ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY ANNA ORLIK

IGNACIO PRESNO

STERN SCHOOL OF BUSINESS, NEW YORK UNIVERSITY AND NEW YORK UNIVERSITY

JOB MARKET PAPER

Abstract. This paper studies the design of optimal time-consistent monetary policy in an economy where the planner trusts his own model, while a representative household uses a set of alternative distorted probability distributions governing the evolution of the exogenous state of the economy. In such environments, unlike in the original studies of time-consistent monetary policy, management of households’ expectations becomes an active channel of optimal policymaking per se; a feature that our paternalistic government seeks to exploit. We adapt recursive methods in the spirit of Abreu, Pearce and Stacchetti (1990) as well as computational algorithms based on Judd, Yeltekin and Conklin (2003) to fully characterize the equilibrium outcomes for a class of policy games between the government and a representative household who distrusts the model used by the government.

1. Introduction Undoubtedly, inflation expectations of the public influence greatly actual inflation, and, therefore, central bank’s ability to achieve price stability which is central to good monetary policy. But what do we mean precisely by the "state of inflation expectations"? And, most importantly, what role does the optimal monetary policy play in shaping or managing inflation expectations1? As noted by Sargent (2008), under rational expectations paradigm, a government strategy plays two roles; first, as an actual decision rule, and second, as a system of private sector expectations about that very government action. The theory is silent about who chooses a particular equilibrium system of beliefs. Is it the government, by following a particular decision rule? Or is it the public, by forming a particular expectation about the course of policy actions to be undertaken Date: November 18, 2010. Corresponding author: [email protected]. We are grateful to David Backus, Timothy Cogley, Thomas J. Sargent and Stanley Zin. 1

These questions are subject of Bernanke (2007). 1

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by the government? The concept of rational expectations equilibrium precludes this type of deliberation. Instead, the theory is about the government coming into a given period confronting the private sector’s expectations concerning his own actions, which he is bound to confirm. In that sense, the government can be thought of as being trapped by public expectations about what he would do2. In this paper, instead, the management of private beliefs by a central banker becomes an integral part of the theory of optimal monetary policy making. In our modeled economy, which we construct in the tradition of monetary models of Calvo (1978) and Chang (1998), a representative household derives utility from consumption and real money holdings. Government uses the newly printed money to finance transfers or taxes to households. Taxes and transfers are distortionary. The only source of uncertainty in this economy affects the degree of tax distortions through its influence on households’ income. At the heart of this paper lies the assumption that the government has a single approximating model that describes the evolution of the underlying shock while a representative household fears it might be misspecified. To confront that concern, a representative household contemplates a set of alternative probability distributions governing the evolution of the underlying shock and seeks decision rules that would work well across the models. The fact that private agents are unable to assign unique probability distribution to alternative outcomes has been demonstrated in Ellsberg (1961) and similar experimental studies3. Moreover, lack of confidence in the models seems to have become apparent during the recent financial crisis4. Below we present the quote from Bernanke (2010) "Most fundamentally, and perhaps most challenging for researchers, the crisis should motivate economists to think further about modeling human behavior. Most economic researchers continue to work within the classical paradigm that assumes rational, self-interested behavior and the maximization of "expected utility" (...). An important assumption of that framework is that, in making decisions under uncertainty, economic agents can assign meaningful probabilities to alternative outcomes. However, during the worst phase of the financial crisis, many economic actors–including investors, employers, and consumers– metaphorically threw up their hands and admitted that, given the extreme and, in some ways, unprecedented nature of the crisis, they did not know what they did not know."

The government in our model follows the above advice; he recognizes that households are not able or willing to assign meaningful probabilities to alternative realizations of the random state of the economy. He wants to design optimal policy that explicitly accounts for the fact that households’ 2

Chari, Christiano, and Eichenbaum (1998) and Albanesi (2002) interpret the inflation of the 1970s and its sta-

bilization in the 1980s as actions undertaken by a benevolent government who administered bad policies because he was trapped by the public expectations. 3

See, e.g. Halevy (2007).

4

See e.g. Caballero and Krishnamurthy (2008), and Uhlig (2010).

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

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allocation rules are determined by their distorted beliefs; as summarized by our notion of competitive equilibrium with model uncertainty. To solve for that optimal policy, first we work under the assumption that the government can commit at time zero to a policy specifying its actions for all current and future dates and states of nature. Under this assumption, a government chooses at time zero the best competitive equilibrium from the set of competitive equilibria with model uncertainty, i.e. one that maximizes the expected households’ lifetime utility but under his own unique belief. We will refer to such a government as paternalistic Ramsey planner. The key insight that emerges from our recursive approach to that problem is that there are essentially two channels (two instruments) through which the planner affects equilibrium outcomes. First, by announcing the policy for tomorrow (in this paper, it is the promised marginal value of money), the planner affects the intertemporal allocation decision of households for consumption and real money holdings. This role of government’s strategy has been studied extensively following Kydland and Prescott (1980). But this paper features a second novel channel; by choosing continuation values to the household, the planner in our economy influences households’ expectations of his own policy directly. This is due to the fact that, in equilibrium, the degree of distortion in households’ beliefs (with respect to the approximating model) is a function of households’ continuation values. Once we abstract from the assumption that the government has the power to commit, and, instead, chooses sequentially, time inconsistency problem may arise, as first noted by Kydland and Prescott (1977) and Calvo (1978). As a consequence, it is urgent to check whether the optimal policies derived by our paternalistic Ramsey planner are time consistent, and, more generally, to characterize the set of sustainable plans with model uncertainty. This latter notion should be thought of as an extension of Chari and Kehoe (1990, 1993). Characterizing time-consistent outcomes is a challenging task. This is because any time consistent solution must include a description of government and market behavior such that the continuation of such behavior after any history is a competitive equilibrium and it is optimal for the government to follow that policy. Consequently, checking whether a given policy is time consistent requires solving a nontrivial infinite horizon problem for each one of the infinite number of histories. In this paper, we use insights from the work by Abreu, Pearce, and Stacchetti (1990), Chang (1998), Phelan and Stacchetti (2001) to develop recursive methods to study sets of equilibrium payoffs associated with a sustainable plan with model uncertainty. We provide the series of algorithms in the spirit of Judd, Yeltekin and Conklin (2003) based on hyperplane approximation methods that let us compute the

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sets in question. The characterization of the entire set of sustainable equilibrium values facilitates the examination of practical questions. Our numerical examples suggest that policies that account for the fact that households contemplate a set of probability distributions may lead to better outcomes. Although in this paper we restrict attention to the type of models of monetary policymaking which can be cast in the spirit of Calvo (1978), hopefully it will become clear that our approach could be applicable to many repeated or dynamic games between a government and a representative household who distrusts the model used by the government. To our knowledge, there are two papers that try to explore the role of policy maker in managing households’ expectations. Karantounias, Hansen and Sargent (2009) study the optimal fiscal policy problem of Lucas and Stokey (1983) but in an environment where a representative household distrusts the model governing the evolution of exogenous government expenditure. The authors apply techniques of Marcet and Marimon (1998) to characterize the optimal policies when the government has power to commit. Woodford (2010) discusses the optimal monetary policy under commitment in an economy where while both the government and the private sector fully trust their own models, the government distrusts its knowledge of the private sector’s beliefs about prices. The remainder of this paper is organized as follows. Section 2 sets up the model and outlines the assumptions made. In Section 3 we introduce the notion of competitive equilibrium with model uncertainty. In Section 4 we discuss the recursive formulation of the Ramsey problem for the paternalistic government. Section 5 contains the discussion of sustainable plans with model uncertainty. In Section 6 we describe the computational algorithms we have implemented to determine the set of all the equilibrium values to the government and to the representative household, and promised marginal utilities. Also, we present some numerical results. Section 7 briefly discusses an alternative hypothesis with both the government and households using, possibly distinct, sets of models. Finally, Section 8 concludes.

2. Model Uncertainty in the Problem of A Representative Household Two infinitely-lived players - a representative household, H, and a government or a planner, G, interact in a dynamic environment at discrete dates indexed as t = 0, 1, .... The only source of risk and uncertainty in the modeled economy is given by the shock, st . Formally, let (Ω, F, Pr) be the

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

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underlying probability space. The shock process, (st )t , is an exogenous stochastic process such that s0 ∈ S is given (i.e. known to a representative household and to the planner) and st : Ω → S for all t > 0. Let st ≡ (s0 , s1 , ..., st ) ∈ S × S × ... × S ≡ St+1 describe the history of the realizations of
productivity shock up to and including time t. Let S t ≡ F st be the sigma-algebra generated by the history st .

