Journal of Monetary Economics 59 (2012) 488–492
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Discussion
Comments on ‘Robustly optimal monetary policy in a microfounded New Keynesian model’ by K. Adam and M. Woodford Burton Hollifield Tepper School of Business, Carnegie Mellon University, United States
1. Introduction Economic agents need to forecast the future to make current decisions. Where do those forecasts come from? Models where agents have rational expectations impose that agents’ expectations must be consistent with the distributions produced by the model—expectations are endogenous. What happens if economic agents do not have rational expectations, but instead have beliefs that are close to rational? How does the equilibrium change relative to the rational expectations outcome? Can we say anything about optimal policy in such an environment? Adam and Woodford provide a general approach to solving for robustly optimal policy commitments when private sector agents have such near-rational beliefs. Adam and Woodford provide an upper bound of what a policy commitment may achieve in such a fully specified economic environment, they provide conditions under which that upper bound may be implemented by a policy rule, and they also compute the policy and linear approximations to the dynamics around the deterministic steady state in an example economy. Adam and Woodford apply their approach to a microfounded New Keynesian model by replacing the rational expectations assumption with the assumption that the private sector beliefs are close to satisfying rational expectations. Adam and Woodford show that the optimal policy is a commitment on the paths of inflation and output gaps, similar in form to the familiar optimal policy commitment under rational expectations for this class of models. In the linearized dynamics around the steady state, the short term inflation response to cost push costs is smaller than under rational expectations with the subsequent dynamics the same as in the rational expectations economy. A difference between the fully microfounded model and to the rational expectations solution and the model in which the robust policy is assumed to be the linear in Woodford (2010) is that in the long run the price level does not return to its original level so that in the long term inflation ‘under-reacts’ to the shock. The overall conclusion, however, is that the policy prescriptions arising from the microfounded New Keynesian model under rational expectations are robust to the monetary authority’s concern for robustness.
2. A simplified example I use a simplified version of the Adam and Woodford economy to frame my discussion. The monetary authority does not know the private sector’s belief formation process, but does know that private beliefs are not too far away from the monetary authority’s beliefs. The private sector’s distorted beliefs are measured by the change of measure process fMt g1 t¼0 defined in the paper. The one step change in measure process mt
Mt þ 1 Mt
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B. Hollifield / Journal of Monetary Economics 59 (2012) 488–492
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is used to compute one step conditional expectations and to measure differences in beliefs. The one step change of measure is a strictly positive random variable with conditional mean equal to one, but is otherwise unrestricted. If the monetary authority knew how the private sector’s beliefs were formed, then they would know the change of measure process and how it might depend on the current states and the policy commitment. Since the monetary authority does not know the private sector’s belief formation process, the monetary authority does not know how the change of measure process is formed by the private sector. The desire for robustness is captured by having the monetary authority maximize against ‘worst case’ beliefs as coded by the change of measure process. Let zt denote the natural rate of output in the economy and let xt denote the logarithm of the output gap. Log output is yt ¼ xt þzt :
ð1Þ
The representative agent consumes aggregate output Yt and has expected utility preferences " # 1 X bt UðY t Þ , E^ 0
ð2Þ
t¼0
with b the time discount factor and U the period utility function. The notation E^ 0 denotes expectations taken with respect to the private sectors’ beliefs. I abstract from labor supply here, Adam and Woodford include labor supply in their richer model. Using the change of measure notation introduced earlier and for random variables Wt satisfying suitable regularity conditions, private sector conditional expectations are computed as E^ t ½W t þ 1 ¼ Et ½mt þ 1 W t þ 1 ,
ð3Þ
with Et denoting expectations taken with respect to the monetary policy authority’s beliefs. Loosely speaking, Eq. (3) requires that the monetary authority and private sector agree on the events that have probability zero. The random variables mt þ 1 are strictly positive and satisfy Et ½mt þ 1 ¼ 1, but are otherwise unrestricted. Let Pt be inflation between time t1 and time t. Staggered price setting and monopolistic competition results in a nonlinear expectations augmented Phillips curve, here written as a nonlinear function of current output gap, expectations of future inflation, and cost-push shocks
