On the Measure of Distortions Hugo Hopenhayn May 11, 2012 1 / 38
Introduction � Recent literature emphasis on inter- fi rm distortions in the allocation of inputs � Firm-speci fi c wedges � Understand mapping between distortions and aggregate productivity? � On the inference from data � Identi fi cation of distortions and structure: role of curvature 2 / 38
Understanding fi rm level distortions: The undistorted economy � Simpli fi ed Lucas style model � Production function: y i = e i n η i � N total labor endowment inelastically supplied � Optimal allocation: � � 1 � ln n i = a 0 + e i 1 − η � y i / n i = y / n = a � (except if fi xed costs as in Bartelsman, Haltiwanger and Scarpetta, 2011) 3 / 38
Aggregation � Aggregate production function � Homogeneous of degree one in fi rms (given distribution) and labor AM 1 − η N η y = � 1 − η 1 � 1 − η A = Ee i 4 / 38
The Distorted Economy � y i / n i not equated across fi rms � Two types of distortions: � uncorrelated � correlated 5 / 38
Wedges as primitives: Restuccia-Rogerson � Let a fi rm’s pro fi ts be ( 1 − τ i ) y i − wl i − rk i ,where τ i denotes an implicit sales tax. % Estab. taxed Uncorrelated Correlated τ t τ t 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 6 / 38
Mapping Distortions into Aggregate TFP: Example � two type of fi rms e 1 = 2, e 2 = 1. � 16 fi rms each type � η = 1 2 � N = 2000 1 1 − 1/2 = 4. � Optimal: n 1 / n 2 = 2 � n 1 = 100 and n 2 = 25 16 e 1 n 1/2 + 16 e 2 n 1/2 y = 1 2 = 16 × 2 × 10 + 16 × 1 × 5 = 400 7 / 38
Example: distorted economy 1 � 12 of type 2 fi rms excluded from production ( τ = 1 ) and 4 get 100 workers each ( τ = − 1 ) � Total employment doesn’t change 16 e 1 n 1/2 + 4 e 2 n 1/2 y = 1 2 360 = 8 / 38
Example: distorted economy 2 � 3 type 1 fi rms excluded from production ( τ = 1 ) and one gets 400 workers ( τ = − 1 ) � Total employment doesn’t change 12 e 1 n 1/2 + 16 e 2 n 1/2 + e 1 400 1/2 y = 1 2 360 = 9 / 38
Example: distorted economy 3 � 12 fi rms of type 2 fi rms excluded from production � 1 fi rm of type 2 given 400 workers. � No change in total employment 15 e 1 n 1/2 + 4 e 2 n 1/2 + e 1 400 1/2 y = 1 2 360 = � All three distortions have same effect on TFP (10% drop) 10 / 38
Example: conclusion � What is common to all examples? � Some fi rms excluded from production (hit by τ = 1 ) � Some fi rms 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 11 / 38
A measure of distortions � Distortions result in deviations of output from optimal: 1 1 − η n ( e ) n ( τ , e ) = ( 1 − τ ) 1 1 − η distorted employment is θ n ( e ) . � Using θ = ( 1 − τ ) � N ( θ ) is the measure of total original employment that was distorted with some θ � ≤ θ . � It is silent about the productivity of the fi rms underlying these distortions. � It integrates to total employment ˆ N = dN ( θ ) . 12 / 38
Examples N 2000 1900 300 � 0 1 4 13 / 38
The measure of distortions and TFP e ( θ n ( e )) η d µ ( θ , e ) � First de fi ne total output: y = ´ � recall y ( e ) = an ( e ) � y ( θ , e ) = e ( θ n ( e )) η = θ η y ( e ) = a θ η n ( e ) ˆ θ η n ( e ) d µ ( θ , e ) y a = ˆ θ η dN ( θ ) a = 14 / 38
TFP and the measure of distortions � We obtained our formula: ˆ θ η dN ( θ ) . y = a � Undistorted economy has N ( θ ) mass point at one. y e f f = aN � It follows that: TFP y = 1 ˆ θ η dN ( θ ) = TFP e f f y e f f N � The effect of distortions depends on η and the distribution of distortions. 15 / 38
TFP and the concentration of distortions TFP y = 1 ˆ θ η dN ( θ ) = TFP e f f y e f f N � dN ( θ ) / N is a probability measure � Mean preserving spreads in this measure reduce TFP / TFP e f f � And mean preserving spreads give rise to the same aggregate employment! ˆ N = θ dN ( θ ) � Mean preserving spreads ⇐ ⇒ more concentrated distortions. 16 / 38
