as high (low) as possible.212121This may not generalize to some settings with
endogenous separations in which the firm has the opportunity to lay off the
worker ex-post. In turn, it is optimal for each party to make offers barely
acceptable to the other.222222If indifferent, a party is assumed to accept.
Hence, slightly abusing notation, if a firm gets to make an wage offer
$w^{f}_{j,\Delta,t}$ to a worker in a sub-period $j<M$, this offer should
fulfill

$\displaystyle
V\left((1-\tau^{y})[(M-j+1)w^{f}_{j,\Delta,t}+(j-1)\tilde{b}_{\Delta,t}(s)],\cdots\right)$
$\displaystyle=$ $\displaystyle
V\left((1-\tau^{y})[(M-j)(1-\tau^{w})w^{w}_{j+1,\Delta,t}(s)+j\tilde{b}_{\Delta,t}(s),\cdots]\right)$
(57)

with $V$ being a value functions as in (21) or (22), having added $y_{t}$ as
additional input. The left-hand side is the value a worker would obtain from
accepting the offer, while the right-hand side is the value of not accepting
and making the equilibrium counter-offer $w^{w}_{j+1,\Delta,t}(s)$ in the next
sub-period (which will be accepted). Since $V$ as in (21), (22) is strictly
increasing in income (additional resources can always be consumed), (57)
implies

$\displaystyle(M-j+1)w^{f}_{j,\Delta,t}(s)+(j-1)\tilde{b}_{\Delta,t}(s)=(M-j)w^{w}_{j+1,\Delta,t}(s)+j\tilde{b}_{\Delta,t}(s)~{}~{}.$
(58)

Intuitively, worker wealth does not matter for in indifference condition (57),
as any worker prefers higher period income in our setting. Similarly, if a
worker gets to make an wage offer $w^{f}_{j,\Delta,t}$ to a firm in a
subperiod $j<M$, this offer should fulfill

$\displaystyle(M-j+1)(h_{\Delta,t}s-w^{w}_{j,\Delta,t}(s))=-\gamma_{\Delta}(s)+(M-j)(h_{\Delta,t}s-w^{f}_{j+1,\Delta,t}(s))~{}~{}.$
(59)

Finally, if no wage is accepted until period $j=M$, the indifference condition
for an agency contemplating a worker’s offer $w^{w}_{M,\Delta,t}$ would be

$\displaystyle
h_{\Delta,t}s-w^{w}_{M,\Delta,t}(s)+(1-\zeta)(1-\delta(s_{t}))\beta\mathbb{E}_{t}J(s_{t+1},\Gamma_{t})=0$
(60)

as, if rejecting the offer, the firm would have to look for a new worker, the
value of which is $0$ due to free entry. We note that worker wealth does not
enter indifference condition (60) either, as any worker would like to claim
the maximum possible amount of income during the final bargaining period.
Since the equilibrium wage outcome can be characterized using equations (58),
(59) and (60), it follows that our AOB bargaining scheme delivers wages that
are independent of worker wealth:

######  Proposition 5.

The per-period wage of a matched worker with labor productivity $s$ will be
given by

$\displaystyle
w_{t}(s)=\frac{1}{2}\left(h_{t}s+\tilde{b}_{t}(s)\right)+\frac{M-2}{2M}\gamma(s)+(1-\zeta)(1-\delta(s_{t}))\beta\mathbb{E}_{t}J(s_{t+1},\Gamma_{t})~{}~{}.$
(61)

###### Proof.

Re-arranging indifference conditions (58) and (59) yields

$\displaystyle
w^{f}_{j,\Delta,t}(s)=\frac{\tilde{b}_{\Delta,t}(s)}{(M-j+1)}+\frac{M-j}{M-j+1}w^{w}_{j+1,\Delta,t}(s)$
(62)

and

$\displaystyle
w^{w}_{j,\Delta,t}(s)=\frac{\gamma_{\Delta}(s)+h_{\Delta,t}s}{M-J+1}+\frac{M-j}{M-j+1}w^{f}_{j+1,\Delta,t}(s)$
(63)

which we can combine to obtain

$\displaystyle
w^{w}_{j,\Delta,t}(s)=\frac{\gamma_{\Delta}(s)+h_{\Delta,t}s+\tilde{b}_{\Delta,t}(s)}{M-j+1}+\frac{M-j-1}{M-j+1}w^{w}_{j+2,\Delta,t}(s)~{}~{}.$
(64)

