Control Theory
Useful after all

Since the 1960s, economists have sought to apply the techniques of control theory, first used in engineering, to design 'optimal' economic policies. Attempts to apply control theory have been complicated, however, by the issue of time inconsistency. If the government's policy actions are governed by a constitutional provision, then they are set once and for all in the initial time period, and a straightforward application of control theory leads to a fully optimal policy. Without such precommitment, however, such an optimal policy may be time-inconsistent: policy actions which are optimal when announced may cease to be optimal after the passage of time and the government will be tempted to renege on its announced policies. This property makes the design of policy by a government facing private agents very different from the case of a single 'controller' playing against nature. As a result it has been argued that control theory cannot in practice be used for policy design. In Discussion Paper No. 141, Research Fellow Daniel Cohen and Philippe Michel demonstrate how control theory can be used for policy design even when there is no precommitment and the optimal time-inconsistent policy cannot be pursued.

Cohen and Michel first consider a situation with no precommitment, in which they define a time-consistent policy as one which a government will find optimal to follow at each point of time, given that it is expected to do so then and at all succeeding points in time (Bellman's dynamic programming principle). The authors then define a time-consistent policy with 'instantaneous' precommitment. This is a sequence of policy actions which both the private sector and the government expect will be followed at later dates, and which the government finds it optimal to follow at each point of time provided that it can always select its policy before the private sector chooses its own decision. Even though this amounts to an infinitely small precommitment by the government in the model they consider, it need not coincide with the time-consistent policy with no precommitment.

Full precommitment, which would allow the pursuit of the optimal time-inconsistent policy, generally yields higher welfare than policies which are constrained to be time-consistent. The equilibrium with 'instantaneous' precommitment, however, will not necessarily yield a more desirable outcome than the time- consistent equilibrium. Cohen and Michel find that this is so only when the government uses a policy instrument such as the inflation rate or a distortionary tax which has undesirable side- effects on the economy. In this case, the ability of the government to 'announce' its policy in advance allows it to reduce its intervention in the economy and proves beneficial. In other situations, this is not so: playing first forces the government to increase its intervention in the economy. Leadership is not always beneficial and can reduce policy-makers' welfare, Cohen and Michel conclude.
The authors demonstrate how control theory can be applied to calculate these equilibria. In order to calculate the optimal time-consistent policy, the government should ignore the dependence of the private sector's decision rule on its own policy decisions. If it took this rule into account, it would merely arrive at the optimal time-inconsistent solution, which relies on the private sector's response to government policy. The government should assume instead that the behaviour of the private sector is unaffected by the government's choice of policies. It is this 'deliberate myopia' which makes it possible for the government to apply control theory: its problem is then in effect a standard engineering control problem which can be solved in an iterative fashion. Cohen and Michel note that this procedure resembles conventional macroeconometric modelling practice, which has often treated the feedback rules adopted by the private sector as independent of the government's actions. Econometric practice, insofar as it resembles Cohen and Michel's solution procedure, may actually lead the economy towards the time-consistent equilibrium.

Cohen and Michel also demonstrate how control theory can be used to calculate an equilibrium with instantaneous precommitment. Here, the government should ignore the equation governing the private sector, and substitute for it a postulated feedback rule for the private sector. This procedure, although complicated, seems to be an efficient way to compute an explicit optimal policy.


How Should Control Theory be Used by a Time-Consistent Government?
Daniel Cohen and Philippe Michel

Discussion Paper No. 141, December 1986 (IM)