Misallocation or Mismeasurement?

Mark Bils, University of Rochester and NBERPete Klenow, Stanford University and NBER

Cian Ruane, International Monetary Fund

January 2020

Any opinions and conclusions expressed herein are those of the author(s) and do notnecessarily represent the views of the U.S. Census Bureau. This research wasperformed at a Federal Statistical Research Data Center under FSRDC Project 1440.All results have been reviewed to ensure that no confidential information is disclosed.The views expressed in this paper are those of the authors and should not be attributedto the International Monetary Fund, its Executive Board, or its management.

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Motivation

Widespread perception that resource misallocation an importantcause of low aggregate productivity (TFP) in EMDEs

Motivates structural reforms as a source of growth / convergence

Gains from resource reallocation are difficult to quantify, butgains are potentially huge

I Banerjee & Duflo (2005)I Restuccia & Rogerson (2008)I Hsieh & Klenow (2009, 2014)I Baqaee & Farhi (2019)

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Hsieh and Klenow (2009)

Large dispersion in revenue/inputs (TFPR) across plants in India,China and U.S.

Interpret dispersion in TFPR as dispersion in marginal products⇒ quantify misallocation

Reducing resource misallocation in India and China to U.S.levels⇒ increase TFP by 40% and 50%

But dispersion in measured average products need not reflectdispersion in true marginal products

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U.S. manufacturing in recent decades

Kehrig (2015) noticed rising TFPR dispersion

I Suggests falling allocative efficiency

For 1978–2013 we find it would imply:

I A drag on TFP growth of 1.7 percentage points per year

I 45 percent lower TFP (cumulatively by 2013)

Has misallocation really increased dramatically?

Or has mismeasurement and/or misspecification worsened?

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U.S. allocative efficiency

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What we do

Propose a way to estimate misallocation allowing for:

I Measurement error in revenue and inputs

I Misspecification due to overhead costs

Apply to:

I manufacturing plants in the U.S. 1978–2013

I manufacturing plants in India 1985–2013

Preview of results:

I No longer a severe decline in U.S. allocative efficiency

I For U.S. potential gains ∼ 49% rather than ∼ 123%

I For India, potential gains ∼ 89% rather than ∼ 111%

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U.S. vs India corrected allocative efficiency

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Others skeptical of misallocation

Adjustment costs

I Asker, Collard-Wexler and De Loecker (2014)

I Kehrig and Vincent (2019)

Overhead costs

I Bartelsman, Haltiwanger and Scarpetta (2013)

Variable production elasticities

I David and Venkateswaran (2019)

Measurement error and imputation

I Rotemberg and White (2019)

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Outline

1 Illustrative example

2 Full model

3 TFPR dispersion in U.S. and Indian data

4 Estimating measurement and specification error

5 Corrected measures of misallocation

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Simple model setup

Y =

(∑i Y

1− 1ε

i

) 1

1− 1ε , P =

(∑i P

1−εi

) 11−ε

Yi = Ai · Li

max(1− τYi

)PiYi − wLi

I Monopolistic competitor takes w, Y , and P as given

PiYi = PiYi + gi

Li = Li + fi

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Simple model TFPR

Pi =

(ε

ε− 1

)×(τi ·

w

Ai

), where τi ≡

1

1− τYi

PiYiLi

∝ τi

TFPRi ≡PiYi

Li∝ τi ·

PiYiPiYi

· LiLi

Let ∆Xt ≡Xt −Xt−1

Xt−1for variable X

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Identifying misallocation vs. dispersion in TFPR

∆P Y i = ∆Li ·τi

TFPRi

I assuming distortions and measurement error are fixed over time

I in which case ∆PiYi = ∆Li = (ε− 1) ∆Ai

We will generalize to allow:

I Sales R and a composite input I

I Shocks to distortions and to measurement errors

And regress ∆R on ∆I in different deciles of TFPR

I Measurement error should make coefficients fall with TFPR

I Will use this to estimate E (ln τi | ln TFPRi)

