A two-sector model of endogenous distributive cycles and overproduction
Author
John Cajas Guijarro
Title
A two-sector model of endogenous distributive cycles and overproduction
Description
Mathematical supplement to accompany Cajas Guijarro (2024): "A two-sector model of endogenous distributive cycles and overproduction" (El Trimestre Económico, in press)
Category
Working Material
Keywords
sectoral capacity utilization, bargaining power, long run cycles, overproduction
URL
http://www.notebookarchive.org/2024-02-6yy60je/
DOI
https://notebookarchive.org/2024-02-6yy60je
Date Added
2024-02-15
Date Last Modified
2024-02-15
File Size
6.69 megabytes
Supplements
Rights
CC BY-NC-SA 4.0
Download
Open in Wolfram Cloud
Mathematical supplement to accompany Cajas Guijarro (2024): “A two-sector model of endogenous distributive cycles and overproduction” (El Trimestre Económico, in press)
A two-sector model of endogenous distributive cycles and overproduction
A two-sector model of endogenous distributive cycles and overproduction
John Cajas Guijarro
This notebook introduces a two-sector model of endogenous distributive cycles and overproduction, using elements from Marxian and Kaleckian perspectives. The model combines the two-sector model with market power presented by Dutt (1988) with insights from the cyclical model of Goodwin (1967) and the model of growth and distribution of Dutt (1987). The notebook establishes the analytical conditions for a short-run equilibrium point for sectoral capacity utilization rates. Furthermore, for the long run, considering capital accumulation, the notebook demonstrates the existence of stable cycles in four dimensions (sectoral distribution of capital, employment rate, and markups of sectors producing capital and consumption goods). These findings are used to reinterpret overproduction within capitalist cycles, where the relative magnitudes of overaccumulation of capital and underconsumption change at different phases of the cycle, while full employment is not a relevant case.
(***********************************************************************)(*Limpiamostodo*)ClearAll["Global`*"]
In[]:=
(**********************)(***Esquemainicial***)(**********************)
In[]:=
ec[1]={p1[t](1+z1[t])*w[t]/q1[t],p2[t](1+z2[t])*w[t]/q2[t]};
In[]:=
ec[2]={q1[t]Q1[t]/L1[t],q2[t]Q2[t]/L2[t]};
In[]:=
ec[3]={g1[t]α1+β1*r1[t]+τ1*u1[t],g2[t]α2+β2*r2[t]+τ2*u2[t]};
In[]:=
ec[4]={u1[t]Q1[t]/K1[t],u2[t]Q2[t]/K2[t]};
In[]:=
ec[5]={σ1[t]Q1M[t]/K1[t],σ2[t]Q2M[t]/K2[t]};
In[]:=
ec[6]={p1[t]*Q1[t]w[t]*L1[t]+r1[t]*p1[t]*K1[t],p2[t]*Q2[t]w[t]*L2[t]+r2[t]*p1[t]*K2[t]};
In[]:=
ec[7]=ED1[t]g1[t]*K1[t]+g2[t]*K2[t]-Q1[t];
In[]:=
ec[8]=ED2[t](w[t]/p2[t])*(L1[t]+L2[t])+(p1[t]/p2[t])*(1-s)*(r1[t]*K1[t]+r2[t]*K2[t])-Q2[t];
In[]:=
ec[9]=k[t]K1[t]/K2[t];
In[]:=
deduccion1=Solve[ec[1],{w[t],p1[t]}]
Out[]=
w[t],p1[t]
p2[t]q2[t]
1+z2[t]
p2[t]q2[t](1+z1[t])
q1[t](1+z2[t])
In[]:=
{ec[10],ec[11]}=deduccion1[[1]]/.{RuleEqual}
Out[]=
w[t],p1[t]
p2[t]q2[t]
1+z2[t]
p2[t]q2[t](1+z1[t])
q1[t](1+z2[t])
In[]:=
deduccion2=Solve[Flatten[{ec[1],ec[2],ec[4],ec[6]}],{r1[t],r2[t],L1[t],L2[t],Q1[t],Q2[t],p1[t],p2[t]}]
Out[]=
r1[t],r2[t],L1[t],L2[t],Q1[t]K1[t]u1[t],Q2[t]K2[t]u2[t],p1[t],p2[t]
u1[t]z1[t]
1+z1[t]
q1[t]u2[t]z2[t]
q2[t](1+z1[t])
K1[t]u1[t]
q1[t]
K2[t]u2[t]
q2[t]
w[t](1+z1[t])
q1[t]
w[t](1+z2[t])
q2[t]
In[]:=
{ec[12],ec[13]}=deduccion2[[1,1;;2]]/.{RuleEqual}
Out[]=
r1[t],r2[t]
u1[t]z1[t]
1+z1[t]
q1[t]u2[t]z2[t]
q2[t](1+z1[t])
In[]:=
ec[17]=δ1[t](β1*z1[t]/(1+z1[t]))+τ1;
In[]:=
ec[18]=δ2[t](β2*z2[t]*q1[t]/(q2[t]*(1+z1[t])))+τ2;
In[]:=
ec[14]={u1'[t]f1*(ED1[t]/K1[t]),u2'[t]f2*(ED2[t]/K2[t])};
In[]:=
