Credit Constraints and Quantitative Easing

Credit Constraints and Quantitative Easing

Salem Abo-Zaid Ahmed KamaraUMBC TAMU-Corpus Christi

American Economic AssociationJanuary 7-9, 2022

Credit Constraints and Quantitative Easing AEA 2022 1 / 20

Motivation

Studying the effectiveness of Quantitative Easing (QE).

Since the start of the Covid-19 pandemic, the Federal Reserve hasimplemented QE policies.

The pandemic has disproportionately affected low(er)-incomehouseholds.

The Paper

A New Keynesian model with borrowing constraints on one set ofhouseholds (“impatient/constrained”).

Agents: unconstrained Households, constrained Households,intermediate-good firms, final-good firms, a government and amonetary authority.

QE is introduced using the portfolio rebalancing channel.

Characterizing the effect of QE on output analytically andnumerically.


The Portfolio Rebalancing Channel

Assets of different maturities are imperfect substitutes.

If assets are imperfect substitutes, changes in the demand for oneasset would change the relative price of that asset. As such, theCentral Bank can use QE asset purchases to alter the relative priceof long-term assets, thereby changing the yields of these assets.

This could lead to effects on real economic activity.


Main Result

Credit constraints make QE less effective.


Patient (Unconstrained) Households

The problem of the representative household:

maxcP,t ,nP,t ,bS

t ,bLt

∞t=0

E0

∞

∑

t=0

βtP

⎛

⎝

c1−σP,t

1 − σ− χ

P

n1+νP,t

1 + ν

⎞

⎠

(1)

subject to the budget constraint:

cP,t+bSt

RSt Pt+

bLtRLt Pt

⎡⎢⎢⎢⎢⎣

1 +ψ

2(φ

bStbLt− 1)

2⎤⎥⎥⎥⎥⎦

≤ wP,tnP,t+bSt−1Pt+bLt−1Pt−TP,t+ΠP,t (2)

The solution gives:

βEt (cP,t+1

cP,t)

−σ

= Etπt+1 [1

RSt

+ψφ

RLt

(φbStbLt− 1)] (3)

βEt (cP,t+1

cP,t)

−σ

=Etπt+1

RLt

⎡⎢⎢⎢⎢⎣

1 +ψ

2(φ

bStbLt− 1)

2

− ψφ(φbStbLt− 1)

bStbLt

⎤⎥⎥⎥⎥⎦

(4)

χnνP,tcσP,t = wP,t (5)


Credit Constrained (Hand-to-Mouth) Households

The problem of the representative household:

maxcI,t ,nI,t∞t=0

E0

∞

∑

t=0

βtI

⎛

⎝

c1−σI ,t

1 − σ− χ

I

n1+νI ,t

1 + ν

⎞

⎠

(6)

subject to the following budget constraint:cI ,t ≤ wI ,tnI ,t −TI ,t (7)

Solving the household problem gives the following labor supply condition.χnνI ,tc

σI ,t = wI ,t . (8)


Intermediate-Good Firms/Entrepreneurs

The intermediate-good sector is perfectly competitive.

Each firm j employs labor from both household sectors, and producesan intermediate good xj ,t .

Production function: xj ,t = N1−αP,j ,tN

αI ,j ,t .

Each period, a firm j has the probability 1− η of changing its price asin Calvo (1983, JME).

Labor demand conditions:

(1 − α)N−αP,tNαI ,tmct = wP,t (9)

αN1−αP,t N

α−1I ,t mct = wI ,t (10)


Final-Good Firms

Monopolistically competitive.

They transform intermediate goods into final goods.

Demand for variety j :

yj,t = (

pj,t

pt)

−ε

yt = xj,t (11)

The New Keynesian Phillips Curve (in log deviations):

πt = βEt πt+1 + κmct (12)


The Financial Intermediary

A representative perfectly competitive financial intermediary (FI) sellsshort term bonds (to the Ricardian agents) that pay RS

t in return.

It gets liquidity bCBt from the central bank in the form of long termassets with a return RCB

t .

The FI buys long term bonds from the government that pay RLG ,t in

return, and keeps reserves qt at the central bank that return RQt .

The FI also faces adjustment costs of changing the mix of short termand long term assets.


