“A Disequilibrium Growth Cycle Model
with Differential Savings”***
Serena Sordi
DEPFID-Department of Political Economy,
Finance and Development
University of Siena
E-Mail: [email protected]
Workshop MDEF (Modelli Dinamici in Economia e Finanza)
Urbino, 25-27 September 2008
*** Downlodable at http://www.depfid.unisi.it/WorkingPapers/
1
Abstract: This paper extends Goodwin’s growth cycle model by
assuming both differential savings propensities and disequilibrium
in the goods market. It is shown that both modifications entail an
increase in the dimensionality of the dynamical system of the
model. By applying the existence part of the Hopf bifurcation
theorem, the possibility of persistent and bounded cyclical paths
for the resulting 4-dimensional dynamical system is then
established. With the help of numerical simulation some evidence
is finally given that the limit cycle emerging from the Hopf
bifurcation is stable.
2
Relations with the existing literature:
Goodwin (1967)
Atkinson (1969, RES), Izzo (1971), Desai (1973,
JET), Gandolfo (1973), Maresi & Ricci (1976), Vercelli (1977, EN), , , ,
, , , , , , , , , , , , , , , , , , , , , ,
, , , , , , , Desai (2006, JEDC), Velupillai (2006, JMacro), , 
, 
Velupillai (1979, 1982a, 1982b, 1983 and 2006)
• ‘classical’ assumption about savings behaviour
• equilibrium in the goods market
Sordi (2001, 2003)
3
Notation and basic assumptions:
for any variable x, x  dx / dt , xˆ  x / x,
q, output
q e , expected output
l , employment
q / l  a  a0e t ,   0, labour productivity
Sordi (2001)
n  n0 e  t ,   0, labour force
g n     , natural rate of growth
g , rate of growth of output
w, real wage
Sordi (2003)
u  wl / q, share of wages
v  l / n, employment rate
k  kc  k w , capital stock
kc   k , capital stock held by capitalists    kc / k
kw  1    k , capital stock held by workers
k d , desired capital stock
  k / q, capital-output ratio
Pc , capitalists’ profits
Pw , workers’ profits
P  Pc  Pw
r  P / k  1  u  q / k  1  u  /  , rate of profit
sw , S w , workers’ propensity to save and workers’ savings respectively
sc , Sc , capitalists’ propensity to save and capitalists’ savings respectively
0  sw  sc  1
S  Sc  S w
4
The Model
Four building blocks:
1. Real wages dynamics:
wˆ  f  v, vˆ   h  v    vˆ
h  v   0, h  v   0 v, h  0   0, lim h  v   ,
v1
 0
2. Savings behaviour:
Cugno & Montrucchio (1982)
Sportelli (1995)
Sordi (2001, 2003)
Sc  sc Pc  sc 1  u  q
S w  sw  wl  Pw   sw q  sw 1  u  q
5
3. Investment equations:
where
k   k d  k ,   0
 1
k d   qe    q   q  ,   0
 1
kw  Sw  sw q  sw 1  u  q
kc  k  k w
4. Goods market adjustment mechanism:


q  gn q   k  S ,   0
qˆ  gn      qˆ    sw   sc  sw  1  u   
qˆ 
g n     s w 
1  


   sc  sw  1  u   
1  
6
Dynamical system of the model:
We obtain the following complete 4D-dynamical system in the four endogenous
variables v, u, ε and σ:
v

  sw   g n     sc  sw  1  u    v

1  
u   h  v    vˆ    u  G  v, u ,  ,     u
   1      1    g     s    s  s 1  u  

 c w    

n
w


 1   
 sw  sw 1  u   







   

      
 g n     sw      sc  sw  1  u   
1  
7
Local stability analysis:
*
The model has a unique positive equilibrium point E *   v , u * ,  * , v*  such
that:
sc  g n  sw  *
g n
*
v  h   , u  1 
, 
,  
sc
 sc  sw  g n
*
1
*
where we must have:
0h
1
   1
g
0  1 n  1
sc
0
sc  g n  sw 
 sc  sw  g n
1
0
0  sw   g n  sc  1
8
At E*:
• the output and the capital stock grow at a rate equal to the natural rate:

 qˆ   kˆ 
*
*
gn
 g n

 gn
1   1  
• the Cambridge equation is satisfied
r 
*
1  u*
*
gn

sc
• the real wage grows at the same rate as labour productivity:
 ŵ
*
 h  v*   
• the ‘Pasinetti case’ holds
9
Moreover:
0
j
J |E*   21
0

