Conflict and Uncertainty: A Dynamic Approach

other agents in conflict. Therefore this is the reaction curve for effort. It can be easily shown that
the optimal effort level of agent i is increasing in the effort level of agent j (i / j).

Note that this reaction curve does not depend on ɪπⁱ- Then the dissipation in this conflict is
independent of the exogenous valuation of the good. Also, the choice of effort is an intra-temporal
problem. Then the static model of conflict partially describes this part of the conflict.

The conditions (2) and (3) summarize all the dynamic choices in the conflict. They just say that
the expected cost (with information available to agent i) of investing a marginal unit of the valuable
resource (given by the current loss of utility) must be equal to the expected profit of this investment,
which is the additional resource that can be expected to be obtained the next period thanks to the
marginal investment in the conflict technology.

We have developed a dynamic stochastic general equilibrium model of a simple conflict with
endowments. The equilibrium of the model is an optimal strategy that takes into account the
present and expected future of all variables. Our conflict agents solve a well defined micro-founded
problem, fully consistent with the static conflict model, but with a true dynamic behaviour.

4.2 Deterministic steady state

We define the “deterministic steady state” as the equilibrium in which the variables would not
change with the time, being all stochastic variables in their unconditional mean. In order for the
deterministic steady state to exist, we require some properties for the functions. First, we define the
steady state effort levels e¹ as the solution to the system given by the deterministic version of the
optimal rule (1):

∂fpⁱ (_e^i,_eI∖{ⁱ}^,αl) ^r = ɪgⁱ (eⁱ; ^θ

where R is the unconditional mean of the resource.

From (2) and (3) we get that in the deterministic steady state

βi∂Ti"i ⁽xi, "i, zi, ^si) ∂~iPi (^βi, ^eI^XW, ^αI) R   ¹ — βi∂T~i"i ^(æi, "i, Z ^si)

(√X                        ad∣                       a                 ad∣

where z is the long-run value of the stochastic shock to investment. In the steady state, the following
must also hold

αⁱ / αⁱ (xⁱ, αⁱ, zⁱ, sⁱ)

and we are able to solve for αⁱ and xⁱ.

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