Expectations and the Rate of Inflation*
Iván Werning, MIT
July 2022
Preliminary and Incomplete

What is the effect of higher expectations of future inflation on current inflation? I
compute this passthrough for a series of canonical firm-pricing models, but allowing for arbitrary (non-rational) expectations. In the Calvo model, the expectationalpassthrough can be made arbitrarily close to zero for sufficiently high stickiness, but
in practice, for reasonable parameters, passthrough is close to its upper bound of 1. In
the Taylor model, in contrast, the upper bound for passthrough is ½ instead of 1. For ageneral time-dependent model I show that: (i) passthrough is given by a measurable
sufficient statistic: the ratio of the average duration of ongoing price spells to that ofcompleted price spells; (ii) the lowest theoretically possible passthrough equals ½ byTaylor pricing; and (iii) passthrough can be theoretically greater than 1 with hazards
that decrease over time; (iv) breaking down the passthrough across horizons, it is expectations in the near future that matters the most, expectations of long-run inflation
are completely irrelevant; (v) I provide a generalized Phillips curve for current inflation as a linear function of expectations of future inflation and realized past inflations;(vi) I show that the sum of all coefficients, both past and future, sums to one, so thatthe long-run Phillips curve is vertical. Finally, I study state-dependent “menu cost”
models and show that passthrough in these models can be extremely low or extremely
high, depending on the exact specification and inflation rate. I suggest a model wherefirms must pay a fixed cost for changing their sS pricing policy bands. This extension
gives a passthrough of 0 for small enough changes in expectations.

* For excellent research assistance I thank Tomás Caravello and Pedro Martı́nez-Bruera. I am grateful for

useful conversations and comments from Marc de la Barrera, Martin Beraja, Jordi Galı́, Guido Lorenzoni,
Helene Rey, Raphael Schoenle, Jón Steinsson, Ludwig Straub, Silvana Tenreyro and Christian Wolf.

11

Introduction

It is widely believed that inflation is strongly affected by the expectations of future inflation, perhaps in a near one-to-one relationship. This tight reverse feedback mechanism
justifies considering the management of expectations a crucial part of monetary policy
playbook. As the story goes, if inflation expectations are stable or so-called “well anchored” at low levels, then inflation may remain stable and low; on the other hand, if bad
news or high inflation rates raise the specter of future inflation, then inflation inevitably
follows from the rise in expectations. The tail wags the dog. These are stories, but what
do we really know?1
Expectations no doubt matter to some degree, but just how direct and powerful is thecausal link from inflation expectations to current inflation? Is it long-run or short-run
inflation expectations that matters the most? Does the magnitude of this passthrough depend on details of the economy or the way prices are set? Finally, how does past inflation
impact current inflation? Must past inflation matter more when expectations matter less?
I explore these questions through the lens of canonical economic models of firm pricesetting. I calculate the passthrough for the ubiquitous Calvo-pricing setting first (the right
expression cannot simply be read off the usual Phillips curve), contrast this to Taylorpricing and then develop a result for more general time-dependent models. Finally, I also
explore a variety of state dependent “sS menu cost” models. As is well appreciated, it
is notoriously challenging to identify the effect of expectations on inflation empirically.
Studying the theoretically predicted values for passthrough is complementary to empirical approaches to this question.
Although the backbone of the models I employ are standard, the way I solve themis less standard. I solve for the impact of a change in inflation expectations, holding
all other relevant variables fixed. Changing expectations in this way requires parting
ways with rational expectations, or, indeed, any other particular model of expectations
formation. Thus, a key element of my analysis is flexibly allowing for any arbitrary set of
expectations and solving for the “temporary equilibrium” at that point in time.
Allowing for arbitrary expectations in this way has additional advantage: the solution
for inflation as a function of expectations can always be combined, if desired, with any
model of expectations formation, including rational expectations as a special case. Even
if this is the ultimate goal, it is useful not to leap to this combination of assumptions in
one step: the intermediate temporary equilibrium that does not impose any particular
1 Theoretically, early rational expectations models relating inflation and output such as Lucas Jr. (1972)

justified such a belief and captured the ideas from Friedman (1968) and Phelps (1967). As pointed out
below, a cursory approach to the Calvo model also reinforces such a belief.

2expectation formation theory sheds light into the economic mechanisms at play.2
My stated goal is to shed light on the short-run impact of inflation expectations on
current inflation. I go from inflation expectations to inflation. This complements an extensive and important body of theoretical and empirical work going in the opposite direction, studying the formation of expectations, often drawing a link from past inflation
to expectations of future inflation.
The metric I focus on is the impact effect on inflation of a change in expectations of future inflation. To keep things simple, I first start by assuming that firms expect a constantinflation rate. This allows me to focus on a single “passthrough” coefficient to measure
the immediate impact of a change in expectations. I later complement this by studying
the effect of past realized inflation.
My results cast doubt on the firmly held view that the short-run passthrough frominflation expectations to inflation is nearly one-for-one. In particular, I show that thepassthrough can take on a wide range of values and is plausibly much lower than unity.
On the other hand, for plausible values, the passthrough is greater or equal to 1/2 intime-dependent setups, so one might also say that these theories put a non-trivial lower
bound on the effects of expectations. However, I also show that state-dependent setups
are capable of lower values.
I start with the widely adopted Calvo pricing model, where firms have a constantprobability of getting a chance to reset their price. For this model, it is extremely tempting
and common to read the passthrough off the linearized “Phillips curve” equation as being
equal to the discount factor, which then leads one to conclude that passthrough is close
to one. The logic is correct for computing the “long run slope of the Phillips curve”, i.e.
the steady solution with constant inflation, but is not right for deriving the independentrole of expectations themselves. One way to see this is that solving forward we can also
write inflation as a function of current and future output gaps. Should we conclude thatthe passthrough is zero or one?
The paradox is resolved by recalling that both of these expressions for the Phillips
curve impose rational expectations. As a result, inflation expectations are not free andtied down to the future evolution of other variables. With rational expectations one simply cannot consider the thought experiment of modifying expectations without also ma2 This general idea is embraced in many contexts by economists.

To take a microeconomic example,
the notions of supply and demand curves are beloved concepts, and nobody insist on leaping instead to
only solving for the equilibrium. Instead, one often pauses first to think about each curve’s determinants,
how elastic they are, their shape, etc. This is especially useful if we then wish to move from a competitive
analysis to, say, that of a monopolist—we now use the demand curve differently. This is akin to considering
different models of expectations formation.

