Learning and Endogenous Business Cycles in a Standard Growth Model

characteristic polynomial rewitten as follows: Q(z)=(z- 1+η) z² + ε(η)z + κ(η) = 0 whose coefficients
are linear with respect to η and are given by ε(η) = -(1 — η) — Gi < 0, κ(η) = ηa +(1 — η)Gι > 0 with
dε(η)∕dη > 0, dκ(η)∕dη< 0. Therefore Q(1) = 0 and Q(—1) = 0 correspond to equations 1 + ε + κ = 0,
1 — ε + κ = 0 respectively. In the rest of this section, I discuss the qualitatively local stability of the
system (23), (24), (25) by locating the line segment M₀M₁ in the (ε, κ) plane where M_η =(ε(η),κ(η))
and η ∈ (0,1]. For η = 0, the point M₀ = (—1 — G₁, G₁) belongs to the straight line associated with
equation 1 + ε + κ = 0 and satisfying Q(1) = 0. For η = 1, the point Mi = (—Gi, a) is located anywhere
above the horizontal axis and to the left of the vertical axis.

Proposition 7 Under constant gain learning with h =1and 0 < Gi ≤ 1, the fol lowing two config-
urations occur; i) if a<1, then the non-trivial steady state is locally stable for any η ∈ (0, 1]; ii) if
a> 1, then the non-trivial steady state is locally stable for any η ∈ (0, eη_H), undergoes a Hopf-Neimark
bifurcation at ен = (1 — Gi)/(a — Gi), and becomes locally unstable for any η ∈ (г/н, 1].

Proof. If0 < Gi ≤ 1, then M0 lies either anywhere between the points (—1, 0) and (—2, 1) or corresponds
to the point (—2, 1) on the line segment associated with equation 1+ε + κ =0.

i) If a< 1, then Mi lies below the horizontal line associated with κ =1east from M0 . For any
г ∈ (0, 1], eigenvalues z2, z3 are either complex conjugate with modulus less than 1, or real with identical
sign in the interval (—1,1). Since z₁ = 1 — η < 1, then (k, w, r) is a sink. (see figure A2)

ii) If a> 1, then Mi lies above the horizontal line associated with κ = 1 northeast from M0 . Therefore
the line segment M0Mi lies above the straight line corresponding to the equation 1+ε+κ = 0, and crosses
the horizontal line κ = 1 at η = ηH. For any η ∈ (0,eH), eigenvalues z₂, z₃ are either complex conjugate
with modulus less than 1, or real with identical sign in the interval (—1, 1). Since z_i =1— г<1, then,
(k, w, r) is a sink. At ен = (1 — G₁)∕(a — Gi), eigenvalues z₂, z3 are complex conjugate with modulus
equals to 1. For any η ∈ (ен, 1], eigenvalues z₂, z3 are complex conjugate with modulus greater than 1.
Since zi = 1 — η < 1, then, (k, w, r) is a saddle. (see figure A5) ■

Proposition 8 If 0 < Gi ≤ 1, local stability under constant gain learning when h = 1 implies local
stability when h =0. However the reverse is not true.

Proof. If 0 < Gi ≤ 1, then local stability under constant gain learning when h = 1 requires according
to proposition 7 that a < 1 implying that a < 1 + G_i which is according to proposition 4 i) a necessary
condition for local stability under constant gain learning when h =0. However the reverse is not true
since condition a< 1+ Gi may imply a> 1 which is according to proposition 7 a necessary condition
for local instability under constant gain learning when h = 1. (see figures A2, A3) ■

Proposition 9 If 0 < Gi ≤ 1, and a > 1 + Gi, then the set of constant gain parameters η associated
with local stability is larger for h =0 than for h =1.

Proof. If 0 < Gi ≤ 1, and a > 1 + Gi, local stability under constant gain learning requires η ∈ (0,bF)
for h = 0, and η ∈ (0,ен) for h = 1 where bF = 2(1 + Gi)∕(1 + a + Gi) and ен = (1 — Gi)∕(a — Gi).
Under above conditions on parameters Gi, a, we can easily show that bF > ен. (see figure A4) ■

Proposition 10 Under constant gain learning with h = 1 and 1 < Gi < 1 + a, the following two
configurations occur; i) if a < 1, then the non-trivial steady state is locally unstable for any е ∈ (0, ен),
undergoes a Hopf-Neimark bifurcation at ен = (1 — Gi)∕(a — Gi), and becomes locally stable for any
е ∈ (ен, 1]; ii) if a > 1, then the non-trivial steady state is locally unstable stable for any е ∈ (0,1].

Proof. If 1 < Gi < 1 + a, then Mo lies anywhere between the points (—2,1) and (—2 — a, 1 + a) on the
line segment associated with equation 1+ε + κ =0.

i) If a< 1, then Mi lies south east from point M0 . Therefore the line segment M0 Mi lies above the
straight line corresponding to the equation 1 + ε + κ = 0, and crosses the horizontal line κ = 1 at е = e _н.
For any е ∈ (0, е_н), eigenvalues z₂, z₃ are either complex conjugate with modulus greater than 1, or real
with identical sign in the intervals (~∞, —1) or (1, ∞). Since z₁ = 1 — е < 1, then, (k, w, r) is a saddle.
At ен = (1 — Gi)∕(a — Gi), eigenvalues z₂, z3 are complex conjugate with modulus equals to 1. For
any е ∈ (ен, 1], eigenvalues z₂, z3 are either complex conjugate with modulus less than 1, or real with
identical sign in the interval (—1, 1). Since z_i = 1 — е < 1, then, (k, w, r) is a sink. (see figure A7)

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