Misallocation or Mismeasurement? Mark Bils, University of Rochester - - PowerPoint PPT Presentation

Misallocation or Mismeasurement?

Mark Bils, University of Rochester and NBER Pete Klenow, Stanford University and NBER Cian Ruane, Stanford University

February 2017

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Motivation

Large gaps in average revenue products (TFPR) across plants

◮ Syverson (2011)

Huge purported gains from reallocation of inputs

◮ Banerjee & Duflo (2005) ◮ Restuccia & Rogerson (2008) ◮ Hsieh & Klenow (2009, 2014)

But big differences in measured average products need not imply big differences in true marginal products

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U.S. manufacturing since the 1970s

Major increase in TFPR dispersion (Kehrig, 2015)

◮ Implies falling allocative efficiency ◮ If true, lowered TFP growth by about 2.5 percent per year ◮ Cumulated to 55 percent lower TFP by late 2000s ◮ Given measured TFP growth was about 1.7 percent per year,

would imply residual TFP growth of 4.2 percent per year

Real, or measurement error getting worse?

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U.S. Allocative Efficiency & Residual TFP

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What we do

Propose way to estimate marginal products under:

◮ Measurement error ◮ Misspecification due to overhead costs

Apply to:

◮ manufacturing plants in the U.S. 1978–2007 ◮ manufacturing plants in India 1985–2011

Preview of results:

◮ Eliminates the severe decline in U.S. allocative efficiency ◮ Reduces potential gains from reallocation in India by 40% ◮ Leaves U.S. a stable 30% higher allocative efficiency than India 5 / 68

Measurement error in the Indian data

Book values instead of market values for capital Number of contract workers not known End-of-previous-year = beginning-of-current-year stocks Jumps in reported age across years for many plant

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Measurement error in the U.S. data

Book values instead of market values for capital Census is frequently forced to impute data

◮ SSA, IRS data on a subset of plants, variables ◮ Sometimes impute based on other plants ◮ See White, Reiter and Petrin (2016) for a critique ◮ And Petrin, Rotemberg and White (2017, in progress) 7 / 68

Table of Contents

1

Simple model

2

Full Model

3

Data & Measurement

4

Patterns in the data

5

Test for measurement & specification error

6

Correcting Aggregates

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Simple model setup

Y =

• i Y

1− 1

ǫ

i

• 1

1− 1 ǫ , P =

• i P 1−ǫ

i

• 1

1−ǫ

Yi = AiLi max (1 − τ Y

i )PiYi − wLi

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

• PiYi ≡ PiYi + gi
• Li ≡ Li + fi

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Simple model TFPR

Pi = markup × marginal cost Pi =

• ǫ

ǫ − 1

• ×
• τi · w

Ai

• , where τi ≡

1 1 − τ Y

i

PiYi ∝ τi · Li TFPRi ≡

• PiYi
• Li

∝

• τi × 1 + gi/(PiYi)

1 + fi/Li

• 10 / 68

Numerical example

τi — so the true distortion is fixed over time gi, fi — so additive measurement error is fixed over time Ait — so productivity is time-varying PY L

PY L

• PY
• L
• PY
• L

PY L

PY L

Firm 1 100 50 2 120 50 2.4 50 25 2 Firm 2 50 50 1 40 50 0.8 25 25 1

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Lessons from the numerical example

• PitYit/

Lit = τi when constant measurement error, distortions Regressing ln

• PitYit/

Lit

• n ln (TFPR) yields:

◮ 1 if there is no measurement error in TFPR ◮ 0 if all TFPR dispersion is due to measurement error ◮ ∼ 2/3 in the numerical example above

Later we generalize in order to:

◮ Allow shocks to measurement error and distortions ◮ Infer the signal from covariance b/w levels, first differences 12 / 68

