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

F (k, l) = Ôa,

where c = δα and μ = (1 + β + δ) φ. Equations (2.14a, b) are the steady-state versions
of the Hrst order conditions (2.10a, b), while equation (2.14c) is the standard long-run

representation of the Euler equation, which implies, given our technological assumptions,
that the marginal physical product of capital, and, thus, the capital-labor ratio
is pinned-down by the exogenous rate of time preference β. Finally, due to the fact that
there is no depreciation of physical capital in this model, equation (2.14d) states that
steady-state output equals steady-state durable consumption, which corresponds to the
long-run level of durable goods deprecation.

Linearizing the differential equation system (2.13a)-(2.13d) about the steady state
(2.14a)-(2.14d), we calculate the fourth-order matrix equation:

Z = Jz =

where z = (φ, a, μ, k, )^,and J denotes the Jacobian matrix of (2.15) in the case of endoge-
nous employment. Observe that functions of variables are evaluated at the steady-state
equilibrium (2.14a)-(2.14d). To determine the stability properties of the equilibrium, we
first consider the trace and determinant of the Jacobian matrix

^{tr (J)} = ^ω∣ ^{+ ω}2 ^{+ ω}3 ^{+ ω}4 = ^{2β - l}μ^μ^^kl ^{+ F}l^lk = ^2β > ^0,

det (J) = ω1ω2ω3ω4
= (β + δ-)δc_μμ (Fkk + Fkilk) - (1 + δ)(1+ β + δ)l_μμFkk (у/l) > 0

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