The role of interaction

It is interesting to investigate from a game-theoretic perspective why the decentral-ization of authority over policy decisions allows for the emergence of a new MPE

19In deriving (1.24), we have exploited the rational expectations consistency condition which links the initial interest rate (1 +R₀) to the initial rate of money expansion (1 +µ₀). The impli-cation is that the ratio _(1+µ^z0

0) = _M^B0

0

(1+R0)

(1+µ₀) does not depend on the initial rate of money growth.

Consequently, (1.24) holds for any stationary allocation, irrespective of the stationary rate of money growth (1 +µ).

which was not sustainable in a single-agency economy where the policy maker con-trols both policy instruments, τ^c and M⁰. Specifically, it may not be immediately clear why the quasi-indexation MPE with interacting authorities is not an equilib-rium with a single policy maker. The point is that a single decision maker should in principle be able to replicate anything that could be implemented as an equilibrium with interaction among separate government authorities. This issue is of particular relevance because the quasi-indexation equilibrium which becomes available under interaction dominates the single-agency equilibrium both in welfare terms and in a Pareto sense from the two authorities’ perspective.

So why is the stationary equilibrium under interaction not an equilibrium with a single decision maker? The answer to this question lies in the underlying time in-consistency problem of the monetary authority and the properties of the respective policy instruments. With a monolithic policy maker, it is the case that, once private expectations are fixed, whatever is done via the consumption tax τ^c can be done more efficiently via the inflation tax induced by money injections M⁰. The reason is that, while both policy instruments operate on the same distortionary margin, the inflation tax has the additional benefit of relaxing future implementability con-straints by deflating the inelastically supplied real stock of debt. Thus, the single policy authority has always an incentive to substitute M⁰ for τ^c and is thus sub-ject to a time inconsistency problem which makes a non-inflationary equilibrium unsustainable.

Conversely, with interacting authorities, the quasi-indexation equilibrium is sus-tainable due to the asymmetry of the dynamic constraints faced by the two author-ities. While the monetary authority is still subject to a time inconsistency problem, the fiscal authority is not subject to such a problem because, even when private ex-pectations are fixed, its policy instrument does not allow for non-distortionary rev-enue generation. Indeed, with decentralized decision power, each authority makes its current policy choice not only for given private expectations, but - as an implica-tion of the Nash equilibrium prevailing in any stage game - also for a given current policy choice by the respective other authority. That is, unlike in standard optimal policy models, decentralized decision making does not allow to substitute one pol-icy instrument for another because the respective other authority’s polpol-icy rule is a given constraint rather than a free choice variable. In other words, the reason why a time consistent policy ruleϕimplyingε_µ(z;ϕ) = 0 can be sustained is an instance of coordination failure among the two interacting authorities. The consequence is that there is scope for a favorable coordination of private expectations under de-centralized decision power over policies: A fiscal policy which keeps the level of real debt constant without the need to recur to the inflation tax makes zero-inflation expectations rational.²⁰ But given such expectations, the extra liability costs of

20As indicated by (1.8), what ultimately matters is the composite distortion (1 +R)(1 +τ^c)

nominal debt vanish, and consequently the rationale for an erosion of the stock of outstanding debt via surprise inflation disappears. The point is that the monetary authority is not willing to subject the economy to additional distortions on top of those already caused by the fiscal authority in order to drive down the stock of real debt.²¹

Crucial for this conclusion is that not only the current fiscal authority’s play is taken as given (i.e. τ^c(z) is a given number), but that the same is true also for the continuation play (i.e., in the proposed equilibrium, τ^c(z⁰;ϕ) and µ(z⁰;ϕ) are given nondecreasing functions). This makes the current monetary authority’s problem different from the situation faced by a Ramsey planner who can simultaneously control future policies when deciding about the current allocation. Indeed, even when constrained by current taxes being fixed at a level that, absent a monetary expansion, would keep the aggregate state z constant, a Ramsey planner would decide to additionally use the inflation tax in order to relax the implementability constraint she will face in the subsequent period. In contrast, under sequential policy implementation, there is an additional feedback induced via the given future policy rule ϕ; this feedback makes the current monetary authority prefer to refrain from using the inflation tax.²²

The stationarity of the allocation implemented via the quasi-indexation MPE suggests that the time inconsistency problem of optimal policy is not necessarily relevant with interacting authorities. To clarify this statement, two remarks are expedient: First, as a consequence of the fact that the two policy instruments con-sidered are equivalent with respect to the margins that they distort, there may be multiple combinations of time invariant policy rules for τ^c and M⁰ which decentral-ize the same allocation. This is true also for the quasi-single agency MPE, because what ultimately matters for the determination of a MPE is the value taken by the variableε_µ(z⁰;ϕ) and not the specific policy rule ϕthat decentralizes the allocation.

rather than the interest rate distortion in isolation. Indeed, there are also inflationary (in the sense of (1 +µ(z))>1) policy rules which implement the quasi-indexation allocation. The critical property of these policy rules is that monetary policy does not respond to marginal variations in the stock of real debt, i.e. ε_µ(z;ϕ) = 0.

21This is a consequence of the intertemporal elasticity of substitution of one implied by log-utility. Specifically, the same forces that make a single policy maker implement policies that asymptotically drive the stock of real debt to zero (and not to negative or positive values) imply that the monetary authority, when faced with a budget balancing fiscal policy, does not want to inflate the economy any more. For a general specification of preferences, the described effect would emerge in a modified version, where the MPE outcome involves the authorities implementing a negative (positive) long-run level of real debt if the intertemporal elasticity of substitution is larger (smaller) than one.

22In this context, D´ıaz-Gim´enez et al. (2006) discuss an interesting constellation where the periodt fiscal authority commits its policy (as a fixed number) one period in advance such that the feedback via the reaction of the period t+ 1 fiscal policy is absent; then, the quasi-indexation equilibrium collapses.

Hence, the key difference between the two classes of MPE is that, in the quasi-single agency MPE, inflation is always systematically related to the dynamic inconsistency in the policy problem which results in an inflation bias as long as there are posi-tive amounts of outstanding public liabilities; in contrast, in the quasi-indexation MPE, the responsiveness of inflation to the stock of public liabilities z is always zero. Second, it has been established that the decentralization of authority over macroeconomic policies facilitates the existence of a MPE implementing a superior outcome because the interest rate distortions stemming from the dynamic inconsis-tency of optimal policies are absent. However, the inferior allocation implemented by the quasi-single agency MPE cannot be ruled out neither such that the econ-omy is in a situation of multiple equilibria. Although the quasi-indexation MPE is welfare-superior and even Pareto-dominant from the two authorities’ perspective, its selection is not automatically guaranteed because the authorities’ period-by-period incentive to coordinate their policy choices - cutting back onτ^c and substituting via an increase in M⁰ - makes the time consistency problem potentially reappear and thereby undermines the sustainability of the superior MPE.²³ Indeed, one might want to argue that communication between monetary and fiscal authorities ren-ders the non-cooperative outcomes of Nash play implausible. Note however that, as long as formal contracts are unavailable, the set of self-enforcing plans (in the sense of correlated equilibria) under direct communication between the two authorities would still be given by randomizations among the Nash equilibria of the original non-cooperative game.²⁴ Hence, without further arguments, the quasi-indexation MPE cannot be dismissed.