Using Power Laws to Identify the Structural Parameters of Trade Models with Firm Heterogeneity
Zeynep Akgul

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Using Power Laws to Identify the Structural Parameters of Trade Models with Firm Heterogeneity

Oﬃce of Economics Working Paper 2017–05–F
May 2017

Abstract

While various ﬁrm heterogeneity models of trade have recently emerged in the CGE literature, their mainstream adoption in trade policy analysis has been limited partly due to lack of available parameter estimates at the disaggregated sector level. In particular, the productivity dispersion and substitution elasticity parameters need to be estimated in a manner consistent with the theoretical underpinnings of the ﬁrm heterogeneity framework. In this paper we address this gap by estimating the productivity dispersion parameter by using ORBIS ﬁrm-level data and imputing substitution elasticities by ﬁtting the ﬁrm size distribution and productivity distribution to the Pareto distribution.

Oﬃce of Economics, Research Division

Zeynep Akgul
Center for Global Trade Analysis, Purdue University
Oﬃce of Economics, Research Division
zakgul@purdue.edu

### 1 Introduction

Since the seminal work of Melitz (2003), a growing trend in trade policy analysis is to incorporate observed ﬁrm characteristics such as heterogeneity in productivity and size in the underlying theoretical framework. A reasonable argument can actually be made that the absence of heterogeneous ﬁrms in trade dynamics leads to an incomplete picture of overall welfare from changes in the trade environment. As a consequence, several ﬁrm heterogeneity models of trade in the Computable General Equilibrium (CGE) framework have emerged in recent years with the ability to explain trade and welfare eﬀects of economic integration in greater precision (Zhai2008Balistreri et al.2011Balistreri and Rutherford2013Dixon et al.2016Akgul et al.2016). As a result, this growing literature on ﬁrm heterogeneity is generating new economic insights for policy analysis.

As is the case with traditional trade models, chosen parameter values play a critical role in determining the welfare predictions from ﬁrm heterogeneity models. Thus, establishing the appropriate parameter values for use in these models is critical for accurate policy analysis. While there are a number of methodologies prevalent in the current literature for obtaining these parameters, they often lack consistency with the underlying ﬁrm heterogeneity theory, indicating a clear need for continued eﬀorts towards theory-consistent parameterization of ﬁrm heterogeneity models (Akgul et al.2015). Indeed, the lack of a general and theoretically-sound approach for obtaining parameters in ﬁrm heterogeneity models remain one of the main challenges in advancing their widespread adoption for policy analysis.

One of the reasons why identiﬁcation of the structural parameters in the ﬁrm heterogeneity model is such a challenge is that there are more parameters to consider in the Melitz (2003) framework than in the traditional Armington trade models. The key parameter in Armington models is the trade elasticity term (Armington elasticity), which is generally estimated using gravity models.1 On the other hand, ﬁrm heterogeneity models have two structural parameters: the shape parameter of Pareto distribution for ﬁrms’ productivity ($\gamma$) along with the elasticity of substitution across varieties ($\sigma$). Quantitative results of trade cost reductions on trade ﬂows and welfare are sensitive to these structural parameters. The importance of the value of $\sigma$ on trade patterns and welfare is quite established (Hertel et al.2007Kancs2010Hillberry and Hummels2013Feenstra2014). The value of $\gamma$ is equally important in Melitz models. For example, di Giovanni and Levchenko (2013) show that in the case when the ﬁrm size distribution is fat-tailed (small shape parameter), the incumbent ﬁrms in the industry are large and have a disproportionate share of overall sales compared to the small marginal ﬁrms and the welfare impact of trade is driven by incumbent ﬁrms rather than the marginal ones. Therefore, the contribution of the extensive margin to trade is found to be negligible when the shape parameter is small. Moreover, in order for the Melitz model to be well-deﬁned and the ﬁrm size distribution to have a ﬁnite mean, parameter values need to satisfy the constraint, $\gamma >\sigma -1$, adding another layer of restriction in choosing appropriate parameter values.

To simplify some of the complexities, the majority of the Melitz implementations (CGE and non-CGE) adopt a set of trade elasticities from the existing literature (Balistreri et al.2011Eaton et al.2011di Giovanni and Levchenko2013Melitz and Redding2013). There have also been some attempts to estimate these parameters directly from a structural model. For example, Crozet and Koenig (2010) rely on gravity equations and use French ﬁrm-level data to estimate the structural parameters in Chaney (2008), which are the shape parameter of productivity distribution, elasticity of substitution, and the distance elasticity of trade costs. They estimate three equations to identify the three parameters. Their ﬁrst equation is a gravity equation that determines the intensive margin of trade by estimating ﬁrm export values. This estimation yields a combination of substitution elasticity and distance elasticity of trade costs. The second equation is another gravity equation that determines the extensive margin of trade, where the probability of ﬁrm export participation in a bilateral trade link is estimated. This yields a combination of the shape parameter and distance elasticity. The last equation is a rank-size distribution of productivity, where the ﬁrm-level TFP is estimated based on Olley and Pakes (1996). Finally, this estimation yields a combination of the shape parameter and substitution elasticity. They identify each parameter by solving the coeﬃcient estimates in three equations. The resulting structural parameter values show considerable variation across the manufacturing sectors in their database.

Following the methodology in Crozet and Koenig (2010), Akgul et al. (2015) estimate a combination of ﬁrm heterogeneity parameters for manufacturings sectors using a two-stage estimation method with country and industry level data. Since aggregate data bases do not allow for individual identiﬁcation of parameters, Akgul et al. (2015) rely on the shape parameter estimates of Spearot (2016) to impute the values of elasticity of substitution. While they provide signiﬁcant improvement on the existing methods to obtain elasticity values that can be used in CGE models that incorporate Melitz (2003), they rely on other studies for the shape parameter. We extend their idea to provide parameter values that are consistent with the data base used, eliminating the reliance on outside sources for the shape parameter that may not be fully consistent with the Melitz (2003) framework.2

In this paper, we propose a simpler method to estimate the structural parameters of ﬁrm heterogeneity models that relies on the theoretical relationship between the size distributions of ﬁrms and the $\gamma$ and $\sigma$ parameters. The Pareto assumption for ﬁrm sales is equivalent to assuming that ﬁrm productivity is Pareto-distributed, though with a diﬀerent shape parameter. In general, when ﬁrm productivity is from a Pareto distribution with shape parameter $\gamma$, then the ﬁrm size also follows a Pareto distribution, however, with a diﬀerent shape parameter, $\zeta$, which in fact is a ratio based on the two structural parameters: $\zeta =\frac{\gamma }{\sigma -1}$ (di Giovanni and Levchenko2013). Since $\zeta$ can be estimated directly from ﬁrm-level data, it provides a useful way to infer the ratio of structural parameters in the ﬁrm heterogeneity model. Empirical studies such as di Giovanni et al. (2011) use this property to consistently estimate $\zeta$ from ﬁrm-level sales.3 However, since this expression is a combination of $\gamma$ and $\sigma$, it is not possible to estimate the individual structural parameters in these studies. Therefore, more information is needed for separate identiﬁcation of both of these parameters. One approach is to use existing estimates of elasticity of substitution (Broda and Weinstein2006) and then impute the shape parameter (Chaney2008). While this method circumvents some of the diﬃculties associated with parameter identiﬁcation, it has two drawbacks: (i) Estimates for elasticity of substitution are often obtained from traditional gravity equations that depend on the Armington assumption, which is fundamentally inconsistent with ﬁrm heterogeneity theory and reﬂects only the demand-side heterogeneity in the model (ii) The resulting values for the shape parameter typically are not sector and region-speciﬁc and therefore do not capture the signiﬁcant variation along these dimensions. Not accounting for these drawbacks is likely to lead to biased estimates of the parameters in the calibrated model.

