DERIVATION OF THE PHILLIPS CURVE
Let us start from the log–linear version of the optimal price setting condition "∞ # X α τ ? (βξp ) pbt = (1 − βξp )Et sbt+τ + pbt+τ α + θ(1 − α) τ =0
Define Θ =
α α+θ(1−α) ,
pb?t
then the optimal price can be rewritten as
= (1 − βξp )ΘEt
"∞ X
#
"
τ
(βξp ) sbt+τ + (1 − βξp )Et
τ =0
∞ X (βξp )τ pbt+τ
# (1)
τ =0
Given that the gross inflation rate is defined as πt =
Pt Pt−1 ,
the price in period t + τ is given by
Pt+τ = πt+τ × πt+τ −1 × . . . πt × Pt−1 The log-linear version of the preceding equation is given by pbt+τ = π bt+τ + π bt+τ −1 + . . . + π bt + pbt−1 = pbt−1 +
τ X
π bt+s
s=0
Plugging this last result in equation (1), we get "∞ # "∞ X X ? τ pbt = (1 − βξp )ΘEt (βξp ) sbt+τ + (1 − βξp )Et (βξp )τ τ =0
τ =0
pbt−1 +
τ X
!# π bt+s
s=0
or " pb?t = (1−βξp )ΘEt
∞ X
#
"
(βξp )τ sbt+τ +(1 − βξp )Et
τ =0
|
∞ X
#
"
(βξp )τ pbt−1 + (1 − βξp )Et
τ =0 {z
∞ τ X X τ (βξp ) π bt+s τ =0
}
(a)
|
s=0
{z
(b)
Let us first focus on component (a) of this sum. It shall be clear that pbt−1 is independent from τ and can therefore be taken out of the sum. Furthermore, since pbt−1 is predetermined, it belongs to the information set with respect to which expectations are taken. Therefore (a) can be rewritten as (1 − βξp )b pt−1
∞ X
(βξp )τ
τ =0
Since β ∈ (0, 1) and ξp ∈ (0, 1), the infinite series converges to (1 − βξp )−1 , such that term (a) reduces to pbt−1 .
#
}
The Calvo Fairy: Derivation of the Phillips Curve
2
Let us now focus on the term (b). The main trouble with this term is the double summation. The following lemma proves useful to deal with it. Lemma 1 The following inversion summation formula holds p X
aτ
τ =0
τ X
bs cτ s =
p X
s=0
s=0
bs
p X
aτ cτ s
τ =s
Proof: The proof simply involves rewriting the summation as p X
aτ
τ =0
τ X
X
bs cτ s =
s=0
X
aτ bs cτ s =
06τ 6p 06s6τ
aτ bs cτ s =
p X p X
aτ bs cτ s =
s=0 s=τ
06s6τ 6p
p X
bs
s=0
p X
aτ cτ s
s=τ
q.e.d.
In our case, aτ = (βξp )τ , bs = π bt+s and cτ s = 1, and p = +∞, hence we have "∞ # "∞ # τ ∞ X X X X τ τ (βξp ) π bt+s = Et π bt+s (βξp ) Et τ =0
s=0
τ =s
s=0
which rewrites " Et
∞ X
(βξp )τ
τ =0
τ X
#
"
∞ X
"
∞ ∞ X X (βξp )s π bt+s (βξp )τ −s
π bt+s = Et
s=0
∞ X π bt+s (βξp )τ −s+s τ =s
s=0
= Et
#
#
τ =s
s=0
∞ ∞ X X = Et (βξp )s π bt+s (βξp )j s=0
1 Et = 1 − βξp
j=0
"∞ X
# (βξp )s π bt+s
s=0
An alternative, simple, way to deal with the double summation is just to expand it as ∞ X
(βξp )τ
τ X
τ =0
π bt+s =
π bt
s=0
+ (βξp ) π bt + (βξp ) π bt+1 + (βξp )2 π bt + (βξp )2 π bt+1 + (βξp )2 π bt+2 + (βξp )3 π bt + (βξp )3 π bt+1 + (βξp )3 π bt+2 + (βξp )3 π bt+3 + (βξp )4 π bt + (βξp )4 π bt+1 + (βξp )4 π bt+2 + (βξp )4 π bt+3 + (βξp )4 π bt+4 .. . Summing each column, we see that the sum can be rewritten as ∞ X j=0
j
(βξp ) π bt +
∞ X j=0
j
(βξp ) (βξp )b πt+1 +
∞ X j=0
j
2
(βξp ) (βξp ) π bt+2 +
∞ X j=0
(βξp )j (βξp )3 π bt+3 + . . .