2.1. Planner. Planner in this economy chooses how much money, Mt (st ), to supply; to create or Mt−1 (st−1 ) to withdraw from circulation. Let ht (st ) = Mt (st ) denote the inverse rate of money growth, and qt (st ) the price of money, and assume [A1] ht (st ) ∈ Π ≡ [π, π] with 0 < π <

1 β

≤π

where β ∈ (0, 1) denotes the time discount factor of the planner (and that of a representative household). [A1] constitutes a bound on admissible rates of money creation. The lower bound can be thought of as corresponding to the assumption that the supply of money be positive. The upper bound is set for technical reasons. The planner uses the newly printed money to finance the transfers (subsidies) to households, xt , in line with the following budget constraint 
 xt (st ) = qt (st ) Mt−1 st−1 − Mt (st )

(1)

2.2. Technology and Uncertainty. Following Chang (1998), we assume that taxes and transfers are distortionary. To that end, we model household’s income, yt , as a function, yt (st ) ≡ f (xt (st ), st ), of taxes

5

collected in period t, and of the exogenous shock, st , where f : X × S → R is assumed to

be at least twice continuously differentiable with respect to its first argument and [A2] f (0, st ) > 0, f1 (0, st ) = 0, f11 (xt , st ) < 0 [A3] f (xt , st ) = f (−xt , st ) for all xt ∈ X. where f1 and f11 denote, respectively, the first and second derivative of function f with respect to its first argument, and where X is a compact subset of R, and S is finite with cardinality S. [A2] indicates that it is increasingly costly (in terms of consumption good) to impose taxes or to give transfers. The symmetry of f in [A3] implies that taxes and subsidies are equally distortionary. 5The alternative setup could have a more elaborated model of production and, for example, distortionary labor

taxation.

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The second argument of function f , st , denotes the shock which is the only source of risk and
uncertainty in our modeled economy. We assume that the kernel of st t is Markov.
[A4] πt st+1 |st = π (st+1 |st ) We refer to π(st ) as the approximating model. At the heart of this paper lies the assumption that while the planner fully trusts the approximating model, a representative household suspects that this model might have been misspecified. To confront that concern, he considers a set of probability distributions instead, and seeks decision rules that would work well across this set of alternative models. Each of the perturbed probability distri
butions that a representative household contemplates, π e st , is assumed to be absolutely continuous with respect to the approximating model, i.e.



[A5] π e st  π st i.e. π st = 0 ⇒ π e st = 0, ∀st ∈ St+1 Assumption [A5] allows us to invoke Radon-Nikodym theorem and to express the discrepancy in beliefs of the government and a representative household by means of multiplicative pertur
π e ( st ) bations, Dt st = π(st ) ≥ 0, which are martingales with respect to the approximating model,


P i.e. π st+1 |st Dt+1 st+1 = Dt st . Moreover, we define the conditional likelihood ratio as st+1


dt+1 st+1 |st ≡

Dt+1 (st ,st+1 ) Dt (st )

=

π e(st+1 |st ) π(st+1 |st ) .

Note that the expectation of the conditional likelihood
P ratio under the approximating model is always equal to 1, i.e. π st+1 |st dt+1 = 1. st+1

2.3. Formalizing A Representative Household’s Taste for Robustness. Households derive their utility from consumption of a single consumption good, ct (st ), and real money balances, mt (st ). Once we have departed from the assumption that households and a government share common beliefs, we need a good theory for discriminating among the models from the set of probability models used by a representative household. Following Hansen and Sargent (2008), we endow households with multiplier preferences V

H

=

max

min

∞ X

t t ∞ {ct (st ),mt (st )}∞ t=0 {Dt (s ),dt+1 (st+1 |s )}t=0

(2)

βt

X

n


 u ct st + v(mt st )

st

t=0

+θβ

π(st )Dt (st )

X

o π(st+1 |st )dt+1 (st+1 |st ) log dt+1 (st+1 |st )

st+1

where θ ∈ (θ, +∞] is a parameter that describes a representative household’s preference for robustness. In the setting with θ → +∞ the fears of private agents disappear; households fully trust the approximating model of the planner.

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

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In the above specification of preferences, a representative household is thought of as playing a zerosum game against a fictitious evil agent who is just a metaphor for household’s fears. Following are the assumptions concerning functions u and v [A6] u : R+ → R is twice continuously differentiable, strictly increasing, and strictly concave [A7] v : R+ → R is twice continuously differentiable, and strictly concave [A8] limc→0 u0 (c) = limm→0 v 0 (m) = +∞ [A9] ∃m < +∞ such that v 0 (m) = 0 The assumptions [A6]-[A8] are standard. In [A9] we define the satiation level for real money balances. Following Chang (1998), we assume that [A10] mt ∈ M ≡ [0, m] We can then rewrite the planner’s budget constraint (1) as   xt (st ) = mt (st ) ht (st ) − 1

(3)

Notice that from (3) xt (st ) ∈ X ≡ [(π − 1) m, (π − 1) m]. A representative household maximizes (2) given the initial level M−1 , s0 and D0 = 1, subject to the following constraints (4) (5)







qt st Mt st ≤ yt st − xt st − ct st + qt st Mt−1 (st−1 )

qt st Mt st ≤ m

(6) (7)

Dt+1 (st+1 ) = dt+1 (st+1 |st )Dt (st ) X π(st+1 |st )dt+1 (st+1 |st ) = 1 st+1

(4) is the budget constraint which states that for all t ≥ 0, and all st endowment of the consumption good in period t, yt , together with the value of money holdings have to be sufficient to cover the period-t expenditures: consumption, taxes and new purchases of money. (5) imposes an upper bound on real money holdings. Conditions (6) and (7) discipline the choices of the malevolent alter ego. In (6) we define recursively the evolution of the likelihood ratio, Dt , that our representative household contemplates. Condition (7) ensures that every distorted probability is a well defined probability measure.
∞ 
∞  A representative household takes sequences of prices, qt st t=0 , income, yt st t=0 , taxes
∞  or subsidies, xt st t=0 , and conditional likelihood ratio chosen by his malevolent alter ego, 
∞ dt+1 st+1 |st t=0 , as given.

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2.4. Within Period Timing Protocol: A Game in A Game. In this paper we are interested in characterizing time-consistent optimal policies. To that end, we study a repeated game in which two players - a representative household (H) and the government (G) - choose sequentially. At the beginning of a given period t = j, the productivity shock realizes. Then, planner announces
its money growth policy {ht st }∞ t=j . After that, households construct the optimal allocation rule
{mt st }∞ t=j . Finally, taxes, xt , are collected to balance the planner’s budget (given by (1)), and consumption, ct , takes place in the amount resulting from (4). The main game, Γ, between the planner and a representative household can be, therefore, defined formally in the following way   Γ = {G, H} , {Π, [0, m]} , V G , V H where V H denotes the payoff to a representative household as defined in (2), while V G the corresponding payoff to the government (see (12) below). When choosing the allocation rule for real money holdings, for any given choice of governments’ policy, our representative household effectively plays a zero-sum game against his malevolent alter ego (AE). Therefore, there is a subgame, Γh , in the main game defined as   Γh = {H, AE} , {[0, m] , D} , V H , −V H Notice that irrespective of the timing protocol for the actions taken by the maximizing and minimizing players in (2), the solution to Γh will be the same since the players’ incentives are perfectly misaligned6, and the resulting payoffs are equal to V H and −V H , respectively. In this modeled economy, it is clearly desirable to bring the real value of money to the satiation level, m. In equilibrium, however, this can only be done by deflating the economy at some optimal rate. Once households have made their corresponding choice for real money holdings, the government has an incentive to deviate from the previously announced deflation rate and to deflate the economy at the slower pace. This is due to the fact that the marginal tax distortion is convex; bringing about deflation is the less costly in terms of consumption the more prolonged the deflation process is (equivalently, the smaller the period-by-period deflation rates). This time-inconsistency problem is present also in the original model of Chang (1998).

6For further details see Hansen and Sargent (2008), chapter 7.