I Validate approach using simulations

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Outline

1 Illustrative example

2 Full model

3 TFPR dispersion in U.S. and Indian data

4 Identifying measurement and specification error

5 Corrected misallocation

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Full Model (Setup)

Closed economy, S sectors, Ns firms, L workers, K capital

Q =∏Ss=1Q

θss , Qs =

(∑Nsi Q

1− 1ε

si

) 1

1− 1ε

Qsi = Asi(Kαssi L

1−αssi )γsX1−γs

si

max Rsi − (1 + τLsi)wLsi − (1 + τKsi )rKsi − (1 + τXsi )Xsi

I Rsi ≡ PsiQsiI Monopolistic competitor takes input prices as given

C = Q−X , X =∑S

s

∑Nsi Xsi

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Model (Aggregate TFP)

TFP ≡ C

L1−αKα

I where α ≡∑Ss=1 αsγsθs∑Ss=1 γsθs

TFP = T ×S∏s=1

TFP

θs∑Ss=1 γsθs

s

I TFPs ≡Qs

(Kαss L1−αs

s )γsX1−γss

I T = reflects sectoral distortions (set aside)

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Model (Sectoral TFP)

Suppressing s here and whenever possible:

TFP =

[N∑i

Aε−1i

(τiτ

)1−ε] 1ε−1

τi ≡[(

1 + τLi)1−α (

1 + τKi)α]γ (

1 + τXi)1−γ

τ ≡[(

1 + τL)1−α (

1 + τK)α]γ (

1 + τX)1−γ

where 1 + τL ≡[∑N

i=1RiR

11+τLi

]−1and so on

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Model (Sectoral TFP Decomposition)

TFP = AE · PD ·A ·N1ε−1

AE ≡ Allocative Efficiency

PD ≡ Productivity Dispersion

A ≡ Average productivity

N1ε−1 ≡ Variety

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Model (Sectoral TFP Decomposition)

TFP =

[1

N

N∑i

(Ai

A

)ε−1 (τiτ

)1−ε] 1ε−1

︸ ︷︷ ︸AE=Allocative Efficiency

×

[1

N

N∑i

(Ai

A

)ε−1] 1ε−1

︸ ︷︷ ︸PD=Productivity Dispersion

×N1ε−1︸ ︷︷ ︸

Variety

× A︸︷︷︸Average Productivity

A =[

1N

∑Ni (Ai)

ε−1] 1ε−1 (power mean)

A =∏Ni=1A

1Ni (geometric mean)

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Full Model (Sectoral TFP Decomposition)

TFP ≡ Q

(L1−αKα)γX1−γ

=

[1

N

N∑i

(Ai

A

)ε−1 (τiτ

)1−ε] 1ε−1

︸ ︷︷ ︸AE=Allocative Efficiency

×

[N∑i

(Ai)ε−1

] 1ε−1

︸ ︷︷ ︸Residual Productivity

τi ≡[(

1 + τLi)1−α (

1 + τKi)α]γ (

1 + τXi)1−γ

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Aside: Intuition in a special case

If Ai and τi are jointly lognormal and τLi = τKi = τXi then

AE =ε

2· Var [ln(τi)]

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Outline

1 Illustrative example

2 Full model

3 TFPR dispersion in U.S. and Indian data

4 Identifying measurement and specification error

5 Corrected misallocation

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Indian Annual Survey of Industries (ASI)

Survey of Indian manufacturing plants

I Long panel 1985–2013

I Used in Hsieh and Klenow (2009, 2014)

Sampling frame

I All plants > 100 or 200 workers (45% of plant-years)

I Probabilistic if > 10 or 20 workers (55% of plant-years)

I ≈ 43,000 plants per year

Variables used

I Gross output (Ri), intermediate inputs (Xi), labor (Li), wage bill(wLi), and capital (Ki)

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U.S. Longitudinal Research Database (LRD)

U.S. Census Bureau data on manufacturing plants

I Long panel, 1978–2013

I Used in Hsieh and Klenow (2009, 2014)