Iterating (64) forward $M/2-1$ times, we obtain

$\displaystyle
w^{w}_{2,\Delta,t}(s)=\frac{M-2}{2}\frac{\gamma_{\Delta}(s)+h_{\Delta,t}s+\tilde{b}_{\Delta,t}(s)}{M-1}+\frac{1}{M-1}w^{w}_{M,\Delta,t}(s)$

which we can use in (62) for $j=1$ to get

$\displaystyle
w^{f}_{1,\Delta,t}(s)=\tilde{b}_{\Delta,t}(s)+\frac{M-2}{2}\frac{\gamma_{\Delta}(s)+h_{\Delta,t}s+\tilde{b}_{\Delta,t}(s)}{M}+\frac{1}{M}w^{w}_{M,\Delta,t}(s)~{}~{}.$
(65)

Substituting (60) and re-arranging, we obtain the equilibrium subperiod 1
offer extended by the firm

$\displaystyle
w^{f}_{1,\Delta,t}(s)=\frac{1}{2}\left(h_{\Delta,t}s+\tilde{b}_{\Delta,t}(s)\right)+\frac{M-2}{2M}\gamma_{\Delta}(s)+\frac{1}{M}(1-\zeta)(1-\delta(s_{t}))\beta\mathbb{E}_{t}J(s_{t+1},\Gamma_{t})$
(66)

which will be accepted. In turn, the period wage is this wage times $M$, i.e.

$\displaystyle
w_{t}(s)=\frac{1}{2}\left(h_{t}s+\tilde{b}_{t}(s)\right)+\frac{M-2}{2M}\gamma(s)+(1-\zeta)(1-\delta(s_{t}))\beta\mathbb{E}_{t}J(s_{t+1},\Gamma_{t})~{}~{}.$

∎

###  B.2 Definition of equilibrium

######  Definition 1.

A Recursive Equilibrium of the model consists of

  * •

value functions $V^{a}(a_{it},k_{it},e_{it},s_{it},\Psi_{it};\Gamma_{t})$,
$V^{na}(a_{it},k_{it},e_{it},s_{it},\Psi_{it};\Gamma_{t})$ and
$J(s_{it},\Gamma_{t})$,

  * •

household policies $a^{a}(a_{it},k_{it},e_{it},s_{it},\Psi_{it};\Gamma_{t})$,
$a^{na}(a_{it},k_{it},e_{it},s_{it},\Psi_{it};\Gamma_{t})$,
$k(a_{it},k_{it},e_{it},s_{it},\Psi_{it};\Gamma_{t})$ and
$c^{a}(a_{it},k_{it},e_{it},s_{it},\Psi_{it};\Gamma_{t})$,
$c^{na}(a_{it},k_{it},e_{it},s_{it},\Psi_{it};\Gamma_{t})$,

  * •

firm sector policies $I_{t}$, $K_{t}$, $H_{t}$, $Y_{t}$, $u_{t}$,
$\theta_{t}$,$B_{t}^{l}$, $\Pi_{t}$,$y_{jt}\forall j\in[0,1]$

  * •

prices $h_{t}$, $r_{t}$, $q_{t}$, $r_{t}^{l}$, $mc_{t}$

  * •

a wage schedule $w_{t}(s)\forall s\in\mathcal{S}$,

  * •

government policies $b_{t}(s),G_{t},B_{t+1},\tau_{t}^{y},R_{t+1}^{B}$,

  * •

measures $m_{t}(\cdot)$,

so that

  1. 1.