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

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3. Competitive Equilibrium With Model Uncertainty In this section we define the notion of a competitive equilibrium with model uncertainty. For the rest of the paper, bold letters will be used to denote sequences. Definition 1 Competitive equilibrium with model uncertainty is given by a sequence of alloca∞ ∞ tions (m, x, h) = {mt , xt , ht }∞ t=0 , prices q = {qt }t=0 , beliefs distortions d = {dt+1 }t=0 , and the H }∞ such that for all t and all corresponding continuation values of households utility VH = {Vt+1 t=0

st
(i) given q and the belief distortions d, m, x, h, VH solves households’ maximization problem;
(ii) given q and m, x, h, VH , d solves the alter ego’s minimization problem; (iii) government’s budget constraint holds; (iv) markets clear, i.e. ct (st ) = yt (st ) and mt (st ) = qt (st )Mt (st ). Under assumptions [A1-A10] we can prove the following proposition Proposition 1. A competitive equilibrium is completely characterized by sequences m, x, h, d, VH



such that for all t and all st , mt st ∈ M, xt st ∈ X, ht st ∈ Π, dt+1 st+1 ∈ D, and H (st+1 ) ∈ V and Vt+1

(8)




mt st u0 (f (xt st , st )) − v 0 (mt st ) = X 


β π(st+1 |st )dt+1 (st+1 |st ) u0 (f (xt+1 st+1 , st+1 )ht+1 st+1 mt+1 st+1 , ≤ if mt = m st+1

  V H (st+1 ) exp − t+1 θ  V H (st+1 )  dt+1 (st+1 |st ) = P π(s |s ) exp − t+1 θ t+1 t st+1

(9)

(10)

VtH

X


= u f (xt s , st ) + v mt st − βθ log π (st+1 |st ) exp t




st+1

(11)

Proof. See Appendix A.1.-A.2.

−xt s


t

= mt s


t

1 − ht st

H −Vt+1 st+1 θ


!










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Eq. (8) is an Euler equation for real money balances. Eq. (9) defines the equilibrium distortion of households’ beliefs; it shows how the evil agent tilts probabilities towards states with lower continuation values7. Eq. (10) uses the equilibrium condition (9) to substitute out the continuation values in the definition of households’ lifetime utility value, eq. (2). Finally, (11) is the governments’ budget constraint. In Appendix A.3. we demonstrate why households’ transversality condition does not have to be included in the list of conditions characterizing competitive equilibrium in our setup. Notice how, by Proposition 1, the set of competitive equilibria with model uncertainty can be characterized by infinite sequence of conditions, each of which connecting at most two periods. Formally, let E ≡ M × X × Π × D × V and E∞ ≡ M∞ × X∞ × Π∞ × D∞ × V∞ . We define a set of competitive equilibria for each of the realizations of the initial state s0 
CEs = m, x, h, d, VH ∈ E∞ | (8)-(11) hold and s0 = s In Appendix A.4. we present an example of a competitive equilibrium sequence. Corollary 1. CEs for all s ∈ S is nonempty. 

Proof. See Appendix A.4. Corollary 2. CEs for all s ∈ S is compact.



Proof. See Appendix A.5.

Corollary 3. A continuation of a competitive equilibrium with model uncertainty is a compet
 H ∞ ∈ CE for all t and itive equilibrium, i.e. if m, x, h, d, VH ∈ CEs0 then mt , xt , ht , dt , Vt+1 st j=t all s0 , st ∈ S. 

Proof. Follows immediately from Proposition 1.

4. Ramsey Problem for a Paternalistic Government: Recursive Formulation Although in this paper we are primarily interested in the study of optimal policies when the planner chooses the optimal policy sequentially, it will be useful to formulate first a time-zero Ramsey problem for two reasons. First, it will allow us to introduce a notion of a paternalistic government and to determine the best equilibrium that the planner can achieve in this economy with households who 7To

see this, construct the ratio of conditional likelihood ratios for two realizations of the state tomorrow, st+1 = 1

H H and st+1 = 2 such that Vt+1 (st+1 = 1) > Vt+1 (st+1 = 2) given st . Then

dt+1 (st+1 =2|st ) dt+1 (st+1 =1|st )

> 1.

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

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distrust the model used by the planner. Second, it will allow us to discuss our recursive description of the set of competitive equilibria cast in two state variables. These state variables are crucial both for the recursive formulation of the time-zero Ramsey problem as well as of the problem of the planner who chooses sequentially. Assume then for a moment that the government can credibly commit to its policy at time 0. In the usual setting where both households and a planner consider a model like our approximating model π(st ) to be a good description of the underlying uncertainty, the Ramsey problem of a benevolent planner consists in choosing a state contingent policy ht (st ) associated with the best competitive equilibrium, i.e. one that maximizes households’ expected lifetime utility. Our paper, instead, features a notion of a paternalistic planner who does strive to choose the best competitive equilibrium but in accordance with his own beliefs π(st ) at every node st , i.e. (12)

VtG

=

max

∞ X

(m,x,h,d,VH )

βt

t=0

X




 πt (st ) u ct st + v(mt st )

s.t. (8) - (11)

st

Variety of methods can be used to solve the above problem. Below we extend the procedure proposed originally by Kydland and Prescott (1980). The key to the procedure is to construct a recursive description of the set of competitive equilibria. Following Kydland and Prescott (1980), and Chang (1998), we designate a pseudo-state variable with the essential feature of the "promise"
of future policies made by the equilibrium outcome in period t. Formally, define µt+1 st+1 ≡


u0 (f (xt+1 st+1 , st+1 )(ht+1 st+1 mt+1 st+1 )8. We can think of µt+1 as the period-t+1 "promised" marginal utility of money. By the nature of our problem, µt s alone are not sufficient for the recursive description of the set of competitive equilibria with model uncertainty. To see this, substitute (9) into the Euler equation (8) and use the definition of µt+1
 0

u (f (xt st , st )) − v 0 (mt st ) =   H Vt+1 (st+1 ) exp − θ X
β π(st+1 |st ) P  V H (st+1 )  µt+1 st+1 , ≤ if mt = m t+1 st+1 st+1 π(st+1 |st ) exp − θ mt st

For a given realization st , and tax revenue xt , the future paths of ht+1 , mt+1 , and, hence, xt+1 influence today’s choice of real money balances by a representative household, mt , not only through their effect on µt+1 but also through the effect they have on the degree of distortion in households’ beliefs, as given in equilibrium by (9). Therefore, our recursive description of a set of competitive 8To be precise, Kydland and Prescott (1980) choose the Lagrange multiplier on the implementability constraint as

their pseudo-state.

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H as state variables. Together, equilibria under model uncertainty has to rely on both µt+1 and Vt+1

they can be thought of as device used to ensure that the effects of future policies on agents’ behavior in earlier periods are accounted for. Let <2 be the space of all the subsets of R2 . Moreover, let Ω : S → <2 be the value correspondence such that n


µs , VsH ∈ R × R| µs ≡ u0 [f (x0 (s0 ) , s0 )] [x0 (s0 ) + m0 (s0 )] and  H  P −V1 (s1 ) VsH = u (f (x (s0 ) , s0 ) + v (m (s0 )) − βθ log π (s1 |s0 ) exp θ os1  H ∞ ∈ CE . for some mt , xt , ht , dt+1 , Vt+1 s t=0 Ω (s = s0 ) =

For each initial state s0 , the correspondence Ω finds the set Ω(s) of current promised marginal utilities and values to the households that can be delivered by a competitive equilibrium. It is straightforward to check that Ω(s) is non-empty and compact.
n
Also, let Ψ s, µs , VsH = m, x, h, d, VH ∈ CEs |µs = u0 [f (x0 (s0 ) , s0 )] [x0 (s0 ) + m0 (s0 )] and VsH =
o
P u (f (x (s0 ) , s0 )+v (m (s0 ))−βθ log π (s1 |s0 ) exp −V1H (s1 ) /θ . If we knew sets Ω(s) and Ψ s, µs , VsH , s1

we could solve the Ramsey problem for our paternalistic government in two steps as follows. In the first step (13) V G∗ (s, µs , VsH ) =

max

∞ X

(m,x,h,d,VH )

βt

X

t=0

Let µ = [µ1 , µ2 , ..., µS ] and V H




 πt (st ) u ct st + v(mt st ) s.t.



m, x, h, d, VH ∈ Ψ s, µs , VsH

st

  = V1H , V2H , ..., VSH . The primes are used to denote next-period

values. In the below proposition we recast (13) in a way that lets us solve the Ramsey problem using dynamic programming techniques.
Proposition 2. V G∗ s, µs , VsH satisfies the following Bellman equation (14)


V G s, µs , VsH =

max

(m,x,h,µ0 ,V H0 )

[u (f (x, s)) + v(m)] + β




µ0s0 , VsH0 0




∈ Ω (s0 ) for all s0

µs = u0 [f (x, s)] [x + m]

(15)

VsH = u (f (x, s)) + v (m) − βθ log

X s0

π (s0 |s) exp



−VsH0 0 θ



−x = m [1 − h]

(17) (18)

π(s0 |s)ws0 s0 , µ0s0 , VsH0 0

s0

(m, x, h) ∈ M × X × Π and

(16)

X

m {u0 (f (x, s)) − v 0 (m)} = β

X s0

  V H0 exp − sθ0  V H0  µ0s0 , ≤ if m = m π(s0 |s) P s0 0 s0 π(s |s) exp − θ

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

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Conversely, if a bounded function V G : S × Ω(s) → R satisfies the above equation, then it solves (13). Proof. Based on the Bellman principle of optimality, straightforward extension of Chang (1998), p. 