Sampling frame

I Annual Survey of Manufacturing (ASM) plants

I ∼ 50k plants per year with at least one employee

I Quinquennial sample for ∼ 34k plants, certainty for other ∼ 16k

Variables used

I Gross output (Ri), intermediate inputs (Xi), labor (Li), wage bill(wLi), and capital (Ki)

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Data cleaning steps

Step Cleaning

1 Starting sample of plant-years2 Missing no key variables3 Common sector concordance4 Trim 1% of each left and right TFPR and TFPQ tail

1,806,000 plant-years for U.S. LRD

943,186 plant-years for Indian ASI

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Inferring allocative efficiency a la Hsieh & Klenow (2009)

AE =

[N∑i

(TFPQi

TFPQ

)ε−1(TFPRiTFPR

)1−ε] 1ε−1

Ai = TFPQi =(Ri)

εε−1

(Kαi L

1−αi )γX1−γ

i

TFPRi =Ri

(Kαi L

1−αi )γX1−γ

i

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Inferring allocative efficiency as in Hsieh & Klenow (2009)

AE =

[N∑i

(TFPQi

TFPQ

)ε−1(TFPRiTFPR

)1−ε] 1ε−1

TFPQ =[∑N

i TFPQε−1i

] 1ε−1

TFPR =(

εε−1

) [MRPL

(1−α)γ

](1−α)γ [MRPKαγ

]αγ [MRPX1−γ

]1−γ

I MRPK =

[∑i

RiR

1

MRPKi

]−1

and so on

I MRPKi =

(ε− 1

ε

)αγ

RiKi

and so on

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Inferring aggregate AE

Aggregating within-sector allocative efficiencies:

AEt =

S∏s=1

AE

θst∑Ss=1 γsθst

st

Parameterization:

ε = 4 based on Redding and Weinstein (2016)

αs and γs inferred from sectoral cost-shares (r = .2)

θst inferred from sectoral shares of aggregate output

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Indian allocative efficiency

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U.S. allocative efficiency

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Allocative efficiency in the U.S. relative to India

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TFPR Dispersion in the U.S. and India over time

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TFPQ Dispersion in the U.S. and India over time

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Elasticity of TFPR wrt TFPQ in the U.S and India

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Outline

1 Illustrative example

2 Full model

3 TFPR dispersion in U.S. and Indian data

4 Identifying measurement and specification error

5 Corrected misallocation

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Measurement error and ∆TFPR

Recall TFPRi = τi ·RiRi· IiIi

∆TFPRi = ∆τi + ∆

(RiRi

)− ∆

(IiIi

)

∆ is the growth rate of a variable relative to the sector s mean.

Increases in Ri and Ii reduce the role of additive measurement error

But have no impact on the role of multiplicative measurement error

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Previewing our empirical specification

We will regress revenue growth on input growth for a panel of plants:

∆Ri = λk + βk∆Ii + ei

i denotes plant

k denotes one of K bins of TFPR

∆ is the growth rate of a variable relative to the sector s mean.

Additive measurement error shows up as lower βk at higher TFPR’s

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Measurement error: key assumptions

Only additive measurement error

Ii = Ii + fi; Ri = Ri + gi

I Conservative, as multiplicative also overstates TFPR differences

I Analogous to heterogeneous overhead costs

Common measurement error across inputs

Zero covariance between ln (τi) and ln

(RiRi· IiIi

)

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Look at elasticity of revenue wrt inputs

Defineβk ≡

Covk

(∆R,∆I

)V ark

(∆I)

k indexes the level of TFPR (plant index i implicit)

βk is the population projection of ∆R on ∆I , at k–level TFPR

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βk in the absence of measurement error

If f = 0 and g = 0, then:

∆I = ∆I = (ε− 1) ∆A − ε∆τ

∆R = ∆R = (ε− 1)(

∆A−∆τ)

βk = 1 + φk

where φk ≡Covk (∆τ,∆I)

V ark (∆I)

=−εV ark (∆τ) + (ε− 1)Covk (∆τ,∆A)

V ark (∆I)

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βk in the absence of measurement error cont.