Projection of First Differences on Levels

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Full model vs. Simple model

Capital, labor, and intermediates Distortions hitting each input Multiple sectors Shocks to τ and shocks to measurement error Key assumption: measurement error is orthogonal to τ

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Table of Contents

1

Simple model

2

Full Model

3

Data & Measurement

4

Patterns in the data

5

Test for measurement & specification error

6

Correcting Aggregates

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Model (Setup)

Closed economy, S sectors, Ns firms, L workers, K capital Q = C + X Q = S

s=1 Q θs s

Qs = Ns

i

Q

1− 1

ǫ

si

• 1

1− 1 ǫ

Qsi = Asi(Kαs

si L1−αs si

)γsX1−γs

si

max Rsi − (1 + τ L

si)wLsi − (1 + τ K si )rKsi − (1 + τ X si )Xsi

◮ Rsi ≡ PsiQsi ◮ Monopolistic competitor takes input prices as given 16 / 68

Model (Aggregate TFP)

TFP ≡ C L1−˜

αK ˜ α

◮ where ˜

α ≡ S

s=1 αsγsθs

S

s=1 γsθs

TFP = T ×

S

• s=1

TFP

θs S s=1 γsθs

s

◮ T = reflects sectoral distortions (set aside) ◮ TFPs ≡

Qs (Kαs

s L1−αs s

)γsX1−γs

s

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Model (Sectoral TFP)

Suppressing s here and whenever possible: TFP = N

• i

Aǫ−1

i

τi τ 1−ǫ

• 1

ǫ−1

τi ≡

• 1 + τ L

i

1−α 1 + τ K

i

αγ 1 + τ X

i

1−γ τ ≡

• 1 + τ L1−α

1 + τ Kαγ 1 + τ X1−γ where 1 + τ L ≡ N

i=1 Ri R 1 1+τ L

i

−1 and so on

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Model (Sectoral TFP Decomposition)

TFP = AE · PD · ¯ A · N

1 ǫ−1

AE ≡ Allocative Efficiency PD ≡ Productivity Dispersion ¯ A ≡ Average productivity N

1 ǫ−1 ≡ Variety 19 / 68

Model (Sectoral TFP Decomposition)

TFP =

• 1

N

N

• i

Ai ˜ A ǫ−1 τi τ 1−ǫ

• 1

ǫ−1

• AE=Allocative Efficiency

×

• 1

N

N

• i

Ai ¯ A ǫ−1

1 ǫ−1

• PDs=Productivity Dispersion

× N

1 ǫ−1

Variety

× ¯ A

• Average Productivity

˜ A =

• 1

N

N

i (Ai)ǫ−1

1 ǫ−1

(power mean) ¯ A = N

i=1 A

1 N

i

(geometric mean)

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Table of Contents

1

Simple model

2

Full Model

3

Data & Measurement

4

Patterns in the data

5

Test for measurement & specification error

6

Correcting Aggregates

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Indian Annual Survey of Industries (ASI)

Survey of Indian Manufacturing Plants

◮ Long panel 1985–2011 ◮ Used in Hsieh and Klenow (2009, 2014)

Sampling Frame:

◮ ∼ 43,000 plants per year ◮ All plants > 100 or 200 workers (45% of plant-years) ◮ Probabilistic if > 10 or 20 workers (55% of plant-years)

Variables used:

◮ Gross output (Ri), intermediate inputs (Xi), labor (Li), labor cost

(wLi), and capital (Ki)

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U.S. Longitudinal Research Database (LRD)

U.S. Census Bureau data on manufacturing plants

◮ Long panel, 1978–2007 analyzed so far ◮ Used in Hsieh and Klenow (2009, 2014)

Sampling Frame:

◮ Annual Survey of Manufacturing (ASM) plants ◮ ∼ 50k plants with at least one employee ◮ Probabilistic sampling for ∼ 34k plants, certainty for other ∼ 16k

Variables used:

◮ Gross output (Ri), intermediate inputs (Xi), labor (Li), labor cost

(wLi), and capital (Ki)

