αt1+1 = (1 - δ1 (st1) ) αt1 + xtzt
αt2+1 = (1 - δ2 (st2)) αt2 + 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 Ut = ut - u, where u is the steady state
value of variable ut.
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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