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
(η)∕dη > 0, (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 segmen
t M0M1 in the (ε, κ) plane where Mη =(ε(η)(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, eηH), 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
w
ith modulus less than 1, or real with identical sign in the interval (1, 1). Since zi =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 + Gi which is according to proposition 4 i) a necessary
condition for local stabil
ity 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 (0н) 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, ен), 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 zi = 1 — е < 1, then, (k, w, r) is a sink. (see figure A7)

10



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