F (k, l) = Ôa,
(2.14d)
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.
(k/k) = k,
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 =
|
φ ■ ʌ |
( (1 + β + δ) 0 -1 0 |
^ φ — φ ^ | |||
|
a |
- cμ - (1 + δ) cμ 0 |
a — a | |||
|
μ |
0 0 -^k^kl —jk(Fkk + Fkllk) |
μ — μ | |||
|
(2.15) | |||||
|
∖ k У |
Cμ 1 Fllμ - Cμ β + Fllk J |
y k — k y |
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∣ + ω2 + ω3 + ω4 = 2β - lμμ^^kl + Fllk = 2β > 0,
det (J) = ω1ω2ω3ω4
= (β + δ-)δcμμ (Fkk + Fkilk) - (1 + δ)(1+ β + δ)lμμFkk (у/l) > 0