3

nipulating future primitives. My goal is, instead, to express inflation as a function of the
expectation of inflation when treated as free variables, holding constant the expectationof future real marginal costs (which can be expressed in terms of output gaps) and anyother determinants of inflation.
Solving for the passthrough from expected inflation to inflation in the Calvo model, I
show that it has the potential for being very low. Indeed, for any given discount factor,
the passthrough approaches zero as prices become fully rigid. This theoretical possibility
notwithstanding, I show that, in practice, for plausible parameter values, the passthroughis close to one. I show this by studying the limit without discounting and arguing that itprovides a good approximation.
We find ourselves back with the same conclusion of a near one-for-one passthroughfrom inflation expectations to realized inflation, but this time on better logical footing.
However, my next results show that the one-for-one passthrough is special to Calvo andnot a robust economic conclusion.
I turn first to the Taylor-pricing case where firms set prices every fixed number of periods N. This form of price rigidity was initially popular in influential macroeconomic studies of monetary policy and nominal rigidities, introduced by Fischer (1977) and Taylor(1980) for wages and prices. However, these fell out of favor after Calvo (1983) provided
a more tractable setup, with a constant probability of changing prices, which simplifies
the aggregation and dynamics. Models with nominal rigidities have a large state variable: the entire distribution of prices, but this state can be dispensed with in the special
Calvo-pricing model.
Adopting the no discounting limit as an approximation, I show the passthrough offuture inflation expectations to current inflation in the Taylor model equals 1/2, rather
than 1; discounting gives lower values, but once again in practice this effect is minimal.
What is the economic mechanism behind these results and what explains this difference? Intuitively, when firm plan to have prices fixed for some time they want to set themso that they average out to an ideal price. When firms anticipate positive inflation thisideal price is rising. As a result, to get things right on average, they set their price initially above the ideal price and over time their price ends up below their ideal price. The
greater the expected inflation, the greater must be this initial price “overshoot”.
This overshooting mechanism is at the heart of the transmission mechanism from expectations of future inflation to current inflation. Overshooting is more aggressive in the
Calvo-pricing setup because the constant probability of a price change creates a righttailed risk, where prices remaining unchanged for very long periods of time. One manifestation of this is that the median price spell duration lower than the mean. In contrast, in4the Taylor setup there is no such right-tailed risk; the median duration equals the averageage. Overshooting is less aggressive.
In both Calvo and Taylor, I show that the passthrough coefficient does not depend on
the frequency of price changes. Lower frequency makes firms overshoot inflation more,
but this is exactly offset by the having fewer firms changing prices at any point in time.
The Calvo and Taylor model constitute two historically important benchmarks, yetthey are both special. To dive deeper I study a general time-dependent pricing model,
with an arbitrary hazard rate schedule. The hazard gives the probability of a price change
as a function of the time elapsed since the last price change.
For this relatively general setup, I obtain a surprisingly simple result: passthroughequals the ratio of two duration measures. The numerator is the average duration of
ongoing price spells. The denominator is the average duration of completed spells. Revisiting Taylor, it is easy to see that this ratio is 1/2 since the average age of ongoing prices
is N/2. In the Calvo case, both averages are equal to each other. More generally, if one
thinks of the average duration of ongoing prices as a proxy for that of completed prices
there are two sources of “bias”. On the one hand, for a given spell, the age of an ongoing
price is lower than its eventual duration. On the other hand, short spells are underrepresented relative to longer spells. From the duration literature, it is well known that withan exponential distribution these two effects offset each other, which is why passthroughequals one in Calvo.
I then show that across all time-dependent hazard rates (in the limit with no discounting) the lowest possible passthrough is 1/2 attained by Taylor. Intuitively, all other specifications have greater right-tail risk for firms. Passthroughs above 1 are also possible for
distributions with fatter tails than the exponential, such as a Pareto distribution, obtainedwith falling hazard rates.
I then extend the analysis to allow for general expectations of future inflation, that depend on the horizon of the forecast, as well for nonzero past realized inflation. I produce
a general Phillips curve, with two sets of coefficients, those on expected future inflation
rates and on past realized inflation rates. I show that the coefficients on expectations arelargest at shorter horizons. Thus, expectations of short run inflation dominate. Indeed,
expectations of inflation for the very long run are shown to be irrelevant.
The coefficients on past inflation are generally non-zero, except in the Calvo case. Indeed, the sum of both sets of coefficients, past and future, add up to one. In this sense, the“long-run Phillips curve” can be said to be vertical: any steady state inflation is possible(at a “natural” real marginal cost). Although long-run neutrality is sometimes taken for
granted, it is not immediately apparent in this general setup.
5This more general analysis highlights the spirit of my analysis, as well as its advantage and limits. I am purposefully seeking to elaborate on the determination of inflation,
focusing on price setting firms and allowing for flexible expectations of future inflation as
well as past inflation, and other real determinants. In this way, my analysis can be seen as
characterizing a more flexible Phillips curve. The spirit of the analysis is not to “close themodel” and combine this condition with a theory of expectation formation, or with otherpieces of a greater macroeconomic model and a specification of policy. Any such exercise
would be special or open up a plethora of options and can be carried out in other studies.
Put starkly, I seek to further our understanding of one important equilibrium conditionand leave using it in combination with other equilibrium conditions for another day.
Finally, I turn to state dependent “menu cost” models. I first show that the standard
set of these models produce extreme results: inflation can jump discontinuously up or
down when expected inflation rises! This has the potential for making passthrough verysmall (even negative) or very large. However, behind this result is the prediction thatthe frequency of price changes dramatically in the very short run. In my view, this is an
unrealistic feature of these basic models.
Thus, I consider two variations. In the first, the frequency of price changes is assumed
fixed in the very short run. The motivation is that the resources of goods and time tochange prices is difficult to adjust in the short run. I show that this can produce a morereasonable passthrough, potentially below 1/2.
In the second extension, I elevate the main feature of menu cost models to anotherlevel: I consider fixed costs of changing the sS pricing bands. I argue that it is difficult
to entertain costs to changing prices, for given pricing rules, and not also consider thecosts of changing the bands themselves. If the change in expectations is not too large, thefirm will not find it profitable enough to pay the fixed cost to change the bands. If the
bands do not change, then there is no change in inflation and the passthrough is zero.
Numerical explorations show that reasonable values of the fixed cost produce relatively
wide ranges of inaction. For example, if the fixed cost of changing bands is 5 times thatof a price change then it takes a 12% rise in expected inflation to trigger a change in thepricing bands.
Related Literature. There is a vast empirical literature on the determinants of inflation
related to the estimation or testing of so-called Phillips curve. In the context of a hybrid
New Keynesian Phillips Curve Galı́ and Gertler (1999) estimate time-series regressions
coefficients on future inflation and past inflation and find a non trivial role for future
inflation (about 0.6–0.7). These findings and their interpretation have been debated; see
6example Rudd and Whelan, 2005 and a response by Gali, Gertler and David Lopez-Salido(2005).3 More important, however, for the purposes of the present paper, is that these estimates do not use data on actual expectations, but instead use future realized inflation, as
justified under the rational-expectations model they lay out. Thus, they cannot separate
the role of expectations of inflation from that of other determinants.
There is a large body of empirical and theoretical work on the formation of expectations, as well as a growing recent body of work on the implications of these expectations.4
Among the later, the most pertinent attempt to estimate causal impacts of inflation expectation based on surveys of firms. These papers exploit randomized information provisionand find relatively small passthrough (Coibion et al., 2020, 2018b) or zero (Rosolia, 2021).
There is a large literature studying departures from rational expectations, of various
particular kinds. Perhaps closest in spirit to the approach of the present paper are Preston(2005), Garcı́a-Schmidt and Woodford (2019) and Farhi and Werning (2019) and others,
who use the notion of a temporary equilibrium to allow for arbitrary beliefs. However,these papers do so mostly in passing, as a stepping stone on their way to explore particular departures from rational expectations (adaptive, reflective and level-k expectations,
respectively), combining Calvo pricing and consumption decisions to study particularfeatures of monetary policy. They do not summarize features of the passthrough of inflation expectations to inflation or how it depends on the pricing model outside of Calvo.

2

Calvo 1 vs. Taylor ½

It is useful to understand the general spirit of my exercise starting with the most familiar
models of price setting. The Calvo model features a constant hazard probability of a pricechanges each period; it is very tractable and for this reason often used as part of the New
Keynesian model. The Taylor model, introduced before Calvo, has a constant intervalof time between price changes. The assumption is relatively natural, especially for somegoods and labor, but it is somewhat less tractable. For both these models we shall reachvery simple and stark conclusions.

2.1

Calvo Pricing

I start with the most familiar form for price stickiness, the Calvo-pricing setup. Before
jumping into the calculations it is useful to define what the question is and is not.
3 Rudd (2021) is a recent paper collecting arguments against the impact of expectations of inflation.
4 For the former, see For examples, see the surveys by D’acunto, Malmendier and Weber (2022) andCoibion, Gorodnichenko and Kamdar (2018a).

7

The Passthrough Question. The goal is to compute the passthrough going from expectations of future inflation to current actual inflation. I hold fixed current and future real
marginal costs and do not impose rational expectations or any other particular form of
expectations (adaptive, learning, inattentive, level-k, imperfect information, etc.). I will
also start by imposing that firms expect the inflation rate to be constant over time. This
allows me to focus on a single passthrough coefficient. I later extend the analysis relaxing
this assumption.
Taking expectations as given, at any point in time, I study the best response of firms
and aggregate them to compute a “temporary equilibrium”. This allows one to contemplate the impact effect at that point in time of a change in expectations, a comparative
static exercise.
Studying a dynamic response may require adopting a particular model of expectations
formation, as well as specifying the household and policy side of the model. But these
are not required to study the impact effect which is my focus. In any case the Phillips
curve I develop must hold at any point in time, even as expectations evolve. Thus, incharacterizing this object, I am providing an essential input into any dynamic analysis
that must be carried out with a fuller model.
NK Phillips Curve Cannot Provide Answer. To see why this distinction is important,let us review standard practice, which does not separately condition on expectations andother determinants of inflation. With Calvo pricing one can show that in equilibrium
inflation satisfies
πt = κxt + βEt πt+1 .
Here πt is aggregate inflation and xt is the “output gap” (departure from the flexibleprice value of output), κ is a parameter summarizing price stickiness, and β is the time
discount factor. This equilibrium condition is often labeled the “New Keynesian Phillips
Curve”. From this equation, it is tempting to conclude the the discount factor β ∈ (0, 1) isthe sought after passthrough going from inflation expectations to inflation. Furthermore,assuming β near 1 one is then led to conclude a near one-for-one passthrough.
However, this reasoning is misleading in that it does not answer the passthrough question as stated above. The reason is that the Phillips curve equilibrium condition is derived
under the assumption of rational expectations. Thus, the term Et πt+1 is doing double
duty: it is capturing expected inflation but also the effect of future output gaps and thesemust both be related. Indeed, one can solve forward to express inflation as being propor-

