Learning and Endogenous Business Cycles in a Standard Growth Model

The case where h =0

In the situation in which one-period ahead constant gain adaptive input price forecasts include the
beginning of the period announcements made by the ‘Walrasian Auctioneer’, the eigenvalues denoted
by zi for i =1, 2, 3 of the corresponding Jacobian matrix are the roots of the following characteristic
polynomial: Q(z) = z³ - T r(η)z² + ω(η)z - Det(η) = 0 whose coefficients are the trace: Tr(η) =
2(1-η)+Gι-ηa< 0, the sum of the principal minors of order two: ω(η) = (1—η) [1 + 2G_i — η(1 + a)] > 0,
and the determinant: Det(η) = Gi (1 — η)2 > 0 of the Jacobian matrix with dTr(η)∕dη < 0, dω(η)∕dη< 0,
dDet(η)∕dη < 0. Since one eigenvalue corresponds to the weight affected last period’s predictions:
z1 =1- η, the other two: z2 , z3 are solution to the 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) — G₁ + ηα>0, κ(η) = G₁(1 — η) > 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) is located on the straight line associated with equation 1+ε + κ =0and satisfying
Q(1) = 0.Forη = 1, the point M1 =(a — G1, 0) lies on the horizontal axis.

Proposition 4 Under constant gain learning with h = 0 and 0 < Gi ≤ 1, the following two configu-
rationsoccur;i)ifa<1+G1, then the non-trivial steady state is local ly stable for any η ∈ (0, 1]; ii)
if a>1+Gi, then the non-trivial steady state is locally stable for any η ∈ (0, bη_F), undergoes a Flip
bifurcation at ηb_F = 2(1 + Gi)/(1 + a + Gi), and becomes local ly unstable for any η ∈ (ηb_F , 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+G_i , then M_i lies on the horizontal axis anywhere between the points (1, 0) and (—1, 0).
Therefore the line segment M₀M_i lies above the straight line corresponding to the equation 1+ε+ κ =0,
above the straight line corresponding to the equation 1 — ε + κ =0, and ends on the horizontal axis. For
any η ∈ (0, 1], eigenvalues z2, z3 are either complex conjugate with modulus less than 1, or real in the
interval (—1, 1) with identical sign. Since z₁ = 1 — η < 1, then (k, w, r) is a sink (see figure A1)

ii) If a> 1+Gi , then Mi lies on the horizontal axis anywhere to the left of the point (1, 0). Therefore
the line segment M₀M_i lies above the straight line corresponding to the equation 1+ε + κ = 0, crosses
the straight line corresponding to the equation 1 — ε + κ =0 at η = ηb_F , and ends up on the horizontal
axis. For any η ∈ (0, bη_F),eigenvaluesz2, z3 are either complex conjugate with modulus less than 1,or
real with the same sign in the interval (—1, 1) and since zi = 1 — η < 1, then (k, w, r) is a sink. At
ηb_F = 2(1 + Gi)/(1 + a + Gi), eigenvalues z2, z3 are real with one equals to —1, and the other in the
interval (—1, 0). For any η ∈ (bη_F , 1], eigenvalues z₂, z₃ are real with identical sign: one in the interval
(—1, 1), and the other in the interval ^∞, —1). Since zi = 1 — η < 1, then (k, w, r) is a saddle (see
figure A4) ■

Proposition 5 Under constant gain learning with h =0and 1 < Gi < 1+a,thefollowingthree
configurations occur, i) if a < 1 + Gi, then the non-trivial steady state is locally unstable for any η ∈
(0, ηb_H), undergoes a Hopf-Neimark bifurcation at ηb_H = —(1 — Gi)/Gi and becomes locally stable for any
η ∈ (ηb_H, 1];ii)If1+Gi <a<Gi + χ with χ = —(1 + 3Gi)/(1 — Gi), then the non-trivial steady state
is locally unstable for any η ∈ (0, bη_H), undergoes a Hopf-Neimark bifurcation at bη_H = —(1 — Gi)/Gi;
becomes locally stable for any η ∈ ('ηH, bF), undergoes a Flip bifurcation at b_F = 2(1 + Gi)∕(1 + a + Gi),
and becomes locally unstable for any η ∈ (rj_F, 1]; iii) if Gi + χ < a with χ = —(1 + 3Gi)∕(1 — Gi), then
the non-trivial steady state is locally unstable for any η ∈ (0, 1].

Proof. If 1< G_i < 1+a,thenM₀ lies either anywhere between the points (—2, 1) and (—2 — a, 1+a)
on the line segment associated with the equation 1+ε + κ =0.

i) If a < 1 + Gi, then Mi lies on the horizontal axis anywhere between the points (—1, 0) and (1, 0).
Therefore the line segment M0Mi lies above the straight line corresponding to the equation 1+ε+ κ =0,
crosses the horizontal line associated with equation κ =1at η = ηb_- , and ends on the horizontal axis
above the straight line associated with equation 1 — ε + κ =0 . For any η ∈ (0, ηb_- ), eigenvalues z₂,

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