To overcome these methodological issues, we instead use the actual distribution of ﬁrm productivity to get estimates of the $\gamma$ parameter directly from the data. Using TFP and ﬁrm size distribution together makes a potentially useful tool for empirical research on estimating the structural parameters of ﬁrm heterogeneity, since it can be used on the same ﬁrm-level dataset and without needing a number of model-based equations. In a broad outline, our methodology includes the following steps to identify the structural parameters of ﬁrm heterogeneity:

1.
Estimate the power law exponent of the ﬁrm size distribution by ﬁtting it to the Pareto distribution. This yields an estimate of $\zeta =\frac{\gamma }{\sigma -1}$.
2.
Calculate a productivity measure (TFP) for each ﬁrm in the data.
3.
Fit the ﬁrm productivity distribution to the Pareto distribution and obtain the $\gamma$ parameter.
4.
Finally use the estimates of $\zeta =\frac{\gamma }{\sigma -1}$ and $\gamma$ to identify $\sigma$.

We illustrate this methodology using the ORBIS ﬁrm-level database and focusing on the U.S. Motor Vehicles and Parts Sector. We select this sector as there is considerable heterogeneity across ﬁrms and diﬀerentiation across the available varieties, which can better reﬂect the characteristics of the underlying Melitz theory.

This empirical methodology can be performed for diﬀerent sectors and countries and thereby allows for obtaining structural parameter of ﬁrm heterogeneity estimates not only at the sectoral level but also for the country level depending on data availability. There is, in fact, considerable variation in parameter values across sectors, and even across countries. For example, estimation results in Spearot (2016) suggest that while some sectors, such as motor vehicles, electronic equipment and machinery, can be characterized by highly heterogeneous, others are relatively less heterogeneous, such as oil, wheat, processed rice. The parameters at the sectoral level obtained with our methodology can then be used as inputs in CGE models incorporating ﬁrm heterogeneity at a global scale and thus better quantify and determine the trade pattern and welfare eﬀects of economic integration.

The remainder of this paper is organized as follows. We begin our appraisal in Section 2 with a review of the literature on ﬁrm size and total factor productivity as well as a discussion on the current empirical challenges to parameterize ﬁrm heterogeneity models. Section 3 goes over the empirical methodology while Section 4 discusses the ﬁrm-level data. In Section 5 we move on to the estimation results. Section 6 concludes the paper.

### 2 Related Literature

A brief overview of parameter values used in the ﬁrm heterogeneity literature is presented in Table 1 based on Akgul et al. (2015). This table summarizes the mainstream approach in obtaining parameter estimates and compares the parameter values in relevant studies.

Table 1: Summary Statistics of U.S. Firms in the Motor Vehicles and Parts Sector.
 Author (Year) Country ζ$\zeta$ γ$\gamma$ σ$\sigma$ US ﬁrm-level 1.06 - - OECD cross-section - [3.6, 12.86] - US plant-level - 3.6 3.79 Costa Rica - 5.3 6.0 French ﬁrm-level - [1.65-7.31] [1.15-6.01] Cross-section - [3.92-5.17] 3.8 French ﬁrm-level 2.46 4.87 2.98 French ﬁrm-level 1.06 - - Cross-section 1.06 5.3 6 US 1.42 4.25 4 Cross-section - [1.76-6.29] -
Note: This table summarizes the values of structural parameters used in the ﬁrm heterogeneity literature. The table is adapted from Akgul et al. (2015).

In the literature, the shape parameter is often calibrated using the Power Law exponent of ﬁrm size based on existing substitution elasticities (di Giovanni and Levchenko2013Eaton et al.2011Melitz and Redding2013). The calibrated values of shape parameters are often higher compared to the estimated values. In particular, the calibrated values are in the range of 4-8 and are based on aggregated sectors. On the other hand, the shape parameter estimates show substantial sectoral variation at the more disaggregate level. Notably, the ﬁndings of Crozet and Koenig (2010) show that the shape parameter values are in the range of 1.65-7.31. Similarly, the ﬁndings of Spearot (2016) indicate signiﬁcant variation across sectors in the range of 1.76-6.29. The diﬀerences in values are important as using calibrated values of shape parameters would attribute lower productivity dispersion to the industry, while there could, in fact, be much higher productivity dispersion across ﬁrms. Unfortunately, there is very little work or guidance in the literature on how to estimate the shape parameter in a way that is consistent with the underlying ﬁrm heterogeneity theory.

A similar argument can be made for the substitution elasticity. Even for Armington elasticity values, there is a lack of consistency in the literature. McDaniel and Balistreri (2003) highlight this point by stating that “The estimates from the literature provide a wide range of point estimates to apply to a given commodity in a given model for a given aggregation.” This is an accurate picture of not only Armington elasticities, but also Melitz elasticities. The elasticity values presented in Table 1 indicate that several of these studies adopt a value around 4 based on Bernard et al. (2003). This value applies to the manufacturing sector; however, when the manufacturing sector is disaggregated further, there is more variation in the elasticity estimates, especially when the underlying theory is consistent with ﬁrm heterogeneity. For example, elasticity estimates in Crozet and Koenig (2010) are in the range of 1.15-6.01, reﬂecting a wide range of demand-side heterogeneity compared to the more aggregated studies.

#### 2.1 Firm Size Distribution in a Melitz World

A stylized fact in the empirical trade literature with heterogeneous ﬁrms is that the tail behavior of the ﬁrm size distribution follows a Power Law, speciﬁcally Pareto distribution (Axtell2001). In fact, it is argued that the tail behavior is well approximated by a Zipf Law, where full granularity is realized with the power law exponent near unity.

The relationship between ﬁrm rank and ﬁrm size are analyzed in the literature through the power law distribution. The cumulative distribution function (CDF) of the power law distribution is described by a non-negative random variable X satisfying:

 Pr[X≥x]≈Cx−ζ$Pr\left[X\ge x\right]\approx C{x}^{-\zeta }$ (1)

where $C>0$ is a constant and $\zeta$ is the power law exponent, i.e. shape parameter. Thus the Pareto distribution is considered to be a power law since it can be expressed as

 Pr[X≥x]=(xB)−ζ$Pr\left[X\ge x\right]={\left(\frac{x}{B}\right)}^{-\zeta }$ (2)

where $B>0$ is the minimum level of $x$, which can be described by $C={B}^{\zeta }$. The value of the shape parameter $\zeta$ has signiﬁcant implications for the Pareto distribution. If $\zeta =1$, a special case of the Pareto distribution is achieved which is known as the Zipf’s Law (Zipf1950). This special case is also referred to as the rank size rule because it implies that ﬁrm size is inversely proportional to the rank of the ﬁrm size (Segarra and Teruel2012).

In the heterogeneous ﬁrms model, if the ﬁrm productivity distribution can be described by the Pareto distribution, then the ﬁrm size also follows the Pareto distribution, but with a diﬀerent power law exponent. We provide the link under autarky following di Giovanni et al. (2011).

Let’s assume that ﬁrm productivity has the Pareto CDF according to Equation (2) as follows

 Pr[φ≥z]=(zB)−γ$Pr\left[\phi \ge z\right]={\left(\frac{z}{B}\right)}^{-\gamma }$ (3)

where $\phi$ is the productivity of the ﬁrm and $\gamma$ is the shape parameter of the productivity distribution, and $B$ is the minimum level of productivity for which the Pareto distribution holds (also known as the scale factor). The optimal demand and price for each variety in the ﬁrm heterogeneity model yields the following domestic sales by ﬁrm:

 Si=piqi=YP1−σpi1−σ=YP1−σ(σσ−1W)1−σφiσ−1=Aφσ−1$\begin{array}{cc}\begin{array}{rl}{S}_{i}& ={p}_{i}{q}_{i}\\ & =\frac{Y}{{P}^{1-\sigma }}{{p}_{i}}^{1-\sigma }\\ & =\frac{Y}{{P}^{1-\sigma }}{\left(\frac{\sigma }{\sigma -1}W\right)}^{1-\sigma }{{\phi }_{i}}^{\sigma -1}\\ & =A{\phi }^{\sigma -1}\end{array}& \end{array}$ (4)

where ${p}_{i}$ is the price charged by the heterogeneous ﬁrm $i$ in the monopolistically competitive sector, ${q}_{i}$ is the demand for ﬁrm $i$’s variety, $Y$ is the aggregate demand, $P$ is the aggregate price index, $W$ is the cost of factor payments, and $A=\frac{Y}{{P}^{1-\sigma }}{\left(\frac{\sigma }{\sigma -1}W\right)}^{1-\sigma }$.