deduccion3=FullSimplify[Solve[Flatten[{ec[2],ec[3],ec[4],Table[ec[x],{x,7,14}],ec[17],ec[18]}],{u1'[t],u2'[t],g1[t],g2[t],L1[t],L2[t],Q1[t],Q2[t],ED1[t],ED2[t],K1[t],w[t],p1[t],r1[t],r2[t],τ1,τ2}]]
Out[]=
[t],[t],g1[t]α1+u1[t]δ1[t],g2[t]α2+u2[t]δ2[t],L1[t],L2[t],Q1[t]k[t]K2[t]u1[t],Q2[t]K2[t]u2[t],ED1[t]K2[t](α2+k[t](α1+u1[t](-1+δ1[t]))+u2[t]δ2[t]),ED2[t],K1[t]k[t]K2[t],w[t],p1[t],r1[t],r2[t],τ1-+δ1[t],τ2-+δ2[t]
′
u1
f1(α2+k[t](α1+u1[t](-1+δ1[t]))+u2[t]δ2[t])
k[t]
′
u2
f2(k[t]q2[t]u1[t](1+z1[t]-sz1[t])-sq1[t]u2[t]z2[t])
q1[t](1+z2[t])
k[t]K2[t]u1[t]
q1[t]
K2[t]u2[t]
q2[t]
K2[t](k[t]q2[t]u1[t](1+z1[t]-sz1[t])-sq1[t]u2[t]z2[t])
q1[t](1+z2[t])
p2[t]q2[t]
1+z2[t]
p2[t]q2[t](1+z1[t])
q1[t](1+z2[t])
u1[t]z1[t]
1+z1[t]
q1[t]u2[t]z2[t]
q2[t]+q2[t]z1[t]
β1z1[t]
1+z1[t]
β2q1[t]z2[t]
q2[t]+q2[t]z1[t]
In[]:=
{ec[15],ec[16]}=deduccion3[[1,1;;2]]/.{RuleEqual}
Out[]=
[t],[t]
′
u1
f1(α2+k[t](α1+u1[t](-1+δ1[t]))+u2[t]δ2[t])
k[t]
′
u2
f2(k[t]q2[t]u1[t](1+z1[t]-sz1[t])-sq1[t]u2[t]z2[t])
q1[t](1+z2[t])
In[]:=
ec[19]={u1'[t]0,u2'[t]0};
In[]:=
ec[22]=Ω[t]s*((1-τ1)*(1+z1[t])-β1*z1[t])-(1+(1-s)*z1[t])*(β2+τ2*(q2[t]/q1[t])*((1+z1[t])/z2[t]));
In[]:=
deduccion4=FullSimplify[Solve[Flatten[{ec[15],ec[16],ec[17],ec[18],ec[19],ec[22]}],{u1[t],u2[t],u1'[t],u2'[t],δ1[t],δ2[t],β1}]]
Out[]=
u1[t],u2[t]-,[t]0,[t]0,δ1[t]+,δ2[t]τ2+,β1(τ2q2[t](1+z1[t])(-1+(-1+s)z1[t])+q1[t]z2[t](s-β2-sτ1+(s-β2+sβ2-sτ1)z1[t]-Ω[t]))
s(α2+α1k[t])(1+z1[t])
k[t]Ω[t]
(α2+α1k[t])q2[t](1+z1[t])(-1+(-1+s)z1[t])
q1[t]z2[t]Ω[t]
′
u1
′
u2
τ2q2[t](-1+(-1+s)z1[t])
q1[t]z2[t]
s-β2+(s+(-1+s)β2)z1[t]-Ω[t]
1+z1[t]
s
β2q1[t]z2[t]
q2[t]+q2[t]z1[t]
1
sq1[t]z1[t]z2[t]
In[]:=
{ec[20],ec[21]}=deduccion4[[1,1;;2]]/.{RuleEqual}
Out[]=
u1[t],u2[t]-
s(α2+α1k[t])(1+z1[t])
k[t]Ω[t]
(α2+α1k[t])q2[t](1+z1[t])(-1+(-1+s)z1[t])
q1[t]z2[t]Ω[t]
In[]:=
deduccion5={{D[ec[15][[2]],u1[t]],D[ec[15][[2]],u2[t]]},{D[ec[16][[2]],u1[t]],D[ec[16][[2]],u2[t]]}}
Out[]=
f1(-1+δ1[t]),,,-
f1δ2[t]
k[t]
f2k[t]q2[t](1+z1[t]-sz1[t])
q1[t](1+z2[t])
f2sz2[t]
1+z2[t]
In[]:=
ec[23]=ΤCP[t]Tr[deduccion5]
Out[]=
ΤCP[t]-+f1(-1+δ1[t])
f2sz2[t]
1+z2[t]
In[]:=
deduccion6=ΔCP[t]FullSimplify[Det[deduccion5]]
Out[]=
ΔCP[t]-
f1f2(sq1[t]z2[t](-1+δ1[t])+q2[t](1+z1[t]-sz1[t])δ2[t])
q1[t](1+z2[t])
In[]:=
deduccion7=FullSimplify[Solve[{deduccion6,ec[17],ec[18],ec[22]},{ΔCP[t],δ1[t],δ2[t],β1}]]
Out[]=
ΔCP[t],δ1[t]1--,δ2[t]τ2+,β1-
f1f2z2[t]Ω[t]
(1+z1[t])(1+z2[t])
(1+z1[t]-sz1[t])β2+
τ2q2[t](1+z1[t])
q1[t]z2[t]
s(1+z1[t])
Ω[t]
s+sz1[t]
β2q1[t]z2[t]
q2[t]+q2[t]z1[t]
s(-1+τ1)(1+z1[t])+(1+z1[t]-sz1[t])β2++Ω[t]
τ2q2[t](1+z1[t])
q1[t]z2[t]
sz1[t]
In[]:=
ec[24]=deduccion7[[1,1]]/.{RuleEqual}
Out[]=
ΔCP[t]
f1f2z2[t]Ω[t]
(1+z1[t])(1+z2[t])
In[]:=
deduccion8=Solve[{ec[12],ec[13],ec[20],ec[21]},{r1[t],r2[t],u1[t],u2[t]}]
Out[]=
r1[t],r2[t]-,u1[t],u2[t]-
s(α2+α1k[t])z1[t]
k[t]Ω[t]
(α2+α1k[t])(-1-z1[t]+sz1[t])
Ω[t]
s(α2+α1k[t])(1+z1[t])
k[t]Ω[t]
(α2+α1k[t])q2[t](1+z1[t])(-1-z1[t]+sz1[t])
q1[t]z2[t]Ω[t]
In[]:=
{ec[25],ec[26]}=deduccion8[[1,1;;2]]/.{RuleEqual}
Out[]=
r1[t],r2[t]-
s(α2+α1k[t])z1[t]
k[t]Ω[t]
(α2+α1k[t])(-1-z1[t]+sz1[t])
Ω[t]
In[]:=
ec[27]=k'[t]/k[t]g1[t]-g2[t];
In[]:=
ec[29]=γ1[t]β1*z1[t]+τ1*(1+z1[t]);
In[]:=
ec[30]=γ2[t](1+(1-s)*z1[t])*(β2+τ2*(q2[t]/q1[t])*((1+z1[t])/z2[t]));
In[]:=
deduccion9=FullSimplify[Solve[Flatten[{ec[3],ec[20],ec[21],ec[25],ec[26],ec[27],ec[29],ec[30]}],{k'[t],g1[t],g2[t],u1[t],u2[t],r1[t],r2[t],τ1,τ2}]]
Out[]=
[t](α1-α2)k[t]+,g1[t]α1+,g2[t]α2+,u1[t],u2[t]-,r1[t],r2[t]-,τ1,τ2-
′
k
(α2+α1k[t])(sγ1[t]-k[t]γ2[t])
Ω[t]
s(α2+α1k[t])γ1[t]
k[t]Ω[t]
(α2+α1k[t])γ2[t]
Ω[t]
s(α2+α1k[t])(1+z1[t])
k[t]Ω[t]
(α2+α1k[t])q2[t](1+z1[t])(-1+(-1+s)z1[t])
q1[t]z2[t]Ω[t]
s(α2+α1k[t])z1[t]
k[t]Ω[t]
(α2+α1k[t])(-1+(-1+s)z1[t])
Ω[t]
-β1z1[t]+γ1[t]
1+z1[t]
q1[t]z2[t](-β2+(-1+s)β2z1[t]+γ2[t])
