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The Macroeconomic Impact of Microeconomic Shocks: Beyond Hulten’s Theorem David Baqaee Emmanuel Farhi LSE Harvard June 19, 2017

Transcript of The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For...

Page 1: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

The Macroeconomic Impact of MicroeconomicShocks: Beyond Hulten’s Theorem

David Baqaee Emmanuel Farhi

LSE Harvard

June 19, 2017

Page 2: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Macroeconomic Impact of Shocks

For economy with efficient equilibrium, Hulten (1978):

d logC/d logAi = salesi/GDP = λi .

First-order approximation or log-linearization.

Foundation for Domar aggregation:

sales approximate sufficient statistics.

details of production structure are irrelevant.

“Bugbear” for production networks literature.(shocks to Walmart and electricity equally important)

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What We Do

Extend Hulten to second order to capture nonlinearities.

General formula: reduced-form macro-elasticities of substitution.

Mapping from micro to macro using a general structural model:structural micro elasticites of substitution.returns to scale.factor market reallocation.network linkages.

Nonlinearities lead to asymmetric responses of output to shocks.amplification of negative shocks, attenuation of positive shocks.lower mean, negative skewness, excess kurtosis.

Nonlinearities quantitatively important:×10 welfare costs of business cycles from 0.05% to 0.6% of GDP.×4 impact of 70’s oil price shocks from −0.7% to −2.4% of GDP.

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Related Literature

Long and Plosser (1983), Horvath (2000), Gomme and Rupert(2007).

Jovanovic (1987), Durlauf (1993), Scheinkman and Woodford(1994), Horvath (1998), Dupor (1999).

Gabaix (2011), Carvalho and Gabaix (2013), Acemoglu et al.(2012), Carvalho (2010), Acemoglu et al. (2017), Foerster et al.(2011), Atalay (2016), Bigio and La’O (2016), Baqaee (2016),Di Giovanni et al. (2014), Pasten et al. (2017).

Kremer (1993), Jones (2011), Jones (2013).

Houthakker (1955), Jones (2005), Oberfield and Raval (2014),Boehm and Oberfield (2017), Beraja et al. (2016).

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Agenda

Framework

Structural ModelMacro-SubstitutionInput-output MutlipliersNetworks

Quantitative Examples

Conclusion

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Agenda

Framework

Structural ModelMacro-SubstitutionInput-output MutlipliersNetworks

Quantitative Examples

Conclusion

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General Framework

Perfectly competitive economy, representative consumer.

Preferences represented by CRS consumption-bundle metric

C = C (c1, . . . ,cN) ,

where ci is consumption of good i .

Consumer budget constraint

∑i

pici =M

∑i=1

wi li +N

∑i=1

πi ,

where pi , wi , and πi are prices, wages, and profits.

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General Framework

Profits earned by the producer of good i :

πi = piyi −M

∑k=1

wk lik −N

∑j=1

pjxij .

Each good i is produced using production function:

yi = AiFi(li1, . . . , liM ,xi1, . . . ,xiN),

Ai Hicks-neutral technology (Harrod-neutral as special case).

xij intermediate inputs of good j used in the production of good i .

lik labor of type k used by i .

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Hulten’s Theorem

Define C(A1, . . . ,AN) to be competitive equilibrium aggregateconsumption function interpreted as output.

Theorem 1.1 (Hulten)

Let λi denote industry i ’s sales as a share of output, then

d logCd logAi

= λi .

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Elasticity of SubstitutionDefinition 1.2

For general CRS function f (A1, . . . ,AN) define Morishimaelasticity of substitution:

1ρij

=−d log(MRSij)

d log(Ai/Aj)=− d log(fi/fj)

d log(Ai/Aj)

where fi = ∂ f/∂Ai .

For output function C(A1, . . . ,AN), define macro-elasticity ofsubstitution:

1ρij≡−d log(MRSij)

d log(Ai)=−d log(Ci/Cj)

d log(Ai),

where Ci = ∂C/∂Ai .

Note that d log(λi/λj )d logAi

=d log[(Ci Ai )/(Cj Aj )]

d logAi=

d log(Ci/Cj )d logAi

+ 1 = 1− 1ρij.

