persistence theory applied to keen's model a link between...
TRANSCRIPT
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 ModellingYork University, Toronto
Persistence theory applied to Keen’s model First Previous Next Last 1
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
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, 2011.
• Which species, in a mathematical model of interacting species, will survive overthe long term?
• In a mathematical model of an epidemic, will the disease drive a hostpopulation 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
Persistence definitions
Let Φ(t, x) : R+×X → X be the semiflow generated by a differential system withinitial values x ∈ X. For a nonnegative functional ρ from X to R+, we say
• Φ is ρ - uniformly strongly persistent (USP) if lim inft→∞ ρ(Φ(t, x)) > ε forany x ∈ X with ρ(x) > 0.
• Φ is ρ - uniformly weakly persistent (UWP) if lim supt→∞ ρ(Φ(t, x)) > ε forany x ∈ X with ρ(x) > 0.
• Φ is ρ - strongly persistent (SP) if lim inft→∞ ρ(Φ(t, x)) > 0 for any x ∈ Xwith ρ(x) > 0.
• Φ is ρ - weakly persistent (WP) if lim supt→∞ ρ(Φ(t, x)) > 0 for any x ∈ Xwith ρ(x) > 0.
Persistence theory applied to Keen’s model First Previous Next Last 4
Example: Goodwin’s model
• (Goodwin 1967) Predator-prey system of wage share (ω) and employment rate(λ):
ω′ = ω[Φ(λ)− α];
λ′ = λ[π/ν − α− β − δ],
where π = 1− ω is the profit share.
• Lyapunov functional
V (ω, λ) =
∫ λ Φ(s)− αs
ds−∫ ω (1− s)/ν − α− β − δ
sds.
• The Goodwin’s model is eπ - UWP and eπ - SP but not eπ - USP.
Persistence theory applied to Keen’s model First Previous Next Last 5
Example: Goodwin’s model
The Goodwin’s model is eπ - UWP and eπ - SP but not eπ - USP.
0.7 0.8 0.9 1 1.1 1.20.7
0.75
0.8
0.85
0.9
0.95
1
wage
empl
oym
ent
Persistence theory applied to Keen’s model First Previous Next Last 6
Notations
• Variables: ω is the wage share, λ is the employment rate, d is the debt ofcapitalists, gS is the government spending, gT is the tax share, and π is theprofit share.
• Parameters: α is the growth rate of productivity, β is the growth rate of totallabor 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) isthe government spending function, and Ξ(π) is the tax function.
Persistence theory applied to Keen’s model First Previous Next Last 7
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 < rand r > 0.
• ω = 0, λ = 0 and d = −∞ is locally asymptotically stable if κ(−∞)/ν − δ < rand r < 0.
Persistence theory applied to Keen’s model First Previous Next Last 8
Persistence results
• If κ(−∞)/ν − δ > r, then (1) is ed - USP and eπ - UWP:
lim inft→∞
d(t) > −M and lim supt→∞
π(t) > −M.
• If κ(−∞)/ν − δ > r and r > 0, then (1) is e−π - USP and e−d - UWP:
lim supt→∞
π(t) < M and lim inft→∞
d(t) < M.
• If r 6= 0, then (1) is e−π - UWP: lim inft→∞ π(t) < M.
• If r = 0, then (1) is e−π - USP, eπ - UWP, eπ - SP, e−d - UWP and ed - UWP.
Persistence theory applied to Keen’s model First Previous Next Last 9
Keen’s model with government intervention I
(Keen 1995) Five-dimensional system of wage share (ω), employment rate (λ),capital debt (d), government spending (gS) and tax (gT ):
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)− gS[κ(π)/ν − d];
g′T = Ξ(π)− gT [κ(π)/ν − d],
where π = 1−ω− rd+ gS − gT is the profit share. Let u = rd− gS + gT , 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
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
Persistence results
• If κ(−∞)/ν − δ > r and r ≥ 0, then (3) is eu - USP, e−π - USP, eπ - UWPand e−u - UWP:
−M < lim supt→∞
π(t) < M and −M < lim inft→∞
u(t) < M.
• If κ(−∞)/ν − δ > r and r < 0, then (3) is e−u - USP, e−π - UWP and eπ -UWP:
lim supt→∞
u(t) < M, lim inft→∞
π(t) < M and lim supt→∞
π(t) > −M.
• (3) is always e−π - UWP: lim inft→∞ π(t) < M.
• If η(λ) = O(1/λ) as λ→ 0 and α+ β > r, then (3) is eπ - UWP:lim supt→∞ π(t) > −M.
Persistence theory applied to Keen’s model First Previous Next Last 12
Keen’s model with government intervention II
(Keen 1995, modified by Grasselli and Costa Lima 2012) Five-dimensional systemof wage share (ω), employment rate (λ), capital debt (d), government spending(gS) and tax (gT ):
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 = gS{η(λ)− [κ(π)/ν − d]};g′T = gT{Ξ(π)− [κ(π)/ν − d]},
where π = 1− ω − rd+ gS − gT is the profit share. We assumeΞ(−∞) < κ(−∞)/ν − δ.
