Advances in Mathematical Economics by Toru Maruyama

Author:Toru Maruyama
Language: eng
Format: epub
ISBN: 9789811507137
Publisher: Springer Singapore

(15)

Since the RSS model (ρ, a, d) has the optimal policy function h = H, (15) implies

Since H[1∕(1 − d)] = 1, we have h 3(1) = h(h 2(1)) = 1, and we obtain the period-three cycle

Next, we turn to the necessity result

Proposition 7

Let (ρ, a, d) be the parameters of a two-sector RSS model such that there is an optimal policy function h which generates a period-three cycle from some initial stock. Then a < (1∕3).

Proof

Denote the optimal policy function by h, and the period-three cycle stocks by α, β, γ. Without loss of generality we may suppose that α < β < γ. There are then two possibilities to consider: (1) β = h(α), (2) γ = h(α).

In case (i), we must have α ∈ A, and since β > α. Consequently, β = (1∕a) − ξα, and γ ≠ h(α). Thus, we must have γ = h(β), and since γ > β, we must have β ∈ A. But, since β = (1∕a) − ξα with α ∈ A, we must have β ∈ B ∪ C, a contradiction. Thus, case (i) cannot occur.

Thus case (ii) must occur. In this case, since γ > α, we must have α ∈ A, and Consequently, γ = (1∕a) − ξα, and β ≠ h(α). Thus, we must have β = h(γ); it also follows that we must have α = h(β). Since β < γ, we must have γ ∈ B ∪ C; similarly, since α < β, we must have β ∈ B ∪ C. Also, note that β cannot be or k, and similarly γ cannot be or k.

We claim now that γ > k. For if γ ≤ k, then, we must have But, then, and so β = h(γ) implies by (8) that Since we must have β ∈ A, a contradiction. Thus, the claim that γ > k is established. But, then, by (8), we can infer that β = (1 − d)γ.

We claim, next, that Since β ∈ B ∪ C, and β cannot be or k, we must have β > k if the claim is false. But, then, by (8), we can infer that a contradiction, since α ∈ A. Thus, our claim that is established.

Since and α = h(β), we can infer from (8) that α ≥ (1 − d). To summarize our findings so far, we have:

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