Appendix: Additional Proofs and Some Closed Forms

9 Appendix: Additional Proofs and Some Closed Forms

9.1 Additional Proofs

Proof of Proposition 2.1 For consumption, this is simply a re-expression of Lemma 4.2 in Gabaix (2016), adapting the notations and the timing.

Labor supply is g −γ (N t )=ω t u ′ (c t ), i.e. N t =ω t c t . Taking logs and deviations from the

ω ˆ constant values, φ t = −γ ˆ c t

Proof of Proposition 2.4 Derivation of (16). This comes naturally for the general formalism. Call z s = (B s ,d s ,d s+1 ,d s+2 , ...) the state vector (more properly, the part of it that concerns deficits). Under the rational model, z s+1 = Hz s for a matrix H: (Hz) (1) = z (1) + Rz (2) and (Hz) (i) = z (i + 1) for i > 0, where z (i) is the i−th component of vector z. Under the cognitive discounting

model simulated by the agent at time t, set z d t = (B t , 0, 0, ...) and the subjective model z s = z d t +¯ mH z s −z d t . This captures that the agent “sees” the debt B t , but more dimly the deficits

d t .We also have

T s =− B s +d s =e z s with e := − , 1, 0, 0, ... .

So,

BR

[T s ]=E t

BR

e ·z s =e ·E t [z s ]=e ·z t +(¯ mA)

BR

d s−t

z s −z t

d s−t rat

=e ·z t +¯ m E t z s −z t =− B t +¯ m

s−t

=− B t +¯ m

s−t

d s −r

u=t

Derivation of (17). We have:

We see that the impact of B t cancels out, a form of Ricardian equivalence. Old debt B t does not make the agent feel richer. But a new deficit today (d t ) does.

This implies:

x t =m y (x t + Ad t )+ x t+1

with

R (1 − ¯ m) A=1−r

X m m ¯ ¯ s−t R

r¯ m

s−t =1−r

m ¯ =1−

s≥t+1 R

So, rearranging as in the derivation leading up to Proposition 2.3,

(Arm y d t +¯ mx t+1 )=b d d t +

rm y R(1− ¯ with b m)

d = R−m y r R− ¯ m

Proof of Proposition 2.5 The proof follows the steps and notations of Gal´ı (2015, Chapter 3). I simplify matters by assuming constant return to scale (α = 0 in Gal´ı’s notations). So, the marginal cost at t + k is simply ψ t+k , not ψ t+k|k .

Notations. When referring to equation 10 of Chapter 3 in Gal´ı (2015), I write “equation (G10)” and do the same for (G11) and other equations. Lower-case letters denote logs. I replace the coefficient of relative risk aversion (σ in his notations) by γ (as in u ′ (C) = C −γ ).

The law of motion is correctly perceived, but in the profit function, firms see only a fraction m f of the (variable component) of the markup: they replace the real markup ψ f

t+ −p t by m (ψ t+k −p t )+

f d 1−m d ψ t+k −p t . Firms can reset their price with probability 1−θ. Mimicking Gal´ı’s calculations, with behavioral f d 1−m d ψ t+k −p t . Firms can reset their price with probability 1−θ. Mimicking Gal´ı’s calculations, with behavioral

t satisfies:

X k BR

p t −p t = (1 − βθ)

(βθ) E t [ψ t+k −p t ].

k≥0

Given our assumption that firms underperceive the departure of the markup from the baseline:

p t −p t = (1 − βθ)

(βθ) m ¯ m f E t [ψ t+k −p t ].

k≥0

Equation (G15) still holds, with µ t := p t −ψ t , and becomes simply ψ t+k =p t+k −µ t+k , so

i.e., using π 1−θ = (1 − θ) (p ∗

t −p t−1 )= θ (p ∗ t −p t ),

(βθ ¯ m) m f E t [p t+k −p t −µ t+k ] (64) 1−θ

which is a close cousin of the equation right before (G16).

