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

(2.10a) is the first order condition for own consumption in which the agent takes the
average level of durable consumption in the economy, C + A, as given in performing
his optimization. This is also the case in equation (2.10c), which describes the dynamics
shadow value φ when the stock of own durable consumption goods a is chosen optimally.
Equation (2.10b) is a standard first order condition for work effort in the neoclassical
context, while equation (2.10d) defines the dynamics of the shadow value μ when physical
capital к is chosen optimally. Our specification of preferences in equations (2.3a)-(2.3b)
guarantees that the Hamiltonian (2.9) is jointly concave in the control variables c and I
and the state variables a and к. This implies that if the limiting transversality conditions
lim_t→∞ aφe^-βt = limt→∞ kμe^-βt = 0 hold, then necessary conditions (2.10a)-(2.10d) are
sufficient for optimality.

As is the usual practise in models of the type, we restrict our subsequent analysis to
symmetric equilibria in which identical agents make identical choices. This is the procedure
followed by Gali (1994), Persson (1995), Harbaugh (1996), Rauscher (1997b), Grossmann
(1998), Ljungqvist and Uhlig (2000), and Fisher and Hof (2000a, b), among others. In
the context of our model, we specify that the individual quantities of durable consump-
tion (current flows and aggregate stocks) equal their average levels, i.e., c + a ≡ C + A.
Substituting this relationship into (2.10a) and combining with (2.10c), we obtain:

U_c [c + a, s (1)] + ^{Us [C +} α+ ⁽J^)] *' ⁽¹⁾ = μ — φ = (β + δ)φ - φ,       (2.11)

where the optimality conditions for work effort and capital accumulation remain un-
changed.

Using (2.11) and (2.10b), it is straightforward to calculate the following instantaneous
solutions for consumption and work effort in terms of the state and costate variables:

c = c (α,μ, φ) ;     c_a < 0,     c_μ = -cφ < 0,                (2.12a)

I = I (k,μ);      l_k > 0,      l_μ > 0,

where the expressions for the partial derivatives are given in the appendix. The partial

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