notes on technology, demand, and the size distribution of firms

SALES AND MARKUP DISPERSION:THEORY AND EMPIRICS

Monika Mrazova

Geneva and CEPR

J. Peter Neary

Oxford, CEPR and CESifo

Mathieu Parenti

ULB and CEPR

DEGIT XXIUniversity of Nottingham

September 1, 2016

MNP (Geneva, Oxford, ULB) Sales and Markup Dispersion DEGIT XXI: Sept. 1, 2016 1 / 62

Introduction

Models of Heterogeneous Agents

Distributionof Agent

Characteristics

Distributionof

Outcomes

Model ofIndividualBehaviour

Income distribution theory

Optimal income taxation

Urban economics

International trade: Heterogeneous firms

Introduction

Characteristics

Distributionof

Outcomes

Urban economics

Introduction

Characteristics

Distributionof

Outcomes

Urban economics

Introduction

Models of Heterogeneous Firms

ProductivityDistribution

SalesDistribution

Demands

Technology

Introduction

SalesDistribution

Demands

Technology“Dixit-Stiglitz”

(f, c)

Introduction

SalesDistributionDemands

Introduction

The Canonical Model of Heterogeneous Firms

ParetoProductivities

ParetoSales

CESDemands

Introduction

The Canonical Result: HMY-Chaney

ParetoSales

CESDemands+

Introduction

Problems with The Canonical Model

Canonical Model:

Pareto distribution of productivities

CES demands

Pareto distribution of sales

Justified on the grounds that:

Distribution of firm sales is close to Pareto ... in the upper tail

Theoretical tractability

Distribution of firm sales is not necessarily Pareto

Head, Mayer and Thoenig (2014) show the result holds for log-normal

CES implies a Dirac distribution of mark-ups (no competition effects)

Introduction

Canonical Model:

CES demands

Introduction

Canonical Model:

CES demands

Introduction

Beyond the Canonical Result: Head-Mayer-Thoenig

Log-normalProductivities

Log-normalSales

CESDemands+

Introduction

Canonical Model:

CES demands

Introduction

Empirical Evidence on Mark-Up Distributions I

Figure 3: Distribution of Prices in 1989 and 1997

−.4 −.2 0 .2 .4Log Prices

1989 1997

Sample only includes firm−product pairs present in 1989 and 1997.Outliers above and below the 3rd and 97th percentiles are trimmed.

Distribution of Prices

Figure 4: Distribution of Markups and Marginal Costs in 1989 and 1997

−1 −.5 0 .5 1Log Markups

1989 1997

Distribution of Markups

−1 −.5 0 .5 1Log Marginal Costs

1989 1997

Distribution of Marginal Costs

Appendix

A A Formal Model of Input Price Variation

This appendix provides a formal economic model that rationalizes the use of a exible polynomial

in output price, market share and product dummies to control for input prices. The model is a

more general version of the models considered in Kremer (1993) and Verhoogen (2008).

We proceed in the following steps. We rst show that under the assumptions of the model, the

From: de Loecker, Goldberg, Khandelwal and Pavcnik (2014)

Introduction

Empirical Evidence on Mark-Up Distributions II

From: Lamorgese, Linarello and Warzynski (2014)

Theory

Introduction

Our Contribution

Characterize exact implications of alternative assumptions about:Demand

Distribution of firm productivities

. . . For:Distribution of firm sales

Distribution of mark-ups

Quantify the “goodness of fit” of inexact assumptions using:

Indian data on sales and markups (Thanks to Jan de Loecker)

French data on exports (Thanks to Julien Martin)

Applications:New family of distributions that nests Pareto, log-normal and Frechet

New demand function: “CREMR” - “Constant Revenue Elasticity ofMarginal Revenue”

Summary of Results

Introduction

Our Contribution

Summary of Results

Introduction

Our Contribution

Summary of Results

Introduction

Why Should We Care?

Distributions of sales and mark-ups matter for:

Interpretation of trade elasticities

Chaney (2008), Arkolakis et al. (2013)

Granular origins of aggregate fluctuations

Gabaix (2011), Di Giovanni and Levchenko (2015)

Quantifying the misallocation of resources

Hsieh and Klenow (2009)

Introduction

Related Work I

Canonical Model:

Melitz (2003), Helpman-Melitz-Yeaple (2004), Chaney (2006)

Mark-Ups Increasing in Firm Size/Competition Effects:

ZKPT (2013), Bertoletti-Epifani (2014): Only with subconvex demand

Proliferation of subconvex alternatives to CES:

Quadratic preferences: Melitz-Ottaviano (2008)Stone-Geary LES: Simonovska (2010)Translog: Feenstra-Weinstein (2010)Negative exponential/CARA: Behrens-Murata (2007)Bulow-Pfleiderer: Atkin-Donaldson (2012)QMOR: Feenstra (2014)APT and Laplace Transform: Fabinger-Weyl (2015)Mrazova-Neary (2013)

Introduction

Related Work II

Size Distribution of Sales - Theory:

Pareto: Helpman-Melitz-Yeaple (2004), Chaney (2006)Log-normal: Head-Mayer-Thoenig (2014)Frechet: Eaton-Kortum (2002), Tintelnot (2014)

Size Distribution of Sales - Empirics:

Pareto: Axtell (2001), Corcos-del Gatto-Mion-Ottaviano (2012)Mixture of thin- and fat-tailed Pareto: Edmonds et al. (2015)Log-normal: Bee-Schiavo (2014)Piecewise log-normal-Pareto: Luttmer (2007), Eaton et al. (2011)Mixture of log-normal and Pareto: Combes et al. (2014)

Distribution of Mark-Ups:

Lamorgese, Linarello and Warzynski (2014)Edmonds et al. (2015)De Loecker, Goldberg, Khandelwal and Pavcnik (2015)

Introduction

Related Work III

Information theory: Shannon (1948), Kullback-Leibler (1951)

Economic applications of Shannon Entropy:

Inequality: Theil (1967)

Macro: Sims (2003) on rational inattention

Decision theory: Maccheroni-Marinacci-Rustichini (2006),Cabrales-Gossner-Serrano (2013)

Trade: Dasgupta-Mondria (2014)

Economic applications of Kullback-Leibler Divergence:

Econometrics: Vuong (1989), Cameron-Windmeijer (1997), Ullah(2002)

Consumer economics: Adams (2014)

Introduction

Outline

1 Introduction

2 General Results

3 Backing Out Demands

4 Inferring Sales and Mark-Up Distributions

5 From Theory to Calibration

6 Empirics

7 Conclusion

General Results

Outline

1 Introduction

2 General Results

6 Empirics

7 Conclusion

General Results

Proposition 1: A General Characterization Result

Characteristicx G(x)

Characteristicy F(y)

Firm Behaviourx = v(y)

Proposition 1: Any two imply the third

Proposition 2 Section 3 Section 4 Sections 5 and 6

General Results

Proposition 1: Formal Statement

Assumption A1: i, x(i), y(i) ∈[Ω× (x, x)× (y, y)

], where Ω is the set

of firms, with both x(i) and y(i) monotonically increasing functions of i.

