Ejecutar un Golpe

This subsection gives the canonical state space form of the non-linear Calvo model. It also spells out the mathematical assumptions that will be in place throughout the paper to support the formal analysis.

Assumption 1. The Calvo model described by equations (2.1)-(2.43) and the two

alternative structural models subsequently in Section 2.7 presented throughout this paper possess a canonical form EtZt+1= f (Zt, γ, UtS) with the following properties

i Z and US are respectively k × 1 and k0× 1 dimensional vectors that can be de-

composed into jump and state variables Z = (Z_kj

0×1, Z

s

k1×1), U = (U

j

k0₀×1, Uks0₁×1)

such that k0 ≤ k00 and k10 = k00

ii The interior of Z and US forms smooth manifolds

iii Zj is stochastically non-singular so that for any 1 ≤ i ≤ k₀ there is no non- stochastic function ˜f such that Zi,t= ˜f (Z−i,t). Uj will be mutually independent

iv P(Zt+1j ∈ B|(Zt, UtS)) > 0 for all B ∈ Σj where Σj is the σ-algebra of the vector

of jump variables Zj

v Local to any candidate steady state Z consistent with (i)-(iv) there is an Anosov diffeomorphsim Z × U_tS → f × U_t+1S

The last assumption is the non-linear equivalent of the familiar rank condition of linear algebra and will be met if each variable Zt contributes a single error. Part

(i) is a non-negligible restriction on the error process {U_tS_{} just because EC}−σ exists for some σ > 0 does not imply EC exists but I do not explore it here. The penul- timate restriction is an accessibility condition it will mean limiting distributions are defined on the whole space. This will rule out hysteresis and multiple equilibria. It is uniformly adopted in informal formulations of the benchmark New Keynesian model. Proposition 1. Abstracting from policy shocks the Calvo model can be characterized

by a canonical form with Zt= (πt−1, πt, Yt, ∆t) and

U_tS= (φt−1, φt, ϕt−1, ϕt, At−1, At)

Proof. The main complexity arises with the derivation of the non-linear Phillips curve.

This is because inflation via the reset price depends on the future of output, the discount factor and marginal costs - in a non-recursive fashion through ℵ and i. Note that I have drawn equivalence between inflation and the reset price since the relationship is a diffeomorphism via the inverse function theorem since

dπt d(p∗_t/Pt) = (1 − α) α _p∗ t Pt −θ (1 + πt)2−θ > 0

The proof consists of using the lagged inflation relationship to eliminate the denom- inator it from the present inflation relationship and solve for ℵt in terms of present

inflation πt and then using the recursion of ℵt to provide the recursion for inflation

that forms the Phillips curve. Identical results would appear if I eliminated ℵ_t and used the recursion of it, the crucial point is that I need to use a lagged inflation

relationship in addition to the present inflation relation to create a simultaneous equation system that allows me to solve out for the two summation terms ℵtand it.

It will be essential to use inverse function arguments admitted by the differentiabil- ity assumptions in place, as non-linear stochastic systems rarely permit closed form expressions.

h(πt−1) =

uc(Ct−1)Yt−1M Ct−1+ αβ(1 + πt)θℵt+ ˜f0(ujt)

uc(Ct−1)Yt−1+ αβ(1 + πt)θ−1it+ ˜f1(ujt)

πt)θℵt and Et−1(1 + πt)θ−1it which follows from the non-negativity of expectations

of positive random variables proved in the course of lemma 2.

˜

f0(ujt) > −αβ(1 + πt)θℵt

˜

f1(ujt) > −αβ(1 + πt)θ−1it

It is convenient to simplify the model at this point. Note that from the cost mini- mization and labor supply conditions we know that

uc(Ct)M Ct=

νl(lt)

At

Now using the market clearing conditions (12) and (24) note that

uc(Ct−1)M Ct−1=

ν0(∆_t−1Yt−1/At−1)

At−1

To obtain the required form it is necessary to remove Yt−1 and ∆t−1. First writing

the lagged Euler as

u0(Yt−1) − βRt−1(Yt−1, πt−1)Et−1 u0(Y_t) 1 + πt φt φt−1 = 0

By the implicit function theorem I can solve out for Yt−1 = ˜f3(πt−1, πt, Yt, φt−1, ujt)

with ˜f3,1< 0, ˜f3,2 > 0, ˜f3,3 > 0 and ˜f3,4 > 0. Note that I have used the properties of

the Taylor rule discussed in Section 2.4. I can do likewise for lagged price dispersion using the stochastic analog of (40) shown in remark 2 ∆t−1= ˜f4(πt, u∆t−1) with ˜f4,1 < 0