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Measurement error in the Indian ASI

Frequency Magnitude Age 12.4% 4 years EOY & BOY capital stocks 25.7% 15.4% EOY & BOY goods inventories 22.0% 24.8% EOY & BOY materials inventories 22.3% 20.2%

There is measurement error in age if age in year t is not equal to 1 + age in year t − 1. The magnitude of this measurement error is the median absolute deviation. There is measurement error in stocks and inventories if the deviation of the BOY value in year t from the EOY value in year t−1 is greater than 1%. The magnitude

• f this measurement error is the standard deviation of the absolute value of the

percentage measurement error.

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Data cleaning steps

Indian ASI U.S. LRD Step Cleaning Remaining Obs Remaining Obs 1 Starting sample of plant-years 1,159,641 1,767,000 2 Missing no key variables 924,547 1,589,000 3 Common Sector Concordance 899,793 1523,000 4 Trimming extreme TFPR & TFPQ 844,875 1,428,000

The last step trims 1% tails of MRP & TFPQ deviations from sector-year averages

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Inferring allocative efficiency from the data

• AE =

N

• i

TFPQi TFPQ ǫ−1 TFPRi TFPR 1−ǫ

1 ǫ−1

• Ai = TFPQi =
• Ri
• ǫ

ǫ−1

( Kα

i

L1−α

i

)γ X1−γ

i

TFPRi =

• Ri

( Kα

i

L1−α

i

)γ X1−γ

i

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Inferring allocative efficiency from the data

• AE =

N

• i

TFPQi TFPQ ǫ−1 TFPRi TFPR 1−ǫ

1 ǫ−1

TFPQ = N

i TFPQǫ−1 i

• 1

ǫ−1

TFPR =

• ǫ

ǫ−1 MRP L (1−α)γ

(1−α)γ

MRP K αγ

αγ

MRP X 1−γ

1−γ

◮ MRPK =

• i
• Ri
• R

1 MRPKi −1 and so on

◮ MRPKi =

ǫ − 1 ǫ

• αγ
• Ri
• Ki

and so on

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Inferring aggregate AE from the data

Aggregating within-sector allocative efficiencies:

• AEt =

S

• s=1
• AE

θst S s=1 γsθst

st

Parameterization: ǫ = 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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Table of Contents

1

Simple model

2

Full Model

3

Data & Measurement

4

Patterns in the data

5

Test for measurement & specification error

6

Correcting Aggregates

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Indian potential gains from reallocation (102% on average)

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U.S. potential gains from reallocation

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India vs. U.S. potential gains from reallocation

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Table of Contents

1

Simple model

2

Full Model

3

Data & Measurement

4

Patterns in the data

5

Test for measurement & specification error

6

Correcting Aggregates

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Measurement error in revenue and inputs

• Ii ≡ φi · Ii + fi
• Ri ≡ χi · Ri + gi

Ii and Ri = true inputs and revenues

• Ii and

Ri = measured inputs and revenues fi and gi = additive measurement errors φi and χi = multiplicative measurement errors

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Measurement Error and TFPR

TFPRi ≡

• Ri
• Ii

∝

• ǫ

ǫ − 1

• τi

Ri Ri Ii

• Ii
• Marginal product =
• Ri
• Ri
• Ii

Ii

• × Average product

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Measurement error and ∆TFPR

∆TFPRi = ∆τi + ∆ Ri Ri

• − ∆

Ii Ii

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

If only multiplicative measurement error: ∆TFPRi = ∆τi + ∆χi − ∆φi

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Measurement error and ∆TFPR

∆TFPRi = ∆τi + ∆ Ri Ri

• − ∆

Ii Ii

• If only additive measurement error:

∆TFPRi = ∆τi

• Ri/Ri

− Ri − Ri

• Ri

−

• Ii − Ii
• Ii
• ∆Ii

+ gi

• Ri

− fi

• Ii

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Measurement error: key assumptions