0
j12
j13
j22
j23
j32
j33
0
0
j14 
j24 

j34 

j44 
where:
v
j12 
u
 sc  g n  sw  v*
v

j13 

1    g n
E*
u
j21 
v
E*
u

E*
j23 
j32 
j34 
j44 

u



E*


*
j22 
sw sc  g n  sw  sc  g n 
1    sc  sw  g n2 2
sw  sc   g n 

1    sc  sw  g n 2
  1  g n 
E
E*
 v*

1  
  g n  sw  sc  g n 
u

sc
u E*
1    g n
  sc  sw  g n  sc  g n 
  sc  g n 
u
 u *

j





24
 E*
1  
1    sc2
1    sc

E*
h  v*   sc  g n 
E*
  sc  sw  g n v*
v

j14 

1    sc
j33 



E*
 sc  sw  g n   sc  g n  sw 
1    sc
10
Under the following two assumptions − which can be shown to hold for a
wide range of plausible parameter values,
Assumption 1: The desired capital-output ratio is high enough and such that:
  1/ 
Assumption 2: The natural growth rate is high enough to satisfy:
 sw sc
gn  
0
sc 1     sw
the signs of all non-zero elements of J | * are uniquely determined
E
j12  0
j13  0
j14  0
j21  0
j22  0
j23  0
j32  0
j33  0
j34  0
j24  0
j44  0
11
Simple calculation shows that the characteristic equation is given by:
   j44   3  A 2  B  C   0
A    j22  j33   0
B  j22 j33  j23 j32  j12 j21  0
C  j12 j21 j33  j13 j21 j32  0
1  j44  0
 3  A 2  B  C  0  2 , 3 and 4
A, B, and C are all positive
2 , 3 and 4are all negative if real or have a negative real part
if complex iff
AB  C  0
12
Selecting the propensity to save out of wages as the bifurcation parameter:
A  sw  B  sw   C  s w 
1


  sw  sc  g n  1  u *     g n  sw  u *
1    g 
 1    g n2  sc  g n  sw  u *  g n  v*h  v*  
2
2
n
2
 1    g n2 sc  g n  sw  v*h  v*  u *
such that:

A  sw  B  s w   C  s w   0

F  sw     sw  sc  g n  1  u *     g n  sw  u * 

  g n  v*h  v*    1    g n3  0

F   sw     sc  g n  1  u *    u *   0

13
Two propositions:
Proposition 1: Under Assumptions 1 and 2, if the propensity to save
out of wages is sufficiently low and such that
F  sw   0
the positive equilibrium E* of the dynamical system (15)-(18) is locally
asymptotically stable.
Proposition 2: Under Assumptions 1 and 2, there exists a value of the
propensity to save out of wages:
swH 
g n u *  g n  v*h  v*    1    g n3
 g n  v h  v    sc  g n  1  u    u 



*
*
*
*
0
at which the dynamical system (15)-(18) undergoes an Hopf bifurcation.
14
Some numerical simulations of the model:
wˆ  h  v    vˆ     v   vˆ
  1 sc  0.8   2.57 
*
*
 v  0.9421 u  0.9171
  0.0221   0.0037    *
*


0.5389

 2.57


  0.92   1   0.1 
swH 
g n u *  g n  v*h  v*    1    g n3
 g n  v h  v    sc  g n  1  u    u 



*
*
*
*
 0.0349
15
sw1  0.32  swH  sw2  0.0354
16
sw1  0.032  swH
17
swH  sw2  0.0354
18
Summing-up and conclusions:
• We have investigated a generalisation of the growth cycle model and
arrived at a version of the model with a four-dimensional dynamical system
in the employment rate, the share of wages, the proportion of capital
held by capitalists and the capital-output ratio
•
•
We have utilized the HBT to prove the existence of persistent and
self-sustained growth cycles and given numerical evidence that the
emerging limit cycle is stable
Much remains to be done:
 what we have done in this paper provides only a basic
understanding of the dynamics of the model
 we have negletected many other possible extensions of the model,
well established in the literature on the topic
19