Multi-Product Markups
John CairncrossPeter Morrow

University of Toronto

University of Toronto

john.cairncross@mail.utoronto.ca

peter.morrow@utoronto.ca

Scott Orr

Swapnika Rachapalli∗

UBC Sauder School of Business

UBC Sauder School of Businessscott.orr@sauder.ubc.ca

s.rachapalli@sauder.ubc.ca

June 27, 2023

Abstract
We examine the identification of firm-product markups in multi-product firms using
production-side data. Identifying within-firm markup differences relies on identifying
the marginal rates of transformation across goods. Since marginal rates of transformation are generally functions of (i) the degree of joint production and (ii) the magnitude of within-firm productivity differences, we explore whether markup estimates
are sensitive to misspecification in these factors. Monte Carlo exercises indicate thatmisspecification of within-firm productivity differences is particularly likely to generatebias. However, a weighted average of firm-product level markups (the “firm markup”)
can be identified without information on these factors, and nests the firm markup of
De Loecker and Warzynski (2012). The firm markup can be estimated using standard
empirical methods. Standard parametric restrictions on the marginal rates of transformation across goods often deliver implausible estimates of plant-product markups.
Firm markups, which do not require these restrictions, are more well behaved. We also
discuss the welfare properties of the firm markup.

∗ We

thank Matthew Backus, Germán Bet, Bernardo Blum, Matilde Bombardini, Cecile
Gaubert, Gunnar Heins, Volodymyr Lugovsky, Devesh Raval, David Rivers, Valerie Smeets, Frederic Warzynski, as well as seminar participants at Aarhus, Berkeley, Columbia GSB, University of
British Columbia-Sauder, University of Florida, University of Toronto, the Canadian Economics
Association Meetings (2021), EARIE (2022), and EIIT (2022), IIOC (2023) and NAPW (2023) for
helpful feedback. Any remaining errors are our own.

1

Introduction

In an extremely influential paper, De Loecker and Warzynski (2012) build on Hall (1986)
to develop an approach for estimating firm-level markups. This “production-side” approachis valuable because it does not require a researcher to estimate demand nor impose market conduct assumptions, as is common elsewhere in applied industrial organization (e.g.
Bresnahan 1989, Berry et al. 1995). While most exercises using De Loecker and Warzynski (2012) provide estimates of markups at the firm or plant level, textbook treatments ofmarket power emphasize market power at the product level. While this distinction makes nodifference for single-product firms, multi-product firms account for a majority of economic
activity in many countries.1
Given the importance of multi-product production in modern economies, this paper aims
to answer two key questions. First, what features of the production technology does an
econometrician need to identify to estimate firm-product markups in multi-product firms?
Second, how might we interpret firm-level markups estimated for a multi-product producer
using the De Loecker and Warzynski (2012) methodology? This second question is important, as a recent literature asking whether market power has increased relies cruciallyon this production side approach to estimating markups (De Loecker and Eeckhout, 2018;
De Loecker et al., 2020).
To help us answer both questions, we first consider the problem of identifying firmproduct-level markups when output data is available at the firm-product level but input
data is only available at the firm-level. We refer to such data as standard production data.2
De Loecker et al. (2016) provide a popular solution to this problem based on two key restrictions. First, they require production to be non-joint, meaning that each multi-productproducer allocates all of their inputs in a mutually exclusive and exhaustive way across theirproduct-line-specific production functions. Second, they require that there are no within-firm
productivity differences, such that a firm is equally productive at producing all products.
We consider identification of product-level markups in a setting that allows for joint or
non-joint production, as well as within-firm productivity heterogeneity. We characterize eachfirm’s technology through an output distance function (e.g. Caves et al. 1982). This approach allows us to represent non-joint technologies, as well as joint production technologies
that incorporate economies of scope, a potentially important rationale for multi-product pro1

For example, Bernard et al. (2011) document that 12% of U.S. firms in 2000 that exported more than
five products to more than five destinations accounted for more than 90% of export value, and multi-productfirms produce roughly two-thirds of aggregate output in our data for India that we discuss below.
2
While the data used in this paper is reported at the plant-level, we use the terms firms and plantsinterchangeably in the paper unless otherwise stated.