Given prices $r_{t}^{l}$, $r_{t}$, $q_{t}$, wage schedule $w_{t}(s)$ and
profits $\Pi_{t}$ as well as labor market tightness $\theta_{t}$, the value
functions $V^{a}(a_{it},k_{it},e_{it},s_{it},\Xi_{it};\Gamma_{t})$,
$V^{na}(a_{it},k_{it},e_{it},s_{it},\Xi_{it};\Gamma_{t})$ solve the
households’ Bellman equations in (21) and (22) and
$a(a_{it},k_{it},e_{it},s_{it},\Xi_{it};\Gamma_{t})$,
$k(a_{it},k_{it},e_{it},s_{it},\Xi_{it};\Gamma_{t})$,
$c(a_{it},k_{it},e_{it},s_{it},\Xi_{it};\Gamma_{t})$ are the resulting optimal
policy functions.

  2. 2.

$y_{jt}\in[0,1]$ are consistent with demand schedule (4) and final output
$Y_{t}$ given by (3).

  3. 3.

Inflation $\pi_{t}$ is consistent with Phillips curve (23).

  4. 4.

Given prices $h_{t}$, $r_{t}$, $q_{t}$, $mc_{t}$ and technology shock $Z_{t}$
the intermediate goods producers choices $K_{t},H_{t},u_{t}$ are consistent
with optimality conditions (25)-(27).

  5. 5.

Given price $q_{t}$, the intermediate goods producers choices $I_{t}$ are
consistent with optimality condition (28).

  6. 6.

Given prices $h_{t}$ and wage schedule $w_{t}(s)$, labor agency value
functions $J(s_{it},\Gamma_{t})$ are consistent with (29).

  7. 7.

The wage schedule $w_{t}(s)$ is consistent with bargaining outcome (33).

  8. 8.

Labor market tightness $\theta_{t}$ is consistent with free-entry condition
(32).

  9. 9.

The Liquid Asset Funds’ portfolio choice is consistent with (35).

  10. 10.

The return of liquid assets is given by (71).

  11. 11.

Given inflation $\pi_{t}$ and unemployment $u_{t}$, the monetary authority set
$R_{t+1}^{B}$ according to (37).

  12. 12.

Taking the remaining values as given, the government sets taxes according to
(LABEL:eq:tax_rule) and issues debt $B_{t+1}$ so that (38) holds.

  13. 13.

The market for liquid asset clears, i.e.

$\displaystyle
A^{l}_{t}=\int^{\infty}_{\underline{a}}a_{it}m_{t}(a_{it})da_{it}~{}~{}.$

  14. 14.

The government bond market clears, i.e.

$\displaystyle B^{l}_{t}=B^{g}_{t}~{}~{}.$

  15. 15.

Capital market clearing requires, i.e.

$\displaystyle
K_{t}=\frac{A^{l}_{t}-B^{l}_{t}}{q_{t-1}}+\int^{\infty}_{0}k_{it}m_{t}(k_{it})dk_{it}~{}~{}.$

  16. 16.

The market for investment good clears, i.e.

$\displaystyle
K_{t+1}=(1-\delta(u_{t}))K_{t}+\left[1-\frac{\phi}{2}\left(\frac{I_{t}}{I_{t-1}}-1\right)^{2}\right]I_{t}$

  17. 17.

The market for labor services clears, i.e.

$\displaystyle H_{t}=\sum_{s\in\mathcal{S}}sm_{t}^{e}(s)~{}~{}.$

  18. 18.

The market for intermediate goods clears, i.e.

$\displaystyle\int^{1}_{0}y_{t}(j)dj=F_{t}(u_{t}K_{t},H_{t})$

  19. 19.

The final good market clears, i.e.

$\displaystyle Y_{t}=$ $\displaystyle
C_{t}+G_{t}+I_{t}+\frac{\phi}{2}\left[\frac{I_{t}}{I_{t-1}}-1\right]^{2}+\kappa
V_{t}+\bar{R}\int^{0}_{\underline{a}}a_{it}m_{t}(a_{it})da_{it}$
$\displaystyle-\left[\phi+\frac{\Psi}{2}\left(1-\frac{B_{t}^{l}}{A_{t}^{l}}\right)^{2}\right]A_{t}^{l}~{}~{}.$

  20. 20.

