the intertemporal keynesian cross - bank of canada...• recent literature: mpcs are crucial for pe...

The Intertemporal Keynesian Cross

Adrien Auclert, Matt Rognlie and Ludwig Straub

Bank of Canada Annual ConferenceNovember 2018

1

This paper

Q: How does the macroeconomy propagate shocks?

• what micro moments are important?

• Recent literature: MPCs are crucial for PE e�ects

• Idea: for PE impact response to shocks, want models to beconsistent with evidence on C response to Y

• Applications: �scal policy [Kaplan-Violante], monetarypolicy [Auclert], house prices [Berger et al], . . .

• In GE, C now and in future a�ects everyone’s Y• Here: “intertemporal MPCs” (iMPCs) are crucial for the GEimpulse response

2

Application: When is the �scal multiplier large?

• Lots of theory + empirical work. Two workhorse models:

1. Representative-agent (RA) models• response of monetary policy is key• large when at ZLB

[Eggertsson 2004; Christiano-Eichenbaum-Rebelo 2011]

2. Two-agent (TA) models• aggregate MPC is key• large when de�cit �nanced, e�ects not persistent

[Galí-López-Salido-Vallés 2007; Coenen et al 2012; Farhi-Werning 2017]

New: Heterogeneous-agent (HA)models→ iMPCs are key, can be used for calibration→ large and persistent Y e�ect when de�cit �nanced

3

Our contribution: Interaction of iMPCs and de�cit-�nancing

1. Benchmark model, allows for RA, TA, HA• without capital & constant-real-rate monetary policy• multiplier = function of iMPCs and de�cits only

= 1 if zero de�cits or �at iMPCs (RA) [Woodford 2011]

> 1 if de�cit-�nanced and realistic iMPCs (HA, TA?)

2. Quantitative model with capital & Taylor rule

• large & persistent Y e�ects, despite these extra elements

• iMPCs still crucial for Y response

3. Role of iMPCs for the GE e�ects of other shocks• Today (not in paper): monetary policy

4

Outline

1 The intertemporal Keynesian Cross

2 iMPCs in models vs. data

3 Fiscal policy in the benchmark model

4 Fiscal policy in the quantitative model

5 Other shocks

5

The intertemporal Keynesian Cross

Model overview Unions

• GE, discrete time t = 0...∞, no aggregate risk (MIT shocks)

• Mass 1 of households:• idiosyncratic shocks to skills eit, various market structures• real pre-tax income yit ≡ Wt/Pteitnit

nit = Nt

• after tax income zit ≡ yit − Tt(yit) ≡ τty1−λit [Bénabou, HSV]

• Government sets:• tax revenues Tt =

∫(yit − zit)di

rationing

• government spending Gt• monetary policy: �xed real rate = r

• Supply side:

relax later

• linear production function Yt = Nt• �exible prices ⇒ Pt = Wt• sticky Wt ⇒ πwt = κw

∫Nt(v′ (nit)− ∂zit

∂nitu′ (cit))di+ βπwt+1

6



• Mass 1 of households:• idiosyncratic shocks to skills eit, various market structures• real pre-tax income yit ≡ Wt/Pteitnit

nit = Nt

• after tax income zit ≡ yit − Tt(yit) ≡ τty1−λit [Bénabou, HSV]


∫(yit − zit)di

rationing


• Supply side:

relax later



∂nitu′ (cit))di+ βπwt+1 6



• Mass 1 of households:• idiosyncratic shocks to skills eit, various market structures• real pre-tax income yit ≡ Wt/Pteitnit nit = Nt• after tax income zit ≡ yit − Tt(yit) ≡ τty1−λit [Bénabou, HSV]


∫(yit − zit)di rationing


• Supply side:

relax later






• Mass 1 of households:• idiosyncratic shocks to skills eit, various market structures• real pre-tax income yit ≡ Wt/Pteitnit nit = Nt• after tax income zit ≡ yit − Tt(yit) ≡ τty1−λit [Bénabou, HSV]