Persistence theory applied to Keen’s model First Previous Next Last 13
Persistence results
• ω = 0, λ = 0, d =∞, gS = 0 and gT = 0 is locally asymptotically stable ifη(0) < κ(−∞)/ν − δ < r.
• ω = 0, λ = 0, d =∞, gS =∞ (with gS � d) and gT = 0 is locallyasymptotically stable if κ(−∞)/ν − δ < η(0) < r.
• (4) is always e−π - UWP: lim inft→∞ π(t) < M.
• If κ(−∞)/ν − δ > r and r ≥ 0, then (4) is e−π - USP: lim supt→∞ π(t) < M.
• If κ(−∞)/ν − δ > r, then (4) is eπ - UWP: lim supt→∞ π(t) > −M.
• If η(0) > r, then (4) is eπ - UWP: lim supt→∞ π(t) > −M.
Persistence theory applied to Keen’s model First Previous Next Last 14
Brief summary
• Keen’s model is always e−π - UWP: lim inft→∞ π(t) < M.
• If κ(−∞)/ν − δ > r and r ≥ 0, then Keen’s model is e−π - USP:lim supt→∞ π(t) < M.
• If κ(−∞)/ν − δ > r, then Keen’s model is eπ - UWP: lim supt→∞ π(t) > −M.
• For the Keen’s model with government intervention I, if η(λ) = O(1/λ) asλ→ 0 and α+ β > r, then it is still eπ - UWP: lim supt→∞ π(t) > −M.
• For the Keen’s model with government intervention II, if η(0) > r, then it isstill eπ - UWP: lim supt→∞ π(t) > −M.
• Question: when do we have eπ - USP: lim inft→∞ π(t) > −M?
Persistence theory applied to Keen’s model First Previous Next Last 15
Discussion: from UWP to USP
• The Goodwin’s model is eπ - UWP and eπ - SP but not eπ - USP.
ω′ = ω[Φ(λ)− α];
λ′ = λ[π/ν − α− β − δ],
where π = 1− ω is the profit share.
• We will have eπ - USP by introducing self-adjustment for the wage share andemployment rate:
ω′ = ω[Φ(λ)− α−A1(ω)];
λ′ = λ[π/ν − α− β − δ −A2(λ)].
Persistence theory applied to Keen’s model First Previous Next Last 16
Discussion: from UWP to USP
• For the Keen’s 3D model with self-adjustment, we also have eπ - USP providedκ(−∞)/ν − δ > r and r ≥ 0.
ω′ = ω[Φ(λ)− α−A1(ω)];
λ′ = λ[κ(π)/ν − α− β − δ −A2(λ)];
d′ = [κ(π)− π]− d[κ(π)/ν − δ].
• The proof is organized as follows:
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, eπ - USP.
• Remark: it is strange but most likely the case that in the proof of USP werequire eventually uniform boundedness (certain compactness) of the semiflow.
Persistence theory applied to Keen’s model First Previous Next Last 17
Discussion: sustained periodic orbits
Keen’s 3D model without government intervention:
0 5 10 15 200.7
0.8
0.9
1
1.1
1.2
wag
e
2
2.05
2.1
2.15
2.2
2.25
2.3
debt
0 5 10 15 200.75
0.8
0.85
0.9
0.95
1
empl
oym
ent
year
0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.150.75
0.8
0.85
0.9
0.95
1
wage
em
plo
ym
en
t
Persistence theory applied to Keen’s model First Previous Next Last 18
Discussion: sustained periodic orbits
Keen’s 5D model with government intervention I:
0 2 4 6 8 100.74
0.76
0.78
0.8
wag
e
0.075
0.08
0.085
0.09
0.095
0.1
0.105
0.11
debt
0 2 4 6 8 100.95
0.96
0.97
0.98
empl
oym
ent
year
0.75 0.755 0.76 0.765 0.77 0.775 0.78 0.7850.958
0.96
0.962
0.964
0.966
0.968
0.97
0.972
0.974
0.976
0.978
wage
em
plo
ym
en
t
Persistence theory applied to Keen’s model First Previous Next Last 19
Discussion: sustained periodic orbits
Keen’s 5D model with government intervention II:
0 0.5 1 1.5 20.6
0.8
1
1.2
1.4
wag
e
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
debt
0 0.5 1 1.5 20.8
0.85
0.9
0.95
1
empl
oym
ent
year
0.6 0.7 0.8 0.9 1 1.1 1.20.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
wage
em
plo
ym
en
t
Persistence theory applied to Keen’s model First Previous Next Last 20
Thank you!
Persistence theory applied to Keen’s model First Previous Next Last 21