Detecting misallocation: an empirical investigation through firm-level data

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NIVERSITÀ DI

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HESIS October 5, 2020

Detecting misallocation

An empirical investigation through firm-level data

Author:

Leonardo FLORI

Advisor: Prof. Federico TAMAGNI Co-Advisor: Prof. Andrea MINA

Master of Science in Economics Dipartimento di Economia e Management

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UNIVERSITÀ DI PISA

Abstract

Dipartimento di Economia e Management

Detecting misallocation

An empirical investigation through firm-level data

by Leonardo FLORI

The present work is centered on the economic idea of an optimal allocation of re-sources across productive plants. In the recent years, the academic debate paved the way for and explored new methods to empirically analyse this topic, starting from the seminal contribution by Hsieh and Klenow (2009). After a short literature re-view, mainly focused on the criticisms of such framework, we introduce the covari-ance between firms’ productivity and size as an alternative measure for allocative efficiency. Our main reference here is the work by Bartelsman, Haltiwanger, and Scarpetta (2013). Then, we present our own estimation of such covariance within manufacturing sector in France, Italy and Spain. The empirical exercise has been carried out through firm-level data at four-digit sectors. Finally, a robustness check with detailed French customs data is presented.

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List of Figures

1.1 Optimal allocation of capital subsequent to productivity shock. . . 6

1.2 Marginal productivity of capital in presence of totally fixed capital. . . 7

1.3 Productivity shock in HK framework. . . 9

1.4 Productivity shock with non-constant marginal costs and not

isoelas-tic demand . . . 10

2.1 French manufacturing firms’ employment and turnover distribution. . 14

2.2 Italian manufacturing firms’ employment and turnover distribution. . 15

2.3 Spanish manufacturing firms’ employment and turnover distribution. 16

3.1 French manufacturing firms’ employment and turnover distribution.

Export considered. . . 22

F.1 Distribution of not deflated turnover, 2012. . . 42

G.1 Effect of changing factors elasticity of output on average TFPR

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List of Tables

2.1 Firms’ turnover (thousands). Whole sample vs firms with missing

employment data. . . 18

2.2 Average LPR interquartile ratio in manufacturing. . . 18

2.3 Average TFPR interquartile ratio in manufacturing. Productivity has

been computed both through revenues as output and value added as

output. . . 18

2.4 Input allocation. Covariance and correlation between labor

produc-tivity and employment shares, and between total factor producproduc-tivity

and composite input shares. Average over the years 2009-2019. . . 20

2.5 Output allocation. Covariance and correlation between productivity

and market shares. Average over the years 2009-2019. . . 20

3.1 Output allocation in France. Covariance and correlation between

pro-ductivity and market shares computed only for domestic market.

Av-erage over the years 2009-2019. . . 23

F.1 Our sample’s coverage against Eurostat data - France, 2012. Empty

fields represent missing values. Nace Rev. 2 classification is used for sectors. Firms are divided according to their size class (number of employees). Sectors with only one reported value or no values at all

have been removed.. . . 37

F.2 Our sample’s coverage against Eurostat data - Italy, 2012. Empty

fields represent missing values. Nace Rev. 2 classification is used for sectors. Firms are divided according to their size class (number of employees). Sectors with only one reported value or no values at

all have been removed. . . 38

F.3 Our sample’s coverage against Eurostat data - Spain, 2012. Empty

fields represent missing values. Nace Rev. 2 classification is used for sectors. Firms are divided according to their size class (number of employees). Sectors with only one reported value or no values at all

have been removed.. . . 40

F.4 Number of sectors (at four digits level) with no data available for price

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1

Introduction

Economics deals, inter alia, with the optimal allocation of limited resources across the agents of an economic system. What “optimal” should mean though is a limit-less source of debate, and even the very notion of “optimum” in a social context can be questioned. Nonetheless, putting aside the deepest philosophical concerns about the definition of social welfare, the efforts toward a better understanding of the eco-nomic system are familiar to those who aim to ameliorate the society in which they live. Therefore, whoever wants to approach the study of social problems can not disregard the traditional and purely economic idea of optimum. As we were saying, finding the best allocation of resources in a context of scarcity is an authentically economic problem, and in this work we will discuss in particular the allocation of capital and labor across producers in light of the existing and more up-to-date (to our best knowledge) economic literature.

Obviously, such a wide argument would deserve entire volumes to be treated exhaustively, and this is not the case. The present work will focus on two specific ways of modeling the phenomenon of misallocation: the first one builds on the ap-proach initiated by the seminal papers by Restuccia and Rogerson (2008) and Hsieh and Klenow (2009). This approach has been called “indirect”1 _{in that it should}

al-low to detect the extent of the overall misallocation in an economy without caring about its sources. The second one refers to the works by Olley and Pakes (1992) and Bartelsman, Haltiwanger, and Scarpetta (2013), and it takes into account the relation between productivity and size of the firms.

The dissertation is organized as follows: the first chapter presents an overview of the recent literature with a specific focus on the basic model and on the criticisms that have been raised against it. The second chapter illustrates an alternative model to de-tect misallocation and it includes our empirical estimates on three samples (French, Italian and Spanish data). The third chapter presents a robustness check through detailed French customs data. Final remarks will conclude.

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3

Chapter 1

Economic modeling for

misallocation effects

1.1 The indirect approach

The literature about misallocation effects on TFP stems from the wider debate on GDP per capita differences across countries, which in recent years has been increas-ingly gaining attention. In fact, as more and more authors agree on the importance of TFP discrepancies in explaining countries inequalities, asking what could cause such dissimilarity has rightfully become a key question.

The idea that a non-optimal allocation of the productive factors might signifi-cantly affect the growth of an economy is not new: for example, Banerjee and Duflo (2005), along with a wide criticism to the standard aggregative approach of growth theory1, stress the role of inefficiencies spread all over the market, and point out the huge role that a nonoptimal resource allocation plays in economic modeling -in particular, they focus on the role of credit market imperfections. However, our analysis links directly to the literature branch initiated by Hsieh and Klenow (2009) - henceforth, we will refer to their framework using the acronym “HK”. Their semi-nal paper deals notably with the effects of capital misallocation on TFP - thus putting aside the traditional focus on cross-country technology divergence as an explanation for TFP differences. They basically develop a model of monopolistic competition, introducing firm-level distortions in the cost of factors and the price or quantity of output. The underlying idea is somewhat simple: assuming a marginalist view, in an ideal, competitive and frictionless environment the cost of capital should equate its marginal return, and it should be the same across all the firms within an econ-omy. However, if some kind of distortion exists, the firms in the economy could face different marginal costs, and therefore exhibit different marginal productivity levels. In the end, this will result in a lower TFP compared to that obtainable by redirecting financial investments (or workers) from less productive firms to those with higher potentiality. A similar approach has been undertaken by Restuccia and Rogerson (2008) in those same years, yet for the most of the following literature the HK model has become the starting point on which extensions and modifications can be built.