∫(yit − zit)di rationing


• Supply side: relax later• linear production function Yt = Nt• �exible prices ⇒ Pt = Wt• sticky Wt ⇒ πwt = κw



Nesting four types of households

Household i solves

max E[∑

βt {u (cit)− v (nit)}]

cit + ait = (1+ r)ait−1 + zit

• RA: no risk in e (or complete markets)

• TA: share µ of agents with cit = zit• HA-std: one asset model, with constraint ait ≥ 0• HA-illiq: simpli�ed two asset model

• illiquid account ailliq = �xed no. of bonds ( + capital)• liquid account ait = all remaining bonds + railliq

7

The aggregate consumption function Eq de�nition

• Given {ai0} and r, aggregate consumption function is

Ct =∫citdi = Ct ({Zs}∞s=0)

[Farhi Werning 2017, Kaplan Moll Violante 2017, ...]

with Zt ≡aggregate after-tax labor income

Zt ≡∫zitdi = Yt − Tt

• C summarizes the heterogeneity and market structure

• Equilibrium de�ned as usual

8

Intertemporal MPCs

• An output path {Yt}∞t=0 is part of equilibrium⇔

Yt = Gt + Ct ({Ys − Ts}) ∀t ≥ 0

• Impulse response to shock {dGt,dTt}

dYt = dGt +∞∑s=0

∂Ct∂Zs︸︷︷︸≡Mt,s

· (dYs − dTs) (1)

→ Response {dYt} entirely characterized by {Mt,s}!

• partial equilibrium derivatives, “intertemporal MPCs”

• how much of income change at date s is spent at date t

•∑∞

t=0(1+ r)s−tMt,s = 1

9

The intertemporal Keynesian cross

• Stack objects: M = {Mt,s} ={∂Ct∂Zs

}, dY = {dYt}, etc

• Rewrite equation (1) as

dY = dG−MdT+MdY

• This is an intertemporal Keynesian cross• entire complexity of model is in M

• with M from data, could get dY without model simulations!

• When unique, solution is

dY =M · (dG−MdT)

whereM is (essentially) (I−M)−1

10

iMPCs in models vs. data

Measuring aggregate iMPCs using individual iMPCs

• Object of interest: (aggregate) iMPCs

Mt,s =∂Ct∂Zs

where Ct =∫citdi and Zs =

∫zisdi

• Direct evidence on Mt,s is hard to come by for general s

• More work on column s = 0 (unanticipated income shock)

• Can write

Mt,0 =

∫ zi0Z0︸︷︷︸

income weight

· ∂cit∂zi0︸︷︷︸

individual iMPC

di

→ aggregate iMPCs are weighted individual iMPCs

11

Obtain date-0 iMPCs from cross-sectional microdata

• Two sources of evidence on ∂cit∂zi0

:

1. Fagereng Holm Natvik (2018) measure in Norwegian data

cit = αi + τt +5∑

k=0γklotteryi,t−k + θxit + εit

• Weighting by income in year of lottery receipt⇒ Mt,0

2. Italian survey data (SHIW 2016) on ∂ci0∂zi0

• Lower bound for Mt,0 using distribution of MPCs• Example: income-weighted average of(1−MPCi)MPCi ⇒ lower bound for M1,0

12

iMPCs in the data

0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

Year (t)

Data from Fagereng et al (2018)Lower bound from SHIW 2016

• Annual M0,0 consistent with evidence from other sources13

Compare iMPCs across models Calibration

• RA

• TA: share of hand-to-mouth calibrated to match M0,0

• HA-std: one-asset HA, standard calibration

• HA-illiq: two-asset HA calibrated to match M0,0

• . . . and for fun:• BU: bonds-in-utility model, calibrated to match M0,0[Michaillat Saez 2018; Hagedorn 2018; Kaplan Violante 2018]