Examples of mean preserving spreads Uncorrelated taxes to larger fi rms are worse for productivity than for smaller fi rms (holding the number of fi rms affected constant) � Increasing the variance of the θ � s for large fi rms will put more employment at the tails than if done for small fi rms � But might not be so when taking into account that there are more small fi rms. � It all depends on the share of employment of small vs. large fi rms. 17 / 38
Explaining results in Restuccia and Rogerson � Let N t employment affected by θ t < 1 and rest ( N s ) with subsidy θ s N t θ t + N s θ s N = N t N θ t + N s 1 = N θ s αθ t + ( 1 − α ) θ s 1 = � Increasing α corresponds to a mean preserving spread : 1- ' α � t s 18 / 38
Inference from the size distribution of fi rms � Size distribution of fi rms vary a lot across economies � "missing middle" of underdeveloped economies � Too many small fi rms 19 / 38
Size distribution and distortions � What inferences can be made by comparing size distributions? � Nothing in general: distortions might not be revealed in size distribution � Can generate the ef fi cient distribution by taxing all ef fi cient fi rms out of the market and a distribution of taxes across the most inef fi cient ones that replicates the ef fi cient 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. 20 / 38
A lower bound on distortions � Take ef fi cient distribution of fi rm sizes with cdf F ( n ) � Distorted economy with cdf D ( m ) � Equivalent formulation consider P ( dm , dn ) and θ = m n � Optimization problem: 1 ˆ � m � η max nP ( dm , dn ) ¯ n n P ( dm , dn ) subject to: D ( dm ) = P ( dm , N ) F ( dn ) P ( N , dn ) = � Objective same as: m η n 1 − η 21 / 38
Solution � Assortative matching – no rank reversals � Identi fi es unique solution function θ ( n ) de fi ned implictly by: F ( n ) = D ( θ n ) TFP = 1 ˆ θ ( n ) η ndF ( n ) ¯ TFP e f f n θ ( n ) 1 − η ndF ( n ) � Note: reversing distributions get 1 ´ ¯ n 22 / 38
Computing lower bound on distortions � 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: 23 / 38
Scaling � Question asked: lower bound on how far India is from its frontier given its number of fi rms M and N � Y i = A i M 1 − η N η n η i = A i M i ¯ i i � Let γ = ¯ n u / ¯ n d TFP e = γ η TFP d ˆ m ( z ) η z 1 − η dF ( z ) n u ¯ � G ( m ( z )) = F ( z ) 24 / 38
India, China, Mexico vs US � average sizes: Mexico=15, India=50, US=272, China=558. 25 / 38
Measures of distortion 26 / 38
Results: TFP ratios 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 27 / 38
Size Distribution with distortions in R&R 28 / 38
Measure of distortions and lower bound on distortions 29 / 38
Results for R&R and rank reversals 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 fi rm with 2 employees in the undistorted economy has approximately 1,000 in the distorted one � a fi rm with an original employment of 9,000 employees ends up with less than 300 � ln θ 10 − ln θ 90 = 9.6 in RR and only 4 in lower bound. 30 / 38
Dispersion of Ln θ India (94) China (98) US Percentiles 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 31 / 38
Wedges, curvature and productivity � HK use η = 0.5 others η = 0.85. How does curvature affect the impact of distortions? � Relationship subtle � Two perspectives: � Curvature for fi xed measure of distortions � Curvature and the inference of the measure of distortions 32 / 38
Curvature for fi xed measure of distortions θ η dN ( θ ) ´ TFP d = TFP e N � Ratio <1 (by Jensen’s inequality) � Ratio =1 when η = 0 and η = 1. � Non-monotonic relationship � India: 1/8 1/4 1/3 1/2 2/3 3/4 7/8 η ratio 0.967 0.944 0.934 0.928 0.937 0.947 0.969 33 / 38
Curvature and the measurement of distortions (H-K) � Distribution of productivities depends on η � Implicit distortions also vary with η � Data: ( n 1 , y 1 , n 2 , y 2 , ...., n M y M ) � Production function y i = e i n η i � Given parameter η , solve for e i and do counterfactuals. 34 / 38
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