8

tional to the discounted sum of expected output gaps∞

π t = κ ∑ β s Et [ x t + s ].
s =0which might now, equally misleadingly, suggest a zero passthrough from expected inflation to inflation.5
These observations simply help underscore the spirit of our exercise: to separate therole of future primitives from the expectation of future inflation. To do so requires abandoning rational expectations and deriving equilibrium conditions for inflation that allow
for any expectation of inflation and other determinants. I now turn to these calculationsin the Calvo setting.
Optimal Pricing In the Calvo model firms have a constant hazard probability 1 − λ ofgetting a price reset opportunity each period. We will be approximating around a zero1
inflation steady state with a constant (nominal and real) interest rate of 1+r = β. The log
linearized reset price is then given by∞p∗t = µ + (1 − βλ)Et−1 ∑ ( βλ)s ( Pt+s + mct+s )s =0

The interpretation of this condition is that firms set their price so that they are at theirideal value on average; note that ωs = ( βλ)s (1 − βλ) is indeed a weighted average with∑∞
s=0 ωs = 1. The ideal price, in turn, equals a constant markup µ over the nominal
marginal cost Pt+s + mct+s . In this section, I have jumped directly to this relatively standard and familiar condition. But I later justify this log-linearized expression and generalize it in the context of a general time-dependent model, while also allowing for moregeneral shocks, such as shocks to markups.
In what follows, Et−1 must not be interpreted as an objective expectation, but rather
5 Hazell et al. (2022) use the NK Phillips curve, with rational expectations, and decompose it in a waythat a casual reader may interpret as a passthrough of 1, but it is important to understand why this is
not the case. They assume inflation and output gaps are stationary and define long run expected inflationand output gaps as π̄ = Et πt+s and x̄ = Et xt+s (which are independent of t because of stationarity).
They then write the demeaned condition πt − π̄ = κ ( xt − x̄ ) + βEt (πt+1 − π̄ ) and solve it forward πt =s
π̄ + κ ∑∞s=0 β Et [ x̂t+s ] where x̂t = xt+s − x̄ is the deviation of the output gap from its long run average.
This suggests a zero passthrough for short- or medium-run expectations, but a one-to-one passthrough forlong-run inflation. However, here once again a change in expected inflation π̄ requires a change in future
output gaps x̄. In other words, this is a correct decomposition that is useful for some purposes, but it does
not isolate the expectational passthrough.
In the present paper I explore the impact of changes in expectations even when the expectation of very
long-run inflation is constant. Indeed, I find that expectations for the very long-run are entirely irrelevant,but that expectations for the short run do matter.

9

firms’ own subjective expectations. As discussed above, I will purposefully stay awayfrom placing restrictions on expectations or modeling their evolution. Taking expectations as given, at any point in time, I study the best response of firms and aggregate themto compute a “temporary equilibrium”. This allows one to contemplate the impact effectat that point in time of a change in expectations, a comparative static exercise.
A brief word about the conditioning information Et−1 instead of Et : this notation ismeant as a reminder that firms do not have the price level Pt in their information set
when they set prices: the fraction of firms that are resetting prices for period t are doing
so simultaneously and do not observe the prices set by others until after they have set
their own. Thus, they know the previous period price level Pt−1 and have an expectation
for the period price level Pte = Pt−1 + πte . For these reasons, it is useful to imagine pricereseting happening at the end of a period and selecting the price that will be in place theat the beginning of the next period.
To focus on the contribution from inflation expectations let I write∞p∗t − Pt−1 = (1 − βλ)Et−1 ∑ ( βλ)s ( Pt+s − Pt−1 ) + ats =0s
where at ≡ µ + Et−1 ∑∞s=0 ( βλ ) mct+s collects all the non inflation expectation items.

Constant Expectations. To simplify, we first consider expectation of future expectationthat are constant at some π e across all horizons6Et−1 Pt+s − Pt−1 = π e (1 + s)
We later relax this assumption to break down inflation expectations across different horizons. Substituting this constant expectation into the price setting condition and carryingout the calculations givesp∗t − Pt−1 = (1 − βλ)∞1∑ ( βλ)s (1 + s)π e + at = 1 − βλ π e + at .

s =0

Our focus is on the first term involving expected inflation π e . Thus, from now on whenever it simplifies the discussion set at = 0; it is trivial to bring back any sequence of { at }.
Note that as long as prices are sticky (λ > 0) the coefficient on π e is greater than one: thefirm overshoots its price, relative to the static optimum price at t. The economic rationale
for overshooting is that future inflation erodes its price p∗t relative to the rising ideal price.
6 This is implied but weaker than assuming the one period ahead inflation expectation is constant andunchanging over time: Et+s Pt+s+1 − Pt+s = π e and that the law of iterated expectations holds.

10

In the flexible price limit as λ → 0 or as firms become myopic β → 0 we see the coefficient
becomes one: firms do not overshoot their price, and simply set it at the static optimump∗t = Pte = Pt−1 + π e . Conversely, as βλ → 1 the overshooting becomes infinitely large.
Inflation. Inflation is the weighted average of inflation across firms that cannot change
their prices 0 with weight λ and those that do reset prices p∗t − Pt−1 with weight 1 − λ,πt ≡ Pt − Pt−1 = (1 − λ)( p∗t − Pt−1 )Combining the two previous equations then gives
πt = φπ e + (1 − λ) at
where
φ=

1−λ
1 − βλAs is well known, inflation is entirely forward-looking in the Calvo model; thus, past
inflation does not appear in the above expression. This reflects the fact that reset-pricing
behavior is forward looking and firms that get to reset prices are randomly drawn fromthe pool of all firms, so there is no selection effect. This is a special property of the Calvo
assumption of a constant hazard rate.
The next proposition is based on the above expression for passthrough.
Proposition 1. In the Calvo model passthrough φ satisfiesφ ∈ (0, 1)
and any value can be attained in this interval for some parameters. Indeed, in the limit of no
discounting β → 1 or flexible prices λ → 0: passthrough attains its upper bound φ → 1;
conversely, in the limit of rigid prices λ → 1 then passthrough attains its lower bound φ → 0.
Any value for φ between zero and one can be obtained by varying the degree of pricestickiness only, for fixed given β. This once again dispels the notion, that passthroughequals β, as in the standard expression for the New Keynesian Phillips curve.
However, we next argue that in practice, for reasonable parameter values, passthroughφ is quite close to its upper bound of 1. It is simplest to make this argument in the continuous time limit, to which we now turn.
Continuous Time. The expression for passthrough becomes particularly simple in thecontinuous-time version of the model. This can be done by using the discrete time ex11

pression and setting β = e−ρ∆ and λ = e−δ∆ and then taking the limit as the period length
shrink ∆ → 0 or by setting up the model in continuous time directly.7 Either way one
arrives at
1
φ=,ρ/δ + 1so that passthrough only depends on the ratio ρ/δ.
Using the continuous time condition we easily see that for reasonable values of parameters φ will be relatively close to 1. For example, for ρ ≤ 0.05 and δ ≥ 1 (i.e. average
duration of a year or less) then φ ≥ 0.95. Although theoretically φ can be very low, inpractice this requires significant amounts of impatience or stickiness. We conclude thatthe limit with no discounting giving φ = 1 is a good approximation in the Calvo model.
This conclusion also applies to the discrete time version of the model.8

2.2

Taylor Pricing

In the Taylor setting prices are changed every N periods, with firms staggered over timeso that a fraction 1/N changes prices each period. Once again, we write the reset price asa weighted average of the expected prices∑sN=−01 βs Pte+s∑sN=−01 βs (s + 1) e∗
+ at = Pt−1 +π + atpt =∑sN=−01 βs∑sN=−01 βs
where at ≡ µ + Et−1 ∑sN=−01 βs mct+s / ∑sN=−01 βs . Combining this with
πt =

1 ∗
( p − Pt−1 )N t

we arrive at
πt = φπ e +1atN

7 In continuous time, price setters setp∗t − Pt =R ∞ −(δ+ρ)ses ds
1
R0 ∞ −(δ+ρ)s π e + at =π e + at
δ
+
ρ
e
ds
0

R∞
R∞
where at = 0 e−(δ+ρ)s mct ds/ 0 e−(δ+ρ)s ds and the result for φ then follows by computing πt = δ( p∗t −Pt ).
8 To see this, set a period to a year and use β = 0.95 and λ = 1/2 (giving a relatively large average−0.5
duration of 2 years): then φ = 1−10.95
×0.5 = 0.5/0.525 = 0.952. Setting a period to a quarter instead with
−0.75
β = 0.99 and λ = 0.75 (i.e. average duration of 4 quarters) gives φ = 1−10.99
×0.75 = 0.25/0.2575 = 0.97.