q2[t](1+z1[t])(-1+(-1+s)z1[t])
In[]:=
ec[28]=deduccion9[[1,1]]/.{RuleEqual}
Out[]=
′
k
(α2+α1k[t])(sγ1[t]-k[t]γ2[t])
Ω[t]
In[]:=
ec[31]=k'[t]0;
In[]:=
deduccion10=Solve[{ec[28],ec[31]},{k[t],k'[t]}]
Out[]=
k[t]sα1γ1[t]-α2γ2[t]+α1Ω[t]-α2Ω[t]+[t]0,k[t]--sα1γ1[t]+α2γ2[t]-α1Ω[t]+α2Ω[t]+[t]0
1
2α1γ2[t]
4sα1α2γ1[t]γ2[t]+
,2
(sα1γ1[t]-α2γ2[t]+α1Ω[t]-α2Ω[t])
′
k
1
2α1γ2[t]
4sα1α2γ1[t]γ2[t]+
,2
(sα1γ1[t]-α2γ2[t]+α1Ω[t]-α2Ω[t])
′
k
In[]:=
ec[32]={deduccion10[[1,1]],deduccion10[[2,1]]}/.{RuleEqual}
Out[]=
k[t]sα1γ1[t]-α2γ2[t]+α1Ω[t]-α2Ω[t]+-sα1γ1[t]+α2γ2[t]-α1Ω[t]+α2Ω[t]+
1
2α1γ2[t]
4sα1α2γ1[t]γ2[t]+
,k[t]-2
(sα1γ1[t]-α2γ2[t]+α1Ω[t]-α2Ω[t])
1
2α1γ2[t]
4sα1α2γ1[t]γ2[t]+
2
(sα1γ1[t]-α2γ2[t]+α1Ω[t]-α2Ω[t])
In[]:=
ec[33]=dgkdkFullSimplify[D[ec[28][[2]]/k[t],k[t]]]
Out[]=
dgkdk-+α1γ2[t]
sα2γ1[t]
2
k[t]
Ω[t]
In[]:=
(********************************************)(***Modeloconsalariosypreciosvariables***)(********************************************)
In[]:=
ec[34]=l[t](L1[t]+L2[t])/NN[t];
In[]:=
deduccion11=FullSimplify[Solve[Flatten[{ec[34],ec[2],ec[4],ec[9]}],{l[t],L1[t],L2[t],Q1[t],Q2[t],K1[t]}]]
Out[]=
l[t],L1[t],L2[t],Q1[t]k[t]K2[t]u1[t],Q2[t]K2[t]u2[t],K1[t]k[t]K2[t]
K2[t](k[t]q2[t]u1[t]+q1[t]u2[t])
NN[t]q1[t]q2[t]
k[t]K2[t]u1[t]
q1[t]
K2[t]u2[t]
q2[t]
In[]:=
ec[35]=deduccion11[[1,1]]/.{RuleEqual}
Out[]=
l[t]
K2[t](k[t]q2[t]u1[t]+q1[t]u2[t])
NN[t]q1[t]q2[t]
In[]:=
deduccion12=Solve[{ec[20],ec[21],ec[35]},{l[t],u1[t],u2[t]}]
Out[]=
l[t]-,u1[t],u2[t]-
(α2+α1k[t])K2[t](1+z1[t])(-1-z1[t]+sz1[t]-sz2[t])
NN[t]q1[t]z2[t]Ω[t]
s(α2+α1k[t])(1+z1[t])
k[t]Ω[t]
(α2+α1k[t])q2[t](1+z1[t])(-1-z1[t]+sz1[t])
q1[t]z2[t]Ω[t]
In[]:=
ec[36]=deduccion12[[1,1]]/.{RuleEqual}
Out[]=
l[t]-
(α2+α1k[t])K2[t](1+z1[t])(-1-z1[t]+sz1[t]-sz2[t])
NN[t]q1[t]z2[t]Ω[t]
In[]:=
ec[38]={g1[t]K1'[t]/K1[t],g2[t]K2'[t]/K2[t]};
In[]:=
ec[39]=NN'[t]/NN[t]n;
In[]:=
ec[40]={q1'[t]/q1[t]θ1,q2'[t]/q2[t]θ2};
In[]:=
deduccion13=FullSimplify[Solve[Flatten[{D[Log[ec[36][[1]]]Log[ec[36][[2]]],t],ec[38],ec[39],ec[40]}],{l'[t],K1'[t],K2'[t],NN'[t],q1'[t],q2'[t]}]]
Out[]=
[t](l[t](-(α2+α1k[t])(1+z1[t])(-1+(-1+s)z1[t])Ω[t][t]+s(Ω[t]((1+z1[t])((n+θ1-g2[t])(α2+α1k[t])-α1[t])-(α2+α1k[t])[t])+(α2+α1k[t])(1+z1[t])[t])+z2[t](Ω[t]((1+z1[t])(-1+(-1+s)z1[t])(-(n+θ1-g2[t])(α2+α1k[t])+α1[t])+(α2+α1k[t])(-2+s+2(-1+s)z1[t])[t])-(α2+α1k[t])(1+z1[t])(-1+(-1+s)z1[t])[t])))((α2+α1k[t])(1+z1[t])z2[t](-1+(-1+s)z1[t]-sz2[t])Ω[t]),[t]g1[t]K1[t],[t]g2[t]K2[t],[t]nNN[t],[t]θ1q1[t],[t]θ2q2[t]
′
l
′
z2
2
z2[t]
′
k
′
z1
′
Ω
′
k
′
z1
′
Ω
′
K1
′
K2
′
NN
′
q1
′
q2
In[]:=
ec[37]=deduccion13[[1,1]]/.{RuleEqual}
Out[]=
′
l
′
z2
2
z2[t]
′
k
′
z1
′
Ω
′
k
′
z1
′
Ω
In[]:=
deduccion14=FullSimplify[Solve[{ec[3][[2]],ec[21],ec[26]},{g2[t],u2[t],r2[t]}]]
Out[]=
g2[t]α2-,u2[t]-,r2[t]-
(α2+α1k[t])(-1+(-1+s)z1[t])(τ2q2[t](1+z1[t])+β2q1[t]z2[t])
q1[t]z2[t]Ω[t]
(α2+α1k[t])q2[t](1+z1[t])(-1+(-1+s)z1[t])
q1[t]z2[t]Ω[t]
(α2+α1k[t])(-1+(-1+s)z1[t])
Ω[t]
In[]:=
ec[41]=deduccion14[[1,1]]/.{RuleEqual}
Out[]=
g2[t]α2-
(α2+α1k[t])(-1+(-1+s)z1[t])(τ2q2[t](1+z1[t])+β2q1[t]z2[t])
q1[t]z2[t]Ω[t]
In[]:=
ec[43]=q21q2[t]/q1[t];
In[]:=
ec[42]=FullSimplify[D[Log[ec[22][[1]]]Log[ec[22][[2]]],t]/.Solve[Flatten[{ec[40],ec[43]}],{q1'[t],q2'[t],q2[t]}]][[1]]
Out[]=
′
Ω
Ω[t]
′
z1
(-1+s)(q21τ2(1+z1[t])+β2z2[t])[t]
′
z1
z2[t]
q21τ2(-1+(-1+s)z1[t])(z2[t]((θ1-θ2)(1+z1[t])-[t])+(1+z1[t])[t])
′
z1
′
z2
2
z2[t]
q21τ2(1+z1[t])
z2[t]
In[]:=
ec[44]=w'[t]/w[t]-p2'[t]/p2[t]-ϕ+ρ*l[t]-μ*(p2'[t]/p2[t]);
In[]:=
deduccion15={FullSimplify[D[Log[ec[1][[1,1]]]Log[ec[1][[1,2]]],t]]/.Solve[ec[40][[1]],q1'[t]],FullSimplify[D[Log[ec[1][[2,1]]]Log[ec[1][[2,2]]],t]]/.Solve[ec[40][[2]],q2'[t]]}
Out[]=
[t]+[t]θ1+[t],[t]+[t]θ2+[t]
′
w
w[t]
′
z1
1+z1[t]
′
p1
p1[t]
′
w
w[t]
′
z2
1+z2[t]
′
p2
