On the Measure of Distortions Hugo Hopenhayn May 11, 2012 1 / 38 - - PowerPoint PPT Presentation

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On the Measure of Distortions

Hugo Hopenhayn

May 11, 2012

Introduction

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Recent literature emphasis on inter-firm distortions in the allocation of

inputs

Firm-specific wedges Understand mapping between distortions and aggregate productivity? On the inference from data Identification of distortions and structure: role of curvature

Understanding firm level distortions: The undistorted economy

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Simplified Lucas style model Production function: yi = einη

i

N total labor endowment inelastically supplied Optimal allocation:

ln ni = a0+

• 1

1−η

• ei

yi/ni = y/n = a (except if fixed costs as in Bartelsman, Haltiwanger and Scarpetta,

2011)

Aggregation

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Aggregate production function Homogeneous of degree one in firms (given distribution) and labor

y

=

AM1−ηNη A

=

• Ee

1 1−η

i

1−η

The Distorted Economy

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yi/ni not equated across firms Two types of distortions:

uncorrelated correlated

Wedges as primitives: Restuccia-Rogerson

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Let a firm’s profits be (1 − τi) yi − wli − rki,where τi denotes an implicit

sales tax. % Estab. taxed Uncorrelated Correlated τt τt 0.2 0.4 0.2 0.4 90% 0.84 0.74 0.66 0.51 50% 0.96 0.92 0.80 0.69 10% 0.99 0.99 0.92 0.86

potentially large effects More when larger fraction taxed More when correlated

Mapping Distortions into Aggregate TFP: Example

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two type of firms e1 = 2, e2 = 1. 16 firms each type η = 1

2

N = 2000 Optimal: n1/n2 = 2

1 1−1/2 = 4.

n1 = 100 and n2 = 25

y

=

16e1n1/2

1

+ 16e2n1/2

2

=

16 × 2 × 10 + 16 × 1 × 5 = 400

Example: distorted economy 1

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12 of type 2 firms excluded from production (τ = 1) and 4 get 100 workers

each (τ = −1)

Total employment doesn’t change

y

=

16e1n1/2

1

+ 4e2n1/2

2

=

360

Example: distorted economy 2

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3 type 1 firms excluded from production (τ = 1) and one gets 400 workers

(τ = −1)

Total employment doesn’t change

y

=

12e1n1/2

1

+ 16e2n1/2

2

+ e14001/2 =

360

Example: distorted economy 3

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12 firms of type 2 firms excluded from production 1 firm of type 2 given 400 workers. No change in total employment

y

=

15e1n1/2

1

+ 4e2n1/2

2

+ e14001/2 =

360

All three distortions have same effect on TFP (10% drop)

Example: conclusion

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What is common to all examples?

Some firms excluded from production (hit by τ = 1) Some firms had employment multiplied by 4 (τ = −1)

Fraction of original employment hit by τ = 1 and by τ = −1 is the same

accross examples:

Example 1: τ = 1 : 12 ∗ 25 = 300 and τ = −1 : 4 ∗ 25 = 100 Example 2: τ = 1 : 3 ∗ 100 = 300 and τ = −1 : 1 ∗ 100 = 100 Example 3: τ = 1 : 12 ∗ 25 = 300 and τ = −1 : 1 ∗ 100 = 100

A measure of distortions

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Distortions result in deviations of output from optimal:

n (τ, e) = (1 − τ)

1 1−η n (e)

Using θ = (1 − τ)

1 1−η distorted employment is θn (e).

N (θ) is the measure of total original employment that was distorted with

some θ ≤ θ.

It is silent about the productivity of the firms underlying these distortions. It integrates to total employment

N = ˆ dN (θ) .

Examples

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300 1900 2000 1 4

• N

The measure of distortions and TFP

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First define total output: y =

´ e (θn (e))η dµ (θ, e)

recall y (e) = an (e) y (θ, e) = e (θn (e))η = θηy (e) = aθηn (e)

y

=

a ˆ θηn (e) dµ (θ, e)

=

a ˆ θηdN (θ)

TFP and the measure of distortions

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We obtained our formula:

y = a ˆ θηdN (θ) .

Undistorted economy has N (θ) mass point at one.

ye f f = aN

It follows that:

TFP TFPe f f

=

y ye f f

= 1

N ˆ θηdN (θ)

The effect of distortions depends on η and the distribution of distortions.

TFP and the concentration of distortions

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TFP TFPe f f

=

y ye f f

= 1

N ˆ θηdN (θ)

dN (θ) /N is a probability measure Mean preserving spreads in this measure reduce TFP/TFPe f f And mean preserving spreads give rise to the same aggregate employment!

N = ˆ θdN (θ)

Mean preserving spreads ⇐

⇒ more concentrated distortions.

Examples of mean preserving spreads

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Uncorrelated taxes to larger firms are worse for productivity than for smaller firms (holding the number of firms affected constant)

Increasing the variance of the θs for large firms will put more employment

at the tails than if done for small firms

But might not be so when taking into account that there are more small

firms.

It all depends on the share of employment of small vs. large firms.