14

iMPCs across models

0 1 2 3 4 50

0.2

0.4

0.6

Year (t)

iMPC

Mt,0

(a) Data and model �t

DataHA-illiq

0 1 2 3 4 50

0.2

0.4

0.6

Year (t)

(b) Alternative models

DataRATA

HA-stdBU

15

iMPCs across models including TABU

0 1 2 3 4 50

0.2

0.4

0.6

Year (t)

iMPC

Mt,0

(a) Data and model �t

DataHA-illiqTABU

0 1 2 3 4 50

0.2

0.4

0.6

Year (t)

(b) Alternative models

DataRATA

HA-stdBU

16

What about non-date-0 iMPCs?

• Existing evidence useful for response to date-0 incomeshocks, {Mt,0}

• What about response to future shocks?

→ rely on calibrated HA-illiqmodel to �ll in the blanks!

17

Response of HA-illiq to other income shocks

0 5 10 15 20 25 300

0.2

0.4

0.6

Year t

iMPC

Mt,s

s = 0s = 5s = 10s = 15s = 20

18

Not entirely arbitrary→ TABU is very similar! Durables

0 5 10 15 20 25 300

0.2

0.4

0.6

Year t

iMPC

Mt,s

s = 0s = 5s = 10s = 15s = 20TABU

19

Fiscal policy in the benchmark model

Fiscal policy in the benchmark model

• Recall intertemporal Keynesian cross:

dY = dG−M · dT+M · dY

• dY entirely determined by iMPCs M and �scal policy(dG,dT)

• Next: Characterize role of iMPCs for

1. balanced budget policies, dG = dT

2. de�cit-�nanced policies

20

The balanced-budget unit multiplier

• With balanced budget, dG = dT⇒ multiplier of 1:

dY = dG

• Similar reasoning already in Haavelmo (1945)

• Generalizes Woodford’s RA results

• heterogeneity irrelevant for balanced budget �scal policy

• similar to Werning (2015)’s result for monetary policy

• Proof: dY = dG is unique solution to

dY = (I−M) · dG+M · dY

21

De�cit-�nanced �scal policy

• With de�cit �nancing dG 6= dT we have

dY = dG+M ·M · (dG− dT)︸︷︷︸dC

Consumption dC depends on primary de�cits dG− dT

• Example: TA model with de�cit �nancing

dY = dG+µ

1− µ (dG− dT)

• consumption dC depends only on current de�cits

• initial multiplier can be large ∈[1, 1

1−µ

]. . .

• but cumulative multiplier is = 1 !∑(1+ r)−tdYt∑(1+ r)−tdGt

= 1

22

De�cit-�nanced �scal policy

• With de�cit �nancing dG 6= dT we have

dY = dG+M ·M · (dG− dT)︸︷︷︸dC

Consumption dC depends on primary de�cits dG− dT

• Example: TA model with de�cit �nancing

dY = dG+µ

1− µ (dG− dT)

• consumption dC depends only on current de�cits

• initial multiplier can be large ∈[1, 1

1−µ

]. . .

• but cumulative multiplier is = 1 !∑(1+ r)−tdYt∑(1+ r)−tdGt

= 122

Simulate model responses for more general shocks Impulse

• Parametrize: dGt = ρGdGt−1 and dBt = ρB (dBt−1 + dGt)• vary degree of de�cit-�nancing ρB

0 0.2 0.4 0.6 0.8

1

1.5

2

2.5

3

Persistence of debt ρB

dY0/dG

0

Initial multiplier

HA-illiqHA-stdRATATABU

0 0.2 0.4 0.6 0.8

1

1.5

2

2.5

3


∑ (1+r)

−t dY t

∑ (1+r)

−t dG t

Cumulative multiplier

Calibration: ρG = 0.7

23

Simulate model responses for more general shocks Impulse

• Parametrize: dGt = ρGdGt−1 and dBt = ρB (dBt−1 + dGt)• vary degree of de�cit-�nancing ρB