The dynamics of measures $m_{t}(\cdot)$ is consistent as described in Appendix
B.3

###  B.3 Details on measures $m$

Formally, $m_{t}$ describes a probability measure on the measurable space
$(\mathcal{X},\mathcal{A})$, with
$\mathcal{X}:=[\underline{a},\infty]\times\mathbb{R}_{+}\times[0,1]\times\mathcal{S}\times[0,1]$
and
$\mathcal{A}:=\mathbf{B}([\underline{a},\infty])\times\mathbf{B}(\mathbb{R}_{+})\times\mathbf{P}([0,1])\times\mathbf{P}(\mathcal{S})\times\mathbf{P}([0,1])$,
where $\mathbf{P}(\cdot)$ denotes the power set and $\mathbf{B}(\cdot)$ the
Borel $\sigma$-algebra of a given set.
Practically, with some abuse of notation, we have $m_{t}(\cdot)$ describe the
masses of households in a particular state at the beginning at period $t$, i.e
$m_{t}(a=a_{i},k=k_{i},e=e_{i},s=s_{i},\Xi=\Xi_{i})$ is the mass of households
with assets $a_{i},k_{i}$, employment status $e_{i}$, skill $s_{i}$ and
“entrepreneur status” $\Xi_{i}$. For ease of notation above, I suppress states
that are fully integrated over, e.g.

$\displaystyle
m_{t}(a=a_{i},e=e_{i})=\sum_{\Xi_{i}\in\\{0,1\\}}\sum_{s_{i}\in\mathcal{S}}\int^{\infty}_{0}m_{t}(a=a_{i},k=k_{i},e=e_{i},s=s_{i},\Xi=\Xi_{i})dk_{i}$
(67)

denotes the mass of households with employment status $e_{i}$ and bond
holdings $a_{i}$. Additionally, I suppress the annotation of non-suppressed
inputs whenever it does not cause any confusion, i.e. we may write
$m_{t}(e_{i})=m_{t}(e=e_{i})$.
Naturally, to be consistent with a unit mass of households, we require

$\displaystyle\sum_{e_{i}\in\\{0,1\\}}\sum_{\Xi_{i}\in\\{0,1\\}}\sum_{s_{i}\in\mathcal{S}}\int^{\infty}_{0}\int^{\infty}_{\underline{a}}m_{t}(a=a_{i},k=k_{i},e=e_{i},s=s_{i},\Xi=\Xi_{i})da_{i}dk_{i}=1~{}~{}.$

Additionally, the evolution of measures also need to be consistent with
household choices. Defining

$\displaystyle\tilde{X}^{na}(a^{\prime},k,e,s,\Psi;\Gamma_{t})$
$\displaystyle:=\left\\{a\in[\underline{a},\infty):~{}a^{na}(a,k_{it},e_{it},s_{it},\Psi_{it};\Gamma_{t})=a^{\prime}\right\\}$
$\displaystyle\tilde{X}^{a}(\\{a^{\prime},k^{\prime}\\},e,s,\Psi;\Gamma_{t})$
$\displaystyle:=\left\\{\\{a,k\\}\in[\underline{a},\infty)\times\mathbb{R}_{+}:~{}a^{a}(a,k,s,\Psi;\Gamma_{t})=a^{\prime}~{}~{}\text{and}~{}~{}\right.$
$\displaystyle\left.~{}~{}k(a,k,e,s,\Psi;\Gamma_{t})=k^{\prime}\right\\}$

as well as the “middle of period” measure $\tilde{m}_{t}(a,k,e,s,\Xi)$
fulfilling