p2[t]
In[]:=
ec[45]=Flatten[deduccion15]
Out[]=
[t]+[t]θ1+[t],[t]+[t]θ2+[t]
′
w
w[t]
′
z1
1+z1[t]
′
p1
p1[t]
′
w
w[t]
′
z2
1+z2[t]
′
p2
p2[t]
In[]:=
ec[46]={p1'[t]/p1[t]λ1*(ζ1-z1[t]),p2'[t]/p2[t]λ2*(ζ2-z2[t])};
In[]:=
deduccion16=FullSimplify[ec[45]/.Solve[Flatten[{ec[44],ec[46]}],{p1'[t],p2'[t],w'[t]}]]
Out[]=
θ1+ζ1λ1+ζ2λ2(-1+μ)+ϕρl[t]+λ1z1[t]+λ2(-1+μ)z2[t]+[t],θ2+ζ2λ2μ+ϕρl[t]+λ2μz2[t]+[t]
′
z1
1+z1[t]
′
z2
1+z2[t]
In[]:=
ec[47]=deduccion16[[1,1]]
Out[]=
θ1+ζ1λ1+ζ2λ2(-1+μ)+ϕρl[t]+λ1z1[t]+λ2(-1+μ)z2[t]+[t]
′
z1
1+z1[t]
In[]:=
ec[48]=deduccion16[[1,2]]
Out[]=
θ2+ζ2λ2μ+ϕρl[t]+λ2μz2[t]+[t]
′
z2
1+z2[t]
In[]:=
sistema1=Flatten[{ec[22],ec[28],ec[29],ec[30],ec[37],ec[41],ec[42],ec[43],ec[47],ec[48]}]
Out[]=
Ω[t]s(-β1z1[t]+(1-τ1)(1+z1[t]))-(1+(1-s)z1[t])β2+,[t](α1-α2)k[t]+,γ1[t]β1z1[t]+τ1(1+z1[t]),γ2[t](1+(1-s)z1[t])β2+,[t](l[t](-(α2+α1k[t])(1+z1[t])(-1+(-1+s)z1[t])Ω[t][t]+s(Ω[t]((1+z1[t])((n+θ1-g2[t])(α2+α1k[t])-α1[t])-(α2+α1k[t])[t])+(α2+α1k[t])(1+z1[t])[t])+z2[t](Ω[t]((1+z1[t])(-1+(-1+s)z1[t])(-(n+θ1-g2[t])(α2+α1k[t])+α1[t])+(α2+α1k[t])(-2+s+2(-1+s)z1[t])[t])-(α2+α1k[t])(1+z1[t])(-1+(-1+s)z1[t])[t])))((α2+α1k[t])(1+z1[t])z2[t](-1+(-1+s)z1[t]-sz2[t])Ω[t]),g2[t]α2-,[t]-s(-1+β1+τ1)[t]+-s(1-τ1-(-1+β1+τ1)z1[t])-(1+z1[t]-sz1[t])β2+,q21,θ1+ζ1λ1+ζ2λ2(-1+μ)+ϕρl[t]+λ1z1[t]+λ2(-1+μ)z2[t]+[t],θ2+ζ2λ2μ+ϕρl[t]+λ2μz2[t]+[t]
τ2q2[t](1+z1[t])
q1[t]z2[t]
′
k
(α2+α1k[t])(sγ1[t]-k[t]γ2[t])
Ω[t]
τ2q2[t](1+z1[t])
q1[t]z2[t]
′
l
′
z2
2
z2[t]
′
k
′
z1
′
Ω
′
k
′
z1
′
Ω
(α2+α1k[t])(-1+(-1+s)z1[t])(τ2q2[t](1+z1[t])+β2q1[t]z2[t])
q1[t]z2[t]Ω[t]
′
Ω
Ω[t]
′
z1
(-1+s)(q21τ2(1+z1[t])+β2z2[t])[t]
′
z1
z2[t]
q21τ2(-1+(-1+s)z1[t])(z2[t]((θ1-θ2)(1+z1[t])-[t])+(1+z1[t])[t])
′
z1
′
z2
2
z2[t]
q21τ2(1+z1[t])
z2[t]
q2[t]
q1[t]
′
z1
1+z1[t]
′
z2
1+z2[t]
In[]:=
deduccion17=Solve[sistema1,{k'[t],l'[t],z1'[t],z2'[t],g2[t],Ω[t],Ω'[t],q2[t],γ1[t],γ2[t]}];
In[]:=
ec[48]=deduccion17[[1,1;;4]]/.{RuleEqual};
In[]:=
(*{s1,α1α,α2α,β1β,β2β,τ1τ,τ2τ,θ1θ,θ2θ,λ1λ,λ2λ}*)
In[]:=
simplificacion={s1,α1α,α2α,β11,β2β2,τ10,τ20,θ1θ,θ2θ,λ10,λ2λ2,n0};
In[]:=
sistema2=Solve[FullSimplify[ec[48]/.simplificacion],{k'[t],l'[t],z1'[t],z2'[t]}][[1]]/.{RuleEqual}
Out[]=
[t],[t]-l[t]+++λ2(-1+μ)z2[t],[t](1+z1[t])(θ-ζ2λ2+ζ2λ2μ+ϕ-ρl[t]+λ2z2[t]-λ2μz2[t]),[t]-(1+z2[t])(-θ-ζ2λ2μ-ϕ+ρl[t]+λ2μz2[t])
′
k
α(1+k[t])(β2k[t]-z1[t])
-1+β2
′
l
α-(-1+β2)(λ2(ζ2(-1+μ)+μ)+ϕ)+αz1[t]
-1+β2
θ+ζ2λ2μ+ϕ
z2[t]
ρl[t](-1+z2[t])
z2[t]
′
z1
′
z2
In[]:=
{ec[50],ec[51],ec[52],ec[53]}=sistema2;
In[]:=
equilibrios=FullSimplify[Solve[sistema2/.{k'[t]0,l'[t]0,z1'[t]0,z2'[t]0},{k[t],l[t],z1[t],z2[t]}]]/.{RuleEqual}
Out[]=
k[t]-1,l[t]0,z1[t]-1,z2[t],k[t]-,l[t]0,z1[t]-1,z2[t],k[t]-,l[t]0,z1[t]-1,z2[t]-1,k[t]-1,l[t],z1[t]-1-,z2[t]-1,k[t]-,l[t],z1[t]-1-,z2[t]-1,k[t]-1,l[t],z1[t]-1,z2[t]-1,k[t]-,l[t],z1[t]-1,z2[t]-1,{k[t]-1,l[t]0,z1[t]-1,z2[t]-1},k[t]-1,l[t],z1[t]-1+,z2[t]ζ2,k[t]-,l[t],z1[t]-1+,z2[t]ζ2,k[t]-1,l[t],z1[t]-1,z2[t]ζ2+,k[t]-,l[t],z1[t]-1,z2[t]ζ2+
θ+ζ2λ2μ+ϕ
λ2μ
1
β2
θ+ζ2λ2μ+ϕ
λ2μ
1
β2
θ+(1+ζ2)λ2(-1+μ)+ϕ
ρ
(-1+β2)(θ-(1+ζ2)λ2)
α
α+(-1+β2)(θ-(1+ζ2)λ2)
αβ2
θ+(1+ζ2)λ2(-1+μ)+ϕ
ρ
(-1+β2)(θ-(1+ζ2)λ2)
α
θ+(1+ζ2)λ2(-1+2μ)+2ϕ
2ρ
1
β2
θ+(1+ζ2)λ2(-1+2μ)+2ϕ
2ρ
θ+ϕ
ρ
θ-β2θ
α
α-θ+β2θ
αβ2
θ+ϕ
ρ
θ-β2θ
α
θ-θμ+ϕ
ρ
θ
λ2
1
β2
θ-θμ+ϕ
ρ
θ
λ2
In[]:=
Length[equilibrios]
Out[]=
12
In[]:=
{ec[54],ec[55],ec[56],ec[57]}=equilibrios[[10]]
Out[]=
k[t]-,l[t],z1[t]-1+,z2[t]ζ2
α-θ+β2θ
αβ2
θ+ϕ
ρ
θ-β2θ
α
In[]:=
(*********************************************************)(**********ANEXOS:pruebasdeestabilidadyciclos**********)(*********************************************************)
In[]:=
(************Anexo1:Estabilidaddelmodelo***********)
In[]:=
(*Equilibriodinamico*)
In[]:=
equilibriodin=equilibrios[[10]]
Out[]=
k[t]-,l[t],z1[t]-1+,z2[t]ζ2
α-θ+β2θ
αβ2
θ+ϕ
ρ