1

duction (Baumol et al. 1982). Because within-firm productivity dispersion can be difficult
to define in settings involving joint production, we consider technologies that allow for heterogeneity in marginal rates of transformation; a natural metric for productivity differencesin multi-product firms.
We show that identification of product-level markups using standard production datarequires that the researcher know the extent of joint production, as well as the degree ofheterogeneity in marginal rates of transformation across firms (“MRT heterogeneity”). We
use Monte Carlo simulations to show that abstracting from joint production and (especially) MRT heterogeneity can lead to noticeable differences in estimated markups. Unfortunately, past research in this area has shown that identification of MRT heterogeneity may
require that researchers know how firms compete with one another, which already disciplines
product-level markups within the firm.3
This motivates the second part of our paper which derives an alternate firm-level markup
that does not require the researcher to know the magnitude of MRT heterogeneity or joint
production. This object is a weighted average of firm-product markups within the firm,with weights equivalent to product-level cost shares, where costs are allocated to products
by considering how much costs would rise (locally) when increasing one output, relative toincreasing all outputs by the same proportion. This object is welfare relevant, in that it canbe interpreted as a firm-level output wedge which scales down the production of each productwithin the firm uniformly, decreasing overall welfare relative to what a social planner would
choose. Therefore, our firm-markup is a symptom of across-firm misallocation as in Edmondet al. (2015, 2022).
Empirically, the firm-markup is recovered by the De Loecker and Warzynski (2012) formula under reasonable restrictions on the technology. This is important for two reasons.
First, our definition of the firm-markup, which is based on a general representation of a firm’s
technology, provides a coherent lens to interpret the De Loecker and Warzynski (2012) firm
markup formula. Second, by building a model of multi-product producers to rationalize the
De Loecker and Warzynski (2012) markup when quantity data is available, we address recent
criticisms by Bond et al. (2021) that if production technologies are estimated with revenue,
then the approach is unlikely to recover the firm’s markup. In particular, we show that theprocedure developed in De Loecker et al. (2016), which first selects the subsample of singleproduct firms to estimate a product-level production function, can be used to identify thefirm-markup even if there is joint production and within-firm productivity heterogeneity.4
3

See Valmari (2016), Gong and Sickles (2021), Orr (2022). These papers only considered non-joint
production settings where MRT heterogeneity ends up being analogous to identifying within-firm productivity
dispersion.
4
We also show that their selection correction can be extended to settings with joint-production as well.

2This is important as this approach allows researchers to use quantity data to estimate the relevant technology parameters, while avoiding the complications generated by multi-productfirms in estimation. We also show that a cost share estimator (Syverson 2004, Foster et
al. 2008, Backus 2020, Raval 2023) can be used to recover the firm markup. Importantly,
this approach can be used even when quantity data is not available, and the estimator canbe applied to all firms in the sample, including multi-product producers.5 Consistent withpast work in this area (e.g. De Loecker and Syverson 2021), these two approaches rely on
non-overlapping restrictions to the specification of a firm’s technology and environment.
We close by using Indian Annual Survey of Industries (ASI) data provided by the Indian
Ministry of Statistics to compare product-level markups, estimated under various assumptions on MRT heterogeneity and joint production, to plant-level markups. We documentthat restricting MRT heterogeneity tends to generate a large mass of estimated markups
where MPC = 0. We call this a zeros puzzle. We show that this puzzle can only partly
be resolved by relying on a productivity ladders model following Mayer et al. (2014) thatallows for within-firm productivity differences. We also show that the puzzle can be almost
completely resolved by relying on revenue shares to allocate inputs across product lines.
However, since revenue share input allocation rules impose constant within-firm markups asin Orr (2022), these two results suggest that further work quantifying the degree of MRT
heterogeneity is needed to fully pin down reasonable estimates of product-level markups inmulti-product plants.
We then compare these results to our plant markups. We find that plant markups are
generally quite well behaved, and do not suffer from the zeros puzzle. We also find thatrelying on cost share estimator, or an estimator based on single product plants with aselection correction, yield remarkably similar estimates of the distribution of plant-markups.

Contribution to the LiteratureFirst and foremost, our paper contributes to the vast and growing literature on estimating
markups using production data. The paper’s core contributions are to two separate strandsof this literature. First, we build on past work by De Loecker et al. (2016), Grieco et al.
(2016), and Dhyne et al. (2022) on how to approach estimation of markups using productionside data when firms produce many products. On the theoretical front, we propose and study
a class of technologies that nest both single- and multi-product production, which allows us
5This also allows our approach to deal with critiques raised by Flynn et al. (2019), Doraszelski and
Jaumandreu (2019), and Bond et al. (2021) concerning the internal consistency of proxy-variable approachto production function estimation in the presence of market power, and also allows us to construct estimatorsthat can deal with the issues documented in Raval (2022, 2023).