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Input-Output Multiplier

Definition 1.3Define input-output mutliplier

N

∑i=1

d logCd logAi

=N

∑i=1

λi = ξ .

“Macro returns to scale”: ξ > 1 implies reproducibility.

ξ constant if and only if C homogenous of degree ξ .

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Extending Hulten: Idiosyncratic Shocks

Theorem

d2 logCd(logAi)2 =

λi

ξ∑j 6=i

λj

(1− 1

ρij

)+ λi

∂ logξ

∂ logAi.

General formula for second-order terms (nonlinearities) in termsof reduced-form macro-elasticities of substitution.

Sales distribution not sufficient statistic.

ρij = 1, ξ constant: knife-edge case where effect disappears.

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Extending Hulten: Common Shocks

Proposition 1.4

d2 logCd logAi d logAj

= λiξ

∑k 6=j λk

(1− 1

ρjk

)+ λi

∂ logξ

∂ logAj−λi

(1− 1

ρji

)(i 6= j)

Shocks not additive.

ρij = 1, ξ constant: knife-edge case where effect disappears.

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Macro Moments

Proposition 1.5Suppose that logAi are subject to idiosyncratic shocks with variances2

i . Then we have the following formula for the mean of output:

E(log(C/C))≈ 1ξ

∑i

s2i

2ξλi ∑

j 6=i

λj

(1− 1

ρij

)+∑

i

s2i

2λi

d logξ

d logAi.

See paper for:

more general mean formula for correlated shocks.

beyond mean, formulas for skewness and excess kurtosis.

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Welfare Costs of Business Cycles

Proposition 1.6

Let u : R→ R be a CRRA with parameter γ . Suppose TFP A hasidiosyncratic shocks with variance s2

k . Then the welfare costs ofbusiness cycles are given by:

C[E(u(C))−u(C)]

u′(C)≈ −1

N

∑k

λ2k s2

k︸ ︷︷ ︸Consumption nonlinearities

+C2

N

∑k

∂ 2C∂A2

ks2

k︸ ︷︷ ︸Production nonlinearities

,

where recall C = C(A).

Nonlinearities in consumption: small cost in Lucas (1987).

Nonlinearities in production: can be order of magnitude larger.

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Mapping Micro Parameters to Macro Elasticities

Proposition 1.7

ρij and d logξ/d logA can be solved for explicitly as a function ofobservable expenditure shares and micro elasticities of substitution.

See paper for explicit characterization of reduced-formmacro-elasticities in terms of micro primitives.

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Agenda

Framework

Structural ModelMacro-SubstitutionInput-output MutlipliersNetworks

Quantitative Examples

Conclusion

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Household

Preferences given by consumption-bundle metric:

C

C=

(∑k

bk

(ck

ck

) σ−1σ

) σ

σ−1

.

Consumer budget constraint:

∑k

pk ck = ∑k

wLk +∑k

wk lk +∑k

πk .

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Firms

Industry k ’s production function given by

yk

yk= Ak

ak

((Lk

Lk

)βk(

lklk

)1−βk) θk−1

θk

+ (1−ak )

(Xk

X k

) θk−1θk

θk−1

θk

Xk composite intermediate input given by

Xk

X k=

(∑

lωklx

εk−1εk

lk

) εk−1εk

,

where xkl intermediate inputs from industry l used by industry k .

Lk mobile generic labor and lk fixed specific labor.

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Agenda

Framework

Structural ModelMacro-SubstitutionInput-output MutlipliersNetworks

Quantitative Examples

Conclusion

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Substitution and Reallocation

Proposition 2.1

Suppose each good is produced using only labor. Assume uniformlabor reallocation/returns to scale β ∈ [0,1] for every k. Then

ρij =σ(1−β ) + β

σ(1−β ) + β + (1−σ), λi = bi , ξ = 1,

d logξ

d logAi= 0.

To build intuition, consider polar cases with β = 1 and β = 0.

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Lesson #1: Micro-Elasticity of Substitution Matters

β = 0 =⇒ ρ = σ .

−0.15 −0.1 −0.05 0 0.05 0.1 0.15−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

log(A)

log(

C/C

)

LeontiefHulten/Cobb-DouglasPerfect Substitutes

d2 logCd logA2

i= bi(1−bi)

(1− 1

σ

).

