Sustainable monetary policy and inflation expectations

A.1 Environment and sustainable-equilibrium definition

This is an infinite horizon economy t=1, 2, … Monetary policy is characterized as the outcome of a repeated game between the monetary authority and a continuum I of agents, which are denoted as a whole as the private sector.

At the beginning of each period t≥1, the exogenous state s_t becomes common knowledge. I assume that the exogenous state s_t∈S is governed by a stochastic first-order Markov probability process μ.

After the exogenous state is observed, private-sector agents choose some variable, z_it∈Z. For example, some firms may set nominal prices in advance or unions may agree on nominal wages for the period. Let zt=∫zitdi be the average of all agents decisions. I will assume that the private-sector agent’s and monetary authority’s payoffs depend on actions {z_it}_I only through its average z_t. This implies that a private-sector competitive equilibrium can be thought of as the average action zt∗ solving the optimality conditions zt∗=ζ(zt∗, π^t), where π^t is the private-sector inflation expectation. Furthermore, I assume that, given an inflation expectation π^t, the private-sector competitive equilibrium is unique. Hence, the private-sector competitive equilibrium can be summarized by π^t, which is treated as the “ action” of the private sector.

Finally, the monetary authority sets the actual inflation rate π_t taking as given the private sector’s inflation expectations π^t. Inflation is bounded below by the zero-lower bound in the nominal interest rate, π_t∈Π≡{π≥δ} where δ>0.

The economy considered is very simple. Output and inflation are linked by a classic Phillips curve,

yt=y(πt−π^t; st)

where y(πt−π^; st) is strictly positive, increasing in the first argument, and bounded above. Let y^*(s_t)≡y(0, s_t). Output is transformed one-to-one into a non-storable consumption good. The aggregate resource constraint is then c_t≤y_t.

The household period payoff is given by u(y_t)–g(π_t), where u is a strictly increasing, concave, and twice differentiable function; and g is a strictly increasing and twice differentiable equation. Function u is assumed to be bounded below and g bounded above. Finally, I assume that the monetary authority is benevolent so its period payoff is given by u(y_t)–g(π_t) as well.^[24]

I briefly introduce the time-inconsistency problem in a one-period version of this economy. First, I will define an one-period Nash equilibrium – the discretionary equilibrium in the language of Barro and Gordon (1983b). I will also characterize the one-period optimal monetary policy.

Definition 1:A one-period Nash equilibrium at state s is π≥δsuch that

u(y∗(s))−g(π)≥u(y(π′−π; s))−g(π′)

for allπ′≥δ.

I will assume that there exists a one-period Nash equilibrium for each state s∈S.

Definition 2:The optimal monetary policy at state s is π≥δ such that

u(y∗(s))−g(π)≥u(y∗(s))−g(π′)

for allπ′≥δ.

The optimal monetary policy is trivially given by π=δ for all s∈S.^[25] The time-inconsistency problem arises when the two equilibrium concepts are not equivalent, i.e. the optimal monetary policy is not the unique one-period Nash equilibrium. If this is the case, some commitment technology is required to implement the optimal monetary policy as the unique rational expectations equilibrium.

The set of Nash equilibrium is usually enlarged in the infinite horizon economy. In particular, history-dependent equilibria arise, and some of the equilibria feature trigger strategies that resemble reputation.

I start by introducing the notation for histories.^[26] Let π^t={π₁, π₂…, π_t}∈Π^t denote the history of the monetary authority’s actions up to date t≥1. In a similar fashion define π^t∈Πt and s^t∈S^t for t≥1. To denote the set of continuation histories at s^t, I use S^j|s^t, j≥t. The payoff-relevant node is ht={πt, π^t, st}∈Ht≡Π2t×St for t≥1, with h⁰={∅}. Node h^t is also the end-of-the-period history.

A consumption plan c={c(h^t):h^t∈H^t, t≥1} specifies a level of consumption at every node h^t∈H^t, t≥1. Let F be a probability measure over all game nodes H^t, t≥1. The private-sector welfare at any h^t, t≥1, is defined over c given F

where 0<δ<1 is the intertemporal discount rate. I stretch the definition of v to ex-ante welfare

Finally, the resource constraint is given by

for all h^t∈H^t, t≥1.^[27]

Now I describe the strategy space. For the private sector, the relevant decision node is (h^t−¹, s_t) as the present state of the economy is known in addition to the history of past actions.^[28] The period strategy is denoted by σ^(ht−1, st)∈Δ(Π). The strategy space contains mixed strategies in order to allow equilibria where private-sector expectations are driven by a sunspot variable. The private-sector strategy is then

σ^={σ^(ht−1, st):(ht−1, st)∈Ht−1×St, t≥1}.

