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

φ n		( (1 + β + δ) O —1 O ʌ	^Г φ — φ ^
12	—	^-cμ ^-^{(1 +}^δ) ^cμ ^o	a — a	(A8)
^μ		O OO —μf ''	μ — μ
∖^k '		C Cμ 1 —Cμ β у	к к — к у

where z = (φ, a, μ, к, )^,and J^z denotes the Jacobian matrix of (A8) in the Hxed employment
case. To determine the stability properties of the equilibrium, we Hrst consider the trace
and determinant of the Jacobian matrix

tr (jɔ = ψι + ψ₂ + ψ3 + ' 'ɪ = 2β > O,

det (jɔ = ψιψ₂ψ₃⅜ = (β + δ)δc_μμf" > O,

where ψ = 1

— 4 are the eigenvalues of J. The condition tr

> O rules out the case

of four negative eigenvalues, while det

(J^f)

> O eliminates, respectively, the cases of one

negative and three positive eigenvalues, and three negative and one positive eigenvalue.

As in the more general framework with endogenous employment, we must directly eval-
uate the characteristic equation to determine whether J^z has two negative and positive
eigenvalues or four positive eigenvalues. In this case the characteristic equation, denoted
by det (j^z-ψl) = O equals:

det (j^r-ψl) =

^ψ4^-^tr (^j^Z) ^ψ3⁺ [^²^-^{(1 + δ}X^{1 +}^β⁺^δ) — ^cμ^μf "]^ψ2⁺^β[(^{1 + δ}X^{1 +}^β⁺^δ) ⁺^cμ^μf "]^ψ∙

+ det (J^r) = O

(A9)

Matching the coefficients of (2.16a) and (A9), we observe that

Ψ1Ψ2Ψ3 + Ψ1Ψ2⅜ + Ψ1ψ3⅜ + Ψ2ψ3ψ4 = β[(ι + δ)(ι + β + δ) + c_μμf"] < O,

which implies that we can rule out the case in which all the eigenvalues are positive.

Thus, the Hxed-employment equilibrium of (A9) is a saddlepoint with two negative and

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