Therefore, when ﬁrm productivity follows Pareto distribution with $\phi \sim Pareto\left(B,\gamma \right)$, ﬁrm size is also described by the Pareto distribution with ${S}_{i}\sim Pareto\left({B}^{1-\sigma }A,\frac{\gamma }{\sigma -1}\right)$.

#### 2.2 Power Law Exponents for Firm Size

One of the canonical studies that estimates ﬁrm size distribution is Axtell (2001). Using the US Census Bureau data for 1992 and 1997, he establishes that $\zeta$ is close to 1 for all of the US ﬁrms in the sample (more than 5 million ﬁrms). Two alternative measurements of ﬁrm size are considered in his study: ﬁrm revenue and number of employees. For ﬁrm revenue in 1997, his estimates provide $\zeta =0.994$ with standard error 0.064 and ${R}^{2}=0.976$. For the number of employees in 1997, he ﬁnds $\zeta =1.059$ with standard error 0.054 and ${R}^{2}=0.992$. Thus, both ﬁrm revenue and the number of employees yield estimates of $\zeta$ that are close to 1, in support of Zipf’s Law.

A more recent study by di Giovanni et al. (2011) estimates the ﬁrm size distribution based on French ﬁrm-level data in 2006. Their dataset includes more than 2 million ﬁrms, of which approximately 9% are exporters. Since the power law may not be a good ﬁt for the small ﬁrms under a minimum size threshold, they follow the common practice of truncating their dataset based on graphical exposition, where the regression of log-rank on log-size provides a better ﬁt above the threshold. While this cutoﬀ point is selected based on a visual inspection, they also report that it also corresponds to an institutional standard, where the reporting requirements are diﬀerent below the cutoﬀ of 750,000 Euro annual sales. When this truncation is applied, the number of observations reduce to 150,000 ﬁrms. Similar to Axtell (2001), di Giovanni et al. (2011) use sales and number of employees as the ﬁrm size proxy and ﬁnd estimates of $\zeta$ that are close to 1.4

di Giovanni et al. (2011) also report estimates at the sectoral level, which illustrates that there is considerable variation in $\zeta$ values across sectors. For tradable sectors, which includes food, manufactures, and select services, they report values ranging between 0.422 and 1.233, implying diﬀerent levels of ﬁrm size heterogeneity across sectors. For non-tradable sectors which include services sectors, they report values ranging between 0.548 and 1.473. These results imply that the mathematical constraint of $\gamma >\sigma -1$ is not satisﬁed for several sectors since $\zeta <1$ means that $\gamma <\sigma -1$. This has important implications for generalization of the Zipf’s Law that has found strong support in the empirical literature. While country-level estimates yield values close to unity, this may not be the case when sectoral-level estimates are concerned.

### 3 Empirical Methodology

#### 3.1 Estimation of Shape Parameter for a Power Law Distribution

There are two main methods in the literature to estimate the power law exponent of the ﬁrm size distribution, $\zeta$. The ﬁrst method relies on Maximum Likelihood approach to estimate the shape parameter. The second class of estimators is based on Least Squares, with regressions applied to log-log transformations of the data.5 The LS estimation can be done in three diﬀerent ways to obtain the shape parameter: (i) estimation of CDF based on the deﬁnition of the power law, (ii) estimation of PDF based on the deﬁnition of the power law, and (iii) estimation of rank with a correction for small sample bias. In this section, we give a brief overview of each of these estimation methods.

##### 3.1.1 Maximum Likelihood Approach

The PDF of a random variable that follows a power law is:

 f(s)=−ζb[sb]−(1+ζ)$f\left(s\right)=\frac{-\zeta }{b}{\left[\frac{s}{b}\right]}^{-\left(1+\zeta \right)}$ (5)

The likelihood function for a sample $\left({s}_{1},...,{s}_{n}\right)$ is then given as:

 L=n∏i=1ζb[sib]−(1+ζ)$\mathsc{ℒ}=\prod _{i=1}^{n}\frac{\zeta }{b}{\left[\frac{{s}_{i}}{b}\right]}^{-\left(1+\zeta \right)}$ (6)

The MLE estimate of $\zeta$ is then just the value of $\zeta$ that maximizes the likelihood function. Taking logs and setting $\frac{\partial \mathsc{ℒ}}{\partial \zeta }=0$ we get

 ˆζ=n⎡⎢ ⎢⎣n∑i=1lnsib⎤⎥ ⎥⎦−1$\stackrel{̂}{\zeta }=n{\left[\sum _{i=1}^{n}ln\frac{{s}_{i}}{b}\right]}^{-1}$ (7)

##### 3.1.2 Cumulative Distribution Function

The ﬁrst LS method is based on the deﬁnition of the power law. We start with Equation (2) and take the natural logarithm of both sides. The probability of sales of ﬁrm $i$, ${S}_{i}$, being greater than the target sales $s$ is then regressed on the target sales to obtain the following relationship:

 ln(Pr(Si>s))=lnC−ζlns+εi$ln\left(Pr\left({S}_{i}>s\right)\right)=lnC-\zeta lns+{𝜀}_{i}$ (8)

The probability $Pr\left({S}_{i}>s\right)$ measures the proportion of ﬁrms that have higher sales than the target. Therefore, the dependent variable is calculated as the log of the ratio of ﬁrms in the sample with higher sales than s to the total number of ﬁrms. The standard approach in the literature is to organize the sales of each ﬁrm into classes or bins. This is often done in cases where single observations for each ﬁrm in the dataset are not available (Bottazzi et al.2015).

##### 3.1.3 Probability Density Function

Alternatively, we can use the deﬁnition of the PDF in Equation (5). This method ﬁrst divides the sample of observations into $N$ bins. The widths of the bins are often selected such that their bounds are distributed equidistantly in the logarithmic space. The fraction of observation within each bin is then calculated as the number of ﬁrms in each bin divided by the width of the bin.

One important disadvantage of the binned regression is that there are signiﬁcant data restrictions when observations are grouped into bins. The estimation has to be performed with a considerably smaller number of observations when bins are used, which increases the noise in the estimate. This is especially signiﬁcant in small samples. Since our sample size is small, we do not perform the binned PDF-LS regression using this database6.

##### 3.1.4 Rank Estimator

Another popular way to estimate $\zeta$ is the rank estimator, which regresses the ﬁrm’s rank to its size using LS:

 ln(Ranki−δ)=lnC−ζlnSi+εi$ln\left(Ran{k}_{i}-\delta \right)=lnC-\zeta ln{S}_{i}+{𝜀}_{i}$ (9)

where $Ran{k}_{i}$ is the rank of ﬁrm $i$, ${S}_{i}$ is the size of the corresponding ﬁrm, and $\delta =0$ is assumed. This regression equation is motivated by Equation (2) when $N$ is the total number of observations and the following holds:

 ln(Ranki∕N−δ)=ln(Pr(Si>x))=lnC−ζlnx+εi$ln\left(Ran{k}_{i}∕N-\delta \right)=ln\left(Pr\left({S}_{i}>x\right)\right)=lnC-\zeta lnx+{𝜀}_{i}$ (10)

Despite its widespread use due to simplicity and robustness, the performance of OLS log-log rank-size regression has been subject to scrutiny, especially when the sample size is small. One of the arguments against the OLS log-log rank-size regression in small samples is that the coeﬃcient estimates are biased downwards (Gabaix and Ioannides2004). In order to address this issue, a correction is proposed by Gabaix and Ibragimov (2011). They show that the bias is reduced when the rank is corrected by a $1∕2$ shift. As such, they assume $\delta =1∕2$ and regress the natural logarithm of $Rank-1∕2$ of each ﬁrm on that of its sales as follows:

 ln(Ranki−12)=C+ζlnSi+εi$ln\left(Ran{k}_{i}-\frac{1}{2}\right)=C+\zeta ln{S}_{i}+{𝜀}_{i}$ (11)

#### 3.2 Background on Total Factor Productivity Distribution

Recent theoretical and empirical studies show that a ﬁrm’s productivity is the most important characteristic in determining its place in the global economy, with a more productive ﬁrm better able to deal with changes in the trade environment. 7 First, as shown in the seminal work of Melitz (2003), there is a strong selection eﬀect into exporting so that only the most productive ﬁrms are able to incur the ﬁxed costs required to sell across diﬀerent markets. Empirical studies such as Bernard et al. (2007) have found strong support in favor of a self-selection eﬀect, with exporters more productive than non-exporters in the years before entering the export market. 8 Second, ﬁrms’ foreign activities can increase their productivity at home through a learning by exporting eﬀect, where learning is just the knowledge and expertise a ﬁrm gains from serving international markets. 9 Finally, trade liberalization leads to more competition and an increase in varieties available in the domestic market, forcing domestic ﬁrms to decrease output and accept lower operating proﬁts. Firms with lower productivity will be unable to aﬀord the ﬁxed costs of production and thus exit the domestic market Melitz (2003).

