Each electoral period the incumbent receives r units of resources with electoral value. The total
amount of revenues accruing to the incumbent during the period is assumed to be known.8 If
she spends this amount, she can get a probability p(r) of re-election whereas if she spends x =
xe she will be certain of being re-elected, that is p(xe) = 1. It is assumed that r < xe. For the
case r >= xe, the solution is trivial: the incumbent spends x = xe in every election and remains
in power forever.
Additionally, there is a dictator who is concerned about the depletion of the country's stock of
reserves. The probability that a dictator takes over power at time t is a function of the level of
reserves Wt . In other words, the probability of his intervention increases at the pace of the
depletion of the stock of reserves. Once the dictator intervenes, he stays in power forever. This
probability can be expressed as:
p (dictator /1 ) = d ( Wt )
It is assumed that d(Wt ) is a convex function with respect to Wt. That is to say:
∂d( Wt ) < 0 ;
∂ Wt ’
∂ 2d(Wt) > 0
∂ 2 Wt
The total discount factor is defined as the inverse of the probability of a dictator (political
discount) times a constant term μ accounting for the incumbent's pure myopia (which is not
related to the probability of a take-over).
δt = δ (Wt, μ) = [1 - pt (dictator )] μ
Here, a distinction is made between two components of the discount factor. The first one finds
its justification and meaning in the political arena, reflecting the chances the incumbent party has
to be again in office. In this model, a future enjoyment of power depends on the intervention of
the dictator, but it also can be thought as accounting for the intensity of political competition,
where the entry of new competitors reduces the future chances each party has to be in office.
The second component is included to measure parties’ time preferences. It ranges from μ = 0
8 This assumption is necessary to make the interaction between the parties tractable. As a consequence, I
will abstract from the issue of volatility of external revenues, which is a key feature of many developing
economies, and particularly in mineral economies.