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Journal of Monetary Economics

Journal of Monetary Economics 59 (2012) 446–448

0304-39

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journal homepage: www.elsevier.com/locate/jme

Discussion

Comment on ‘‘Three Types of Ambiguity’’ by Lars Hansen andThomas Sargent

S-evin Yeltekin

Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, United States

1. Introduction

Optimal policymaking is typically formulated as a dynamic mechanism design problem in a Bayesian environment.Hansen and Sargent study dynamic monetary policy design in a setting where the Bayesian approach is replaced by onethat incorporates concerns for robustness against model misspecification. More precisely, they analyze Ramsey policyunder three different robustness configurations, each associated with a different type of ambiguity the Ramsey plannermight have.

In all three models, it is the policymaker who has a concern for robustness. In Model I, the policymaker has noconfidence in his beliefs about the shocks that perturb the underlying economy or in his beliefs about the beliefs of agents,which are taken to be correct. In Model II, the policymaker has no confidence in his beliefs about the shocks, but does haveconfidence in his beliefs about the beliefs of agents, which are taken to be wrong. In Model III, which mirrors thespecification adopted by Woodford (2010), the policymaker has confidence in his beliefs about the shocks, but does nothave confidence in his beliefs about the beliefs of agents. The first model is one of endogenously homogeneous beliefs,while the next two deliver endogenously heterogeneous beliefs. Hansen and Sargent then proceed to analyze howdifferent misspecification assumptions and concern for robustness shape policy. In doing so, they make several notablecontributions to the literature. First, they clarify robustness interpretations (mis)used in the literature. Second, they makecloser contact with the monetary policy literature.1 Finally, they push the analysis forward on the technical front bydelivering continuous-time formulations of these policy problems, which help deliver sharper results due to theiranalytical tractability compared to their discrete time counterparts.

Despite this improved analytical tractability however, the paper remains highly technical. In a series of simple one andtwo-period environments, I illustrate the main tradeoffs that shape robust policy in two of the models Hansen and Sargentconsider and compare these tradeoffs to the ones that arise in an environment without model misspecification. Iconcentrate on Models I and II to keep the discussion brief. In addition to simplifying some of the insights of the main textfor non-experts, these problems display how concerns for robustness can affect policy choices and the sensitivity of thesechoices to the particular robustness configuration assumed.

2. A static example and its implications

First I begin with the one-period policy problem. As in the main text, yn is the output target, yt is the log of aggregateoutput, ct denotes the stochastic cost-push shock, lt is inflation and k and cn are positive constants. Beliefs are denoted by

32/$ - see front matter & 2012 Elsevier B.V. All rights reserved.

x.doi.org/10.1016/j.jmoneco.2012.06.001

ail address: sevin@andrew.cmu.edu

ansen and Sargent (2008, Ch.16) features a model with a robustness configuration similar to Model II, but the analysis focuses on the interactions

n a monopolist and competitive fringe of firms.

S- . Yeltekin / Journal of Monetary Economics 59 (2012) 446–448 447

m. The firms choose prices in a way that can be described by a New Keynesian Phillips curve as in Woodford (2010). Fromthe firm’s static optimization problem, one can obtain the FOC:

l1 ¼ ky1þc1þcn,

which states that the inflation rate in period 1 is a linear combination of output and the cost-push shock in period 1.Substituting this FOC into the Ramsey planner’s objective, the planner’s problem can be expressed as choosing output, y1,to minimize expected inflation and deviation of output from its target, yn2:

maxy1

�1

2E½mððky1þc1þcnÞ

2þðy1�ynÞ

2Þ�:

The first order condition for y1 to the above problem yields:

y1 ¼yn

1þk2�

k1þk2

ðc1þcnÞ:

The above equation shows how the planner resolves the intratemporal tradeoff between moving y1 closer to the desiredoutput yn or depressing it to fight inflation. The choice of y1 is independent of beliefs, m. This intratemporal stabilizationmotive will be present in the two-period models discussed in the next section, but an additional, intertemporal tradeoffthat emerges will alter the output choice relative to the static case.

3. Two-period problems

I now outline a group of two-period problems. One has no model misspecification and the other two represent two-period, discrete time versions of the Hansen and Sargent Model I and II.

No model misspecification. In this case, the Ramsey planner’s problem is:

maxy�

1

2ððky0þc0þcnþbE½ðky1þc1þcnÞ�Þ

2þðy0�ynÞ

2Þ�

b2

E½ððky1þc1þcnÞ2þðy1�ynÞ

2Þ�:

Note that the two-period version of the representative firm’s FOC implies

l0 ¼ ðky0þc0þcnþbE½ðky1þc1þcnÞ�,

which states that the inflation rate in period 0 is a linear combination of output and the cost-push shock in period 0 andalso of expected period 1 inflation. From the FOC for y0 of the planner’s problem, we get :

y0 ¼1

1þk2yn�

k1þk2

ðc0þcnþbE½ðky1þc1þcnÞ�Þ:

As in the static case, the planner choses period 0 output to stabilize the economy in that period. Similarly, the FOC for y1

implies:

y1 ¼1

ð1þk2Þ2

yn�k

1þk2ðc1þcnÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

period 1 stabilization

�k

ð1þk2Þ2ððc0þcnÞþbE½ðky1þc1þcnÞ�Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

extra deflation motive

:

Although the above equation does not contain an explicit solution for y1, it is useful for decomposing the choice of y1 intothe intratemporal and the intertemporal tradeoffs the Ramsey planner faces. The first grouped term on the right-hand sideis intratemporal tradeoff present in the static problem: y1 is used to stabilize the economy in period 1. The second term isassociated with the use of y1 to depress inflation in period 0. To see that, notice that expected period 1 inflation,E½ðky1þc1þcnÞ�Þ, affects period 0 inflation. If firms expect the cost-push shock to be high and hence inflation to be high inperiod 1, they would choose to incur the cost of price adjustment and set their prices high in period 0. The choice of y1

depends both on the planner’s desire to stabilize the economy in the second period and to damp inflationary expectationsin the first.