Proposition 1: Given A1, any two of (1)–(3) imply the third: Proof

1 x is distributed with CDF G(x), where g(x) = G′(x) > 0;

2 y is distributed with CDF F (y), where f(y) = F ′(y) > 0;

3 x = v(y), v′(y) > 0;

where:

(i) (1) + (3) ⇒ (2) with F (y) = G[v(y)] and f(y) = g[v(y)]v′(y).

(ii) (1) + (2) ⇒ (3) with v(y) = G−1[F (y)].

Part (i) is standard; part (ii) is not, and requires Assumption 1.[Matzkin (2003)]

General Results

3 x = v(y), v′(y) > 0;

where:

(ii) (1) + (2) ⇒ (3) with v(y) = G−1[F (y)].

General Results

3 x = v(y), v′(y) > 0;

where:

(ii) (1) + (2) ⇒ (3) with v(y) = G−1[F (y)].

General Results

3 x = v(y), v′(y) > 0;

where:

(ii) (1) + (2) ⇒ (3) with v(y) = G−1[F (y)].

General Results

Proposition 2

Characteristicx G(x; )

Characteristicy G[h(y); ]

Firm Behaviourx = x0h(y)E

Proposition 2: Any two imply the third

Proposition 1 Section 3 Section 4 Sections 5 and 6

General Results

1 x has CDF: G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0

2 y has CDF: F (y; θ′) = G [h(y); θ′] = H(θ0 +

θ′1θ′2h(y)θ

), Fy > 0

3 x = x0h(y)E

where:

(i) (1) + (3) ⇒ (2) with θ′1 = Eθ1xθ20 and θ′2 = Eθ2.

(ii) (1) + (2) ⇒ (3) with E =θ′2θ2

and x0 =(θ2θ1

θ′1θ′2

) 1θ2 . KLD

“Generalized Power Function” [“GPF”] Family of Distributions:

Includes: Pareto, truncated Pareto, log-normal, Frechet, etc. Details

All with 2 free parameters

H(·), h(·): Completely general, except sgn(H ′) = sgn(θ1) and h′ > 0

General Results

1 x has CDF: G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0

θ′1θ′2h(y)θ

), Fy > 0

3 x = x0h(y)E

where:

(ii) (1) + (2) ⇒ (3) with E =θ′2θ2

and x0 =(θ2θ1

θ′1θ′2

) 1θ2 . KLD

General Results

1 x has CDF: G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0

θ′1θ′2h(y)θ

), Fy > 0

3 x = x0h(y)E

where:

(ii) (1) + (2) ⇒ (3) with E =θ′2θ2

and x0 =(θ2θ1

θ′1θ′2

) 1θ2 . KLD

General Results

1 x has CDF: G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0

θ′1θ′2h(y)θ

), Fy > 0

3 x = x0h(y)E

where:

(ii) (1) + (2) ⇒ (3) with E =θ′2θ2

and x0 =(θ2θ1

θ′1θ′2

) 1θ2 . KLD

General Results

1 x has CDF: G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0

θ′1θ′2h(y)θ

), Fy > 0

3 x = x0h(y)E

where:

(ii) (1) + (2) ⇒ (3) with E =θ′2θ2

and x0 =(θ2θ1

θ′1θ′2

) 1θ2 . KLD

General Results

1 x has CDF: G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0

θ′1θ′2h(y)θ

), Fy > 0

3 x = x0h(y)E

where:

(ii) (1) + (2) ⇒ (3) with E =θ′2θ2

and x0 =(θ2θ1

θ′1θ′2

) 1θ2 . KLD

General Results

Proposition 2: Special Cases

Proposition 2:

1 x has CDF: G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0

θ′1θ′2h(y)θ

), Fy > 0

3 x = x0h(y)E

Two special cases of h(y):

h(y) = y: “Self-reflection” → Distributions of sales and output

h(y) = y1−y : “Odds reflection” → Distributions of mark-ups Details

General Results

Proposition 2:

1 x has CDF: G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0

θ′1θ′2h(y)θ

), Fy > 0

3 x = x0h(y)E

General Results

Proposition 2:

1 x has CDF: G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0

θ′1θ′2h(y)θ

), Fy > 0

3 x = x0h(y)E

Backing Out Demands

Outline

1 Introduction

2 General Results

6 Empirics

7 Conclusion

Backing Out Demands

Section 3: Backing Out Demands

Productivity G(; )

Salesr G[r; ]

Demand

Proposition 1 Proposition 2 Section 4 Sections 5 and 6

Backing Out Demands

Which Demands are Consistent with Self-Reflection?

GPFProductivities

GPFSales

CREMR: = 0rE

Corollary 1: Any two imply the third

Compare Corollary 3

Backing Out Demands

CREMR Demands

From Proposition 2: Self-reflection IFF ϕ = ϕ0rE

Implications for demand? We need more assumptions:

MR=MC ⇒ ϕ ≡ c−1 = (r′)−1

All firms face the same residual demand function: r(x) = xp(x)

These imply a simple differential equation:

ϕ = ϕ0rE ⇒ [r′(x)]−1 = ϕ0r(x)E

Hence “CREMR”: “Constant Revenue Elasticity of Marginal Revenue”

Integrate to get CREMR demand function: Details

p(x) =β

x(x− γ)

σ−1σ

Restrictions: 1 < σ <∞, x > γσ, β > 0

CES a special case: γ = 0 ⇒ p(x) = βx−1/σ

CREMR elasticity of demand: ε(x) = x−γx−γσσ

Compare CEMR Demands

Backing Out Demands

CREMR Demands

MR=MC ⇒ ϕ ≡ c−1 = (r′)−1

ϕ = ϕ0rE ⇒ [r′(x)]−1 = ϕ0r(x)E

p(x) =β

x(x− γ)

σ−1σ

Backing Out Demands

CREMR Demands

MR=MC ⇒ ϕ ≡ c−1 = (r′)−1

ϕ = ϕ0rE ⇒ [r′(x)]−1 = ϕ0r(x)E

p(x) =β

x(x− γ)

σ−1σ

Backing Out Demands

CREMR Demands

MR=MC ⇒ ϕ ≡ c−1 = (r′)−1

ϕ = ϕ0rE ⇒ [r′(x)]−1 = ϕ0r(x)E

p(x) =β

x(x− γ)

σ−1σ

Backing Out Demands

CREMR Demands

MR=MC ⇒ ϕ ≡ c−1 = (r′)−1

ϕ = ϕ0rE ⇒ [r′(x)]−1 = ϕ0r(x)E

p(x) =β

x(x− γ)

σ−1σ

Backing Out Demands

Properties of CREMR Demand Functions

σ: Elasticity parameter

Lower bound for demand elasticity (when γ > 0)

“REMR” E depends only on σ: E = 1σ−1 [Chaney: CES]

