Learning and Endogenous Business Cycles in a Standard Growth Model

z3 are either complex conjugate with modulus greater than 1, or real with identical sign either in the
interval (-∞, -1) or (1, ∞). Since z₁ = 1 — η < 1, then (k, w, r) is a saddle. At ηH = -(1 — G_i)∕G_i,
eigenvalues z₂, z₃ are complex conjugate with modulus equals to 1. For any η ∈ (bη_H, 1], eigenvalues z₂,
z₃ are complex conjugate with modulus less than 1, or real with identical sign in the interval (-1, 1) and
since z₁ = 1 — η < 1, then (k, w, r) is a sink, (see figure A6).

ii) If 1+G1 <a<G1 + χ, then M1 lies on the horizontal axis anywhere between the points (1, 0)
and (χ, 0) with χ = -(1 + 3G1)/(1 - G1). Therefore the line segment M0M1 lies above the straight line
corresponding to the equation 1+ε + κ =0, crosses both the horizontal line associated with equation
κ =1at η = η_H and the straight line associated with equation 1 - ε + κ =0at η = bη_F , then ends on the
horizontal axis. For any η ∈ (0, bη_H), eigenvalues z2 , z3 are either complex conjugate with modulus greater
than 1, or real with identical sign either in the interval (-∞, -1) or (1, ∞). Since z1 =1-η<1,then(k,
w, r) is a saddle. At Ьн = —(1 — Gι)∕Gι, eigenvalues z₂, z3 are complex conjugate with modulus equals
to 1. For any η ∈ (bη_H, bη_F), eigenvalues z₂, z₃ are complex conjugate with modulus less than 1, or real in
the interval (—1,1) and since zi = 1 — η < 1, then (k, w, r) is a sink. At bF = 2(1 + Gι)∕(1 + a + Gι),
eigenvalues z2 , z3 are real with identical sign 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, 0), and the other
in the interval ^∞, —1). Since z₁ = 1 — η < 1, then (k, w, r) is a saddle (see figure A9).

iii) If a>G1 + χ, then M1 lies on the horizontal axis anywhere to the left of point (χ, 0) with
χ = —(1 + 3G1)∕(1 — G1). Therefore the line segment M0M1 lies above the straight line corresponding
to the equation 1+ε + κ =0, crosses to the left of the point (2, 1) both the straight line associated with
equation 1 — ε + κ =0at η = ηb_F and the horizontal line associated with equation κ =1at η = bη_H where
ηb_F < ηb_H . For any η ∈ (0, ηb_F), eigenvalues z2 , z3 are either complex conjugate with modulus greater than
1, or real with identical sign either in the intervals ^∞, —1) or (1, ∞). Since z₁ = 1 — η < 1, then (k,
w, r) is a saddle. At bF = 2(1 + Gι)∕(1 + a + Gι), eigenvalues z₂, z3 are real with one equals to —1,
and the other in the interval ^∞, —1). For any η ∈ (b_F,Ьн), eigenvalues z₂, z3 are real and outside the
unit circle either in the intervals ^∞, —1) or (1, ∞). Since zi = 1 — η < 1, then (k, w, r) is a saddle.
At Ьн = —(1 — G1)∕G1, eigenvalues z2, z3 are real with one equals to —1, and the other in the interval
^∞, —1). For any η ∈ (ηH, 1], eigenvalues z₂, z₃ are real with identical sign: one in the interval (—1,1)
and the other one in the intervals ^∞, —1) or (1, ∞). Since zi = 1 — η < 1, then (k, w, r) is a saddle.
(see figure A10). ■

Proposition 6 Under constant gain learning with h =0and G1 ≥ 1+a, then the non-trivial steady
state is locally unstable for any b ∈ (0, 1].

Proof. If G1 ≥ 1+a,thenM0 lies either anywhere to the right of the point (—2 — a, 1+a) or at the
point (—2 — a, 1+a) on the line segment associated with the equation 1+ε + κ =0,andM1 lies either
anywhere to the right of the point (—1, 0) or at the point (—1, 0) on the horizontal axis. For Gi > 1 + a,
the line segment M0M1 lies below the straight line corresponding to the equation 1+ε+ κ =0. Therefore
eigenvalues z2 , z3 are real with identical sign: one in the interval (0, 1) and the other in the interval
(1, ∞). Since zi = 1 — η < 1, then (k, w, r) is a saddle. For Gi = 1 + a, the line segment MoMi coincide
with the straight line corresponding to the equation 1+ε + κ =0. Therefore eigenvalues z2, z3 are real
with one equals to 1, and the other in the interval (1, ∞). Since zɪ = 1 — η < 1, then (k, w, r) is a saddle.
(see figure A11). ■

Under the different restrictions imposed on the values of Gi and a when h = 0, figure A12 summarizes
in the Gi — a plane the local qualitative properties of the model around the non-trivial steady state.

Thecase where h =1

In the situation in which one-period ahead constant gain adaptive input price forecasts do not 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(b)z² + ω(b)z — Det(b) = 0 whose coefficients are the trace: Tr(b) =
2(1 — η) + G₁ > 0, the sum of the principal minors of order two: ω(η) = (1 — η) [(1 — η) + 2G_i] + ηa > 0,
and the determinant: Det(η) = (1 — η) [ηa +(1 — η)G_i] > 0 of the Jacobian matrix with dTr(η)∕dη < 0,
dω(b)∕db _>^< 0, dDet(b)∕db ^<_>0. Since one eigenvalue zi =1— b, the other two: z2, z3 are solution to the

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