Given these considerations, ﬁrm productivity is a key component in any thorough analysis of trade impacts, and so there is a strong need to measure it in an accurate and consistent manner. However, several complications arise in the measurement of ﬁrm productivity including the fact that the ﬁrm’s output is often captured by aggregate revenue with little information on prices, many ﬁrms produce multiple products, and that ﬁrm inputs like labor and capital are often not distinguished by quality (skilled vs. unskilled) and reporting measures (cost vs. market value). These concerns are accentuated with data constraints like missing or inconsistent values. The general trend in the literature has been to compute broad measures of productivity while recognizing that each measure has some drawbacks associated with it.

A popular and simple measure of ﬁrm productivity is labor productivity $\left(Y∕L\right)$ where $Y$ is often total revenue or value added.10 Labor productivity is a popular measure of ﬁrm productivity as most ﬁrms provide information on revenues and amount of labor used, and thus there is adequate ﬁrm coverage for meaningful policy analysis. However, labor productivity does not consider the intensity in the use of the excluded inputs such as capital, which may act as a substitute to labor in certain production processes. So the ﬁrm’s total factor productivity (TFP), which controls for other inputs, should also be considered as a measure of ﬁrm productivity. Generally, TFP is calculated using either an index number approach or estimation based methods. Under the index number or Solow residual technique, the TFP relates output to a weighted sum of inputs with the weights determined from aggregate (sector or economy wide) sources on labor and capital shares of income. Estimation based TFPs are the residuals from an estimated production function using ﬁrm-level (or plant-level) data. We next provide more details on both these methods.

##### 3.2.1 Index-number based TFP

Our initial TFP measure is based on an index number approach, ﬁrst suggested by Solow (1957) to account for productivity growth due to technological progress. Despite its longevity, it still remains one of the more popular ways to determine TFP at both aggregate and sectoral levels (Del Gatto et al.2011). Given a standard Cobb-Douglas production function, the index-number TFP for each ﬁrm is computed as: ${y}_{it}-\alpha {l}_{it}+\left(1-\alpha \right){k}_{it}$ where all variables are in natural log terms and $\alpha$ is equal to the labor share of income. The labor cost share for the U.S. motor vehicle and parts sector is obtained from the BEA’s GDP by Industry database.11

##### 3.2.2 Estimation-based TFP

The residuals of the production function estimated with ﬁrm-level variables serve as our estimation-based TFP measure. We assume a production function such that:

 yit=β0+βkkit+βllit+μit${y}_{it}={\beta }_{0}+{\beta }_{k}{k}_{it}+{\beta }_{l}{l}_{it}+{\mu }_{it}$ (12)

where all variables are expressed in natural logs, ${y}_{it}$ is the ﬁrm’s deﬂated revenue, ${l}_{it}$ is the amount of labor used and ${k}_{it}$ the capital stock of ﬁrm $i$ at time $t$. ${\mu }_{it}$ captures the unexplained shock to a ﬁrm’s productivity. This error term can be further decomposed into two components:

 μit=ωit+ηit${\mu }_{it}={\omega }_{it}+{\eta }_{it}$ (13)

where ${\omega }_{it}$ is the productivity innovation that is only observed by the ﬁrm while ${\eta }_{it}$ is the i.i.d. component representing unexpected shocks. Thus, in this framework ${\eta }_{it}$ has no eﬀect on the ﬁrm’s decisions but ${\omega }_{it}$ can impact a ﬁrm’s choice of inputs, and whether it continues production.

Although OLS can be used to obtain the residuals of (1), the generated TFP measure will suﬀer from simultaneity and also possibly selection issues. Simultaneity occurs as ${\omega }_{it}$ is observed by the ﬁrm before making a decision on inputs. So the input variables and ${\omega }_{it}$ will be positively correlated, with a ﬁrm observing a high productivity shock more likely to purchase more inputs. OLS will thus provide biased estimates of the labor and capital coeﬃcients in (13) . 12

Along with simultaneity, OLS estimates may also suﬀer from selection problems if there is substantial attrition in the sample due to ﬁrms that are no longer producing. Since ${\omega }_{it}$ likely inﬂuences the exit decision of the ﬁrm, ﬁrms that continue to produce will be a selected sample, with the selection criteria partially determined by the ﬁrm’s inputs like capital stock. For instance, ﬁrms with larger capital stock will be able to stay in the market even for low realizations of productivity shocks. As a result, selection implies a negative correlation in the observed sample between productivity shocks and capital stock, leading to a downward bias in the estimated capital coeﬃcient.

A traditional approach in dealing with the simultaneity and selection issues is to assume ${\omega }_{it}$ as constant over time. Consistent estimates of the coeﬃcients ${l}_{it}$ and ${k}_{it}$ can then be obtained by ﬁxed eﬀects estimation using either within or ﬁrst-diﬀerencing techniques. However, as Ackerberg et al. (2007) notes that there are a number of issues with ﬁxed eﬀects estimation including the fact that ${\omega }_{it}$ is not likely to be constant over extended periods of time due to changes in the ﬁrm’s environment. Moreover, measurement error in inputs can cause ﬁxed eﬀect estimators to perform worse than OLS estimators and is one of the reasons why these estimators can give unreasonably low estimates of capital coeﬃcients in applied work.

In light of these issues, Olley and Pakes (1996) (henceforth OP) propose an alternate three-stage methodology for estimating the production function. They account for the unobserved productivity by examining the ﬁrm’s investment behavior which subsequently depends on capital and productivity, and so can be treated as the state variable in the ﬁrm’s dynamic optimization problem. OP also address sample selection issues by using an exit rule to estimate survival probabilities conditional on ﬁrm’s available information. These probabilities are then used in the productivity estimation to correct for selection. Overall, the OP methodology allows ${\omega }_{it}$ to vary over time, controls for potential selection bias, and deals with the endogeneity of input variables.

We next brieﬂy discuss the OP framework for obtaining TFP estimates.13 As mentioned above, OP use the ﬁrm’s investment as a proxy for unobserved productivity and impose the requirement that it be monotonically increasing in productivity, conditional on the rest of state variables. Thus, this approach requires that the sample has enough non-zero positive investment observations for adequate estimation. 14 OP further assume that ${\omega }_{it}$ follows an exogenous ﬁrst order Markov process with future productivity strictly increasing in ${\omega }_{it}$ so that a ﬁrm with a high ${\omega }_{it}$ today has a greater chance of getting a high ${\omega }_{it+1}$ in the next period. Capital stock is assumed to accumulate in a deterministic manner with ${k}_{it}$ taken as the sum of the non-depreciated capital stock $\left(1-\delta \right){k}_{it-1}$ and the ﬁrm’s chosen investment level ${i}_{it-1}$. Within this framework, OP show that investment depends only on capital and productivity: ${i}_{it}=f\left({k}_{it},{\omega }_{it}\right)$. If ${i}_{it}$ is strictly increasing in ${\omega }_{it}$, then the function $h\left(.\right)={f}^{-1}\left(.\right)$ exists and so ${\omega }_{it}$ can be expressed as a function of observables:

 ωit=h(kit,iit)${\omega }_{it}=h\left({k}_{it},{i}_{it}\right)$ (14)

In the First Stage, OP then substitute (14) and (13) into (12) to get :

 yit=βllit+ϕ(kit,iit)+ηit${y}_{it}={\beta }_{l}{l}_{it}+\varphi \left({k}_{it},{i}_{it}\right)+{\eta }_{it}$ (15)

where $\varphi \left({k}_{it},{i}_{it}\right)={\beta }_{0}+{\beta }_{k}{k}_{it}+h\left({k}_{it},{i}_{it}\right)$ controls for unobserved productivity. OP treat $h\left({k}_{it},{i}_{it}\right)$ nonparametrically and so ${\beta }_{l}$ is the only coeﬃcient to be estimated consistently in the First Stage of the estimation. Following Yasar et al. (2008), we use a second-order polynomial in investment and capital to approximate $\varphi \left({k}_{it},{i}_{it}\right)$ in the First Stage estimation.