The monetary authority is aware of the history of actions, the state of the economy, and the realization of the private-sector strategy. The relevant node for the monetary authority is then xt≡(ht−1, st, π^t)∈Xt≡Ht−1×St×Π and its period strategy is σ(x^t)∈Π. Let

σ={σ(xt):xt∈Xt, t≥1}

denote the monetary authority strategy.

The chosen equilibrium concept is the sequential equilibrium.

Definition 3:A sequential equilibrium (SE) is a strategy profile {σ^, σ}, a probability distribution F and a consumption plan c such that:

1. For all x^t∈X^t, t≥1

   v(c; {xt, σ(xt)}, F)≥v(c; {xt, π′}, F)

   for all π′≥δ;

2. For all (h^t−¹, s_t)∈H^t−¹×S, t≥1, σ^is such that

   π^t=σ(xt)

   almost everywhere in H^t|X^t;

3. For all h^t∈H^t, t≥1, the resource constraint (7) holds;

4. Probability measure F is generated from{σ^, σ}.

The key elements of the sequential-equilibrium definition are the subgame perfection requirement and rational expectations. The former ensures that the monetary authority’s decision maximizes private-sector welfare at every decision node – this is Condition 1 in the definition. Condition 2 imposes rational expectations: it implies that the private sector has the correct beliefs with respect to the actual inflation rate along the equilibrium path. Note that Conditions 1 and 2 are very different. The private-sector “ player,” actually a continuum of atomistic agents, does not behave strategically.

It is straightforward to show that the infinitely repeated version of a one-period Nash equilibrium is a SE. Usually there will be many SE with distinct welfare properties. The SEs that deliver the highest welfare possible are of obvious interest.

Definition 4:A best sequential equilibrium (BSE) is a sequential equilibrium {σ, σ^, F, c} such that

v(c; F)≥v(c′; F′)

for all sequential equilibria{σ′, σ^′, F′, c′}.

Before proceeding to the discussion of the equilibrium properties, I define a monetary policy plan as a probability measure M over Π^t×S^t. Clearly, for a given SE {σ, σ^, F, c}, the probability measure F defines a monetary policy plan. Following Chari and Kehoe (1990), a monetary policy plan will be sustainable if there exists a SE generating the policy plan.

Definition 5:A probability measure M over Π^t×S^tis a sustainable monetary policy plan if there exists a sequential equilibrium {σ, σ^, F, c} such that M is generated by F.

I will often use sustainable plan for short.

A.2 Sustainable-equilibrium properties

A key object in the analysis of SE is the equilibrium value set, i.e. the set of private-sector welfare associated with a SE. It will prove instrumental to characterizing the BSE. For convenience, I first define the equilibrium value set conditional on the state s,

V(s)≡{v(c; s, F)|{σ, σ′, F, c} is a SE}

and then the ex-ante welfare set

V≡{v(c; F)|{σ, σ′, F, c} is a SE}.

As mentioned earlier, the infinitely repeated version of a one-period Nash equilibrium is a SE. Hence V(s) is not empty for all s∈S.

The remainder of this section builds on the seminal work of Abreu, Pearce and Stacchetti (1990) to characterize the set V(s) and then the BSE. The private-sector welfare can be written recursively. From (4), the end of period welfare can be decomposed in the period payoff and the continuation value,

v(c; ht, F)=u(c(ht))−g(πt)+δ∑st+1μ(st+1|st)∫Ht+1v(c; ht+1, F)F(dht+1|ht).

The key observation is that every subgame in a SE constitutes a SE as well. This implies that, at any node of the game, the continuation payoffs must belong to V(s) as well, i.e.

∫Htv(c; ht, F)F(dht|ht−1, st)∈V(st)

for all (h^t−¹, s_t)∈H^t−¹×S, t≥1.

The next step is to define a sustainable action π∈Π. Sustainable actions are the building blocks of a SE. Loosely speaking, for an action π_t to be sustainable, there should be subgame-perfect strategies such that the monetary authority validates the private-sector expectations π^t=πt. These strategies must specify a plan of action for every possible monetary authority action, and each of these plans must be sustainable.

To characterize an action as sustainable seems potentially complicated. It is possible, though, to easily characterize sustainable actions in terms of continuation values: for an action π to be sustainable, there should be a schedule of continuation values in {V(s)}_s∈_S such that the monetary authority validates the private-sector expectations π^=π.

