PAOLA ANTONELLOLUISS - GUIDO CARLI §Istituto di Studi EconomiciViale Pola, 12I-00198 ROME Tel.: +39 +6 85225739Fax : +39 +6 8845252E-Mail pantonel@lgc-net.luiss.it

Simultaneous Adjustment of Quantities and Prices:an Example of Hamiltonian Dynamics.

(Provisional)

Index:1. Introduction.2. Goodwin's "Walrasian" model as an optimal control model.3. Symplectic changes of coordinates.4. Consumer demand functions.5. Further analysis of the dynamics of a two degree of freedom Hamiltonian.6. Conclusions.

§ This paper was written largely in 1993-94, during a five-monthstay at the Faculty of Economics of the University of California,Santa Cruz. I wish to thank Professor Nirvikar Singh and all theFaculty for their friendly hospitality, and Professor Ralph Abrahamfor his invaluable help. Responsibility for errors, of course,remains mine.

Abstract

In a well-know essay, first published in 1953, ("Static and DynamicLinear General Equilibrium Models"), Richard Goodwin analyses thedynamic adjustment of quantities and prices to long periodequilibrium, in a set of n "Walrasian" markets. He treats the crossadjustment of prices and quantities as a linear Hamiltonian vectorfield. In more recent works Goodwin has introduced non-linearperturbations in his multisectorial adjustment model. He assumesthat real consumption depends non-linearly on relative prices. Thispaper shows that: 1) Goodwin's behavioural hypotheses arecompatible with the assumption that agents maximize; 2) if thedynamic process is Hamiltonian, symplectic coordinates changes areessential tools of analysis; 3) if real wage is rigid and returnsto scale are not constant, the Hamiltonian model can generatechaotic transients or, in extreme cases, pure chaotic motions.

2

1. Introduction

In a pioneering work of 1953 ("Static and Dynamic Linear GeneralEquilibrium Models") Richard Goodwin presents a leading analysis ofthe dynamic adjustments of quantities and prices to their longperiod equilibrium values. He takes two different kinds ofadjustment processes into consideration: the uncoupled adjustmentof prices and quantities, i.e. a gradient vector field in whichprices react to excess profits per unit of output, and quantitiesreact to excess demands per unit of output, and the crossedadjustment of prices and quantities, i.e. a Hamiltonian vectorfield in which the time derivative of prices depends on excessdemands per unit of output, and the time derivative of grossquantities depends on excess profits per unit of output. In bothcases the properties of the dynamic processes are analysed bytreating the set of intersectoral demand coefficients as a linearoperator on its generalized eigenspace.

In more recent works (for example "Swinging along the Autostrada:Cyclical Fluctuations along the Von Neumann Ray", (1989) or in"Chaotic Economic Dynamics", (1990)), Richard Goodwin introducessome elements of non-linearity in his basis multisectoral model; byassuming that the wage is fixed in nominal terms, he lets realconsumption and investment demand depend non-linearly on relativeprices. Moreover, in "The Dynamics of a Capitalist Economy",(1987), the linear crossed-dual adjustment process is representedby a Hamiltonian vector field in symplectic coordinates. This opensa wide range of fascinating potential developments for the analysisof adjustment processes in multisectoral systems subject to realperturbations.

This paper has two aims: firstly, it aims to investigate whetherthe behavioural hypotheses adopted by Goodwin are compatible withthe assumption that agents maximize and, if so, what is the natureof their respective objective functionals. Secondly, it aims toanalyse the dynamic behaviour of his 1987 Hamiltonian model,subject to the non-linear perturbation he suggests in "Chaotic

3

Economic Dynamics", and to point out some of the fascinatingdevelopments this approach may lead to.

In paragraph 2, Goodwin's 1953 cross-dual model is derived from anoptimal control model, the objective functional of which is givenby the aggregate excess profits function, i.e. by the sum ofsectoral excess profits. In paragraph 3, the resulting Hamiltonianvector field is transformed, by a symplectic change of coordinates,into Goodwin's 1987 linear model. This model generates simpleharmonic motions, which are illustrated by a two sector numericalexample. Hence the long period equilibrium solution is stable, butthe model is structurally unstable. In order to simplify thesubsequent analysis of the linear model subject to non-linearperturbations, a further symplectic transformation to action anglecoordinates (non-linear polar coordinates) is performed.

In paragraph 4, the following assumptions are introduced: a) thenominal wage is fixed and entirely spent for consumption; b)consumer utility functions are Cobb-Douglas; c) consumers aim tomaximize their current utility. Under these hypotheses, Goodwin's1990 consumer demand functions are easily derived. Since prices andthe real wage are flexible, the resulting perturbed Hamiltonian mayhave equilibria which are sinks. Hence, in the numerical example,the closed orbits solution disappears and the long periodequilibrium solution becomes asymptotically stable.

In the last paragraph, the analysis is restricted to a two sectormodel. Goodwin's basis hypotheses are slightly modified. It isassumed that the unperturbed two degree of freedom Hamiltonian isnon-linear and has a homoclinic orbit. It is further assumed thatthe perturbation is itself periodic. By applying Melnikov's method,it is then proved that the perturbed Hamiltonian system hastransverse homoclinic orbits and, therefore, Smale horseshoes.

axqJ(p,q)'m

t1

t0

F(t,p,q)dt'mt1

t0

p t(t)(I&A)q(t)d

0p(t)'&(I&A)q(t)

H(t,p,q,8)'80F(t,p,q)&8t(t)(I&A)q(t)'

'80p t(t)(I&A)q(t)&8t(t)(I&A)q(t)

4

(2.1)

(2.2)

(2.3)

2. Goodwin's "Walrasian" model as an optimal control model.

Basic assumptions:

a) Technology in the n-sectors economic system can be representedas a linear operator in a Euclidean n-space E. The associatedmatrix in standard coordinates is A=[a ], i=1...n, j=1...n.ij

b) The economic system as a whole pursues the aim of maximizing excess profits in the time interval t -t , t0ú, i.e.0 1

where p (t)0E and q(t)0E are the n-dimensional vectors of thet *

deviations of relative prices and gross quantities from their longperiod equilibrium values and E is the dual space of E.*

c) Relative prices are flexible and their time rates of change areincreasing linear functions of sectoral excess demands:

Equations 2.1 and 2.2 represent an example of a dynamic controlproblem where p(t) are the state variables, q(t) are the controlvariables, 2.1 is the objective functional and 2.2 are theequations of motion, i.e. the n-dimensional set of constraints onthe objective functional.

The Hamiltonian function is

0p(t)'D8tH'&(I&A)q(t)

08(t)%Dp tH'08(t)%80(I&A)q(t)'0

08(t)'&80(I&A)q(t)

DqH'8

0p t(t)(I&A)&8t(t)(I&A)'0

8t(t)'p t(t)80

8t(t)'p t(t)

5

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

where 8 , 8 (t) are conjugate or costate variables. Necessary0t

conditions for a maximum are

where D tH and D tH are vectors of first derivatives of H. Hence8 p

. . 8 (t) is a scalar multiple of p(t). On the other hand, fromequation 2.5 we have

hence

and, setting 8 =10

Necessary conditions tell us that, when the objective functional ismaximized with respect to the control variable q(t), shadow prices8(t) coincide with excess prices. In this case, H(t) is identically0 at all t, therefore equation 2.6 does not help in determining theoptimal path of q(t).

We shall therefore turn our attention to the transversalitycondition. If the planning horizon is unbounded and the right handside boundary is free, we can construct a one-parameter family ofadmissible trajectories of the state variables p(t;ß), withp(t;0)=p(t,q*(t)) where q*(t) is an optimum control; thetransversality condition then requires that

limt64

p t(t;0)D$p(t;0)'0

limt64

p t(t;0)'0

P t(t)'mt

0& (I&A)q(s)ds%k'&(I&A)m

t

0q(s)ds%k

&(I&A) limt64 m

t

0q(s)ds % (y

0&y e

0)'0

limt64 m

t

0q(s)ds ' (I&A)&1 (y

0&y e

0)

6

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

where D p(t;0) is the derivative of p(t;ß) at p(t,q*(t)). Aß

solution to 2.10 is

i.e the transversality condition is satisfied if long periodequilibrium is globally asymptotically stable. Since p(t) dependson q(t), equation (2.10) is in fact a constraint on the dynamicbehaviour of q(t). From equation 2.4 we have

Setting k=(y - y ), i.e equal to excess final demand at t=0, weo oe

have

hence the constraint on the dynamic behaviour of q(t) is

This means that long period equilibrium prices are globallyasymptotically stable if gross outputs converge asymptotically toany initial level of gross demand. Long period equilibrium pricesare therefore independent of output or, equivalently, they are notassociated with any particular level of final demand or ofemployment. On the other hand, the assumption that gross outputsare rapidly adjusted to demand and equation 2.4 necessarily implythat equation 2.10 (the transversality condition) is verified. Wemight call this assumption the "Keynesian multiplier asymptoticstability assumption".