I now introduce a concern for robustness, as Hansen and Sargent do in the models they study.Concern for robustness: Model I. In this case, the Ramsey planner believes the firms know the true model of the economy

and distrusts his own approximating model. His preferences (with the FOC of the firms substituted in for inflation asbefore) can be expressed as:

SIðy,mÞ ¼ �

1

2ððky0þc0þcnþbE½mðky1þc1þcnÞ�Þ

2þðy0�ynÞ

2Þ�

b2

E½mððky1þc1þcnÞ2þðy1�ynÞ

2Þ�:

2 In this formulation of the policymaker’s problem, the planner is not jointly picking output and inflation. Instead he picks output, and inflation is

determined by the firms’ beliefs.

S- . Yeltekin / Journal of Monetary Economics 59 (2012) 446–448448

where now the beliefs m affect the planner’s objective directly and also indirectly through the expected period 1 inflationterm E½mðky1þc1þcnÞ� obtained from the firms’ Euler equation. In this case, the robust policy is the solution to:

maxy

minm

SIðy,mÞþbyE½m log m� s:t: E½m� ¼ 1,

where bE½m log m� is the discounted relative entropy between the true probability distribution and the approximatingmodel the planner uses. The parameter y penalizes the alternative probability models considered by the planner. Aftersolving the inner min problem and substituting the solution into the outer max problem, the FOC with respect to y1 yieldsthe following expression:

y1 ¼1

ð1þk2Þ2

yn�k

1þk2ðc1þcnÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

period 1 stabilization

�k

ð1þk2Þ2ððc0þcnÞþbE½mðky1þc1þcnÞ�Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

extra deflation motive

:

The FOC for y1 is similar to the one associated with the no misspecification model; there is an intratemporal andintertemporal tradeoff component, except the latter now includes the beliefs m. Compared to the model without anymisspecification, the planner is relatively more ’’concerned’’ about using deviations in period 1 output to stabilize period 0inflation.

Concern for robustness: Model II. In this case, the Ramsey planner and the firms use the same approximating model,which the planner distrusts. His preferences (with the FOC of the firms substituted in for inflation) can be expressed as:

SIIðy,mÞ ¼ �

1

2ððky0þc0þcnþbE½ðky1þc1þcnÞ�Þ

2þðy0�ynÞ

2Þ�

b2

E½mððky1þc1þcnÞ2þðy1�ynÞ

2Þ�:

The robust policy can be found as the solution to:

maxy

minm

SIIðy,mÞþbyE½m log m� s:t: E½m� ¼ 1:

After solving the inner min problem and substituting the solution into the outer max problem, the FOC with respect to y1

yields the following expression:

y1 ¼1

ð1þk2Þ2

yn�k

1þk2ðc1þcnÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

period 1 stabilization

�1

mðy1,c1Þ

kð1þk2Þ

2ððc0þcnÞþbE½ðky1þc1þcnÞ�Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

extra deflation motive

:

The FOC for y1 has, once again, an intratemporal and an intertemporal tradeoff expression, similar to the nomisspecification model, except the latter has 1=mðy1,c1Þ in front of it. It is easy to show that mðy1,c1Þ is increasing in c1:a higher cost push shock in period 1 leads to a smaller 1=mðy1,c1Þ, hence in higher c1 states, the extra deflationary motivearising from the need to suppress inflation in period 0 has less prominence. In other words, in high c1 states, the Ramseyplanner is relatively more ‘‘concerned’’, compared to the model with no misspecification, about period 1 stabilization, andhence less willing to use deviations in period 1 output to stabilize period 0 inflation.

The two period models illustrate how the policymaker’s concerns for robustness affect policy and how the exact form ofthe policy can be tied to the specific robustness configuration. Both in Models I and II, a concern for robustness, inparticular the concern that high cost push shocks are likely in the second period (and especially if the firms think they arelikely), leads the policymaker to adopt a more active policy of using period 1 output: to control period 0 inflation (Model I)and to stabilize the period 1 economy (Model II). The numerical example displayed in Hansen and Sargent, however,suggests that the ambiguity specification does not affect the policy choices. Whether this is an artifact of the particularnumerical example or the underlying reduced form model is unclear.3 Identifying the elements of the environment thatlead to different policy choices coupled with careful quantitative applications should be objects of study as the growingliterature on robust policy design moves forward.

References

Hansen, Lars Peter, Sargent, Thomas J., 2008. Robustness. Princeton University Press, Princeton, New Jersey.Kocherlakota, Narayana, Phelan, Christopher, 2009. On the robustness of laissez-faire. Journal of Economic Theory 144 (6), 2372–2387.Woodford, Michael, 2010. Robustly optimal monetary policy with near-rational expectations. American Economic Review 100 (1), 27–303.

3 In contrast, an alternative robustness configuration adopted by Kocherlakota and Phelan (2009), where the policymaker has no confidence in his

beliefs about shocks or the equilibrium that will be played, delivers laissez faire, or passive policy, as the robust policy.