Explaining results in Restuccia and Rogerson

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Let Nt employment affected by θt < 1 and rest (Ns) with subsidy θs

Ntθt + Nsθs

=

N Nt N θt + Ns N θs

=

1 αθt + (1 − α) θs

=

1

Increasing α corresponds to a mean preserving spread :

• s

t

α 1-

'

Inference from the size distribution of firms

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Size distribution of firms vary a lot across economies "missing middle" of underdeveloped economies Too many small firms

Size distribution and distortions

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What inferences can be made by comparing size distributions? Nothing in general: distortions might not be revealed in size distribution

Can generate the efficient distribution by taxing all efficient firms out

• f the market and a distribution of taxes across the most inefficient
• nes that replicates the efficient size distribution.

Special case:

Both economies with same underlying G (de) One of the economies with no distortions Can easily calculate lower bound on distortions for other economy.

A lower bound on distortions

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Take efficient distribution of firm sizes with cdf F (n) Distorted economy with cdf D (m) Equivalent formulation consider P (dm, dn) and θ = m

n

Optimization problem:

max

P(dm,dn)

1 ¯ n ˆ m n η nP (dm, dn) subject to: D (dm)

=

P (dm, N) F (dn)

=

P (N, dn)

Objective same as: mηn1−η

Solution

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Assortative matching – no rank reversals Identifies unique solution function θ (n) defined implictly by:

F (n) = D (θn) TFP TFPe f f

= 1

¯ n ˆ θ (n)η ndF (n)

Note: reversing distributions get 1

¯ n

´ θ (n)1−η ndF (n)

Computing lower bound on distortions

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Issue: countries might not have same employment If average size is the same, not a problem (homogeneous degree one in

N, M)

But average size might not be the same:

Scaling

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Question asked: lower bound on how far India is from its frontier given its

number of firms M and N

Yi = AiM1−η

i

Nη

i = AiMi ¯

nη

i

Let γ = ¯

nu/ ¯ nd TFPd TFPe = γη ¯ nu ˆ m (z)η z1−ηdF (z)

G (m (z)) = F (z)

India, China, Mexico vs US

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average sizes: Mexico=15, India=50, US=272, China=558.

Measures of distortion

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Results: TFP ratios

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η = 1/2 η = 2/3 η = 0.85 China 0.991 0.992 0.995 India 0.928 0.937 0.964 Mexico 0.931 0.939 0.966 USA mean 0.655 0.693 0.817

Size Distribution with distortions in R&R

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Measure of distortions and lower bound on distortions

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Results for R&R and rank reversals

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η = 1/2 η = 2/3 η = 0.85 India 0.928 0.937 0.964 R&R bound 0.851 0.873 0.929 R&R 0.51

a firm with 2 employees in the undistorted economy has approximately

1,000 in the distorted one

a firm with an original employment of 9,000 employees ends up with less

than 300

ln θ10 − ln θ90 = 9.6 in RR and only 4 in lower bound.

Dispersion of Ln θ

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Percentiles India (94) China (98) US H-K Bound H-K Bound RR Bound SD 4.5 0.5 4.9 0.3 75-25 5.4 0.7 6.3 0.4 90-10 10.7 1.1 12.4 0.5 9.6 4

Wedges, curvature and productivity

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HK use η = 0.5 others η = 0.85. How does curvature affect the impact of

distortions?

Relationship subtle Two perspectives:

Curvature for fixed measure of distortions Curvature and the inference of the measure of distortions

Curvature for fixed measure of distortions

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TFPd TFPe

=

´ θηdN (θ) N

Ratio <1 (by Jensen’s inequality) Ratio =1 when η = 0 and η = 1. Non-monotonic relationship India:

η 1/8 1/4 1/3 1/2 2/3 3/4 7/8 ratio 0.967 0.944 0.934 0.928 0.937 0.947 0.969

Curvature and the measurement of distortions (H-K)

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Distribution of productivities depends on η Implicit distortions also vary with η Data: (n1, y1, n2, y2, ...., nMyM) Production function yi = einη

i

Given parameter η, solve for ei and do counterfactuals.

TFP gains

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Aggregate TFP in economy: TFP =

y nη

Efficient: TFPe = ∑

• e

1 1−η

i

1−η Substitute measured ei TFPe

=

 ∑

• yi

nη

i

• 1

1−η

 

1−η

TFPe TFP

=

  ∑  

yi nη

i

y nη

 

1 1−η 

 

1−η

=

 ∑ ni n yi

ni y n

• 1

1−η 



1−η

TFP gains

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TFPe TFP

=

ni n (LPRi)

1 1−η

1−η where LPRi

=

yi/ni y/n TFPe TFP

• 1

1−η

= ∑

ni n (LPR1i)

1 1−η

• Proposition. TFPe/TFP is the certainty equivalent of the lottery

ni

N , LPRi

• with

CRRA −η

1−η. It is thus incresing in η.

Extreme: equal to one when η = 0.

Example: sensitivity to curvature

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Suppose n1/n = n2/n = 1/2

η 0.2 0.5 0.8 0.95 relative yi/ni 0.2 1.09 1.28 1.57 1.74 0.4 1.05 1.17 1.39 1.55 0.6 1.02 1.08 1.22 1.35 0.8 1.01 1.02 1.07 1.16 1 1 1 1 1 .

Final remarks

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What matters for aggregate TFP is the concentration of distortions Correlation with size/efficiency not as important Develop a method to recover lower bound of distortions from size

distributions

Lower bounds very small Effects of distortions sensitive to curvature In HK effects increase with curvature