ϕ Pareto with shape k

⇒ r Pareto with shape k

σ−1

“Normal” selection effects: Profit function “supermodular” IFF σ ≥ 2

γ: Markup parameter

CREMR nests CES when γ = 0

For any γ, CREMR converges to CES as x→∞Variable mark-ups: For any γ 6= 0

Competition effects: “Subconvex” IFF γ ≥ 0

Utility function is analytic and can be simulated Utility Function

Very different from standard demand functions

Illustrations CREMR Demand Manifold Other Corollaries

Backing Out Demands

σ−1

Backing Out Demands

σ−1

Backing Out Demands

σ−1

Inferring Sales and Mark-Up Distributions

Outline

1 Introduction

2 General Results

4 Inferring Sales and Mark-Up DistributionsMarkups with CREMR DemandsOther Sales and Mark-Up Distributions

6 Empirics

7 Conclusion

Inferring Sales and Mark-Up Distributions

Section 4: Inferring Sales and Mark-Up Distributions

Productivity G()

Sales: r Fr(r)Markups: m Fm(m)

Demandp = p(x)

Proposition 1 Proposition 2 Section 3 Sections 5 and 6

Inferring Sales and Mark-Up Distributions Markups with CREMR Demands

Markups with CREMR Demands

Mark-ups in general: m(x) ≡ pc = ε(x)

ε(x)−1

CREMR mark-up: ε(x) = x−γx−γσσ ⇒ m(x) = x−γ

xσσ−1

With subconvex demands (γ > 0), larger firms have higher markups:

m(x) ∈[m, σ

σ−1

]as x ∈ [x,∞]

Define the relative mark-up: m ≡ mm = σ−1

σ m ∈ [m, 1]

So: m(x) = x−γx

Productivity and markups: Details

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

Satisfies Proposition 2’s conditions for “Odds Reflection” Recall

ε(x)−1

xσσ−1

m(x) ∈[m, σ

σ−1

]as x ∈ [x,∞]

σ m ∈ [m, 1]

So: m(x) = x−γx

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

ε(x)−1

xσσ−1

m(x) ∈[m, σ

σ−1

]as x ∈ [x,∞]

σ m ∈ [m, 1]

So: m(x) = x−γx

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

ε(x)−1

xσσ−1

m(x) ∈[m, σ

σ−1

]as x ∈ [x,∞]

σ m ∈ [m, 1]

So: m(x) = x−γx

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

ε(x)−1

xσσ−1

m(x) ∈[m, σ

σ−1

]as x ∈ [x,∞]

σ m ∈ [m, 1]

So: m(x) = x−γx

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

ε(x)−1

xσσ−1

m(x) ∈[m, σ

σ−1

]as x ∈ [x,∞]

σ m ∈ [m, 1]

So: m(x) = x−γx

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

Markups with CREMR and GPF Productivity

From Proposition 2:

GPF Productivity + CREMR demand ⇒ “GPF-Odds” markups

Examples:

1 Pareto productivity G (ϕ) = 1−(ϕ

)k⇒ “Pareto-Odds” markups: F (m) = 1−

)n′ (m

)−n′2 Log-Normal productivity G(ϕ) = Φ

[1s logϕ− µ

]⇒ “Log-Normal-Odds” markups: F (m) = Φ

[1s′

1−m − µ′]

3 Frechet productivity ⇒ “Frechet-Odds” markups

From Proposition 2:

Examples:

)n′ (m

)−n′

2 Log-Normal productivity G(ϕ) = Φ[

1s logϕ− µ

[1s′

1−m − µ′]

From Proposition 2:

Examples:

)n′ (m

[1s logϕ− µ

[1s′

1−m − µ′]

From Proposition 2:

Examples:

)n′ (m

[1s logϕ− µ

[1s′

1−m − µ′]

The Log-Normal-Odds Distribution of Mark-UpsIntroduction From theory to data Empirics

Logit-normal

a.k.a. “Logit-Normal” [Johnson (1949), Mead (1965)]

Matches empirical mark-up distribution Recall

Inferring Sales and Mark-Up Distributions Other Sales and Mark-Up Distributions

Other Sales and Markup Distributions

p(x) or x(p) ϕ(r) or ϕ(r) ϕ(m) or ϕ(m)

CREMR βx (x− γ)

σ−1σ ϕ0r

1σ−1 ϕ0

Linear α− βx 1α

) 12 2m−1

LES δx+γ γδ

)2γδm

Translog 1p (γ − η log p) ϕ0 exp

)m exp

(m− η+γ

Productivity as a Function of Sales and MarkupsFor Selected Demand Functions

Empirics 1 Empirics 2

Inferring Sales and Mark-Up Distributions Other Sales and Mark-Up Distributions

Other Sales and Markup Distributions

p(x) or x(p) ϕ(r) or ϕ(r) ϕ(m) or ϕ(m)

CREMR βx (x− γ)

σ−1σ ϕ0r

1σ−1 ϕ0

Linear α− βx 1α

) 12 2m−1

LES δx+γ γδ

)2γδm

Translog 1p (γ − η log p) ϕ0 exp

)m exp

(m− η+γ

Productivity as a Function of Sales and MarkupsFor Selected Demand Functions

Empirics 1 Empirics 2

From Theory to Calibration

Outline

1 Introduction

2 General Results

5 From Theory to Calibration“Goodness of Fit” for DistributionsThe Kullback-Leibler Divergence

6 Empirics

7 Conclusion

From Theory to Calibration

Sections 5 and 6: Confronting Theory and Data

Data:Sales: r Fr(r)

Markups: m Fm(m)

Sales: r Fr(r)Markups: m Fm(m)

Productivity G()

Demandp = p(x)

Proposition 1 Proposition 2 Section 3 Section 4

From Theory to Calibration “Goodness of Fit” for Distributions

“Goodness of Fit” for Distributions

So far: Exact characterizations linking distributions and demands

BUT: Does this matter?

..... quantitatively?

How do we quantify the cost of predicting the wrong distribution?

