2 Model formulation
Time is modelled as a continuous variable. Consider a common property asset with constant
rate of return R. We interpret this asset as the aggregate capital stock in an economy with
insecure property rights. There are n identical agents with access to this asset. Each agent
represents a group of individuals who - because they are organized and because property rights
are poorly defined or not fully enforced - can influence the allocation of the capital stock among
the n groups. We do not explicitly mo del how the groups can redistribute capital: this can
happen, for example, through their influence on the fiscal process, through lobbying activities
or corruption, or through forcible misappropriation.4 Adopting the simplifications made in the
literature cited at the opening of the introduction, we assume that the agents can withdraw from
the public asset stock but that they cannot invest into it. The withdrawal rate (or extraction
rate) of agent i at time t from the common property asset will be denoted by xi (t), the asset
stock itself at time t is denoted by z(t), and the initial value at time 0 is denoted by z0. The
aggregate capital stock evolves therefore according to the differential equation
n
Z(t) = Rz(t) - ∑xi(t), z(0) = zо. (1)
i=1
Resources that the agents extract from the aggregate capital stock can be either consumed or
invested into private and secure assets. These private assets can be interpreted as safe bank
accounts in foreign countries, where property rights are fully enforced. The rate of return on
the private asset is constant and given by r. Denoting the rate of consumption of agent i at
time t by ci(t), it follows that investment into the private asset holdings of agent i at time t is
given by xi(t) - ci(t). Let us denote the private asset stock of agent i at time t by yi(t) and its
initial value at time 0 is yi0 . It follows that
Уi(t) = ryi(t) + Xi(t) - Ci(t), yi(0) = yiо (2)
for all i and all t. Consumption must be non-negative at all times. We also assume that
the agents cannot incur debt, that is, the private asset holdings must be non-negative. The
feasibility conditions can therefore be summarized by the requirement that for all t and all i the
inequalities
Xi(t) ≥ 0,ci(t) ≥ 0 , yi(t) ≥ 0,z(t) ≥ 0 (3)
are satisfied.
Let us define
Ai(t)=yi(t)+γz(t), (4)
where γ is a non-negative parameter. We may interpret Ai(t) as the total wealth of agent i
at time t, where γ measures the weight given to public asset holdings relative to private asset
holdings. The weight γ can be smaller or larger than 1, depending on whether the agents attach
more or less importance to their private asset stock. For example, it may be that γ =1/n ,
4Tornell and Lane [10] develop a detailed model of how powerful groups can redistribute capital through the
fiscal process.