The present dissertation does not have the purpose to provide a thorough and complete review of the literature regarding the relation between misallocation and TFP, therefore we will not go through a detailed list of authors who dealt with it. It will be enough to recall the classification proposed by Restuccia and Rogerson (2017) about the two methodological approaches generally adopted in this field. They label

1_{Their contribution is neither the first to analyze the limits of the standard aggregative growth}

model nor the most critical of the neoclassical approach. It is nevertheless an important overview paper of high relevance within the specific strand of research we will consider - see for instance Hsieh and Klenow (2009) and Restuccia and Rogerson (2017).

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as direct approach those studies which try to identify specific sources of misallocation and to assess their impact on productivity - a longstanding tradition in public finance models; on the other hand, the indirect approach avoids seeking the exact distortion in the economy, while focusing instead on the estimates of the losses due to such misallocation. The HK framework employs this second methodology: in fact, the focus is not on why such misallocation arises, but rather on trying to detect the extent of this phenomenon and its effect on the aggregate level.

The strand of the literature that has followed mostly built on this approach, either extending the basic model or modifying some marginal assumptions. At the same time, however, many objections have been raised against it.

1.2 The HK model

Hsieh and Klenow (2009) outlined a theoretical model to infer and support a new way to detect allocative inefficiencies. In particular, they proposed to estimate the ex-tent of market distortions through the variance of TFPR, that is the revenue product in terms of units of the “composite” input of production. The idea behind this strat-egy is that the revenue product of an input is proportional to its marginal revenue product: in other words, if distortions are present, marginal products are different across firms, and this will result in a non-zero variance of the revenue productivity. Such distortions are of two types: the first is an inter-firms difference in the cost of inputs, τk, which affects the capital-labor ratio. The second is related to revenues,

τy, and we can think of it as a tax or a quota on production.

The production technology is assumed to be a Cobb-Douglas with constant re-turns to scale: TFPRi= PiYi Kαs i L 1−αs i (1.1) where Piis firm-level price, Yiindividual production, Ki and Lithe classic input of production capital and labor, αs the capital share common to all firms within a sector. Using this functional form, it can be shown that:

TFPRi∝(MRPKi)αs(MRPLi)1−αs (1.2) where MRPKi and MRPLi are respectively the marginal products of capital and labor - the full model is illustrated in detail in AppendixA.

There are two major reasons why this literature uses TFPR as main variable un-der study. The first reason traces back to Foster, Haltiwanger, and Syverson’s (2008) important contribution: they find an empirical confirmation of the predictable neg-ative correlation between prices (Pi) and physical productivity (Ai, which is often referred to as TFPQ). Basically, in an undistorted HK environment, firms with differ-ent levels of productivity are allowed to coexist, but more productive firms will set lower prices. The fundamental assumption is that in absence of distortions prices should completely offset deviations in productivity, leading to the same TFPR for all the firms.

The second reason why TFPR plays such a big role is practical: plant-level price data are difficult to recover, so that researchers are forced either to make troublesome assumptions or to deal with revenues related measures.

The seminal paper by Hsieh and Klenow (2009) has undoubtedly had the merit of initiating an alternative approach for the study of misallocation effects, but as we will point out in the following sections it displays many limitations, and as of

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1.3. First reactions to the HK model 5 today, after more than a decade, it seems close to be abandoned in favor of its more sophisticated descendants.

1.3 First reactions to the HK model

The work by Hsieh and Klenow (2009) has attracted many reactions and criticisms. This section is dedicated to two papers which provided some first examples of the failure of the explanatory potential of HK framework. Right after its publication, scholars realized that the simple model above outlined was lacking the important dimension of industrial dynamics over time, and it turned out soon that problems like firms’ choice under uncertainty or operating fixed costs could have not been easily neglected.

1.3.1 Labor allocation with overhead costs

In their paper, Bartelsman, Haltiwanger, and Scarpetta (2013) developed a model in which overhead labor is introduced, whose simplest version is reported in Appendix C. The main implication of assuming that a fixed part ( f ) of labor does not contribute to the proper production process entails that even though revenues per effective unit of labor are equal across firms in absence of distortions, the measure generally used to detect productivity could still be varying. As we show in AppendixC, for instance, labor productivity would be:

LPR_i = PiYi n∗_i = C 1−τi − C f (1−τi)ni (1.3) which means that even when τi (individual distortion) is zero for each firm, the variance of productivity could be not. The same reasoning naturally applies also to HK measure for misallocation, that is the variance of TFPR: in the presence of overhead labor, this measure could still be different across firms, even when the actual productivity is the same.

It is worth noting that the result of this model does not invalidate the theoretical construction of HK framework per se. Rather, it questions the eligibility of TFPR variance as a proper measure for detecting inefficiencies.

1.3.2 Capital allocation with adjustment costs

Asker, Collard-Wexler, and De Loecker (2014) set up a model which shows how variation in TFPR could not necessarily reflect the presence of distortions within the economy in the presence of a dynamic choice of capital by firms which face the problem of adjustment shocks. Rather than illustrate the whole content of their study, we prefer to summarize the main takeaway from their model. In order to do this, we refer to Figures1.1and1.2for clarity.

Let us imagine an environment in which firms have the same production technol-ogy, with decreasing returns for capital, but in which they could face an exogenous shock in productivity (A) every period. In a context of perfect and free mobility of capital (Figure1.1) this would not create problems, as the economy would reallocate more capital to the most productive firms, and viceversa. So far, HK framework could still hold, since we would not observe any dispersion in the marginal product of capital across firms.

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(A) Production function

(B) Marginal product of capital

FIGURE1.1: Optimal allocation of capital subsequent to productivity shock.

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1.3. First reactions to the HK model 7

(A) Production function

(B) Marginal product of capital

FIGURE 1.2: Marginal productivity of capital in presence of totally fixed capital.

However, if we imagine that adjustment costs are not negligible, in a dynamic perspective the optimal problem of each firm must take into account this fact. Since firms make their investment decisions ex-ante with respect to their productivity shock, the resulting allocation in the next period could be far from optimal, leading to dis-persion in the variance of the marginal products of capital. In the extreme case of totally fixed capital (Figure1.2) we have the highest variance of MPK. Can we even talk about misallocation in this framework? Of course, this equilibrium could be improved if adjustment costs were eliminated, but this sounds somewhat an ideal rather than a concrete economic situation. More importantly, this model predicts a positive relation between the variance of productivity and the variance of marginal products, in that the more uncertain the future is, the harder for productive plants to guess right the suitable level of capital. In the end, this model unveils that adopting the HK measure of TFPR dispersion could detect different levels of uncertainty of the economic environment instead of proper misallocation.