We focus on additive measurement error

◮ Conservative, as multiplicative also overstates TFPR differences ◮ Analogous to heterogeneous overhead costs

fi and gi are mean zero and orthogonal to lnτi and lnAi ∆fi and ∆gi are orthogonal to each other No skewness in logs of Ai, τi, Ri/Ri, and Ii/Ii Constant relative variances in logs of Ai, τi, Ri/Ri, Ii/Ii

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Measurement error and revenue growth

To simplify exposition for awhile: no measurement error in inputs ∆ Ii = ∆Ii = (ǫ − 1) ∆Ai − ǫ∆τi Revenue growth is: ∆ Ri = Ri

• Ri

[(ǫ − 1)(∆Ai − ∆τi) + ∆gi]

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Measurement error and the elasticity of revenue wrt inputs

Define βi ≡ σ∆

R,∆I

σ2

∆I

If ∆τ = 0 then E {βi | ln(TFPRi)} = E

• Ri
• Ri | ln(TFPRi)
• If ∆τ = 0 then

E {βi | ln(TFPRi)} = E Ri

• Ri
• 1 − Ω∆τ

i

ǫ

• | ln(TFPRi)
• where

Ω∆τ

i

≡ σ∆τ,∆I σ2

∆I

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Elasticity in a simplified case

If ∆τi and ∆Ai are i.i.d. then Ω∆τ

i

does not depend on ln(TFPRi). In this case: E {βi | ln(TFPRi)} =

• 1 − Ω∆τ

ǫ

• E

Ri

• Ri

| ln(TFPRi)

• 41 / 68

Elasticity in a simplified case

E Ri

• Ri

| ln(TFPRi)

• ≈ 1 −

σ2

ln

R R + σlnτ, ln R R

σ2

ln T F P R

ln(TFPRi) ⇒ E {βi | ln(TFPRi)} ≈

• 1 − Ω∆τ

ǫ  1 − σ2

ln

R R + σlnτ, ln R R

σ2

ln T F P R

ln(TFPRi)   ≡

• 1 − Ω∆τ

ǫ

• [1 − (1 − λ) · ln(TFPRi)]

where λ ≡ σ2

lnτ + σlnτ, ln

R R

σ2

ln T F P R

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Compression factor λ

λ can be used to estimate the variance of true distortions τsi: E {ln τi | ln(TFPRi)} = λ · ln(TFPRi) σ2

lnτ = λ · σ2 ln(TFPR) − σlnτ, ln

R R

Assuming mean-zero measurement error orthogonal to A, τ ⇒ σ2

lnτ = λ · σ2 ln(TFPR)

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Measurement error in both revenue and inputs

E {βi | ln(TFPRi)} =

• 1 − Ω∆τ

i

ǫ − Ω∆f

i

• E
• Ri

Ii

• RiIi

| ln(TFPRi)

• ≡
• 1 − Ω∆τ

i

ǫ − Ω∆f

i

• [1 − (1 − λ) · ln(TFPRi)]

where λ ≡ σ2

ln τ + σln τ, ln[( RI)/(R I)]

σ2

ln T F P R

and Ω∆f

i

= σ∆f,∆

I

σ2

∆ I

Still get σ2

ln τ = λ · σ2 ln(TFPR) when σln τ, ln [( RI)/(R I)] = 0

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Baseline specification

∆ Rit = Φ · ln(TFPRit) + Ψ · ∆ Iit −Ψ · (1 − λ) · ln(TFPRit) · ∆ Iit + Dt + ξit λ = σ2

ln τ

σ2

ln(TFPR)