However, this is not the case in Goodwin's "Walrasian" model. Inorder to find Goodwin's solution to the optimal control problem,let us first assume that the planning horizon is bounded, i.e. thatthe time interval over which the objective functional is to bemaximized ends at t , which is finite.1

Call M the smooth manifold generated by the intersection of s1

hypersurfaces S (t,p,q)=0, u=1...s, s$1, M the manifold generateduf

by equation t-t =0 and set M= M 1M . Then, if {p,q} ends at the1 1 f

point {t ,p(t ),q(t )}0M, and q(t ) is an optimal control, the1 1 1 1

transversality condition requires that vector {p,q} must be

S(t,p,q)'q t(t)M tM q(t)%p t(t)M M t p(t)&k'0

M 'B t 00 B

0q(t)'(I&A t)p(t)

S(t,p,q)'q t(t)q(t)%p t(t)p(t)&k''j

iq 2i(t)%j

ip 2i(t)&k'0

.'D>>̄(>&>̄)

>̄t(t

1).(t

1)'>̄t(t

1)D

>>̄(t

1) >(t

1)&>̄(t

1) '0

7

(2.15)

(2.16)

(2.17)

(2.18)

(2.19)

(2.20)

orthogonal to M (t ) or, equivalently, normal to any vector in the1 1

tangent hyperplane at {p(t ),q(t )}. We shall prove that: 1 1

1) the transversality condition is satisfied if M is the manifold1

generated by the bilinear form

where k…0 and M is a non-singular, block diagonal matrix

2) If B=(I-A) and the time behaviour of prices is given by theequations of motion 2.4, the optimal control is given by

Proof:

Suppose that B=I. Equation 2.15 then becomes

The equation of a vector . in the tangent hyperplane through >̄={p,q}0M is 1

where D >̄ is the Jacobian matrix of > at >̄. The transversality>

condition is

2q t(t)0q(t)%2p t(t)0p(t)'0

q t(t)0q(t)&q t(t)p(t)'0

q t(t)(0q(t)&p(t))'0

0q(t)'p(t)

t(t)(I&A)(I&A t)p(t)% q t(t)(I&A t)(I&A)q(t)&

(t1)'6p̄ t(t

1),q̄ t(t

1)>M tD

ex>ex̄>M6(p(t

1)&p̄(t

1)),(q

8

See appendix 2.A.1

See appendix 2.B.2

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

(2.26)

which is always satisfied by any >̄(t )0M (t ) . Hence, if B=I, an1 1 1

1

optimal trajectory which satisfies the transversality condition att is any point {p̄(t ), q̄(t )} which is a solution of 2.18.1 1 1

Differentiating 2.18 with respect to time we obtain:

If we set A=0 in the equations of motion 2.4 and substitute $p(t) in2.21, we obtain

hence

and

Now, assume that B=(I-A). S becomes

Setting ex> =(I-A )p, ex> =(I-A)q and ex>={ex> , ex> }, proposition1 2 1 2t

(1) can be proved as in the previous case . The transversality2

condition now becomes

which is verified by any {p̄,q̄}0M. Since M D (e$x>)M is the imagetex>

of D (p̄,q̄) in the space spanned by p,q

M '(I&A t) 0

0 (I&A)

q t(t)(I&A t)(I&A)0q(t)%2p t(t)(I&A)(I&A t)0p(t)'

0p t(t)(I&A)[&0q(t)%(I&A t)p(t)]'0

0q(t)'(I&A t)p(t)

limt64

>(t)

S 1(t,p,q)'p t(t)(I&A)(I&A t)p(t)%q t(t)(I&A t

9

See Hadley and Kemp (1971), theorem 4.3.2, p. 246.3

(2.27)

(2.28)

(2.29)

(2.30)

(2.31)

(2.32)

2.26 and 2.20 are the same equation in different coordinates. Thetime derivative of S is now

which, together with equation 2.4, implies

Hence, the dynamic behaviour of the optimal control is given inthis case by

which proves proposition (2).

We will now release the assumption that the planning horizon isfinite. Our model merely requires that

lies in a bounded manifold M, therefore, if lim >(t) exists, ittv4

is possible to choose >(t) so that 2.31 is orthogonal to M at theterminal point. When this is done, any necessary conditions at theterminal point of the trajectory will be satisfied .3

Consider the following smooth hypersurfaces

where k…0 is a finite real constant, and

S 2(t,p,q)'t&t1'0

M ' _u'1,2

S u

10

E and E are dual, Euclidean n-spaces and ú is the real4 *

line.

See appendix 2.C.5

(2.33)

(2.34)

where t $w, and w is a sufficiently large real number. Their1

intersection

is a (2n-1) dimensional smooth manifold in E x E x ú. Equation4 *

2.32 states that both lim p(t) and lim q(t) must lie in M ,tv4 tv4 1

i.e. in the manifold generated by S . If k…0 is finite, M is11

bounded . From equation 2.4 we know that, since lim q(t) exists,5tv4

lim &p(t) also exists. The intersection of S and S is the set oftv41 2

points {p(t ),q(t )} which satisfy the bilinear map 2.32 at t=t ,1 1 1

hence {p(t ), q(t )}0M, and {p(t ),q(t )} is orthogonal to M (t ).1 1 1 1 1 1

This holds for any t $ w, with sufficiently large w, hence lim1 tv4

{p(t),q(t)} is orthogonal to M at the terminal point and thedynamic behaviour of the optimal control is again given by equation2.30.

We may then conclude that Goodwin's "Walrasian" prices andquantities adjustment equations can be regarded as the solutions toan optimal control problem with an unbounded planning horizon,provided that the terminal point lies in a bounded smooth manifoldgenerated by equation 2.32. This equation states that the the sumof the squared modules of the two n-vectors p(t)(I-A) and (I-A)q(t), which respectively represent excess profits and excessfinal demands per unit of output, must remain constant during thewhole adjustment process; i.e. that all solution curves in the 2n-Euclidean space of excess profits and excess final demands per unitof output lie on a hypersphere of radius %k, centred at the origin.In physics, this condition means that the total energy of the

(p ,t&p (t(I&A)(q ,&q ()

8(t)'p(t)'p ,t(t)%p ((t)

q ,'q (

(p ,t&p (t(I&A)(I&A t)(p ,&p ()'k…0

0q(t)'(I&A t)p(t)

11

See Hirsch and Smale (1974), p. 292.6

(2.35)

(2.36)

(2.37)

(2.38)

(2.39)

system is conserved . How shall we interpret this condition in the6

context of a model aimed at representing the process by which a setof n "Walrasian" markets reacts to disequilibrium?

We know that the necessary conditions for a maximum of theobjective functional

where p', p* are actual and equilibrium relative pricesrespectively, and q', q* are actual and equilibrium gross productsrespectively, require that

i.e. that shadow prices are equal to current market prices. If

the objective functional is equal to 0. However the terminalcondition requires that

and the transversality condition

tells us that quantities will change as long as p'…p*. Hence theterminal condition excludes both the possibility that prices andquantities converge asymptotically to their long period values, andthe possibility that prices and quantities increase or decreaseinfinitely.

In standard Cartesian coordinates the origin of axes representslong period equilibrium. Hence %k - the distance from the origin ofvector {(I-A)q, (I-A )p } - can be interpreted as a measure of thet

"degree of disequilibrium" which exists in the economic system. Itthen becomes evident that the terminal condition 2.15, by keeping

12

Of course, this condition constrains the dynamic behaviour7

of the model rather arbitrarily. I leave it to the reader todecide whether it is more or less legitimate than the previouslymentioned "Keynesian multiplier stability assumption" or thelocal asymptotic stability assumption, which is a common featureof a large part of the recent new-classical literature oneconomic growth. On this point see, for example, Serena Sordi(1990).

i.e., the unperturbed and the perturbed vector fields are8

not topologically equivalent. See Guckenheimer and Holmes (1990),p. 39.

the existing degree of disequilibrium constant, serves the purposeof assuring that the model is globally stable, although it is notglobally asymptotically stable . We might then call the terminal7

condition 2.15 "Goodwin's Walrasian stability assumption". On theother hand, as we shall soon see, Goodwin's Hamiltonian model isstructurally unstable .8

>̄'

(k&jn

i'2p̄ 2i& j

n

i'1q̄ 2i

p̄2

.

