the time-invariant feedback laws xi (t)=φix(y(t),z(t)) and ci (t)=φic(y(t),z(t)), respectively.
A strategy profile is a n-tuple of Markovian strategies, one for each agent. A strategy profile
(φ1 ,φ2 ,... ,φn ) is called symmetric, if φi = φj holds for all i and j . A strategy profile is a
Markov-perfect Nash equilibrium if, for all z0 ∈ (0, +∞), all y0 =(y10,y20,...,yn0) ∈ [0, +∞)n,
and all i the following is true: the optimal control problem of maximizing (5) subject to (1)-(4),
xj (t)=φjx(y(t),z(t)), and cj (t)=φjc(y(t),z(t)) for all j = i has an optimal solution which
satisfies xi(t) = φix (y(t), z(t)) and ci(t) = φic(y(t), z(t)).
In the above definition we have assumed that Markovian strategies can depend on the common
property asset stock and on all private asset stocks. In all equilibria discussed in the present
paper, the appropriation and consumption rates of agent i will only depend on the public capital
stock and on agent i’s own private asset holdings. Thus, nothing would be lost if one would
restrict the domain of admissible strategies in such a way. In particular, it is irrelevant for the
results of this paper whether the players can actually observe their opponents’ private asset
stocks or not.
3 Main results
Consider the equation
(ρ - r)u'(q) = w((?). (6)
From the assumptions mentioned in section 2 it follows that the left-hand side decreases strictly
from +∞ to0asq?goes from 0 to +∞. The right-hand side is a strictly positive and strictly
increasing function of q?. Since both sides depend continuously on q?, there must exist a unique
positive solution of equation (6). To understand the relevance of this solution, suppose that
the consumption/wealth ratio is constant and equal to q. The cost (loss of utility) of reducing
consumption at time t by an infinitesimally small amount dc is u'(q)dc. The gain from doing
so, on the other hand, is that the private asset holdings at time t increase by dc. This implies
that there are er(τ -t)dc additional units of the private asset available at time τ ≥ t. The
additional discounted utility that can be derived from these additional asset holdings is given
by w(q)dc/(ρ — r). Equation (6) therefore says that, at q = ?, the marginal cost of a reduction
of consumption equals its marginal gain. Note that equation (6) defines q? as a function of ρ - r
but that q? is independent of γ and κ.
Equation (6) may also be written as
w(q?)
r + u ( ?) ρ,
(7)
which can be interpreted as a wealth-adjusted modified golden rule. Recall that the usual
modified golden rule says that the optimal steady-state holdings of a capital stock must be
such that the return (marginal productivity) is equal to the rate of time preference. In our
model, because wealth appears in the utility function, the rate of return to private capital is
r plus the marginal rate of substitution of private capital for consumption which, in turn, is
[∂U(c,A)/∂c] / [∂U(c,A)/∂y] = w(q)/uf(q). Here, we have used the fact that ∂U(c,A)/∂y =