1

““A Disequilibrium Growth Cycle ModelA Disequilibrium Growth Cycle Modelwith Differential Savingswith Differential Savings””

Serena SordiDEPFID-Department of Political Economy,

Finance and Development

University of Siena

E-Mail: sordi@unisi.it

Workshop MDEF (Modelli Dinamici in Economia e Finanza)Workshop MDEF (Modelli Dinamici in Economia e Finanza)Urbino, 25-27 September 2008 Urbino, 25-27 September 2008

*** Downlodable at http://www.depfid.unisi.it/WorkingPapers/ *** Downlodable at http://www.depfid.unisi.it/WorkingPapers/

******

2

AbstractAbstract:: 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.

3

Velupillai (1979, 1982a, 1982b, 1983 and 2006)

Sordi (2001, 2003)

Relations with the existing literature: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), , ,

• ‘classical’ assumption about savings behaviour• equilibrium in the goods market

4

Notation and basic assumptions:

0

0

ˆfor any variable , / , / ,, output, expected output

, employment/ , 0, labour productivity

, 0, labour force, natural rate of growth

, rate of growth of output,

e

t

t

n

x x dx dt x x xqqlq l a a en n eggw

real wage/ , share of wages

/ , employment rate, capital stock

, capital stock held by capitalists /1 , capital stock held by workers

, desired capital stock/ , capital

c w

c c

wd

u wl qv l nk k kk k k kk kk

k q

-output ratio, capitalists’ profits, workers’ profits

/ 1 / 1 / , rate of profit, , workers’ propensity to save and workers’ savings respectively, , capitalists’ propensity t

c

w

c w

w w

c c

PPP P Pr P k u q k us Ss S

o save and capitalists’ savings respectively0 1w c

c w

s sS S S

Sordi (2001)

Sordi (2003)

5

The ModelThe ModelFour building blocks:

1. Real wages dynamics:

1

ˆ ˆ ˆ,

0, 0 , 0 0, lim ,

0v

w f v v h v v

h v h v v h h v

2. Savings behaviour:

1

1c c c c

w w w w w

S s P s u q

S s wl P s q s u q

Cugno & Montrucchio (1982)Sportelli (1995)Sordi (2001, 2003)

6

3. Investment equations:

, 0dk k k where

, 0d ek q q q

4. Goods market adjustment mechanism:

, 0nq g q k S

1

ˆ ˆ 1n w c wq g q s s s u

ˆ 11 1

n wc w

g sq s s u

1

1w w w wk S s q s u q

c wk k k

7

Dynamical system of the model:Dynamical system of the model:

We obtain the following complete 4D-dynamical system in the four endogenousvariables v, u, ε and σ:

11

ˆ , , ,

1 11

1

1

11

w n c w

n w c w

w w

n w c w

v s g s s u v

u h v v u G v u u

g s s s u

s s u

g s s s u

8

Local stability analysis:Local stability analysis:

The model has a unique positive equilibrium point suchthat:

** * * *, , ,E v u v

1* * * *, 1 , , c n wn

c c w n

vs g

su

sgh

s s g

where we must have:

10 1 0 1 1 0 1 0c n wn

c c w n

s g sghs s s g

0 1w n cs g s

9

At E*:

• the output and the capital stock grow at a rate equal to the natural rate:

• the Cambridge equation is satisfied

• the real wage grows at the same rate as labour productivity:

• the ‘Pasinetti case’ holds

** ˆˆ1 1

n nn

g gq k g

**

*

1 n

c

gurs

* *w h v

10

Moreover:

*

12 13 14

21 22 23 24

32 33 34

44

0

|00 0 0

E

j j jj j j j

j j jj

J

where:

* * *

* *

* *

* * *

*

12 13 14

21 2

23 22

2

4

1 1 1

1

1

E E

c n w c w n

n c

c n n w c n

c n

c w n c

E

E E

E

n

c E

s g s v s s g v vg s

h v s g g s s gs gs s g

v v vj j ju

u uj j

s gv uu u

sj j

* *

*

*

*

2 232 33

34

44

2

1 1

1 1

1

1

c n

c

w c n w c n c w n c n w

c w n c

w c n

c

E

n

E

E

wE n

s gus

s s g s s g s s g s g ss s g s

s s gs

j ju

j

j

s g

g

11

Under the following two assumptions − which can be shown to hold for awide 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:

0

1w c

nc w

s sg

s s

the signs of all non-zero elements of are uniquely determined

12 13 14

21 22 23 24

32 33 34

44

0 0 00 0 0 00 0 00

j j jj j j jj j jj

*|E

J

12

Simple calculation shows that the characteristic equation is given by:

3 244

22 33

22 33 23 32 12 21

12 21 33 13 21 32

0

0

00

j A B C

A j j

B j j j j j jC j j j j j j

1 44

3 22 3 4

0

0 , and

j

A B C

A, B, and C are all positive

2 3 4, and are all negative if real or have a negative real part

if complex iffiff 0AB C

13

Selecting the propensity to save out of wages as the bifurcation parameterthe bifurcation parameter:

* *2 2 2

2 * * *

2 * * *

1 11

1

1

w w w

w c n n w

n

n c n w n

n c n w

A s B s C s

s s g u g s ug

g s g s u g v h v

g s g s v h v u

such that: 0w w wA s B s C s

* *

* * 3

* *

1

1 0

1 0

w w c n n w

n n

w c n

F s s s g u g s u

g v h v g

F s s g u u

14

Proposition 1: Under Assumptions 1 and 2, if the propensity to saveout of wages is sufficiently low and such that

the positive equilibrium E* of the dynamical system (15)-(18) is locallyasymptotically stable.

0wF s

Proposition 2: Under Assumptions 1 and 2, there exists a value of thepropensity to save out of wages:

at which the dynamical system (15)-(18) undergoes an Hopf bifurcation.

* * * 3

* * * *

10

1n n n

wHn c n

g u g v h v gs

g v h v s g u u

Two propositions:

15

Some numerical simulations of the model:Some numerical simulations of the model:

ˆ ˆ ˆw h v v v v

* *

* *

1 0.8 2.570.9421 0.9171

0.0221 0.00370.5389 2.57

0.92 1 0.1

cs v u

* * * 3

* * * *

10.0349

1n n n

wH

n c n

g u g v h v gs

g v h v s g u u

16

1 20.32 0.0354w wH ws s s

17

1 0.032w wHs s

18

2 0.0354wH ws s

19

Summing-up and conclusions: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