5 Conclusions

sink. Moreover,T(k^a₂)<2

D(k^a₂),therefore paths approaching this steady state display damped oscillation.

Example 7. Forσ = 1.0222, κ = 0.9682, GDP₁ = 1.861, k₁^a = 3.939, k₁^b = 0.337, RER₁= 0.548, s₁ =−0.377,whileGDP₂ = 1.889, k₂^a = 4.062, k₂^b = 0.342, RER₂ = 0.542, s₂ = −0.392.Furthermore, D(k^a₂) = 0.955 < 1,and T(k₂^a) = 1.908<1 +D(k₂^a)hold. Thus the high-output-low-RER steady state is a sink. Moreover,T(k^a₂)<2

D(k^a₂),therefore paths approaching this steady state display locally monotone dynamics.

For this set of examples, low levels of money growth rates result in an indeterminate steady state, but locally monotone dynamics. As the money growth rate and the steady state inflation rate increase, eventually endogenous volatility emerges. For even higher rates of money growth, the high-output-low-RER steady state becomes a source and can no longer be approached.

policy in response to the sharp increase in world interest rates could have caused the banking system to collapse. In our model banks face no aggregate risk and thus the fragility of the banking system is not an issue when evaluating the effectiveness of monetary policy. Our model could be extended to incorporate that type of financial vulnerability in the analysis.

Finally, we noted that Calvo and Mendoza [10], and Cole and Kehoe [14], Chang and Velasco [12], [13], all have stressed the importance of self-fulfilling expectations, besides conditions in international and domestic financial markets, in the analysis of financial crises. An interesting extension of our model would consider the interaction between self-fulfilling equilibria and dynamics of the type we have described.

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Appendix

A. Proof of Lemma 2

It follows from 45 thatT(k^a)>(<)1 +D(k^a)iff:

T(k^a) = 1 +D(k^a) +^{w(ka)[1−r∗}(1−δ)(1−β)][^r∗−θσ]{^w(ka)fa(k^a)+f_a(k^a)[q−w(k^a)]}

{f_a(k^a)w(k^a)[1−r^∗(1−δ)(1−β)]−f_a(k^a)w(k^a)}^θ_σ[q−w(k^a)]

>(<)1 +D(k^a). This condition is equivalent to

w(k^a)[1−r^∗(1−δ)(1−β)]

[

^r∗−θσ

]{

^w(ka)fa(k^a)+f_a(k^a)[q−w(k^a)]

}

{f_a(k^a)w(k^a)[1−r^∗(1−δ)(1−β)]−f_a(k^a)w(k^a)}^θ_σ[q−w(k^a)]

> ( < )0 .

Under the assumption that1−r^∗(1−δ)(1−β)>0(which is equivalent toκ <1), this is true when

w(k^a)f_a(k^a) +f_a(k^a) [q−w(k^a)]<(>)0. (A.1) Noting thatw(k^a) =−k^af(k^a),w(k^a) =f_a(k^a)−k^af(k^a), and rearranging terms, equation (A.1) is equivalent to

q >(<)f_a(k^a).

B. Proof of Lemma 3

For the Cobb-Douglas economy,

d(r^∗, σ, θ) = αr^∗(1−δ)(1−β)

(1−α) [1−r^∗(1−δ)(1−β)] +α+ r^∗(1−α) [1−r^∗(1−δ)(1−β)]

{(1−α) [1−r^∗(1−δ)(1−β)] +α}^θ_σk^aH(k^a)

It is easy to show that the derivative with respect tor^∗of the first term in the sum is positive. Furthermore, the derivative of the same term with respect toσis equal to zero. Now, observing that^dk_dr^2a∗ >0andH(k^a)>0, straightforward differentiation of the second term ind(r^∗, σ, θ)with respect toσshows thatd₂(r^∗, σ, θ)>0.

Differentiating with respect tor^∗the second term, it is easy to see that a sufficient condition for the derivative to be positive given that ^dk_dr^a2∗ > 0andH(k^a) > 0is 1−2r^∗(1−δ)(1−β)>0.

C. Proof of Proposition 4 D(k₂^a)<(>)1iff

r^∗(1−δ)(1−β)w(k^a₂)f_a(k₂^a)^θ_σ[q−w(k₂^a)] +r^∗w(k₂^a)w(k^a₂)f_a(k₂^a) [1−r^∗(1−δ)(1−β)]<(>)

{w(k₂^a)f_a(k^a₂) [1−r^∗(1−δ)(1−β)]−w(k^a₂)f_a(k₂^a)}^θ_σ[q−w(k^a₂)]. Rearranging terms shows that this condition is equivalent to

θ

ασr^∗ >(<) f_a(k^a₂)w(k₂^a)w(k₂^a)

[q−w(k^a₂)] [f_a(k₂^a)w(k^a₂)−f_a(k^a₂)w(k^a₂)], which in turn can be simplified to

θ

ασr^∗ >(<) w(k^a₂) [q−w(k₂^a)]

by noticing thatf_a(k₂^a)w(k^a₂) =f_a(k₂^a)w(k^a₂)−f_a(k^a₂)w(k^a₂). D. Proof of Proposition 5

(a) Consider an equilibrium of the economy such thatd(r^∗, σ, θ)<1.It is easy to show that lim

r^∗→∞d(r^∗, σ, θ) = ∞,whenk^a is an increasing function ofr^∗.From monotonicity and continuity ofdit follows that there exists a value r_c^∗ such that d(r^∗_c, σ, θ) = 1.

(b) Whenr^∗ =r^∗_c,D(k₂^a) = 1and

T(k^a₂) = 2 +^{w(k2a)[1−r∗c}(1−δ)(1−β)]{^w(ka2)fa(k^a₂)+fa(k^a₂)[q−w(k^a₂)]}^(rc^∗−_σ^θ)

{^fa(k^a₂)w(k^a₂)[1−r^∗_c(1−δ)(1−β)]−f_a(k^a₂)w(k^a₂)}σ^θ[^q−w(ka2)] . Thus, whenr^∗=r^∗_c, T(k^a₂)²<4D(k₂^a) = 4holds iff

w(k^a₂)[1−r^∗_c(1−δ)(1−β)]{^w(k2^a)f_a(k^a₂)+f_a(k₂^a)[q−w(k^a₂)]}^(r∗c−_σ^θ)

{^fa(k₂^a)w(k^a₂)[1−r^∗_c(1−δ)(1−β)]−f_a(k₂^a)w(k^a₂)}^θσ[^q−w(ka2)] <0. This relationship holds iff

w(k₂^a)f_a(k^a₂) +f_a(k^a₂) [q−w(k^a₂)]<0.

Noticing that whenD(k₂^a) = 1,[q−w(k₂^a)] = w(k₂^a)^ασr_θ^∗c,the above inequality is satisfied when(1−α)σr^∗_c > θ.Under this condition,T(k^a₂)<2atr_c^∗,and the result follows from continuity.

Inflation, growth and exchange rate regimes in