.

.p̄nq̄1

.

.q̄n

0M1

D>>̄ '

0*>̄

1

*>̄j

*j'2...n

0 I2n&1

>̄1

>̄j

*j'2...n '&p̄2p̄1

,p̄3

p̄ 1,...,

p̄np̄1

,q̄1p̄1

,q̄2p̄1

,...,

>̄t.'>̄

tD>>̄(>&>̄)'

(p̄1,...,p̄n,q̄1,...,q̄n)@0

*>̄1

*>̄j

*j'2...n

0 I(2n&1)

.1'&

1p̄1

jn

i'2pip̄i& j

n

i'2p̄ 2i% j

n

i'1qiq̄i& j

n

i'1q̄ 2i

jn

i'2p̄ 2i% j

n

i'1q̄ 2i' k & p̄ 2

1.1' &

1p̄1

jn

i'2pip̄i% j

n

i'1qiq̄i& k%p̄ 2

1

[.i]'[>

i&>̄

i],i'2...2

n

(>̄t.)' & jn

i'2pip̄ i % j

n

i'1qiq̄i& k % p̄ 2

1%

% jn

1'2pip̄ 2i% j

n

i'1qiq̄ 2i& j

n

i'2p̄ 2i& j

n

i'1q̄ 2i

'0

13

(2.A.1)(2.A.2)

(2.A.3)

(2.A.4)

(2.A.5)

(2.A.6)(2.A.7)

(2.A.8)

(2.A.9)

Appendix 2.A

Set

The Jacobian matrix of >̄ is

where

The transversality condition at t=t is 1

where

Sincewe have

and

Hence

14

ex> 'ex>

1

ex>2

' Mpq

S 1(ex>1, ex>

2) ' j

n

i'1(ex>

1i)2 % j

n

i'1(ex>

2i)2 & k '

' p t(I&A)(I&A t)p % q t(I&A t)(I&A)q & k ' 0

ex>

0*ex>

11

*ex>1j

*j'2...n*ex>

11

*ex>2j

*j'1...n

0 I(2n&1)

*ex>11

*ex>1j* j'2...n ' &

ex>1j

ex>11* j'2...n

*ex>11

*ex>2j* j'1...n ' &

ex>2j

ex>11* j'1...n

ex>11 ' k & jn

j'2(ex>1j)

2 & jn

j'1(ex>2j)

2

jn

j'2(ex>

1j)2 % j

n

j'1(ex>

2j)2 ' k & (ex>

11)2

(ex>)t. ' (ex>)t Dex>

ex> (ex>&ex>) ' 6p̄ t(I

@0 &

ex>1j

ex>11* j'2...n &

ex>2j

ex>1i* j'1...n)

0 I(2n&1)

ex>11

' (1&a11)p̄

1&j

n

j'2aj1p̄j'

' k&jn

j'2(ex>

1j)2&j

n

j'1(ex>

2j)2

15

(2.B.1)

(2.B.2)

(2.B.3)

(2.B.4)

(2.B.5)

(2.B.6)(2.B.7)

(2.B.8)

(2.B.9)

Appendix 2.B

Set

Then

where

and

The transversality condition then becomes

Setting

Dex>

ex>'0 &

1

ex>11

p̄ t(n&1)(I&A)(n&1), q̄ t(I&A t)

0 I(2n&1)

p̄ tn&1

(I&A)(n&1)

'(ex>12...ex>

1n)

16

(2.B.10)

(2.B.11)

we have

where

and

Dex>ex>(ex>&ex>)'

'

&1

ex>11

[&k%(ex>11)2%p̄ t

(n&1)(I&A)(n&1)(I&A)t(n&1)p

ex>12&ex>

12

. .

. .

. .

ex>2n

&ex>2n

k%(ex>11)2%p̄ t

n&1(I&A)

(n&1)(I&A t)

(n&1)p(n&1)

%q̄ t(I&

>11)2%p̄ t

(n&1)(I&A)(n&1)(I&At)(n&1)p(n&1)%q̄

t(I&A t)(

17

(2.B.12)

(2.B.13)

hence

S 1(t,p,q)'j2n

i'1(ex>

i)2'k

ex>i'± k

ex>(t)0M1

18

(2.C.1)

2.C.2)

(2.C.3)

Appendix 2.C

Equation 2.32 can be reformulated as follows

where k is any finite real constant …0. If ex> =0, j=1...i-1,j

i+1...2n, 2.C.1 implies that

Since this holds for any i=1...2n, +%k, -%k are the upper and lowerbounds of each ex> (t)0ex>(t). Hence ex>(t) is bounded. Buti

hence M is bounded.1

0>'0p

0q'

&(I&A)q

(I&A t)p'J

exp

exq'Jex>

J'0 &I

I O

0B

0x'00'Q0>'QJgrad

ex>k(ex>)'QJQ tgrad

ex0k(ex>(ex0)

h(ex0)'k(ex>(ex0))

19

See Abraham and Marsden (1987) pp. XXI, XXII. 9

(3.1)

(3.2)

(3.I.1)

(3.I.2)

3. A symplectic change of coordinates.

Let us call ex q the i-th element of vector (I-A)q, equal to thei

excess demand of the i-th commodity, and ex p the i-th element ofi

vector (I-A )p, equal to the excess profit of the i-th sector.t

Hamilton's equations 2.4 and 2.30 can be written as

where

DEFINITION 3.I: A canonical or symplectic coordinatestransformation of 3.1 is a transformation 0=f(>), where f: ú x ú 6 ú x ú is smooth, which satisfies the followingn n n n

conditions:

a) if >(t) is the solution of Hamilton's equations 3.1.,0(t)=f(>(t)) is the solution of equations

*0i

where Q =[))))] is the Jacobian of f and Q is the transpose of Q.ij t

*> j

b) if and only if QJQ =J, the equations for 0 will be Hamiltoniant

with energy 9

In order to simplify argument, let us make the following

HYPOTHESIS 3.i : (I-A) has n distinct real eigenvalues 8 ,...,81 n

. ^

X(I&A)X &1'8̂'(X &1)t(I&A t)X t

0'Q>'(X &1)t 0

0 X

p

q'B

x

t'(X &1)t 0

0 X

0 &I

I 0

X &1 0

0 X t'0 &I

I 0'J

00'Q0>'QJ Lex>

k(ex>)'J Lex0

h(ex0)

ex0'QMQ &1Q>'8̂ 0

0 8̂

B

x'8̂B

8̂xk(ex>)'k(ex>(ex0))'

'12(ex>)t(ex>)'

12(ex>(ex0))t(ex>(ex0))

(ex>)t(ex>)'(ex0)t(ex0)

(ex0)t ' (ex>)tQ &1

20

See, for example, Eisele and Mason (1970), part II.10

i.e. (ex>) = (ex0) Q.11 t t

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

Call 8 the diagonal matrix of the eigenvalues of (I-A) and X thematrix of the eigenvectors belonging to them. Then we have

We shall prove that the transformation

is a symplectic trasformation, i.e. it satisfies conditions a) andb) of Definition 3.I. Condition b) can be easily checked, since

Condition a) is satisfied if

From 3.4 we have

Setting

from condition b) we have

Since the column vector ex> is trasformed contravariantly, thescalar product k(ex>) is invariant to the coordinatestransformation 3.7 only if the row vector ex> is transformedcovariantly , i.e. if10 11

Thus equation 3.8 becomes

ex> (ex0))'(ex>(ex0))t(ex>(ex0))'(ex0)tQQ &1(ex

Lex0

k(ex>(ex0))'*k(ex>(ex0))

*ex0

t

'Q &1tQ t(ex0)'

00'Q0>'QJ Lex>

k(ex>)'QJ[(ex>)t]t'QJ[(ex0)tQ]t'

QJQ tex0'QJQ t Lex0

k(ex>(ex0))'Jex0

0B'&8̂x

0x'8̂B

0>'J Lex>

k(ex>)

Q0>'QJ Lex>

k(ex>)

ex0 ' Q ex>

*0

*>'

*(ex0)*(ex>)

' Q

*k(ex>)*ex>

'*k(ex>(ex0))

*ex0.