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1.4 The “constant returns to scale” assumption

As it is shown in AppendixA, the HK model assumes that firms are endowed with a constant returns to scale (CRS henceforward) technology. How much does this as-sumption affect the soundness of the model? This point has been discussed by Gong and Hu (2016): they find that when this assumption is proven wrong - and this hap-pens in most cases - taking the dispersion of TFPR as a measure for misallocation brings about an overestimation of the phenomenon. They show that the TFPR of the single company depends both on factor distortions and on its own productiv-ity, therefore its variance is not generally a good measure for misallocation. As a consequence, they propose to estimate the marginal revenue product of each factor as an alternative - see AppendixBfor the mathematical evidence. This conclusion looks more intuitive than that of adopting the variance of TFPR from a conceptual point of view, since theoretically, whereas marginal returns should be equalized in an optimal allocation, the identity of revenue productivity across firms is not so straightforward.

Even though this contribution does not question the Cobb-Douglas functional form, it represents a step forward along the direction indicated by the seminal pa-per by Hsieh and Klenow (2009), since it provides a more general framework for the analysis. Nonetheless, the contribution by Gong and Hu (2016) deserves to be mentioned for a further reason. Apparently, relaxing the CRS assumption does not have a severe impact on the HK model: at first glance, it only leads to reformulate the equation for the efficiency gap of the final output. However, leaving aside this assumption would have been much more disruptive for the consistency of the orig-inal paper. In fact, among the robustness checks, Hsieh and Klenow (2009) also try to address the problem of possible varying capital-to-labor ratios: since they assume CRS, they are able to calculate firm specific capital shares on total output and then recalculate TFPR. It turns out that output distortion still plays a major role, according to their interpretation. Unfortunately, when CRS are no longer a plausible assump-tion, we are not allowed anymore to use capital shares on total output as parameters of the production function. The problem of firms’ technology heterogeneity arises, and it is anything but trivial.

1.5 Functional form and heterogeneity

The most glaring issue that shows up while reading the paper by Hsieh and Klenow (2009) is probably that of functional form. Their whole construction is basically based on the assumption of a Cobb-Douglas production function, equal for all the firms. Moreover, the aggregation of sectors is made through a Cobb-Douglas as well. While the latter is addressed by the author themselves2, the assumption about firm level production function is hardly touched upon. Yet, as noted by Restuccia and Rogerson (2017), this is crucial to interpret the whole dispersion of TFPR as misallo-cation. What if producers actually differ in terms of production function? What if the efficient allocation does not imply equality of capital-to-labor ratios across produc-ers? As we can see, the problem of firms heterogeneity is strictly tied to the choice of a functional form.

In this section we will report the main content of the work by Haltiwanger, Kulick, and Syverson (2018), which definitively dismantles the general applicability

2_{They essentially argue that using a CES function modestly change the estimates, but does not affect}

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1.5. Functional form and heterogeneity 9

FIGURE1.3: Productivity shock in HK framework. The absolute price variation is exactly equal to the productivity shift. Notice that this happens because we have simultaneously constant marginal costs

and isoelastic demand.

of the HK framework, revealing the huge misspecification problems by which such an approach is affected. As a matter of fact, their criticism is twofold, as they point out that very strict assumptions on both sides of the market (demand and supply) are needed by the HK model to be valid.

In a nutshell, the core argument by Haltiwanger, Kulick, and Syverson (2018) is that, in order to interpret TFPR dispersion as misallocation, we need to have a negative unit elasticity of producer’s price to its own productivity, and in turn for this condition to be satisfied we need the following two assumptions:

• isoelastic demand

• marginal costs that are jointly constant and negative unit elastic with respect to TFPQ

The analytical proof of these crucial results is reported in Appendix D. Below we prefer to limit ourselves to a graphical representation for sake of readability. In order to figure out what these two assumptions involve, let us consider an isoelastic demand

yD_i =Dp−ε

i

whereby, given a production technology yS_i, we can obtain: revenues piySi =D 1 εy ε−1 ε ⇒ log MR=logε −1 ε + 1 εlog D− 1 εlog yi

where MR stands for marginal revenues. Combining this together with constant marginal costs, it is easy to see that changes in productivity are totally offset by price movements, as shown in Figure1.3.

However, when even only one of these two conditions is not satisfied, this com-pensation does not fully occur, as Figure1.4shows. Analogously, as Haltiwanger, Kulick, and Syverson (2018) show in the appendix, a demand shock does not change TFPR in HK setting, but when isoelastic demand or constant marginal costs are aban-doned TFPR is affected.

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FIGURE 1.4: Productivity shock when marginal costs are non-constant (left-hand side) and demand is not isoelastic (right-hand

side).

1.6 Measurement issues

The most recent contribution that questions a broad applicability of HK approach comes from one of its authors himself. Admittedly, the concern about mismeasure-ment was already present in the paper by Hsieh and Klenow (2009), but only ten years later it has been treated as a major issue. Bils, Klenow, and Ruane (2017) pro-pose a method to correct the estimates of efficiency gaps when potential mismea-surement is taken into account. This concern stems from the concrete difficulty to rely on book value measures or other statistics reported by private parties.

Basically, they assume additive measurement errors across plants that hinder the correct detection of productivity shocks. For instance, the presence of a systematic overstatement of revenues makes its measured percentage growth, in response to input growth, lower than it is in reality. Therefore, their strategy is to regress revenue growth on input growth for each decile of average product, gathering all different elasticity estimates and inferring mismeasurement if they decrease with higher level of average product.

Needless to say, this strategy rests on heavy assumptions as well. However, it provided its authors with a correction of the original estimates and stressed the im-portance of measurement issue for this strand of research.

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Chapter 2

An alternative approach for

detecting misallocation

In the previous chapter, we have reviewed the idea and the vulnerability behind the HK framework. In order to avoid many of its strict assumptions, it would be desirable to turn to an alternative approach. In what follows, we will present both the contribution by Bartelsman, Haltiwanger, and Scarpetta (2013), who actually drew on a previous clue by Olley and Pakes (1992), and our empirical exercise, based on such approach.

2.1 The OP covariance

In an important contribution, Olley and Pakes (1992) - henceforth OP - propose to analyse the allocative efficiency of the US telecommunication industry through the following index:

N

∑

i=1

(ω_it−ω¯t)(sit−¯st) (2.1)

where ω stands for some measure of productivity, s to each firm’s weight within the industry, a bar over a variable its unweighted mean, and finally the subscripts i and t refer respectively to the i-th firm operating in year t. In particular, these authors chose to consider TFP as a measure of productivity and market shares as firms’ weights. The underlying idea is that an efficient allocation should guaran-tee that more productive firms gain larger production quotas. Quoting Bartelsman, Haltiwanger, and Scarpetta (2013), the OP covariance represents

the extent to which firms with higher than average productivity have a higher than average share of activity.

Also, the OP covariance can be easily proven to be the difference between weighted and unweighted productivity of a sector: in other words, how far we are from as-signing to each firm, regardless of its efficiency, the same share of inputs or output.

In addition to being intuitive and not excessively demanding in terms of assump-tions, Bartelsman, Haltiwanger, and Scarpetta (2009) recall another advantage of this method:

Measurement problems make comparisons of the levels of either of these measures across sectors or countries very problematic, but taking the dif-ference between these two measures reflects a form of a difdif-ference-in- difference-in-difference approach.

