Abstract
No doubt part of the reason why the solution and simulation of dynamic games between the government and the private sector is not yet standard practice among students of business cycles is that there is no standard algorithm available which (at least approximately) solves models of this kind. Ambler and Paquet (1995) is the first attempt to present such an algorithm. The algorithm presented in this paper is an extension of Ambler and Paquet.
The present paper, extends the results of Ambler and Paquet (1995) mainly by allowing for government-issued securities with unknown price processes. It also shows how to apply Gaussian exponential control theory as suggested by Hansen and Sargent (1995) in order to make the decision rules risk sensitive.
The game between the government and the private sector is modelled as
follows. Time is modelled as discrete, so that 
and the
number of periods is assumed to be infinite. At the beginning of each period
t, the government takes its actions. Having observed these actions, all
(atomistic) private agents (who are qualitatively identical) take their
actions simultaneously.
The solution concept employed is subgame perfect Markov equilibrium. Subgame
perfection is guranteed by requiring the strategies of all agents to satisfy
a Bellman equation in each period and state of the world. The qualification
`Markov' is a refinement concept advocated by, among others, Maskin and
Tirole (1988). The point of using this refinement is to avoid folk-theorem
style results and get a unique equilibrium. For more philosophical arguments
in favour of the Markov refinement, see Maskin and Tirole (1995). In the
present context, Markov equilibrium means that the equilibrium strategies
can be represented as feedback rules of the form 
where 
is a vector of decision variables and 
is a vector of state
variables of the same dimension in each period. Indeed, because of the
infinite time horizon, the equilibrium strategies are time invariant, so
that we may write 
Finally, because all the
problems involved are linear-quadratic, the equilibrium strategies are
linear, so that we may write 
. Similarly, asset prices can be
written 
.
Especially when asset pricing is involved, the certainty equivalence property of linear-quadratic control is embarrassing. In particular, it means that all risk premia are identically zero. In order to deal with this, I show how to apply discounted linear exponential quadratic Gaussian control, as developed by Hansen and Sargent (1995).
Having presented the algorithm, I discuss practical problems that might and do arise in implementation, focussing especially on a specific model with endogenous taxes and government spending.