Pt ¼ Kðxt , E^ t Pt þ 1 ,ut Þ ¼ Kðxt ,Et mt þ 1 Pt þ 1 ,ut Þ,
ð4Þ fut g1 t¼0
where K is a nonlinear function, and the process is the exogenous cost-push shock process. Let it be the one period nominal interest rate. The representative agent’s first order condition for one period nominal bond holdings is 1 U 0 ðY t þ 1 Þ 1 U 0 ðY t þ 1 Þ 1 ¼ E^ t b P m b P ¼ E ð5Þ t tþ1 tþ1 tþ1 : 1 þit U 0 ðY t Þ U 0 ðY t Þ The model is solved under the assumption of complete financial markets. Together, the Phillips curve and the bond pricing equation imply that the inflation rate and output gap in the model will depend on the process fmt g1 t ¼ 1 and the policy chosen by the monetary authority given the remaining parameters of the economy. For notational convenience, let Pt ðmÞ, xt(m) and Yt(m) be the inflation, output gap, and output processes as a function of the private sector’s beliefs as parameterized by the change in beliefs process. The monetary authority does not know the private sector’s belief formation process, and therefore the monetary authority cannot completely predict the equilibrium outcomes given a policy commitment, since different changes of beliefs will lead to different predicted aggregate output and inflation processes. The monetary authority evaluates policies according to the representative agent’s utility function evaluated using the monetary authority’s beliefs—policy is evaluated by " # 1 X E0 bt UðY t ðmÞÞ : ð6Þ t¼0
Robustness is achieved by using the max–min framework to choose the optimal policy commitment—the monetary authority’s objective function is the payoff where the private sectors belief formation is chosen to minimize the representative agent’s payoff with the expected utility penalized for the discounted one-period entropy, computed as " # 1 X tþ1 b mt þ 1 log mt þ 1 : E0 ð7Þ t¼0
The upper bound policy commitment comes from the optimization " # " # 1 1 X X t tþ1 b UðY t ðmÞÞ þ yE0 b mt þ 1 ln mt þ 1 min max E0 m2M
c2C
t¼0
Z max min E0 c2C
m2M
"
1 X t¼0
# t
t¼0
b UðY t ðmÞÞ þ yE0
"
1 X
t¼0
#
b
tþ1
mt þ 1 ln mt þ 1 ,
ð8Þ
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where c is a policy commitment, and M is a set of change of measures. The minimization is how the model allows for robustness: the more concern that the monetary authority has for robustness to the private sector’s belief formation process, the smaller the parameter y is. The model recovers the familiar rational expectations solution in the limit as y-1. Adam and Woodford provide a first-order-condition approach to compute the upper bound allocation, and provide policy rules that can implement the optimal commitments. The first order conditions are nonlinear, leading to a system of nonlinear difference equations that characterize the upper bound solutions. They provide a characterization of the optimal steady state. In order to analyze the dynamics of the optimal commitment, they study a local linear approximation to the system assumed to be stationary, in which the approximation is taken around the non-stochastic steady state. They provide characterizations of the interest rates or targeting rules that implement the optimal commitment under the assumption that the equilibrium is local unique, and also provide many additional analytical results. 3. Intuition for the results A log-normal approximation provides intuition for why the optimal commitment in the economy with robustness concerns has inflation reacting less to shocks than in the rational expectations economy. Approximate the conditional distribution of the worst case beliefs as log-normal: ln mt þ 1 N ðmm,t , s2m,t Þ:
ð9Þ 2 m,t
Under log-normality, the condition Et ½mt þ 1 ¼ 1 implies that mm,t þ 0:5s
¼ 0 and the one period relative entropy is
2 m,t ,
Et ½mt þ 1 ln mt þ 1 ¼ 0:5s
and the discounted sum of relative entropy is " # 1 X tþ1 2 E0 b 0:5sm,t :
ð10Þ
ð11Þ
t¼0
The penalty function in the beliefs minimization in Eq. (8) determining the worst case beliefs penalizes volatility in the change of measure process. The Phillips curve is a supply curve relating the current output gap to current inflation, the cost-push shock and the private sectors’ expectations about future inflation. Approximating the conditional distribution inflation by a log-normal random variable ln Pt þ 1 N ðmp,t , s2p,t Þ
ð12Þ
and using the log-normal approximation for the change measure, expected inflation under the worst case beliefs is E^ t ½Pt þ 1 ¼ Et ½mt þ 1 Pt þ 1 ¼ expðmm,t þ 0:5s2m:t þ mp,t þ 0:5s2p,t þ rm, p,t sm,t sp,t Þ ¼ expðmp,t þ0:5s2p,t þ rm, p,t sm:t sp,t Þ, E^ t ½Pt þ 1 ¼ expðmp,t þ 0:5s2p,t Þ exp ðrm, p,t sm:t sp,t Þ,
ð13Þ ð14Þ