12

with

1 ∑sN=−01 βs (s + 1)
φ=N
∑sN=−01 βs

and φ is increasing in β.9 In the limiting case without discounting β → 1 then φ =1
1 10
Note that if we separate expectations current and future inflation then we
2 (1 + N ).
could write
111
πt = π0e + π e + at .
N
2
N
e
e
where π0 = Pt − Pt−1 . Thus, the value on expected future, rather than current, inflation isalways 1/2. Relatedly, if we think of periods as being very short and increase N accordingly to maintain the same calendar time, then as we take the continuous time limit with
N → ∞ we see that φ → 1/2.
Just as in the Calvo case, the continuous time limit offers some advantages. As we
see next, in continuous time 1/2 serves as an upper bound on φ, one that also provides
a very good approximation in practice. Taking the limit ∆ → 0 to continuous time bysetting β = e−ρ∆ and Nδ = 1δ where 1/δ denotes the time interval between price changes(δ is the frequency of price changes) or directly solving the model in continuous time
gives11
π=φ

ρ
δ

πe

with the decreasing function φ given byφ(ρ/δ) =

1
1
.
− ρ/δ
ρ/δ e − 1

Here 1/δ denotes the time interval between price changes and δ is the frequency of pricechanges. Note that only the ratio ρ/δ matters, just as in the Calvo case.
N −1s

9 The expression ∑s=0 β (s+1) is a weighted average of the sequence 0, 1, 2, . . . , N − 1 and an increase in∑sN=−01 βs

β puts relative more weight on higher values.

N −1
10 Note that with β = 1 we have φ = 1 ∑s=0 (s+1) and N −1 ( s + 1) = 1 N ( N + 1) so φ = 1 (1 + 1 )∑ s =0NN
2
2N
11 In the continuous time model price setting is given by

R 1/δ −ρse sds e1 (1 − e−ρ/δ (1 + ρ/δ)) e
p − P = R01/δπ =πρ1 − e−ρ/δ
e−ρs ds0∗

where I am using
Z ∆
0

and

1
1
1
e−ρs sds = − e−ρ∆ ∆ + 2 (1 − e−ρ∆ ) = 2 (1 − e−ρ∆ (1 + ρ∆))ρρρR ∆ −ρsds = 1ρ (1 − e−ρ∆ ). Inflation is then π = δ( p∗ − P).
0 e

13Proposition 2. In the Taylor-pricing caseφ ∈ (0, 12 )and φ → 12 in the limit of no discounting or flexible prices ρ/δ → 0; whereas φ → 0 in the limitof rigid prices ρ/δ → ∞.
Although the result shows that φ near zero is a theoretical possibility and that 12 is
an upper bound, for reasonable values of parameters, the passthrough φ once again lies
close to this upper bound. For example, if ρ < 0.05 and δ ≤ 1 then one calculates thatφ ≥ 0.495. Once again, as with Calvo, we conclude that the no-discounting case is a good
approximation for reasonable parameters.
The coefficient of 21 on future inflation stands in contrast with some well known derivations for the Taylor model in the New Keynesian literature. For the special case of N = 2
Roberts (1995) works out a condition for inflation under rational expectations. The expression features the expectation of inflation in the next period with a unit coefficient.
In addition to this derivation being carried out under rational expectations, unlike myanalysis, the coefficient in Roberts (1995) cannot be easily interpreted to think about thereaction of inflation to shocks, including shocks that affect the expectations term. The
reason is that the equation also contains an expectation error term that will systematicallyco-move with inflation.12

2.3

Intuition: Calvo vs Taylor

Why is expectational passthrough lower in the Taylor price setting relative to Calvo?
Firms set their price initially above their ideal price, but over time their price ends
up below their ideal price. The greater the expected inflation, the greater must be theprice over the currently ideal price. This “overshooting” mechanism is at the heart of thetransmission mechanism from expectations of future inflation to current inflation.
12 Roberts (1995) equation (8) can be written as

π t = Et π t +1 + a t + η t
where xt collects terms related to the real economy and ηt is an expectations error given byηt = Et−1 Pt − Pt = Et−1 πt − πt
It follows that
πt =

1
1
1
Et π t +1 + a t + Et −1 π t .
2
2
2

14

Intuitively, price overshooting is more aggressive in the Calvo-pricing setup. The reason is that with a constant price-change probability the distribution of price ages is exponential. This creates a non-trivial right-tailed risk of prices remaining unchanged forvery long periods of time. In contrast, in the Taylor setup where firms change their prices
every fixed number of periods, there is no such right-tailed risk. Firms overshoots theirideal price, but do so less aggressively. Indeed, the price will be too high exactly half thetime, too low the rest. This leads to a lower passthrough, that exactly equals 1/2.
It is interesting to note that in both cases the passthrough coefficient does not dependon the frequency of price changes. Lower frequency makes firms overshoot inflation proportionally more, but this is exactly offset by the fact that there are fewer firms changing
prices.

3

Sufficient Statistics for General Time Dependent Pricing

I now consider a general time dependent model. As we shall see, the more general results
I obtain can still be stated rather simply in terms of two sufficient statistics. The moregeneral formulation also helps sheds further light on the two previous special cases.

3.1

Preliminaries

We take as given a hazard function hs giving the probability of getting a price reset s + 1
periods since the previous reset (e.g. h0 denotes the probability of resetting a price if
the price was also reset in the previous period). It is useful to imagine price reseting
happening at the end of a period and selecting the price that will be in place the at the
beginning of the next period.
The hazard rate determines the survival probability Ss for each age s = 0, 1, . . .
S s +1 = S s (1 − h s )with S0 = 1. Note that Fs = 1 − Ss+1 (F−1 = 0) represents the cumulative distributionfunction for the duration of completed spells i.e. the probability a spell will be s or less isgiven by Fs . The associated density is f s = Fs − Fs−1 = Ss − Ss+1 = Ss hs .
Rescale the survival probability so that it adds up to one defines the distribution of
ongoing spellsSs,ωs = ∞∑ s =0 S s
where I assume ∑∞s=0 Ss < ∞. This distribution has two economic interpretations. First, it
15

represents the unique invariant distribution under the Markov process for age s, definedby s0 = s + 1 with probability 1 − hs and s0 = 0 with probability hs .13 Under this interpretation ωs represents the fraction of firms with age s in a cross-section of firms as wellas the “long run” average time spent at age s for a given firm. A second interpretation is
also possible, one that holds for a single firm and a single price spell: ωs represents the
expected amount of time spent at age s divided by the expected time spent at all otherages. In this way, it captures the relative importance of age s relative to all other ages fora given spell.
Using the distributions of completed and ongoing spells f s and ωs I define the averagehazard h̄ ≡ ∑s hs ωs , the average duration of completed spells d¯ ≡ ∑∞s=0 f s ( s + 1) and the∞14
average duration of ongoing spells dˆ ≡ ∑s=0 ωs (s + 1). I assume all these averages are
finite valued. One can show that15
1h̄ = ω0 = ¯.
dThese relations are intuitive. The average frequency of price changes h̄ must equal thedensity of firms that are resetting their price ω0 . Likewise, the average duration of completed price spells d¯ equals the reciprocal of the frequency of price changes 1/h̄, a relation
familiar in the special Calvo case with hs = h̄.
Calvo and Taylor Again. For reference, in the Calvo model the probability of a pricechange is constant so that hs = h̄ and Ss = (1 − h̄)s , yielding ωs = h̄(1 − h̄)s . In the Taylor
model, instead, prices are stuck for N periods (over t = 0, 1, . . . N − 1) so that hs = 0 for
s < N − 1 and hs = 1 for s ≥ N − 1 so that Ss = 1 for s ≤ N − 1 and Ss = 0 for s ≥ N,
yielding ωs = N1 for s ≤ N − 1 and h̄ = N1 .
Note that in the Calvo case the distribution of completed spells is the same as that of∞s+1 = ωs (1 − hs ) and ω0 = ∑s=0 hs ωs . The invariant distribution is unique0
because s = 0 is a recurrent point. Under relatively weak conditions, so that Ss ∈ (0, 1) for some s, the13 This follows because ω

unique invariant distribution is also stable, i.e. we converge to it starting from any other distribution. The
Taylor case is a knife-edged case lacking stability, starting from any distribution we cycle endlessly every
N periods.
14 We take the expectation of s + 1 not s because a spell that is reset at s = 0 is a spell of duration 1.
T
15 Using the definitions above we have that ω = h ω + ωss s
s+1 implying ω0 = ∑s=0 hs ωs + ω T +1 =∞∑s=0 hs ωs = h̄ where I have used that ωT +1 → 0 because ST +1 → 0 is implied by the assumption that∑∞
s=0 Ss < ∞. Next
∞
∞
∞
∞
∞
1
.
d¯ ≡ ∑ f s (s + 1) = ∑ (Ss − Ss+1 )(s + 1) = ∑ Ss (s + 1) − ∑ Ss s = ∑ Ss =
ω
0
s =0
s =0
s =0
s =0
s =1

Note that these same calculations also justify the second interpretation for ωs mentioned above, since ωs =Ss /d.¯

16

ongoing spells, since both are exponential. In contrast, in the Taylor case the distributionof ongoing spells is uniform, whereas that of completed spells has full mass at N. In
particular, the average duration of completed spells is greater than that of ongoing pricespells in Taylor but identical in Calvo. This will play a role in interpreting our previous
results on expectational passthrough.
Prices and Inflation. The price level (in logs) is defined as the average across firms,
which equals the weighted average of past reset prices∞

Pt = ∑ ωs p∗t−s .
s =0Inflation is then given by
πt = Pt − Pt−1 .
A bit of algebra shows that∞πt = ∑ ωs hs ( p∗t − p∗t−1−s )s =0

inflation is a weighted average of the inflation rate associated with each firm (note that,mechanically, a fraction 1 − h̄ firms produce zero inflation).