$\displaystyle\tilde{m}_{t}(a,k,e=1,s,\Xi=0)$
$\displaystyle=p_{t}^{UE}m_{t}(a,k,e=0,s,\Xi=0)+(1-\delta(s)+\delta(s)p_{t}^{UE})m_{t}(a,k,e=1,s,\Xi=0)$
$\displaystyle\tilde{m}_{t}(a,k,e=0,s,\Xi=0)$
$\displaystyle=(1-p_{t}^{UE})m_{t}(a,k,e=0,s,\Xi=0)+\delta(s)(1-p_{t}^{UE})m_{t}(a,k,e=1,s,\Xi=0)$
$\displaystyle\tilde{m}_{t}(a,k,e,s,\Xi=1)$
$\displaystyle=m_{t}(a,k,e,s,\Xi=1)$

means they must follow

$\displaystyle m_{t+1}(a,k,e,s,\psi=0)=$
$\displaystyle(1-\zeta)\sum_{s_{t}\in\mathcal{S}}\Pi^{s}(s,|s_{t},e)\left(\lambda\int_{\tilde{X}^{a}(\\{a,k\\},e,s_{t},\Xi=0;\Gamma_{t})}d\tilde{m}_{t}(a_{it},k_{it},s_{t},\Xi_{t}=0)\right.$
$\displaystyle+\left.(1-\lambda)\int_{\tilde{X}^{na}(a,k,e,s_{t},\Xi=0;\Gamma_{t})}d\tilde{m}_{t}(a_{it},k,s_{t},\Xi_{t}=0)\right)$
$\displaystyle+\iota
p_{s}\left(\lambda\int_{\tilde{X}^{a}(\\{a,k\\},e,s,\Xi=1;\Gamma_{t})}d\tilde{m}_{t}(a_{it},k_{it},\Xi_{t}=1)+(1-\lambda)\int_{\tilde{X}^{na}(a,k,e,s,\Xi=1;\Gamma_{t})}d\tilde{m}_{t}(a_{it},k,\Xi_{t}=1)\right)$

and

$\displaystyle m_{t+1}(a,k,e=0,\psi=1)=$
$\displaystyle\zeta\left(\lambda\int_{\tilde{X}^{a}(\\{a,k\\},e,s,\Xi=0;\Gamma_{t})}d\tilde{m}_{t}(a_{it},k_{it},\Xi_{t}=0)+(1-\lambda)\int_{\tilde{X}^{na}(a,k,e,s,\Xi=0;\Gamma_{t})}d\tilde{m}_{t}(a_{it},k,\Xi_{t}=0)\right)$
$\displaystyle+$
$\displaystyle(1-\iota)\left(\lambda\int_{\tilde{X}^{a}(\\{a,k\\},e,s,\Xi=1;\Gamma_{t})}d\tilde{m}_{t}(a_{it},k_{it},\Xi_{t}=1)+(1-\lambda)\int_{\tilde{X}^{na}(a,k,e,s,\Xi=1;\Gamma_{t})}d\tilde{m}_{t}(a_{it},k,\Xi_{t}=1)\right)$

Finally measures $m_{t}^{e},m_{t}^{u}$ and $m_{t}^{\Psi}$ will fulfill

$\displaystyle
m_{t}^{e}=\sum_{s\in\mathcal{S}}\left[(1-\delta(s)+\delta(s)p_{t}^{UE})m_{t}(e=1,s)+p_{t}^{UE}m_{t}(e=0,s)\right]$

as well as

$\displaystyle
m_{t}^{u}=\sum_{s\in\mathcal{S}}\left[\delta(s)(1-p_{t}^{UE})m_{t}(e=1,s)+(1-p_{t}^{UE})m_{t}(e=0,s)\right]~{}~{}.$

###  B.4 Details on numerical implementation

####  B.4.1 Details on Steady State Solution

The household problem needs to be solved on a discretization of the state
space: I choose $70$ grid points for both $a$ and $k$, either of which are
non-linearily spaced as household decision functions tend to be more non-
linear for lower levels of assets. In particular, the grid points for both $a$
for $k$ are spaced according to the “double exponential” rule, i.e.

