Conflict and Uncertainty: A Dynamic Approach

^αt¹+1 = (^{1 - δ1} (st¹) ) ^αt¹ + xtzt

^αt²+1 = ^{(1 - δ2} (st²)) ^αt^{2 +} xt²zt²

δ (s¹ ) = δ¹ + α¹ (/- - 1) + b¹ (/- - s¹)2
∖st-1      /         ∖st-1        /

δ (s2) = δ² + α² (- 1) + b² (■/- - 1)²
∖st-1      /         ∖st-1      /

This is a standard dynamic model. If we assume rational expectations we can use any solution
method proposed by the macroeconomic theory to solve this model.

5.2 Solving a rational expectations dynamic model

The equilibrium in this conflict is in correspondence with a highly non-linear system of dynamic
stochastic equations, which cannot easily be solved. A linear approximation around the non-
stochastic steady state is used. The method consists in linearizing each first order and/or equilibrium
condition around the steady state by a simple first order application of Taylor’s Theorem. A linear
system is obtained, with transformed variables of the form U_t = u_t - u, where u is the steady state
value of variable u_t.

The linear system of differential equations is solved by the method explained by Schmitt-Grohe
and Uribe (2004). A general description of the methodology can be found in Heer and Maussner
(2005). The solution is given by the H, M and R matrices that generate the dynamic solution by
the iteration of the equations:

ut = Hst

^st+1  = Mst+Rεt+1

where u is the vector of forward-looking variables (controls, со-states, flow variables), s is the
vector of endogenous and exogenous backward-looking state variables, H characterizes the optimal
policy function, M is the state transition matrix, ε is the innovation vector and R specifies how the
exogenous shocks (innovations) affect the dynamic system.

Note that the uniqueness of the solution can be assured for the stable case. Henceforth a solution
to our conflict model a la macroeconomic theory needs to be, at a first instance, a stable solution.

The stable solution rules out, by definition, the chaotic behaviour of the endogenous variables⁹.
But even with this strong constraint we can obtain several conclusions about the dynamic of the
⁹If the model is stable, eventually the endogenous variables return to the steady state. We could then predict the
future value of the variables, and therefore they would not be chaotic.

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