We use the Kullback-Leibler Divergence

Plus the QQ estimator as a robustness check

From Theory to Calibration The Kullback-Leibler Divergence

Kullback-Leibler Divergence

KLD: Measures the “information loss” or “relative entropy” when onedistribution, F , is used to approximate another, F :

DKL(F ||F

)≡∫ r

)f(r)dr

In our case:

F (r): Actual distribution of firm sales Empirics

F (r; θ): Distribution of firm sales implied by:G(ϕ): Distribution of firm productivities

ϕ(r; θ): Model of firm behaviour, parameterized by θ

Desirable features:

Axiomatic foundation in information theory Background

Statistical interpretation: Expected value of log-likelihood ratio

Links with Proposition 2: KLD can be decomposed to show that onecomponent depends on variations in the REMR Details

DKL(F ||F

)≡∫ r

)f(r)dr

In our case:

Desirable features:

DKL(F ||F

)≡∫ r

)f(r)dr

In our case:

Desirable features:

Empirics

Outline

1 Introduction

2 General Results

6 EmpiricsOperationalizing the KLDFrench Exports to GermanyIndian Sales and MarkupsRobustness: The QQ EstimatorRobustness to Truncation

7 ConclusionMNP (Geneva, Oxford, ULB) Sales and Markup Dispersion DEGIT XXI: Sept. 1, 2016 40 / 62

Empirics Operationalizing the KLD

Operationalizing the KLD

Discrete counterpart of continuous KLD: Compare QQ Estimator

DKL(F || F (· ;θ)) =

nb∑i=1

(∆Fi∆Fi

)∆Fi

∆Fi ≡ F (r+i∗b)−F (r+(i− 1)∗b) ∆Fi ≡ F (r+i∗b)−F (r+(i− 1)∗b)

nb: Number of bins of width b, dividing the support of F , [r, r]nb = 1, 000 in all cases (except nb = 40 in draft paper)French results not very sensitive to bin sizeIndian results are more sensitive

We report below DKL(F || F (· ; θ))

θ: Parameter vector that minimizes DKL(F || F (· ;θ))

Units of measurement:KLD usually measured in “bits” (log to base 2) or “nats” (natural logs)Here we present results relative to a uniform benchmark Details

Analogous to “dartboard” (Ellison-Glaeser (1997)) or “balls and bins”approaches (Armenter-Koren (2014))

DKL(F || F (· ;θ)) =

nb∑i=1

(∆Fi∆Fi

)∆Fi

DKL(F || F (· ;θ)) =

nb∑i=1

(∆Fi∆Fi

)∆Fi

DKL(F || F (· ;θ)) =

nb∑i=1

(∆Fi∆Fi

)∆Fi

Empirics French Exports to Germany

French Exports to Germany

French exports to Germany in 2005

Similar to data used by Head, Mayer, and Thoenig (2014)

We work with 161,191 firm-product observations on export sales

Produced by 27,550 firms: 5.85 products per firm

Results

A First Look at the Data: Obviously Pareto?

A Second Look at the Data: Obviously Log-normal?

(1) 50% of observations, 0.1% of sales

(3) 99.8% of firms, 50% of sales; (4) 411 ex 161,191 firms.

A Third Look at the Data: Pareto where it Matters?

(1) 50% of observations, 0.1% of sales

(1) (2)

(1) 50% of observations, 0.1% of sales; (2) 77.9% of observations, 1.0% of sales

(1) (2) (3)

(3) 99.8% of firms, 50% of sales

(1) (2) (3)

(3) 99.8% of observations, 50% of sales; (4) 411 ex 161,191 observations.

Best-Fit Pareto and Log-Normal

0 0.5 1 1.5 2 2.5 3 3.5 4

(a) All Observations

0 1 2 3 4 5 6

(b) All Bar Top 89 Observations

KLD: Pareto (green): 0.0012; Log-normal (red): 0.0001Top 89 firms: 0.05% of observations, 32% of sales

KLD for Different Specifications: French Exports

CREMR/CES Translog Linear and LES

Pareto 0.0012 0.3819 0.4711

[0.0012, 0.0013] [0.1600, 0.3826] [0.4704, 0.4715]

Log-Normal 0.0001 0.7315 0.8304

[0.0001, 0.0001] [0.4891, 0.7320] [0.7679, 0.8310]

(Bootstrapped 95% Confidence Intervals in Parentheses)

Recall Derivations KLD for Indian Data Bootstrapped Confidence Intervals Conclusion

KLD for Different Specifications: French Exports

CREMR/CES Translog Linear and LES

Pareto 0.0012 0.3819 0.4711

[0.0012, 0.0013] [0.1600, 0.3826] [0.4704, 0.4715]

Log-Normal 0.0001 0.7315 0.8304

[0.0001, 0.0001] [0.4891, 0.7320] [0.7679, 0.8310]

(Bootstrapped 95% Confidence Intervals in Parentheses)

Recall Derivations KLD for Indian Data Bootstrapped Confidence Intervals Conclusion

Empirics Indian Sales and Markups

Indian Sales and Markups

Indian sales and markups in 2001

As used by de Loecker, Goldberg, Khandelwal and Pavcnik (2015)

2,457 firm-product observations

Markups are estimated at firm-product level

Using the “production approach”: Assumes cost-minimization only

Measures markups by the deviation between the output elasticity of avariable input and that input’s share of total revenue

Does not impose assumptions on demand or market structure

KLD: Indian Sales and Markups

KLD (Sales)

KLD(Markups)

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

All results relative to the uniform benchmark

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

Log-Normal Productivity

Pareto Productivity

KLD (Sales)

KLD(Markups)

Conclusion

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

Pareto Productivity

KLD (Sales)

KLD(Markups)

TranslogTranslog

LESLES

Linear

Recall Derivations KLD for French Exports Compare QQ Conclusion

Empirics Robustness: The QQ Estimator

Robustness Check: The QQ Estimator

[Kratz-Resnick (1996), Head-Mayer-Thoenig (2014), Nigai (2016)]

QQ Estimator: Compare actual and predicted quantiles:

QQ(F || F (·;θ)) =

n∑i=1

(log qi − log qi(θ))2

n = 100 quantiles

qi = F−1(i/n): i’th quantile observed in our data

qi(θ) = F−1(i/n;θ): i’th quantile predicted by the theory Details

We report below QQ(F || F (·; θ))

θ: Parameter vector that minimizes QQ(F || F (·;θ))

Uniform benchmark: 2, 420 (sales), 90.1328 (markups)

Compare KLD

Robustness Check: The QQ Estimator

[Kratz-Resnick (1996), Head-Mayer-Thoenig (2014), Nigai (2016)]

QQ Estimator: Compare actual and predicted quantiles:

QQ(F || F (·;θ)) =

n∑i=1

(log qi − log qi(θ))2

n = 100 quantiles

qi = F−1(i/n): i’th quantile observed in our data

qi(θ) = F−1(i/n;θ): i’th quantile predicted by the theory Details

We report below QQ(F || F (·; θ))

θ: Parameter vector that minimizes QQ(F || F (·;θ))

Uniform benchmark: 2, 420 (sales), 90.1328 (markups)

Compare KLD

QQ Estimator: Indian Sales and Markups

0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0

Pareto Productivity

QQ (Sales)

QQ(Markups)

Translog

Linear

Compare KLD Conclusion

Empirics Robustness to Truncation

Robustness to Truncation

Recall French versus Indian Results:

Results with French and Indian data very similar . . .