here rm, p,t is the correlation between the change of beliefs and future inflation. The second line follows from the lognormality of mt þ 1 and Pt þ 1 , and the third line from the condition Et ½mt þ 1 ¼ 1. Plugging in the worst-case expected inflation into the Phillips curve in Eq. (4)
Pt ¼ Kðxt ,expðmp,t þ0:5s2p,t Þ expðrm, p,t sm,t sp,t Þ,ut Þ:
ð15Þ
The rational expectations Phillips curve is obtained with the restriction that rm, p,t sm:t ¼ 0 meaning that expðrm, p,t sm,t sp,t Þ ¼ 1. In the optimal commitment, the monetary authority commits to future state-contingent inflation given the current precommitted inflation and the current cost-push shock. The monetary authority’s concern for robustness means that it chooses the policy commitment fearing that the private sector will pick the worst-case beliefs. Using the log-normal approximations, the monetary authority chooses mp,t and s2p,t to maximize the objective function in Eq. (8), and the parameters of the worst case beliefs rp,m,t and sm,t are chosen to minimize the objective function in Eq. (8). The monetary authority’s objective is to set the output gap as close to zero as possible, and to set inflation as close to its efficient level as possible. The worse case beliefs tradeoff the entropy cost of increasing volatility of the change of measure against the benefits of worsening the covariance of inflation and the change for measure for the output gap in Eq. (15) above. From Eq. (15), increases in the inflation volatility sp increase the sensitivity of the output gap and inflation moments to the worst case sector beliefs: the larger in absolute value sp,t is, the more sensitive the objective function is to the worst case beliefs. The monetary policy authority therefore has an incentive to decrease the sensitivity of inflation to cost-push shocks relative to what she would choose in the rational expectations economy. As a consequence, inflation is less sensitive to cost-push shocks than in the rational expectations solution.
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If the monetary authority knew exactly how the private sector formed their beliefs, then it would be possible to solve for the optimal policy commitment by adjusting the sensitivity of inflation to shocks. For example, if the change of measure was given exogenously as a function of the cost push shocks, then optimal policy would be of the same form as in the rational expectations solution, but would lead to inflation having a different sensitivity to the cost push shocks. If the change in beliefs depended on other exogenous variables, then perhaps the inflation commitment could be adjusted to offset such effects. My discussion is much simplified relative to the analysis by Adam and Woodford, who relate the parameters of the Phillips curve to the structural parameters of the economy including labor supply and a richer production side while moving beyond the log-normal approximations presented above. Nonetheless, the simplified model captures the main economic forces at work here: robustness in the New Keynesian model implies that worst-case beliefs are chosen to make inflation reactions costlier to the monetary authority than under rational expectations so that with the optimal commitments inflation is less responsive to shocks than it is under rational expectations. 4. What can the monetary authority observe? In Adam and Woodford’s analysis, the monetary authority does not know the private sectors’ belief formation process and therefore chooses the policy commitment against the worst-case beliefs. But the assumption is ignoring information available to the monetary authority in real time. The model assumes complete financial markets, meaning that there are ^ t þ 1 be the actual change of measure used the private sector in forming their observable prices for financial assets. Letting m beliefs as opposed to the worst case change of measure used to determine the optimal policy commitment, the one-period bond price that the monetary authority observes in the market is 1 U 0 ðY t þ 1 Þ 1 U 0 ðY t þ 1 Þ 1 U 0 ðY t þ 1 Þ 1 ^ t þ 1b ^ t þ 1 Et b ^ t þ 1,b ¼ Et m Pt þ 1 ¼ Et m Pt þ 1 þcovt m Pt þ 1 0 0 0 1 þit U ðY t Þ U ðY t Þ U ðY t Þ U 0 ðY t þ 1 Þ 1 U 0 ðY t þ 1 Þ 1 ^ t þ 1,b Pt þ 1 þ covt m Pt þ 1 , ð16Þ ¼ Et b U 0 ðY t Þ U 0 ðY t Þ where the last line follows because the conditional mean of the change of measure must equal one. The expectations here are computed using the monetary authority’s beliefs. The first term is the bond price that would prevail if the private sector’s beliefs were the same as the monetary authority’s beliefs, and the second term captures how the change in measure is correlated to marginal utility and inflation. Similar expressions are available for longer horizon nominal bonds, for inflation protected bonds, and for stock prices; such expressions would involve additional conditional moments for the change of measure process used by the private sector. The monetary authority can learn about how the private sector is forming its beliefs from observing market prices since the monetary authority can compute the covariance term from market prices. This opens the