*ex0*ex>

Lex>

k(ex>(ex0)) '*k(ex>)(ex0))

*ex0.

*ex0*ex>

t

'

'*ex0*ex>

t

Lex0k(ex>(ex0))'Qt Lex0h(ex0)00'Q0>'QJQ tgrad

ex0h(ex0)

21

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

and

Substituting expression 3.12 in equation 3.6 we obtain

Hence Hamilton's equations in the new symplectic coordinates are

as in Goodwin and Punzo (1987), pp. 79-82.

Alternatively, condition a) can be proved as follows: set

hence

From

we have

Since QJQ =J and 3.9 is verified, we havet

and

Hence 3.16 becomes

Integral curves of system 3.14 can be easily found : taking thetime derivative of the first equation of 3.14 and substituting x0from the second equation we obtain a system of n second order linear differential equations

B̈'&8̂2B

µj'± &82

j' ±i8

j

e±i8jt ' cos 8

jt ± i sin 8

jt

0B1'&8

1x1

0x1'8

1B1

22

Goodwin and Punzo, (1987), p. 73.12

(3.22)

(3.23)

(3.25)

(3.26.1)

(3.26.2)

which, under hypothesis 3.i, has only imaginary roots with zeroreal parts

By Euler's formula

hence, in symplectic coordinates, all integral curves of theHamiltonian system 3.1 lie on a manifold given by the Cartesianproduct of n circles of radius 1. In Goodwin's own words: "Themotion is dynamically stable, in the sense of bounded, but is notasymptotically stable towards equilibrium. It is structurallyunstable (...) so that a slight error in, or perturbance of, theparameters would lead either to the disappearance of the cycle orto its explosion without limit." Figure 1 is the phase portrait12

of a numerical example of system 3.14 (Navajo), in the case wheren=2. Figure 2 is the graph of B vs. time. The parameters and theinitial conditions of this simple two-sector system are given intable 1. This numerical example provides the foundations on whichthe more complex examples of the following paragraphs will bebuilt.

In order to have a greater insight into the dynamic behaviour ofthe vector field 3.1 and to simplify the analysis of the perturbedHamiltonian, which will be introduced in the next paragraph, it isuseful to perform a further, symplectic change of coordinates. Forthe moment, we will go on assuming that n=2; equations 3.14 canthen be written as

0B2' &8

2x2

0x2' 8

2B1

B̈ ' &821B1

B̈ ' &822B2

k1'12(82

1B21%82

1x 22)

k2'12(82

2B22%82

2x 22)

v1' 0B

1v2' 0B

2

k1'12(82

1B21%v 2

1)

k2'12(82

2B22%v 2

2)

23

See Guckenheimer and Holmes (1983), pp.212-215.13

(3.26.3)(3.26.4)

(3.27.1)(3.27.2)

(3.28.1)(3.28.2)

(3.29.1)(3.29.2)

(3.30.1)(3.30.2)

and equation 3.22 can be written as

Equations 3.26 and 3.27 make clear that vector field 3.14 can bedivided into two uncoupled Hamiltonians:

Setting

we have

In this case it is possible to further transform the linear non-orthogonal coordinates of system 3.14 into non-linear polarcoordinates (action angle coordidates ). Set13

B2'

2I82

sinN

v2' & 2I8

2cosN

k'82I%

821B21%v 2

1

2'k(I,N,B

1,v

1)

I'182

k&821B21%v 2

1

2'I(k,N,B

1,v

1)

B)

1'dB

1

dN'0B1

N@

v )

1'dv1

d0N'0v1N@

N@'*k*I

'82

B)

1'

182

v1

v )

1' &

82182

B1(N)'BE

1cos SN%

<E1

81

sin SN

v1(N)'&BE

181sin SN%<E

1cos SN

PEk'

cos2BS181

sin2BS

&81sin2BS cos 2BS

B1

v1µ

1,2' cos 2BS ±i sin 2BS

B1(t)'BE

1(cos2BSt&*)

24

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

(3.36)

(3.37)

(3.38)

(3.39)

(3.40)

hence

The Hamiltonian becomes

hence

Setting

the reduced Hamiltonian system becomes:

Since k does not depend explicitly on N, the reduced system 3.36 isautonomous. The linearized Poincaré map can thus be easily obtainedby solving system 3.36 based at {( , B }. The general solution isE E

1 1

where S=(8 /8 ). 3.37 is 2B periodic in N. Thus the linearized1 2

Poincaré map is

Its eigenvalues are again complex conjugate, with unit modules

By applying De Moivre's theorem we obtain the solution for B , 1

which is periodic with period 2BS. Thus the long period equilibriumsolution {0,0} is "an elliptic centre surrounded by a family ofclosed curves filled with periodic points if S is rational, and

B1'

2J81

sin R

v1'& 2J8

1cos R

k1'8

1J

0J'&*k

1

*R'0

R@'*k1*J

'81

25

See Guckenheimer and Holmes (1983) p. 215.14

(3.41)

(3.42)

(3.43)

(3.44)

with dense orbits if S is irrational" . In our numerical example14

S is an irrational number, hence flows are dense orbits, as can beeasily seen from figures 1 and 2.

Also variables < and B can be transformed into a second set of1 1

action angle variables. Set

and

The second Hamiltonian becomes

and its vector field is

0J'0;R@'81; 0I'0; N@'8

2

J'JE I'IER(t)'8

1t%RE N(t)'8

2t%NE

(t)'2J81

sin (81t%RE) x

1(t)'

2J81

cos (81t%R

(t)'2I82

sin (82t N )

2t)'

2I82

cos (82t%N

0Bj

0xj

'

0 8j

j 0

j

x

26

is given by the Cartesian15 2

product of two circular phase spaces S x S . 1 1

Guckenheimer and Holmes (1983) p. 59.16

In the two-dimensional example we can set 7=R =8 . The non17 ·1

degeneracy condition requires that 7'(J)…0. Since, in this case,R is a real constant, this condition is violated. (Guckenheimer·

and Holmes, 1983, p.219).

See Guckenheimer and Holmes (1983), p. 219.18

(3.45)

(3.46)

(3.47)

(3.48)

The complete dynamic system is then reduced to

which, for initial conditions {JE, IE, RE, NE}, has the solution

and, in terms of the original variables

"Thus the four dimensional phase space is filled with two-dimensional tori , given by J=JE, I=IE and each torus carries15

rational or irrational flows depending on the ratio 8 /8 . In1 2

general, n degree of freedom integrable Hamiltonian systems giverise to flows on n-dimensional tori" . In our case the j-th vector16

field can be written as

In the two sector case (equation 3.45) the linearized Poincaré mapis degenerated . Hence, the Kolmogorov-Arnold-Moser theorem ,17 18

which asserts that, if the period of R is a function of J, most ofthe closed irrational orbits of the unperturbed Poincaré map arepreserved for sufficiently small perturbations, cannot be applied.The system is structurally unstable.

q dj

' wsj

pj

l )

jj sj ' 1

27

In this book Goodwin represents the adjustment process of19

prices and quantities by a gradient vector field.

See, for example, Varian (1984) pp.128-129.20

(4.1)

(4.2)

4. Consumer demand functions.

In this paragraph we will add a slightly non-linear perturbation tothe linear dynamic adjustment model analysed in paragraphs 2 -3. Inthe first chapter of Chaotic Economic Dynamics Goodwin assumes thatonly wage earners spend for consumption, that wage income isentirely spent, that the uniform, nominal wage rate w remainsconstant during the whole adjustment process , and that consumers19

expenditure is distributed among different goods and services infixed proportions. Under these assumptions, the Marshallian demandfunction for the j-th consumption good is given by

where q is the demand of good j, p' is its price, l' is totaldj j

employment and s is the proportion of total wage income spent inj

purchasing good j. Of course

As is well known, constant expenditure shares are a feature ofCobb-Douglas utility functions. Consumer demand functions like 4.1can be easily derived from a constrained maximum problem (consumersmaximize their current utility under the constraint of a givennominal income), provided the utility function is Cobb-Douglas .20

On the other hand, demand functions like 4.1 imply that the realwage income is flexible and inversely related to prices.