0 0.2 0.4 0.6 0.8

1

1.5

2

2.5

3


dY0/dG

0

Initial multiplier


0 0.2 0.4 0.6 0.8

1

1.5

2

2.5

3


∑ (1+r)

−t dY t

∑ (1+r)

−t dG t


Calibration: ρG = 0.7 23

Fiscal policy in the quantitativemodel

Adding new elements to the HA-illiq model . . . Calibration

• Government:

• gov spending shock, dGt = ρGdGt−1

• �scal rule, dBt = ρB (dBt−1 + dGt)

• Taylor rule, it = rss + φπt, φ > 1

• Supply side:

• Cobb-Douglas production, Yt = Kαt N1−αt

• Kt subject to quadratic capital adjustment costs

• sticky prices à la Calvo, πt = κpmct + 11+rtπt+1

• Two reasons for lower multipliers:

• distortionary taxation & crowding-out of investment

24

iMPCs still a crucial determinant of response! Impulse

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2


dY0/dG

0

Initial multiplier

HA-illiqHA-stdRATA

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

Persistence of debt ρB∑ (1

+r)

−t dY t

∑ (1+r)

−t dG t


Calibration: ρG = 0.7, κw = κp = 0.1, φ = 1.5

25

Other shocks

What can we learn for other shocks? – back to benchmark

• Aggregate consumption may depend on other shocks θ,

Ct = Ct ({Zs}, θ)

[e.g. deleveraging, inequality, preferences, mon. policy]

• Can generalize intertemporal Keynesian cross as

dY = dG−MdT+ Cθdθ︸︷︷︸≡∂C

+MdY

→ iMPCs also determine propagation of other shocks

dY = dG+MM(dG− dT) +M∂C

26

Monetary policy experiment

• Economy starts in steady state

• Monetary policy sets {rt} according to

rt =

r t 6= Tr − dr t = T

with shock at horizon t = T

• Next: Compare responses• RA vs HA-illiq (matching iMPCs)• investment vs no investment (δ = 0,∞ adj. costs)

27

No investment: RA ∼ HA (Werning 2015)

0 5 10 15 20

0

0.5

1

1.5

Quarter

Percentofs.s.output

RA, no investment

Consumption

0 5 10 15 20

0

0.5

1

1.5

Quarter

HA, no investment

Consumption

28

With investment: HA is ampli�ed,� RA

0 5 10 15 20

0

0.5

1

1.5

Quarter

Percentofs.s.output

RA, with investment

ConsumptionInvestment

0 5 10 15 20

0

0.5

1

1.5

Quarter

HA, with investment

ConsumptionInvestment

→ “Forward guidance is more powerful than you think!”

29

Conclusion

M matters for Macro !

→ crucial for GE propagation→ new insights for �scal policy

New avenues:

more evidence on Mimplications for other shocks

30

Extra slides

Unions back

• Mass 1 of unions. Each union k• employs every individual, ni ≡

∫nikdk

• produces task Nk =∫einikdi from member hours

• pays common wage wk per e�cient unit of work e• requires that all individuals work nik = Nk

• Final good �rms aggregate N ≡(∫ 1

0 Nε−1ε

k dk) εε−1

• Union k sets wkt each period to maximize

maxwkt

∑τ≥0

βτ{∫{u (cit+τ )− v (nit+τ )}di−

ψ

2

(wkt+τwkt+τ−1

)2}• ⇒ nonlinear wage Phillips curve

(1+ πwt )πwt =

ε

ψ

∫Nt(v′ (nit)−

ε− 1ε

∂zit∂nit

u′ (cit))di

+ βπwt+1(1+ πwt+1

)31

Equilibrium de�nition back

• Given {Gt, Tt}, a general equilibrium is a set of prices,household decision rules and quantities s.t. at all t:1. �rms optimize2. households optimize3. �scal and monetary policy rules are satis�ed4. the goods market clears

32

iMPCs for model with durables back

0 5 10 15 20 25 300

0.2

0.4

0.6

Year t

iMPC

Mt,s

s = 0s = 5s = 10s = 15s = 20Durab.