457, and is left to the reader.

To complete the solution of the paternalistic Ramsey problem, the following second step has to be undertaken V G∗ (s) =

(19)

max

V G∗ s, µs , VsH




(µs ,VsH )∈Ω(s)

In the first period the government is not bounded by previous promises of marginal utilities nor
values to be delivered to households. Therefore, it is free to choose the initial vector µs , VsH . To make Proposition 2 operational, in what follows we present a procedure for computing the set Ω(s) as a fixed point of an appropriately constructed value correspondence operator, in the spirit of Abreu, Pearce and Stacchetti (1990). Let G be the space of all the correspondences Ω. Also, let Q ∈ G. Construct the operator B : G → G such that B (Q) (s) =





µs , VsH ∈ R × R| ∃ m, x, h, µ0 , V H0 ∈ M × X × Π × Q such that (15)-(18)hold

Thus, B(Q)(s) is a set of current-period’s µs ’s and VsH ’s consistent with the existence of some triple (m, x, h) ∈ M × X×Π and with the next-period’s vectors of µ0 and V H0 . Notice that the definition of B implies that B is a monotone operator in the sense that Q(s) ⊆ Q0 (s) implies B(Q)(s) ⊆ B(Q0 )(s). The following proposition establishes that the set in question Ω(s) is the largest fixed point of the operator B and that it can be found as a limit of iteration on this operator B provided that the iteration process starts with a sufficiently large initial set Q0 (s). h i H ∞ , V Let Q0 (s) = [0, µs ] × V H s s , Qn (s) = B (Qn−1 ) (s) and Q∞ (s) = ∩n=0 Qn (s) for n = 1, 2, .... Proposition 3. (i) Q(s) ⊆ B (Q) (s) ⇒ B (Q) (s) ⊆ Ω(s); (ii) Ω(s) = B (Ω) (s) (iii) Ω(s) = limn→∞ B ∞ (Q0 )(s). Proof. Simple extension of the arguments in Chang (1998).



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The resulting Ramsey plan for the paternalistic planner will have the recursive structure. Along the equilibrium path, given the state variables µ and V H following are the optimal policies
ht = h st , µt (st ), VtH (st )
xt = x st , µt (st ), VtH (st )
mt = m st , µt (st ), VtH (st )
µt+1 = ψ st , µt (st ), VtH (st )
H Vt+1 = $ st , µt (st ), VtH (st )

5. Sustainable Plans with Model Uncertainty For the rest of the paper, we assume that the planner does not have the ability to commit to an infinite sequence of the money growth rates at time t = 0. Instead, he will be choosing statecontingent sequences of money growth rates sequentially. Under this assumption, the government faces a credibility problem for the reasons explained in the section 2.4. The characterization of optimal time consistent policies in the economy populated by households who distrust planner’s model and contemplate instead a set of probability distributions requires a well-defined equilibrium concept. In this section, we present a notion of sustainable plans with model uncertainty which should be seen as an extension of the concept of sustainable equilibria introduced by Chari and Kehoe (1990). Let ht = (h0 , h1 , ..., ht ) describe the history of the money growth rates in all the periods up to t. The strategy for the government, σ G , can be defined as a sequence of functions {σtG }∞ t=0 such that σ0G : S → Π and σtG : St × Πt−1 → Π for all t ≥ 19. We restrict the government’s choice of strategy to the following space 

CEsΠ = h ∈ Π∞ | there is some m, x, d, VH such that m, x, h, d, VH ∈ CEs CEsΠ is the set of money growth rates consistent with the existence of competitive equilibria. Clearly, this set is nonempty, and compact. The strategy for a representative household, σ H , can be defined as a sequence of functions {σtH }∞ t=0 such that σtH : St × Πt → M for all t ≥ 0.  ∞ The strategy for the alter ego, σ AE , can be defined as a sequence of functions σtAE ≡ σtAE t=0 such 9The reader might wonder why the relevant histories do not include households’ actions. This is due to the fact

that households’ problem is convex and households are atomistic.

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

15

that σ0AE = d0 = 1 and σtAE : St × Πt → D for all t ≥ 1. The allocation rule for tax revenues, αx , can be defined as a sequence of functions αx ≡ {αtx }∞ t=0 such that αtx : St × Πt → X for all t ≥ 0. Definition 2. A government strategy, σ G , a representative households’ strategy, σ H , an alter ego strategy, σ AE , and an allocation rule, αx , are said to constitute a sustainable plan with model uncertainty (SP) if after any history st and ht−1 (i) the continuation future induced by σ G belongs to CEsΠ ; (ii) given government strategy σ G , the continuation of σ G together with σ H , σ AE , and αx induce a competitive equilibrium sequence; (iii) given σ H , σ AE , and αx , the continuation of σ G is optimal from the point of view of the paternalistic government, i.e. the sequence of continuation future induced by σ G maximizes 


 P∞ t P t t + v(mt st ) over the set CEsΠ . t=0 β st πt (s ) u ct s Sustainable equilibrium requires that both the monetary policy strategy and households’ reaction function be sequentially rational (Chari, Kehoe and Prescott (1989)). Note that in the definition we require that the representative household, the alter ego and the government act optimally for every history of policies, even for the histories which are not induced by government strategy. This implies, in particular, that after any history ht−1 the government’s choice is further restricted to the following set  CEs0 = h ∈ Π| there is some h ∈ CEsΠ with h0 = h Associated with every sustainable plan with model uncertainty for every period t is a current-period action profile, a = (b h, m(h), x(h), d0 (h)), a vector of promised marginal utilities, µ0 , and a vector of continuation values to the government, V G0 , and to a representative household, V H0 . The action profile a prescribes a recommended (inverse) money growth rate b h to be chosen by the government, and the corresponding reaction function m(h) for the real money holdings chosen by the households. Let h be any other (inverse) money growth rate that the government is free to choose. The reaction function m(·) will assign a level of money holdings not only for b h but also for all possible h. In case the government follows the recommended action b h, households will choose m(b h). But if the b government deviates by choosing some h 6= h, households respond by playing m(h). Let A(s) =





m, x, h, d, VH ∈ CEs | there is a SP whose outcome is m, x, h, d, VH .

Let <3 be the space of all the subsets of R3 . Moreover, let Λ : S −→ R3 be the value correspondence n

such that Λ(s) = VsG , VsH , µs | there is a m, x, h, d, VH ∈ A(s) with 


 P P t t t value to the government VsG = ∞ + v(mt st ) , t=0 β st πt (s ) u ct s

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 P P t t t t value to the household VsH = ∞ + v(mt st ) + t=0 β st πt (s )Dt (s ) u ct s P θβ st+1 π(st+1 |st )dt+1 (st+1 |st ) log dt+1 (st+1 |st ) , o and the promised marginal value µs = u0 [f (x0 (s0 ) , s0 )] [x0 (s0 ) + m0 (s0 )] . Λ(s) is a set of equilibrium values to the government, to the household and of the promised marginal utilities for a given initial state s0 = s that can be associated with outcome of a sustainable plan. We denote as Gb the space of all such correspondences.

b (a, V G0 (·), V H0 (·), µ0 (·)) is said to be admissible Definition 3. For any correspondence Z ⊂ G, with respect to Z at state s if (i) a = (b h, m(h), x(h), d0 (h)) ∈ Π × [0, m]Π × X Π × RΠ ; H0 0 0 (ii) (VsG0 0 (h), Vs0 (h), µs0 (h)) ∈ Z(s )

∀ h ∈ CEs0 , s0 ∈ S;

(iii) (17)-(18) are satisfied; P b (iv) u(f (x(b h), s)) + v(m(b h)) + β s0 ∈S π(s0 |s)VsG0 0 (h) ≥ P ∀h ∈ CEs0 . u(f (x(h), s)) + v(m(h)) + β s0 ∈S π(s0 |s)VsG0 0 (h) Condition (i) guarantees a belongs to the action space. Condition (ii) ensures that continuation values and next period’s (adjusted) marginal utilities for tomorrow’s state s0 when action h is played in the current period have to belong to the set Z(s0 ). Condition (iii) implies that competitive equilibrium conditions hold in the current period. Condition (iv) is the government’s incentive constraint10 which rules out profitable deviations by the government. b Let B b : Gb −→ Gb be defined as follows Let Z ⊂ G. n b B(Z)(s) = co (VsG , VsH , µs )|∃ admissible (a, V G0 (·), V H0 (·), µ0 (·)) with respect to Z at s with:

VsG

a = (b h, m(h), x(h), d0 (h)) X b = u(f (x(b h), s)) + v(m(b h)) + β π(s0 |s)VsG0 0 (h) s0 ∈S

(

VsH

b V H0 0 (h) = u(f (x(b h), s)) + v(m(b h)) − βθ log π(s |s) exp − s θ s0 ∈S o µs = u(f (x(b h), s))(x(b h) + m(b h)) X

)

0

10The One-Period Deviation Principle guarantees that there is no profitable deviations at all if one-period deviations

can be ruled out.