Bartelsman, Haltiwanger, and Scarpetta (2013) draw on this intuitive measure to intervene in the misallocation debate and advocate its superiority over the HK

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approach. Along with a theoretical criticism of TFPR variance as a measure for mis-allocation (cf. Section1.3.1), they simulate the effects of changing the dispersion of firm-specific distortions on different indicators. They take into account the stan-dard deviation as well as the OP covariance for labor and total factor productivity. It turns out that OP covariance seems to be a more robust measure to infer distor-tions in the economy, especially when a positive correlation between idiosyncratic productivity and distortions is assumed. Their findings are encouraging as for the hypothesis that country-level distortions could hinder an optimal allocation of labor input across firms, reflected by the difference in OP covariances that various coun-tries display.

Henceforth, our aim is to reconnect with this strand of the literature providing a tentative, empirical analysis of misallocation through the OP covariance within the manufacturing sector of France, Italy, and Spain. We also define “OP correlation” the Pearson correlation index between productivity and activity shares. Ideally, this index should give us a clue about how much the two variables are correlated, while the covariance only tells us about the contribution of allocative efficiency to the ag-gregate productivity.

2.2 Data cleaning

As it should have emerged from the previous paragraph, the measure of misallo-cation on which the present work will build needs to define two basic elements: a measure of productivity and the right weight to assign to each firm as compared to the whole industry. The choice of each of them has been naturally constrained by data availability, so that a preliminary description of the data set is necessary. We downloaded from Orbis database balance sheet data for French, Italian and Spanish firms. The reason why we have chosen these countries hinges upon the concerns about the representativeness of the samples. In order to construct the data set we needed, we followed some hints contained in Kalemli-Ozcan et al. (2015): by com-paring the statistics gathered in this paper with Eurostat data coverage, it turned out that France, Italy and Spain were the three big EU countries with the highest sample representativeness.

Orbis database has some non trivial limitations, since the download of big pan-els is terribly time consuming and only a limited number of observations can be downloaded at a time. Therefore, our choice has been necessarily conservative, try-ing to maintain the data set as representative as possible while eliminattry-ing those not strictly required observations from the very first moment. As a consequence, first we selected only firms operating in the manufacturing sector; this is actually in line with many previous studies, including Hsieh and Klenow (2009) and Gopinath et al. (2017). The reason is that other sectors, such as agriculture, not always report infor-mation about some inputs of production like materials or number of employees (it is mandatory only for listed companies)1.

Another criterion of selection has been the consolidation code available in Orbis: we decided not to take into consideration firms with a limited number of financial

1_{In our particular case, Italian firms required to file accounts are the following: S.p.A. (Società per}

Azioni), S.r.l. (Società a responsabilità limitata), Sapa (Società in accomandita per azioni), Società Co-operative, Società Consortili, G.e.i.e, Società di persone (only consolidated accounts), Consorzi con qualifica di Confidi, Società a responsabilità a socio unico and società per azioni a socio unico. French firms are sociétés à responsabilité limitée (SARL et EURL), sociétés de personnes (sociétés en nom collectif and sociétés en commandite simple), under certain conditions; sociétés par actions (sociétés anonymes, sociétés par actions simplifiées et sociétés en commandite par actions), foreign firms whose

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2.2. Data cleaning 13 items (these are precisely classified by LF or NF - see the Orbis Internet User Guide for more details). Furthermore, we only included unconsolidated accounts (consoli-dation codes U1 and U2) or consolidated accounts with no unconsolidated available (C1), in order not to repeat information for holding companies.

We selected the time stamp “Relative years” instead of “Absolute years”, because there could be firms lacking some records over the years, and this method allows us to have more available information2_{. We also dropped those firms whose fixed}

assets, labor cost and value added were not available.

Being mindful that Orbis database often lacks information about small compa-nies, we cared about this issue by comparing the national distribution of employ-ment and turnover3 with our data. In Figures 2.1, 2.2 and2.3 we report an illus-trative graphical representation: it can be noticed, as expected, that the sample is highly unbalanced, especially when it comes to small-sized enterprises. Moreover, since a reduction in the amount of revenues came up after having deflated the vari-ables (see FigureF.1for a preliminary distribution), we suspected that there could be some sectors (at four digit level) whose price index is not reported, thereby dis-appearing from the sample. This is exactly the case in our data - see SectionF.1in the appendix for more details. This means that our final estimates will not cover the whole manufacturing sector since some industries are not included.

Another remark is also due: Figure2.1a,2.2aand2.3ashow that big firms in our sample employ more than actually reported by Eurostat - this is particularly straight-forward for France. Similarly, FigureF.1aexhibits revenues that are higher in our sample than those reported by Eurostat, as regards big firms. This has to do with the fact that Orbis database collects information from firms’ financial statements, which means that in case of multinational companies data for plants all over the world are displayed. This problem could be faced with either plant level or export data, but in the absence of both we had to decide whether to discard some companies from the sample or to settle for this flawed data set. We opted for the latter, for two rea-sons: first, whatever decision about elimination of firms from our sample would be dangerously arbitrary due to the lack of information about companies’ cross-border activity. Second, our aim is to detect performance-wise measures, so that we can as-sume, admittedly with some brave approximation, that the within firm productivity is the same across all its own plants. Rather, a serious concern involves input and output shares: these will be undoubtedly inflated for multinational companies.

Finally, a sector-by-sector coverage investigation is reported in TablesF.2,F.1and F.3. In the end, looking at the share of each size class in our samples, it seems that Spanish data are the most representative, while French ones the worst, both in terms of labor inputs and revenues.

venue is abroad but have at least one subsidiary in France, sociétés d’exercice libéral (SELARL, SE-LAFA, SELCA, SELAS), sociétés coopératives and unions under certain conditions. Spanish firms are Societad Limitada and Societad Anonima

-2_{We have converted the relative into the absolute year through the “closing date” variable. If the}

closing date was before the 1st of June, we assigned the year preceding that of the closing date; other-wise, the same of the closing date.

3_{Eurostat provides information about five size classes: 0 to 9 employees, 10 to 19, 20 to 49, 50 to 249}

and more than 250. For each category we have data about the number of employees and the total sales value.

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1.183% 10.9% 1.098% 7.1% 2.669% 12.2% 7.025% 23.7% 88.024% 46.0% 0e+00 1e+06 2e+06 0−9 10−19 20−49 50−249 250+

Class size (number of employees)

Number of emplo y ees Sample Eurostat Orbis (A) Employment distribution 0.749% 6.7% 0.778% 3.7% 2.410% 8.4% 6.599% 19.1% 89.465% 62.1% 0e+00 2e+05 4e+05 6e+05 0−9 10−19 20−49 50−249 250+

Re v en ues (thousands) Sample Eurostat Orbis

(B) Turnover distribution. Orbis data have been deflated.

FIGURE2.1: French manufacturing firms’ employment and turnover distribution. Comparison between Eurostat data and our sample.

