Initialization

> restart;

> with( DEtools ):

> with( plots ):

> with( linalg ):

> with( PDEtools ):

> cobweb:=proc(f,p0,n,minp,maxp,legend1,legend2)
local fk,list1,list2,list3,list4,web,lines;
fk:=(p,k)->simplify((f@@k)(p));
list1:=transpose(array([[seq(fk(p0,k),k=0..n)],
[seq(fk(p0,k),k=0..n) ] ] ) ):
list2:=convert(convert(list1,vector),list):
list3:=convert(transpose(array([ list2[1..nops(list2)-1],
list2[2..nops(list2)] ]) ),listlist):
list4:=[list3[2..nops(list3)]]:
web:=plot(list4):
lines:=plot({f(p),p},p=minp..maxp,colour=blue):
display(web,lines);
end:

>

Parameters and Descriptions

Skott adopts two hybrids variables in his analysis of economic fluctuations:

- F represents the degree of financial fragility

- T denotes the degree of financial tranquility

According to Nasica(1992, p. 65):

"The financial system is fragile if small disruptions - such as an unanticipated drop in income or an unforeseen rise in the interest rate - make it difficult, if not impossible, for a significan percentage of agents to meet their contractual obligations ... tranquility - the capacity of agents to meet their commitments - depends on realized cash flows and on the amounts of contractual repaymets. It is thus the result of the interaction between real and financial factors"

Equations

Saving function:

(2.4)

where stands for the actual output-capital ratio and

s denotes the propensity to consume

> s:=0.4:

The linear version of the investment function is:

(2.5a)

where, the rate of capacity utilization, tranquility and fragility have a positive effect on accumulation: a > 0, b > 0 and c > 0.

> a:=0.02:

> b:=0.1:

> c:=0.1:

This is an autonomous component of investment, sort of animal spirits of entrepreneurs

> d:=0.01:

The financial tranquility is determined as:

(2.2a)

so that, T depends positively on the output-capital ratio as well as on financial fragility

> A:=0.2:

The equilibrium value of for which I/K = S/K is given by:

(2.6)

The relation that describes the endogenous evolution of fragility and tranquility is:

(2.1a)

and combining the equations above we have that

(2.7)

where,

> alpha1:=1+(A*b)/(s-a-c*A);

> beta1:=1+(A*c)/(s-a-c*A);

> gamma1:=(A*d)/(s-a-c*A);

The non-linearity of function f may be specified by way of a quadratic function:

(2.8)

where,

> B:=1.2:

This parameter determines the convegence (or not) to equilibrium.

> C:=1:

here, De = D

> De:=0.3:

Thus, from (2.7) we have

(2.9)

where,

> alpha2:=alpha1+beta1*B;

> beta2:=-beta1*C;

> gamma2:=gamma1+beta1*De;

The Dynamics

Time of Analysis

> tempo:=50:

Initial Condition

> F_0:=0.1:

Stable Equilibrium

C:=1:

> beta2:=-beta1*C;

> f:=proc(p)

> alpha2*p+beta2*p^2+gamma2

> end proc:

> cobweb(f,F_0,tempo,0,2);

It is not possible to use the Rsolve to find the dynamics equation since (2.9) is a second-order equation.

Thus, we will calculate the value for each period

> x1:=array(0..tempo):
x1[0]:=F_0:
for i from 0 to tempo-1 do
x1[i+1]:=alpha2*x1[i]+beta2*(x1[i])^2+gamma2:
end do:

> x1points:=[seq([t,x1[t]],t=0..tempo)]:

> plot(x1points,labels=["Time","Fragility Level"],labeldirections=[HORIZONTAL,VERTICAL]);

Perpetual Fluctuation

C:=2:

> beta2:=-beta1*C;

> f:=proc(p)

> alpha2*p+beta2*p^2+gamma2

> end proc:

> cobweb(f,F_0,tempo,0,1);

> x1:=array(0..tempo):
x1[0]:=F_0:
for i from 0 to tempo-1 do
x1[i+1]:=alpha2*x1[i]+beta2*(x1[i])^2+gamma2:
end do:

> x1points:=[seq([t,x1[t]],t=0..tempo)]:

> plot(x1points,labels=["Time","Fragility Level"],labeldirections=[HORIZONTAL,VERTICAL]);

Chaotic Dynamics

C:=3.9:

> beta2:=-beta1*C;

> f:=proc(p)

> alpha2*p+beta2*p^2+gamma2

> end proc:

> cobweb(f,F_0,tempo,0.09,0.68);

> x1:=array(0..tempo):
x1[0]:=F_0:
for i from 0 to tempo-1 do
x1[i+1]:=alpha2*x1[i]+beta2*(x1[i])^2+gamma2:
end do:

> x1points:=[seq([t,x1[t]],t=0..tempo)]:

> plot(x1points,labels=["Time","Fragility Level"],labeldirections=[HORIZONTAL,VERTICAL]);

The Bifurcation Diagram

> unassign('C','j');

> beta2:=-beta1*C;

Redefining the Equation, so that C varies.

> f:=proc(p)

> alpha2*p+beta2*p^2+gamma2

> end proc:

Scaling the changes in C:

> C:=0.001*j:

> points9:=evalf(seq(seq([C,(f@@k)((f@@8)(0.5))],k=0..8),j=1000..4000),4):

> pointplot({points9},symbol=point, labels=["Parameter C","Equilibrium Value"],labeldirections=[HORIZONTAL,VERTICAL]);

SKOTT, P. (1992) 'On the modelling of systemc financial fragility' in A. K. Dutt (ed.) New Directions in Analytical Political Economy . Aldershot: Edward Elgar pp. 49-76

NASICA, E. (2000) Finance, investment and economic fluctuations: an analysis in the tradition of Hyman P. Minsky. Cheltenham: Edward Elgar pp. 51-75