As a ﬁrst-order Markov process, ${\omega }_{it}=E\left[{\omega }_{it}|{\omega }_{it-1}\right]+{\zeta }_{it}=g\left({\omega }_{it-1}\right)+{\zeta }_{it}$ where g is some unknown function and ${\zeta }_{it}$ is an unexpected innovation that is uncorrelated with productivity and capital in period $t$. This results in the following equation:

 yit−βllit=β0+βkkit+ωit+ηit=β0+βkkit+g(ωit−1)+ζit+ηit=β0+βkkit+g(ϕit−1−β0−βkkit−1)+ζit+ηit$\begin{array}{cc}\begin{array}{rl}{y}_{it}-{\beta }_{l}{l}_{it}& ={\beta }_{0}+{\beta }_{k}{k}_{it}+{\omega }_{it}+{\eta }_{it}\\ & ={\beta }_{0}+{\beta }_{k}{k}_{it}+g\left({\omega }_{it-1}\right)+{\zeta }_{it}+{\eta }_{it}\\ & ={\beta }_{0}+{\beta }_{k}{k}_{it}+g\left({\varphi }_{it-1}-{\beta }_{0}-{\beta }_{k}{k}_{it-1}\right)+{\zeta }_{it}+{\eta }_{it}\\ \end{array}& \end{array}$ (16)

While ${\beta }_{l}l$ and ${\varphi }_{it}$ are not observable, their estimated values can be obtained from the First Stage with ${\stackrel{̂}{\varphi }}_{it}={ŷ}_{it}-{\stackrel{̂}{\beta }}_{l}{l}_{it}$. Thus, OP in the Second Stage substitute these predicted values in (16) to obtain:

 yit−ˆβllit=βkkit+˜g(ˆϕit−1−βkkit−1)+ζit+ηit${y}_{it}-{\stackrel{̂}{\beta }}_{l}{l}_{it}={\beta }_{k}{k}_{it}+\stackrel{̃}{g}\left({\stackrel{̂}{\varphi }}_{it-1}-{\beta }_{k}{k}_{it-1}\right)+{\zeta }_{it}+{\eta }_{it}$ (17)

where $\stackrel{̃}{g}$ accounts for the ${\beta }_{0}$ terms in (17).15 As in the First Stage, $\stackrel{̃}{g}$ is treated by OP as a nonparametric term. In our estimations, we approximate $\stackrel{̃}{g}$ with a second-order polynomial and then estimate (6) by NLS to obtain a consistent estimate of ${\beta }_{k}$. The estimated TFP is then just given by: $TF{P}_{it}=$ exp$\left({y}_{it}-{\stackrel{̂}{\beta }}_{0}-{\stackrel{̂}{\beta }}_{l}{l}_{it}-\stackrel{̂}{{\beta }_{k}}{k}_{it}\right)$

Alternatively, Wooldridge (2009) combines both the OP stages into a single set of moments, which are then estimated under a General Method of Moments (GMM) framework in just one-step. The GMM approach is more eﬃcient than OP as it controls for potential serial correlation or heteroskedasticity in the error terms. Further, one-step GMM estimation allows for robust standard errors without the need for bootstrapping methods. We will use the GMM method as a robustness check for our OP based TFP estimates.

### 4 Data

We rely on the ORBIS database to obtain annual ﬁrm-level ﬁnancial data on the U.S. motor vehicles and parts sector. We restrict the time frame of our study to the 2010-2016 period.16 ORBIS uses both administrative and public data to provide ﬁrm-level information for over 200 million companies worldwide. Several procedures have been undertaken in ORBIS to verify the quality of reported data, including an indexation strategy to ensure the uniqueness of individual ﬁrms as well as an analysis to detect unusual variations of ﬁnancial values between years. Detailed ownership information for ﬁrms is also provided in this database.

While ORBIS provides information about a wide variety of ﬁrms, some limitations are encountered when using it for productivity analysis. Key ﬁnancial variables are often missing for a number of ﬁrms in the database, which can sharply reduce the sample of ﬁrms available for analysis. This is especially a concern for the U.S. data with only a small number of ﬁrms providing adequate information on the variables required for TFP computations. Notably, there is limited coverage on proﬁtability variables such as EBITDA (earnings before interest, taxes, depreciation and amortization), while labor and material costs are not provided at all by U.S. ﬁrms. Thus we are unable to determine each ﬁrm’s value added during production, and instead rely on total revenue as the measure of the ﬁrm’s output in all of our productivity computations for the U.S. motor vehicles and parts sector. 17

With these issues in mind, we clean the initial sample from ORBIS by dropping observations that had missing or inconsistent information for key ﬁnancial variables. Our ﬁrst criteria for exclusion is if ﬁrms, in a given year, reported negative and zero values for either employment or total revenue, the proxies for ﬁrm size in our analysis. This reduced the sample of ﬁrms in the U.S motor and vehicle parts sector from an initial count of 150 ﬁrms to around 70. For the TFP calculations, we further drop ﬁrm-year observations that had no information on the capital stock, with tangible ﬁxed assets serving as the measure of capital stock at the ﬁrm-level. Following the Perpetual Inventory Method, annual ﬁrm investment is calculated as the diﬀerence between the current and lagged book value of ﬁxed tangible assets plus any depreciation expenses. Firm-year observations with negative values for investment are then dropped to be consistent with the OP methodological framework. These steps lead to a sample of 250 ﬁrm-year observations over a span of 6 years.

Table 2 reports descriptive statistics for total revenue, number of employees, capital, and investment values for ﬁrms in the U.S. Motor Vehicles and Parts Sector compiled from the ORBIS database. Except for employment, all entries in the table are in millions of dollars. In order to compare values over time, ﬁrm revenue is deﬂated using the U.S motor vehicle and parts sector’s price index for gross output. Similarly, capital stock and investment are deﬂated using the sector’s price index for value added. These price indexes are obtained from the BEA’s GDP by Industry database.The mean indicates the average of ﬁrm values across available years. Descriptive statistics for these variables by year is provided in the Appendix.

Table 2: Summary Statistics of U.S. Firms in the Motor Vehicles and Parts Sector.
 Variable Observation Mean Std. Dev. Min. Max. Median Revenue (mil. USD) 70 7,775 25,922 0 155,167 837 Employees 70 16,843 39,188 5 215,833 4,008 Capital (mil. USD) 70 1,709 6,017 0 39,239 105 Investment (mil. USD) 68 556 2,264 -15 15,836 26
Note: All values except employees are in millions USD.

The total number of observations for ﬁrm revenue, employees, and capital is 70, when the zero values and non-applicable values are dropped from the original database. We use the total revenue and the number of employees to analyze the ﬁrm size distribution. While the maximum total revenue available in the database is approximately 155 billion USD, the minimum value is reported as 0. Since values are in millions of USD, the 0 value corresponds to a very small revenue of $14,000. The average ﬁrm revenue pooled across years is 7,775 million USD. The yearly average revenue in Table 6 show that the majority of the observations are reported between the years 2011-2015, which generate average revenue around$8,000+. Similar observations apply to the other variables.