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Lesson #2: Reallocation Matters

β = 1 =⇒ ρ = 12−σ

.

−0.15 −0.1 −0.05 0 0.05 0.1 0.15−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

log(A)

log(

C/C

)

LeontiefHulten/Cobb-DouglasPerfect Substitutes

d2 logCd logA2

i= bi(1−bi)(σ −1) .

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Network Irrelevance Result

Proposition 2.2

Let σ = θi = εi , and consider Harrod-neutral (labor-augmenting)shocks. Then for any arbitrary network

ρij = ρ, ξ = 1,d logξ

d logAi= 0,

where

ρ =

σ if labor cannot be reallocated1

2−σif labor can be reallocated

.

d2 logCd logA2

i= λi(1−λi)

(1− 1

ρ

).

Extends Hulten network irrelevance to second-order.

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Taking Stock

General formula for second-order nonlinear effects of shocks interms of macro-elasticities of substitution.

Reduced-form macro-elasticities of substitution shaped by:

structural micro-elasticities of substitution.

factor reallocation and returns to scale.

Network irrelevance if (1) uniform micro-elasticities, and (2)Harrod-neutral (labor-augmenting) shocks.

Can break network irrelevance with (1) heterogenous productionelasticities, and (2) Hicks-neutral (TFP) shocks.

Page 26: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Agenda

Framework

Structural ModelMacro-SubstitutionInput-output MutlipliersNetworks

Quantitative Examples

Conclusion

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The role of ξ

So far, ξ = 1, constant macro returns to scale.

For most applications, ξ > 1: intermediate goods, capital, trade.

In many applications, ξ restrictted to be constant: Gomme andRupert (2007), Aghion and Howitt (2008), Jones (2011), Gabaix(2011), Acemoglu et al. (2012), Kim et al. (2013), Bartelme andGorodnichenko (2015).

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Variable ξ

Assume

Y

Y= A

(a

(L

L

) θ−1θ

+ (1−a)

(X

X

) θ−1θ

) θ

θ−1

,

whereC + X = Y .

Proposition 2.3

d2 logCd logA2 =

(1a−1

)(θ −1) = (ξ −1)(θ −1).

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Variable input-output multiplier

−0.15 −0.1 −0.05 0 0.05 0.1−10

−8

−6

−4

−2

0

2

4

log(A)

log(

C/C

)

LeontiefHulten/Cobb-Douglasθ = 2

For this calibration, a = 0.1.

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Agenda

Framework

Structural ModelMacro-SubstitutionInput-output MutlipliersNetworks

Quantitative Examples

Conclusion

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General Networks

Definition 2.4The N×N input-output matrix Ω is the the matrix whose ij th elementis equal to the steady-state value of

Ωij =pjxij

piyi.

The Leontief inverse isΨ = (I−Ω)−1.

Ψij measures i ’s reliance on j .

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Networks

Proposition 2.5

Assume βi = 1 (CRS, full reallocation), and εi = θi for every i(w.l.o.g.), and Hicks-neuitral (TFP) shocks. Then

d2 logCd logA2

k= (σ −1)Varb(Ψ(k)) +∑

j(εj −1)λjVarΩ(j)(Ψ(k)).

Weighted variances of Leontief inverse:

Varb(Ψ(k)) = ∑i biΨ2ik − (∑i biΨik )2 ,

VarΩ(j)(Ψ(k)) = ∑i ΩjiΨ2ik − (∑i ΩjiΨik )2 .

Centrality measure mixing network and elasticities.

Generalization to DRS and multiple factors.

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Example: Universal Inputs

M...1 M+1 ...

E

HH

N

ε

σ

d2 logCd logA2

E= (σ −1)λE

(NM−1

)λE + (ε−1)λE

(1− N

MλE

),

= λE (1−λE )(σ −1) + (σ − ε)λE

(NM

λE −1

).

Page 34: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Direction of Diffusion

Proposition 2.6Consider two industries k and l that sell the same share to all otherindustries and the household ωik = ωil for each i and bk = bl . Thenthese industries are equivalent up to the second order:

d logCd logAk

=d logCd logAl

,

andd2 logCd logA2

k=

d2 logCd logA2

l.