To investigate some of the possible effects of the adoption ofthese behavioural hypotheses on the dynamic process analysed in theprevious paragraphs, we will restate them as follows:

HYPOTHESIS 4.i The nominal wage per unit of uniform labour is

w ' 1

pj' p )

j&1

w ' 'jsj

F '

F1

F2

'

s1

1%p1

1&s1

1%p2

28

(4.3)

(4.4)

(4.5)

(4.6)

constant and equal to 1; i.e.

HYPOTHESIS 4.ii Current prices are expressed in index numbers,based at their respective long period equilibrium values; i.e.

HYPOTHESIS 4.iii The wage is entirely spent for consumption infixed proportions; i.e.

Finally, we will assume that n=2.

From hypotheses 4.ii and 4.iii we have that s and (1-s ) represent1 1

equilibrium real consumption per unit of labour, i.e equilibriumcommodity wage. Hence current commodity wage is:

q ( ' (I&A&sal)&1y (

l ( ' alq (

(I&A)q&Fl )%sl ('(I&A)q&l(I%p̂)&1s%l (p̂(I%p̂)&1s

p̂ ' Diag(p)

Fi&s

i' &p

i

si

1%pi

0q' (I&A t)p0p' &(I&A)q%l(I%p̂)&1&l (p̂(I%p̂)&1s

0l' al0q' a

l(I&A t)p

0x ' 8̂B

0B ' &8̂x%lX[I%Diag(X tB)]&1s&l (X[Diag(X tB)]@@[I%Diag(X tB)]&1s0l ' alX

&1X0q ' "l8̂B

29

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

Define as y*(p*) the two-dimensional vector of equilibrium final demands. We know that

and therefore

where a is the row vector of constant labour inputs coefficients.l

Excess supply is now given by

where l=(l'-l*)=a (q'-q*) is current excess employment, l

and

The dimension of the phase space is now (2n+1)=5, and the vectorfield is now

In symplectic coordinates 4.12 becomes

Although the number of sectors is only two, the number ofparameters involved is relatively high (eight), hence a wide rangeof dynamic behaviours and bifurcations becomes possible. This makesthe stability analysis of the flows on the five dimensional phasespace rather difficult. Analysis will therefore be restricted to abrief discussion of the outcomes of the simple two-sector numericalexample presented in table 1 (Sioux). The phase portait of thismodel is given in figure 3. Figures 4, 5 and 6 are the graphs ofB , B , and l', i.e. of the deviations of current prices from their1 2

equilibrium values and of total employment respectively, vs. time.

Since the deviations of current prices from their equilibrium

30

values tend to vanish and the same holds for l, equilibrium is, inthis example, locally asymptotically stable. However, it may beasked whether the zero solution of this model can be regarded as along period equilibrium. Total employment (figure 5) does not tendto an exogenously given constant level, but to an endogenouslydetermined value, which depends on both the behavioural hypotheseson which the model is built and the initial conditions. This meansthat excess gross products q tend to vanish, i.e that currentgross outputs adjust to their respective equilibrium values which,however, are not independent of the adjustment process itself.

This result is, of course, a consequence of the assumption of alinear technology. In a linear production model, long periodequilibrium prices are unequivocally defined once technology andincome distribution are defined. The same holds for the structureof gross outputs, once technology and the structure of final demandis given, as is the case in Goodwin's 1990 model. However, theabsolute level of outputs in each sector (the module of the grossoutputs vector) remains indeterminate. Hence, in a dynamicadjustment model like the one presented in this paragraph, anylevel of gross production can be regarded as an equilibrium level,provided it is stationary.

Fl )&sl ( ' l(I%p̂)&1s&l (p̂(I%p̂)&1s

[I%Diag(X tB)]&1s&lX[Diag(X tB)][I%Diag(X tB)]

k 1 'g

2Ft[l (Diag(BX)X t&lX t][l (X Diag(BX)&lX]

0<g#1

X '

1 b12

b21 1

31

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

5. Further analysis of the dynamics of a two degree of freedomHamiltonian.

The aim of this paragraph is only to provide an example of theinteresting potential results which the Hamiltonian approach maylead to, if the assumptions of real wage flexibility and constantreturns to scale are released. For this purpose, some rathersubstantial modifications of Goodwin's basis Hamiltonian model needto be introduced. Formally, these will take the shape ofmodifications of both the non-linear perturbation introduced inparagaph 4 and the linear unperturbed Hamiltonian analysed inparagraphs 2 and 3.

Firstly, we will assume that the commodity wage F changes if priceschange, but these changes will not be sufficient to entirely cancelthe initial "degree of disequilibrium" affecting incomedistribution. As in the case of prices, the "degree ofdisequilibrium" is measured by the length of vector

or, in symplectic coordinates

The perturbation then becomes

where

The two-matrix of right eigenvectors of (I-A) can be written in theform

Setting

a ' s 21(1%b 2

21)

b ' s1(1&s

1)(b

12%b

21)

c ' (1&s1)(1%b 2

12)

k g'12[82

1(B2

1%x 2

1)%82

2(B2

2%x 2

2)]%

%g6 l (2h2

%12[

l ) ap1

2

%l ) cp2

2

]&

&al (l )

p1&cl (l )

p2&bl ) l (

p1%l (

p2&

l )

p1p2>'kE%gk 1

h ' a%2b%c

p )

1' 1%B

1%b

21B

p )

2' 1%b

21B1%B

2

l )'"lx%l (

(p )&p ()(I&A)

(I&A)(q )&q ()

0B'&Lex x

kE(ex B,ex x)&gLex x

k 1(ex B,ex x)

0x'Lex B

kE(ex B,ex x)%gLex B

k 1(ex B,ex x)0l '"

l0x0B ' &8̂x%8̂&1

"tlFtX t[X Diag(BX)&lX]F

' 8̂B%l )8̂&1[X(I%Diag(X tB)&1F̂X t][X Diag(BX)&lX

0l ' "l0x

32

(5.6)

(5.7)(5.8)

(5.9)

(5.10)

(5.11)

(5.12)

(5.13)

(5.14)

the perturbed Hamiltonian can now be written as

where

and

From equation 5.1 one can see that the "degree of disequilibrium"affecting income distribution depends on the deviations of currentprices from their equilibrium values and on the deviations of totalemployment from its equilibrium level l*. Since the nominal wage isconstant,

can still be used as a satisfactory measure of excess profits. Onthe other hand, excess final goods supply

can still be used as a measure of excess supply. Thus, insymplectic coordinates, the Hamiltonian vector field can be writtenas

Setting l*=1 and ,=1, system 5.13 becomes

.

Since k =0, a third conserved quantity has been added to k and,1

k . However, integral curves are still flows on a two-dimensional2

manifold. A numerical example of this model (Paiute) is alsopresented in table 1; its phase portrait is presented in figure 7.

0q1' D

11sin f

1(p

1,p

2)%D

12f2(p

1,p

2)

0q2' D

21sin f

1(p

1,p

2)%D

22f2(p

1,p

2)

C'D11

D12

D21 D22' X t

f1(p

1,p

2)' B

1

f2(p

1,p

2)' 8

2B2

33

(5.15)

(5.16)

(5.17)

Figures 8, 9 and 10 are the graphs of B , B and l' vs. time.1 2

In order to simplify the discussion of the second substantialmodification of the basis Hamiltonian system, I assume, for themoment, that ,=0. The hypotheses that the time derivatives ofsectoral outputs are linear increasing functions of sectoral excessprofits and that the time derivatives of prices are linearincreasing functions of sectoral excess demands are now replaced bythe assumptions that:

- the changes of the gross output of each sector depend linearlyand non-linearly on excess profits in all sectors;- the changes of the price of each product depend linearly onexcess demands in all sectors.