It is important to note that power laws ﬁt the ﬁrm size distribution at the right tail of the distribution which corresponds to ﬁrms with higher revenues (Axtell2001). In other words, there is a certain minimum size threshold below which the power law may not be a good choice. This issue is addressed in the literature by selecting a low cutoﬀ based on the visual inspection of the ﬁt (Gabaix2009di Giovanni et al.2011). In line with the literature, we identify a minimum size cutoﬀ based on graphical inspection of the ﬁrm-level data. We truncate our sample at the cutoﬀ below which ﬁrm revenue and number of employees are considerably low relative to the rest of the ﬁrms in the database. This corresponds to a cutoﬀ value of $1{0}^{8}$ for ﬁrm revenue and 1000 for the number of employees. Even though the revenue cutoﬀ level seems rather high, it does not result in the removal of the ﬁrms that have a larger share of the total revenue. In fact, the twelve ﬁrms that are dropped from the dataset account for only 0.02% of total revenue.

### 5 Results

#### 5.1 Firm Size

We begin by estimating the power law in ﬁrm size using the two mainstream methodologies in the literature outlined above. For ﬁrm size we use two alternative proxies. It is measured in terms of both total revenue and the number of employees. Data are pooled over six years for all U.S. ﬁrms in the Motor Vehicles sector.

Table 3 uses total revenue as the measure of ﬁrm size. The results show that the OLS-CDF ﬁt in this sample (${R}^{2}=0.945$) is better than the OLS-Rank ﬁt (${R}^{2}=0.676$). The point estimates of $\zeta$ are similar in OLS-CDF, 0.578, and in OLS-Rank, 0.5. However, we should note that domestic sales follow a power law with exponent close to 1 in the U.S. data reported in the literature. Both our point estimates are below one, which is what the ﬁrm size estimates in the literature ﬁnd when a comprehensive dataset of ﬁrms in the entire country is pooled for the analysis. Often, these studies base their estimates on a sample of thousands of ﬁrms if not millions. Comparatively, our dataset only focuses on the U.S. Motor Vehicles sector with 70 observations in the untruncated sample and 53-54 observations in the truncated sample. Thus, given our small sample, it is not surprising that the point estimates of $\zeta$ are below one.

Indeed, a fairer comparison of point estimates would be at the sectoral level, focusing on the Motor Vehicles and Parts sector. While the estimates of $\zeta$ based on the sales distributions in di Giovanni et al. (2011) report values close to one when all ﬁrms in the dataset are pooled, the sector-level estimates of $\zeta$ based on all sales reveal a diﬀerent picture with estimates varying between 0.422 to 1.279, for all sectors. In particular, for the automotive sector, their estimates are in line with our ﬁndings. When all sales are considered, the value of the power law exponent for the automotive sector is 0.538. When only domestic sales are considered, the value is 0.588, where the number of ﬁrms is 955. The estimate varies across exporters and non-exporters as well. For exporters, the value is found as 0.531 with 608 observations, while for non-exporters it is slightly higher, 0.651 with 347 observations. di Giovanni et al. (2011) compare power law estimates based on total and domestic sales using the French ﬁrm level data. They show that the bias introduced by selection into exporting is typically not large for France. Thus empirical power law estimates based on total sales probably give a reasonable estimate of the degree of dispersion in domestic sales as well. Table 4 reports estimates on the number of employees as the measure of ﬁrm size. The ﬁt is similar and the point estimates are slightly higher than those for revenue.

Table 3: Power Law Estimates of Firm Size by Revenue
 Method: CDF ln(Rank-0.5) MLE ζ$\zeta$ 0.578*** 0.500*** 0.334*** (0.0320) (0.083) (0.046) Constant 11.37*** -7.709*** (0.684) (1.787) Observations 53 54 54 R-squared 0.945 0.676
Note: Robust standard errors in parentheses. *** $p<0.01$, ** $p<0.05$, * $p<0.10$

Table 4: Power Law Estimates of Firm Size by Employees
 Method: CDF ln(Rank-0.5) MLE ζ$\zeta$ 0.668*** 0.601*** 0.471*** (0.0461) (0.091) (0.065) Constant 4.967*** -2.377*** (0.407) (0.833) Observations 53 54 54 R-squared 0.934 0.703
Note: Robust standard errors in parentheses. *** $p<0.01$, ** $p<0.05$, * $p<0.10$

Figure 1 presents the revenue results graphically. Panel A and Panel B show the empirical CDF and Power Law ﬁts in linear and logarithmic spaces, respectively, when total revenue is used as the ﬁrm size proxy. The Power law ﬁt is found to be slightly heavier tailed than the empirical CDF.

Panel C plots the eﬀect of threshold selection on the shape parameter estimates for ﬁrm size. We ﬁnd that the estimates are sensitive to the threshold with higher values of threshold associated with higher values of $\zeta$. The threshold we select, \$100,000,000 ($1×1{0}^{8}$, i.e. $log\left(threshold\right)=8$), corresponds to a plateau, after which the shape parameter estimates are not stable.

It is also important to note that higher threshold values signiﬁcantly decrease the sample size, which is plotted in Panel D of Figure 1. The eﬀect of the sample size on the ﬁrm size distribution is tested by Segarra and Teruel (2012) using Spanish manufacturing ﬁrms for the years 2001 and 2006. Their ﬁndings indicate that the sample size inversely aﬀects the estimated coeﬃcient of ﬁrm size distribution. They ﬁnd that estimates of $\zeta$ tend to be larger ($\zeta >1$) with small samples of large ﬁrms than compared to large samples that include smaller ﬁrms. The same ﬁnding applies for both sales and employees as the ﬁrm size proxy. In particular, in 2006, they ﬁnd that the estimated $\zeta$ of the largest 100 ﬁrms is 1.36 for sales and 1.66 for employees, respectively. However, when the whole sample of 60,000+ ﬁrms are considered, the parameter estimates decrease to 0.68 for sales and 0.97 for employees. They argue that increasing sample size has a negative eﬀect on the power law parameter.

#### 5.2 Firm Productivity

##### 5.2.1 Total Factor Productivity Estimates

Table 5 shows the estimates of the production function given in Equation (12), using ﬁrm-level data from the U.S. Motor Vehicles and Parts sector. As discussed earlier, OLS and Fixed Eﬀect estimates do not control for biases caused by simultaneity, and so we also consider OP and GMM estimates of the production function. We see that the OLS estimates for labor (capital) elasticity are slightly higher (lower) than the OP estimates. This is to be expected as coeﬃcients associated with variable inputs (e.g., labor and materials) are expected to have an upward bias, and the coeﬃcients associated with quasiﬁxed inputs (e.g., capital) are expected to be biased downward (Olley and Pakes1996). Further, the Fixed Eﬀect coeﬃcient for capital elasticity is unreasonably low, which has been a known source of concern in the productivity literature Ackerberg et al. (2007).

Table 5: Production Function Estimates for U.S. Motor and Parts Sector
 Method: N βl${\beta }_{l}$ SE βk${\beta }_{k}$ SE OLS 246 0.83*** 0.05 0.25*** 0.05 Fixed Eﬀects 246 0.47* 0.27 0.003 0.02 Olley-Pakes 189 0.80*** 0.05 0.30*** 0.03 GMM 189 0.70*** 0.07 0.20 0.15
Note: A second order polynomial is used in the OP estimation. *** $p<0.01$, ** $p<0.05$, * $p<0.10$

Firm-level studies generally show that there is a large and persistent diﬀerence in productivity between ﬁrms even within narrowly deﬁned industries. Using all of our measures of productivity, we ﬁnd that dispersion in the U.S. Motor Vehicles and Parts sector ranges from a low of 1.35 (OLS) to a high of 1.93 (LP), where dispersion is calculated as the diﬀerence between the 90th and the 10th percentile of log TFP. For the four-digit U.S. manufacturing industries, Syverson (2004) found a mean dispersion in the range of $\left[0.65-1.42\right]$, and so the calculated dispersion in this sector is at the high end of these estimates. Our results thus suggest that there is more room for aggregate productivity increase in the U.S. Motor Vehicles and Parts sector as resources get reallocated from the less to the more productive ﬁrms.

