Slides for Piotr's talk.

1. Asset Supply and Liquidity Transformation in
HANK
Yu-Ting Chiang 1 Piotr Zoch 2
1FRB of St Louis
2University of Warsaw and GRAPE
May 9, 2023

2. Motivation
Question: role of the financial sector in the transmission of
macroeconomic policies.
Framework: General model of the financial sector + HANK.
- banks: issue deposits, hold illiquid capital and liquid asset
- nesting a large class of models of financial intermediation
Result: aggregate responses depend on the financial sector through
a liquid asset supply function
Dt
rB
s , rK
s
∞
s=0
:= deposits - liquid assets held by banks
Key elasticities: ∂Dt/∂rK
s
Dt
(cross-price)
- Low cross-price: disturbance in the liquid asset market →
large changes in capital price and aggregate demand.
1 / 17

3. Results Overview
Liquid asset supply elasticities are quantitatively important
Policy experiment:
illiquid asset purchases financed by liquid government debt
Model comparison:
liquid asset supply ranging from perfectly inelastic (HANK) to
perfectly elastic, with Gertler-Karadi-Kiyotaki in between
Result: Aggregate output responses are by two orders of magnitude
larger in HANK than with perfectly elastic supply.
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4. Model

5. Household
A continuum of households choose consumption and savings in
liquid and illiquid asset to maximize utility:
max
ai,t,bi,t,ci,t
E
X
t≥0
βt
u (ci,t, hi,t) , s.t.
ai,t + bi,t + ci,t + Φi,t(ai,t, ai,t−1)
= (1 + rA
t )ai,t−1 + (1 + rB
t )bi,t−1 + (1 − τt)
Wt
Pt
zi,thi,t
1−λ
where zi,t is idiosyncratic shocks, Φi,t is portfolio adjustment costs,
and
ai,t ≥ a, bi,t ≥ b
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6. Production and Labor Supply
I Firms maximize profit with technology yt = kα
t−1h1−α
t :
max
kt−1,{h`,t}
Ptyt − Rtkt−1 −
Z
W`,th`,td`.
ht: CES aggregator; union ` supplies h`,t =
R
zi,thi,`,tdi
I Union ` sets nominal wage growth πW,`,t = W`,t/W`,t−1 to
maximize household welfare with Rotemberg adjustment cost.
I Capital: kt = (1 − δ) kt−1 + Γ (ιt) kt−1, ιt = xt
kt−1
.
I Return on capital:
1 + rK
t+1 = max
ι̂t+1
Rt+1
Pt+1
+ qt+1 (1 + Γ (ι̂t+1) − δ) − ι̂t+1
qt
.
4 / 17

7. The Financial Sector
I A representative bank issues deposits to hold capital and gov.
debt. Both deposits (d̃t) and gov. debt (bB
t ) are liquid.
I Let dt := d̃t − bB
t denote the liquid asset supply. The bank’s
problem:
νt+1nt = max
kB
t ,dt
rK
t+1qtkB
t − rB
t+1dt
subject to their balance sheet and a financial constraint:
qtkB
t = dt + nt, qtkB
t ≤ Θ
rK
s+1, rB
s+1 s≥t
nt
I Net worth of banks follows:
nt+1 = (1 − f )(1 + νt+1)nt + m.
I Illiquid assets consist of capital and net worth of banks:
at = qtkF
t + nt, rA
t+1 =
1
at
(rK
t+1qtkF
t + νt+1nt).
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8. Government
I Government issues liquid liabilities bG
t to purchase illiquid
assets aG
t and goods
bG
t − aG
t = (1 + rB
t )bG
t−1 − (1 + rA
t )aG
t−1 + gt − Tt.
I Tax revenue collected by the government is
Tt =
Wt
Pt
ht −
Z
(1 − τt)
Wt
Pt
zi,thi,t
1−λ
di.
I Government sets the nominal interest rate iB
t to keep the real
interest rate equal to its desired level rB
t .
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9. Definition of Equilibrium
Given {rB
t , gt, bG
t , τt}, an equilibrium consists of prices {Pt, qt, Rt,
iB
t , rK
t , rA
t , W`,t} and allocations {yt, kt, xt, dt, nt, kF
t , kB
t , aG
t , ai,t,
bi,t, ci,t, hi,`,t} s.t. agents optimize given constraints, gov. budget
constraints hold, and markets clear:
Z
ci,t + Φi,t(ai,t, ai,t−1)di + xt + gt = yt,
Z
bi,tdi = dt + bG
t ,
Z
ai,tdi + aG
t = qtkt − dt,
and kt = kF
t + kB
t . Labor and capital rental market clearing are
embedded in the notation.
We focus on an equilibrium in which the financial constraint is
always binding.
7 / 17