θ-β2θ
α
In[]:=
(*Sistemadinamico*)
In[]:=
sistemadin=sistema2
Out[]=
[t],[t]-l[t]+++λ2(-1+μ)z2[t],[t](1+z1[t])(θ-ζ2λ2+ζ2λ2μ+ϕ-ρl[t]+λ2z2[t]-λ2μz2[t]),[t]-(1+z2[t])(-θ-ζ2λ2μ-ϕ+ρl[t]+λ2μz2[t])
′
k
α(1+k[t])(β2k[t]-z1[t])
-1+β2
′
l
α-(-1+β2)(λ2(ζ2(-1+μ)+μ)+ϕ)+αz1[t]
-1+β2
θ+ζ2λ2μ+ϕ
z2[t]
ρl[t](-1+z2[t])
z2[t]
′
z1
′
z2
In[]:=
(*Seobtienelamatrizjacobianadelsistemadinamicoevaluadaenelpuntodeequilibrio*)
In[]:=
jacobiana=D[{sistemadin[[1,2]],sistemadin[[2,2]],sistemadin[[3,2]],sistemadin[[4,2]]},{{k[t],l[t],z1[t],z2[t]}}]/.(equilibriodin/.{EqualRule})//FullSimplify
Out[]=
α-θ,0,,0,0,-,,-,0,,0,,{0,-(1+ζ2)ρ,0,-(1+ζ2)λ2μ}
-α+θ
β2
(-1+ζ2)(θ+ϕ)
ζ2
α(θ+ϕ)
ρ-β2ρ
λ2(ζ2(-1+μ)-μ)(θ+ϕ)
ζ2ρ
(-1+β2)θρ
α
(-1+β2)θλ2(-1+μ)
α
In[]:=
jacobiana//MatrixForm
Out[]//MatrixForm=
α-θ | 0 | -α+θ β2 | 0 |
0 | - (-1+ζ2)(θ+ϕ) ζ2 | α(θ+ϕ) ρ-β2ρ | - λ2(ζ2(-1+μ)-μ)(θ+ϕ) ζ2ρ |
0 | (-1+β2)θρ α | 0 | (-1+β2)θλ2(-1+μ) α |
0 | -(1+ζ2)ρ | 0 | -(1+ζ2)λ2μ |
In[]:=
(*Polinomiocaracteristicodelamatrizjacobiana*)
In[]:=
(*polinomiocar=Collect[FullSimplify[CharacteristicPolynomial[jacobiana,x]],x]*)
In[]:=
(*Deducciondecadacomponentedelpolinomiocaracteristico*)
In[]:=
ecb1=b1(-Tr[jacobiana]//FullSimplify)
Out[]=
b1-α+θ+(1+ζ2)λ2μ+
(-1+ζ2)(θ+ϕ)
ζ2
In[]:=
ecb2=b2(Tr[Minors[jacobiana,2]]//FullSimplify)
Out[]=
b2θλ2+ζ2θλ2-(1+ζ2)(α-θ)λ2μ+λ2ϕ+ζ2λ2ϕ-+θ(θ+ϕ)
(-1+ζ2)(α-θ)(θ+ϕ)
ζ2
In[]:=
ecb3=b3-(Tr[Minors[jacobiana,3]]//FullSimplify)
Out[]=
b3-((α-θ)θ+(1+ζ2)(α-2θ)λ2)(θ+ϕ)
In[]:=
ecb4=b4(Det[jacobiana]//FullSimplify)
Out[]=
b4-(1+ζ2)(α-θ)θλ2(θ+ϕ)
In[]:=
ecb1234=b1*b2*b3-(b1^2)*b4-(b3^2)(((ecb1[[2]]*ecb2[[2]]*ecb3[[2]])-(ecb1[[2]]^2)*ecb4[[2]]-ecb3[[2]]^2)//FullSimplify)
Out[]=
b1b2b3--b4(θ+ϕ)-(θ+ϕ)+(1+ζ2)(α-θ)θλ2-((α-θ)θ+(1+ζ2)(α-2θ)λ2)-α+θ+(1+ζ2)λ2μ+θλ2+ζ2θλ2-(1+ζ2)(α-θ)λ2μ+λ2ϕ+ζ2λ2ϕ-+θ(θ+ϕ)
2
b3
2
b1
2
((α-θ)θ+(1+ζ2)(α-2θ)λ2)
2
-α+θ+(1+ζ2)λ2μ+
(-1+ζ2)(θ+ϕ)
ζ2
(-1+ζ2)(θ+ϕ)
ζ2
(-1+ζ2)(α-θ)(θ+ϕ)
ζ2
In[]:=
(************Anexo2:Ciclos***********)
In[]:=
condicion1=(Solve[ecb1/.{b10},μ]//FullSimplify)
Out[]=
μ
αζ2+θ-2ζ2θ+ϕ-ζ2ϕ
ζ2λ2+λ2
2
ζ2
In[]:=
condicion2=(Solve[ecb2/.{b20},μ]//FullSimplify)
Out[]=
μ
(α-αζ2+(-1+2ζ2)θ+ζ2(1+ζ2)λ2)(θ+ϕ)
ζ2(1+ζ2)(α-θ)λ2
In[]:=
condicion3=(Solve[ecb3/.{b30},μ]//FullSimplify)
Out[]=
{}
In[]:=
condicion4=(Solve[ecb4/.{b40},μ]//FullSimplify)
Out[]=
{}
In[]:=
condicion5=(Solve[ecb1234[[2]]0,μ]//FullSimplify)
Out[]=
μ,μ(ζ2+αθ((-2+6ζ2)θ+ζ2(1+ζ2)λ2)+α((-2+3ζ2)θ+ζ2(1+ζ2)λ2)ϕ+(θ-4ζ2θ+ϕ-ζ2ϕ)+θ((1-3ζ2)-2ζ2(1+ζ2)λ2ϕ+θ(ϕ-2ζ2(λ2+ζ2λ2+ϕ))))
θ(θ-ζ2θ+λ2+ζ2λ2)-(-1+ζ2)(θ+λ2+ζ2λ2)ϕ
ζ2(1+ζ2)λ2(θ+λ2+ζ2λ2)
1
ζ2(1+ζ2)λ2
2
(α-θ)
3
α
2
α
2
θ
In[]:=
Collect[condicion5[[2,1,2]],{ζ2,(1+ζ2),(α-θ),λ2}]
Out[]=
-++θ-2α++ϕ-2αθϕ+ϕ+
2ζ2
2
θ
(1+ζ2)
2
(α-θ)
αθ+αϕ-2θϕ
2
(α-θ)
2
α
2
θ
3
θ
2
α
2
θ
ζ2(1+ζ2)λ2
2
(α-θ)
-2+-4θ+6α-3-ϕ+3αθϕ-2ϕ
2
θ
3
α
2
α
2
θ
3
θ
2
α
2
θ
λ2
(1+ζ2)
2
(α-θ)
In[]:=
vc1=μHB1condicion5[[1,1,2]]
Out[]=
μHB1
θ(θ-ζ2θ+λ2+ζ2λ2)-(-1+ζ2)(θ+λ2+ζ2λ2)ϕ
ζ2(1+ζ2)λ2(θ+λ2+ζ2λ2)
In[]:=
vc2=μHB2condicion5[[2,1,2]]
Out[]=
μHB2(ζ2+αθ((-2+6ζ2)θ+ζ2(1+ζ2)λ2)+α((-2+3ζ2)θ+ζ2(1+ζ2)λ2)ϕ+(θ-4ζ2θ+ϕ-ζ2ϕ)+θ((1-3ζ2)-2ζ2(1+ζ2)λ2ϕ+θ(ϕ-2ζ2(λ2+ζ2λ2+ϕ))))
1
ζ2(1+ζ2)λ2
2
(α-θ)
3
α
2
α
2
θ
In[]:=
dy1=FullSimplify[FullSimplify[D[ecb1234[[2]],μ]]/.condicion5[[1,1]]]
Out[]=
(1+ζ2)λ2(-α(θ+λ2+ζ2λ2)+θ(θ+2(1+ζ2)λ2))(θ+ϕ)(-2αθ+θ(2θ+λ2+ζ2λ2)+(θ+λ2+ζ2λ2)ϕ)
2
α
In[]:=
FullSimplify[Solve[dy10,α]]
Out[]=
α,αθ-
θ(θ+2(1+ζ2)λ2)
θ+λ2+ζ2λ2
-(θ+λ2+ζ2λ2)(θ+ϕ)
,αθ+-(θ+λ2+ζ2λ2)(θ+ϕ)
In[]:=
dy2=FullSimplify[FullSimplify[D[ecb1234[[2]],μ]]/.condicion5[[2,1]]]
Out[]=
(1+ζ2)λ2((α-θ)θ+(1+ζ2)(α-2θ)λ2)(θ+ϕ)(-2αθ+θ(2θ+λ2+ζ2λ2)+(θ+λ2+ζ2λ2)ϕ)
2
α
In[]:=
(******************************************)(**********SIMULACIONESNUMERICAS**********)(******************************************)
In[]:=
(*Simulacion1(Franciasimplificado)*)
In[]:=