Definition 6:An action π∈Π is a sustainable action at s∈S if there exists vectors w₁, w₂ ∈{V(s′)}_s′∈_Ssuch that

u(y∗(s))−g(π)+δ∑s′μ(s′|s)w1(s′)≥ u(y(π′−π; s))−g(π′)+δ∑s′μ(s′|s)w2(s′)

for allπ′≥δ.

Vector w₁ details the continuation value for each state s′ if the monetary authority validates the private-sector expectations; loosely speaking, it is the “ reward.” The “ punishment” vector w₂ states the continuation value if the monetary authority does not validate the private-sector expectations. Because continuation values belong to {V(s)}_s∈_S, they correspond to some SE so there exist sub-game perfect strategies sustaining the action π. Note also the sustainable actions are defined with a simple punishment rule: any deviation π′ has the same punishment, i.e. the same continuation value vector w₂. These simple punishment rules are without loss of generality, as first pointed out in Abreu (1988).

The next theorem states the key results from Abreu, Pearce and Stacchetti (1990) used in this paper.

Theorem 1:For all s∈S, V(s) is a non-empty and compact set in ℜ. Moreover, for any sequential equilibrium {σ, σ^, F, c} and for all nodes h^t−¹∈H^t−¹, t≥1,

1. The private-sector welfare satisfies

   v(c; hj, F)∈V(s)

   almost everywhere in H^j|h^t−¹, j≥t–1;

2. Strategy σ specifies sustainable actions almost everywhere X^j|h^t−¹, j≥t–1.

Finally, for any profile of sustainable actions {π(s)}_s∈_Sand a date t≥1, there exists a sequential equilibrium such thatF assigns probability one to {π(s)}_s∈_Sat date t.

Proof: See Abreu, Pearce and Stacchetti (1990). For a simpler proof restricted to perfect monitoring games such as the present one, see Cronshaw and Luenberger (1994) or Ljungqvist and Sargent (2000)■

It is useful to rephrase Theorem 1 above in terms of sustainable plans. First, it implies that a sustainable plan M assigns positive probability only to sustainable actions. Moreover, for a given profile of sustainable actions and a given date, there exists at least one sustainable plan implementing these actions at the given date with probability 1. However, Theorem 1 does not imply that any probability measure M over sustainable actions is sustainable.

Everything is in place to characterize the BSE and the most important result of this section. Using the results in Theorem 1, I show that the BSE features the lowest sustainable inflation rate along the whole equilibrium path. Hence, while strategies are history dependent in a potentially complicated way, the best sustainable plan M^* is unique and strikingly simple: along the equilibrium path inflation depends only on the current state s.

Proposition 1:A sequential equilibrium is a best sequential equilibrium if and only if σ(x^t)=π^*(s_t) almost everywhere in X^tfor all t≥1 where

π∗(s)=inf{π|π is sustainable at s}.

Proof: Using the fact that V(s) is compact, π is sustainable if and only if

for all π′≥δ.

I will use this to prove that the set of sustainable actions is non-empty and compact for all s∈S. First, the set always includes any one-period Nash equilibrium and (8) follows trivially. Consider a convergent sequence {πj}j=0∞→π˜ of sustainable actions at s. Since the right-hand side of 8 is a measurable function for all π′≥δ, it follows that π˜ satisfies (8). Hence π^*(s) exists and it is a sustainable action at s.

The second part of the proof shows that π^*(s) effectively characterizes the BSE. First, I prove that V(s) is bounded above by

v¯(s)=u(y(π∗(s)))−g(π∗(s))+δ∑s′μ(s′|s1)sup{V(s′)}.

To prove this, assume there exists a SE with strictly larger value v(c˜; s1, F˜)>v¯(s1) for s₁∈S. Because all actions with positive probability in a SE are sustainable actions, v(c˜; h2, F˜)≤sup{V(s2)} almost everywhere in H². Hence v(c˜; s1, F˜)>v¯(s1) implies that with positive probability

u(y(π∗(s1)))−g(π∗(s1))<u(y(π∗(s1)))−g(π˜(h1))

and thus g(π˜(h1))<g(π∗(s1)). But g(π) is strictly increasing in π. Therefore it would imply that there exists a sustainable inflation rate π˜(h1) strictly lower than π^*(s₁), which violates the definition of π^*(s₁).

It follows that v̅(s₁)=sup{V(s₁)} – from Theorem 1 v̅(s₁)∈V(s₁) as there exists a SE featuring π^*(s₁) at date t=1.

The same construct for any s∈S implies

for all s∈S. The system of equations described by (9) proves the if and only if part of the proposition.■

Next I present a simple corollary of Proposition 1 that plays an important part in the result. It implies that the BSE must be sustained by promising the highest reward and threatening with the utmost punishment. Hence, if at date t=1 the best sustainable inflation π^*(s₁) is achieved for all realizations of s₁, it must be that the BSE follows. The corollary highlights the importance of the recursive structure of the SE.^[29]

Corollary 1:If π^*(s)>δ, a sequential equilibrium with σ(x¹)=π^*(s₁) almost everywhere in X¹is a best sequential equilibrium.