I now define: λ := 1

θ (1 − θ) (1 − βθ) m f , so that

(βθ ¯ m) µ t+k . (65) λ

I use the forward operator F (F y t := y t+1 ), which allows me to evaluate infinite sums compactly, as in (with δ = βθ ¯ m):

δ k F k µ t = (1 − δF ) µ t . (66)

I also drop the expectation operator to simplify the notation. Hence (65) becomes:

1 −1 π t = A − (1 − βθ ¯ mF ) µ t λ 1 −1 π t = A − (1 − βθ ¯ mF ) µ t λ

Multiplying both sides by λ (1 − βθ ¯ λ mF ) gives, defining C :=

1−βθ ¯ m :

(1 − βθ ¯ mF ) π t = Cβ ¯ mθF π t − λµ t

i.e. π t = (1 + C) θβ ¯ mF π t − λµ t , and reintroducing the expectation operator:

π t =β¯ mθ (1 + C) E t [π t+1 ] − λµ t .

Thus we obtain the key equation (which is a behavioral version of (G17)):

We finally need to express the desired markup µ t as a function of primitives. Recall that ζ t

is log productivity. The labor supply is still (49), N −γ t =ω t C t , and as the resource constraint is

C (γ+φ) t =e ζ N ,ω

C t . Then, recall the definition µ t =p t −ψ t , we obtain

. The real marginal cost is then Ψ t /P t =

Next, if the pricing frictions disappeared, the markup would be 0 (recall that the government has Next, if the pricing frictions disappeared, the markup would be 0 (recall that the government has

t s.t.

0 = (1 + φ) ζ n

t − (γ + φ) c t

which gives the efficient level of consumption:

So, the output gap is x n

t := c t −c t satisfies:

µ t = − (γ + φ) x t

Plugging this in (68), we obtain the behavioral version of (G22):

t = βM E t [π t+1 ] + κx t

with κ = λ (γ + φ), i.e.

Proof of Proposition 3.2 We have: ˙x t = ξx t − σ (r + π t ). To solve for the system, note:

¨ x t = ξ ˙x t − σ ˙π t = ξ ˙x t − σ (ρπ t − κx t ) = ξ ˙x t + σκx t − ρσπ t = ξ ˙x t + σκx t + ρ ( ˙x t − ξx t + σr) = (ρ + ξ) ˙x t + (σκ − ρξ) x t + ρσr

so that:

(72) and the boundary conditions are: x T =π T = 0, hence (taking the left derivative):

x ¨ t − (ρ + ξ) ˙x t + (ρξ − σκ) x t = ρσr

x T = 0, ˙x T = −σr.

To analyze (72), we look for solutions of the type x ′ t =e . Call λ ≤ λ the two roots of:

λt

(74) Then, with D = ρσr

ρξ−σκ the solution is:

(Dλ − σr) e ′ λ (t−T ) − (Dλ ′ − σr) e λ(t−T )

In the traditional case, ξ = 0, so that λ < 0 < λ ′ . As D > 0, this implies that, as t → −∞, x t → −∞. We obtain an unboundedly large recession. This is the logic that Werning (2012)

analyzes. However, take the case where cognitive myopia is strong enough, ξ > σκ

ρ . Then, both roots of (74) are positive. Hence, we have a bounded recession. Indeed, as D < 0 in that case, x t is

Proof of Lemma 4.1 The proof mimics the ones in Woodford (2003) and Gali (2015). We have

W=− 2 u

c cE 0 β (γ + φ) x t + ǫvar i (p t (i))

2 t=0

where var i (p t (i)) is the dispersion of prices at time t. As in Woodford (2003, Chapt. 6),

β var i (p t (i)) =

using (71), and calling v −1 := var i (p −1 (i)). Hence,

W=− u c cE 0 β (γ + φ) x t +ǫ

π t − u c cǫ

v −1

2 t=0

2 κ ¯ 2 =− u c c (γ + φ) E 0 β π t + x t +W −

2 κ ¯ t=0

2 =− 2 KE

0 β π t + ϑx t +W −

2 t=0 2 t=0

I used κ = ¯ f κm from equation (70). Note that K and ϑ are independent of behavioral factors, when expressed in terms of primitives including the components of ¯ κ, ε. However, when they’re expressed

in terms of κ, the behavioral term m f intervenes.

Proof of Proposition 4.3 The Lagrangian is

L=E 0 β − π t + ϑx t +Ξ t βM π t+1 + κx t −π t

t=0

where Ξ t are Lagrange multipliers. The first order conditions are: L x t = 0 and L π t = 0 which give respectively

−ϑx t + κΞ t =0 −π t −Ξ t +M f Ξ t−1 =0

i.e. Ξ t = ϑ κ x t and

−ϑ π f

x t −M x t−1 .