simulacion1={s1,α10.004,α20.004,β11,τ10,τ20,θ10.022,θ20.022,λ10,n0,ζ10,q210.5,ρ0.549,ϕ0.491,θ0.022,β20.792,α0.004,λ22,ζ20.1};
In[]:=
vc1/.simulacion1
Out[]=
vc1
In[]:=
simulacion1=Append[simulacion1,μ%[[2]]]
Out[]=
{s1,α10.004,α20.004,β11,τ10,τ20,θ10.022,θ20.022,λ10,n0,ζ10,q210.5,ρ0.549,ϕ0.491,θ0.022,β20.792,α0.004,λ22,ζ20.1,μvc1〚2〛}
In[]:=
{ecb1,ecb2,ecb3,ecb4,ecb1234}/.simulacion1
Out[]=
{ecb1,ecb2,ecb3,ecb4,ecb1234}
In[]:=
simulacion1[[-1]]=μ2.1096;
In[]:=
equilibrios/.simulacion1
Out[]=
{{k[t]-1,l[t]0,z1[t]-1,z2[t]0.221587},{k[t]-1.26263,l[t]0,z1[t]-1,z2[t]0.221587},{k[t]-1.26263,l[t]0,z1[t]-1,z2[t]-1},{k[t]-1,l[t]5.38091,z1[t]-114.256,z2[t]-1},{k[t]-144.263,l[t]5.38091,z1[t]-114.256,z2[t]-1},{k[t]-1,l[t]7.36452,z1[t]-1,z2[t]-1},{k[t]-1.26263,l[t]7.36452,z1[t]-1,z2[t]-1},{k[t]-1,l[t]0,z1[t]-1,z2[t]-1},{k[t]-1,l[t]0.934426,z1[t]0.144,z2[t]0.1},{k[t]0.181818,l[t]0.934426,z1[t]0.144,z2[t]0.1},{k[t]-1,l[t]0.849889,z1[t]-1,z2[t]0.111},{k[t]-1.26263,l[t]0.849889,z1[t]-1,z2[t]0.111}}
In[]:=
solucionnum1=NDSolve[Flatten[{sistema2,k[0]0.182,l[0]0.915,z1[0]0.1,z2[0]0.104}]/.simulacion1,{k,l,z1,z2},{t,0,2000}]
Out[]=
kInterpolatingFunction
,lInterpolatingFunction
,z1InterpolatingFunction
,z2InterpolatingFunction
 |  |
|
|  |
|
|  |
|
|  |
|
In[]:=
solnk1=solucionnum1[[1,1,2]];
In[]:=
solnl1=solucionnum1[[1,2,2]];
In[]:=
solnz11=solucionnum1[[1,3,2]];
In[]:=
solnz21=solucionnum1[[1,4,2]];
In[]:=
fig3a=Plot[solnk1[t],{t,0,500},PlotLabel"Distribución sectorial del capital k[t]",PlotStyle{Black,Thickness[0.001]}];
In[]:=
fig3b=Plot[solnl1[t],{t,0,500},PlotLabel"Tasa de empleo l[t]",PlotRange{{0,500},{0.85,1}},PlotStyle{Black,Thickness[0.001]}];
In[]:=
fig3c=Plot[solnz11[t],{t,0,500},PlotLabel"Margen del sector 1 z1[t]",PlotRange{{0,500},{0.06,0.16}},PlotStyle{Black,Thickness[0.001]}];
In[]:=
fig3d=Plot[solnz21[t],{t,0,500},PlotLabel"Margen del sector 2 z2[t]",PlotStyle{Black,Thickness[0.001]}];
In[]:=
figura3=GraphicsGrid[{{fig3a,fig3b},{fig3c,fig3d}},FrameAll]
Out[]=
In[]:=
fig4a=ParametricPlot[{solnk1[t],solnl1[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"k[t]","l[t]"}]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,200]}]],Arrow[x]};
In[]:=
fig4b=ParametricPlot[{solnk1[t],solnz11[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"k[t]","z1[t]"},PlotRangeAll]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,200]}]],Arrow[x]};
In[]:=
fig4c=ParametricPlot[{solnk1[t],solnz21[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"k[t]","z2[t]"},PlotRangeAll]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,200]}]],Arrow[x]};
In[]:=
fig4d=ParametricPlot[{solnl1[t],solnz11[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"l[t]","z1[t]"},PlotRangeAll]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,200]}]],Arrow[x]};
In[]:=
fig4e=ParametricPlot[{solnl1[t],solnz21[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"l[t]","z2[t]"},PlotRangeAll]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,200]}]],Arrow[x]};
In[]:=
fig4f=ParametricPlot[{solnz11[t],solnz21[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"z1[t]","z2[t]"},PlotRangeAll]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,200]}]],Arrow[x]};
In[]:=
figura4=GraphicsGrid[{{fig4a,fig4b},{fig4c,fig4d},{fig4e,fig4f}},FrameAll]
Out[]=
In[]:=
fig5a=ParametricPlot3D[{solnk1[t],solnz11[t],solnz21[t]},{t,0,1000},AxesLabel{"k","z1","z2"},PlotRangeAll,PlotStyle{Black,Thickness[0.001]}]
Out[]=
In[]:=
fig5b=ParametricPlot3D[{solnl1[t],solnz11[t],solnz21[t]},{t,0,1000},PlotRangeAll,PlotStyle{Black,Thickness[0.001]},AxesLabel{"l","z1","z2"},AspectRatio1]
Out[]=
In[]:=
fig5c=ParametricPlot3D[{solnk1[t],solnl1[t],solnz11[t]},{t,0,1000},PlotRangeAll,PlotStyle{Black,Thickness[0.001]},AxesLabel{"k","l","z1"},AspectRatio1]