The Calvo Fairy: Derivation of the Phillips Curve
or,
3
∞ X (βξp )j π bt + (βξp )b πt+1 + (βξp )2 π bt+2 + (βξp )3 π bt+3 + . . . j=0
Since the term into parenthesis is independent from j, this rewrites as ∞ X 1 1 2 3 π bt + (βξp )b πt+1 + (βξp ) π bt+2 + (βξp ) π bt+3 + . . . = (βξp )τ π bt+τ 1 − βξp 1 − βξp τ =0
Henceforth, component (b) of the sum reduces to ∞ X (βξp )τ π bt+τ τ =0
Therefore, the optimal price setting decision can be written as "∞ # "∞ # X X τ ? τ (βξp ) π bt+τ pbt − pbt−1 = (1 − βξp )ΘEt (βξp ) sbt+τ + Et
(2)
τ =0
τ =0
let us denote ω bt = pb?t − pbt−1 , we have "∞ # "∞ # X X ω bt = (1 − βξp )ΘEt (βξp )τ sbt+τ + Et (βξp )τ π bt+τ τ =0
τ =0
= (1 − βξp )Θb st + π bt + (1 − βξp )ΘEt
"∞ X
# (βξp )τ sbt+τ + Et
τ =1
"∞ X
# (βξp )τ π bt+τ
τ =1
using the change of index j = τ − 1, this rewrites ∞ ∞ X X = (1 − βξp )Θb st + π bt + (1 − βξp )ΘEt (βξp )j+1 sbt+j+1 + Et (βξp )j+1 π bt+j+1 j=0
= (1 − βξp )Θb st + π bt + βξp Et
j=0
! ∞ ∞ X X (1 − βξp )ΘEt+1 (βξp )j sbt+j+1 + Et+1 (βξp )j π bt+j+1 j=0
j=0
{z
|
ω bt+1
ω bt = (1 − βξp )Θb st + π bt + βξp Et ω bt+1
} (3)
We now have the price setting behavior of the firm. Since we are interesting in getting the relationship between inflation and the marginal cost, we need to define aggregate inflation more properly. In particular, we would like to get the link between the aggregate price level and the optimal price setting behavior. This will be achieved by looking at the aggregate price level. First remember that the aggregate price level is given by Z Pt = 0
1
Pit1−θ
1 1−θ
Some firms just reset their price in period t, an event that affects —thanks to the use of the law of large numbers— a fraction (1 − ξp ) of the firms. Some firms set their price in period t − 1 and are still stuck with it in period t. These firms are in number (1 − ξp )ξp . Some firms
The Calvo Fairy: Derivation of the Phillips Curve
4
set their price in period t − 2 and are still stuck with it in period t. These firms are in number (1 − ξp )ξp2 . More generally some firms set their price in period t − j, j = 0, . . . , ∞. There is a fraction (1 − ξp )ξpj of such firms in the economy. In other words the aggregate price level can be rewritten as
1
1 1−θ ∞ X j ? 1−θ Pt = (1 − ξp )ξp Pt−j j=0
It is possible to write the price level recursively as Pt1−θ
=
∞ X
? 1−θ (1 − ξp )ξpj Pt−j
j=0
= (1 − ξp )Pt? 1−θ + (1 − ξp )
∞ X
? 1−θ ξpj Pt−j
j=1
let us make the change of indice k = j − 1 = (1 −
ξp )Pt? 1−θ
+ (1 − ξp )
∞ X
1−θ ? ξpk+1 Pt−k−1
k=0 ∞ X
= (1 − ξp )Pt? 1−θ + ξp (1 − ξp )
1−θ ? ξpk Pt−k−1
k=0
Pt1−θ
= (1 −
ξp )Pt? 1−θ
+
1−θ ξp Pt−1
It is then possible to log–linearize the previous equation as pbt = (1 − ξp )b p?t + ξp pbt−1 which rewrites as pbt − pbt−1 = (1 − ξp )(b p?t − pbt−1 ) ⇐⇒ π bt = (1 − ξp )(b p?t − pbt−1 ) = (1 − ξp )b ωt Multiplying equation (3) by (1 − ξp ) and using the last result we have (1 − ξp )b ωt = (1 − ξp )(1 − βξp )Θb st + (1 − ξp )b πt + (1 − ξp )βξp Et ω bt+1 Using the previous relation, we get π bt = (1 − ξp )(1 − βξp )Θb st + (1 − ξp )b πt + βξp Et π bt+1 which solves as π bt = Denoting κ =
(1−ξp )(1−βξp ) , ξp
(1 − ξp )(1 − βξp ) Θb st + βEt π bt+1 ξp
we obtain π bt = κΘb st + βEt π bt+1
1
Note that in the following formula, we make use of zero steady state inflation.
The Calvo Fairy: Derivation of the Phillips Curve
5
Note that, from the real block, we get sbt =
1+ν (b yt − b at ) α
and from the flexible price economy, we have ybtf = b at . Therefore, defining x bt = ybt − ybtf , we can finally write π bt = κΘχb xt + βEt π bt+1 where χ =
1+ν α .