Key: CRS and one factor.

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Agenda

Framework

Structural ModelMacro-SubstitutionInput-output MutlipliersNetworks

Quantitative Examples

Conclusion

Page 36: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Simulation

Set θj = θ = 0.3,εi = ε ≈ 0, and σ = 0.4 drawing on Atalay(2016), Boehm et al. (2015), Barrot and Sauvagnat (2016),Comin et al. (2015).

Set (σ ,ε,θ) to match ∑i λ iσλi= 0.0197.

Impose no-movement in labor for benchmark (Acemoglu et al.(2016), Autor et al. (2016), Notowidigdo (2011)).

Use the 88-sector US KLEMS annual input-output data from1960-2005, with sector-level TFP data constructed usingJorgenson et al. (1987) methodology by Carvalho and Gabaix(2013).

Set sectoral TFP shocks to be logN (−Σii/2,Σii), where Σii issample variance of ∆ logTFP for industry i .

Page 37: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Simulation Results

Mean Standard Deviation Skewness

GDP Data – 0.0238 -0.6190

TFP Data – 0.0147 -0.2888

Benchmark -0.0057 0.0117 -0.5229Full reallocation -0.0026 0.0110 -0.0745Log Linear Hulten -0.0010 0.0110 0.0000Linear Hulten 0.0000 0.0110 0.0432No Network, no reallocation -0.0014 0.0053 -0.0420No Network, full reallocation 0.0000 0.0053 0.0301(θ ,σ) = (0.1,0.3) -0.0102 0.0138 -1.2864High Volatility Benchmark -0.0117 0.0180 -0.8821High Volatility Hulten 0.0000 0.0155 0.0422

Welfare costs of business cycles 0.57%, order of magnitudelarger than those of 0.05% identified by Lucas (1987).

Page 38: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Histograms

0.94 0.96 0.98 1 1.02 1.04 1.060

10

20

30

40pd

f

BenchmarkHulten

0.94 0.96 0.98 1 1.02 1.04 1.060

5

10

15

20

25

30

pdf

Benchmark High VolatilityHulten High Volatility

Figure: The left panel shows the distribution of GDP for the benchmark modeland log-linearized model. The right panel shows these for shocks whosevariance is twice as high.

Excess kurtosis of 1 in benchmark model, increases with volatility.

Endogenous and asymmetric fat tails (”rare disasters”).

Page 39: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Oil v. Retail

0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

−0.1

−0.05

0

log(TFP)

log(

GD

P)

OilRetailHulten OilHulten Retail

Intuition: low micro-elasticity of substitution, universal input.

Consistent with large asymmetric effects of oil shocks (Hamilton,2003), even without frictions.

Page 40: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Reduced-form Impact of Oil Shocks

Proposition 3.1

Up to the second order in the vector ∆, we have

log(C(A + ∆)/C(A)) =12

[λ (A + ∆) + λ (A)]′ log(∆).

Page 41: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Reduced-form Impact of Oil Shocks

1970 1980 1990 2000 20100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure: Global expenditures on crude oil as a fraction of world GDP.

First-order effect: 5%×−13% = 0.65%.

Second-order effect: 12 (5% + 31%)×−13% = 2.34%.

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Agenda

Framework

Structural ModelMacro-SubstitutionInput-output MutlipliersNetworks

Quantitative Examples

Conclusion

Page 43: The Macroeconomic Impact of Microeconomic Shocks: Beyond ... · Macroeconomic Impact of Shocks For economy with efficient equilibrium, Hulten (1978): dlogC=dlogAi = salesi=GDP =

Conclusion

For empirically relevant cases, nonlinearities missed by first-orderapproximation are important.

Second-order terms depend on macro-elasticities of substitution:macro-objects, not identified by micro-variation.

Micro-elasticities of substitution, factor reallocation, micro-returnsto scale, and networks play an important role in shaping thesesecond order terms.

Ongoing work to: allow for RBC channels (elastic labor supply,capital accumulation); dynamics (reallocation); frictions(markups/wedges); co-movement; macro-elasticities ofsubstitution between factors; trade and gains from trade.

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