Notes on an Endogenous Growth Model with two Capital Stocks II: The Stochastic Case

That is, π ∈ Π(s0) if and only if condition (50) holds. In addition, we know from (38)
and (39) that for all st and any π ∈ Π(st) for all t ∈ N

lim β^tE₀∣^ln π1_-1[A^t-1, ha-¹]^∣ ≤ 0 and lim βtE₀ ∣"ln π2_-1[A^t~¹, h^t_a~¹]l ≤ 0       (51)

t→∞                           t→∞

must hold. Now suppose that π ∈ ΠΠ(s₀), i.e. (44) fails. It follows from the inequality in
(46) that

∞

∑ βt (ln At + (1 - α + γ)ln ∏t¹-i [A^t-1 ] + α ln ∏2,ι[A^t-1f)
t=0

Since (44) fails, the conditions in (45) imply that this series must diverge to minus infinity:
u (π, h₀,k₀,A₀) = -∞; in this case π¹* and π²* dominate π¹ and π². Thus condition
(b) is satisfied, and Theorem 3 applies. That is, v is indeed the value function and the
policy rules are given by (48) and (49).

References

[1] Benhabib, Jess and Roberto Perli (1994): “Uniqueness and Indeterminacy: On the
Dynamics of Endogenous Growth”, Journal of Economic Theory 63, 113-142.

[2] Bethmann, Dirk (2002): “Notes on an Endogenous Growth Model with two Capital
Stocks I: The Deterministic Case”, SFB Discussionpaper 65, Humboldt-Universitat
zu Berlin.

[3] Bethmann, Dirk and Markus Reiβ (2003): “Transitional Dynamics in the Uzawa-
Lucas Model of Endogenous Growth”, SFB Discussionpaper 17, Humboldt-
Universitat zu Berlin.

[4] Chamley, Christophe (1993): “Externalities And Dynamics In Models Of ‘Learning
Or Doing’ ”, International Economic Review 34(3), 583-609.

[5] Lucas, Robert E. (1978): “Asset prices in an exchange economy”, Econometrica 46,
1429-1445.

[6] Lucas, Robert E. (1988): “On The Mechanics Of Economic Development”, Journal
of Monetary Economics 22(1), 3-42.

[7] McCallum, Bennett T. (1989): “Real Business Cycles”, in: Robert Barro (ed), Mod-
ern Business Cycle Theory, Cambridge: Harvard University Press, 16-50.

[8] Mulligan, Casey B. and Xavier Sala-i-Martin (1993): “Transitional Dynamics in Two-
Sector Models of Endogenous Growth”, Quarterly Journal of Economics 108(3), 739-
773.

[9] Stokey, Nancy L. and Robert E. Lucas with Edward C. Prescott (1989): Recursive
Methods in Economic Dynamics, Cambridge (Massachusetts) and London (England):
Harvard University Press.

[10] Uhlig, Harald and Noriyuki Yanagawa (1996): “Increasing the capital income tax
may lead to faster growth”, European Economic Review 40, 1521-1540.

[11] Uzawa, Hirofumi (1965): “Optimum Technical Change In An Aggregate Model Of
Economic Growth”, International Economic Review 6(1), 18-31.

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