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

homogeneous system:

where

/	(1 + β + S) — ωi	0	—1	0		A Ai	∖
	-cμ	— [(1 + δ) + ωi∖	cμ	0		Bi
	0	0	d33 — ωi	⅛4		Ci
∖	cμ	1	d43	d44 — ωi /		Di

0
0	(A3a)
∖0 J

⅛3 = -l_μμF_kl > 0, ⅛4 = — μ(F_kk + F_kll_k) > 0, ⅛ = F_ll_μ-c_μ > 0,  d₄₄ = β+F_ll_k > 0.

From (A3a), the constants Ai, Bi, Ci and Di, i = 1, 2 are written as:

=      Ci      =  Ci  =   —⅜₄Di

(1 + β + δ) — ωi    Φ(ωi)    (θ33 — ωi)Φ(ωi) ’

B           ^cμd34

(^θ33 ^{— ω}⅛) [(^{1 + δ}) ^{+ ω}i]

(⅛Φγ^) Di = Ω. (⅛≡ Di,
V Φ(ωi) J          V Φ(ωi) J

(A3b)

where

''' ^ω = ^{(1 + β +} δ) - ^ωi,     ⅛ ^ω = (θ33 — ωi)ⁱ[(₁³+ ^)+ ^i] ’

To solve for the constants D_i and complete the solution of (A3a), we use the fact that
the stock of durable goods and physical capital evolve continuously from their initial
conditions, α(0) = ao and к (0) = ko∙ From (A2b, d) this gives us the two equation system

a + Ω₁ (ω₁) f¹ — .^φ^ D₁ + Ω₂ (ω₂) P — D₂ = ao
V Φ(ωι) J            4 Φ(ωi) J

к + Di + D2 = ко

(A4a)

from which we solve for D₁, D2 :

^φ(wJ^φ(^ [—(à — ^ao^{) + ω}2 (ω2) ^¹ _φ^φ2^^(к — М]
Ωι (u1) Φ(ω2) [1 — Φ(ω1)] — Ω2 (ω2) Φ(ω1) [1 — Φ(ω2)]^,

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