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2.2. Data cleaning 15 7.7% 15.29% 11.6% 14.94% 17.4% 17.66% _24.7% 25.00% 38.5% 27.10% 0 250000 500000 750000 0−9 10−19 20−49 50−249 250+

Number of emplo y ees Sample Eurostat Orbis (A) Employment distribution 6.8% 9.49% 10.6% 9.18% 18.3% 13.96% 28.9% 25.71% 35.4% 41.66% 0e+00 1e+05 2e+05 3e+05 0−9 10−19 20−49 50−249 250+

FIGURE2.2: Italian manufacturing firms’ employment and turnover distribution. Comparison between Eurostat data and our sample.

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11.1% 15.29% 9.6% 14.94% 15.0% 17.66% 22.2% 25.00% 42.2% 27.10% 0 250000 500000 750000 0−9 10−19 20−49 50−249 250+

Number of emplo y ees Sample Eurostat Orbis (A) Employment distribution 6.12% 7.1% 6.28% 5.3% 13.26% 11.1% 27.70% 23.3% _46.64% 53.3% 0 50000 100000 150000 200000 250000 0−9 10−19 20−49 50−249 250+

FIGURE2.3: Spanish manufacturing firms’ employment and turnover distribution. Comparison between Eurostat data and our sample.

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2.3. Productivity and misallocation estimates 17

2.3 Productivity and misallocation estimates

2.3.1 Which measure for productivity?

The analysis of allocative efficiency that we are setting up will be heavily shaped by the productivity measure which will be chosen. Ideally, a researcher would like to be able to sort all the companies within a sector according to a reliable proxy of productivity, or production efficiency, in order to establish the extent to which each company is assigned the share of available resources it deserves. Of course, this view presumes that a social optimum exists, and it is not exempt from criticisms. It also presumes that a consistent way of estimating such productivity is at hand, otherwise the empirical exercise would turn out to be meaningless. Many papers dealt with this issue, and a good review is contained in Syverson (2011), who well describes the major difficulties that are in place when one tries to recover a reliable measure of production efficiency. The first concern that one should care about is which measures of output and input should be taken into account. A widely used approach only considers labor productivity (LP), that is how many units of output are produced on average by each unit of labor (for example, an employee). Even if LP owns the valuable characteristic of being easy to calculate and to interpret, on the other hand it completely overlooks the contribution to the final output of other inputs. In other words, if we have in mind a production technology which requires both capital and labor, we would risk to wrongly assess two companies as equally efficient while they are not: behind the same output-labor ratio very different amount of invested capital could hide.

The alternative measure generally considered is Total Factor Productivity (TFP), which takes into consideration inputs other than labor, and their respective weights in determining production. However, the higher accuracy that TFP should grant comes at a non trivial cost, namely the assumption that we have to make about pro-duction technology’s functional form. As neither of these two measures is exempt from risks, we decided to consider both of them. More specifically, as it is common in this literature, we assumed a sector specific Cobb-Douglas function to estimate TFP.

In the following sections we will adopt the acronyms LPR and TFPR, mindful of Foster, Haltiwanger, and Syverson’s (2008) important remarks on the distinction between effective technical efficiency and the more blurred measure based on rev-enues - potentially affected by the confounding effects of idiosyncratic demand (cf. Section1.2).

2.3.2 Productivity estimates

We computed LPR as the ratio between value added (revenues minus cost of inter-mediate materials) and number of employees, after having deflated the variables with 4-digit sector specific price indexes4. A closer look at our data could be worth it: as shown in Table2.1, firms with no reported value for number of employees are generally small, and if a positive correlation between size and productivity exists, a selection bias is likely to be in place. In other words, our misallocation estimates risk to leave aside small and less productive firms when LPR is involved.

Table2.2reports, for each year and country, the mean across sectors of the inter-quartile ratio of the average product of labor: as already noted in Syverson (2004), even within the same four-digit sector, there is a great dispersion in performance

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TABLE2.1: Firms’ turnover (thousands). Whole sample vs firms with missing employment data.

Sample Median Mean

France Whole sample 699 32480

Only firms with no information about employment

529 9505

Italy Whole sample 944 8092

112 1326

Spain Whole sample 496 7344

103.1 1385.3

TABLE2.2: Average LPR interquartile ratio in manufacturing.

2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019

France 2.18 2.29 2.27 2.24 2.35 2.24 2.20 2.29 2.23 2.32 1.80

Italy 2.67 2.65 2.89 3.02 3.13 3.06 3.17 3.27 3.43 3.26 2.56

Spain 2.94 2.96 3.11 3.08 2.97 3.01 3.01 2.91 2.95 2.72 1.68

indicators. On average, in the same industry there are firms that coexist with others that can be twice or three times more productive. This is persistent in the last decade, particularly in Italy.

As far as TFPR is concerned, we estimated it as the residual of the regression of log-revenues on log-capital and log-labor - for more details, see AppendixE.

In Table2.3 we reported, as already done with LPR, the average across sectors of the 75th to 25th percentile ratio of TFPR distribution. Contrasting Table2.3with Table2.2, it can be noticed that LPR dispersion is persistently higher than TFPR one. This is consistent with the evidence found in Syverson (2004) and Bartelsman, Halti-wanger, and Scarpetta (2013). Admittedly, the ratio found in these papers ranges from 1 to 1.5, while ours are larger. In particular, when labor and capital elastic-ity are estimated using revenues as output, the ratio exceeds 2 almost every time,

TABLE2.3: Average TFPR interquartile ratio in manufacturing. Pro-ductivity has been computed both through revenues as output and

value added as output.

Output 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 France Revenues 2.08 2.11 2.15 2.17 2.12 2.13 2.14 2.14 2.08 2.11 1.98 Value Added 1.68 1.73 1.74 1.74 1.75 1.72 1.72 1.72 1.70 1.69 1.46 Italy Revenues 2.02 2.06 2.58 2.58 2.34 2.29 2.32 2.86 2.82 2.35 1.93 Value added 1.73 1.75 2.14 2.09 1.95 1.91 1.91 2.24 2.28 1.99 1.58 Spain Revenues 2.30 2.29 2.40 2.32 2.27 2.27 2.27 2.29 2.29 2.15 1.30 Value Added 1.80 1.78 1.87 1.79 1.75 1.77 1.77 1.76 1.78 1.69 1.21

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2.3. Productivity and misallocation estimates 19 whereas when Value Added is adopted, it ranges from 1.2 to 2, depending on the country - similarly to the LPR case, Italy displays the highest dispersion in produc-tivity. Since the main difference in adopting either one or the other proxy for output is the extent of returns to scale (higher with value addded than with revenues), we tested with a simple simulation whether returns to scale could play a role in de-termining the measured dispersion of TFPR - see AppendixG. The results seem to confirm our impression.

2.3.3 Input allocation

In this section we explore the relation between productivity and size in terms of in-puts. The estimates of the OP covariance between these variables should reflect the extent to which more productive firms either employ more people or have access to more capital. Therefore, the exercise is performed using employment shares when labor productivity is involved, and a composite input when dealing with total factor productivity. In particular, the latter is simply the sum of log-labor and log-capital, each of them multiplied by the associated elasticity - see AppendixEfor the estima-tion.