. . . except for CREMR + Pareto sales

Why? French data are for exports, Indian for production

So: presumptively, smaller firms have been selected out of French data

To explore this, we reestimate KLD for Indian sales dropping observations:

“CREMR vs. CREMR”: Pareto versus log-normal

“CREMR vs. the Rest”: Conditional on Pareto

Pareto Does Better as We Drop Small Observations

Number of Observations Dropped

0 200 400 600 800

Pareto + CREMR

Lognormal + CREMR

KLD for CREMR demands and Indian sales, dropping smaller observations

All KLD values relative to appropriate uniform benchmark

Pareto + CREMR dominates when we drop 663 or more observations

27% of observations, 1.2% of sales

Pareto Does Better as We Drop Small Observations

0 200 400 600 800

Pareto + CREMR

Lognormal + CREMR

KLD for CREMR demands and Indian sales, dropping smaller observations

All KLD values relative to appropriate uniform benchmark

Pareto + CREMR dominates when we drop 663 or more observations

27% of observations, 1.2% of sales

Given Pareto, CREMR Also Does Better

0 20 40 60 80 100 120 140 160 180 200

Pareto + CREMR

Pareto + Linear

Pareto + Translog

Conditional on Pareto, CREMR dominates ...

Linear: When we drop 11 or more observations0.44% of observations, 0.0002% of sales

Translog: When we drop 118 or more observations4.80% of observations, 0.03% of sales

Conclusion

Outline

1 Introduction

2 General Results

6 Empirics

7 Conclusion

Conclusion

We characterize the demands that are consistent with particular distributionsof productivities, sales and markups (including Pareto, Log-normal andFrechet)

Most alternatives to CES are not

CREMR demands: Necessary and sufficient for productivity and salesto have the same distribution

CREMR + Log-normal productivity match empirical evidence ondistribution of mark-ups

Kullback-Leibler divergence allows us quantify how unrealistic is a givencombination of assumptions about productivity distribution and demands

Choice between Pareto and log-normal distributions less importantthan the choice between CREMR and other demands

Back to text

Conclusion

Back to text

Conclusion

Back to text

Conclusion

Back to text

Conclusion

Thanks and Acknowledgements

Thank you for listening. Comments welcome!

The research leading to these results has received funding from the European Research Council under the European Union’sSeventh Framework Programme (FP7/2007-2013), ERC grant agreement no. 295669. The contents reflect only the authors’views and not the views of the ERC or the European Commission, and the European Union is not liable for any use that may bemade of the information contained therein.

Conclusion

Proof of Proposition 1

(i) To show that (1) and (3) imply (2): Back to Text

Let F (y) denote the distribution of y implied by (1) and (3): F (y) = Pr [y ≤ y].

From (1) and (3), G−1 [F (y)] is strictly increasing; so:

F (y) = Pr [y ≤ y] = Pr[x ≤ G−1 [F (y)]

]Therefore:

F (y) = G[G−1 [F (y)]

]= F (y)

So, the implied distribution of y is F (y), as was to be proved.

A similar proof shows that (2) and (3) imply (1).

(ii) To show that (1) and (2) imply (3):

Pick an arbitrary firm i with characteristics x(i) and y(i).

Because x(i) and y(i) are strictly increasing in i, the masses of firms withcharacteristics below x(i) and y(i) are equal:

|Ω|Pr(x ≤ x(i)) = |Ω|Pr(y ≤ y(i)),

where |Ω| is the measure of Ω, i.e. the total mass of firms.

This holds for any firm i, so G(x(i)) = F (y(i)), ∀i ∈ Ω.

Inverting gives x(i) = G−1[F (y(i))] as required.

Conclusion

Proof of Proposition 2

(i) To show that (1) and (3) imply (2): Back to Text

Given G (x; θ) = H(θ0 + θ1

θ2xθ2)

, Gx > 0, and x = x0h(y)E . Then F (y):

= Pr [y ≤ y] = Pr[x ≤ x0h(y)E

[θ0 + θ1

x0h(y)E

θ2] = H[θ0 +

θ′1θ′2h(y)θ

]where: θ′2 = Eθ2 and θ′1 = Eθ1x

(ii) To show that (1) and (2) imply (3):

Given G (x; θ) = H(θ0 + θ1

θ2xθ2)

, F (y; θ′) = H(θ0 +

θ′1θ′2h(y)θ

), Gx, Fy > 0

From Proposition 1, x = G−1 [F (y; θ′) ; θ]

Inverting G (x; θ): θ0 + θ1θ2xθ2 = H−1 (G) → x =

[θ2θ1

H−1 (G)− θ0

] 1θ2

Now substitute F (y; θ′) for G:

x =[θ2θ1

(H(θ0 +

θ′1θ′2h(y)θ

))− θ0

] 1θ2 = x0h(y)E

where: E =θ′2θ2

and x0 =(θ2θ1

θ′1θ′2

) 1θ2

Note: θ0 must be the same in G(x; θ) and F (y; θ′).

Conclusion

Generalized Power Function Family of Distributions

G (x; θ) = H(θ0 + θ1

θ2xθ2)

Back to text

G (x; θ) Support H (z) θ0 θ1 θ2

Pareto 1−(xx

)−k[x,∞) z 1 kxk −k

Truncated Pareto1−xkx−k

1−xkx−k[x, x] z 1

1−xkx−kkxk

1−xkx−k−k

Log-normal Φ(

log x−µs

)[0,∞) Φ [log (z)] 0 1

se−µs 1

Uniformx−xx−x [x, x] z − x

x−x1

x−x 1

Frechet e−(x−µs

)−k[µ,∞) e−z

−k−µs

Gumbel e−e−(x−µs

)(−∞,∞) e−e

−z−µs

Weibull 1− e−(x−µs

)k[µ,∞) 1− e−z

k−µs

Reversed Weibull e−(µ−xs

)k(−∞, µ] e−z

− 1s

Conclusion

Properties of CREMR Demands

Back to text

p(x) = βx

(x− γ)σ−1σ

Proof that CREMR has Constant Revenue Elasticity of Marginal Revenue:

r(x) ≡ xp(x) = β (x− γ)σ−1σ

r′(x) = p(x) + xp′(x) = β σ−1σ

(x− γ)−1σ

⇒ r′(x) = βσσ−1

σ−1

σr(x)−

1σ−1

In terms of proportional changes, r ≡ d log r = drr (r > 0):

r = σ−1σ

xx−γ x

r′ = − 1σ

xx−γ x

⇒ r′ = − 1

σ − 1r

Conclusion

CREMR Preferences

Back to text

CREMR demands can be rationalized by additively separable preferences:

∫i∈Ω

ux(i) di

which implies that p(i) = λ−1u′x(i), where λ is the marginal utility of income.

u (x) =βσ

1− σ xσ−1σ 2F1

(−σ − 1

σ,−σ − 1

where:

2F1 (a, b; c; z), |z| < 1, is the hypergeometric function:

2F1(a, b; c; z) =∞∑n=0

(a)n(b)n(c)n

(q)n is the (rising) Pochhammer symbol:

(q)n =

1 n = 0q(q + 1)...(q + n− 1) n > 0

Conclusion

CREMR Preferences: Marginal Utility of Income

Back to text

In text, the marginal utility of income λ is set equal to one

We can eliminate λ by integrating over all demands

With additively separable preferences in general:∫i∈Ω

p(i)x(i) di = λ−1∫i∈Ω

x(i)u′x(i) di = I

⇒ p(i) = u′x(i)∫j∈Ωx(j)u′x(j) dj

So, with CREMR preferences:

λp (i) = u′ x (i) = β(i)x(i)

[x (i)− γ]σ−1σ

⇒ p(i) = 1x(i)

β(i)[x(i)−γ]σ−1σ∫

j∈Ωβ(j)[x(j)−γ]σ−1σ dj

Conclusion

CREMR Demand Functions

Back to text

(a) γ = 0: CES

(b) γ > 0: Subconvex

(c) γ < 0: Superconvex

Conclusion

Back to text

(a) γ = 0: CES

(c) γ < 0: Superconvex

Conclusion

Back to text

(a) γ = 0: CES

(c) γ < 0: SuperconvexMNP (Geneva, Oxford, ULB) Sales and Markup Dispersion DEGIT XXI: Sept. 1, 2016 62 / 62

Conclusion

CREMR Very Different from Other Demands

-2.0 -1.0 0.0 1.0 2.0 3.0

Linear

CARAStone-Geary

TranslogCESSM

The Demand Manifold

Compare CEMR Demand Manifolds

Back to text

-2.0 -1.0 0.0 1.0 2.0 3.0

= 2 = 6 = 3SM CES

p(x) =β

x(x− γ)

σ−1σ

⇒ ρ(ε) = 2− 1

σ − 1

(ε− 1)2

Conclusion

CREMR Very Different from Other Demands

-2.0 -1.0 0.0 1.0 2.0 3.0

Linear

CARAStone-Geary

TranslogCESSM

The Demand Manifold

Compare CEMR Demand Manifolds

Back to text

-2.0 -1.0 0.0 1.0 2.0 3.0

= 2 = 6 = 3SM CES

p(x) =β

x(x− γ)

σ−1σ

⇒ ρ(ε) = 2− 1

σ − 1

(ε− 1)2

Conclusion

A Firm’s-Eye View of Demand

Perceived inverse demand function:

p = p(x) p′ < 0

Two key demand parameters:

1 Slope/Elasticity:

ε(x) ≡ − p(x)xp′(x) > 0

2 Curvature/Convexity:

ρ(x) ≡ −xp′′(x)p′(x) R 0

Back to Text

0-2 -1 0 1 2 3

Conclusion

p = p(x) p′ < 0

1 Slope/Elasticity:

ε(x) ≡ − p(x)xp′(x) > 0

ρ(x) ≡ −xp′′(x)p′(x) R 0

Back to Text

0-2 -1 0 1 2 3

Conclusion

p = p(x) p′ < 0

1 Slope/Elasticity:

ε(x) ≡ − p(x)xp′(x) > 0

ρ(x) ≡ −xp′′(x)p′(x) R 0

Back to Text

0-2 -1 0 1 2 3

Conclusion

The Admissible Region

For a monopoly firm:

First-order condition:p+ xp′ = c ≥ 0 ⇒ ε ≥ 1

Second-order condition:2p′ + xp′′ < 0 ⇒ ρ < 2

1 01.0

0.0-2.0 -1.0 0.0 1.0 2.0 3.0

Proposition: Any well-behaved demand function can be represented by its“Demand Manifold”: a smooth curve in ε, ρ space

Back to Text

Conclusion

The Admissible Region

For a monopoly firm:

First-order condition:p+ xp′ = c ≥ 0 ⇒ ε ≥ 1

Second-order condition:2p′ + xp′′ < 0 ⇒ ρ < 2

1 01.0

0.0-2.0 -1.0 0.0 1.0 2.0 3.0

Proposition: Any well-behaved demand function can be represented by its“Demand Manifold”: a smooth curve in ε, ρ space

Back to Text

Conclusion

CES Demands

In general, both ε and ρ vary withsales

Exception: CES/iso-elastic case:

p = βx−1/σ

⇒ ε = σ, ρ = σ+1σ > 1

⇒ ε = 1ρ−1 0.0

-2.0 -1.0 0.0 1.0 2.0 3.0

Cobb-Douglas: ε = 1, ρ = 2; just on boundary of both FOC and SOC

Back to Text

Conclusion

CES Demands

In general, both ε and ρ vary withsales

Exception: CES/iso-elastic case:

p = βx−1/σ

⇒ ε = σ, ρ = σ+1σ > 1

⇒ ε = 1ρ−1 0.0

-2.0 -1.0 0.0 1.0 2.0 3.0

Cobb-Douglas

Cobb-Douglas: ε = 1, ρ = 2; just on boundary of both FOC and SOC

Back to Text

Conclusion

Other Corollaries of Proposition 2

Self-Reflection Corollaries of Proposition 2: Back to text

1 Pareto productivity and sales, CREMR demands

2 Log-normal productivity and sales, CREMR demands Details

Strictly: Range must include r = 0, which restricts CREMR to CES

3 Frechet productivity and sales, CREMR demands [Tintelnot (2014)]

4 Pareto productivity and output, CEMR demands Details

5 Log-normal productivity and output, CEMR demands, with restrictions

6 Pareto/Log-Normal/Frechet output and sales, CES demandsMNP (Geneva, Oxford, ULB) Sales and Markup Dispersion DEGIT XXI: Sept. 1, 2016 62 / 62

Conclusion

Corollary 2: Log-Normal Productivity and Sales

Log-NormalProductivities

Log-NormalSales

CREMRDemands

Back to Text

Conclusion

Corollary 3: Output Rather than Sales

ParetoOutputs

CEMRDemands

Back to Text Recall Corollary 1

Conclusion

CEMR Demands

“Constant (Output) Elasticity of Marginal Revenue” demand function:

p(x) =1

(α+ βx

σ−1σ

CES a special case: α = 0 ⇒ p(x) = βx−1σ

“CEMR”:

r′(x) = βσ − 1

σx−

1σ ⇒ r′ = − 1

Same as inverse “PIGL” demands[Muellbauer (1975), Mrazova-Neary (2011, 2013)]

σ → 1 ⇒ Inverse Translog: p(x) = 1x (α′ + β′ log x)

Back to Text Recall CREMR Demands

Conclusion

CEMR Demands

“Constant (Output) Elasticity of Marginal Revenue” demand function:

p(x) =1

(α+ βx

σ−1σ

CES a special case: α = 0 ⇒ p(x) = βx−1σ

“CEMR”:

r′(x) = βσ − 1

σx−

1σ ⇒ r′ = − 1

Same as inverse “PIGL” demands[Muellbauer (1975), Mrazova-Neary (2011, 2013)]

σ → 1 ⇒ Inverse Translog: p(x) = 1x (α′ + β′ log x)

Back to Text Recall CREMR Demands

Conclusion

CEMR Demand Manifolds

-2.0 -1.0 0.0 1.0 2.0 3.0

SPT SC

= 0.25 = 1 = 0.5

p(x) =1

(α+ βx

σ−1σ

)⇒ ρ(ε) = 2− 1

σ(ε− 1)