possibility that the monetary authority could do better than the robustly optimal commitment reported in the paper by using the information in market asset prices. That is, the monetary authority would optimize not against worst case beliefs, but instead would optimize using the information they inferred from the observable asset prices. Such a model would require solving a non-trivial fixed point problem, because future output and future inflation themselves depend on future private sector beliefs about the economic environment. But the results would be interesting, and would perhaps allow for a systematic exploration into what information in asset prices should be incorporated into monetary policy. Is it quantitatively feasible to learn more about the private sector belief formation process in the New Keynesian model studied here? Is the resulting welfare gain large with plausibly calibrated parameters? What asset prices should the monetary authority focus on? What would the dynamics of learning look like in such an environment? 5. Multiple equilibria Adam and Woodford solve for the optimal commitment and also provide policy rules that can implement the optimal commitment under a local uniqueness assumption. New Keynesian models of the sort studied by Adam and Woodford have the possibility of multiple equilibria under different policy rules, in which beliefs could become ‘unanchored’. See, for example, Cochrane (2011). Suppose that the private sectors’s actual beliefs start to depend on time or extraneous state variables when the monetary authority implements policy using a Taylor rule. Eq. (16) shows that such deviations might be detectable in asset prices—in this case, the covariance term in Eq. (16) would depend on those state variables. Is it possible to adjust the optimal commitment to deal with such deviations? Could the implementation exploit such information? How would the resulting policy functions change if the private sector beliefs become correlated with exogenous variables? Which asset prices should the monetary authority look at to detect such deviations? The framework developed by Adam and Woodford provides a useful starting point for such an analysis. 6. What is the correct objective function here? Adam and Woodford assume that the monetary authority is dogmatic about its knowledge about structure of the economy, while the monetary authority acknowledges that the private section may have an incorrect model of
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the economy. This is an extreme assumption. Do we really believe that the monetary authority has better knowledge of the economy than the private sector? An alternative assumption is a model in which both the monetary authority and the private sector share the same incorrect beliefs and the same desire for robustness. That is, the monetary authority’s objective function replaces the expected utility term in the maximand in Eq. (8) with the worst-case private sector’s beliefs in Eq. (2). In the class of New Keynesian models considered here, Adam and Woodford argue that optimal policy would be the same in such an economy as in an economy under rational expectations. My sense is that a more plausible model is one where both the monetary authority and the private sector each have a desire for robustness, but that they differ in their robustness parameter, or they differ in their information about the economy or about each others’ beliefs or preferences. Solving for the optimal commitments in such an environment is a challenge for future research, and should generate interesting interactions between the monetary authority’s beliefs, the private sector’s beliefs, each agent’s desire for robustness, and the optimal policy. 7. Conclusions Adam and Woodford provides an analytical approach to compute optimal policy commitments where the policy maker is uncertain about how the private sector forms beliefs. The approach here is quite general, and should prove to be useful to many economic models. Woodford (2010) solves for the optimal commitment in a more reduced form New Keynesian model with restrictions on the form of the optimal policy commitment, and here Adam and Woodford have extended that analysis to a microfounded New Keynesian economy with a general policy commitment. The main analytical finding is that optimal policy commitments are similar in form to the familiar rational expectations optimal commitments even when the monetary authority is concerned about robustness to near-rational expectations. While the current analysis makes strong assumptions on the monetary authority’s knowledge of the economy and on the structure of the economy, the analytical techniques developed are valuable in computing optimal policies in many different economic environments. References Cochrane, J.H., 2011. Determinacy and identification with Taylor rules. Journal of Political Economy 119, 565–615. Woodford, M., 2010. Robustly optimal monetary policy with near-rational expectations. American Economic Review 100, 274–303.