$\displaystyle\mathcal{X}=x_{min}+\exp(\exp(\mathbf{u}(\log(1+\log(1+x_{max}-x_{min})),n_{i}))-1)-1$

where $x_{min}$ is the minimum value on the grid for variable $x$, $x_{max}$
the maximum value and $\mathbf{u}(0,x_{max})$ a vector of equidistant points
on the interval $[0,x_{max}]$. Since household value- and policy functions
will feature and additional kink around $a=0$ when the borrowing penalty kicks
in, I add 5 additional grid points in the immediate vicinity of that point.
Given that individual labor productivity is discretized to $13$ points, this
means that the household problem is solved on a tensor grid of $70\times
70\times(2\times 13+1)=132300$ points (the “entrepreneur” status adds an
additional “income” state to the $2\times 13$ for employed and unemployed
workers). The discretization of the individual labor productivity process is
described in the main text, Section 4. Whenever interpolation is needed off
the grid, I use linear interpolation
For the implementation of the multidimensional EGM algorithm, I follow the
replication codes for Bayer et al. (forthcoming) closely.232323As of January
2024, these replication codes are available under
https://github.com/BASEforHANK/BASEtoolbox.jl. Given the random illiquid asset
adjustment, the EGM scheme only iterates over marginal value functions (i.e.
the derivatives of $V$ with respect to $m$ and $k$) and does not compute $V$
directly.For finding the steady, I iterate over $r^{l}_{ss}$ and $r^{k}_{ss}$:
Given these values, the remaining steady variables can be backed out and the
household-problem solved. I then use a heuristic updating procedure to search
for $r^{l}_{ss}$ and $r^{k}_{ss}$ so that the asset markets clear.

####  B.4.2 Details on Perturbation

As already indicated in the main text, the model’s dynamic equilibrium is
approximated using First-Order Perturbation around the its non-stochastic
steady state.
For the perturbation, note that when using the discretized representations of
the marginal value functions as well as the joint income/asset distribution,
the equilibrium can be represented as the solution to a non-linear difference
equation of the form

$\displaystyle\mathbb{E}_{t}F(\mathbf{y_{t}},\mathbf{x_{t}},\mathbf{y_{t+1}},\mathbf{x_{t+1}})=0$
(68)

as e.g. used by Schmitt-Grohe and Uribe (2004). $\mathbf{y}$ denotes a vector
of control variables, which includes the households’ marginal value functions
on the grid and $\mathbf{x}$ a vector of state variables, which includes the
discretized distribution.
In theory, one could find the linearized equilibrium using the standard
approach of computing the Jacobians of $F$ as in (68) and then solve the
resulting linear difference equation relying on methods such as Klein (2000).
In practice, however, such an approach would involve very high computational
costs for the 2-asset HANK model, given that the full $\mathbf{y}$ and
$\mathbf{x}$ have a combined length exceeding $300,000$.
To overcome this problem, I use the approach by Bayer and Luetticke (2020),
which conducts dimension reduction steps before computing the Jacobians.
Specifically, it first uses a Discrete Cosine Transform (DCT) to dimension-
reduce the marginal value functions: The values resulting from such a DCT are
coefficients of the fitted multidimensional Chebychev polynomial, and,
importantly, their absolute values have an interpretation as measuring the
“importance” of the corresponding polynomial for fitting the data. In turn,
instead of perturbing the discretized marginal value functions directly, one
perturbs only the largest of these DCT coefficients, with the others fixed at
their steady state values. For this application, I perturb the DCT
coefficients accounting for 99.995% of their total Euclidean norm.
For reducing the dimensionality of the joint distribution, Bayer and Luetticke
(2020) furthermore suggest to split it into marginals and a copula, where the
latter is in effect treated as a interpolator mapping the steady state
marginal distributions into the joint distribution. That “interpolator” can
also be dimension-reduced through a DCT or just kept fixed, so one only
perturbs the marginals as well as selected coefficients of the copula, which
have substantially lower dimension.
In the presented set-up featuring the binary employment status, a potential
problem is that there is no unique way to define a marginal CDF over the
income states for the split (or, equivalently, that the copula is not unique).
To sidestep this issue, I dimension-reduce the joint distribution of employed
and unemployed agents separately, which somewhat limits the amount of
dimension reduction that can be achieved. However, even despite that, the
procedure manages to shrink the effective dimensionality of the system to a
manageable number of approx. 1,800.

