Working Paper Series no. 902: Time-Consistent Implementation in Macroeconomic Games

Multiple equilibria plague many macroeconomic situations involving monetary policy (monetary vs non-monetary equilibria, price level determination), financial regulation (bailouts of banks that took excessive risks), or fiscal policy (capital taxation). This paper suggests a new approach for policymakers to avoid multiplicity without requiring implausibly large commitment ability.

The literature on bailouts of banks pointed out that multiplicity arises because governments cannot fully tie their hands in advance. The commitment not to bail out would avoid sub-optimal equilibria in which banks take too much risk. But keeping promises is costly and when risk materializes, the government may prefer to bail out banks to avoid costly financial crises. Unfortunately, such a preference triggers the existence of a bad equilibrium in which banks take excessive risks forcing a government’s bailout. In response, this literature argues that governments have no other means than to use tools such as financial regulation to solve equilibrium multiplicity.

Monetary economics took another route. In this literature, equilibrium multiplicity is perceived as resulting from the commitment to bad rules and the literature has been searching for the “good” rule. The objective of studying interest rate peg, the Taylor principle, the sophisticated policies, or the Fiscal Theory of the Price Level is to find a rule that, when the government commits to it, rules out equilibrium multiplicity and pins down a unique allocation.

To recap: while the bailout literature supposes no commitment technology, and hence no role for rules, monetary economics takes as granted the capacity of public authorities to stick to rules.

In this paper, we try to reconcile the two approaches:  (i) a government cannot fully tie its hands in advance to follow rules (ii) deviating from a pre-announced rule (or any type of commitments) may still be costly. To this aim, we introduce imperfect commitment to rules, we explicitly deal with the potential incentives to stick or deviate from rules and we investigate implications on the design of rules.

More precisely, the government announces a rule before the private sector plays. Then, the government incurs a cost in case of ex-post deviation from the rule. This cost measures its commitment ability. When zero, it is the discretionary case. When infinite, it is the full commitment case.

In a large class of static games, we show that a very small commitment ability is sufficient. Why? Because in general, a small policy change is sufficient to deter a private sector action. So the multiplicity issue is easy to overcome.

Let us take a concrete example. In the capital taxation problem, the government may decide to tax savings ex post, which can discourage investment ex ante. As a result, there may be three Nash equilibria in the absence of commitment ability. In the figure below, the three Nash equilibria correspond to the three intersections between the competitive outcomes curve in red – along which the aggregate capital is optimal given capital taxation – and the ex-post optimal capital taxation in blue. To rule out the two suboptimal outcomes with too low capital accumulation (the first two Nash equilibria), the government can commit to lower than expected tax (black diamonds). These commitments do not require large commitment ability.

To come back to the initial motivation about banks’ bailout, our result suggests that ruling out equilibrium multiplicity only requires a low commitment ability. In this example, the government (or the central bank) should commit to a partial bailout such that it is always optimal for banks to take less risk than the aggregate. In the end, the only remaining equilibrium is the one without banks’ bailout and optimal level of risk-taking.

But sometimes implementation requires more commitment ability: when bold public actions are needed to deter some private sector actions - e.g. when private agents are not fully rational or the actions set is discrete - or when information is imperfect. More commitment ability is also needed when the "game is repeated" and the private sector can react to past policy decisions. Then, government’s patience may be an obstacle for solving equilibrium multiplicity and more commitment ability does not necessarily increases the social welfare.