Proof: Assume there exists a SE {σ˜, σ˜′, c˜, F˜} with σ˜(x1)=π∗(s1) a.e. in X¹ but v(c˜; s1, F˜)<sup{V(s1)}. It then implies that there exists a reward continuation value w₁which renders π^*(s₁) sustainable with δ∑s′μ(s′|s1)(sup{V(s′)}−w1(s′))>0. Since π^*(s₁)>δ, then there exists π^**<π^*(s₁) satisfying (8) and contradicting the definition of π^*(s₁)■

A.3 Date t=0 asset markets

In this section, I introduce trade in nominal and real assets. I lay out a trade structure that provides asset prices but preserves the monetary policy game exactly as described above. Neither payoffs nor action sets are modified; in a precise sense to be made clear later, the equilibrium value set is left intact by asset trade.

A.4 Asset prices and the best sustainable equilibrium

In this section I analyze the mapping between two components of a date t=0 asset market equilibrium: the asset prices and the sustainable equilibrium. This mapping is found to be a correspondence, but the key result is actually characterized by an exception: the relationship between the short-term nominal interest rate and the best sustainable equilibrium is one-to-one.

I start by proving the claim that asset trading has not changed the equilibrium value set. Of course, I disregard the trivial difference induced by the inclusion of period t=0. I define the equilibrium value set for date t=0 asset market equilibrium (AME) as

V˜={v(c; F)|{F, c} are part of a AME}.

By definition, an AME contains a SE; thus V˜⊂V. Proposition 2 proves that V⊂V˜ as well.

Proposition 2:Let {σ, σ^, F, c} be a sequential equilibrium. Then there exists a unique date t=0 market equilibrium with {σ, σ^, F, c}.

Proof: Given a SE, the price level is trivially given by (10). Private-sector bond holdings are given from the market clearing conditions, b^=−b and B^=−B, and the lump-sum taxes are solved from the government budget constraints (11) and (12). The only non-trivial objects in a candidate AME are asset prices. The necessary first-order conditions associated with (13) are

uc(c0)q(st)=δtμ(st|s0)∫Htuc(c(ht))F(dht|st),uc(c0)Q(st)=δtμ(st|s0)∫Htuc(c(ht))P(ht)F(dht|st)

for all s^t∈S^t, t≥1. The only candidate for c₀ is y₀, as imposed by feasibility. Hence, for a given SE, asset prices are uniquely pinned down by the above necessary first-order conditions. Since P(h^t)>0 for all h^t∈H^t, and u^c(c)>0, asset prices are positive. Finally, consumption decisions satisfy the household budget constraints by Walras’ Law■

Asset prices in an AME are characterized by the necessary first-order conditions associated with (13). Since a SE requires πt=π^t almost everywhere, c(h^t)=y^*(s_t) with probability one along the equilibrium path. Therefore real and nominal asset prices satisfy

for all s^t∈S^t, t≥1. Thus the real interest rate is exogenous. The price level P(h^t) is a function only of π^t. This allows writing asset prices in terms of the associated sustainable plan M. It is useful to combine the asset price conditions:

Finally I price a short-term nominal bond. It pays one nominal unit in all states of the world at date t=1, and thus its discount price is

R0−1=∑s1∈SQ(s1).

The two key outputs from an AME are the sustainable plan and the asset prices. I want to analyze the mapping between these two equilibrium objects. It turns out to be convenient to think of the mapping in terms of value sets. Let

Γ({q, Q})={v(c; F)|{F, c, q, Q} are a part of an AME}

defined over any positive asset price schedule {q, Q}. Proposition 2 has established that, given a sustainable plan M, there exists a unique set of asset prices that constitute an AME generating M. However, the converse is not true. Given asset prices {q, Q} there may be no SE conforming to an AME. Or there may be multiple SE. This is important in order to understand the results on coordination.

First I show that there may be no AME for given asset prices {q, Q}. An extreme example is convenient. Assume there is a unique one-period Nash equilibrium, π˜, and, for simplicity, assume it is state invariant. It can be shown that, for low enough δ>0, only π˜ is a sustainable action and hence all SEs must feature π˜ with probability one. Therefore, for any asset prices {q, Q} such that

Q(st)=q(st)1π˜t

there would be no AME with such prices. In this example, Γ({q, Q})=∅ for all but one price schedule.