In Table 2.4, our main results are displayed. We produced estimates for OP covariance and correlation at 4-digit level, and then we aggregated all subsectors through their respective share of turnover on the total - exact values of sectors’ turnover has been taken from Eurostat5. Thus, we averaged such values over the years. What we obtain is hardly comparable with other papers: indeed, both Bartels-man, Haltiwanger, and Scarpetta (2009) and BartelsBartels-man, Haltiwanger, and Scarpetta (2013) find that the covariance between labor productivity and employment share in Western European countries ranges from 0.15 to 0.4. In particular, France seem to be above 0.2 in both papers, while our estimate is lower. Nonetheless, we have to bear in mind that these computations are all from different years.

Possibly more interesting here, recalling that OP covariance is nothing but the difference between the weighted and unweighted average of log-productivity within manufacturing, we can interpret these values as follows: OP covariance represents the percent contribution to sector aggregate productivity with respect to the theoret-ical situation in which each firm is assigned the same share of activity, regardless of its productivity. Therefore, as an example, Spain seems to be the country in which allocative efficiency plays the biggest role, considering both productivity measures. As regards the correlation between size and productivity, we can notice that overall it is quite low when LPR and employment shares are considered, while it is relatively high in the case of TFPR and the composite input.

2.3.4 Output allocation

In this section, the same estimates are performed, but using market shares instead of input shares. The idea is to see whether more productive firms gain bigger portions of the market. Once again, we aggregated the 4-digit level estimates by weighing each sector with its share on total turnover (Eurostat data on manufacturing annual turnover).

Table2.5shows our results. Again, to our knowledge, these values are difficult to compare with the existing literature, as the data we are dealing with are different from those used by other authors.

5_{Clearly, as we lack many 4-digit sectors, shares have been computed not on the whole}

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TABLE2.4: Input allocation. Covariance and correlation between la-bor productivity and employment shares, and between total factor productivity and composite input shares. Average over the years

2009-2019. OP covariance OP correlation France LPR 0.11 0.10 TFPR 0.15 0.57 Italy LPR 0.20 0.10 TFPR 0.20 0.62 Spain LPR 0.41 0.08 TFPR 0.22 0.70

TABLE 2.5: Output allocation. Covariance and correlation between productivity and market shares. Average over the years 2009-2019.

OP covariance OP correlation France LPR 0.33 0.21 TFPR 0.23 0.77 Italy LPR 0.54 0.20 TFPR 0.22 0.89 Spain LPR 0.69 0.11 TFPR 0.30 0.91

Anyway, once again Spain displays the highest difference between weighted and unweighted productivity, confirming a higher contribution of resource allocation to the aggregate productivity with respect to the other two countries. Another remark-able feature is again that correlation between market shares and labor productivity is low, but correlation between market shares and labor productivity is definitely high.

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21

Chapter 3

Check through detailed French

data

As we brought out in the previous chapter, our analysis could be flawed because of manifold data imperfections. One of these is the lack of information about firms’ foreign activity: the possibility that a company operates in more than a country -and in more than a market - might heavily affect the reliability of our estimates. In fact, let us think to market shares: so far, we considered firms within the same sector as competing all into the same market, but this is clearly not always valid.

In this chapter, we will present a simple robustness check made by means of detailed French customs data, trying to distinguish which part of activity each com-pany runs within the national boundaries.

3.1 French customs data

In order to get a cleaner view of firms’ national activity, we gained access to French customs data, whose fields are compulsorily filled by each company involved in an international transaction. Therefore, they are highly reliable. In particular, we ex-tracted firm-level export data from two legal frameworks, DEB (Déclaration d’Echange de Biens, intra-EU trade) and DAU (Document Administratif Unique, extra-EU trade). For more detailed information about this kind of data, we refer to Bergounhon, Lenoir, and Mejean’s (2018) useful guidelines.

Once we have aggregated the amount of export (in Euros) from a monthly to a yearly base, we merged this data set with Orbis data. The match was possible thanks to the SIREN identifier, the 9-digit code assigned to each French company. This is a specific variable in French customs data, while it can be extrapolated from Orbis data by simply removing the country code in the BvD ID number.

The following step involved reducing each firm’s annual turnover by the amount reported by customs data. In particular, we considered as foreign operating rev-enues those proceeds classified as “Binding purchase/sell”1. In so doing, we can recalculate market shares as only domestic.

Once data have been treated in such a way, the size distribution of our sample of French firms look as in Figure 3.1. Even though we have now a more precise measure of revenues, French data are still affected by the fact that SMEs are under-represented. Thus, the conjectures that can be carried out through Orbis data should mostly be confined to considerations about large-sized firms.

1_{In this data set the variable NATR stands for “Nature of transaction”. For simplicity, we only took}

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0.27% 6.7% 0.56% 3.7% 2.29% 8.4% 6.86% 19.1% 90.02% 62.1% 0e+00 2e+05 4e+05 0−9 10−19 20−49 50−249 250+

FIGURE 3.1: French manufacturing firms’ turnover distribution. Comparison between Eurostat data and our sample as modified by

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3.2. Output allocation 23

3.2 Output allocation

After these preliminary modifications, we did exactly the same estimates as before, to verify or prove wrong our previous results. Therefore, we will not go over the definitions and the procedures again - cf. Section 2.3. The only difference is that market shares have been recalculated only on the domestic market.

In Table 3.1 the new estimates of OP covariance and correlation are reported. When the exercise is performed with labor productivity, we obtain values lower than when export was included - cf. Table2.5. However, when TFPR is used as a measure of productivity, the results seems to be robust to the change in market shares.

TABLE3.1: Output allocation in France. Covariance and correlation between productivity and market shares computed only for domestic

market. Average over the years 2009-2019.

OP covariance OP correlation

LPR 0.11 0.07

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25

Conclusions

We started our dissertation by retracing roughly the last ten years of studies about economic allocation of resources, trying to provide the reader with an accurate yet synthetic overview of the theoretical construct behind them. The literature about the so-called “indirect” approach to estimate misallocation has undoubtedly had the merit of having faced the problem from a new perspective. However, the HK framework is far from being applicable to whatever economy. We did not review all the existing extensions of such model, since we preferred to focus on the limitations stemming directly from the basic assumptions. As we deem the allocation topic to be of utmost relevance, we hope for further studies aiming to dig deeper into the theoretical grounds of such a model, both with thorough microfoundation and em-pirical confirmation of the assumptions adopted. For now, the work by Haltiwanger, Kulick, and Syverson (2018) seems to undertake the right direction in this sense, and such an approach would deserve further investigation.

For our part, we decided to follow a different approach, the one originally put forth by Olley and Pakes (1992) and then exploited by Bartelsman, Haltiwanger, and Scarpetta (2013): the covariance between productivity and size within a sector is a more intuitive way to analyse the ability of an economy of “giving more to the best”, provided that we are still able to determine who the best is. More importantly, it requires less theoretical assumptions than the HK measure. This does not mean that we can lightly leave behind the primal objective that the traditional economic theory points out, namely the equalization of marginal products. Until we move in such horizon, we must be aware that these estimates could be not conclusive, as the OP covariance tells nothing about marginal products. Again, a huge role is played by the functional form we consciously assume.