The Financial Intermediary, Contd.

maxcI,t ,nI,t∞t=0

E0

∞∑

t=0βtP

⎛

⎜

⎝

c−σP,tc−σP,0

⎞

⎟

⎠

⎧⎪⎪⎪⎨⎪⎪⎪⎩

RLG,tb

LG,t + R

Qt qt − (1 −ω)R

St b

St − R

CBt b

CBt

⎡⎢⎢⎢⎢⎣

1 +Θ

2

⎛

⎝

ΓbSt

bCBt− 1

⎞

⎠

2⎤⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(13)

subject to the following constraint:

bLG ,t + qt ≤ (1 − ω)bSt + bCBt (14)

The solution to this problem gives:

Λt = RSt + R

CBt ΘΓ (Γ

bStbCBt− 1) (15)

Λt = RCBt

⎧⎪⎪⎨⎪⎪⎩

1 −ΘΓ (ΓbS

bCBt− 1)

bStbCBt+Θ

2(Γ

bStbCBt− 1)

2⎫⎪⎪⎬⎪⎪⎭

(16)


The Financial Intermediary, Contd.

We let bCBt transition as follows:

ln(bCBt

bCB

) = ρMln(

bCBt−1

bCB

) + ξM,t

(17)


The Government

The government budget constraint:

gt + bLG ,t−1 + b

Lt−1 =

bLG ,t

RLG ,t

+bLtRLt

+Tt (18)


Market Clearing

The resource constraint in the economy reads as follows:

yt = ωcI ,t + (1 − ω) cP,t + gt (19)

Aggregate output in the economy is determined as

yt = N1−αP,t N

αI ,t (20)

The financial market also clears:

bLG ,t + qt = (1 − ω)bSt + bCBt (21)

The total demand for labor from both types of households and by allfirms are ∫

10 NI ,j ,tdj = NI ,t and ∫

10 NP,j ,tdj = NP,t . As such, the

clearing conditions in the labor market are:

NI ,t = ωnI ,t (22)

NP,t = (1 − ω)nP,t (23)


The Transmission of QE: Analytical Results

We characterize the transmission of QE asset purchases in theeconomy by determining the response of output to a rise in bCBt .

We assume: RCBt = 0.

Define:yt = AM bCBt (24)


The Transmission of QE: Analytical Results, Contd.

The solution:

AM =

(1 −ω) (1 − βρM) (1 − g) (1 −ωσΩ) (NM1∆1

)

σ (1 − βρM) (1 − ρM) (1 −ωσΩ)M2 − κ (1 −ω)ρM [σM2 + (1 − g) (v + σωΩ)M1](25)

If ω = 0:

AM =(1 − g) cPN2

σ∆1(26)

where:

N2 =

⎧⎪⎪⎨⎪⎪⎩

[RLX2 + ψφΥRSX1]

RSRL−ψΣbCB

bS

⎫⎪⎪⎬⎪⎪⎭

If ω = 1: AM = 0.


Numerical Results

0 0.05 0.1 0.15 0.2 0.25 0.30.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fraction of HtM Agents ()

Re

sp

on

se

of O

utp

ut to

QE

(A

M)

Figure: Response of output to QE asset purchases (the value of AM). Theshare of the HtM households in the economy, ω, ranges from zero to thebenchmark value of 1/3.


Impulse Response Functions

0 5 10 15 20 25 300

5

x 10-3 y

= 0

= 1/3

5 10 15 20 25 300

0.02

0.04

5 10 15 20 25 30-0.2

-0.1

0

RL / RS

5 10 15 20 25 30-0.02

-0.01

0

bL

5 10 15 20 25 30

-0.01

0

0.01

bS

5 10 15 20 25 300

0.05

0.1

cP

5 10 15 20 25 300

0.05

0.1

cI

5 10 15 20 25 300

0.005

0.01

c

5 10 15 20 25 300

0.005

0.01

np

5 10 15 20 25 300

2

4x 10

-3 nI

5 10 15 20 25 300

0.1

0.2

wp

5 10 15 20 25 300

0.1

0.2

wI

Figure: Impulse response of selected variables to a 1% increase in QE assetpurchases. The first model (ω = 0) abstracts from HtM households. Thesecond model (ω = 1/3) introduces HtM households.


Conclusions

Credit frictions actually reduce the effectiveness of quantitativeeasing (through the portfolio rebalancing channel).


Thank you!


Credit Constraints and Quantitative Easing

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