3.2

Price Setting Approximation

I first provide a simple formal result justifying and generalizing the type of log-linearizedcalculations used in the Calvo and Taylor settings.
The firm faces a path of interest rates qt+s . And a path of θt shocks to its profit function. These shocks can capture changes in their production functions or the demand firms
face—leading to variations in desired prices and markups. A firm resetting its price in period t then solves∞max
Et−1 ∑ qt+s ωs Π( p∗t − Pt+s , θt+s )
∗
pt

s =0

with first-order condition∞

Et−1 ∑ qt+s ωs Π p ( p∗t − Pt+s , θt+s ) = 0
s =0Consider a small variation in the firms’ problem { Pt+s , θt+s , qt+s , ωs }. Assume the approximation is carried out around constant primitives θt = θ̄ with zero inflation Pt+s = Pt
and with perfect foresight expectations.
17

Proposition 3. To a first-order approximation, around a zero inflation steady state, the reset pricesatisfies
p∗t =
where

e
Et −1 ∑ ∞s=0 qt+s ωs Pt+ at
Et −1 ∑ ∞s =0 q t + s ω sΠ pθ Et−1 ∑∞s =0 q t + s ω s θ t
at =.
Π pp Et−1 ∑∞s =0 q t + s ω sFurther setting qt+s = βs gives the desired result.
Proof. Totally differentiating gives∞

0 = Et−1 ∑ qt+s ωs Π pp,t (dp∗t − dPte+s )s =0∞+ Et−1 ∑ Π p,t (dqt+s ωs + qt+s dωs )s =0∞+ Et−1 ∑ qt+s ωs Π pθ,t dθt+ss =0

Around a steady state with zero inflation Π pp,t and Π pθ,t are constant over time andΠ p,t = 0 from the first order condition. Thus, the middle term cancels and rearranging
the remaining terms and setting gives
dp∗t =eEt −1 ∑ ∞s=0 qt+s ωs dPt+ dat
Et −1 ∑ ∞s =0 q t + s ω s

At a zero inflation steady state we also have that the constants p∗ = Pe and a = 0 satisfyeEt −1 ∑ ∞s =0 q t + s ω s P+ a.
p =Et −1 ∑ ∞s =0 q t + s ω s∗Adding these two conditions gives the desired result, where pt = p∗ + dp∗t , Pte = P∗ + dPte
and at = a + dat up to first order.
My analysis takes the sequence for {θt } and hence { at } as given and is focused on theeffects of expectations { Pte }.

3.3

Sufficient Statistics for PassthroughI now calculate the impact of a sudden change in inflation expectations. I do so approximating around a zero inflation rate steady state and also focus on the no discounting limit
β → 1.
18

Without discounting the optimal reset price is a weighted average of the marginal cost.
Thus, abstracting from shifters to real marginal costs to focus on expected inflation, we
have:
∞

∞

s =0

s =0

p∗t = ∑ ωs Pte+s = ∑ ωs π e (1 + s) + Pt−1
This gap between the reset price and the price level is one element that is needed to
calculate inflation.
We assume inflation has been zero in the past (we later study the effects of past inflation) and that all firms have the initial price p∗t−s = Pt−1 for all s = 1, 2, . . . and change
their price to p∗t at t = 0 when the shock to expectations occurs. Inflation, then is simply
πt = h̄( p∗t − Pt−1 ). Combining the above expression then gives the following result.
Proposition 4. Up to a first-order approximation around a zero inflation steady state with zero
past inflation
πt = φπ e + at
where

∞

φ = h̄ ∑ ωs (1 + s) =
s =0

dˆ
∑∞
s =0 ω s (1 + s )= ¯∞d∑ s =0 f s (1 + s )The passthrough φ equals the ratio of the duration of ongoing spells dˆ to that of completed spells d.¯ This is a very simple formula in terms of two sufficient statistics that are
in principle directly observable in the data.
The duration of ongoing spells controls the incentive firms have to overshoot theirprice, relative to their current ideal price, in the face of expected inflation. Indeed, theweights ωs for ongoing spells captures the average time firms their price will be at different durations. This average duration determines how much expected inflation will
impact its pricing decision today. In contrast, the duration of completed spells is not relevant for this decision, but appears in the numerator because its captures the frequency of
price changes.
We now use this general result to revisit the two special cases considered earlier.
Two important cases: Calvo and Taylor. In Calvo d¯ = dˆ = 1h̄ while in Taylor d¯ = N 6=dˆ = N2+1 implying
φCalvo = 1
φTaylor →

1
2

19

N→∞

Extension with Heterogeneity. The result extends easily with heterogeneity in the hazard rates. Let firms of type i have hazard h(s; i ) with associated h̄(i ),dˆ(i ) and d¯(i ) =1/ h̄(i ), then one can show thatπ = φπ e + ātZZdˆ(i )
di = h̄(i )dˆ(i ) did¯(i )R
R
R
dˆ(i )diNote that φ is generally different from R d¯(i)di as well as different from d¯(1i) di · dˆ(i ) di =φ=Rh̄(i )di · dˆ(i ) di. Indeed h̄ and dˆ may be correlated in the population of firms. Indeed,this correlation may be negative so that heterogeneity of h̄ and dˆ cancels out. For example,
heterogeneity in the hazard h̄ in Calvo is irrelevant, we always have φ = 1. Likewise,
heterogeneity in the length of rigidity N within Taylor is irrelevant and always gives
φ = 1/2. So in these cases heterogeneity of the frequency of price changes does notaffect φ. However, other forms of heterogeneity in the hazard function may matter: forexample, if a fraction of firms have a Calvo hazard and another have a Taylor hazard.
R

How Low Can We Go? How High? What is the range of possible passthrough φ? In
particular, how low can we make φ by choice of the hazard function? I now show that thelowest possible passthrough is 1/2 achieved by the Taylor pricing case.
Proposition 5. Let φ be given by Proposition 4 then for any {hs } we have1
φ≥ .
2
Moreover, any value of φ can be attained by some choice of the hazard function. In particular,
φ > 1 and arbitrarily large is possible.
Proof. One can show that∞ dˆ = ∑s =0∑s (1 − Fn )1 − n =0 ¯dNext we show that for any duration d¯ ∈ {0, 1, 2, . . . } the distribution { Fn } that minimizes∗¯dˆ subject to d¯ = ∑∞n=0 (1 − Fs ) is the Dirac distribution { Fn } with full mass at d. Anyalternative distribution { F̃n } with ∑(1 − F̃n )ds = d¯ second order dominates Fn∗ implyingthat ∑sn=0 ( F̃n − Fn∗ ) ≥ 0 for all s. This then implies
1 ∞ s
φF̃ − φF∗ = ¯ ∑ ∑ ( F̃n − Fn∗ ) ≥ 0
d s =0 n =0
Moreover φF∗ = 1/2. The Dirac F ∗ corresponds to the Taylor case. For d¯ ∈/ {0, 1, 2, . . . }20

