Persistence theory applied to Keen s model a link between mathematical biology and mathematical economics

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1 Persistence theory applied to Keen s model a link between mathematical biology and mathematical economics Jianhong Wu and Xiang-Sheng Wang Mprime Centre for Disease Modelling York University, Toronto Persistence theory applied to Keen s model First Previous Next Last 1

2 Outline Introduction on persistence theory Keen s model without government intervention Keen s model with government intervention I Keen s model with government intervention II Discussion on uniformly strongly persistence and sustained oscillations Persistence theory applied to Keen s model First Previous Next Last 2

3 Persistence theory on mathematical biology H. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, Providence, RI, Which species, in a mathematical model of interacting species, will survive over the long term? In a mathematical model of an epidemic, will the disease drive a host population to extinction or will the host persist? Can a disease remain endemic in a population? Persistence theory applied to Keen s model First Previous Next Last 3

4 Persistence definitions Let Φ(t, x) : R + X X be the semiflow generated by a differential system with initial values x X. For a nonnegative functional ρ from X to R +, we say Φ is ρ - uniformly strongly persistent (USP) if lim inf t ρ(φ(t, x)) > ε for any x X with ρ(x) > 0. Φ is ρ - uniformly weakly persistent (UWP) if lim sup t ρ(φ(t, x)) > ε for any x X with ρ(x) > 0. Φ is ρ - strongly persistent (SP) if lim inf t ρ(φ(t, x)) > 0 for any x X with ρ(x) > 0. Φ is ρ - weakly persistent (WP) if lim sup t ρ(φ(t, x)) > 0 for any x X with ρ(x) > 0. Persistence theory applied to Keen s model First Previous Next Last 4

5 Example: Goodwin s model (Goodwin 1967) Predator-prey system of wage share (ω) and employment rate (λ): where π = 1 ω is the profit share. Lyapunov functional ω = ω[φ(λ) α]; λ = λ[π/ν α β δ], V (ω, λ) = λ Φ(s) α ds s ω (1 s)/ν α β δ ds. s The Goodwin s model is e π - UWP and e π - SP but not e π - USP. Persistence theory applied to Keen s model First Previous Next Last 5

6 Example: Goodwin s model The Goodwin s model is e π - UWP and e π - SP but not e π - USP employment wage Persistence theory applied to Keen s model First Previous Next Last 6

7 Notations Variables: ω is the wage share, λ is the employment rate, d is the debt of capitalists, g S is the government spending, g T is the tax share, and π is the profit share. Parameters: α is the growth rate of productivity, β is the growth rate of total labor force, δ is the depreciation rate in capital, ν is the capital-to-output ratio, and r is the interest rate. Functions: Φ(λ) is the Phillips curve, κ(π) is the investment function, η(λ) is the government spending function, and Ξ(π) is the tax function. Persistence theory applied to Keen s model First Previous Next Last 7

8 Keen s model without government intervention (Keen 1995) Three-dimensional system of wage share (ω), employment rate (λ) and capital debt (d): ω = ω[φ(λ) α]; λ = λ[κ(π)/ν α β δ]; (1) d = [κ(π) π] d[κ(π)/ν δ], where π = 1 ω rd is the profit share. ω = 0, λ = 0 and d = is locally asymptotically stable if κ( )/ν δ < r and r > 0. ω = 0, λ = 0 and d = is locally asymptotically stable if κ( )/ν δ < r and r < 0. Persistence theory applied to Keen s model First Previous Next Last 8

9 Persistence results If κ( )/ν δ > r, then (1) is e d - USP and e π - UWP: lim inf t d(t) > M and lim sup t π(t) > M. If κ( )/ν δ > r and r > 0, then (1) is e π - USP and e d - UWP: lim sup t π(t) < M and lim inf t d(t) < M. If r 0, then (1) is e π - UWP: lim inf t π(t) < M. If r = 0, then (1) is e π - USP, e π - UWP, e π - SP, e d - UWP and e d - UWP. Persistence theory applied to Keen s model First Previous Next Last 9

10 Keen s model with government intervention I (Keen 1995) Five-dimensional system of wage share (ω), employment rate (λ), capital debt (d), government spending (g S ) and tax (g T ): ω = ω[φ(λ) α]; λ = λ[κ(π)/ν α β δ]; d = [κ(π) π] d[κ(π)/ν δ]; (2) g S = η(λ) g S [κ(π)/ν d]; g T = Ξ(π) g T [κ(π)/ν d], where π = 1 ω rd + g S g T is the profit share. Let u = rd g S + g T, we have u = r[κ(π) π] u[κ(π)/ν δ] η(λ) + Ξ(π). Persistence theory applied to Keen s model First Previous Next Last 10

