically stable. d) In the case of two positive equilibrium points P1(n1,n1) and P2(n2,n2) with n1<n2, then the origin is asymptotically stable, P1is unstable and P2is asymptotically stable. 19/25 Graphical examples 20/25 Computational experiments Extensive simulations were conducted in Python. The parameter configurations considered for the κ-logistic were: Case 1: M= 2, nc= 0.3, and κ= 4; Case 2: M= 2.6, nc= 0.15, and κ= 2; Case 3: M= 3, nc= 0.35, and κ= 5, The parameter configurations considered for the shifted Cobb-Douglas were: Case A: δ= 0.65, sw= 0.45, sr= 0.25, n= 0.45, A= 100, xc= 44, α= 0.15; Case B: δ= 0.65, sw= 0.45, sr= 0.25, n= 0.45, A= 100, xc= 44, α= 0.275; Case C: δ= 0.65, sw= 0.45, sr= 0.25, n= 0.45, A= 100, xc= 44, α= 0.3. 21/25 Poverty trap Combination of case 2 for the κ-logistic with case A or B for the Shifted Cobb-Douglas 22/25 Coexistence of attractors Combination of case 1 for the κ-logistic with case A or B for the Shifted Cobb-Douglas 23/25 Complex attractors Combination of case 2 for the κ-logistic with case C for the Shifted Cobb-Douglas 24/25 Conclusions We explored a growth model using the modified κ-logistic function, which accounts for rare events, leading to a generalized sigmoidal production function. This model can handle both concave and non-concave production functions, relevant for economies at various development stages or encountering rare events. Theoretical findings were confirmed by computational experiments demonstrating the system’s behavior under significant scenarios. 25/25