Finally, we examine how the diﬀerent productivity measures relate to one another. Table 6 shows the correlation between all of our calculated productivity measures. The correlations between labor productivity and the TFP estimates are generally high, around $0.70$ to $0.80$. Similarly, correlations between the index-based TFP measure and the estimated TFP are also high, ranging from $0.75$ to $0.85$. One exception is the TFP measure estimated with Fixed Eﬀects which has a very low correlation of 22 percent with the index-based TFP measures. Turning to the relationship among estimated TFP measures, correlations are very high, reaching nearly one between OP and GMM estimates.18 Again though, the Fixed Eﬀects based TFP measure shows low correlation with the other estimated TFP measures. Thus, given the earlier concerns over Fixed Eﬀects estates, we only consider the OLS and OP methods for our estimation-based TFP measure in the subsequent analysis.

Table 6: Correlation between Productivity Measures
 LP Solow OLS FE OP GMM LP 1.00 Solow 0.41 1.00 OLS 0.80 0.75 1.00 FE 0.71 0.22 0.46 1.00 OP 0.72 0.74 0.97 0.38 1.00 GMM 0.60 0.84 0.91 0.30 0.95 1.00
Note: Correlation based on observations pooled at the ﬁrm level over years 2010-2015.

##### 5.2.2 Power Law Fit of Productivity Measures

We next estimate the power law for our four chosen ﬁrm productivity measures: LP, Solow, OLS and OP. Table 7 shows the estimates of the shape parameter for each measure (Figure 2 shows the ﬁt for each measure based on the MLE method). We see that there is some heterogeneity in these shape parameters based on the particular productivity measure as well as the estimation method used to identify the shape parameter. Despite these diﬀerences, the range of values $\gamma$ takes is around $\left[0.86-2.40\right]$, a relatively narrow interval. This increases our conﬁdence that the shape parameter for the U.S. Motor Vehicles and Parts sector is relatively robust to various productivity and estimation methods. These estimates are also in line with Spearot (2016) which show a shape parameter, averaged across countries, of $2.36$ for the Motor Vehicles and Parts sector.

A small shape parameter around $2$ implies a large dispersion of productivity among ﬁrms, with low-productivity ﬁrms capturing a small share of the market. On the other hand, in an industry with a large shape parameter, there is a large mass of low-productivity ﬁrms that represent a larger share of industry output. di Giovanni and Levchenko (2013) show that in the case when the ﬁrm size distribution is fat-tailed (small shape parameter), the incumbent ﬁrms in the industry are large and have a disproportionate share of overall sales compared to the small marginal ﬁrms and the welfare impact of trade is driven by incumbent ﬁrms rather than the marginal ones. Therefore, the contribution of the extensive margin to trade in the U.S. Motor Vehicles and Parts sector will be relatively small from reductions in trade costs.

Table 7: Power Law Estimates for Firm Productivity Distribution
 Method: γ$\gamma$(cdf) γ$\gamma$(rank) γ$\gamma$(mle) LP -1.72*** 0.86*** 1.11*** (0.07) (0.04) (0.15) Solow -2.40*** 0.88*** 1.67*** (0.11) (0.04) (0.25) OP -2.00*** 1.67*** 1.28*** (0.12) (0.09) (0.18) GMM -2.26*** 1.70*** 1.79*** (0.13) (0.09) (0.27)
Note: Robust standard errors in parentheses. *** $p<0.01$, ** $p<0.05$, * $p<0.10$

#### 5.3 Structural Parameters of Firm Heterogeneity

It is important to use appropriate values for the parameters in policy analysis because welfare predictions are highly sensitive to the value of the ﬁrm heterogeneity parameters. For instance, the elasticity of substitution translates the price diﬀerences across ﬁrms into diﬀerences in market shares and will have opposite eﬀects on each margin of trade (Kancs2010Hillberry and Hummels2013). As Kancs (2010) states, “the elasticity of substitution magniﬁes the sensitivity of the intensive margin to changes in trade barriers, whereas it dampens the sensitivity of the extensive margin” (Kancs2010, pp. 276). When the elasticity of substitution is high, the intensive margin is more sensitive to changes in trade barriers while the extensive margin is less sensitive. When elasticity is high, low-productivity ﬁrms are at a severe disadvantage because they can only capture a small market share. Therefore, their impact on trade ﬂows is marginal and small. However, with a low substitution elasticity, each ﬁrm has more market power over their unique variety and are in a sense more sheltered from the productivity competition. Therefore, new entrants are able to capture a higher market share and make a larger impact on trade ﬂows as well as welfare. Overall, the export sales by new entrants are largest when there is supply-side homogeneity (high $\gamma$) and demand-side heterogeneity (low $\sigma$) (Hillberry and Hummels2013).

The empirical work in previous sections allow us to ﬁnd the shape parameter of ﬁrm size, $\zeta =\frac{\gamma }{\sigma -1}$, and the shape parameter of productivity distribution, $\gamma$. We use four diﬀerent empirical methods to estimate TFP and three diﬀerent methods to ﬁt the TFP estimates to Pareto distribution. This exercise results in $3×4=12$, possible $\gamma$ estimates. We use two proxies for ﬁrm size and three methods to ﬁt ﬁrm size to a Pareto distribution, and this results in $2×3=6$ possible $\zeta$ estimates. When we use the $\gamma$ and $\zeta$ estimates to impute $\sigma$, we ﬁnd $2×3×4=24$ possible $\sigma$ values for the U.S. motor vehicles and parts sector. These values are provided in Table 8 .

As Table 8 shows, the estimators deliver slightly diﬀerent results. When revenue is used as the ﬁrm size proxy, the average value of $\sigma$ is found as 4.53. Overall, MLE method delivers higher $\sigma$ values compared to CDF and ln(Rank-0.5). The highest $\sigma$ value of 6.06 is obtained when productivity is estimated using GMM and ﬁrm size distribution ﬁt is obtained by MLE. Estimates for $\sigma$ are slightly lower when the number of employees is used as the ﬁrm size proxy. The average value in this case is 3.78. $\sigma$ values across estimation methods vary slightly compared to the revenue columns.

Table 8: Elasticity of Substitution (Firm Revenue)
 Method: CDF ln(Rank-0.5) MLE GMM 4.90 4.40 6.42 LP 3.97 2.72 4.36 OP 4.45 4.34 4.88 Solow 5.14 2.76 6.06
Note: Values based on estimates of shape of ﬁrm size and productivity distribution.

All the $\sigma$ values found in Table 8 are high compared to the corresponding $\gamma$ values. Most importantly, they do not satisfy the mathematical constraint of $\gamma >\sigma -1$, which results from the ﬁrm shape parameter of $\zeta <1$, as discussed in Section 5.1. As a counterfactual calibration analysis, we compare the $\sigma$ values in Table 8 to the benchmark Zipf’s Law where $\zeta =1$, which implies that $\gamma =\sigma -1$. Moreover, we compare them to the case where the mathematical constraint $\gamma >\sigma -1$ is satisﬁed, such as when $\zeta =2$. The resulting parameter values are reported in Table 9 and 10.

Table 9: Counterfactual Elasticity of Substitution ($\zeta =1$)
 Method: CDF ln(Rank-0.5) MLE GMM 3.26 2.70 2.79 LP 2.72 1.86 2.11 OP 3.00 2.67 2.28 Solow 3.40 1.88 2.67

Table 10: Counterfactual Elasticity of Substitution ($\zeta =2$)
 Method: CDF ln(Rank-0.5) MLE GMM 2.13 1.85 1.90 LP 1.86 1.43 1.56 OP 2.00 1.84 1.64 Solow 2.20 1.44 1.84

The counterfactual analysis of $\zeta =1$ delivers lower values for $\sigma$ and $\zeta =2$ deliver even lower values. Since the estimates for $\gamma$ are small for U.S. Motor Vehicles and Parts, the corresponding $\sigma$ values should also be accordingly small. Sectors characterized with high productivity heterogeneity tend to have diﬀerentiated varieties.

### 6 Concluding remarks

A growing literature incorporating ﬁrm heterogeneity in trade models has generated new economic insights on the overall impact of globalization. In the meantime, there still remain a number of empirical challenges for extending this broad framework to applied policy work. Model simulations have been shown to be quite sensitive to parameter values, thus making it paramount that the structural parameters are identiﬁed in a manner consistent with underlying theory, rather than relying on ad hoc values from other strands in the trade literature.