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

17

b Each B(Z)(s) is the convex hull of the set of values to the government, values to a representative household and promised marginal values of money that can be supported by some action profile and continuation values and marginal utilities belonging to the set Z(s0 ) for each state s0 tomorrow. Below we prove some properties of our operator. Together, they guarantee that the equilibrium value correspondence Λ is its largest fixed point and can be found by iterating on this operator. Proposition 4. Monotonicity: Z ⊆ Z 0 implies B(Z) ⊆ B(Z 0 ). Proof. The proof is trivial and left to the reader.



Proposition 5. Self-Generation: If Z(s) is bounded and Z(s) ⊆ B(Z)(s), then B(Z)(s) ⊆ Λ(s). Proof. We need to construct a subgame perfect strategy profile (σ G , σ H ) such that (i) for each s ∈ S it delivers a lifetime utility value VsG to the government, VsH to a representative household with an associated marginal promised utility µs , (ii) the associated outcome of the SP satisfies (17)-(18) (iii) government’s incentive constraint holds for every t and after every history (ht−1 , st ). To that end, fix an initial state s and pick any (VsG , VsH , µs ) ∈ B(Z) (s) . Set (V0G , V0H , µ0 ) = (VsG , VsH , µs ) and define (σ G , σ H ) recursively as follows. Let (VtG (ht−1 , st−1 , st ), VtH (ht−1 , st−1 , st ), µt (ht−1 , st−1 , st )) ∈ Z(st ) be the vector of values and marginal utilities after an arbitrary history (ht−1 , st−1 , st ). Since Z ⊂ B(Z), for each s ∈ S there exists an admissible vector (b h, m(h), x(h), V G0 (h), V H0 (h), µ0 (h)) with respect to Z at s. Define σtG (ht−1 , (st−1 , st )) = b h and σtH (ht−1 , (st−1 , st )) = m(h) (if h ∈ CEs0t and 0 otherwise) G (ht , st , s H t t t t G0 H0 0 Also, define (Vt+1 t+1 ), Vt+1 (h , s , st+1 ), µt+1 (h , s , st+1 )) = (Vst+1 (h), Vst+1 (h), µst+1 (h)) if G (ht , st , s H t t t t G H h ∈ CEs0t+1 , and (Vt+1 t+1 ), Vt+1 (h , s , st+1 ), µt+1 (h , s , st+1 )) = (Vst+1 , Vst+1 , µst+1 ) othG (ht , st , s H t t t t erwise. Clearly, (Vt+1 t+1 ), Vt+1 (h , s , st+1 ), µt+1 (h , s , st+1 )) ∈ Z(st+1 ). By admissibility,

(σ G , σ H ) is unimprovable and, thus, is subgame perfect. Since Z(s) is bounded for every s ∈ S, it is straightforward to show that (σ G , σ H ) delivers (VsG , VsH , µs ). Also, admissibility of vectors (b h, m(h), x(h), V G0 (h), V H0 (h), µ(h)) ensures that the equilibrium conditions are satisfied along the equilibrium path.



Proposition 6. Factorization: Λ = B(Λ). Proof. By the previous proposition, it is sufficient to show that Λ(s) is bounded and that Λ(s) ⊂ B(Λ)(s). The result follows from the fact that the continuation of a sustainable plan is also a sustainable plan. Boundness of Λ(s) follows from (i) the fact that any lifetime utility for the government is the expected discounted sum of one-period bounded payoffs; (ii) any lifetime utility for the household can be bounded by discounted sums of non-stochastic extremal one-period payoffs, (iii) marginal utilities are determined by continuous functions f, u0 over compact sets.



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Proposition 7. If Z(s) is compact for each s ∈ S, then so is B(Z)(s). b Define the Proof. Let us show first that B(Z)(s) is bounded. Let Z be a value correspondence in G. operators Υi,s : Gb −→ R for i = 1, 2, where < is the space of subsets in R,  G Vs : ∃(VsG , VsH , µs ) ∈ Z(s)  Υ2,s (Z) = VsH : ∃(VsG , VsH , µs ) ∈ Z(s) Υ1,s (Z) =

Boundness of B(Z)(s) follows from having Υ1,s (B(Z)) ⊂ Us0 + β

X

π(s0 |s)Υ1,s0 (Z)

s0

Υ2,s (B(Z)) ⊂ Us0 − βθ log

X

π(s0 |s) exp −Υ2,s0 (Z)/θ




s0

where the sets of one-period payoffs Us0 (for current state s), and Υi,s0 (Z) for i = 1, 2 are bounded.  +∞ Let us show now that B(Z)(s) is closed. Consider any sequence (V Gn , V Hn , µn ) n=1 such that (VtGn (st−1 , st ), VtHn (st−1 , st ), µnt (st−1 , st )) ∈ B(Z)(st ) ∀st−1 ∈ St−1 , st ∈ S that converges to some (V G∗ , V H∗ , µ∗ ). Fix an arbitrary sequence of states {st }+∞ t=0 . We need to show that (V G∗ (st−1 , st ), V H∗ (st−1 , st ), µ∗ (st−1 , st )) ∈ B(Z)(st ) ∀st−1 ∈ St , st ∈ S. For each (VtGn (st−1 , st ), VtHn (st−1 , st ), µnt (st−1 , st )), there exists an admissible vector (b hn , mn (h), xn (h), V Gn0 (h), V Hn0 (h), µn0 (h)) with respect to Z at s. This vector should be indexed by histories of shocks st . In particular, b hn (st ) = b hn . Since {st }+∞ is fixed, we slightly abuse the t

t=0

notation and refer to b hnt (st ) as just b hnt . Without loss of generality, we assume that b hnt converges to some b h∗ ∈ CE 0 . In a similar way, for each t

st 0 h ∈ CEst , (mn (h), xn (h), V Gn0 (h), V Hn0 (h), µn0 (h)) −→ (m∗ (h), x∗ (h), V G0 (h)∗ , V H0 (h)∗ , µ0 (h)∗ ) ∗ H0 ∗ 0 ∗ 0 0 where (m∗ (h), x∗ (h)) ∈ [0, m] × X and (VsG0 0 (h) , Vs0 (h) , µs0 (h) ) ∈ Z(s ) ∀s ∈ S, by compactness of [0, m] × X and Z(s0 ) ∀s0 ∈ S. By continuity of functions u, v, f, u0 , v 0 , it is straightforward

to check that (m∗ (h), x∗ (h), V G0 (h)∗ , V H0 (h)∗ , µ0 (h)∗ ) satisfies conditions (17)-(18). It follows then that (V G∗ (st−1 , st ), V H∗ (st−1 , st ), µ∗ (st−1 , st )) ∈ B(Z)(st ).



6. Numerical Solution: Monotone Outer and Inner Hyperplane Approximation b on the computer and how to find the In what follows we show how to implement the operator B equilibrium value correspondence Λ. We will also show how to build strategies that support particular equilibrium triples (VsG , VsH , µs ) for all s ∈ S. Our algorithms are adaptations of procedures described in Judd, Yeltekin and Conklin (2003). Regarding the first task of finding the equilibrium value correspondence, it should be noted that representing a multidimensional correspondence on a computer is a challenging endeavor, as pointed

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

19

out by Fernández-Villaverde and Tsyvinski (2002). A presumably simple way would be to discretize the state space. This technique in our case suffers from a severe curse of dimensionality. Judd, Yeltekin and Conklin (2003) propose an alternative to the discrete state space techniques. A general idea is to use hyperplane approximation techniques. We extend those results to the case of our repeated game with two types of players who compute their expected lifetime payoffs according to different probability models. 6.1. Monotone Outer Hyperplane Approximation. We first discretize the action space. For money holdings m, we define the grid mgrid = [m1 , ..., mNm ], where Nm is the number of gridpoints such that m1 = 0 and mNm = m. For (inverse) money growth rates h, we define hgrid = [h1 , ..., hNh ], where Nh is the number of gridpoints such that h1 = π and hNh = π. Approximating any convex set W (s) ⊂ R3 through outer approximation consists in finding the c (s), generated by a set of hyperplanes, that contains W (s). The convex smallest convex polytope W c (s) is determined as the intersection of half-spaces defined by these hyperplanes. We polytope W use D hyperplanes, each of which are characterized by a subgradient gi = (gl,1 , gl,2 , gl,3 ) ∈ R3 and a hyperplane level cl,s ∈ R for l = 1, ..., D. Let G = {g1 , ..., gD } and let Cs be the vector of hyperplane levels, i.e. Cs = (c1,s , ..., cD,s ) for state s. For the sake of simplicity, we use the same set of search subgradient G to approximate each of the sets W (s), for s = 1, ..., S. The vector c (s) is determined of hyperplane levels, Cs , however, will be state-specific. The convex polytope W  c (s) = ∩l=D w ∈ R3 |gl · w ≤ cl,s . The algorithm used to perform the outer approximation as W l=1 is described in Table 1. We initialize it by choosing as input a candidate correspondence Z 0 and e0 (s) sufficiently large sets Z 0 (s), which we approximate using hyperplanes in Step 0. Denote by Z such approximation of Z 0 (s) and, by Ze0 that of Z 0 . Our choice of Z 0 (s) is given by the hypercube determined by the highest and the lowest expected payoffs for households and government, and the highest and lowest possible value for the (adjusted) marginal utility, at state s, i.e. Z 0 (s) = G