Ψ = 1 − Ω∆τ

i

ǫ − Ω∆f

i

Dt = sector-year fixed effects

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Baseline specification

∆ Rit = Φ · ln(TFPRit) + Ψ · ∆ Iit −Ψ · (1 − λ) · ln(TFPRit) · ∆ Iit + Dt + ξit ln(TFPRit) is a Tornqvist of current and previous year Weight by Tornqvist gross output shares Winsorize 1% tails of ∆ Rit and ∆ Iit

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Baseline estimates for all years

India (1985-2011) U.S. (1978-2007) ˆ Φ 0.052 0.053 (0.005) (0.002) ˆ Ψ 0.967 0.794 (0.005) (0.004) ˆ λ 0.547 0.229 (0.035) (0.026) Observations 277,239 1,141,000 The dependent variable is revenue growth. ˆ Φ is the coefficient on TFPR, ˆ Ψ on composite input growth, and 1 − λ on the product of the two. Standard errors are clustered.

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Baseline estimates in windows

India

1985–1993 1994–2001 2002–2011 ˆ λ 0.562 0.510 0.576 (0.050) (0.080) (0.027)

U.S.

1978–1982 1983–1987 1988–1992 1993–1997 1998–2002 2003–2007 ˆ λ 0.358 0.336 0.326 0.326 0.192 0.095 (0.027) (0.034) (0.031) (0.037) (0.032) (0.070)

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Indian first differences vs. levels ( ˆ R/ˆ I vs. R/I)

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What if measurement error or τ mean revert?

E {βi | ln(TFPRi)} = E RiIi Ri Ii

• 1 − Ω∆τ

i

ǫ − Ω∆f

i

• | ln(TFPRi)
• = E

RiIi Ri Ii | ln(TFPRi)

• E
• 1 − Ω∆τ

i

ǫ − Ω∆f

i

| ln(TFPRi)

• if the covariance term is zero.

If measurement error or τ mean-reverting, then 1 − Ω∆τ

i

ǫ

− Ω∆f

i

is reduced at extemes of TFPR Capture with square of ln(TFPRi)

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Specification allowing mean reversion

∆ Rit = Φ · ln(TFPRit) + Ψ · ∆ Iit − Ψ · (1 − λ) · ln(TFPRit) · ∆ Iit + Γ · ln(TFPRit)2 + Λ · ln(TFPRit)2 · ∆ Iit + Υ · ln(TFPRit)3 − Λ · (1 − λ) · ln(TFPRit)3 · ∆ Iit + Dt + ξit

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Indian estimates allowing for mean reversion

All Years 1985-1993 1994-2001 2002-2011 Baseline ˆ λ 0.547 0.562 0.510 0.576 (0.035) (0.050) (0.080) (0.027) ˆ λ with mean reversion 0.520 0.547 0.465 0.562 (0.041) (0.060) (0.090) (0.029)

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U.S. estimates allowing for mean reversion

All Years 1978– 1983– 1988– 1993– 1998- 2003- 1982 1987 1992 1997 2002 2007 Baseline ˆ λ .229 0.358 0.336 0.326 0.326 0.192 0.095 (.026) (0.027) (0.034) (0.031) (0.037) (0.032) (0.070) ˆ λ with mean reversion 0.205 0.371 0.312 0.318 0.318 0.129 0.020 (0.018) (0.029) (0.037) (0.033) (0.038) (0.041) (0.054)

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Measurement error in relative inputs

1/3 of potential gains reflect relative input distortions measured by differences in factor shares — but this, too, could be mismeasurement. Measurement error need not be the same across inputs. Let:

• Ki = Ki + fKi
• Li = Li + fLi

Differences in Ki/ Li can reflect fKi versus fLi. We ask if differences in Ki/ Li decline when inputs increase. Do the same for differences in intermediates versus value added.

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Allowing for relative measurement error across inputs

∆ Ki − ∆ Li = −

Ki−Ki Ki

• −

Li−Li Li

• ∆

Vi +

• Ki
• Ki · Li
• Li
• · ∆Zi

1 − α

Ki−Ki Ki

• − (1 − α)

Li−Li Li

• where

∆Zi = ∆τLi − ∆τKi + ∆fKi − ∆fLi We assume ∆Zi is orthogonal to

Ki−Ki Ki

• −

Li−Li Li

• .