a similar result holds but with a distribution with mass only at the two closest valuess ∈ {0, 1, 2, . . . }.
Next I show that arbitrarily large value φ > 1 are possible. Fix the average completed¯ s̃ and f 0 = 1 − d/¯ s̃ andduration d.¯ Now pick any whole number s̃ ≥ d¯ and set f s̃ = d/f s = 0 otherwise; this gives a bimodal distribution with average duration of completed¯ s̃
spells d.¯ Next, we compute the average duration of ongoing spells. Then S0 = 1, Ss = d/for s = 1, 2, . . . , s̃ and
¯ s̃
d/
Ss
ωs =
=
>0
1 + d¯∑ Ss
so thatd¯ s̃ + 1.
dˆ = 1 +1 + d¯ 2
The result then follows by choosing s̃ large enough. Note that using the same construction
we havedˆ
1 1 1 + s̃
φ = ¯ = ¯+.
dd 2 1 + d¯This implies that we can attain any value for φ in a limiting sense. Choosing any desired+s̃→ φ. Small perturbations of thisvalue for φ we can send d¯ → ∞ and s̄ → ∞ so that 12 11+
d¯
construction can attain any value of φ without taking limits.
The intuition for this result is as follows. The duration of ongoing spells suffers fromtwo “biases” that make it generally different from that of completed spells. Firstly, for any
given spell the age at which we sample an ongoing spell is by definition below that of thecompleted spell; a downward bias. Secondly, unless spells are all of the same duration
there is also an upward bias because we oversample relatively longer spells. Intuitively,
Taylor minimizes the ratio at 1/2 because it has the downward bias, but not the upwardbias.
The proposition not only rules out φ < 1/2, but shows that any value φ ≥ 1/2 is possible. Since Taylor attains ½ and Calvo 1, it should be intuitive that we can get anything
in between by mixing these two models. But φ > 1 arbitrarily large is perhaps less obvious. The intuitive idea is that we can make average duration of ongoing spells very large
relative to the completed ones by having most spells end as soon as they start, but havea small fraction end after a very long time. Ongoing spells are then the selected sample
of “survivors” with very long durations. Effectively, we can make the second positive“bias” described in the previous paragraph arbitrarily large.
Discounting.
What happens away from the no discounting case, when β < 1? With
discounting, firms set prices that are a weighted average including the discounting. As a21

result, discounting lowers passthrough.
Proposition 6. The passthrough φβ as a function of β is increasing in β and
φβ =

dˆβ
φ β =1dˆ∞sswhere dˆβ = ∑∞s=0 ω̂s,β s and ω̂s,β = β ωs / ∑s=0 β ωs is a weight with the property that ω̂s,β isincreasing in a first order stochastic dominance (FOSD) sense. Thus, dˆβ is increasing and dˆ1 = d.ˆ

For Calvo and Taylor we saw that the no-discounting case provided a very good approximation. Based on this proposition, we now see that this conclusion is more general
as long as dˆβ /dˆ lies close to 1. For reasonable discount rates and hazard rates this must
be the case: for price stickiness lasting a year or so, discounting within that year does notchange the relative weights ω̂s,β significantly, thus it will not change dˆβ and φβ significantly.

4

General Phillips Curve: Short vs Long Run Expectations
and Past Inflation

I now develop the equilibrium condition for current inflation while relaxing the simplifying assumptions imposed previously. Since this equilibrium condition is informallytermed a “Phillips curve” one can see this section as developing a Phillips curve for ageneral-time dependent model.
I relax two assumptions from the previous section. First, expectations for inflation at
different horizons are no longer assumed to be flat: expectations are given by an arbitraryesequence {πte+s }∞s=0 where πt+s represents the expectations of inflation for period t + s
held at time t; previously πte = π e constant. Second, I study the impact of past realized
inflation taking any given any sequence {πt−s }∞s=0 ; previously πt−s = 0 for s > 0.
The next proposition writes current inflation as a linear function of expectations of
future inflation rates and past realized inflation rates. It also provides some properties of
the coefficients on these variables. After stating the result, I discuss the results, provide
the sketch of the proof and its associated economic intuition.
Proposition 7. Suppose the hazard function is strictly positive hs > 0 for all s then to a first-orderapproximation in the limit of no discounting we have
∞

−∞

s =0

s=−1

πt = ∑ φs πte+s + ∑ φs πt+s + at22

with φs decreasing for s > 0 and∞∑ φs = 1s=−∞eand { at } a sequence dependent on the expectation of future real variables given by at = α ∑∞s =0 ω s θ t + s .

The previous section characterized φ = ∑∞s=0 φs the sum of the coefficients on expectations of future inflation. This proposition generalizes this result, breaking down φ overdifferent horizons. It also considers non-zero past inflation, the most interesting aspect ofwhich is that it allows us to see that the sum of both sets of coefficients is unity. This canbe interpreted as saying that the “long-run Phillips curve is vertical”, or more preciselythat steady-state inflation is not determined.
Proof and Intuition. To see how this result is derived we first compute the reset price.
This satisfies the usual condition but this time we do not impose Pte+s = π e (1 + s) (setting
at = 0 to simplify expressions)
∞

∞

s

∞

s =0

s =0

j =0

s =0

p∗t = ∑ ωs Pte+s = ∑ ωs ∑ πte+ j + Pt−1 = ∑ (1 − Ωs−1 )πte+s + Pt−1
where the cumulative distribution is Ωs = ∑sn=0 ωn with the convention that Ω−1 = 0.
∗∗
Using πt = ∑∞s=0 ωs hs ( pt − pt−1−s ) we arrive a
∞

∞

s =0
∞

s =0
∞

s =0

s =0

πt = ∑ ωs hs ( p∗t − Pt−1 ) + ∑ ωs hs ( Pt−1 − p∗t−1−s )= h̄ ∑ (1 − Ωs−1 )πte+s + ∑ ωs (h̄ − hs ) p∗t−1−s
The first set of terms provides the required coefficients φs = h̄(1 − Ωs−1 ) for expectationsof future inflation s ≥ 0. We see that that φs are decreasing since the cumulative distribution Ωs−1 is increasing. Intuitively, when resetting their prices, firms care relatively more
about earlier inflation because it affects the price level over more periods. In fact, if Ss = 0
for s ≥ N for some N then inflation expectations beyond N are completely irrelevant.
The second backward-looking set of terms depend on past reset prices { p∗t−s−1 }. Using ωs as probabilities we have ∑∞s=0 ωs hs = Ehs = h̄, so that∞∑ ωs (h̄ − hs ) p∗t−1−s = −Cov hs , p∗t−1−s



s =0

We see immediately that if hs = h̄ is constant as in Calvo then this backward-looking
term is zero. It is also zero if one starts at a zero inflation steady state with constant past
23

reset prices p∗t−1−s = p∗ ; our working assumption in the previous sections. In contrast, if
the reset price p∗t has been rising in the past and the hazard rate hs is increasing then thecovariance is negative, and the new term contributes towards positive current inflation.
To next step in the proof is to convert the expression in terms of past reset prices into
one involving past inflation rates. The price level is a weighted average of past resetp?rices∞

Pt = ∑ ωs p∗t−s .
s =0

The key idea is to prove that this can be inverted to solve for∞p∗t = ∑ αs Pt−ss =0with ∑∞
s=0 αs = 1. This is possible by applying the Eneström–Kakeya Theorem (Gardner
and Govil, 2014), noting that the hypothesis are satisfied because ωs ≥ 0 and ωs+1 ≤ ωs .
Substituting this expression for p∗ and rearranging then provides a linear expression interms of past inflation. I omit the details here, but collect the relevant expressions in an
Appendix.
Finally, to see that ∑∞s=−∞ φs = 1 we can work through the effects of constant past
inflation π 6= 0, noting that this can only emerge from constant inflation in the reset pricep∗t−s = p∗t − sπ.16 Evaluating the second term then gives
∞

∞

∞

∞

s=−1

s =0

s =0

s =0

π ∑ φs = ∑ ωs (h̄ − hs ) p∗t−1−s = π ∑ ωs (h̄ − hs )s = π h̄ ∑ (ωs s − f s s) = π (1 − φ).
Cancelling π we obtain ∑∞s=−1 φs = 1 − φ and the result follows.
Economically this result can be interpreted as saying that the “long-run Phillips curveis vertical”. That is, if at = 0 for all t then any constant solution for πt = π is possible; if
instead at = ā > 0 then π = ∞ and if āt = ā < 0 then π = −∞. .
16 To see this note that if p∗ − p∗t−st−s−1 = π for all s = 0, 1, . . . then
∞

∞

s =0

s =0

Pt+1 − Pt = ∑ ωs ( p∗t+1−s − p∗t−s ) = ∑ ωs π = πThe same is true for all earlier dates. Thus, constant reset price inflation produces constant past inflation.
But since we can invert p∗ from P this must be the only possible sequence of p∗ consistent with constantinflation in P.

24

5

Menu Costs: State Dependent Models

I now explore the implications of “menu cost” state-dependent models. The key difference between these models and the ones studied earlier is that the frequency of priceadjustments is endogenous. In particular, firms can be seen as using so-called “sS bands”
characterizing regions of inaction where the price is left unchanged.
I first explore the simplest setting introduced by Sheshinski and Weiss (1977), a deterministic economy with positive inflation, where there are only price increases. As I
will show the predictions of this model are quite extreme. In particular, an increase in
expected inflation can actually reduce inflation all the way to zero. This is due to extreme
movements in the frequency of price adjustments brought about by discrete changes inthe sS bands. This may not be realistic, so I study a version of the model where the frequency of price changes cannot be changed in the very short run. This model produces a
passthrough that is positive but lower than the Taylor one of ½.