10. Liquid Asset Supply

11. Nesting Models of Financial Intermediation
Problem P: Given {rK
t , rB
t }t≥0, solve for {dt} such that
νt+1nt = max
dt
rK
t+1nt + (rK
t+1 − rB
t+1)dt, where
dt+nt ≤ Θ {rK
s+1, rB
s+1}s≥t
nt, nt+1 = (1 − f )nt(1 + νt+1) + m.
Models of financial intermediation:
1. asset diversion (Gertler and Kiyotaki (2010), Gertler and
Karadi (2011))
2. costly state verification (Bernanke et al. (1999))
3. costly leverage (Uribe and Yue, Cúrdia and Woodford (2011))
4. regulatory constraints (Van den Heuvel (2008))
8 / 17

12. Nesting Models of Financial Intermediation
Lemma
Suppose that {dM
t } solves the bank’s problem in model
M ∈ {1, . . . , 4}. There exists a function Θ such that {dM
t } is the
solution to Problem P. Moreover, when evaluated at the
stationary equilibrium,
∂Θt
∂rK
s+1
= γs−t
Θ̄rK ,
∂Θt
∂rB
s+1
= −γs−t
Θ̄rB , ∀s ≥ t,
where Θt := Θ({rK
s+1, rB
s+1}s≥t) and Θ̄rK , Θ̄rB ≥ 0, γ ∈ [0, 1) are
determined by parameters of model M and steady-state variables.
9 / 17

13. Liquid Asset Supply Function
Let Dt({rK
s ; rB
s }∞
s=0) be the solution of the bank’s problem.
Cross-price elasticities: ∂Dt/∂rK
s
Dt
depend only on Θ̄rK , γ and steady
state variables.
Why it is useful: reduces infinite-dimensional elasticities to three
parameters and allows for systematic model comparison.
Comparing models:
- Asset diversion: Θ̄rK , Θ̄rB , γ 0 , fully determined by the
steady state.
- Costly state verification costly leverage: Θ̄rK , Θ̄rB 0,
γ = 0, not determined by the steady state
10 / 17

14. Liquid Asset Market and Aggregate Responses

15. A Supply and Demand Representation
Approach: characterize equilibrium as an intertemporal supply and
demand system.
Lemma
There exist functions Ct, Bt, Xt, RA
t such that, given
gs, Ts, rB
s , bG
s
∞
s=0
, the equilibrium
ys, rK
s
∞
s=0
solve:
Ct({ys, rA
s ; rB
s , Ts}∞
s=0) + Xt({ys, rK
s }∞
s=0) + gt = yt,
Bt({ys, rA
s ; rB
s , Ts}∞
s=0) = Dt({rK
s ; rB
s }∞
s=0) + bG
t ,
and rA
t = RA
t {rK
s ; rB
s }∞
s=0; Dt−1({rK
s ; rB
s }∞
s=0)
.
Moreover, functions Ct, Bt, Xt, RA
t do not depend on Θ.
Implication: aggregate responses depends on financial frictions Θ
only through the liquid asset supply function Dt.
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16. Liquid Asset Market and Returns on Capital
Define excess liquid asset supply as
t(y, rK
, rB
, T, bG
) := Dt(·) + bG
t − Bt(·).
Proposition
In equilibrium, returns on capital satisfy
drK
= −−1
rK × [dbG
+ T dT + rB drB
+ ydy]
| {z }
excess liquid asset supply shift
.
Cross-price elasticities of liquid asset supply determine rK :
I affect by how much drK need to adjust for the economy to
absorb an increase in excess liquid asset supply
I lower rK imply larger adjustment in drK in response to an
increase in liquid asset supply
12 / 17