H

H [V G s , V s ] × [V s , V s ] × [µs , µs ] where

VG = u(f (x, s)) + v(0) + β s

X

π(s0 |s)V G s0

s0 G

Vs

= u(f (0, s)) + v(m) + β

X

G

π(s0 |s)V s0

s0

VH s

= u(f (x, s)) + v(0) − βθ log

X

π(s0 |s) exp −V H s0 /θ




s0 H

Vs

= u(f (0, s)) + v(m) − βθ log

X s0

µs = 0 µs = u0 (f (x, s))mπ

  H π(s0 |s) exp −V s0 /θ

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ek−1 to compute the convex polytope Z ek (s) for s = 1, ..., S. The In Step 1, in iteration k we input Z ek (s) contains the vectors of current values (VsG , VsH , µs ) that can be supported by some action set Z H0 0 ek−1 (s0 ) and the profile and continuation values (V G0 , V H0 , µ0 ) such that (VsG0 0 , Vs0 , µs0 ) belongs to Z

respective conditions for competitive equilibrium are satisfied. Formulation of the incentive constraint for the government requires determining value to the government associated with the most profitable deviation. We impose that every such deviation will be followed by the harshest punishment which is to be determined in equilibrium, as shown in Step 1, part (a). In case the government deviates from its announced policy, households will not perceive him as credible anymore, and will adjust the expectations accordingly. Given s and government’s choice of inverse money growth h, select m to minimize the government’s value imposing the competitive equilibrium conditions and ek−1 (s0 ) for each next period shock s0 . Given this vector of picking the continuation values from Z values for each h, we compute the value corresponding to the best deviation for the government by selecting the value that maximizes the government’s value. Once we have computed V G s , we proceed ek (s) for s = 1, ..., S. to compute Z

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

21

We repeat this procedure until convergence. Table 1. Monotone Outer Hyperplane Approximation Algorithm Step 0:

Approximate each Z0 (s) ⊃ Λ(s). For each s = 1, ..., S, and gl ∈ G, l = 1, ..., D, compute c0l,s = max gl,1 VsG + gl,2 VsH + gl,3 µs , (VsG , VsH , µs ) Let

Step 1:

Cs0

Given (a)

=

Csk

such that

∈ Z0 (s)

{c01,s , ..., c0D,s }

for s = 1, ..., S

for s = 1, ..., S, update Csk+1 . For each s = 1, ..., S, and gl ∈ G, l = 1, ..., D,

For each pair (m, h), solve P Psk (m, h) = min(V G0 ,V H 0 ,µ0 ) u[f (x, s)] + v(m) + β s0 ∈S π(s0 |s)VsG0 0 , P m[u0 (f (x, s)) − v 0 (m)] = β s0 ∈S π(s0 |s)d0s0 µ0s0 with ≤ if m = m

such that

x = m(h − 1) 0

H 0 k gl · (VsG0 0 , Vs0 , µs0 ) ≤ cl,s0

for s0 = 1, ..., S, l = 1, ..., D

Let Psk (m, h) = +∞ if no (V G0 , V H0 , µ0 ) satisfies the constraints. k Let Rsk (h) = minm Psk (m, h). Let V G s = maxh∈Π Rs (h)

(b)

For each pair (m, h), solve G H 0 ck+1 such that l,s (m, h) = max(V G0 ,V H ,µ0 ) gl,1 Vs + gl,2 Vs + gl,3 µs , P G 0 G0 Vs = u[f (x, s)] + v(m) + β s0 ∈S π(s |s)Vs0  P VsH = u[f (x, s)] + v(m) − βθ log s0 ∈S π(s0 |s) exp −VsH0 0 /θ

(1)

µs = u0 [f (x, s)] (m + x) m[u0 (f (x, s)) − v 0 (m)] = β

P

s0 ∈S

π(s0 |s)d0s0 µ0s0 with ≤ if m = m

x = m(h − 1)  P  d0s0 = exp −VsH0 / s0 ∈S π(s0 |s) exp −VsH0 0 /θ 0 /θ VsG ≥ V G s 0

H 0 k gl · (VsG0 0 , Vs0 , µs0 ) ≤ cl,s0

for s0 = 1, ..., S, l = 1, ..., D

G0 where ck+1 , V H0 , µ0 ) satisfies the constraints. l,s (m, h) = −∞ if no (V 0

Let (V G0 , V H , µ0 )l,s (m, h) ∈ RS×3 be a vector of values that solves (1). (c)

For each s = 1, ...S, and l = 1, ..., D, define k+1 ck+1 l,s = max(m,h) cl,s (m, h)

(m∗ , h∗ )l,s = arg max(m,h) ck+1 l,s (m, h) k+1 Update Csk+1 as Csk+1 = {ck+1 1,s , ..., cD,s } for s = 1, ..., S

Step 2:

k −6 Stop if maxl,s |ck+1 ; otherwise go to Step 1. l,s − cl,s | < 10

6.2. Monotone Inner Hyperplane Approximation. Once we have attained convergence through the outer approximation method, we apply the inner approximation to obtain optimal actions and continuation values. The steps in the inner approximation are similar to the ones in the outer b Z)(s) e approximation. The difference lies in how we approximate the convex sets B( for some ape While with outer approximation we approximate each B( b Z)(s) e proximated correspondence Z. by

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constructing the hyperplanes tangent to it, in the inner approximation we determine a set of vertices b Z)(s) e Es ⊂ B( and consider the convex hull of Es as our approximation. For each vertice in each state s, we keep track of the action profiles and continuation values that sustain it. These actions and continuation values are used to compute simulated time paths for money growth rates and real money holdings. In case a continuation value for some state s0 next period falls in the interior of the convex hull of Es0 we proceed as follows. We know that continuation value for s0 can be represented as a convex combination of vertices in Es , in each of which a particular action profile is played. We then apply a random number generator to simulate a public randomization device so to assign an equilibrium outcome.

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

23

Table 2. Monotone Inner Hyperplane Approximation Algorithm k k Input: Vertices Esk = {ek1,s , ..., ekM s ,s } ⊂ R3 . Construct Csk = {ck1,s , ..., ckDs ,s } and Gks = {g1,s , ..., gD s ,s } such that s

k G H k G H k co(EsK ) = ∩l=D l=1 {gl,s · (Vs , Vs , µs )|gl,s · (Vs , Vs , µs ) ≤ cl,s }

Step 1:

Compute new vertices Esk+1 . For each s = 1, ..., S, and subgradient gl,s ∈ Gks , l = 1, ..., Ds , (a)

For each pair (m, h), solve P Psk (m, h) = min(V G0 ,V H 0 ,µ0 ) u[f (x, s)] + v(m) + β s0 ∈S π(s0 |s)VsG0 0 , P 0 0 0 0 0 m[u (f (x, s)) − v (m)] = β s0 ∈S π(s |s)ds0 µs0 with ≤ if m = m

such that

x = m(h − 1) 0

k G0 H 0 k gl,s 0 · (Vs0 , Vs0 , µs0 ) ≤ cl,s0

for s0 = 1, ..., S, l = 1, ..., Ds

0

Let Psk (m, h) = +∞ if no (V G0 , V H0 , µ0 ) satisfies the constraints. k Let Rsk (h) = minm Psk (m, h). Let V G s = maxh∈Π Rs (h)

(b)

For each pair (m, h), solve G H ck+1 such that l,s (m, h) = max(V G0 ,V H0 ,µ0 ) gl,1 Vs + gl,2 Vs + gl,3 µs , P G 0 G0 Vs = u[f (x, s)] + v(m) + β s0 ∈S π(s |s)Vs0  P VsH = u[f (x, s)] + v(m) − βθ log s0 ∈S π(s0 |s) exp −VsH0 0 /θ

(1)

µs = u0 [f (x, s)] (m + x) m[u0 (f (x, s)) − v 0 (m)] = β

P

s0 ∈S

π(s0 |s)d0s0 µ0s0 with ≤ if m = m

x = m(h − 1)  P  d0s0 = exp −VsH0 / s0 ∈S π(s0 |s) exp −VsH0 0 /θ 0 /θ VsG ≥ V G s 0

k G0 H 0 k gl,s 0 · (Vs0 , Vs0 , µs0 ) ≤ cl,s0

for s0 = 1, ..., S, l = 1, ..., Ds

0

G0 where ck+1 , V H0 , µ0 ) satisfies the constraints. l,s (m, h) = −∞ if no (V 0

Let (V G0 , V H , µ0 )l,s (m, h) ∈ RS×3 be a vector of continuation values that solves (1). Let (VsG , VsH , µs )l,s (m, h) ∈ R3 be a vector of current values that solves (1) (c)

For each s = 1, ..., S, and l = 1, ..., Ds , define (m∗ , h∗ )l,s = arg max(m,h) ck+1 l,s (m, h) ∗ ∗ H G ek+1 l,s = (Vs , Vs , µs )l,s (m , h )l k+1 Update vertices Esk+1 = {ek+1 1,s , ..., eM s ,s }.