###  B.5 Additional model parameters

The model parameters not explicitly stated in Section 4 are provided in Table
5.

| $s$ | $\gamma(s)$ | $\delta(s)$
---|---|---|---
$s_{1}$ | 0.1384 | 0.1282 | 0.1266
$s_{2}$ | 0.1729 | 0.1601 | 0.123
$s_{3}$ | 0.2157 | 0.1997 | 0.1184
$s_{4}$ | 0.2689 | 0.249 | 0.113
$s_{5}$ | 0.3353 | 0.3105 | 0.1065
$s_{6}$ | 0.4188 | 0.3871 | 0.099
$s_{7}$ | 0.5212 | 0.4826 | 0.0903
$s_{8}$ | 0.6498 | 0.6016 | 0.0806
$s_{9}$ | 0.81 | 0.74 | 0.07
$s_{1}0$ | 1.0096 | 0.9348 | 0.0586
$s_{11}$ | 1.2585 | 1.1652 | 0.0470
$s_{12}$ | 1.5685 | 1.4523 | 0.0357
$s_{13}$ | 1.96 | 1.815 | 0.02533
Table 5: Skill-specific parameters

##  Appendix C Robustness

###  C.1 Alternative fiscal rule

As an alternative to the baseline fiscal rule (40), I consider the case of the
government always keeping taxes constant at its steady state level
$\tau_{t}=\tau_{ss}$ and instead gradually reduces its spending $G$ over time
to reduce its budget. In particular, I assume $G$ to follow the rule

$\displaystyle\frac{G_{t}}{G_{ss}}=\left(\frac{G_{t-1}}{G_{ss}}\right)^{\rho_{G}}\left(\frac{B_{t}}{B_{ss}}\right)^{(1-\rho_{G})\psi_{B}}$
(69)

also used by Bayer et al. (2023b) and use $\rho_{G}=0.94$ and
$\psi_{B}=-0.75$.
To briefly convey the impact of this alternative assumption, Figures 17 and 18
display model responses for the TFP- and government spending shock under this
alternative assumptions.
Under rule (69) and the chosen parameterization, the government debt increase
is somewhat less pronounced and persistent after the shocks. At the same time,
we see milder but still noticeable inflation persistence after the shocks,
affirming the robustness to the alternative assumption on fiscal policy but
providing a case in point for the importance of government debt dynamics for
inflation in the aftermath of macroeconomic shocks.

###  C.2 Long-term government debt

For introducing long-term governments bonds, I follow Bayer et al. (2023a),
who consider a setting featuring a simple geometric maturity structure. Bonds
are long-lived. Every period, they pay one nominal unit of return and a
fraction $\delta^{B}$ of them retire without repaying the
principal.242424Equivalently, such a setting can be interpreted as featuring
infinitely-lived bonds with geometrically declining coupon payments. See
Woodford (2001). Denoting the nominal period $t$ price of a bond as
$Q_{t}^{B}$, their expected nominal return is thus given by

$\displaystyle\mathbb{E}_{t}\frac{Q_{t+1}^{B}(1-\delta^{B})+1}{Q_{t}^{B}}~{}~{}.$

As before, similar to the baseline model, the government bonds are not
directly held by the households but instead by the LAFs whose problem becomes