Following mainly in Bartelsman, Haltiwanger, and Scarpetta’s (2013) footsteps, we tried to estimate a measure of misallocation within the manufacturing sector for three EU countries, France, Italy, and Spain. The biggest threats to the validity of this work are basically two: the impossibility to distinguish at firm level where the revenues come from - within or beyond national borders -, and the impossibility to have firm level information about prices. We addressed the former by taking advan-tage of the availability of detailed French customs data, obtaining new estimates that confirm the role of allocation in France as measured by the covariance between TFPR and market shares. As far as the second issue is concerned instead, unfortunately we have not been able to gain access to unit price data. We are aware of the limitations that this entails, so that we hope for further developments that exploit unit price in-formation to advance this field of industrial studies. Reliable results could turn out to have important implications both from a theoretical and a practical point of view: on the one hand, a rather mainstream idea of optimality is involved, so that such studies could serve as a testing ground either for the support or the refusal of the more traditional theory of the firm. On the other hand, knowing if an industry can potentially produce more by simply better allocating its resources can be undeniably of use to policy makers.

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27

Appendix A

The standard model of

misallocation effects on TFP

This appendix is dedicated to the framework developed by Hsieh and Klenow (2009). Actually, we chose to illustrate it using the notation by Gopinath et al. (2017), be-cause it takes into account also the temporal dimension and a specific term for firm’s goods demand.

A.1 The model

The model is rather standard, with a single aggregate final output for the whole economy: Yt= S

∏

s=1 Yθst st , where S

∑

s=1 θst=1 (A.1)

The subscript s stands for sector, t for time. A second stage of aggregation is at sector level: each industry’s output is a CES production function.

Yst= Nst

∑

i=1 Dist(yist) ε−1 ε !_ε−ε1 (A.2) where Distis the demand for firm i’s good (for simplicity, we imagine each com-pany producing only one good), while ε is the elasticity of substitution across goods. Nsis the number of firms operating in the sector s.

Finally, at the individual firm level we have a classic Cobb-Douglas production function1

yist=Aistkα_istsl_ist1−αs (A.3) kist stands for capital, list for labor, Aist for physical productivity. Each firm’s demand takes the form:

yist= pist

Pst −ε

(Dist)εYist (A.4)

Subsequently, the inverse demand function is:

1_{In Gopinath et al. (2017) factor shares are assumed the same in all the industries. Here we prefer to}

recall the original formulation. Hsieh and Klenow (2009) use US industries factor shares also for China and India since those of the “benchmark” economy are supposed to be undistorted.

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pist= " (Dist)−ε yist Yst #−1 ε Pst=By −1 ε ist (A.5)

where B is a fixed term not dependent on yist.

Now we can write down the profits maximization problem: max

kist,list

Πist= (1−τ_isty )pistyist− (1+τ_istk )(rt+δst)kist−wstlist (A.6) where rt+δst is the sum of real interest rate (assumed to be the same for each sector) and the depreciation rate, wstdenotes the cost of labor (same for all the firms within a sector), τ_isty represents a distortion that affects both capital and labor prod-uct, and finally τ_istk is a capital specific distortion2. To figure out what τ_isty and τ_istk essentially are, we can think to the first as a quota on production and to the second as a subsidy or a tax, depending on the sign. These two wedges should essentially render firm-level divergences, taking for granted that there is no firm heterogeneity at any other level.

Next, we derive the first order conditions:

∂Π ∂list =0⇒ 1−αs µ pistyist list = wst 1−τ_isty (A.7) ∂Π ∂kist =0⇒ αs µ pistyist kist = 1+τ k ist 1−τ_isty (rt+δst) (A.8)

The left hand terms of the equation A.7and A.8are respectively the marginal revenue product of labor and capital (partial derivative of the output times its price, given by the inverse demand function). We have to highlight that, for each factor, the marginal revenue product is proportional to its revenue productivity (pistyistdivided by the factor): this crucial result derives from the functional form we have chosen.

We can now define a measure of productivity at firm-level which should allow to measure the extent of misallocation. The traditional way to infer this traces back to the work by Foster, Haltiwanger, and Syverson (2008), who stress the important distinction between physical and revenue-based productivity3. Revenue total factor productivity (TFPR) is defined as follows:

TFPRist:=pist yist kαs istl1 −αs ist (A.9) But we can notice that this formula is nothing but the geometric mean of the marginal products of the factors times the mark-up:

pist yist kαs istl 1−αs ist = µ µpist yist kist αs µ µpist yist list 1−αs = =µ MRPKist αs αs_MRPL ist 1−αs 1−αs (A.10)

2_{More precisely, it represents a distortion of capital with respect to labor when it comes to the choice}

of inputs for the firm.

3_{In particular, Foster, Haltiwanger, and Syverson (2008) find an inverse correlation of physical}

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A.1. The model 29 This is the core argument on which this literature founds its empirical studies: the higher the distortions across firms, the more dispersed the marginal returns will be (instead of being all the same), and accordingly the more dispersed will be the TFPR. In the end, we are just left with the task of finding a good dispersion measure for TFPR - SectionA.1.1.

If we move on to sector level TFP, we obtain the following:

TFPst= Yst Kαs stL1 −αs st = "

∑

i (Dist) ε ε−1A_istTFPRst TFPRist !ε−1#_ε₋1₁ (A.11) where TFPRst=Pst_KαsYst stL1 −αs st

, Pst=∑i(Dist)ε(pist)1−ε and pist= TFPR_A_istist. From now on,(Dist)

ε

ε−1A_istwill be denoted as Z_ist: this allows us to take into account not only

the proper firm productivity, but also its idiosyncratic demand.

Finally, we can arrive at a measure of TFP loss arising from the misallocation of resources. The idea here is to calculate the efficient sector TFP4 and then the ratio between the aggregate output and its efficient level. Gopinath et al. (2017) follow exactly the same reasoning, they just express this measure of loss as a percentage (difference of logarithms): Λst= 1 ε−1 log_i Zist TFPRst TFPRist ε−1 − 1 ε−1log(EiZ ε−1 ist ) (A.12)

A.1.1 Measuring the extent of misallocation

Once obtained that TFPR is directly dependent on marginal products - and hence on firm-level distortions -, assuming TFPQ (:=Aist) and TFPR to be jointly lognormally distributed, it can be shown that5:

log Ast= 1 ε−1log N_st

∑

i Aist − ε

2Var(log TFPRsti)+

−αs(1−αs)

2 Var(log(1+τkist)) (A.13)

This means that the variance of log TFPR well summarizes the losses due to mis-allocation. 4_TFPe st= h ∑iZistε−1 i_ε−11

, where TFPRist=TFPRst. We need to notice that the efficient level does not

necessarily imply that there is no wedge between the proper cost of capital and its marginal cost for the entrepreneur; it just requires that the distortion is the same for all the firms.