Moreover, as far as gross outputs are concerned, it is assumedthat, if positive (negative) deviations from equilibrium are small,all factors of production can be used more (less) efficiently;therefore, returns to scale are increasing (decreasing). However,if the absolute value of deviations from equilibrium exceed a givenmaximum, the absolute values of efficiency increases (decreases)are progressively reduced to zero. A simple way of stating thesehypotheses is the following:

To minimize the difficulty of analysing the dynamic behaviour ofthe nonlinear production model, it is useful to assume also that

and that

Thus, equations 5.15 in symplectic coordinates become:

0x1

0x2' X0q ' (X &1)t X t

sin B1

82B2

0p1' L

11g1(q

1,q

2) % L

12g2(q

1,q

2)

0p2' L

21g1(q

1,q

2) % L

22g2(q

1,q

2)

K '

L11

L12

L21 L22' X &1

g1(q

1,q

2)' &x

1

g2(q

1,q

2)' &8

2x2

0B1

0B2' (X &1)t0p ' &X X &1

x1

82x2'

&x1

&82x2

0B1' &x

1

0x1' sin B

10B2' &8

2x2

0x2' 8

2x2

B2'

2 I82

sin N

<2' 0B

2' & 2 I 8

2cos N

kE 'x 21

2% (1& cos B

1) % 8

2I ' F(B

1,x

1) % G(I)

k g ' F(B1,x

1) % G(I) % gk 1(B

1,x

1,N,I)

k " ' F(B1,x

1)

34

(5.18)

(5.19)

(5.20)

(5.21)

(5.22)

(5.23.1)(5.23.2)

(5.24)

(5.25)

(5.26)

(5.27)

A similar set of simplifying assumptions can be made for the timederivatives of prices. Assume that

and that

Thus, in symplectic coordinates, system 5.19 becomes

and the unperturbed Hamiltonian vector field is

As in paragraph 2, B and x can be transformed to action angle2 2

variables:

hence, the final expression of the unpertrubed Hamiltonian is

If ,…0, the perturbed Hamiltonian becomes:

F(B , x ) is the Hamiltonian of a simple pendulum with energy1 1

In a cylindrical phase space:

F(B1,x

1): S 1 x R 6 R

B10 [&B, B)

x 21

2' k "&2 ' 0

x1' ± 2(k "&2) ' 0

6BE,xE> ' 6± 2arctan(sinh(t)),± 2sech(t)>

l " ' G &1(k g&k ") '182

(k g&2)

35

See appendix 5.A.21

See Guckenheimer and Holmes, p. 219.22

See Guckenheimer and Holmes, p. 201.23

(5.28)

(5.29)

(163)

(5.31)

(5.32)

(5.33)

and

B and x have two rest points: the origin, which is an elliptic1 1

centre, and {B =-B/B, x =0}, which is a saddle. 1 1

The non-degeneracy condition is satisfied in this case . From KAM21

theory we know that, if , is sufficiently small, the perturbed(area preserving) Poincaré map has a set of closed curves, ofpositive Lebesgue measure, close to the original set, filled withdense irrational orbits . Since, for B =±B, x =0 only if22

1 1

i.e

for F(B , x )=k =2, system 5.23.1 has a pair of homoclinic orbits :1 1" 23

Assume that the total "degree of disequilibrium" of system 5.26 isgreater than 2. Then

is a positive constant. The results of the foregoing analysis canthen be summarized as follows:

- equation 5.26 is a two degree of freedom Hamiltonian system withthree conserved quantities;- F has a pair of homoclinic orbits at the energy level k =2;"

6F,k 1> (t%NE)

6F(BE,xE),k 1(BE,xE,(82t%NE),l ")> '

*F*B1

*k 1

*x1&

*F*x1

*k 1

*B19(NE)'m4

&46F,k 1> (t%NE)dt

9(NE) ' m4

&4±sin(2arctan(sinh(t)))"

l1@

@ al)

p )21

%cl )

p )22

%bl )

p )

1p )

2

&l ((a%b)

p )

1

&l ((c%b)

p )

2

±

sech(t)l )

p )21

l ( a%b &l ) a%b

p )

2

%b12l )

p )22

l ((a%b)&l ) a%

B2'

2(k g&2)82

sin (82t%NE) ' T sin (8

2t%NE)

&182

0B2'

2(k g&2)82

cos (82t%NE) ' T cos (8

2t%

36

See Guckenheimer and Holmes, p.252.24

Guckenheimer and Holmes, p. 252.25

See appendix 5.B.26

(5.34)

(5.35)

(5.36)

(5.37)(5.38)(5.39)

- G'(I)=8 >0, and2

- for k >k , l is a constant., " "

Let

denote the symplectic form

and define

Then , if Melnikov's function 9(NE) has a simple zero and is24

independent of ,, for , sufficiently small, system 5.26 hastransverse homoclinic orbits on every energy surface k >k . By, "

Smale-Birkhoff homoclinic theorem this implies that the associated25

Hamiltonian vector field has Smale horseshoes at every "degree ofdisequilibrium" greater than two. Using 5.32 and 5.7, Melnikov'sfunction 5.36 becomes equal to26

where l', p' and p' are given by equations 5.9; B =B , x =x and1 2 1 1E E

Melnikov's function can then be evaluated by the method of

9(NE)'±4a(l (%2"l1%"

l2TcosNE)@

@(1%TsinNE)2%(1%b21TsinNE)

2

(1%TsinNE)2(1%b21TsinNE)2

NE'arccos &l (%2"

l1

"l2T

37

See appendix 5.C.27

(5.40)

(5.41)

residues to yield27

Since 5.40 has a simple zero for

the perturbed Hamiltonian system has Smale horseshoes at every"degree of disequilibrium" greater than two, for , sufficientlysmall.

F(B1,x

1) % 8

2I ' k E

0I ' 0N@ ' 8

2B)

1'

0B1

N@' &

182

x1(N)

x )

1'

0x1

N@'

182

sin B1(N)

B̃)

k̃ )'

0&182

182

0

B̃

k̃µ1,2

' ± i182

P EkE'

cos 2B182

82sin 2B

182

&182

sin 2B182

cos 2B182

B̃

k̃

B̃)

x̃ )

0 &182

&182

0

B̃

x̃µ1,2

' ±182

38

(5.A.1)

(5.A.2)

(5.A.3)

(5.A.4)

(5.A.5)

(5.A.6)

(5.A.5)

(5.A.6)

Appendix 5.A

The unperturbed Hamiltonian can be written as

and its vector field is

From 5.28 and 5.29 we know that B is a periodic function and that1

B 0[-B,B). If we take the restriction E ={(B ,x ,N,I)0S x ú #1 kE 1 1NE 2 2

I=I , N=N 0[0,2B]} and linearize the system 5.A.3 at (B =0, xE E1 1

=0), we obtain

The roots of 5.A.4 are

i.e., they are imaginary with 0 real part. Hence (0,0) is anelliptic centre, sorrounded by closed orbits filled with denseorbits if 1/8 is irrational. The linear Poincaré map is2

If 5.A.3 is linearized at (B = -B/B, x =0), we obtain

and the roots are now real distinct

Hence (-B,0) is a saddle. Since B is periodic with period 2B, the1

non-degeneracy condition of the unperturbed Poincaré map can bechecked as follows: set B =0 x = x̄ >0; we have1 1 1

F(B1,x

1*B

1'0 x

1'x̄

1)'

x̄ 21

2'k ">0

x̄1' 2 k "

x 21

2%1& cos B

1'

x̄ 21

2J ' x̄

1

x1' J 2&2%2 cos B

1

0J' 00B1'7(J)'J

7)(J) ' 1>0

39

(5.A.7)

(5.A.8)

(5.A.9)

(5.A.10)

(5.A.11)

(5.A.12)

(5.A.13)

hence

and, in general

Therefore we set

and

F(B ,x ) is now equal to (J /2), hence1 12

and

9(NE)'m4

&46F,k 1> (8

2t%NE)dt

*F*B

1

'sin B1' sin (± 2 arctan (sinh (t)) '

± sin (2 arctan (sinh (t)))*F*x

1

' x2' ± 2 sech (t)

*k 1

*x1

'*k 1

*l ).