βk = 1 + φk

φk = elasticity of ∆τ wrt ∆I

If ∆τ is i.i.d., then does not project on TFPR

I ⇒ φk = φ

I ⇒ βk = β

If τ is stationary, then V ark (∆τ) larger at both high and lowvalues of ln(TFPR) = ln(τ), so smaller βk at TFPR extremes

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βk in the presence of measurement error

βk ≈ Ek

(RI

R I

)(1 + φk) + ψk

≈ reflects for “small” changes: ∆A, ∆τ ,df

I,dg

R

Recall TFPR = τ · RR· II

Ek

(RI

R I

)captures role of both measurement errors in TFPR

Ek (τ) = Ek

(RI

R I

)· TFPRk

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Measurement error in revenue and inputs, cont.

Now φk reflects elasticities of ∆τ and df

Iwrt ∆I

φk ≡Covk

(I

I∆τ − f ′−f

I, ∆I

)V ark

(∆I)

Added term ψk

ψk =1

V ark

(∆I)(Covk(RI

R I, ∆I

(∆I +

I

I∆τ − f ′ − f

I

))−

Ek

(∆I)Covk

(RI

RI,(

∆I +I

I∆τ − f ′ − f

I

))+ Covk

(g′ − gR

, ∆I

))

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Recovering Ek (τ)

Ek (τ) = Ek

(RI

R I

)· TFPRk =

(βk − ψk1 + φk

)· TFPRk

If ∆τ , ∆A,df

Iand

dg

Rare i.i.d. =⇒ ψk = 0, φk = φ

Ek (τ) ∝ βk · TFPRk

Model simulations to correct for large shocks and φk, ψk

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Getting dispersion in τ

Ek (τ) ∝ βk · TFPRk is τ dispersion that projects on TFPR

With measurement error, also τ dispersion ⊥ to TFPR

“Add back” dispersion, call term ε, in ln τ

I ε ⊥ ln TFPR

I ε ∼ N(0, σ2)

I σ2 chosen to hit var[ln(τ)] assuming ln(R I

R I

)⊥ ln τ

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Dispersion in τ , continued

ln TFPR = ln τ − ln(R I

R I

)V ar

(ln τ

)=

V ar(

ln TFPR)− V ar

(ln

(RI

RI

))+ 2Cov

(ln τ, ln

(RI

RI

))

Assuming Cov(

ln τ, ln

(RI

RI

))= 0

I Eliminates last term in V ar(

ln τ)

I −V ar

(ln

(R I

R I

))= Cov

(ln TFPRk, Ek

(ln

(R I

R I

)))

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Dispersion in τ , continued

As a result:

V ar

(ln τ

)=

V ar

(ln TFPR

)+ Cov

(ln TFPRk, Ek

(ln

(R I

R I

)))

I First term is data

I Second reflects how our βk estimates covary with TFPR

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TFPR correction implementation

Regress revenue growth on input growth by decile of TFPR:I Divide 25+ year samples into 5/6-year windowsI Unbalanced panel of Indian and U.S. plantsI ≈ 28,000 / 6,000 plants per decile-window in U.S. / India

∆Ri = λk + βk∆Ii + ei

i denotes plant, k denotes decileI ∆Ri, ∆Ii and TFPR are deviations from sector-year averageI Use Tornqvist average of TFPR for constructing decilesI Regressions are cost-share weightedI Trim observations where TFPR changes by factor > 5

Merge βk estimates into non-panel sample by decile-window:

ln (τi) = ln(TFPRi) + ln(βk) + εi

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βk’s wrt TFPRk, India and U.S.

Table: Coefficients βk for revenue growth on input growth by TFPR decile

k→ 1 2 3 4 5 6 7 8 9 10

India 1.09 1.04 1.03 1.00 1.01 0.99 1.00 0.99 0.95 0.88

U.S. 1.05 1.00 0.97 0.96 0.93 0.90 0.86 0.84 0.70 0.54

Source: Indian ASI 1985–2013 and U.S. LRD 1978–2013. TFPR deciles are constructed asTornqvist deviations from the (cost-weighted) sector-year average. Regressions weight by plant’sinput costs. Standard errors are clustered at the industry level, and are uniformly below 0.02.