Next I consider the possibility of multiple SEs associated with given asset prices {q, Q}. For simplicity there is no exogenous uncertainty s_t=s̅ for all t≥1, with y^*(s̅)=y₀. Assume that there exists a SE {σ, σ^, F, c} that places probability one in history h˜t={π˜}t for all t≥0 and its value lies in the interior of V(s̅). Then there exists a SE {σ′, σ^′, F′, c′} that places equal probability to histories {π˜+ε1, π˜, π˜, ..} and {π˜−ε2, π˜, π˜, ..} with

1/2π˜+ε1+1/2π˜−ε2=1π˜

and ε₁ small enough. That {σ′, σ^′, F′, c′} is indeed a SE follows from the fact that, for an arbitrarily small ε₁, both π˜+ε1 and π˜−ε2 are sustainable actions. As long as the costs of inflation g(π) have some curvature, the two SEs have different ex-ante welfare.

By construction, both SEs have the same asset prices,

q(s)=δ,Q(s)=δ1π˜=δ(1/2π˜+ε1+1/2π˜−ε2).

Hence these asset prices {q, Q} do not determine the ex-ante welfare level of an AME. Hence, Γ({q, Q}) is not a singleton.

Hence the mapping from date t=0 market equilibrium asset prices to sustainable plans is a correspondence, Γ:{q, Q}⇉ {V,Ø}. However, asset prices are informative to some extent. A key observation is that the arbitrage condition (18) implies a one-to-one mapping between asset prices and the following moment of the distribution of the price level,

q(st)/Q(st)=[∫Πt1P(ht)M(dπt|st)]−1.

Many SEs or no SE may share this moment, yet it is unlikely that all SEs do. In particular, the moment implies that asset prices constrain the support of the compatible sustainable price levels. For example, a sustainable plan assigning probability one to prices strictly above or strictly below q(s^t)/Q(s^t) will not satisfy the moment.

The next is the key result of the paper. As I just discussed, the mapping between asset prices and sustainable policy plans is generally a correspondence. However, the exception is of utmost interest: a date t=0 market equilibrium features the best sustainable equilibrium if and only if the one-period nominal bond is priced at a certain discount rate.

Theorem 2:A date t=0 asset market equilibrium features a best sequential equilibrium if and only if

Proof: If an AME is a BSE, then (17) implies Q(s1)=δμ(s1|s0)uc(y∗(s1))uc(y0)1π∗(s1) for each s₁∈S and (19) follows trivially.

Consider now a AME satisfying (19). The arbitrage condition (18) implies that

∑s1∈SQ(s1)=∑s1∈Sq(s1)(∫Π1P(h1)F(dπ1|s1))

and combined with (19)

∑s1∈Sq(s1)(1π∗(s1)−∫Π1P(h1)F(dπ1|s1))=0.

By the definition of π^*, P(h¹)≥≥π^*(s₁) for any SE. Therefore

1π∗(s1)−∫Π1P(h1)F(dπ1|s1)≥0

for each s₁∈S. It follows that P(h¹)=π^*(s₁) for all s₁∈S. Finally, Condition 1 implies that Corollary 1 applies and the only SE compatible with (19) is the BSE■

The first thing to note about Theorem 2 is that the price of a single asset for a single date is sufficient to pin down the BSE for all periods. This stands in contrast with practically any other complete price schedule, i.e. it is not possible to pin down uniquely other SEs even if all asset prices can be determined. This strong result is built on the properties of the BSE summarized in Proposition 1. In particular, the BSE belongs to the boundary of both the value set V and the set of sustainable actions. It is straightforward to show that the best sequential equilibrium requires the highest discount rate for the short-term nominal interest rate.

Corollary 2:For any date t=0 asset market equilibrium,

R0−1≤δ∑s1∈Sμ(s1|s0)uc(y∗(s1))uc(y0)1π∗(s1)

The proof of Theorem 2 applies almost step-by-step to other asset trade arrangements. If only the short-term nominal bond were available, then (19) is exactly the associated necessary first-order condition. Asset trade could also be sequential with no change in the logic of the result. Of course, if asset holdings do change the monetary authority’s incentives, then asset prices would also bring cooperative considerations.

It is important to note that Condition 1 is in effect: Theorem 2 does not apply if the zero nominal interest rate bound is binding for the BSE at date t=1. It is easy to find parameters such that (19) would imply a negative interest rate. Interestingly, setting ∑s1∈SQ(s1)=1 does not solve the problem. Because I use Corollary 1 to prove the result, if the BSE is the optimal monetary policy, there could be other sustainable plans that start at the zero nominal interest rate level but inflation eventually takes off.