11 Keen s model with government intervention I The system (2) can be reduced to a three-dimensional system with wage share (ω), employment rate (λ) and normalized debt (u): ω = ω[φ(λ) α]; λ = λ[κ(π)/ν α β δ]; (3) u = r[κ(π) π] u[κ(π)/ν δ] η(λ) + Ξ(π), where π = 1 ω u is the profit share. ω = 0, λ = 0 and u = is locally asymptotically stable if κ( )/ν δ < r. Persistence theory applied to Keen s model First Previous Next Last 11

12 Persistence results If κ( )/ν δ > r and r 0, then (3) is e u - USP, e π - USP, e π - UWP and e u - UWP: M < lim sup t π(t) < M and M < lim inf t u(t) < M. If κ( )/ν δ > r and r < 0, then (3) is e u - USP, e π - UWP and e π - UWP: lim sup t u(t) < M, lim inf t π(t) < M and lim sup t (3) is always e π - UWP: lim inf t π(t) < M. If η(λ) = O(1/λ) as λ 0 and α + β > r, then (3) is e π - UWP: lim sup t π(t) > M. π(t) > M. Persistence theory applied to Keen s model First Previous Next Last 12

13 Keen s model with government intervention II (Keen 1995, modified by Grasselli and Costa Lima 2012) Five-dimensional system of wage share (ω), employment rate (λ), capital debt (d), government spending (g S ) and tax (g T ): ω = ω[φ(λ) α]; λ = λ[κ(π)/ν α β δ]; d = [κ(π) π] d[κ(π)/ν δ]; (4) g S = g S {η(λ) [κ(π)/ν d]}; g T = g T {Ξ(π) [κ(π)/ν d]}, where π = 1 ω rd + g S g T is the profit share. We assume Ξ( ) < κ( )/ν δ. Persistence theory applied to Keen s model First Previous Next Last 13

14 Persistence results ω = 0, λ = 0, d =, g S = 0 and g T = 0 is locally asymptotically stable if η(0) < κ( )/ν δ < r. ω = 0, λ = 0, d =, g S = (with g S d) and g T = 0 is locally asymptotically stable if κ( )/ν δ < η(0) < r. (4) is always e π - UWP: lim inf t π(t) < M. If κ( )/ν δ > r and r 0, then (4) is e π - USP: lim sup t π(t) < M. If κ( )/ν δ > r, then (4) is e π - UWP: lim sup t π(t) > M. If η(0) > r, then (4) is e π - UWP: lim sup t π(t) > M. Persistence theory applied to Keen s model First Previous Next Last 14

15 Brief summary Keen s model is always e π - UWP: lim inf t π(t) < M. If κ( )/ν δ > r and r 0, then Keen s model is e π - USP: lim sup t π(t) < M. If κ( )/ν δ > r, then Keen s model is e π - UWP: lim sup t π(t) > M. For the Keen s model with government intervention I, if η(λ) = O(1/λ) as λ 0 and α + β > r, then it is still e π - UWP: lim sup t π(t) > M. For the Keen s model with government intervention II, if η(0) > r, then it is still e π - UWP: lim sup t π(t) > M. Question: when do we have e π - USP: lim inf t π(t) > M? Persistence theory applied to Keen s model First Previous Next Last 15

16 Discussion: from UWP to USP The Goodwin s model is e π - UWP and e π - SP but not e π - USP. where π = 1 ω is the profit share. ω = ω[φ(λ) α]; λ = λ[π/ν α β δ], We will have e π - USP by introducing self-adjustment for the wage share and employment rate: ω = ω[φ(λ) α A 1 (ω)]; λ = λ[π/ν α β δ A 2 (λ)]. Persistence theory applied to Keen s model First Previous Next Last 16

17 Discussion: from UWP to USP For the Keen s 3D model with self-adjustment, we also have e π - USP provided κ( )/ν δ > r and r 0. The proof is organized as follows: ω = ω[φ(λ) α A 1 (ω)]; λ = λ[κ(π)/ν α β δ A 2 (λ)]; d = [κ(π) π] d[κ(π)/ν δ]. 1. We first show that π is eventually uniformly bounded above. 2. Next, we prove by using self-adjustment terms that ω and λ are also eventually uniformly bounded above. 3. Finally, we have π eventually uniformly bounded below, namely, e π - USP. Remark: it is strange but most likely the case that in the proof of USP we require eventually uniform boundedness (certain compactness) of the semiflow. Persistence theory applied to Keen s model First Previous Next Last 17

18 Discussion: sustained periodic orbits Keen s 3D model without government intervention: wage employment debt employment year wage Persistence theory applied to Keen s model First Previous Next Last 18

19 Discussion: sustained periodic orbits Keen s 5D model with government intervention I: wage employment debt employment year wage Persistence theory applied to Keen s model First Previous Next Last 19

20 Discussion: sustained periodic orbits Keen s 5D model with government intervention II: wage employment debt employment year wage Persistence theory applied to Keen s model First Previous Next Last 20

21 Thank you! Persistence theory applied to Keen s model First Previous Next Last 21