5.1

Setup

It is useful to move to a continuous time setup. At any point in time t a firm has price pit
R
and aggregate price Pt = pit dt. It useful to define the price gap xti = pit − Pt .
Firms profits are a function of their price gap f ( x ). They discounted profits at a constant rate ρ, although we often take the limit as ρ → 0. Firms must pay a fixed cost c eachtime they change their price, as a result they change their price only at discrete intervalsof time and use an (s, S) rule keeping x in the interval [s, S] with s < 0 < S adjusting
the price up to x = S when x = s. We are interested in characterizing the (s, S) rule andderiving its implications for the passthrough from expected inflation to realized inflation.
Consider a steady state where expected inflation is given by π e > 0. Let V ∗ denote thestationary value of a firm that has just optimally changed its price. Then the firm problem
at time t is
Z Te
V ( xt ; π ) ≡ max
e−ρs f ( xt+s )dt + e−ρT (V ∗ (π e ) − c)T0xt+s = xt − π e twith xt given. The first order condition for T givesf ( x ) = ρ (V ∗ ( π ) − c )a condition for the lower bound on the sS policy x = xt e−πT . The upper bound of thesS policy is the optimal reset price x ∗ ∈ arg maxx V ( x; π ). This reset price x ∗ actually25

satisfies a condition akin to the Taylor model: the price is set so that the average marginal
profit is zeroZ ∗1 T −ρs 0 ∗ef ( x − πs)dt = 0.
ρ 0Taking a quadratic approximation of f ( x ) around the maximal value x̂ = arg maxx f ( x )R T∗gives f 0 linear, and in the limit ρ → 0 gives x ∗ = T1 0 πt + x̂ = 12 T ∗ + x̂. The x̂ shifter
plays the same role as at did in the time-dependent model; we set x̂ = 0. The bands arethen symmetric around zero: x = − x ∗ .
(Sheshinski and Weiss, 1977) proved that firms will expand their bands, so that x ∗ (π e ) =− x (π e ) is increasing in π e . Moreover, firms anticipate that prices will be set for a shortereamount of time T = 2x∗π(π e ) is decreasing in π e .17

5.2

A Shock To Expectations with Full Adjustment

Now suppose we are at a steady state with constant expected and actual inflation thatcoincide π0e = π0 . At a steady state we have an invariant cross-sectional distribution ofR
firms distributed with density ω ( x ).18 In equilibrium we must have xω ( x ) = 0. The
invariant distribution is uniform between [ x, x ∗ ] with constant density ω = x∗1− x .19
The rate of inflation pushes firms down to the boundary x making them change prices;the higher is inflation the greater the flow of firms hitting the boundary. The density offirms, or frequency of price changes, equals h̄ = ωπ. These firms change their price by adiscrete positive amount ∆+ = x ∗ − x. Thus, inflation is given by the product
π = h̄∆.
One then observes that π = h̄∆ = ωπ ω1 = π a consistency condition.20
Now from this steady state position, imagine firms anticipate higher inflation π e >π0 . Thus, when expectations change firms will expand their bands immediately. As aresult the distribution of firms is strictly away from the bands and no firms adjust prices,
resulting in zero inflation. A decrease in the inflation expectations shrinks the optimal
bands, calling on a mass of firms to adjust prices.
Proposition 8. In the Sheshinski-Weiss menu cost model, starting from a steady state with π0 ,17 This latter result requires imposing a condition that is satisfied with our quadratic approximation forf.
18 In equilibrium we require the consistency condition thatR

ω (s) xds = 0

19 This is the unique invariant distribution and one can ensure stability by perturbing this model slightly,

e.g. allowing some small Poisson arrival of free price changes.
20 For given bounds inflation is indeterminate, but this conclusion is knife-edged and dependent on thesimplifying assumptions we have adopted; thus, we will not be concerned with it.

26

an increase in inflation expectations at t = 0 π e > π0 lowers realized inflation on impact to zeroπ = 0.
Conversely, a decrease in inflation expectations π e < π0 induces at a mass of firms to change
their price immediately. This results in an upward jump in the price level that is greater the larger
the change in expectations π0 − π e .
For an increase in π e the passthrough is initially negative infinity. Once inflation isat zero further increases in π e have no effect, so the marginal passthrough becomes zero.
On the other hand, for a decrease in inflation expectations the immediate passthrough is
also negative infinity, but even more extreme since it is the price level, not inflation, thatjumps up. These results are obviously extreme, but they illustrate the possibilities when
the fraction of firms changing their price becomes endogenous.

5.3

Adjustment Frictions for Price Frequency

The results above are extreme and probably not realistic. Across steady states one canimagine firms adjusting the frequency of price changes, taking as given a “menu cost” ofeach price change, i.e. the cost is linear in frequency. But in the short run the resources
devoted to changing prices may not be perfectly adjustable. For example, if we think of aretail store, managing many products and prices, then changing prices requires employee
time devoted to this task. Staffing and training of these employees cannot be immediately
adjusted so the retail store cannot suddenly increase the rate at which they change prices
arbitrarily; nor will they want to stop adjusting prices altogether since that would leaveidle time and resources devoted to that task.
On the other hand, given enough time the frequency of price changes can be changed,
staffing rearrangements or hiring can be done.
Now let us revisit the result from the previous section, focusing on a rise in inflation
expectations. If inflation expectations rise, the bands are widened, and the frequency of
price adjustments was predicted to fall to zero. However, let us now instead entertainthat in the very short run this frequency is held constant at its previous value, due tothe notion that resources devoted to price changes are fixed in the very short run. On
the other hand, the firm anticipates that it will be able to adjust the frequency of pricechanges over the medium term. Indeed, suppose the firm anticipates that if it is resetting
its price today, then by the time it has to reset it again, it will have been able to freely adjust
its price adjustment frequency. The next result shows that under these assumptions thepassthrough is positive, but below 1/2 the value with Taylor pricing.

27

Proposition 9. In a Sheshinski-Weiss menu-cost model where frequency of price adjustment isfixed in the very short run, the marginal passthrough from inflation expectations to inflation satisfies
0 < φ < 1/2.
The calculations behind this result are as follows. If the frequency of price adjustment
is fixed then inflation on impact satisfies
π = h̄(π0 )( x ∗ (π e ) − x (π0 ))with h̄ and x held fixed at its previous value. Now this is the same calculation for the
Taylor model except for the value that x ∗ (π e ) = p∗ (π e ) − P. In the Taylor case we havep∗ − P =1
Tπ e2

with T = T ∗ (π0 ) fixed at its original value. Instead, we now have the same formula
but with π e > π0 the anticipated value of T ∗ (π e ) is lower. Intuitively, firms anticipate
that they do not need to overshoot due to inflation as much because they will increase
the frequency of price adjustment in the near future. Thus, the price spell that is juststarting is anticipated to be of lower duration. Since the Taylor case gave 1/2 we now geta passthrough below 1/2.

5.4

Idiosyncratic Uncertainty

In the simple Sheshinski and Weiss (1977) menu cost model, all price changes are promptedby inflation. The desired relative prices are constant and in the absence of inflation there
are no price changes. This is obviously a simplification. These models have been extended in various ways to incorporate idiosyncratic uncertainty at the product level. We
now consider these extensions. Departing from the basic benchmark opens a host of opportunities. These models have been extended not just to include uncertainty, but also toallow for free opportunities to change prices, to consider firms managing multiple products and prices, etc.
We take a small step and keep things simple, discussing the setup in Alvarez et al.
(2019). The model is cast is once again cast in continuous time, except that now due toshocks to marginal costs we postulate that firms keep track of xt = −πt dt + σdWt where
Wt is a Brownian motion process, so that dWt can be interpreted as an iid shock acrossperiods, permanently impacting xt with standard deviation σ. Once again the firm sets
up pricing bands, except that now they are characterized by three numbers: x, x ∗ and a28

new upper bound x̄. the firm adjusts prices whenever xt hits x or x̄, in which case it resets
x to x ∗ .
This model is more general than the one we studied earlier. The Sheshinski-Weisscase sets σ = 0, but, intuitively, similar results obtain when σ is small enough relative to
inflation π0 . However, when π0 = 0 and σ > 0 or, more generally, when σ is large andπ is small then we get different results. First, the distribution is no longer uniform, but itpeaks at x ∗ instead. Secondly and most importantly, an increase in π e does not widen thesS bands, instead: it shifts them to the right (see the proof of Proposition 1 in Alvarez etal., 2019). This induces an increase in price increases from the bottom at x—indeed a massof firms instead of a flow—and a drop to zero in the flow of price adjustments downward,
at the upper bound x̄. As a result, one obtains an extreme result: a discrete upwards jumpin the price level.
Such extreme results can no longer be so easily arrested by freezing the frequency of
price adjustments in the short run. If we assume that both the frequency of price increases
and the frequency of price decreases must remain constant, in the short run, then we canget a passthrough below 1/2 as before. However, one may assume instead that the totalfrequency must remain unchanged in the short run, but that the firm can reallocate thisfrequency between price increases and decreases. It may then be optimal to reallocate all
price changes to the price increases, leading to a discrete jump in inflation in response toa small increase in π e .