17. Aggregate Output Response
Define aggregate demand as
Ψt(y, rK
, rB
, T, g) := Ct(·) + Xt(·) + gt,
Theorem
Given {drB, dT, dbG, dg}, the aggregate output response is:
dy = (I − Ψy − Ω y)−1
| {z }
(iii) modified Keynesian cross
×
dg + ΨT dT + ΨrB drB
| {z }
(i) goods market
+ Ω dbG
+ T dT + rB drB
| {z }
(ii) liquid asset market
,
Ω = −ΨrK (rK )−1: aggregate demand response through the
liquid asset market
13 / 17

18. A Quantitative Study of Asset Purchases

19. Government asset purchase and model comparison
Policy: inject liquid assets and purchase illiquid assets dbG
t = daG
t ,
keep drB
t = 0, adjust dTt to balance budget.
Model comparison: same S.S., change cross-price elasticities
1. Perfectly inelastic (HANK): DrK = DrB = 0
2. Baseline (GKK): Θ̄rK , Θ̄rB , γ from steady-state bank balance sheets.
3. Departure: change Θ̄rK from 5 → 10.
4. Perfectly elastic: Θ̄rK → ∞.
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Calibration

20. Comparative Statics: Aggregate Responses
Figure 1: y-axis: % of steady-state GDP. Light red: low cross-price elasticities.
Dark red: high cross-price elasticities. Blue: inelastic supply. Black: perfectly
elastic supply.
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21. Decomposition of Aggregate Output Response
Figure 2: Decomposition of output response; y-axis: % of steady-state GDP.
The decomposition uses formula from Theorem 1:
dy = (I − Ψy − Ω y)−1
| {z }
(iii) Keynesian cross
× ΨT dT
| {z }
(i) goods market
+ Ω (dbG
+ T dT)
| {z }
(ii) liquid asset market
.
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22. Conclusion
I Framework to analyze the role of the financial sector for
macro policies: focus on liquid asset market
I Key elasticities: own- and cross-price elasticities of liquid asset
supply (empirics coming soon)
I Low cross-price elasticity =⇒ disturbances in liquid asset
market generate strong aggregate responses
I Quantitatively, large differences in output responses depending
on cross-price elasticities.
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23. Appendix

24. Calibration Details
I Returns: liquid asset 1.6% per annum, illiquid assets 4.4%, capital 4.1%
I Preferences:
u (c, h) =
c1−σ
− 1
1 − σ
− ς
h
1+ 1
ϕ
1 + 1
ϕ
with σ = 2 and ϕ = 1
I Income risk: discrete-time version of the income process from Kaplan et
al. (2018), which matches moments from SSA data on male earnings.
I Assets: no borrowing a = b = 0 and
Ψi,t(ai,t, ai,t−1) =
χ1
χ2
ai,t − (1 + rA
t )ai,t−1
(1 + rA
t )ai,t−1 + χ0
χ2 h
(1 + rA
t )ai,t−1 + χ0
i
,
where χ0, χ1, χ2 0 are parameters that characterize the adjustment cost
Back

25. Calibration Details
I Production: The elasticity of output with respect to capital α = 0.35.
Depreciation rate δ is 6% per year. Capital adjustment cost is given by
Γ (ιt) = ῑ1ι1−κI
t + ῑ2
with the elasticity of investment to capital price equal to 2. The slope of
the wage Phillips curve is set to 0.04.
I Government: Net taxes to output: T = 0.15, λ, the parameter governing
the tax system’s progressivity is 0.1. Government debt is set to match the
liquid asset holdings of households.
Back

26. Calibration
Households:
- labor income process: matches moments from SSA data (Guvenen
et al. (2015) ).
- portfolio adjustment cost: generates a fraction of HtM households
≈ 22%
- total holdings of liquid and illiquid assets: target FoF data (balance
sheet items consolidated corresponding to the model)
The financial sector:
- net worth: match Call Report data, adjusted to FoF total.
- “effective” leverage: ratio of assets to net worth, excluding liquid
assets.
Back

27. Balance Sheets: Data v.s. Model
Households
assets liabilities
liquid asset 0.74 (0.50) equity 4.56 (3.80)
illiquid asset 3.82 (3.30)
Private Depository Institutions (Banks)
assets liabilities
liquid asset 0.18 (0.17) liquid liability 0.43 (0.43)
capital 0.37 (0.39) equity 0.12 (0.13)
Data: Financial Accounts of the United States, 2019 Q3. Model moments are
in parenthesis. Items are expressed as fractions of the annual % GDP. Appendix
C of our paper discusses the categorization and consolidation of balance-sheet
items.
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