Out[]=
In[]:=
fig5d=ParametricPlot3D[{solnk1[t],solnl1[t],solnz21[t]},{t,0,1000},PlotRangeAll,PlotStyle{Black,Thickness[0.001]},AxesLabel{"k","l","z2"},AspectRatio1/2.5]
Out[]=
In[]:=
simulacion1u1u2=(FullSimplify[Solve[{ec[20],ec[21],ec[22],ec[43]},{u1[t],u2[t],Ω[t],q1[t]}]]/.{s1,α10.004,α20.004,β11,τ10,τ20,θ10.022,θ20.022,λ10,n0,ζ10,q210.5,ρ0.549,ϕ0.491,θ0.022,β20.792,α0.004,λ22,ζ20.1,μ2.1096})
Out[]=
u1[t],u2[t],Ω[t]0.208,q1[t]2.q2[t]
(0.004+0.004k[t])(1+z1[t])z2[t]
k[t](0.+0.208z2[t])
0.5(0.004+0.004k[t])(1+z1[t])
0.+0.208z2[t]
In[]:=
fig6a=Legended[Plot[{simulacion1u1u2[[1,1,2]]/.solucionnum1,simulacion1u1u2[[1,2,2]]/.solucionnum1},{t,0,500},PlotStyle{{Black,Thickness[0.001]},{Gray,Thickness[0.001]}},AxesLabel{"t","u1[t],u2[t]"}],LineLegend[{Black,Gray},{"u1[t]","u2[t]"}]]
Out[]=
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In[]:=
fig6b=ParametricPlot[{simulacion1u1u2[[1,1,2]],simulacion1u1u2[[1,2,2]]}/.solucionnum1,{t,0,500},PlotStyle{Black,Thickness[0.001]},AxesLabel{"u1[t]","u2[t]"}]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.03,50]}]],Arrow[x]}
Out[]=
In[]:=
fig6c=ParametricPlot[{simulacion1u1u2[[1,1,2]],simulacion1u1u2[[1,2,2]]}/.solucionnum1,{t,400,500},PlotStyle{Black,Thickness[0.001]},AxesLabel{"u1[t]","u2[t]"},PlotRangeFull,AspectRatio1/2]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.03,200]}]],Arrow[x]}
Out[]=
In[]:=
data=Table[Flatten[({solnk1[t],solnl1[t],solnz11[t],solnz21[t],simulacion1u1u2[[1,1,2]],simulacion1u1u2[[1,2,2]]}/.solucionnum1)/.{tx}],{x,400,500}];
In[]:=
(*Simulacion2(Franciaampliado)*)
In[]:=
sistema3=Solve[FullSimplify[ec[48]/.{s1,α10.004,α20.004,β11,β20.792,τ10.05,τ20,θ10.022,θ20.022,λ10.3,λ22,n0.008,ζ10.1,ζ20.1,ρ0.549,ϕ0.491,μ1.6885,q210.5}],{k'[t],l'[t],z1'[t],z2'[t]}][[1]]/.{RuleEqual};
In[]:=
Solve[sistema3/.{k'[t]0,l'[t]0,z1'[t]0,z2'[t]0},{k[t],l[t],z1[t],z2[t]}]
Out[]=
{{k[t]-1.26263,l[t]0.,z1[t]-1.,z2[t]0.25191},{k[t]1.53835,l[t]1.12776×,z1[t]1.11273,z2[t]0.25191},{k[t]9.1565,l[t]0.,z1[t]6.859,z2[t]-1.},{k[t]-2.52202×,l[t]1.75523×,z1[t]-1.90232×,z2[t]-2.85348×},{k[t]-9.52652,l[t]7.70073,z1[t]-7.23333,z2[t]-1.},{k[t]-1.26263,l[t]7.70073,z1[t]-1.,z2[t]-1.},{k[t]-1.26263,l[t]1.8571,z1[t]-1.,z2[t]-0.05},{k[t]-1.,l[t]1.76659,z1[t]-0.801905,z2[t]-0.0352857},{k[t]-0.688131,l[t]1.54954,z1[t]-0.566667,z2[t]-2.95478×},{k[t]-1.26263,l[t]1.54954,z1[t]-1.,z2[t]4.36449×},{k[t]0.241506,l[t]0.902552,z1[t]0.134545,z2[t]0.105182}}
-16
10
13
10
13
10
13
10
12
10
-13
10
-14
10
In[]:=
solucionnum2=NDSolve[Flatten[{sistema3,k[0]0.12,l[0]0.9,z1[0]0.1,z2[0]0.1}],{k,l,z1,z2},{t,0,2000}]
Out[]=
kInterpolatingFunction
,lInterpolatingFunction
,z1InterpolatingFunction
,z2InterpolatingFunction
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In[]:=
solnk2=solucionnum2[[1,1,2]];
In[]:=
solnl2=solucionnum2[[1,2,2]];
In[]:=
solnz12=solucionnum2[[1,3,2]];
In[]:=
solnz22=solucionnum2[[1,4,2]];
In[]:=
fig7a=Plot[solnk2[t],{t,0,500},PlotLabel"Distribución sectorial del capital k[t]",PlotStyle{Black,Thickness[0.001]}];
In[]:=
fig7b=Plot[solnl2[t],{t,0,500},PlotLabel"Tasa de empleo l[t]",PlotRangeAll,PlotStyle{Black,Thickness[0.001]}];
In[]:=
fig7c=Plot[solnz12[t],{t,0,500},PlotLabel"Margen del sector 1 z1[t]",PlotRangeAll,PlotStyle{Black,Thickness[0.001]}];
In[]:=
fig7d=Plot[solnz22[t],{t,0,500},PlotLabel"Margen del sector 2 z2[t]",PlotStyle{Black,Thickness[0.001]}];
In[]:=
figura7=GraphicsGrid[{{fig7a,fig7b},{fig7c,fig7d}},FrameAll]
Out[]=
In[]:=
fig8a=ParametricPlot[{solnk2[t],solnl2[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"k[t]","l[t]"},PlotRangeFull]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,100]}]],Arrow[x]};
