3.Stability of Equilibria

This section analyses the stability of equilibria determined by the system of equations（2.14）-（2.16）.Equilibria are solutions of

From（3.3）, one gets k=0 or

Remark that k=0 is not an asymptotically stable solution, because, in view of assumption（2.5）, y（k）is sufficiently high in the vicinity of k=0, which impliesfrom（2.16）.Hence, k=0 cannot be an asymptotically stable solution.Therefore,（3.4）gives the asymptotically stable solution that satisfies（3.3）.

Rewrite（3.1）and（3.2）by using（3.4）, and one obtains：

Meanwhile, from（2.4）, and w＞0, one has

Hence,（3.5）implies

The condition of full employment（2.10）is rewritten, in view of（2.4）and（3.8）, as

Therefore, in the vicinity of the steady state growth path，

from（3.8）and（3.9）.

The point that g≥n is equivalent to w≤FL, mentioned in Section 2, is clear from the above mathematical manipulation.

Now, in view of（3.6）, one sees that there are 2 equilibria:the one with t=0 and the other with s cr=n.When t=0, one obtains k satisfying（3.4）, which is reduced to swy（k）=n, and then, by substituting k in（3.8）, with the above obtained k, one gets r.A set of pairs of k and r thus obtained gives an equilibrium path, D-path.When s cr=n, one getsk from（3.8）, and thent is obtained from（3.4）.Pairs of k and t thus obtained constitute equilibria, P-path. In the case of D-path, r depends solely on sw, whilst in the case of P-path, r depends on sc and n.

If both of two paths can be asymptotically stable depending on the values of the parameters, the duality theorem is true. If only P-path is asymptotically stable, then Pasinetti's theorem is true.Then, as a first step, we assume the existence of equilibria and check the conditions for asymptotic stability.

By linearizing（2.14）-（2.16）in the vicinity of equilibrium, one gets the following Jacobi matrix：

where，

In the outset, the stability of D-path will be examined.If t=0, then one has

The characteristic equation|λE-J D|=0 is reduced to

{λ-（r-y）（βk-αgr）}{λ-（scr-swy（k））}{λ-sw（fk-y）}=0

One obtains 3 real roots：

λ=（r-y）（βk-αgr）, scr-swy（k）, sw（fk-y）

In order for D-path to be locally stable, all roots should be negative.In view of（2.5）and（3.7）, one obtains equivalent conditions：

Both must be met.Since t=0, from（3.4）, one has

Hence,（3.11）is equivalent to

In the next place, we consider the stability conditions for P-path.In view of（3.4）, the Jacobi matrix of P-path is expressed as：

and its characteristic equation|λE-J P|=0 is reduced to

λ3+γ2λ2+γ1λ+γ0=0

where

with

The Routh-Hurwitz theorem tells that necessary conditions for the local stability of P-path are，

γ2＞0, γ1＞0, γ0＞0, γ2γ1＞γ0

From the angle of the stability conditions of the two equilibrium paths, one obtains the following 3 propositions that clarify the relations between Pasinetti's theorem and the duality theorem.

PROPOSITION 3.1.Suppose If D-path is asymptotically stable, it is necessary that r→∞ and gr→0 in equilibrium. This conclusion is true irrespective of values of sc, sw and n.

Proof.From（3.10）,（3.11）and（3.13）, for the stability of D-path，

should hold.From（2.13）, however, if 0＜r≤r0, then g（r, k）=0＜s cr, so that（3.18）is not true.If（3.18）is true, then one must have r＞r0, irrespective of the value of sc.In view of（3.7）, it follows that

Since y（k）is a decreasing function, the set of k satisfying（3.19）is bounded from the above. denote the supremum, and, in view of the boundedness of k, one has

from（3.12）.

If , gr→0 is necessary.From（2.13）, if r is finite, it is necessary that gr＞0, Hence, gr→0 implies r→∞.

Equilibrium depends on parameters like rates of savings, but the above Proposition 3.1 states the restriction on the profit rate of equilibrium.In order for the rate of profit to be infinite, k→0 should hold from（3.7）.Then, in view of（3.18）, sw→0 and sc→0 should follow.Conversely, however, if sc→0, then it is not certain that D-path is stable even with β→0.This is because in（3.20）- and gr do not depend on parameters sw, sc and n.

One may introduce a new setting on the investment function, for example, as, for all r＞r0

gr（r, k）＞ε＞0

Given this assumption, and the least minimum investment motivation of animal spirits is guaranteed, in so far as the rate of profit is positive.That is, investment will never saturate within the scope of analysis; since capitalists save a fixed part of the profit income, it is hard to imagine that increases in profit will not result in a greater amount of investment.This assumption keeps the Keynesian nature of the model.In this way, if we introduce a lower bound for gr, the right-hand side of（3.20）has a positive lower bound. Therefore, for the duality theorem to hold, the adjustment speed of money wages should surpass a certain level in comparison with the price adjustment speed of the product market.If this value of the lower bound depends on the parameters sc; sw and n, then one can confirm the validity of the duality theorem that asserts the convergence of equilibrium either to t=0 or to s cr=n, depending on the parameters.This is not the case, here.If the price adjustment speed in the product market is relatively high, then for any rates of savings, and for any rate of labour growth, D-path with t=0 can never be asymptotically stable.

Apart from such a new assumption, Proposition 3.1 shows that in order for the profit rate to be finite in equilibrium, the lower bound, that does not depend on parameters, exists for the ratio.This proposition is restricted only by the existence of risk premium and animal spirits. This contrasts much to Darity's model in which gr＜0 plays an important role.

