Asset supply

1. Asset Supply and Liquidity Transformation in
HANK
Yu-Ting Chiang 1 Piotr Zoch 2
1FRB of St Louis
2University of Warsaw
September 16, 2022
Views in this paper do not necessarily represent those of the Federal Reserve
Bank of St. Louis or the Federal Reserve System.

2. Motivation
I How does the financial sector interact with the real sector in
response to macro policies?
I A key market is the “liquid asset” market.
- It is the market where (i) monetary policy takes effect, and (ii)
fiscal policy is financed.
- Crucial for (i) the funding of the financial sector and (ii)
households’ consumption-saving decisions
I Framework: a frictional financial sector embedded in a
standard heterogeneous agent New Keynesian model (HANK)
I This paper: the financial sector’s ability to perform liquidity
transformation is key for aggregate responses to macro policies
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3. Result
Main result: The role of financial sector in a large class of models
summarized by two key elasticities of the liquid asset supply:
Own-price: how much return on liquid assets (monetary
policy) affects the real sector
Cross-price: how much shocks to the real sector affect the
financial sector
Quantitative implications (deficit financed gov. spending):
Small cross-price elasticity → large adjustment in return on
capital: dampens GE amplification and increases direct effect
of debt.
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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, Φ is portfolio adjustment cost,
and
ai,t ≥ ā, bi,t ≥ b̄,
Z
Φi,tdi = 0.
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6. Financial sector
I Illiquid assets consists of capital and net worth of banks:
at = qtkF
t + nt, at =
Z
ai,tdi.
Illiquid assets return: rA
t+1 = 1
at
(rK
t+1qtkF
t + νt+1nt).
I Banks hold capital by issuing liquid assets
νt+1nt = max
kB
t ,dt
rK
t+1qtkB
t − rB
t+1dt
subject to balance sheet and a financial constraint:
qtkB
t = dt + nt, qtkB
t ≤ Θt
rK
s+1, rB
s+1
s≥t
nt
I Net worth of banks follows:
nt+1 = (1 − f )nt(1 + νt+1) + mt+1.
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7. 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
(Rotemberg).
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 + qt+1 (1 + Γ (ι̂t+1) − δ) − ι̂t+1
qt
.
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8. Government
I Government liabilities bG
t are liquid and evolve according to
bG
t = (1 + rB
t )bG
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 The government sets the nominal interest rate iB
t to keep the
real interest rate equal to its desired level
rB
t = rt.
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9. Definition of Equilibrium
Given {rB
t , gt, τt}, an equilibrium are sequences {Pt, rK
t , Rt, W`,t}
and {ai,t, bi,t, ci,t, hi,`,t, nt, kF
t , kB
t , kt, xt, yt, } such that
households, banks, firms, and labor unions optimize subject to
constraints, and markets clear:
Z
ci,tdi + xt + gt = yt,
Z
bi,tdi = dt + bG
t ,
Z
ai,tdi = qtkt − dt,
where kF
t + kB
t = kt and labor and capital rental market clearing is
embedded in the notation.
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10. Aggregate Responses with Financial Friction

11. A Supply and Demand Representation
Lemma
The equilibrium can be characterized by the following supply and
demand system:
yt = Ct({ys, rA
s+1; Ts, rB
s+1}∞
s=0) + Xt({ys, rK
s+1}∞
s=0) + gt,
Bt({ys, rA
s+1; Ts, rB
s+1}∞
s=0) = Dt(y0, {rK
s+1; rB
s+1}∞
s=0) + bG
t ,
where {yt, rK
t+1} are endogenous variables to be solved for, and
rA
t+1 = rA
({ys, rK
s+1, rB
s+1}∞
s=0; Dt).
What’s new: equilibrium response to policies depends on financial
friction Θt(·) through the liquid asset supply Dt.
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12. Liquid Asset Supply
Lemma
Suppose that Θt(·) satisfies
∂Θt
∂rK
s+1
= γs−t
Θ̄rK ,
∂Θt
∂rB
s+1
= −γs−t
Θ̄rB , ∀s ≥ t,
then
DrK = M0 + Θ̄rK M(γ),
DrB = −M0
0 − Θ̄rB M(γ).
Key parameters: Θ̄rK , Θ̄rB govern the cross-price and own-price
elasticities of liquid asset supply
M0, M0
0, M(γ): matrices pinned down by the steady state variables and γ
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13. Equilibrium Characterization
I Solve for first order responses dy, drK to government policies
in two steps:
1. Solve for drK
from liquid asset market, taking dy as given.
2. Using the solution to solve for dy from the goods market
I Some useful notation:
’s: elasticities of excess liquid asset demand, e.g.,
rK := BrA RA
rK − DrK , rB := BrB + BrA RA
rB − DrB , . . .
Ψ’s: elasticities of aggregate demand, e.g.,
ΨrK := CrA RA
rK + XrK , ΨrB := CrB + CrA RA
rB , . . .
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14. Liquidity Transformation and Market Segmentation
Proposition
In equilibrium, returns on capital satisfy
drK
= (−rK )−1
[rB drB
+ ydy + T dT − dbG
].
In particular, if Θ̄rK , Θ̄rB → ∞, then
drK
→
Θ̄rB
Θ̄rK
drB
.
Intuition: when Θ̄rB , Θ̄rK → ∞, the financial sector links the two
markets and liquid asset supply is perfectly elastic given rK
t .
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15. Aggregate Output Response
Theorem
Given {drB, dT, dbG, dg}, the aggregate output response is:
dy =(I − Ψy − Ω y)−1
×
dg − ΩdbG
+ [ΨT + ΩT ]dT + [ΨrB + ΩrB ]drB
,
where
Ω = ΨrK (−rK )−1
.
Key components:
- Ω: output responses due to changes in the liquid asset market
- Total response splits into partial v.s. general equilibrium effect
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16. Quantitative results

17. Policy experiment
I Deficit financed government spending shock:
dgt = ρgdgt−1, dbG
t = dgt + ρbG dbG
t−1
dg → real sector, dbG
→ financial sector
I Calibration:
- HH and production: labor income process, fraction of HtM
households, total household holdings of liquid and illiquid assets
- The financial sector: leverage and net worth
I Comparative static:
- Baseline: Gertler-Karadi-Kiyotaki, where key elasticities
identified by steady state bank balance sheet
- Departure: change ΘrK (cross-price elasticity) from baseline to
twice its size
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18. Comparative statics: Government spending shock
Figure 1: Impulse response functions to government spending shocks;
x-axis: quarters, y-axis: % of GDP.
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19. Decomposition of aggregate output response
Figure 2: Decomposition of output response; x-axis: quarters, y-axis: %
of GDP. The decomposition uses formula from Theorem 1:
dy =(I − Ψy − Ω y)−1
×
dg − ΩdbG
+ [ΨT + ΩT ]dT
,
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20. Conclusion
I Framework to analyze the role of financial sector for macro
policies: focus on liquid asset market
I Key elasticties of liquid asset supply: own- and cross-price
I Low cross-price elasticity → large changes in rK
I Government spending shock:
I Low cross-price elasticity dampens GE amplification, increases
the size of the direct effect
I Output response decreasing in cross-price elasticity
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