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

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:

C_α — ^-1,     Cμ — ^-Cφ — ^jJ_cc + ^jsc^s'^(1)/(c + ɑɔ ^- U_sS^,(1)/⁽C + ɑɔ²]   < ^0,

(A1a)

∂l _—   -F_{1   > 0}     ∂l~ _— _  μF_kl

∂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 + A₁e^ω1t + A₂e^ω2^t                          (A2a)

a — k + B₁e^ω1 + B₂e^ω2^t                        (A2b)

μ — μ + C₁e^ω1^t + C₂e^ω2^t                         (A2c)

k — ^k + D₁e^ω11 + D₂e^ω2^t                         (A2d)

where A_j, B_j, C_j and D_j,i — 1, 2 are constants (eigenvectors) corresponding to the stable
eigenvalues ωι and ω2 and φφ, a, k, k^j 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

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