The Phillips curve yields a relationship between prices and the output gap. This is an intermediate stage before computing the in‡ation impulse re-
21(1 )j
is, in fact, undetermined when = 1andj = 0. Actually, this is calculated assuming in…nitesimally close to1but not equal to 1. The resulting formula is indeed
the same as the one used by Mankiw and Reis (2002) for the sticky-information case. A similar comment applies in other places in this paper.
sponse, since plugging equation (3.2) into the Phillips curve yields a rela- tionship between prices and money supply. This section presents the Phillips curve without yet assuming strategic neutrality.
Appendix I shows that, after some tedious algebra, equations (3.1), (3.3) and (3.4) yield the following Phillips curve:
t Et( t+1) = 2
1 [ yt+"t(pt)]; (4.1)
where "t is an operator that takes the sum of expectation errors made at
t (i.e. the average of expectation errors made by various cohorts weighted by the number of …rms in each cohort). In the hybrid model, the sum of expectation errors made at t on pt, is equal to:
"t(pt) =
1
X
k=1
(1 )k[Et k(pt) pt]: (4.2)
The term Et( t+1) in equation (4.1) is de…ned as follows:
Et( t+1) = " 1 X k=0 (1 )kxt+1;k+1 pt # . (4.3)
The expectation operator Et obviously di¤ers from the expectation Et
established on the basis of the best information available at t (or equiva- lently, made by the best-informed agents at t), since there is no reason why
only the expectations of the best-informed …rms should matter while the in‡ation expectations of …rms setting their price at time based on old infor- mation would be completely neglected. What is perhaps more surprising is that the relevant expectation operator is not simply an average of the var- ious in‡ation expectations.22 Equation (4.3) gives the in‡ation expected to
prevail at timet+ 1 when the aggregate price level att+ 1 is expected to be
1
X
k=0
(1 )kxt+1;k+1 while the aggregate price level at timet is known to be
pt. One interpretation is the following: make a survey asking all …rms23 by
how much they expect to increase their own prices fromt tot+ 1(don’t ask them about their expectations for the increase of the aggregate price level); then Et( t+1)is the sum of these expected price increases. The proof is the
following. Each …rm will answer that it faces a probability 1 of keeping its price constant, and a probability of being able to reset its price. Thus, its expected increase of its own price is times the di¤erence between the price it expects to set if it is able to reset it and its current price (all …rms
22The average in‡ation expectation is
" 1 X k=0 (1 )kEt k(pt+1) pt # . See section 5 for a speci…c example in which the operator Et( t+1) is shown to be di¤erent from the
average in‡ation expectations.
23Ask all …rms once they know if they can reset their price at timet or not (…rms that
are aware of their current prices). Summing all these answers yields equa- tion (4.3): times the di¤erence between the sum
1
X
k=0
(1 )kxt+1;k+1 of
all answers about the prices …rms would expect to set at t+ 1 (if they can) and the sum pt of their current prices.
It is easy to verify that if = 1, equation (4.1) boils down to the sticky- price Phillips curve t Et( t+1) =
2
1 yt (given, for example, in MR). In
fact, when = 1, then "t(pt) = 0 (i.e. there are no expectational errors
at t on pt since all …rms are informed) and Et( t+1) = Et( t+1) (since all
…rms have the same information set). In the other pure case, if = 1, then Et( t+1) disappears (…rms do not need to take account of future in‡ation
when setting their current prices, since they can change their prices in every period), and the Phillips curve is given by yt+"t(pt) = 0, which can be
shown to be equivalent to the Phillips curve computed by MR for the sticky- information model.
The hybrid Phillips curve could be compared with the Phillips curves of other models. Three models would be particularly interesting in this respect. Woodford (2003) assumes that information-updating does not occur with a constant probability but is simply delayed by a …xed number of periods (thus
extending a model he wrote with Rotemberg, in which the delay is always one period). Altig et al. (2005) assume that between two re-optimizations …rms follow simple (non-optimal) indexation rules. Gali and Gertler (1999) assume that a fraction of …rms set prices according to a rule of thumb (they index their prices according to last-period in‡ation) while the other …rms have rational expectations.
Some di¤erences between their Phillips curves and mine are due to a di¤erence in frameworks. But even after adapting their models to MR’s framework (this involves setting the preference for the present to zero and assuming that the real marginal cost is proportional to output) important di¤erences remain. In the Rotemberg-Woodford model, when a …rm sets its price for a given date, it always perfectly anticipates the aggregate price-level that will prevail at that date because all other …rms will be setting their prices for that date on the basis of the same common information set. This is not the case in my hybrid model. An important di¤erence between my hybrid model, on one hand, and the Gali-Gertler model or the model of Altig et al. (2005), on the other hand, is that their Phillips curves do not involve past expectations whereas my hybrid model does (it inherits this feature from the sticky-information model).