The consumer-producer’s problem is, thus, formalized as follows:
max
Γ {u
c + ‘,s Q ɪ a ) + V (ŋ) e βt dt,
(2.8a)
subject to:
a = c — δa,
____, _
к = F(к, I) — c,
(2.8b)
(2.8c)
and the initial stocks of durable consumption and physical capital, a(0) = aŋ > 0 and
к (0) = ко > 0, where β is the exogenous rate of time preference. To solve the consumer-
producer’s problem, we form the current value Hamiltonian, which is given by:
where φ and μ are the costate variables corresponding, respectively, to the constraints
(2.8b) and (2.8c). Maximizing equation (2.9), we calculate the following Hrst order opti-
mality conditions:
c + a, s
+ V(Z) + φ (c — δa) + μ [F(к, Z) — c] ,
(2.9)
и и , .1 U4 [c + a, s (г)] s' (г) , „ .
uc [c + a,s (г)] + = μ - φ, (2.l0a)
e + a
V'(Z) = — μFι (k,Z), (2.10b)
φ = (β + δ) φ — Uc [c + a,s (г)] — u∙ [c +‘,'f ,jl s'(г), (2.10c)
e + a
μ = μ [β — Fk(k,Z)] . (2.10d)
The optimality conditions (2.10a)-(2.10d) have a straightforward interpretation. Equation