Calibration: homothetic durables model with dit = 0.1 · cit and δD = 20%

33

Calibration for benchmark model back

• Preferences: u (c) = c1−1ν

1− 1ν, v (n) = bn

1+ 1φ

1+ 1φ

• Income process: log et = ρe log et−1 + σεt

Parameter Parameter HA-illiq HA-std

ν EIS 0.5φ Frisch 1ρe Log e persistence 0.91σ Log e st dev 0.92λ Tax progressivity 0.181G/Y Spending-to-GDP 0.2A/Z Wealth-to-aftertax income 8.2B/Z Liquid assets to aftertax income 0.15 8.2β Discount factor 0.80 0.92r Real interest rate 0.05κw Wage �exibility 0.1 34

Calibration for quantitative model back

• As in benchmark model, plus:

Parameter Parameter

α Capital share 0.33B/Y Debt-to-GDP 0.7K/Y Capital-to-GDP 2.5µ SS markup 1.015δ Depreciation rate 0.08εI Invest elasticity to q 4κp Price �exibility 0.1κw Wage �exibility 0.1φ Taylor rule coe�cient 1.5

35

Impulse responses in benchmark model back

0 2 4 6 8 100

1

2

Percentofs.s.output

Output


0 2 4 6 8 10

0

0.5

1

1.5

Consumption

0 2 4 6 8 100

0.20.40.60.81

Government spending

0 2 4 6 8 10

0

50

100

Years

In�ation (bps)

0 2 4 6 8 10−100

−50

0

50

100

Years

Real interest rate (bps)

0 2 4 6 8 100.2

0.4

0.6

0.8

1

Years

Government bonds (% of Yss)

Calibration: ρG = ρB = 0.7

36

Impulse responses in quantitative model back

0 2 4 6 8 100

0.5

1

1.5

Percentofs.s.output

Output

HA-illiqHA-stdRATA

0 2 4 6 8 10

0

0.5

1Consumption

0 2 4 6 8 10

−0.2

−0.1

0

Investment

0 2 4 6 8 100

0.20.40.60.81

Government spending

0 2 4 6 8 100

0.51

1.52

2.5

Years

Hours (% of s.s.)

0 2 4 6 8 100

20

40

60

Years

In�ation (bps)

0 2 4 6 8 100

10

20

30

40

Years

Real interest rate (bps)

0 2 4 6 8 100.2

0.4

0.6

0.8

1

Years

Government bonds (% of Yss)

Calibration: ρG = ρB = 0.7, κw = κp = 0.1, φ = 1.5

37

True unless very responsive Taylor rule back

0 2 4 6 8 10

0

0.5

1

1.5

2

Years

Percentofs.s.output

Output

φ = 1φ = 1.5φ = 2

0 2 4 6 8 10

−1

−0.5

0

0.5

Years

Investment

Calibration: ρG = 0.7, κw = κp = 0.1, ρB = 0.5, and vary φ in Taylor rule

38

True even with more �exible prices (unless very �exible) back

0 2 4 6 8 10

0

0.5

1

1.5

2

2.5

Years

Percentofs.s.output

Output

κp = 0κp = 0.1κp = 0.25

0 2 4 6 8 10−1.5

−1

−0.5

0

0.5

Years

Investment

Calibration: ρG = 0.7, κw = 0.1, ρB = 0.5, φ = 1.5, and vary κp in price Phillips curve

39

True even with more �exible wages (unless very �exible) back

0 2 4 6 8 10

0

1

2

3

Years

Percentofs.s.output

Output

κw = 0κw = 0.1κw = 0.25

0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

Years

Investment

Calibration: ρG = 0.7, κp = 0.1, ρB = 0.5, φ = 1.5, and vary κw in wage Phillips curve

40

the intertemporal keynesian cross - bank of canada...• recent literature: mpcs are crucial for pe...

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