Equivalence of Rational Expectations to a Strategic Game
The root of the problem raised by the Lucas critique is that it shows the private-sector behaviour to be invariant to the policy vector itself if rational expectations (REs) are introduced.
In other words, when the private sector has REs of future developments, the policymaker loses control of the economic system, as those expectations deny the existence of an equilibrium influenced by policy action, such as that postulated by the classical theory of economic policy.As a consequence of assuming REs, for a long time mainstream economics has been trapped in Lucas critique. Ways have been suggested for alternative formulations of the agents' conduct in order to explain the economy's fluctuations. One has been pursued by De Grauwe (2012: vii), who shows that the view according to which ‘everything becomes possible when we move into the territory of irrationality' is based on the false idea of what a rational agent is, since there are alternative ways to formulate it. He thus develops a behavioural model based on the limited cognitive ability of agents, by which he shows that fluctuations can be explained not by assuming external shocks to the economy but by accepting that they can be endogenously determined. We are only marginally interested in the issue of explaining busts and booms. Our focus is instead on the possibility for the policymaker to steer an economy that does not comply with his or her goals. We will show that this is possible by maintaining the RE assumption.
It is easy to understand that assuming REs amounts to an implicit change in the nature of the economic system confronting the policymaker. In a parametric setting such as that assumed by the classical theory of economic policy, there are certainly links between the decisions of the government and the private sector. Otherwise, the former could not have any influence on the latter.
However, such links are defined, at least from the point of view of the public sector, in an unchanging context of the rules of the game. In fact, the policymaker perfectly knows the parameters of the latter's choice functions (e.g. the consumption function), and these do not change when it sets its instruments (interest rate, tax rates, etc.). The private sector has objectives that conflict with those of the policymaker but can only change its behaviour in a way known to the public sector and does not neutralise its action.In a RE context, instead, the private sector reacts to actions or changes of regime decided by the policymaker by changing the rules dictating its choice and thus can neutralise the policies in advance due to its REs. This would change not just the outcomes but also the way the system itself behaves. The policymaker then faces a system that is no longer parametric, and this in itself leads, according to Lucas, to a loss of control of the economy, i.e. policy ineffectiveness. In formal terms, this would make the multipliers of policy instruments endogenously determined and conditional on the responses by the private sector.
This implication - though reasonable - might not survive an explicit way of dealing with the underlying conflict between the policymaker and the private sector in terms of a policy game.[46] Put differently, this critique holds for the Tinbergen-Theil theory of economic policy, but that is not to say that it is also true of a revised or new theory of economic policy based on strategic setting. It all depends on whether the private-sector reactions (or anticipations) can be accommodated in the policymaker’s decisions or whether the private-sector reactions are strong enough to exactly offset what the policymaker is trying to do. In general, private agents can neither offset those actions completely nor would they try to do so, as we shall show.
Let us examine now in more detail how the Lucas critique should be expected to apply in a world of rational private agents and/or rational policymakers by discussing how it works in a strategic environment where each agent has his or her reaction function.
It is clear that the reaction function of each player will depend on the policies or decisions made by others. Consequently, as soon as one player adjusts his or her policies or policy rule, the other players whose decisions depend on the decisions of the first player will adjust theirs - thus invalidating the premise of constant parameter values in the response by others that must have underlaid the first player's original calculations.[47] Hence, the critique is clearly true and needs to be taken into account. However, the obvious way to overcome it is to solve the implied policy game directly in extensive form.In other words, using REs amounts to implicitly assuming some kind of reaction of the system to the policy enacted. This assumption, and the underlying conflict between the policymaker and the private sector that it reflects, can be made explicit. To this end, the issue facing the policymaker must be framed in a context (that of games) where the private sector's behaviour is explicitly modelled as having been derived from its preferences and objectives. Strategic interactions between the private sector and the policymaker then ensure that the REs of both are satisfied.
RE models are indeed semi-reduced forms of linear quadratic policy games that transform a two-player optimisation problem into a one-player optimisation problem constrained by some additional condition implying the REs of the rival on the optimising player's policies. Hence, they correspond to a Stackelberg game with the private sector as the leader, implying a discretionary equilibrium.[48] However, notice that RE games are often solved by using the Nash equilibrium. This occurs because the control variable of the forecaster is the same forecast on a target (not an instrument) variable. In addition, in these, the Nash and Stackelberg equilibria with the forecaster leadership coincide.[49] [50] More generally, in models with strategic behaviour by multiple players, these players (one of whom may be the private sector or a representative agent) have well-defined objectives in terms of the endogenous variables of the model. A dynamic game between two or more players (including at least part of the private sector) can always be transformed, although not always uniquely, into a conventional singleplayer RE model.[51] This is convenient; the implication is that a forward-looking RE term can represent not only the strategic behaviour of the private sector but also the implicit shifting responses of the private sector to any policy changes or interventions by policy authorities. This covers the case where the private sector's chief concern is to forecast the outcomes of the economy or government policy (i.e. set expectations) accurately. It also means that the Lucas critique can be overcome if we learn to control a RE model correctly without time-inconsistent policies. The remaining sections of this chapter in particular show how to do this - through explicit policy games. However, this also can be done indirectly - and, in some cases, imperfectly - through single-player RE models. We can deduce that making use of a strategic model is to be preferred not only because the conflict and the strategic interaction are made explicit but also because the RE representation of strategic behaviour is typically not unique. In practice, this needs to be tied down by specifying a theoretical model of the underlying economy or by estimating the parameter coefficients directly. In particular, the strategic behaviour at issue in the Lucas critique can come from any kind of policy game, i.e. a Nash or Stackelberg game. Alternatively, a conjectural variations game could be introduced, with a contraction mapping designed to create Pareto improvements over the simple Nash solution. In this game, if the iterations take time to complete, the conjectural variations formulation shows how the Lucas critique can also be represented as a sequence of reactions by the private sector followed by adjustments by the government or other players. In fact, the only games that do not fit this analysis are cooperative games. Such games are solved by appointing a ‘social planner' and therefore do not require strategic interactions at the policy level, as opposed to what may happen between the players at the negotiating stage beforehand. Hence, policies derived from such cooperative games are Lucas critique proof. As a final note to the Lucas critique, one can say not only that this critique can be overcome in a strategic setting and hence the theory of economic policy can be rehabilitated but also that (1) REs provide additional policy ‘instruments' in the form of policy announcements (see Chapters 5 and 6) and therefore make the classical policy intervention stronger rather than weaker and (2) the scope for time-inconsistent policymaking is substantially reduced, thus taking away one of the major objections to the theory (Hughes Hallett, Di Bartolomeo and Acocella 2012a, 2012b). 4.3