and ˜f4,2 < 0. h(πt−1) = ˜ f5(πt−1, πt, Yt, ∆t, φt−1, At−1, u∆t−1, u j t) + αβ(1 + πt)θℵt ˜ f6(πt−1, πt, Yt, φt−1, ujt) + αβ(1 + πt)θ−1it

where ˜f5 has absorbed ˜f0 and likewise ˜f6 has absorbed ˜f1, both must be strictly

positive by the functional restrictions on the errors and expectations laid out above, now solving for ityields:

it=

˜

f5− h(πt−1) ˜f6+ αβ(1 + πt)θℵt

Note that since itis strictly positive by construction (34) and since the denominator is

strictly positive the numerator must also be strictly positive. Hence, I can substitute into the present period reset price relationship

h(πt) =

αβ(1 + πt)θ−1h(πt−1)ℵt

˜

f5− h(πt−1) ˜f6+ αβ(1 + πt)θℵt

Solving for ℵt yields

ℵt= h(πt) αβ(1 + πt)θ−1 ˜ f5− h(πt−1) ˜f6 h(πt−1) − h(πt)(1 + πt)

We are almost done however, there is the issue of a singularity to contend with. This corresponds to the case where p∗_t = p∗_t−1. Fortunately the singularity is removable since by manipulation of the reset prices I am able to solve for ℵ_t for any initial ℵ_t−1 the expression is as follows

ℵ_t= ℵt−1

(1 + α − α(1 + π_t−1))1/(θ−1)

which lies strictly between 1/(1 + α)θ−1 and (1/α)θ−1thanks to the lower and upper limits on inflation. Moreover, by composition of smooth functions - smoothness is retained. This position of the system corresponds to the stochastic steady states of a zero (trend) inflation economy. To complete the proof substitute into the recursion for ℵ (35). The form is highly non-linear so solving for a closed form is inconvenient.Note that we can form an implicit function

˜

f7(πt+1, Yt+1, ∆t+1, ˜f5(.), · · · , ujt) = 0

. Where the first three terms have entered through Et(1 + π)θℵt+1. The appropriate

vector of jump errors comes about by (ii) (v) and remark 2. Thus yielding the correct desired dimensionality Zt= (πt−1, πt, Yt, ∆t)0 and UtS = (φt−1, At−1, u∆t−1, φt, At, u∆t ).

To complete the derivation of the Phillips curve note that ˜f7(Zt, UtS) can be written

as a composition of continuously differentiable functions that is locally non-constant in every variable it is a corollary of the mean value theorem that its set of stationary points will be measure zero. Therefore, it can be inverted almost everywhere for Zt+1

using the implicit function theorem yields a function ˜f8: (Zt, Yt+1, ∆t+1, UtS) → πt+1

where the expectation exists by assumption 1. This topological justification will be used in all subsequent steps.

Finally inverting the Euler for Yt+1in a fashion analogous to the construction of f3and

substituting in the price dispersion relationship for πt+1 yields ˜f9 : (Zt, πt+1, UtS) →

πt+1finally invert ˜f9 and then integrating over UtS gives Etπt+1= fπ(Zt, UtS) to com-

plete the derivation substitute in ˜f9 in to the Euler equation and integrate over the

jump errors to solve for EtYt+1= fy(Zt, UtS) and likewise for Et∆t+1= f∆then note

Etπt−1= πt hence we have the appropriate structural form EtZt+1 = f (Zt, UtS) that

is valid almost everywhere. By assumption 1 (ii) it must apply everywhere.

For proof. Suppose the converse that there were some other variable Z∗ such that

EtZt= f∗(Zt, Zt∗, UtS) such that EtZt+1= f (Zt∗, .) were a constant for µ almost every

(Z_t, U_tS) but non-constant for some set AZ×U with µ(AZ×U) = 0 then a contradic-

tion would arise for there will be a jump discontinuity of f∗ at,at least one point (Z0, U0, Z0∗) where f (Z∗, .) was behaving as a non-constant function. From a topo-

logical perspective, for some open set AZ the inverse image of EtZt+1 must contain

subsets of the form f−1(AZ) = A0Z ∪ X for some disjoint pair where AZ is an open

interval and X has measure zero. However, this means that X cannot be an open set of a Euclidean and therefore nor can f−1(AZ). This proves f∗ must have a dis-

continuity. Now consider the candidate state space form containing the additional variable Z∗ that takes the form Z_t= (π_t−1, πt, Yt, ∆t, Zt∗) and f = ( ˜f , f∗) where f∗

is the recursion for Z∗ _{such that E}tZt+1∗ = f∗(Zt, UtS). Then f inherit the disconti-

nuity from f∗ which completes the contradiction and shows the representation of the structural form is globally valid.