Step 2:

k+1 k −6 k Stop if M s = Ds for s = 1, ..., S, maxl,s |ck+1 and maxl,s |gl,s − gl,s | < 10−6 ; otherwise Step 1. l,s − cl,s | < 10

6.3. Numerical Results. Assume that S = 2, and that f (x, 1) > f (x, 2). We assume the following functional forms and parameter values: u(c) = log c v(m) =

1 500 (mm

− 0.5m2 )0.5

f (x, s) = 64 − 0.5(1 + s)(0.4x)2 π(s0 = 1|s = 1) = π(s0 = 2|s = 2) = 0.75

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We chose the functional forms to demonstrate a somewhat intriguing example. Both for the case of θ = +∞ (when households use the same approximating model as the government) and θ = 5, the Ramsey outcome is sustainable. However, as it will become clear from simulations of equilibrium paths, the effect of government’s management of households beliefs reveals itself through transition dynamics. Figure 1 depicts the equilibrium sets of values to the household, to the government and of the promised marginal values of money that can be sustained by our paternalistic government. To shed more light on the results, in Figure 2 we present the combinations of households’ and government’s equilibrium values only. As one would expect, with θ = +∞, the government’s and households’ equilibrium values are perfectly aligned along the 45-degree line. In turn, when households are concerned with model misspecification, with θ = 5, a given value to the government can be associated with multiple equilibrium values to the household. By choosing one of them, the government effectively chooses equilibrium system of beliefs of a representative household.

Figure 1. Sets of equilibrium values for state s = 1 and s = 2. Value Set (1)

Value Set (2)

0.8

0.6

0.6

0.4

0.4 

2

1

0.8

0.2

0.2

0

0

-0.2 42 41.5

42

-0.2 42 41.5

41.5

41 VH 1

42 41

41

40.5

40.5 40

40

41

40.5

G

V1

H

40

V2

40

VG 2

Figure 2. Government’s and households’ equilibrium values for θ = +∞ and θ = 5. 42

41.9

41.9

41.8

41.8 VH s

VH s

42

41.7

41.7

41.6

41.6

41.5 41.5

41.6

41.7

41.8 VG s

41.9

42

41.5 41.5

41.6

41.7

41.8 VG s

41.9

42

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

25

Finally, we simulate the time paths for the policy variable, ht , and real money holdings, mt for the same history of shock realizations. We hit the economy with 6 consecutive shocks s = 1, and then 9 consecutive shocks s = 2. The results are depicted in Figure 3. The blue line corresponds to the case of θ = +∞, while the red one to the case of θ = 5. Figure 3. Time paths for θ = +∞ and θ = 5. 24 22

mt

20 18 16 14

2

4

6

8 t

10

12

14

2

4

6

8 t

10

12

14

1.12 1.1

ht

1.08 1.06 1.04 1.02 1

In both cases, the optimal policy prescribes a deflationary process in order to bring the real money holdings to their satiation level. With a policy maker managing households’ beliefs, this process is more gradual. To understand this, think of how households in this economy want to guard themselves against a worst-case scenario by increasing their savings and accumulating more money holdings. For a fixed government policy, this implies higher price of money, and, by government’s budget constraint, higher equilibrium tax distortion. The planner, however, realizes that households oversave (from the perspective of his approximating model) and will have an incentive to manipulate households’ beliefs, trying to convince them not to save so much. How can he do this? By announcing future policies consistent with lower dispersion in continuation values which, in equilibrium, translates into lower beliefs distortion. This additional need for the smoothing of continuation values results in lower rates of deflation, and a more gradual transition towards the steady-state.

7. A Competing Hypothesis: Robust Ramsey Planner We have started working on the extension of our methodology that accounts for the fact that the government himself might also distrust the approximating model and contemplate the set of

26

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alternative models, possibly a distinct set than that of a representative household. That extension is motivated by preliminary findings in the study of Orlik (2010) that tries to determine the taste for robustness of the government and that of the households from surveys of expectations. 8. Conclusion This paper has asked the question: How should the optimal monetary policy be designed when the central bank is faced with a representative household who is not able to assign unique meaningful probabilities to alternative outcomes? In such an environment, the effect of future paths of policy choices by the planner on today’s choice of real money holdings by a representative household is twofold. First, by choosing the rates of money growth, the central bank influences the intertemporal allocation decision of the households. Second, as this paper demonstrates, the planner has a power to influence the beliefs of the households directly, by choosing continuation values which, in equilibrium, determine the degree of disagreement between the planner and a representative household. Consequently, as in the standard models based on rational expectations paradigm, central bank designs a policy rule that confirms households’ expectations concerning that rule. Contrary to those models, however, since at the same time the planner is able to manage households’ expectations (through continuation values) it will choose the policy that confirms a particular system of households’ beliefs. In that sense our planner is not trapped by private sector expectations. As long as he is willing to actively manage households’ beliefs, he can design better policies. The high complexity of the game between the decision maker and a representative household studied in this paper makes it impossible to provide analytical solutions. However, we adapted operators in the spirit of Abreu, Pearce and Stacchetti (1990) with the purpose of providing the full characterization of the equilibrium values set. Finally, we implemented our operators using hyperplane approximation techniques of Judd, Yeltekin and Conklin (2003).

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

27

References [1] Abreu D., Pearce D. and Stacchetti E., 1990, Toward a Theory of Discounted Repeated Games with Imperfect Monitoring, Econometrica, vol. 58, pp. 1041-1063 [2] Albanesi S., Chari V. V., Christiano L. J., 2002, Expectations Traps and Monetary Policy, Review of Economic Studies, vol. 70(4), pp. 715-741 [3] Anderson E., Hansen L. P., McGrattan E. R., Sargent T. J. (1996). Mechanics of Forming and Estimating Dynamic Linear Economies. in Amman H., Kendrick D. A., Rust J. (eds.), Handbook of Computational Economics, vol.1 [4] Bernanke Ben S., 2007, Inflation Expectations and Inflation Forecasting, Speech at the Monetary Economics Workshop of the National Bureau of Economic Research Summer Institute, Cambridge, Massachusetts [5] Bernanke Ben S., 2010, Implications of the Financial Crisis for Economics, Speech at the Conference Co-sponsored by the Center for Economic Policy Studies and the Bendheim Center for Finance, Princeton University, Princeton, New Jersey [6] Caballero, Ricardo J. and Krishnamurthy A., 2008, Collective Risk Management in a Flight to Quality Episode, Journal of Finance, vol. LXIII(5), pp. 2195-2230 [7] Calvo G. A., 1978, On the Time Consistency of Optimal Policy in a Monetary Economy, Econometrica, vol. 46(6), pp. 1411-1428 [8] Clarida R., Galí J. and Gertler M., 1999, The Science of Monetary Policy: A New Keynesian Perspective, Journal of Economic Literature, vol. 37, pp. 1661Ű1707 [9] Chang R., 1998, Credible Monetary Policy in an Infinite Horizon Model: Recursive Approaches, Journal of Economic Theory, 81, pp. 431-461 [10] Chari V. V., Christiano L. J. and Eichenbaum M., 1998, Expectations Traps and Discretion, Journal of Economic Theory, vol. 81(2), pp. 462-492 [11] Chari V. V. and Kehoe P., 1990, Sustainable Plans, Journal of Political Economy, vol. 98(4), pp. 783-802 [12] Chari V. V. and Kehoe P., 1993, Sustainable Plans and Debt, Journal of Economic Theory, vol. 61, pp. 230-261 [13] Chari, V.V., Kehoe P. and Prescott E. C. (1989), Time Consistency and Policy, in Robert J. Barro, ed., Modern Business Cycle Theory, Harvard University Press, Cambridge, MA [14] Ellsberg D., 1961, Risk, Ambiguity and Savage Axioms, Quarterly Journal of Economics, vol. 75(4), pp.643-669 [15] Fernández-Villaverde J. and TsyvinskiA. , 2002, Optimal Fiscal Policy in a Business Cycle Model without Commitment, unpublished manuscript [16] Giannoni M. P., , 2002, Does Model Uncertainty Justify Caution? Robust Optimal Monetary Policy in a Forward Looking Model, Macroeconomic Dynamics, 6(1), pp. 111–144 [17] Giordani P., Söderlind P., 2003, Inflation Forecast Uncertainty, European Economic Review, vol. 47(6), pp. 10371059 [18] Halevy Yoram, 2007, Ellsberg Revisited: An Experimental Study, Econometrica, vol. 75(2), pp. 503-536 [19] Hansen L. P., Sargent T. J, Tallarini T. J., 1999, Robust Permanent Income and Pricing, Review of Economic Studies, vol. 66(4), pp. 873-907 [20] Hansen L. P., Sargent T. J., 2008, Robustness, Princeton, NJ: Princeton University Press [21] Judd K., Yeltekin S., Conklin J., 2003, Computing Supergame Equilibria, Econometrica, vol. 71(4), pp. 1239-1254

