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

The consumer-producer’s problem is, thus, formalized as follows:

subject to:

a = c — δa,

____, _

к = F(к, I) — c,

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:

и и           , .₁ U₄ [c + a, s (г)] s' (г)            ,                           „ .

^uc ^{[c + a,s (г)] +}                        = ^{μ - φ,                (2}.l^0a)

e + a

V'(Z) = — μF_ι (k,Z),                               (2.10b)

φ = (β + δ) φ — U_c [c + a,s (г)] — ^u∙ ^{[c +}‘,'f ^,j^{l s}'^(г),            (2.10c)

e + a

μ = μ [β — F_k(k,Z)] .                             (2.10d)

The optimality conditions (2.10a)-(2.10d) have a straightforward interpretation. Equation

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