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 z
i for i =1, 2, 3 of the corresponding Jacobian matrix are the roots of the following characteristic
polynomi
al: 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) = (1η) [1 + 2Gi 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:
z
1 =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 segm
ent M0M1 in the (ε, κ) plane where Mη = (ε(η), κ(η)) and η (0, 1]. For η = 0, the point
M
0 =(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)if
a<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, bηF), undergoes a Flip
bifurcation at
ηbF = 2(1 + Gi)/(1 + a + Gi), and becomes local ly unstable for any η (ηbF , 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+Gi , then Mi lies on the horizontal axis anywhere between the points (1, 0) and (1, 0).
Therefore the line segment M
0Mi 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 M
0Mi 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 = ηbF , and ends up on the horizontal
axis. For any η
(0, bηF),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
η
bF = 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 (bηF , 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, ηbH), undergoes a Hopf-Neimark bifurcation at ηbH = (1 Gi)/Gi and becomes locally stable for any
η (ηbH, 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, bηH), undergoes a Hopf-Neimark bifurcation at bηH = (1 Gi)/Gi;
becomes locally stable for any
η ('ηH, bF), undergoes a Flip bifurcation at bF = 2(1 + Gi)(1 + a + Gi),
and becomes locally unstable for any η (rjF, 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< Gi 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 M
0Mi 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,



More intriguing information

1. The name is absent
2. DISCUSSION: ASSESSING STRUCTURAL CHANGE IN THE DEMAND FOR FOOD COMMODITIES
3. Analyse des verbraucherorientierten Qualitätsurteils mittels assoziativer Verfahren am Beispiel von Schweinefleisch und Kartoffeln
4. ¿Por qué se privatizan servicios en los municipios (pequeños)? Evidencia empírica sobre residuos sólidos y agua.
5. Financial Markets and International Risk Sharing
6. PEER-REVIEWED FINAL EDITED VERSION OF ARTICLE PRIOR TO PUBLICATION
7. Estimating the Economic Value of Specific Characteristics Associated with Angus Bulls Sold at Auction
8. Quality practices, priorities and performance: an international study
9. The name is absent
10. Conservation Payments, Liquidity Constraints and Off-Farm Labor: Impact of the Grain for Green Program on Rural Households in China
11. Towards a framework for critical citizenship education
12. Wirtschaftslage und Reformprozesse in Estland, Lettland, und Litauen: Bericht 2001
13. The name is absent
14. The name is absent
15. Global Excess Liquidity and House Prices - A VAR Analysis for OECD Countries
16. Trade Liberalization, Firm Performance and Labour Market Outcomes in the Developing World: What Can We Learn from Micro-LevelData?
17. The Effects of Reforming the Chinese Dual-Track Price System
18. The name is absent
19. Stillbirth in a Tertiary Care Referral Hospital in North Bengal - A Review of Causes, Risk Factors and Prevention Strategies
20. BILL 187 - THE AGRICULTURAL EMPLOYEES PROTECTION ACT: A SPECIAL REPORT