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[22] Karantounias A. G., Hansen L. P. and Sargent T. J., 2009, Managing expectations and fiscal policy, unpublished manuscript [23] Kydland F. and Prescott E., 1977, Rules Rather Than Discretion: The Inconsistency of Optimal Plans, Journal of Political Economy, vol. 85(3), pp. 473-491 [24] Kydland F. and Prescott E., 1980, Dynamic Optimal Taxation, Rational Expectations and Optimal Control, Journal of Economic Dynamics and Control, vol. 2, pp. 79-91 [25] Orlik A., 2010, Towards a joint theory of government’s, professionals’ and private expectations formation, mimeo [26] Sargent T. J., 2008, Evolution and Intelligent Design, American Economic Review, vol. 98(1), pp. 5-37 [27] Uhlig H., 2010, A model of a systemic bank run, Journal of Monetary Economics, vol. 57(1), pp. 78-96 [28] Woodford M., 2003, Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press [29] Woodford M., 2010, Robustly Optimal Monetary Policy Under Near-Rational Expectations, American Economic Review, vol. 100(1), pp. 274-303

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

29

Appendix A. Characterization of the competitive equilibrium sequence A.1. Solving a representative households’ maximization problem. LH

=

max

min ∞

∞ X

t {ct (st ),Mt (st )}∞ t=0 {λt (s )}t=0

t


t


X



u ct st







 + v(qt st Mt st ) +

st

t=0

t


π(st )Dt (st )





 qt s Mt s − yt s + xt st + ct st − qt st Mt−1 st−1 + 


st qt st Mt st − m

−λt s − µt

βt

t


FOCs

u0 (ct st ) = λt st 



 Dt (st ) v 0 (mt st )qt st − λt st qt st + X



π(st+1 |st )λt+1 st+1 Dt+1 (st+1 )qt+1 st+1 − Dt (st )µt st qt st = 0 β

(20)

(21)

st+1

Substitute eq. (20) into eq. (21), use eq. (6) and note that 0

0

t


v (mt s ) − u (ct

mt st




qt+1 (st+1 ) qt (st )

=

mt+1 (st+1 )ht+1 (st+1 ) mt (st )


t+1 Dt+1 (st+1 ) 0 t+1
qt+1 s s )+β u (ct+1 s ) π(st+1 |st ) Dt (st ) qt (st ) st+1
= 0 if mt st < m X

t


X





u0 (ct st ) − v 0 (mt st ) − β π(st+1 |st )dt+1 st+1 |st u0 (ct+1 st+1 )mt+1 st+1 ht+1 st+1 st+1


= 0 if mt st < m The above expression is our equilibrium condition, eq. (8). A.2. Solving alter ego’s minimization problem. LAE

=

min

∞ t=0

∞ X X max πt (st )Dt (st ){[u(ct ) + v(mt )] + βt ∞ t+1 t φ s ,ϕ (s ) { t+1 ( ) t }t=0 t=0 st

{Dt (st ),dt+1 (st+1 |st )} X +βθ π(st+1 |st )dt+1 (st+1 |st ) log dt+1 (st+1 |st )} + st+1

−β

X

π(st+1 |st )φt+1 st+1




 Dt+1 (st+1 ) − dt+1 (st+1 |st )Dt (st ) +

st+1

 t


−ϕt s

 X

π(st+1 |st )dt+1 (st+1 |st ) − 1

st+1

FOCs for dt+1 (st+1 |st ) and Dt (st ) are, respectively (22)



βθDt (st ) [log dt+1 (st+1 |st ) + 1] + βφt+1 st+1 Dt (st ) = ϕt st X [u(ct ) + v(mt )] + βθ π(st+1 |st )dt+1 (st+1 |st ) log dt+1 (st+1 |st )+ st+1

(23)

+β

X st+1



π(st+1 |st )φt+1 st+1 dt+1 (st+1 |st ) = φt st

≥

0,

≤

0,

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Rearrange eq. (22)

(24)

log dt+1 (st+1 |st )

=

dt+1 (st+1 |st )

=



φt+1 st+1 ϕt st − −1 + βθDt (st ) θ
!
! t ϕt s φt+1 st+1 exp −1 + exp − βθDt (st ) θ

Notice that by eq. (7) it must be that

(25)


!
! X ϕt st φt+1 st+1 exp −1 + π(st+1 |st ) exp − =1 βθDt (st ) s θ t+1
! ϕt st 1   exp −1 + = P βθDt (st ) φt+1 (st+1 ) π(s |s ) exp − t+1 t st+1 θ

Substitute eq. (25) back into eq. (24) to obtain

(26)

  φt+1 (st+1 ) exp − θ   dt+1 (st+1 |st ) = P φt+1 (st+1 ) π(s |s ) exp − t+1 t st+1 θ

Now solve forward eq. (22) and impose a respective transversality condition, β t

P

st+1


π(st+1 |st )φt+1 st+1 dt+1 (st+1 |st ) →

0 as t → ∞. Then

φt st = VtH st

(27)

Use the above result in eq. (26). This gives us our equilibrium condition eq. (9)   H Vt+1 (st+1 ) exp − θ   dt+1 (st+1 |st ) = H P Vt+1 (st+1 ) st+1 π(st+1 |st ) exp − θ A.3. On transversality condition. We will show that the trasversality condition,


  P β t st+1 π(st+1 |st )dt+1 (st+1 |st )u0 (f (xt st , st ) mt st ht st → 0 as t → ∞ for all t and all st , is satisfied if (m, x, h, d, VH ) ∈ E ∞ .



Since E is compact, for any xt st , mt st , ht st , dt+1 (st+1 |st ) ∈ E, it must be that


  P t 0 t , st ) mt st ht st belongs to a compact interval (due to continuity of u0 st+1 π(st+1 |st )dt+1 (st+1 |s )u (f (xt s 


P and f ) for every t. Hence, st+1 π(st+1 |st )dt+1 (st+1 |st )u0 (f (xt st , st ) mt st ht st is a bounded sequence, and the required sequence indeed converges to zero. A.4. Example of competitive equilibrium sequences. Assume that st = H, L and that the production function ∗ is such that f (0, H) = f (0, L). Set (m, x, h) = {m∗ , 0, 1}∞ t=0 where m satisfies the following condition for all t and

all st u0 (f (0, st )) (1 − β) = v 0 (m∗ ) Then (m, x, h) ∈ CEs .

ON CREDIBLE MONETARY POLICIES WITH MODEL UNCERTAINTY

31

A.5. Proof of Corollary 2. Corollary 2. CEs for all s ∈ S is compact.   n Proof. Let mn , xn , hn , dn , VH be the sequence from CEs=s0 for some s0 ∈ S converging to some sequence
H m, x, h, d, V . We need to show that this limiting sequence belongs to CEs=s0 .
CEs=s0 is a nonempty subset of a compact set E∞ . Since E∞ is compact, it is closed, and, hence, m, x, h, d, VH ∈ E∞ (∗).   n mn , xn , hn , dn , VH ∈ CEs=s0 implies that eqs. (8) - (11) are satisfied for each n. Consequently,
m, x, h, d, VH satisfy eqs. (8) - (11) (∗∗).
By Proposition 1, (∗) and (∗∗) imply that m, x, h, d, VH ∈ CEs=s0 . That means that CEs=s0 is a closed subset of

The fact that

the compact set. Hence, it is compact.