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Indian βk slopes wrt TFPRk

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U.S. βk slopes wrt TFPRk

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τ dispersion vs. TFPR dispersion

India

1985–1991 1992–1996 1997–2001 2002–2007 2008–2013

σ2τ/σ

2TFPR 0.68 0.76 0.74 0.69 0.71

U.S.

1978–1984 1985–1991 1992–1998 1999–2005 2006–2013

σ2τ/σ

2TFPR 0.40 0.43 0.38 0.32 0.28

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Allocative efficiency in India

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Allocative efficiency in the U.S.

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Uncorrected vs. corrected gains from reallocation

INDIA1985–2013

Mean S.D.

Uncorrected gains 110.9% 17.3%

Corrected gains (estimates) 87.8% 13.8%

Shrinkage 21% 20%

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Uncorrected vs. corrected gains from reallocation

U.S.1978–2013

Mean S.D.

Uncorrected gains 123.2% 59.7%

Corrected gains (estimates) 49.1% 12.2%

Shrinkage 60% 80%

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Cumulative change in AE

U.S. INDIA1978–2013 1985–2013

Uncorrected -45% -1.5%

Corrected -16% -0.8%

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Allocative efficiency: U.S. relative to India

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Dispersion of U.S. TFPR

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Elasticity of TFPR wrt TFPQ in the U.S.

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Dispersion of TFPQ in the U.S.

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Simulations to test the validity of our strategy

Ait = Ai · ait where ln(Ai) ∼ N(0, σ2A)

ait and τit follow

ln(xit) = ρx · ln(xit−1) + ηxit where ηxit ∼ N(0, σ2x)

fit follows

fit = ρf · fit−1 + ηfit · Iit where ηfit ∼ N(0, σ2f )

Use ε = 4, ρa = ρτ = ρf = 0.9

Estimate {σA, σa, στ , σf} by window to fit{σTFPR, σTFPQ, σ∆I

, projection of ln(βk) on ln(TFPRk)}

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Simulation Results

India

1985–1991 1992–1996 1997–2001 2002–2007 2008–2013

σ2τ/σ

2TFPR (our correction) 0.64 0.76 0.76 0.68 0.71

σ2τ/σ

2TFPR (truth) 0.60 0.80 0.82 0.68 0.71

U.S.

1978–1984 1985–1991 1992–1998 1999–2005 2006–2013

σ2τ/σ

2TFPR (our correction) 0.36 0.41 0.34 0.32 0.25

σ2τ/σ

2TFPR (truth) 0.17 0.29 0.17 0.13 0.02

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Takeaways from simulations

ln(β) vs. ln(TFPR) approach:

Does well at correcting for additive measurement error

I similar results when measurement error is in revenues

I tends to undercorrect when there is lots of measurement error

And further simulations show:

Does not correct at all for multiplicative measurement error

Does not correct at all for adjustment costs

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Conclusion

Proposed a way to estimate true dispersion of marginal products

I Revenue growth is less sensitive to input growth when averageproducts overstate marginal products

I Requires measurement error be additive and ⊥ to distortions

Implemented on Indian ASI

I Potential gains from reallocation reduced by 15

I Time-series volatility reduced by 15

Implemented on U.S. LRD

I Potential gains from reallocation reduced by 35

I Time-series volatility reduced by 45

I No longer a sharp downward trend in allocative efficiency

I U.S. allocative efficiency predominantly higher than in India

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Should the U.S. Census Bureau have noticed?

Arguably hard to see:

Variance of ln R and ln I rose 7.2% and 7.3% (1978–2013)

Correlation of ln R and ln I fell from 0.993 to 0.979

22–58 times higher variance for ln R and ln I than for ln TFPR

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