which would indicate that each agent thinks that only a fraction of the common asset really
belongs to him. Alternatively, the agents may give more weight to the public asset because the
private asset holdings are illegal and cannot be used to indicate high status. Anyway, in our
general formulation we allow γ to take any non-negative value. The agents derive utility both
from consumption and from wealth. We denote the instantaneous utility of agent i derived at
time t by U (ci (t),Ai(t)). The presence of Ai (t) as an argument of the utility function displays
a wealth effect that has so far been ignored in the literature on growth under insecure property
rights.
We assume that both investment into and withdrawal from the private asset holdings are costless.
Extraction from the common property asset, however, is assumed to be costly. This reflects
the fact that considerable resources have to be spent for lobbying or money laundering. The
marginal cost of appropriation from the common property asset measured in units of utility
is assumed to be constant and will be denoted by κ. Finally, we assume that the common
time-preference rate of all agents is given by ρ. Agent i therefore seeks to maximize
subject to constraints (1)-(4).
+∞
e-ρt
[U(ci(t),Ai(t))
- κxi(t)] dt
(5)
As for the parameters of the model we make the following assumption.
Assumption P: The number of players is an integer n ≥ 2. The parameters R, z0 , r, yi0 , γ,
κ, and ρ are real numbers satisfying z0 > 0, yi0 ≥ 0, γ ≥ 0, ρ>0, and ρ>r.
The instantaneous utility function U is assumed to be homogeneous of degree 1. This implies
that U(c, A) = Au(q), where q = c/A is the ratio of consumption to wealth and where the
function u is defined by the equation u(q) = U(q, 1). The properties of U, which we shall
assume throughout the analysis, are summarized next.
Assumption U: The instantaneous utility function U :[0, +∞) × [0, ∞) → IR is concave and
homogeneous of degree 1 on its domain. The function u :[0, +∞) → IR defined by u(q)=U (q, 1)
is continuous, strictly increasing, and strictly concave on its domain and it satisfies u(0) = 0.
Furthermore, u is assumed to be continuously differentiable on the interior of its domain and
to satisfy the Inada-type conditions limq→0 u'(q) = + ∞ and limq→+∞ u'(q) = 0.
Assumptions P and U will be maintained throughout the rest of the paper without further
mentioning. It is well-known that ∂U(c, A)/∂c = u'(c/A) holds for all (c, A) in the inte-
rior of the domain of U. In other words, the marginal utility of consumption depends only
on the consumption/wealth ratio q. Analogously, the marginal utility of wealth is given by
∂U(c,A)/∂A = w(c/A), where w(q) = u(q) — u'(q)q is a strictly positive and strictly increasing
function of q = c/A > 0.
We are interested in symmetric Markov-perfect Nash equilibria of the differential game specified
above. These equilibria are defined as follows. Let y(t) = (y1 (t), y2 (t),...,yn(t)) ∈ [0, +∞)n
be the n-dimensional vector of private asset stocks. A (stationary) Markovian strategy φi for
player i is a pair of functions φix :[0, +∞)n+1 → IR and φic :[0, +∞)n+1 → IR. We call φix
agent i’s appropriation or extraction strategy and φic his consumption strategy. Applying the
strategy φi means that agent i chooses his appropriation and consumption rates according to