*l )

*x1

'

' "l1

l )

p )21

a%l )

p )22

c%l )

p )

1p )

2

b&l ((a%b)

p )

1

&l ((c%b)

p )

2

*k 1

*B1'

*k 1

*p )

1

*p )

1

*B1%

*k 1

*p )

2

*p )

2

*B1'

l )

p )21

l ((a%b)&l ) a%b

p )

2

%b12l )

p )22

l ((a%b)&l ) a%b

p )

1

9(NE) ' m4

&4± sin(2 arctan(sinh(t)))"

l1.

a %l )

p )22

c %l )

p )

1p )

2

b &l ((a%b)

p )

1

&l ((c%b)

p )

2

±2sech

(a%b) & l ) a %b

p )

2

% b12

l )

p )22

@ l ((a%b) & l ) a%

40

(5.B.1)

(5.B.2)

(5.B.3)

(5.B.4)

(5.B.5)

(5.B.6)

Appendix 5.B

Melnikov's function is

On the homoclinic orbit:

and

We also have

and

Thus

sin(2 arctan (sinh(t)))l )

p )21sin 2 arctan(sinh(t)))

l )

p )

1p )

2sin (2 arctan (sinh(t)))1

p )

1sech (t)l )

p )21sech (t)

l )2

p )21

sech (t)l )2

p )21p

)22

l )'[l (%"l1

2 sech(t)%"l2

T . cos (82t%NE)]

p )

1' [1%2 arctan (sinh(t))%b

21T sin(8

2t)%NE)]

' [1%2b12

arctan (sinh (t)) % T sin (82t % NE)

s ' te in t , R%

0 # n # B

41

(5.C.1)(5.C.2)(5.C.3)(5.C.4)(5.C.5)

(5.C.6)

(5.C.7)(5.C.8)(5.C.9)

(5.C.10)

Appendix 5.C

The evaluation of 9(NE) by the method of residues is ratherlaborious. In this appendix I will therefore state only the mostrelevant aspects of the procedure. The reader may refer to Smirnov,vol. 3.II, and to Dieudonné, chapter IX, for a detailed expositionof the theory of residues and of the theorems mentioned in thisappendix.

9(NE) is a linear combination of six basis integrals:

In expressions 5.C.1, 5.C.3, 5.C.4 and 5.C.5 p' can be replaced1

by p' without changing the integral. On the homoclinic orbit,2

total employment and prices are given by

Replace the real variable t with the complex variable s, defined asfollows

thus

dsdt

'e in ; dt'ds e &in

g ' e in ' cos n % i sin n

dt ' ds g&1

s ' gt

t ' nt% 2kB ; 0# n

t# 2 B ; k ' 0, ±1, ±2, ...

s ' (nt% 2kB)g

sech (s) '2 e s

e 2s%1' cosh&1(s)'

'2 e

gnt%e g2kB

e2gnte g4kB%1

limk6±4

cosh(s)'0

limnt6

B

2nt6

3B2

cosh&1(s) ' limnt6

B

2nt6

3B2

cos&1(nt) '

±2i

(±i)2%1' ±4

sinh(s) 'e s&e &s

2

42

(5.C.11)

(5.C.12)

(5.C.13)(5.C.14)

(5.C.15)

(5.C.16)

(5.C.17)

(5.C.19)(5.C.20)

and setting

we have

We can also set

hence

Now, let us analyse the behaviour of the hyperbolic and circularfunctions of s in expressions 5.C.1-5.C.9. We have

for n =/(B/2), , =/i:

and for n=(B/2), ,=i:

for n =/(B/2), , =/i:

limk64

sinh(s)'4

limk6&4

sinh(s) ' &4

sinh(s) 'e2i(nt%2kB)&1

2ei(nt%2kB)

' isin(nt) ' &1isin(nt)

limnt6

B

2

isin(nt) ' i

limnt6

3B2

isin(nt) ' &i

2arctan(sinh(s)) ' 2R

2R '1iln

i&sinh(gt)i%sinh(gt)

* 1&sinh(gt)1%sinh(gt)

* '&1%sinh2(gt)

&1%sinh2(gt)'1

2R '1iarg ln

i&sinh(gt)i%sinh(gt)

' nR%2k

RB

0# nR#2B; k

R' 0,±1,±2,...

2R'1iln

i&isin(nt)

i%isin(nt)

'1iln

1&sinnt

1%sinnt

43

(5.C.21)

(5.C.21)

(5.C.22)

(5.C.23)

(5.C.24)

(5.C.25)

(5.C.26)

(5.C.27)

(5.C.28)

(5.C.29)

For k=0 and n =0, sinh(0)=0. Hence sinh(s) is a monotonict

increasing function of s. For n=(B/2), ,=i:

and

Set

For n=/(B/2), ,=/i, we have

Since

we have

where

Setting k = 0, R becomes a real one-one function of s. ForR

n = (B/2), ,=i

is a complex one-one function, and

limnt6

B

2

1iln

1&sin nt

1%sinnt

' &4

limnt6

3B2

1iln

1&sinnt

1%sinnt

' %42 arctan(i) ' &4

2 arctan(&i) ' 4

sin(2 arctan(sinh(s))) ' sin(2R)

sin(2R) ' sin(2nR)

44

(5.C.30)

(5.C.31)

(5.C.32)

(5.C.33)

(5.C.34)

hence

Thus

for n =/(B/2), , =/i

is a real periodic function. For n = (B/2), , = i

sin( R) sin &iln1&sinNt1%sinN

t

'

'ei &iln

1&sinNt1%sinNt &e

&i &iln1&sinNsubtt

1%sinNt

2i

sin(2R) '12i

(1&sinnt)2&(1%sinn

t)2

1&sin2nt

'

'1i

sinnt

cos2nt

limnt60

sin(2R) ' 0limnt6

B

2

sin(2R) ' %4limnt6

3B2

sin(2R) ' &4

s't 0 ú%

n'0 ; n'B

s't(cosn%isinn); t0ú%

sin(2arctan(sinh(t)))l )

p )21

2"l1sin2R

cosh(s)[1%2R%b21Tsin(8

2s%NE)]2

45

(5.C.35)

(5.C.36)(5.C.37)

(5.C.38)(5.C.39)

(5.C.40)

(5.C.41)

(5.C.42)

(5.C.43)

(5.C.44)

thus

which implies that

In a similar way it can be proved that if

i.e., for

cos(8 s+N) and sin(8 s+N) are periodic functions with unit module.2 2

If

the modules of cos(8 s+N) and sin(8 s+N) are increasing functions2 2

of t0ú . Expression+

can be set equal to the sum of

and

sin2R[l (%"l2Tcos(8

2s%NE)]

[1%2R%b21Tsin(8

2s%NE)]2

2"l1sin(n

R)

cosh(t)[1%nR%b

21T(sin8

2tcosNE%sinNEcos8

2t)]2

limnt6±4

cosh(t) ' 4

2"l1sinn

R

cosht(cosn%isinn)61%nR%b

21Tsin(8

2t(cosn%isinn)%

1i

sin nt

cos2nt

2"l1

cos nt1%

1ien

1&sin nt

1%sin nt

%b21T sin (8

limt64

sin(i82t%NE)' lim

t64[i cos NE sinh t % sin

F(s) '2"

l1sin 2R

cosh (s) [1%2R%b21T sin (8

2s%NE)]2

lims64

s ne &s ' lims64

1

e s

s n

'0

s'i(B

2%2kB)

s'i(3B

2%2kB)

46

(5.C.45)

(5.C.46)

(5.C.47)

(5.C.48)

(5.C.49)

(5.C.50)

(5.C.51)

(5.C.52)

(5.C.53)

(5.C.54)

For n=o or n=B 5.C.44 becomes

Since

5.C.46 tends uniformly to 0 for t64. For 0<n <B, n=/(B/2), 5.C.44t

becomes

and the foregoing analysis of cosh(s) and sin(8 s+N ) shows that2E

5.C.48 also tends uniformly to 0 for t64. For n =(B/2), , = i andn =/(B/2), n =/(3B/2) we havet t

Since

5.C.49 also tends uniformly to 0 for t64.

Set

then lim sF(s) = 0, becauset64

for any n>0. Moreover, since we have set k =0, F(s) is analyticQ

everywhere in the upper half-plane, except at the poles

and

The two poles of order three in the upper half-plane are uniquelydetermined once k is determined. The corresponding values of theresidues of F(s) are invariant for all values of k0ø ; thus, to+

evaluate the residues, k can safely be set equal to 0. From the

limr64 m

r

&rF(t)dt'2Bij a

12

d 2

dn2t

1i

sinnt

cos3nt

2"l1(n

t&n

0jt)

[1%1iln

1&sinnt1%sinn

t

%b21Tsin (i

j'1,2

F(s) '[l (%2"

l1cosh&1(s)%"

l2T cos(8

2s%NE)]2

cosh(s)[1%2R%b21Tsin(8

2s%NE)]2

lims64

s F(s) ' 0

6l (%2"l1

cos&1 nt%"

l2T cos [i8

2(n

t%2kB)%NE]>2

cos nt6[1% 1

iln

1&sinnt

1%sinnt

%b21T sin[i8

2(n

t%2kB)%NE]

47

See Smirnov, 1982, 3.II, p. 227.28

(5.C.55)

(5.C.56)

(5.C.57)

(5.C.58)

(5.C.59)

theory of residues we know that 28

where Ea is the sum of the residues of F(s) at the poles in theupper half-plane. The residues can be evaluated as the limits for n 6 B/2 and n 6 3B/2 oft t

where n = B/2 and n = 3B/2. Since these limits are respectively1 2t t

equal to +4 and -4, their sum is equal to 0. Hence, the principalvalue of the integral of 5.C.44 is 0.

