5.1. Partial Derivatives of (2.12a, b)
Taking the total derivatives of (2.11) and (2.10b), we obtain the expressions for the partial
derivatives of durable consumption and work effort with respect to the model’s state and
costate variables:
Cα — -1, Cμ — -Cφ — jJcc + jscs'(1)/(c + ɑɔ - UsS,(1)/(C + ɑɔ2] < 0,
(A1a)
∂l — -F1 > 0 ∂l~ — _ μFkl
(A1b)
∂μ у '' + μFu , ∂k V '' + μFu
5.2. Solution for (φ, a, μ, k) in the Variable-Employment Economy
The general stable solution to the differential equation system (2.15) is represented by the
following expressions:
φ — φ + A1eω1t + A2eω2t (A2a)
a — k + B1eω1 + B2eω2t (A2b)
μ — μ + C1eω1t + C2eω2t (A2c)
k — k + D1eω11 + D2eω2t (A2d)
where Aj, Bj, Cj and Dj,i — 1, 2 are constants (eigenvectors) corresponding to the stable
eigenvalues ωι and ω2 and φφ, a, k, kj are the long-run solutions calculated from (2.14a)-
(2.14d). Since only two of these constant are independent, the first step in obtaining the
complete solution is to solve Aj, Bj, Cj in terms of Dj, i — 1, 2. Using (2.15), (A2a)-(A2d)
and letting x — (Aj, Bj, Cj, Dj)', these relationships are calculated from following the
26
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