6mc2

The spirit of state dependent models is that firms are often inactive and do not adjust
prices frequently because there is a “menu cost”. Sometimes this cost is taken literally interms of the goods and time cost of printing menus, catalogs or relabeling sticker prices
on physical goods in supermarkets. However, an important component is also the managerial decision of changing prices.
In this section I take the idea of managerial costs seriously and push it one step further.
During normal times the firm may have converged on certain sS policies that are optimalfor some steady state inflation rate. The bands may simply embed the idea that prices
should be ±5% of some desired markup over marginal costs. It seems natural to think that
there are managerial costs to reconsidering and changing these bands to a new situation.
Doing so requires reviewing the available information, weighing the tradeoffs, holding
meetings, making marketing decisions and communicating them.
Following the menu cost literature, we formalize these ideas in a stylized way, assum29

ing there is a fixed cost c B > 0 that must be paid to modify some preexisting pricing
bands. If the cost is not paid, the firm can continue using its previous pricing bands.
For any arbitrary bands x and x ∗ let us denote by V ( x, x ∗ , π e ) the anticipated value
obtained by a firm with expectations π e . Let V ∗ (π e ) denote the value function using
optimal bands x (π e ) and x ∗ (π e ) given π e . Then if
V ( x ( π0 ), x ∗ ( π0 ), π e ) ≥ V ∗ ( π e ) − c B
the firm will choose to maintain its old bands. If the inequality is violated then the new
bands are implemented. For given π0 this induces a region of inaction for π e around π0 .
Proposition 10. Consider the Sheshinski-Weiss menu cost model extended so that, in addition tomenu costs for changing prices, there are also fixed costs c B for changing the pricing bands ( x, x ∗ ).
Then starting from an steady-state with inflation and expected inflation equal to π0 and associated optimized bands x (π0 ) and x ∗ (π0 ), there is an interval of inaction [π, π̄ ] with π < π0 < π̄such that the the firm maintains its bands unchanged. Then for any π e ∈ I0 inflation remains
unchanged, so the passthrough from inflation expectations to inflation is zero.
Moreover, the inaction region increases in c B and ∂c∂ π → −∞ and ∂c∂ π̄ → ∞ as cb → 0.
B

B

If the bands do not changed then inflation expectations has no effect on firm behavior
and hence no effect on inflation. The last part of the proposition suggests that the bands
can be quite significant even for small fixed costs. The reasoning is the same as in Akerlof
and Yellen (1985) and Mankiw (1985): if the bands were initially optimal the loses fromnot changing them are only second order.
Next, I perform a quantitative exploration. Annual inflation is initially 2% and thereal discount rate is 2%. The size of the menu cost and idiosyncratic shocks is calibrated
to match the observed size and frequency of price adjustments (in their sample from Argentina, those values are 10% and 2.7 adjustments per year during low inflation times).
I benchmark the costs of changing the pricing bands relative to the menu costs. It
seems natural to imagine that the costs of changing the bands may be significantly higher
than changing a single price, following a pre-established rule. Thus, I compute the upper
inaction region π̄ − π0 as a function of c B /c between 1–10. The result with and without
idiosyncratic shocks is plotted in Figure (1).
For example, with idiosyncratic uncertainty, if costs are five times greater than menu
costs, then the change in inflation expectations must be upwards of 12% for the firm tofind it worthwhile to re-optimize the pricing bands. The implied costs of following a
sub-optimal policy are small so that large changes in expectations are required to trigger
changes in the bands for these range of costs.
30

Figure 1: Upper inaction region π̄ − π0 as a function of relative fixed costs c B /c. Leftpanel σ = 0.1641; right panel σ = 0.

7

Conclusions

In this paper I take a step towards understanding the effect of inflation expectations on
pricing and inflation, holding all other determinants of these fixed. I show that the common perception that optimizing models imply a 1-to-1 passthrough is generally misleading. My results uncover that this passthrough depends quite a bit on the pricing model.
Exploring a wide range of cases I try to make the case that much lower values of thepassthrough are possible and plausible.
Casting aside these results, I believe that a side-product of the analysis is that it lends
a greater economic intuition and understanding for the transmission mechanism of theinflationary process, often ignored in formal analyses or the subject of speculation. In
particular, expectations matters to the extent that individual firms “overshoot” their ideal
relative price or if the frequency of price increases rises in the short run (the overall frequency of price changes is not relevant, as I show). Understanding this mechanism suggests new empirical or theoretical directions. Can we measure this overshooting directlyat the microeconomic level? Theoretically, are there other important economic considerations shaping the degree of overshooting such as price complementarities or the shape ofdemand?

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33

A

First Order Approximation for General Time Dependent
Model

Consider a general time-dependent model defined by the hazard rate function {hs }. Letωs+1 = ωs (1 − hs ) (the value of ω0 will be inconsequential). The firm faces a path ofinterest rates qt+s . And a path of θt shocks to its profit function.
The firm then solves∞max
Et−1 ∑ qt+s ωs Π( p∗t − Pt+s , θt+s )
∗
p

s =0

with first-order condition∞

Et−1 ∑ qt+s ωs Π p ( p∗ − Pt+s , θt+s ) = 0
s =0We consider a small variation in the firms’ problem { Pt+s , θt+s , qt+s , ωs }. Totally differentiating gives∞

0 = Et−1 ∑ qt+s ωs Π pp,t (dp∗ − dPte )s =0∞+ Et−1 ∑ Π p,t (dqt+s ωs + qt+s dωs )s =0∞+ Et−1 ∑ qt+s ωs Π pθ,t dθt
s =0

Let us assume the approximation is carried out around a constant θt = θ̄ and with zeroinflation Pt+s = Pt then Π pp,t and Π pθ,t are constant and Π p,t = 0 (from the first ordercondition) so the middle term cancels. We are left withe
Et −1 ∑ ∞s=0 qt+s ωs dPt
+ at
dp =
Et −1 ∑ ∞s =0 q t + s ω s

∗

where

Π pθ Et−1 ∑∞s=0 qt+s ωs dθt
at =.
Π pp Et−1 ∑∞s =0 q t + s ω sFurther setting qt+s = βs gives the desired result.

34

B

Coefficients on Past Inflation

The inverse has coefficients satisfying the recursion1
s=0
α s = ω0− ∑s S α
j =1 j s − j s ≥ 1
Writing p∗t in terms of past inflation rates gives∞p∗t = ∑ αs Pt−s
s =0∞

= Pt − ∑ αs ( Pt − Pt−s )
s =1
!
∞

s −1

s =1

j =0

= Pt − ∑ αs∞

∑ πt− j∞= Pt − ∑

∑ αj

s =1

!
π t − s +1

j=s

Define γs = ∑∞j=s α j .

∞p∗t = Pt − ∑ γs πt−s+1

(1)

s =1

Equivalentlyp∗t−s−1 − Pt−1 = p∗t−s−1 − Pt−s−1 + Pt−s−1 − Pt = −∞∑ γz πt−s−z − ( Pt−1 − Pt−s−1 )z =1Since ∑∞
s=0 ωs h̄ − hs = 0 then∞∞∑ ωs h̄ − hs p∗t−s−1 = ∑ ωs h̄ − hs




s =0

s =0

∞


p∗t−s−1 − Pt−1 = ∑ δs p∗t−s−1 − Pt−1
s =0

some calculations
∞

∞

"

∞

∞

z =1
∞

∞

s =0 z =1s =0∑ δs p∗t−s−1 − Pt = ∑ δs − ∑ γz πt−s−z + Pt−s−1 − Pt−1

s =0



s =0

#

= − ∑ ∑ δs γz πt−s−z − ∑ δs ( Pt−1 − Pt−s−1 )35define θs = − ∑sz=1 δs−z γz and µs = ∑∞z=s δz , then
∞

∞
∞

∗
δ
p
−
P
=
θ
π
−
∑ s t − s −1 t ∑ s t − s ∑ µ s π t − s

s =0

thus

s =1

s =1

∞
 ∗
ω
h̄
−
h
p
=
s
s
∑
∑ φ−s πt−s
t − s −1

s =0

s =1

with
φ−s = θs − µs .
This provides the desired expressions to compute φs for all s < 0.

36(2)