In a similar way it can be proved that the principal value of theintegrals of 5.C.45, 5.C.2, 5.C.3, 5.C.4 and 5.C.6 are all equal to0. In order to evaluate the integral of expression 5.C.5 by themethod of residues, let us again substitute the complex variable sfor the real variable t. We obtain

In this case also:

everywhere in the upper half-plane except at n=B/2, where 5.C.57is equal to

Let us consider the following function:

F1(n, t) ' F(n, t*0 # n <

B

2; t 0 ú

%) % F(

B

2,t/

&F(B

2, t*t 0 ú

%) % F(n, t* B

2< n # B; t 0 ú

%)

lims64

s F1(s) ' 0

limr64 mCr F1(s) ds ' 0

F(0, t* t 0 ú%) % F(

B

2,0)&F(

B

2, 0) %

% F(B, t*t 0 ú%) ' F(t*t 0 ú) & F(

B

2,0)

limr64 m

r

&rF (t*t 0 ú)dt&F B

2,0 % mCr F1(s)ds '0

limr64 m

r

&rF (t)dt&F

B

2,0 '0

limr64 m

r

&rF (t)dt'F

B

2,0 '

[l (%2"l1%"

l2T cos NE]

[1%b21

T sin NE]2

9(NE) ' ± 4"[l (%2"

l1%"

l2T cos NE]2

[1%b21 T sin NE]2%

% b12

[l (%2"l1%"

l2T cos NE]2

[1%T sin NE]2NE ' arccos &

l (%2"l1

T "l2

48

See Smirnov, 1989, 3.II, pp.223-229.29

(5.C.60)

(5.C.61)

(5.C.62)

(5.C.63)

(5.C.64)

(5.C.65)

(5.C.66)

(5.C.67)(5.C.68)

F (s) is continuous and has continuous first derivatives for all1

values of n0(0,B), except at the poles; moreover

uniformly in the same domain. Hence29

where C is the semicircle determined by 0#n#B and r=#s#. On ther

real axis F (s) may be written as 1

Since the residues of F(s) in the upper half-plane cancel out, wemust have

Hence

and therefore

Thus, Melnikov's function is reduced to

which has a simple zero for

49

Goodwin, 1990, p. 1.30

6. Conclusions.

"Why is economics like the weather? Because both are highlyirregular if not chaotic, thus making prediction unreliable or evenimpossible." This statement is probably an effective synthesis of30

Goodwin's views on the dynamics of modern economies. These views,or perhaps this "philosophy", appear to the interested scholar likethe leading thread of Goodwin's lifelong work on dynamics.

In this paper it has been shown that this "philosophy" is notincompatible with the usual economic axioms, which state thatagents aim at maximizing either profits, or utility, or both. Ithas also been shown that symplectic transformations of coordinatesare an essential tool of analysis when a Hamiltonian adjustmentprocess is assumed, as is the case of Goodwin's 1953 "Walrasian"model. Finally, it has been shown that, if real wages are notentirely flexible and returns to scale are not constant, theadjusment process can generate very complex, and in extreme caseschaotic, motions.

50

References

Abraham, R. and Marsden, J.E. (1985) Foundations of Mechanics,Addison Wesley.

Dieudonné, J. (1969) Foundations of Modern Analysis, AcademicPress.

Eisele, J.A. and Mason, R.M. (1970) Applied Matrix and TensorAnalysis, Wiley Interscience.

Goodwin, R.M. (1953) Static and Dynamic Linear General EquilibriumModels, reprinted in Essays in Linear Economic Structures,Macmillan, 1983, pp. 75-120.

-- (1989) Swinging Along the Autostrada: Cyclical FluctuationsAlong the von Neumann Ray, in John von Neumann and ModernEconomics, Oxford, pp. 115-140.

-- (1990) Chaotic Economic Dynamics, Oxford.

Goodwin, R.M. and Punzo, L.F. (1987) The Dynamics of a CapitalistEconomy, Polity Press.

Guckenheimer, J. and Holmes, P. (1990) Non linear Oscillations,Dynamical Systems and Bifurcations of Vector Fields, Springer.

Hadely, G. and Kemp, M.C. (1971) Variational Methods in Economics,North-Holland.

Hirsch, M.W. and Smale, S. (1974) Differential Equations, DynamicalSystems and Linear Algebra, Academic Press.

Smirnov, V.I. (1982) Corso di matematica superiore, vol. 3.II,Editori Riuniti. Italian translation from the original Kurs vyssejmatematiki, Nauka, Moscow.

51

Sordi, S. (1990) Teorie del ciclo economico, CLUEB.

Varian, H.R. (1984) Microeconomic Analysis, Norton.

A'.3683 .0492

.0118 .56588'

.6347 0

0 &.4313X'

1. &.2457

.0589 1.s'

.65

.35

"'aX' .1303 .1080

52

TABLE 1Parameters of the two-sector model.

Initial ConditionsVariables Models Navajo Sioux Paiute B .12 -.1 .12B -.16 .1 -.16x -.15 - -.15x .09 - .09l' - .4402 .4402l' - -.001 -B - .02 -B - -.03 -

Simultaneous Adjustment of Quantities and Prices:an Example of Hamiltonian Dynamics.

Paola AntonelloAbstract

In a well-known 1953 essay, "Static and Dynamic Linear GeneralEquilibrium Models", Richard Goodwin analyses the dynamicadjustment of quantities and prices to long period equilibrium, ina set of n "Walrasian" markets. He treats the crossed adjustment ofprices and quantities as a linear Hamiltonian vector field. Inmore recent works Goodwin has introduced non-linear perturbationsin his multisectoral adjustment models, by assuming that realconsumption depends non-linearly on relative prices. Goodwin's useof Hamiltonian dynamics and of symplectic coordinates changes opensup a wide range of fascinating potential developments for theanalysis of adjustment processes in multisectoral systems subjectto real perturbations. It has, however, not been totally exemptedfrom objections, usually referring to the lack of microfoundationsof his macro dynamic analysis and to his use of non-Cartesiancoordinate systems in economics.

The aim of this paper is threefold: i) to investigate whetherGoodwin's behavioural hypotheses are compatible with the assumptionthat agents maximize. ii) To show that, if the dynamic process isHamiltonian, symplectic coordinates changes are essential tools ofanalysis. iii) To analyse the dynamic behaviour of Goodwin'sHamiltonian model, subject to the non-linear perturbation hesuggests in Chaotic Economic Dynamics (1990), and to point out someof the developments this approach may lead to.

Goodwin's 1953 cross-dual model is at first derived from an optimalcontrol model, the objective functional of which is the aggregateexcess profits function, i.e. the sum of sectoral excess profits.This model generates simple harmonic motions. In the second part ofthe paper, the following assumptions are introduced: a) the nominalwage is fixed and entirely spent for consumption; b) consumerutility functions are Cobb-Douglas; c) consumers aim to maximizetheir current utility. Under these hypotheses, Goodwin's 1990consumer demand functions are easily derived. Since prices and thereal wage are flexible, the closed orbits solutions disappear andlong period equilibrium becomes asymptotically stable.

In the last section of the paper, the analysis is restricted to atwo sector model. Goodwin's basis hypotheses are slightly modified.It is assumed that the unperturbed two degree of freedomHamiltonian is non-linear and has a homoclinic orbit. It is furtherassumed that the perturbation is itself periodic. The economicmeaning of these assumptions is that returns to scale are notconstant and that real wages are partially rigid. By applyingMelnikov's method, it is then proved that the perturbed Hamiltoniansystem has transverse homoclinic orbits and, therefore, Smale

horseshoes. Hence, under the assumptions of variable returns toscale and of real wage rigidity the model can generate chaotictransients or pure chaotic motions.

Author:Paola AntonelloLUISS Guido CarliIstituto di Studi EconomiciViale Pola, 12I-00198 ROME Tel.: +39 +6 85225739Fax : +39 +6 8845252E-Mail pantonel@lgc-net.luiss.it