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Estimating relative measurement error across inputs

∆ Kit

• Lit
• = Φ · ln

Kit

• Lit
• + Π · ∆

Vit −(1 − λKL) · ln Kit

• Lit
• · ∆

Vit + Dt + ξit ln

Kit

• Lit
• is a Tornqvist of current and previous year

∆ Vit is growth of capital and labor inputs We winsorize 1% tails of ∆

Kit

• Lit
• and ∆

Vit And proceed similarly to estimate λV X

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Indian estimates with relative measurement error

All Years 1985-1993 1994-2001 2002-2011 ˆ λ with mean reversion 0.520 0.547 0.465 0.562 (0.041) (0.060) (0.090) (0.029) ˆ λKL 0.927 0.910 0.888 0.976 (0.022) (0.035) (0.039) (0.033) ˆ λV X 0.912 0.895 0.902 0.928 (0.011) (0.014) (0.019) (0.020)

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U.S. estimates with relative measurement error

All Years 1978– 1983– 1988– 1993– 1998- 2003- 1982 1987 1992 1997 2002 2007 ˆ λ with mean reversion 0.205 0.371 0.312 0.318 0.318 0.129 0.020 (0.018) (0.029) (0.037) (0.033) (0.038) (0.041) (0.054) ˆ λKL 0.797 0.822 0.777 0.815 0.780 0.777 0.831 (0.009) (0.020) (0.016) (0.017) (0.026) (0.030) (0.026) ˆ λV X 0.838 0.884 0.883 0.840 0.821 0.839 0.811 (0.006) (0.010) (0.011) (0.011) (0.018) (0.014) (0.021)

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Table of Contents

1

Simple model

2

Full Model

3

Data & Measurement

4

Patterns in the data

5

Test for measurement & specification error

6

Correcting Aggregates

59 / 68

Correcting TFPRi

If all measurement error is common across inputs:

• TFPRi = exp
• ˆ

λ

• ln(TFPRi) − ln(TFPR)
• + ln(TFPR) + ǫi
• ln(TFPR) = within-sector weighted average of ln(TFPRi)

ǫi ∼ N

• 0, (ˆ

λ − ˆ λ2)σ2

ln(TFPR)

• Similar correction in the presence of relative measurement error

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Potential Gains from Reallocation in India

61 / 68

Potential Gains from Reallocation in India (ˆ λ in windows)

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Potential Gains from Reallocation in U.S.

63 / 68

Potential Gains from Reallocation in U.S. (ˆ λ in windows)

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Uncorrected vs. Corrected Gains from Reallocation

India U.S. 1985–2011 1978–2007 Mean S.D. Mean S.D. Uncorrected Gains 102% 13.7% 95.6% 53.5% Corrected Gains (Common) 65% 4.5% 28.1% 2.9% Corrected Gains (Common & Relative) 61% 4.0% 24.4% 3.0% Shrinkage 40% 71% 74% 94%

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Allocative Efficiency: U.S. relative to India

66 / 68

Conclusion

Propose way to estimate true dispersion of marginal products

◮ Projects measured marginal products on average products ◮ Requires measurement error be additive and uncorrelated with

distortions, productivity

Implemented on Indian ASI:

◮ Marginal products are 1

2 as dispersed as average products

◮ Potential gains from reallocation reduced by 2

5

◮ Time-series volatility reduced by 2

3

Implemented on U.S. ASM:

◮ Eliminates sharp downward trend in allocative efficiency ◮ Leaves U.S. allocative efficiency higher than in India 67 / 68

Next Steps

Why did measurement error get worse in the U.S.? Analyze cross-sector distortions Relate corrected wedges to size and policies

68 / 68