$\displaystyle\max_{B^{l}_{t+1}}\left\\{\mathbb{E}_{t}\left[(r_{t+1}^{k}+q_{t+1})\frac{A_{t+1}^{l}-B_{t}^{l}}{q_{t}}+\frac{Q_{t+1}^{B}(1-\delta^{B})+1}{\pi_{t+1}Q_{t}^{B}}B^{l}_{t+1}\right]-A_{t+1}^{l}\left(\varphi+\frac{\Psi}{2}\left(1-\frac{B_{t+1}^{l}}{A_{t+1}^{l}}\right)^{2}\right)\right\\}~{}~{}~{},$
(70)

and the ex-post return to household’s liquid savings will be given by

$\displaystyle
r_{t}^{a}=\frac{(q_{t}+r_{t}^{k})(A_{t}^{l}-B_{t}^{l})+\frac{Q_{t}^{B}(1-\delta^{B})+1}{\pi_{t+1}Q_{t-1}^{B}}B^{l}_{t}}{A_{t}^{l}}-\varphi-\frac{\Psi}{2}\left(1-\frac{B_{t}^{l}}{A_{t}^{l}}\right)^{2}-1~{}~{}.$
(71)

i.e. the LAFs pass on potential valuation losses to the households.
Regarding monetary transmission, it is additionally assumed that the model now
features a very small amount of central bank reserves yielding a nominal
return determined by (37) which can also be held by the LAFs and count as
government bonds for the purposes of the quadratic cost. In turn the following
no-arbitrage condition will have to hold in equilibrium:

$\displaystyle\mathbb{E}_{t}\left[\frac{Q_{t+1}^{B}(1-\delta^{B})+1}{\pi_{t+1}Q_{t}^{B}}\right]=\mathbb{E}_{t}\left[\frac{R_{t+1}^{B}}{\pi_{t+1}}\right]$

Finally, the government budget constraint (in real terms) changes from (38) to

$\displaystyle
B_{t+1}+\tau_{t}^{y}\left(\sum_{s\in\mathcal{S}}w_{t}(s)m^{e}_{t}(s)+\Pi_{t}\right)=G_{t}+\frac{(1-\delta^{B})Q_{t}^{B}+1}{\pi_{t}Q^{B}_{t-1}}B_{t}+(1-\tau_{t}^{y})\sum_{s\in\mathcal{S}}b_{t}(s)m^{u}_{t}(s)~{}~{}.$
(72)

The only additionally parameter this setting requires is $\delta^{B}$, which I
set to $0.05$, implying an average time to maturity of 5 years (20 quarters).
Figures 19 and 20 display model responses for the TFP- and government spending
shock under this alternative assumptions. Except slightly larger initial dips
of (real) public debt at impact of the inflationary shocks, the model dynamics
are not substantially different compared to the main text.

###  C.3 Response to stimulus checks

As an alternative fiscal policy scenario, I re-do some analysis of section 6
and 7 for the case of a one time stimulus shock, at impact of which each
household receives an identical debt-financed transfer worth 1% of steady
state annual GDP per head from the fiscal authority.

Figure 11: Model IRFs to stimulus checks by $\Psi$

Figure 11 displays the equivalent to Figure 9 in the main text. While all the
responses are more short-lived due to the one-time nature of the shock, we
still clearly see that $\Psi$ and thus the pressure of public debt on the
“neutral” liquid rate matters for the inflation dynamics in its aftermath.

Figure 12: Model IRFs to stimulus checks by $\Psi$

Similarly, Figure 12 reveals the debt-targeting rule to provide for a
particular fast disinflation after the shock without imposing substantial real
costs.

##  Appendix D Additional Figures

Figure 13: Model IRFs to 1% TFP shock Figure 14: Model IRFs to Government
Spending shock Figure 15: Model IRFs to gov’t spending shock for different
$\Psi$ Figure 16: Model IRFs to TFP shocks for different policy rules Figure
17: Model IRFs to TFP shocks for different fiscal policy rule Figure 18: Model
IRFs to gov’t spending shock for different fiscal policy rule Figure 19: Model
IRFs to TFP shocks under long-term debt Figure 20: Model IRFs to gov’t
spending shock under long-term deb Figure 21: Model IRFs to TFP shocks under
higher adjustment costs Figure 22: Model IRFs to gov’t spending shock shocks
under higher adjustment costs Figure 23: Model IRFs to TFP shocks for
different policy rules Figure 24: Model IRFs to gov’t spending shock for
different policy rules