(Similar to CREMR for large ε)

-2.0 -1.0 0.0 1.0 2.0 3.0

Linear

CARAStone-Geary

TranslogCESSM

Back to Text Recall CREMR Demand Manifolds

Conclusion

Matching Moments is Not Enough

Predicted Size Distributions of Firms from CES and Linear Demands

Back to Text

Conclusion

CREMR Markups: Details

Back to text

m(x) = x−γx ⇒ x(m) = γ

Profit maximization implies: ϕ(x) = [r′(x)]−1 = 1β

σσ−1 (x− γ)

Combining ϕ(x) and x(m):

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

Conclusion

Back to text

m(x) = x−γx ⇒ x(m) = γ

σσ−1 (x− γ)

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

Conclusion

Back to text

m(x) = x−γx ⇒ x(m) = γ

σσ−1 (x− γ)

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

Conclusion

Back to text

m(x) = x−γx ⇒ x(m) = γ

σσ−1 (x− γ)

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

Conclusion

Back to text

m(x) = x−γx ⇒ x(m) = γ

σσ−1 (x− γ)

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

Conclusion

Back to text

m(x) = x−γx ⇒ x(m) = γ

σσ−1 (x− γ)

ϕ(m) = ϕ0

1− m

ϕ0 ≡1

σ − 1γ

Conclusion

Information and Shannon EntropyBack to Text

What is the information content of one draw from a known F (r)?

Should be: Additive; non-negative; and inversely related to probability

⇒ I(r) = − log(f(r))

Shannon Entropy: The expected value of information from a single draw:

SF ≡ EF [I(r)] = −∫ r

log(f(r)

)f(r)dr

S measures the unpredictability or uncertainty about an individual drawimplied by a known distribution F (r).

S = 0 when F is a Dirac distribution (i.e., all its mass concentrated ata single point): Knowing the distribution tells us everything aboutindividual draws, so an extra draw conveys no new information.

S is arbitrarily large when F is uniform: Knowing the distributionconveys no information whatsoever about individual draws.

Conclusion

⇒ I(r) = − log(f(r))

SF ≡ EF [I(r)] = −∫ r

log(f(r)

)f(r)dr

Conclusion

⇒ I(r) = − log(f(r))

SF ≡ EF [I(r)] = −∫ r

log(f(r)

)f(r)dr

Conclusion

Back to Text

KLD: Measures the “information loss” or “relative entropy” when onedistribution, F , is used to approximate another, F : Illustration

DKL(F ||F

)≡∫ r

)f(r)dr

“Relative entropy”: Relative to Shannon entropy:

DKL(F ||F

)= −

log (f(r)) f(r)dr︸︷︷︸“Cross Entropy”

log(f(r)

)f(r)dr︸︷︷︸

minus Shannon Entropy

Caveats:

Asymmetric: Measures divergence, not distance, of F (r) from F (r)Requires that f(r) > 0 ∀r ∈ [r, r]All draws equally weighted

Conclusion

Back to Text

DKL(F ||F

)≡∫ r

)f(r)dr

DKL(F ||F

)= −

log(f(r)

)f(r)dr︸︷︷︸

Caveats:

Conclusion

Back to Text

DKL(F ||F

)≡∫ r

)f(r)dr

DKL(F ||F

)= −

log(f(r)

)f(r)dr︸︷︷︸

Caveats:

Conclusion

KLD: Example with Normal Distributions

From: “KL-Gauss-Example” by T. Nathan Mundhenk, Ph.D thesis appendix C. Licensed under CC by SA 3.0 via Wikipedia.http://en.wikipedia.org/wiki/File:KL-Gauss-Example.png#mediaviewer/File:KL-Gauss-Example.png

Conclusion

Decomposing the KLD

Back to Text Details Empirics

DKL(F ||F

)= log f (r)− log f (r; θ)︸︷︷︸

1− F (r)

[rf ′(r)

f(r)+ 1

−ϕg′(ϕ)

g(ϕ)+ 1

E(r; θ)︸︷︷︸

− rE′(r; θ)

E(r; θ)︸︷︷︸(3)

Decomposing the information cost of using the wrong F :

1 Over-/under- predicting the mass of the smallest firms

2 Over-/under- predicting elasticities of density and REMR E(r; θ)

3 Non-constant REMR

i.e., failure of Proposition 2 assumption Recall

Estimation: Choose θ to miminize KLD ⇔ MLE (asymptotically)

Conclusion

Decomposing the KLD

DKL(F ||F

)= log f (r)− log f (r; θ)︸︷︷︸

1− F (r)

[rf ′(r)

f(r)+ 1

−ϕg′(ϕ)

g(ϕ)+ 1

E(r; θ)︸︷︷︸

− rE′(r; θ)

E(r; θ)︸︷︷︸(3)

3 Non-constant REMR

Conclusion

Decomposing the KLD

DKL(F ||F

)= log f (r)− log f (r; θ)︸︷︷︸

1− F (r)

[rf ′(r)

f(r)+ 1

−ϕg′(ϕ)

g(ϕ)+ 1

E(r; θ)︸︷︷︸

− rE′(r; θ)

E(r; θ)︸︷︷︸(3)

3 Non-constant REMR

Conclusion

KLD with Pareto and CREMR

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Example: Pareto productivity and sales, CREMR demand: Details

DKL(F ||F ) = logn

n︸︷︷︸(1)

+n− nn︸︷︷︸(2)

, n = Ek =k

σ − 1

[k = 1, n = 2, σ = k+nn

= 1.5 ]

Conclusion

KLD with Pareto and CREMR: Decomposition

‐2.0

‐1.5

‐1.0

‐0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Part 1

Part 2

Assuming a value of σ lower than the true value implies:

1 Over-predicting the mass of the smallest firms2 Under-predicting the mass of larger firms

Conversely if we assume a value of σ greater than the true value

Conclusion

‐2.0

‐1.5

‐1.0

‐0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Part 1

Part 2

Conclusion

‐2.0

‐1.5

‐1.0

‐0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Part 1

Part 2

Conclusion

KLD with Truncated Pareto and CREMR

‐2.0

‐1.5

‐1.0

‐0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Part 1A

Part 2A

Part 1

Part 2

Example: Truncated Pareto productivity and sales, CREMR demand: Details

DKL(F ||F ) = logn

n+ log

1− λn

1− λn︸︷︷︸(1)

+n− nn

+ (n− n)λn

1− λnlog λ︸︷︷︸

[n = kσ−1

, λ ≡ rr∈ [0, 1]; λ = 0.1, k = 1, n = 2, σ = k+n

n= 1.5 ]

Conclusion

Decomposing the KLD: Details

Back to Text

Given:

F (r): Actual distribution of firm salesF (r; θ): Distribution of firm sales implied by:

G(ϕ): Distribution of firm productivitiesϕ(r; θ): Model of firm behaviour, parameterized by θ

We want to use KLD to compare F (r; θ) with F (r):

