derivatives in (2.12a, b) are interpreted as follows: an increase in stock of durable con-
sumption a lowers its current level c. Indeed, as we show in the appendix, an increase in a
lowers c by one-for-one. In addition, current durable consumption depends negatively the
marginal utility of wealth μ and positively on the shadow value of the stock of durable
consumption φ. Regarding the short-run response of work effort, (2.12b) indicates an in-
crease in the shadow value μ raises work effort. Moreover, a given increase in the capital
stock к also encourages labor supply, since Fkι > 0, which is implied by constant returns
to scale production technology.
From the equations (2.11), (2.8b), (2.10d) and (2.8c), we state the independent dy-
namics of the economy:
φ =(1+ β + δ) φ - μ, (2.13a)
cl = c (a, μ, φ) — δa, (2.13b)
μ = μ [β — Fk (к, I (μ, к))] , (2.13c)
к = F [(к, I (μ, к)) — c (a, μ, φ)] , (2.13d)
where we have substituted for I = I (к, μ) in (2.13c, d) and c = c (α,μ, φ) in (2.13b, d),
respectively. Letting φ = a = μ = к = 0, the long-run equilibrium equals:
Uc [(1+ δ) α,s (1)] + U 1(1 + 1∖α,/. wμ'(1) =(β + δ)φ (2.14a)
(1 ∣ δ ) a
V' (Z) = — (1 + β + δ) φFι (k, Zj , (2.14b)
Fk (k, k) = β,
(2.14c)