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Appendix B

The role of constant returns to scale

in the standard model

This Appendix reports the modifications to the HK framework as studied by Gong and Hu (2016).

Removing the assumption of constant returns to scale entails that sector produc-tivity must be rewritten as1:

TFPst=As= ( Nst

∑

i " Aist MRPKst MRPKist !αs MRPLst MRPList !βs#Tst)_Tst1 (B.1) where Tst= _σ σ−1− (αs+βs) −1 , MRPLst= ∑Nst i MRPL −1 sti PistYist PstYst −1 and MRPKst= ∑Nst i MRPK −1 sti PistYist PstYst −1

. It follows that the variance of logTFPR is no longer a suit-able measure for the amount of misallocation. As a matter of fact, this variance can be decomposed in three components; assuming joint lognormality of Aist, MRPKist and MRPList, the formula of log Astbecomes2:

log Ast= 1 Tst log N_st

∑

i ATst ist −1

2γTst[φsVar(log MRPKist)+

+ψsVar(log MRPList) +2µαsβsCov(log MRPKist, log MRPList)] (B.2) where φs=αs[1−βs(ε−_ε1)], ψs=βs[1−αs(ε−_ε1)]and µ= _ε₋ε₁.

As a consequence, the potential gain from reallocation is given by: Y Yefficient = S

∏

s=1 ( Nst

∑

i " Aist Ast MRPKst MRPKist !αs MRPLst MRPList !βs#Tst)θst_Tst (B.3) where θstis a weight given to each sector on the basis of its contribution to manufac-turing total production.

1_{Refer to Gong and Hu (2016), p. 27 . We maintained the notation previously adopted.} 2_Ibid.

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31

Appendix C

Labor productivity with overhead

labor

Below we show the simplest version of the model put forth by Bartelsman, Halti-wanger, and Scarpetta (2013). The original one introduces also quasi-fixed capital with transitory productivity shock, but for our illustrative purpose this feature is better addressed in Section1.3.2.

The most important difference with the HK framework is the following technol-ogy of production1_:

Yit=Ai(nit− f)γ−αkαit, γ<1

where nitis the total amount of labor employed by firm i at time t and f is overhead labor. Inverse individual demand is

Pit=Pt ¯

Yt Yit

1−ρ

Given the same cost for labor and capital units for each firm, the profit function is straightforwardly

πit= (1−τi)PtY¯t1−ρ[Ai(nit− f)γ−αkαit]ρ−wtnit−Rtkit

where τiis the well known revenue distortion. New firms willing to enter the market do not know ex-ante their specific productivity and distortion, they only know their joint distribution G(A, τ). Thus we have a free entry condition

We= Z

A,τ

max[0, W(A, τ)]dG(A, τ) −ce=0 where ceis the sunk entry cost and

W(A, τ) = ∞

∑

t=0 E[π] 1−λ 1+rt t = E[π] 1− 1−λ 1+rt

is the present value of a company given a discount rate r2_{and a probability of exit λ.}

Once idiosyncratic draws of productivity and distortion are known, the true profits

1_{We mantained the same notation from the original paper almost everywhere. Variables and}

pa-rameters which preserve the notation of HK model will not be specified again.

2_{The discount rate r is pinned down by solving a simple problem of representative consumer’s}

utility maximization: ∞

∑

0 βtU(Ct) s.t.

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are unveiled, leading the firm to permanently stay in the market if they are non-negative, or to exit otherwise. Clearly, in a more complex environment with tem-porary shocks of productivity and distortions, this choice is made every period, not once and for all.

Profit maximization involves the following expressions - time subscripts are im-plicit: ∂πi ∂ki =0⇒ki= R (1−τi)Aρ_iαρP ¯Y1−ρ _αρ1₋₁ (ni− f) ρ(γ−α) 1−αρ ∂π_i ∂ni =0⇒ni− f = w (1−τ_i)A_iρρ(γ−α)P ¯Y1−ρ _ρ₍_γ₋1_α₎₋₁ k αρ 1−ρ(γ−α) i After some algebra, we obtain

k 1−γρ 1+αγρ2−α2 ρ2−γρ i =A[ (1−τi)Aρi | {z } idiosyncratic term=Si ] 1 1+αγρ2−α2 ρ2−γρ ⇒k∗ i =AS 1 1−γρ i n∗_i − f=BS 1 1−ρ(γ−α) i S αρ (1−ρ(γ−α))(1−γρ) i =BS 1 1−γρ i

where A and B are constant terms common to all the firms. Finally, it turns out that PiYi n∗_i − f = C 1−τi (C.1) and PiYi n∗_i =LPRi= C 1−τi − C f (1−τi)ni (C.2) where C is constant across all the firms. EquationsC.1andC.2show that even when the revenue product per effective unit of input is equalized (τi=0), the empir-ical measure of productivity (LPR) may vary.

∞

∑

0 pt(Ct+Kt+1− (1−δ)Kt) = ∞

∑

0 pt(wtNt+RtKt+πt)

where βt is the utility discount rate, pt is the price level at period t, Ct is consumption, Nt is the

inelastically supplied total amount of labor which in equilibrium equals the demand. Solving the Bellman equation entails:

Rt−δ= 1 β−1

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33

Appendix D

Misspecification problems in the

HK model

This appendix is devoted to analytically demonstrate the important results shown in the paper by Haltiwanger, Kulick, and Syverson (2018).

D.1 Constant marginal costs

One of the pillars of HK model is the presence of marginal costs that do not vary with the quantity produced. In order to see this, let us set up the ordinary cost-minimization problem - time subscripts have been omitted for ease of exposition:

min Ci=wli+ (1+τ_ik)(r+δ)ki s.t. yi=Ailαik1i−α

We can write down the Lagrangian function and the FOC: L=wli+ (1+τ_ik)(r+δ)ki−λ(Alαk1i−α−yi) ∂L ∂li =0⇒w=λαyi li ∂L ∂ki =0⇒ (1+τ_ik)(r+δ) =λ1−αyi ki From which we obtain

li= r+δ w α 1−α (1+τ_ik)ki ⇒yi= Ai r+δ w α 1−α(1+τ k i ) α ki

Once labor and capital have been expressed as functions of yi, we can rewrite the cost function as:

C(yi) =wα(r+δ)1−α(1+τ_ik)1−α " α 1−α 1−α + α 1−α −α# yi Ai = = " w α α r+_δ 1−α 1−α (1+τ_ik)1−α # yi Ai

Detecting misallocation: an empirical investigation through firm-level data

U

NIVERSITÀ DI

P

ISA

M

’

T

Detecting misallocation

Abstract

Contents

List of Figures

List of Tables

Introduction

Chapter 1

Economic modeling for

misallocation effects

1.1

The indirect approach

1.2

The HK model

1.3

First reactions to the HK model

1.4

The “constant returns to scale” assumption

1.5

Functional form and heterogeneity

1.6

Measurement issues