Conflict and Uncertainty: A Dynamic Approach

4.1 Solution to the share model

The problem faced by each agent i ∈ I during the period t ∈ T is:

XX Etβi {πⁱ (⅛,, uⁱ (pt₊jR_t+1 - g (⅛; <⅜⅛) - ∣⅛))}
j=o

=   = r>ⁱ (_eI∖{ⁱ} _αI .)

^pt+j   ^P p^et+3 ^{, p}t+j ^{, α}t+j J

^αt+j+1 = ^α (,^xt+j^{, α}t+j^{, z}t+j^{, s}t+j )

^pt+j ^Rt+j ^- g (^et+j^{; θ}t+j ) ^{- x}t+j ≥ ⁰

t + j ∈T

where E^z_t is the expectation operator referred to the information set available to the agent i during
period t.

This problem simply states that each agent chooses the effort level given the effort of the other
agents and the uncertainty about the total amount of the resource and investment in order to
maximize the discounted sum of the expected utility.

This is a standard dynamic programming problem and we solve it using the method of Lagrange.
The first order conditions for agent i are:

^d _ni (_pi _pI∖{i} _ni) _r
∂pt^p √^t'^{p    ,αt}) ^Rt

∂    .....∂ ■ , ■          .....

_dr_lπi («;. ut) д-Juⁱ (piR - Sⁱ (Pt; ^θt) -
u u           c ɑt

^αt+1

^i
pt

i     /)iʌ i ʌ ^d i t i      I∖{i}   I ∖ D

^et + 1; θt + 1 J ^{- x}t + 1 J dâï---^p (₄^et+1^{, e}t + 1  ^{, α}t + 1 J Rt+1

= Yl - βiE^ltYt+ι „ ^d_i   a fxⁱ+ι,at+ι,zt+ι,st+ι)

^dαt + 1    ×                        ^j

where γt_t is the Lagrange multiplier associated with the investment technology.

The condition (1) represents the optimal choice of effort level. This effort depends on the tech-
nology parameters (whose values were decided the period before) and the optimal effort of all the

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