1 Reexpress DKL(F ||F ) in terms of elasticities of densities: Derivation

DKL(F ||F

)= log f (r)−log f (r; θ)+

1− F (r)

[rf ′ (r)

f (r)− rf ′ (r; θ)

f (r; θ)

2 Relate F (r; θ) to G (ϕ) and ϕ(r; θ): Derivation

rf ′(r; θ)

f(r; θ)=

[ϕg′(ϕ)

g(ϕ)+ 1

]E(r; θ)− 1 +

rE′(r; θ)

E(r; θ), E(r; θ) ≡ ϕ/r i.e., REMR

Conclusion

Back to Text

Given:

DKL(F ||F

1− F (r)

[rf ′ (r)

f (r)− rf ′ (r; θ)

f (r; θ)

rf ′(r; θ)

f(r; θ)=

[ϕg′(ϕ)

g(ϕ)+ 1

]E(r; θ)− 1 +

rE′(r; θ)

Conclusion

Back to Text

Given:

DKL(F ||F

1− F (r)

[rf ′ (r)

f (r)− rf ′ (r; θ)

f (r; θ)

rf ′(r; θ)

f(r; θ)=

[ϕg′(ϕ)

g(ϕ)+ 1

]E(r; θ)− 1 +

rE′(r; θ)

Conclusion

Back to Text

Given:

DKL(F ||F

1− F (r)

[rf ′ (r)

f (r)− rf ′ (r; θ)

f (r; θ)

rf ′(r; θ)

f(r; θ)=

[ϕg′(ϕ)

g(ϕ)+ 1

]E(r; θ)− 1 +

rE′(r; θ)

Conclusion

Express KLD in terms of Elasticities of Densities

Recall Shannon Entropy: SF ≡ −∫ rrf (r) log f (r) dr; F (r) = 0, F (r) = 1.

Integrate by parts:

u = log f (r) → du = f ′(r)f(r)

dr dv = f (r) dr → v = F (r) + C

→ SF = −[F (r) + C

log f (r)

+∫ rr

F (r) + C

f ′(r)f(r)

= − (1 + C) log f (r) + C log f (r) +∫ rr

F (r)+Cr

rf ′(r)f(r)

Now, set C equal to zero or −1:

SF = − log f (r) +

rf ′ (r)

f (r)dr = − log f (r)−

1− F (r)

rf ′ (r)

f (r)dr

Repeat for the KLD:

DKL(F ||F

)= log f (r)− log f (r)−

[rf ′ (r)

f (r)− rf ′ (r)

= log f (r)− log f (r) +

1− F (r)

[rf ′ (r)

f (r)− rf ′ (r)

Conclusion

Elasticity of Density for a Derived Sales Distribution

From Proposition 1(i):

Pr[ϕ ≤ ϕ < ϕ

]=∫ ϕϕg (ϕ) dϕ and r = r (ϕ)

⇒ Pr[r(ϕ)≤ r < r (ϕ)

]=∫ r(ϕ)

r(ϕ)g (ϕ (r)) dϕ

i.e., f (r) = g (ϕ (r)) dϕdr

Reexpress in terms of elasticities:

rf ′(r)

f(r)=ϕg′(ϕ)

rϕ′(r)

ϕ(r)+rϕ′′(r)

ϕ′(r)

Reexpress in terms of elasticity of marginal revenue with respect to total revenue:

E(r) ≡ rϕ′(r)ϕ(r)

⇒ rϕ′′(r)ϕ′(r) = E(r)− 1 + rE′(r)

⇒ rf ′(r)

[ϕg′(ϕ)

g(ϕ)+ 1

]E(r)− 1 +

rE′(r)

Special case when G(ϕ) is Pareto:

ϕg′(ϕ)g(ϕ)

= −(1 + k) ⇒ rf ′(r)f(r)

= −[kE(r) + 1] + rE′(r)E(r)

Conclusion

KLD with Pareto and CREMR

Back to text

When F and F are both Pareto with parameters n and n, the KLD is:

DKL(F ||F ) = logn

n− 1

When productivity is Pareto with parameter k and demand is CREMR:

σ − 1, E′ = 0, n =

σ − 1

Hence:

DKL(F ||F ) = logn

k+ log(σ − 1) +

σ − 1− 1

dDKLdσ = (σ − 1)2

(σ − k+n

)This is positive, i.e., DKL is increasing, if and only if σ ≥ k+n

d2DKLdσ2 = −(σ − 1)3

(σ − 2k+n

)This is negative, i.e., DKL is concave, if and only if σ ≥ 2k+n

Conclusion

Uniform Benchmark for KLD

Back to text

An attractive candidate for an uninformative prior is the uniform distribution:

FU (r) =r − rr − r

fU (r) =1

r − r

The KLD between this and the observed distribution F is:

DKL(F ||FU ) = −SF + log(r − r)

SF is the Shannon Entropy of the data: SF ≡ −∫ rrf (r) log f (r) dr

Conclusion

Bootstrapped Confidence Intervals for KLD

0 0.005 0.01 0.015 0.02 0.0250

(a) Pareto and CREMR

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

(d) Log-Normal andCREMR

1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46 1.480

(b) Pareto and Linear

3.55 3.6 3.65 3.7 3.75 3.80

(e) Log-Normal andLinear

0.19 0.195 0.2 0.205 0.21 0.215 0.22 0.225 0.23 0.2350

(c) Pareto and Translog

2.3 2.32 2.34 2.36 2.38 2.4 2.42 2.440

(f) Log-Normal andTranslog

Back to text

Conclusion

Quantiles for the QQ Estimator: Details

Quantiles for Sales Back to text

Demand Pareto P(ϕ, k) Lognormal LN (µ, s)

CREMR r (1− y)−σ−1

k exp(µ + s · Φ−1 [y]

)Translog η ·

(W[e · (1− y)

− 1k

]− 1

)η ·(W[exp

+ 1 + s · Φ−1 [y] + µ)]− 1

)Linear/LES r

(1− (1− y)

)r −

exp(−2(µ+s·Φ−1[y]

Quantiles for Markups

Demand Pareto P(ϕ, k) Lognormal LN (µ, s)

CREMRm·(1−y)

− 1k

m−1+(1−y)− 1k

m[1 + exp

(−µ− s · ζ

[y; 1s·(ln(

1m−1

)− µ

)])]−1

Translog W[e · (1− y)

− 1k

]W[exp

[s · ζ

[y;− 1

s·(γη

+ µ)]

+ γη

+ µ + 1]]

Linear 12

+ 12· (1− y)

− 1k 1

[µ + s · ζ

[y;− 1

s· (ln (α) + µ)

]]LES (1− y)

− 12·k

√δγ· exp

+ s2· ζ[y, 1s·(ln[γδ

]− µ

)])W: The Lambert functionΦ[z]: c.d.f. of a standard normalζ[y; z] ≡ Φ−1 [(1− Φ [z]) · y + Φ [z]]

notes on technology, demand, and the size distribution of firms

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