In[]:=
fig8b=ParametricPlot[{solnk2[t],solnz12[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"k[t]","z1[t]"},PlotRangeFull]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,100]}]],Arrow[x]};
In[]:=
fig8c=ParametricPlot[{solnk2[t],solnz22[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"k[t]","z2[t]"},PlotRangeFull]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,100]}]],Arrow[x]};
In[]:=
fig8d=ParametricPlot[{solnl2[t],solnz12[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"l[t]","z1[t]"},PlotRangeFull]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,100]}]],Arrow[x]};
In[]:=
fig8e=ParametricPlot[{solnl2[t],solnz22[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"l[t]","z2[t]"},PlotRangeFull]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,100]}]],Arrow[x]};
In[]:=
fig8f=ParametricPlot[{solnz12[t],solnz22[t]},{t,0,500},PlotStyle{Black,Thickness[0.001]},AspectRatio1,AxesLabel{"z1[t]","z2[t]"},PlotRangeFull]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.04,100]}]],Arrow[x]};
In[]:=
figura8=GraphicsGrid[{{fig8a,fig8b},{fig8c,fig8d},{fig8e,fig8f}},FrameAll]
Out[]=
In[]:=
fig9a=ParametricPlot3D[{solnk2[t],solnz12[t],solnz22[t]},{t,0,1000},AxesLabel{"k","z1","z2"},PlotRangeFull,PlotStyle{Black,Thickness[0.001]}]
Out[]=
In[]:=
fig9b=ParametricPlot3D[{solnl2[t],solnz12[t],solnz22[t]},{t,0,1000},PlotRangeFull,PlotStyle{Black,Thickness[0.001]},AxesLabel{"l","z1","z2"},AspectRatio1]
Out[]=
In[]:=
fig9c=ParametricPlot3D[{solnk2[t],solnl2[t],solnz12[t]},{t,0,1000},PlotRangeFull,PlotStyle{Black,Thickness[0.001]},AxesLabel{"k","l","z1"},AspectRatio1]
Out[]=
In[]:=
fig9d=ParametricPlot3D[{solnk2[t],solnl2[t],solnz22[t]},{t,0,1000},PlotRangeFull,PlotStyle{Black,Thickness[0.001]},AxesLabel{"k","l","z2"},AspectRatio1/2.5]
Out[]=
In[]:=
simulacion2u1u2=(FullSimplify[Solve[{ec[20],ec[21],ec[22],ec[43]},{u1[t],u2[t],Ω[t],q1[t]}]]/.{s1,α10.004,α20.004,β11,β20.792,τ10.05,τ20,θ10.022,θ20.022,λ10.3,λ22,n0.008,ζ10.1,ζ20.1,ρ0.549,ϕ0.491,μ1.6885,q210.5})
Out[]=
u1[t],u2[t],Ω[t]0.158-0.05z1[t],q1[t]2.q2[t]
(0.004+0.004k[t])(1+z1[t])z2[t]
k[t](0.+(0.158-0.05z1[t])z2[t])
0.5(0.004+0.004k[t])(1+z1[t])
0.+(0.158-0.05z1[t])z2[t]
In[]:=
fig10a=Legended[Plot[{simulacion2u1u2[[1,1,2]]/.solucionnum2,simulacion2u1u2[[1,2,2]]/.solucionnum2},{t,0,500},PlotStyle{{Black,Thickness[0.001]},{Gray,Thickness[0.001]}},AxesLabel{"t","u1[t],u2[t]"},PlotRangeFull],LineLegend[{Black,Gray},{"u1[t]","u2[t]"}]]
Out[]=
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In[]:=
fig10b=ParametricPlot[{simulacion2u1u2[[1,1,2]],simulacion2u1u2[[1,2,2]]}/.solucionnum2,{t,0,500},PlotStyle{Black,Thickness[0.001]},AxesLabel{"u1[t]","u2[t]"},PlotRangeFull]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.03,50]}]],Arrow[x]}
Out[]=
In[]:=
fig10c=ParametricPlot[{simulacion2u1u2[[1,1,2]],simulacion2u1u2[[1,2,2]]}/.solucionnum2,{t,400,500},PlotStyle{Black,Thickness[0.001]},AxesLabel{"u1[t]","u2[t]"},PlotRangeFull,AspectRatio1/2]/.Line[x_]{Arrowheads[Flatten[{0,ConstantArray[0.03,50]}]],Arrow[x]}
Out[]=
(**Sobreproduccionyciclos**)
In[]:=
deduccion18=FullSimplify[Solve[{ec[20],ec[21],ec[22]}/.simplificacion,{u1[t],u2[t],Ω[t]}]/.{equilibriodin/.{EqualRule}}]
Out[]=
u1[t]-,u2[t],Ω[t]1-β2
(-1+β2)(α-θ)θ
α+(-1+β2)θ
(-1+β2)(α-θ)θq2[t]
αβ2ζ2q1[t]
In[]:=
{ec[59],ec[60]}=deduccion18[[1,1,1;;2]]/.{RuleEqual}
Out[]=
u1[t]-,u2[t]
(-1+β2)(α-θ)θ
α+(-1+β2)θ
(-1+β2)(α-θ)θq2[t]
αβ2ζ2q1[t]
Cite this as: John Cajas Guijarro, "A two-sector model of endogenous distributive cycles and overproduction" from the Notebook Archive (2024), https://notebookarchive.org/2024-02-6yy60je
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