This paper addresses this gap in the literature by proposing a simple method to estimate the structural parameters of trade models with ﬁrm heterogeneity. When ﬁrm productivity follows a Pareto distribution with the shape parameter $\gamma$, ﬁrm size also follows Pareto distribution with shape parameter $\frac{\gamma }{\sigma -1}$. We can thus estimate both distributions using the established approaches in the literature and then use the estimated $\gamma$ and $\frac{\gamma }{\sigma -1}$ values to impute the corresponding $\sigma$ values. Using the same database and distribution, our proposed method is able to consolidate the estimation of structural parameters within the ﬁrm heterogenity framework, thus ensuring better informed parameter values and more reliable model predictions.

We illustrate this methodology by focusing on the U.S. Motor Vehicles and Parts sector using the ORBIS database. The estimates for $\frac{\gamma }{\sigma -1}$ are found to be in line with the estimates in the literature for motor vehicles sector. However, they are slightly lower than the values found in the literature when all manufacturing ﬁrms are pooled. Similarly, the estimates for $\gamma$ are in line with the literature for the motor vehicles sector. While the resulting values for $\sigma$ are close to the elasticity estimates for manufacturing sectors, they do not satisfy the mathematical constraint for a well-deﬁned ﬁrm heterogeneity model. We ﬁnd that smaller $\sigma$ values are required to satisfy the constraint given the estimates for $\gamma$.

A possible remedy to this ﬁnding could be to increase the sample size in the database. The number of observations for the U.S. Motor Vehicles Sector in our sample is relatively small which is found to result in smaller shape parameters for ﬁrm size distribution. Extending this analysis with a larger sample size is a potential venue for the next step of this study.

### Appendix

Table 11: Revenue (milions USD)
 Year Observations Mean Std. Dev. Min. Max. Median 2008 2 140 198 0 280 140 2009 7 2,692 6,631 3 17,710 18 2010 23 2,150 8,684 0 41,946 221 2011 64 8,107 25,913 0 150,000 797 2012 64 8,299 26,294 0 152,000 866 2013 65 8,584 27,597 0 155,000 984 2014 65 8,920 27,884 0 156,000 980 2015 57 8,707 28,061 0 152,000 1,403 2016 39 12,910 35,158 3 166,000 2,810

Table 12: Employees
 Year Observations Mean Std. Dev. Min. Max. Median 2008 2 823 1,147 12 1,634 823 2009 6 8,173 19,189 16 47,326 94 2010 20 4,330 11,352 8 51,623 1,121 2011 61 17,537 38,683 2 207,000 3,319 2012 59 18,209 40,503 3 213,000 4,400 2013 54 20,518 43,596 2 219,000 5,327 2014 56 21,650 43,690 3 216,000 5,930 2015 49 23,747 47,068 12 215,000 6,700 2016 34 33,159 56,679 1,700 225,000 11,493

Table 13: Capital (millions USD)
 Year Observations Mean Std. Dev. Min. Max. Median 2008 2 41 57 0 81 41 2009 7 2,395 6,236 0 16,536 8 2010 22 765 3,260 0 15,352 39 2011 60 1,495 4,567 0 23,790 126 2012 63 1,556 4,891 0 25,845 142 2013 65 1,659 5,350 0 29,250 142 2014 64 1,826 6,060 0 34,803 153 2015 57 2,085 7,816 0 51,401 185 2016 39 3,866 12,379 0 70,346 366

Table 14: Investment (millions USD)
 Year Observations Mean Std. Dev. Min. Max. Median 2009 2 7 10 0 14 7 2010 7 252 653 0 1,733 1 2011 21 136 558 -4 2,568 4 2012 61 397 1,350 -22 8,057 24 2013 63 445 1,538 -12 9,178 35 2014 63 499 1,992 -783 12,115 26 2015 56 735 3,402 -190 24,288 31 2016 39 1,509 5,210 0 29,025 85

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1There is an extensive literature on estimating trade elasticity using gravity models. See for example Anderson and van Wincoop (2003), Head and Mayer (2014) and Simonovska and Waugh (2014).

2For example, Spearot (2016) does not rely on a constant elasticity of substitution framework in estimating the shape parameters.

3Using French ﬁrm-level production data, di Giovanni et al. (2011) ﬁnd $\zeta$ close to 1 for their full sample of ﬁrms. However, when they separate the ﬁrms into exporting and non-exporting ones, the power law coeﬃcient for exporters is consistently lower than the full sample of ﬁrms.

4In particular, when sales is used as the measure of ﬁrm size, they ﬁnd $\zeta$ to range around $\left[0.825,1.019\right]$, depending on the particular estimation methodology. Results are similar when the number of employees is used as the measure of ﬁrm size.

5The LS estimation has generally been the more popular approach in the ﬁrm size distribution studies. However, in a test simulation, Clauset et al. (2009) show that the MLE has better performance, and that the LS regression methods can give signiﬁcantly biased values. Thus we use both these approaches in our power law estimations.

6An extensive review by Bottazzi et al. (2015) discusses that neither CDF nor PDF log-log estimators have strong properties for unitary tail inference (Zipf’s Law) for ﬁrm size. They report that especially the PDF estimator performs very poorly with pooled data. They argue that $Rank-1∕2$ and Maximum Likelihood estimators such as the Hill estimator prove to perform better, especially in small samples.

7See Hayakawa et al. (2012) for a broader survey on the causal mechanisms that allow productivity to have a key role in determining the eﬀect trade has on the ﬁrm.

9As discussed in Wagner (2007), evidence regarding the learning-by-exporting is somewhat mixed with only few studies showing post-entry diﬀerences in productivity growth between exporters and non-exporters. So exporting, in itself, does not necessarily improve ﬁrm performance.

10To account for intermediate inputs, value added (revenue-cost of inputs) should be used as the measure of output. However, value added is not always available in a ﬁrm’s ﬁnancial data. For example, many countries, including the U.S. do not require ﬁrms to provide information on material and labor costs, leading to insuﬃcient coverage of value added measures in the ORBIS database.

11The labor share is averaged over the 2008-2015 period and does not vary in the subsequent calculations of the TFP.

12Ackerberg et al. (2007) note that if capital is positively correlated with labor and labor has a higher correlation with ${\omega }_{it}$ then ${\beta }_{l}$ will be upward biased while ${\beta }_{k}$ will be underestimated.

13See Van Beveren (2012) for a more detailed discussion.

14Petrin and Levinsohn (2003) extend this framework, by using inputs, such as electricity or materials, instead of investment to control for the ﬁrm’s unobserved productivity. These inputs usually have more non-zero observations than investment, and so increases the eﬃciency of TFP estimates when used with manufacturing surveys of developing countries. However, the ﬁnancial data on the U.S. motor sector in ORBIS does not include a separate account for material costs, and so we continue to follow the OP framework in our analysis.

15To account for selection eﬀects, OP also include a term for predicted probabilities ${\stackrel{̂}{P}}_{it}$ in (16) where ${\stackrel{̂}{P}}_{it}$ are obtained from a probit model on ﬁrm survival with ${i}_{t-1}$ and ${k}_{it-1}$ as the main explanatory variables. However, Levinson and Petrin (2003) show that incorporating survival probability only leads to very small eﬃciency gains and so we do not include a correction for selection bias in our analysis.

16We pool the year-ﬁrm observations for those ﬁrms that have not yet reported for 2016.

17Gal (2013) suggests using external sources to impute a ﬁrm’s missing labor costs. Relying on industry level data such as the OECD STAN database, average labor cost per worker in a particular sector can be obtained by dividing the total labor cost by the number of employees per country, year and 2-digit industry level. This average cost is then multiplied by the number of ﬁrm employees in ORBIS to get the imputed ﬁrm-speciﬁc labor costs. However, this approach will only work if within-industry wage diﬀerentials are not too prominent, a feature unlikely to hold in the U.S. where empirical evidence has generally shown productive ﬁrms paying a greater premium on wages.

18Gal (2012) also shows that in practice there was not a lot of diﬀerence when using these two TFP measures.