Proof of Proposition 5.3 The state vector is z t = (x t ,π t ,π t−1 ). Write the system as E t z t+1 = Bz

CB

t + aπ CB t , for a matrix B. To study stability, we dispense with the forcing term aπ t . We can write E t z t+1 = Bz t , with

 κσ+β f (1+σφ x ) σ(βφ π −1)

Consider also the characteristic polynomial of B, Φ (Λ) = det (ΛI − B) (with I the identity matrix),

which factorizes as Φ (Λ) = , where the Λ i ’s are the eigenvalues of B.

i=1

When α 6= 0, inflation π d t is a predetermined variable, not a jump variable. Hence, for deter- minacy, B needs to have 1 eigenvalue less than 1 in modulus (corresponding to the predetermined When α 6= 0, inflation π d t is a predetermined variable, not a jump variable. Hence, for deter- minacy, B needs to have 1 eigenvalue less than 1 in modulus (corresponding to the predetermined

Also, in this draft I simply state the “central” necessary condition, that Φ (1) > 0. The other, more minor, “Ruth-Hurwitz” conditions (see also Woodford (2003, pp. 670-676)) will be added in

9.2 Microfoundations for the model with backward looking expecta- tions

Here I provide microfoundations and derivations for the model in Section 5. The consumer is as in the main model, so the IS equation (40) is as in the main model. 67 The firms, however, have access to one more “signal” about future inflation, the default inflation π d t .I assume that it follows (42), i.e. that it is a mix of past actual inflation and inflation guidance by the central bank. The key is to derive (41).

I will use the notation:

ρ := βθ ¯ m

I call F is the forward operator, F y t := y t+1 , which gives:

where F is the forward operator, F y t := y t+1 , and I used (66). I also drop the expectation operator to simplify the notation.

Let us start by calculating:

A : = (1 − βθ)

(βθ) (p t+i −p t ) = (1 − βθ)

(βθ) (π t+i + ... + π t+1 )

π t+k (βθ)

t+k (βθ) 1 k>0

k≥0

67 Section 10.2 gives a continuous-time derivation, which is more compact and useful when studying variants of the model.

We call µ ∗

t := p t −ψ t the desired markup. The firm i’s optimal price p t satisfies:

BR X k

p t −p t = (1 − βθ) E t

(βθ) (ψ t+k −p t )

(βθ) (p t+k −µ t+k −p t )

(βθ) µ t+k

(βθ) (π t+k 1 k>0 − (1 − βθ) µ t+k )

k≥1

I assume 68

Plugging this into (79) gives:

p ∗ t −p t =E t

m f (βθm) fπ

t+k 1 k>0 + (1 − f ) π t+k−1 1 k>0 − (1 − βθ) µ t+k

where ρ := βθ ¯ m and

(82) Using π ∗

t := (1 − f ) ρπ t − (1 − βθ) µ t .

=p t −p t−1 = (1 − θ) (p t −p t−1 ) as a fraction 1 − θof firms reset their price

68 The assumption is slightly different than the one for Proposition 2.5, in part to probe the robustness of this mechanism.

so that with C := 1−θ

Multiplying both sides by (1 − ρF ) gives:

Using (82) and reintroducing the expectation operator, π d

fm f βθ ¯ m=β¯ m θ+fm (1 − θ) ,

f β f = βM

f M f =¯ mθ+fm (1 − θ)

α=β¯ f mm (1 − f ) (1 − θ)

Thus we obtain the key equation (which is a behavioral version of (G17)):

t =β E t [π t+1 ] + απ t − λm f µ t

with λ = m f C (1 − βθ).

The rest of the proof is as in Gal´ı. The labor supply is still (49), N −γ t =ω t C t , and as the resource constraint is C −(γ+φ)

t =N t ,ω t =C t , i.e. ˆ ω t = − (γ + φ) x t . This gives µ t =ˆ ω t = − (γ + φ) x t , and we obtain the behavioral version of (G22):

t = βM E t [π t+1 ] + απ t + κx t t = βM E t [π t+1 ] + απ t + κx t