DURABLE CONSUMPTION AS A STATUS GOOD: A STUDY OF NEOCLASSICAL CASES

trace and determinant of the Jacobian matrix

tr (J^soe) = θι + θ₂ + θ₃ + θ ɪ = 2β + α'f > 0,

det (J^soe) = θιθ₂θ3θ₄ = μα' [(1+ β + δ) (1 + δ)F^fl_μ - (β + δ)δc_μ∖ > 0,

where θ = 1 — 4 are the eigenvalues of J^soe. The condition tr (J^soe) > 0 rules out the
case of four negative eigenvalues, while det (J^soe) > 0 eliminates, respectively, the cases
of: i) one negative and three positive eigenvalues; and ii) three negative and one positive
eigenvalue. As in the closed economy framework, we must directly evaluate the character-
istic equation to determine whether J^soe has: i) two negative and positive eigenvalues; or
ii) four positive eigenvalues. In the small open economy case the characteristic equation,
denoted by det (J^soe-θl) = 0 equals:

det (J^soe-θl) =

θ⁴ - tr (J^soe) θ³ + [(β + α'n) β - (1 + δ)(1 + β + δ) - μα^, (F^,l_μ - Cμ)∖ θ²

+ [(1 + β + δ)(1 + δ) (β + α∕f) + μα' (F^fl_μ - c_μ) β∖ θ + det (J^soe) = 0

(3.10)

Matching the coefficients of (2.16a) and (3.10), we observe that θ1θ₂θ3 + θ1θ₂θ4 + θ₁θ3θ4 +
θ2θ3θ4 = - [(1 + β + δ)(1 + δ) (β + α'n) + μα' (F'lμ - Cμ) β] < 0, which implies that we
can rule out the case in which all the eigenvalues are positive. Thus, the fixed-employment
equilibrium of (3.9) is a saddlepoint with two negative and two positive eigenvalues ordered
according to:

θι < θ2 < 0 < θ3 < θ4.

We have established that both closed and open economies display saddlepath dynamics.
As in the case of the closed economy, (3.9), using standard procedures, can be solved for
the solutions paths (φ, a, μ,f, ). This is done in the last part of the appendix.

We close this part of the paper with an analysis of the impact of an increase in the
degree of status consciousness in the small open economy context. Differentiating (2.14a^,)

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