As opposed to the nature of the duality theorem, Pasinetti's theorem holds even if the adjustment speed of money wages is relatively slow.The next proposition states this.

PROPOSITION 3.2.For an arbitrary positive number α, there exists a pair of（sc, sw, n）such that P-path exists and asymptotically stable, even for the caseβ→0.

Proof.（1）The first half of the proof is dedicated to the proof of the existence of equilibrium path satisfying s cr=n when（α, β）=（α*,0）, by choosing an appropriate pair（sc, sw, n）.

For an arbitrarily chosenα*, one may take（α, β）=（α*,0）.Putβ=0 in（3.1）.（3.3）, and the condition for t≠0 becomes

Firstly, consider a special case of sw=0.Then,（3.21）becomes

Take any positive k*, and consider an equation g（r, k*）=s cr with respect to r. From（2.13）, it follows that g（r0, k*）=0＜s cr.This and g rr＜0 enable us to take a value of s c in such a way that g（r, k*）=s cr has at least 2 positive solutions. Let be one such value.Let r1and（r1＜）r2be the minimum and the maximum among solutions, respectively, and it follows

Let r2 be r*.One may choose k* so as to ensure y（k*）＞r*＞fk（k*）.注2Hence, it is obvious that, at the point,（r*, k*,1）is a solution of（3.22）.

注2In fact, the existence of such a k*is guaranteed as in the following:Take k*sufficiently small, and two extremes of（y（k*）, fk（k*））can be bigger.For such a k*, one can take sufficiently small , and hence sufficiently large r*, so that r* lies inside of（y（k*）; fk（k*））.

Next, consider solutions in the vicinity of , which is near（sc, sw, n）= .It is seen that（r, k, t）=（r*, k*, t**）, where （＜1 because of y＞r）, is a solution of（3.21）.In fact, is sufficiently small, t**is sufficiently close to 1; hence, in view of , it follows that

Thus one can choose（sc, sw, n）which ensure the existence of equilibrium path satisfying s cr=n when（α, β）=（α*,0）.

（2）Now, we show the main part of the proo.f Whenβ＞0, the proof above for the caseβ=0 is still valid and the existence of the equilibrium is ascertained for sufficiently small value ofβ.

As for stability, one has to examine the property of the Jacobi matrix of the system evaluated at the solution.Whenβ=0 one get

from（3.15）～（3.17）, and（3.23）.

For a positive value ofβ,（3.23）is still true, if the value is sufficiently smal.l Then, the coefficients ofβin（3.15）～（3.17）are all positive.Thus, positiveness of coefficients of the characteristic equation is fulfilled for sufficiently smallβ.

From（3.15）

γ2=n-swfk+（r-fk）t（sc-sw）+（βk-α（gr-t（sc-sw）））δ2

then one may choose sw in such a way thatγ2＞n, as a solution for sufficiently small sw is considered here.Furthermore, one gets γ1＞βδ5, γ0=β r s c δ5=n βδ5.Hence, γ2 γ1＞n βδ5=γ0.Thus, stability is ensured for sufficiently small values ofβ, if the value is positive.

Proposition 3.2 proved the validity of Pasinetti's theorem in an extreme case with 0 in order to contrast Pasinetti's theorem with the duality theorem.In view of the proof，it is clear that one can choose s c in such a way that r satisfying g（r, k）=s c rs is finite.Recall that the D-path is not asymptotically stable for any parameters.

The last proposition demonstrates that Pasinetti's theorem is more likely to be true than the duality theorem.

PROPOSITION 3.3.Suppose that there exists an asymptotically stable D-path for arbitrary chosen（α, β）and）.Then, for（α, β）and appropriate（sc, sw, n）, there exists an asymptotically stableP-path.

Prof.Let be a solution, and from the stability condition of D-path, namely,（3.12）and（3.14）, one has

Let and it is easy to see rP＞rD.

Next, take a（r, k）fulfilling（3.8）.By differentiating（3.8）, one obtains

where the denominator is positive in view of r-fk＜0, fkk＜0 and gk＜0.At（rD, kD）, from（3.25）.Let kP denote the value of k obtained from（3.8）with r=rD, and it follows that

In fact, if（3.27）does not hold, at an r* such that rD＜r*＜rP βk=αgr on the curve（3.8）.Hence, In addition, .Therefore, k fulfilling（3.8）needs to be a local minimum at r=r*.In the light of

however, the denominator of this at r=r*is negative by assumption.This contradicts that k is a local minimum at r=r*.Therefore,（3.27）is true.One sees also from the above that for rD＜r＜rP, , Hence, kP＜kD.

One may take a sufficiently small in such a way that .If tP is set as , then tP＜1.This is because 0, , and .

Hence, at is a solution of s cr=n.

As for the stability of the solution, it is clear that the coefficients, γ2, γ1andγ0, of the characteristic equation are all positive.As in the proof of the previous proposition, for sufficiently small s w, one has γ2＞n.In addition, from（3.27）, γ1＞βδ5.Hence, γ2 γ1＞nβδ5=γ0.Therefore,（rP, kP, tP）is asymptotically stable.

From this proposition, it is seen that if D-path is stable, then there exists a stable P-path with a smaller sw.In the light of the first 2 propositions, it is obvious that the converse of this proposition cannot be true.

Remark that even ifβis close to 0, there may exist an asymptotically stable P-path;but no stable D-path at all in such a situation.