3

to explore what features of a firm’s technology can be identified by a (selected) sample ofsingle-product firms. This allows us to extend the identification results in De Loecker etal. (2016) to settings with joint production. Our identification results for the firm-markup
also extends insights in De Loecker and Warzynski (2012) on the interpretation of firm
markups. In particular, they note that when firms sell products to many markets, their firmlevel markup can be interpreted as an input expenditure share weighted average of market
specific markups. We provide a generalization of this result to joint production settings,
where input allocations may not be well defined. This generalization relies on the insightthat one can obtain product-level cost shares by considering the ratio of local cost increases
for increasing a given output, relative to the local cost increase of increasing all outputs by
the same proportion. We further show that this firm-level object is welfare relevant, and canbe used to quantify the degree of across-firm misallocation, taking within-firm misallocation
as given. Finally, our Monte Carlo exercises, as well as out empirical results, illustrate that
accounting for MRT heterogeneity in multi-product firms may be of first-order importancefor identifying product-level markups.
We also contribute to a second strand of the literature on production-side markups, focused on clarifying the data requirements and assumptions necessary to properly identify
markups. Bond et al. (2021) point out that standard approaches based on estimating revenue production functions, rather than quantity production functions, are unlikely to identify
markups. In response, Kasahara and Sugita (2020), Kirov and Traina (2021) have proposed
alternative methods that will still identify the markup in settings where only revenue isobserved, although they restrict attention to single-product firms. De Ridder et al. (2022)
also consider a single-product firm environment, and argue that even though revenue production functions generate bias, the recovered markup may still be strongly correlated withtrue markups. Our paper avoids the Bond et al. (2021) critique by embracing the fact that
production-side markups will be correctly estimated as long as one can estimate the relevant
quantity elasticities. This data is becoming increasingly available (e.g. Foster et al. 2008,
Baldwin and Gu 2009, De Loecker et al. 2016, Blum et al. 2018, De Roux et al. 2021, Orr2022, Dhyne et al. 2023), and the techniques developed in this paper provide a useful guidefor how to obtain these relevant elasticities in practice. We do this by addressing a specificdata challenge generated by quantity data — the inability to allocate inputs to product lines— and provide multiple ways to recover firm markups in this setting.
This paper contributes to the literature on specifying multi-product production technologies. We provide what we believe to be a novel articulation, building on insights inSamuelson (1966) and Hall (1973), of how different specifications of a firm’s production possibility set can capture joint or non-joint production. We also provide a representation of a4

firm’s technology through it’s output distance function that is dual to the CES cost function
for multi-product firms studied by Baumol et al. (1982), and estimated in other appliedstudies (e.g. Johnes 1997). We derive this representation of a firm’s technology through an
input allocation problem, where firms can use their inputs to produce private, rival intermediates, as well as public, non-rival intermediates, that contribute to the production of all
products simultaneously.6 This particular specification of a firm’s technology ends up having
a similar structure to the class of production models considered in Eslava et al. (2023) and
De Roux et al. (2021), except that we rely on an entirely production-side focused derivationof the technology, with no reference to consumer preferences. We believe that this particularspecification of a firm’s technology will be useful to further applied work, as it provides aconvenient way to nest both joint and non-joint production, as well as parameterize thedegree of scope economies within a firm.7
The paper is structured as follows. Section 2 describes how we use output distance functions to parameterize a firm’s technology. Section 3 derives expressions for plant-productmarkups using output distance functions, highlighting the key features of a production technology that needs to be known by a researcher to identify these markups. Section 4 presents
our firm-level object and offers welfare based interpretations. Section 5 provides techniques
for identifying the firm-level markup. Section 6 discusses our data and econometric implementation. Section 7 presents results. Section 8 concludes.

2

Production Technologies

We first introduce notation and what is known to the econometrician. Firms are indexed
i = 1, .., N , products j = 1, ..., J, and inputs s = 1, ..., S. We assume that the econometrician
observes what we call standard production data that includes a 1 × J vector of output
quantities Yi ≡ (Yi1 , Yi2 , ...YiJ ) ∈ RJ≥0 and revenues Ri ≡ (Ri1 , Ri2 , ...RiJ ) ∈ RJ≥0 , as well
as a 1 × S vector of input quantities Xi ≡ (X1i , X2i , ...XSi ) ∈ RS≥0 and expenditures Ei ≡(E1i , E2i , ...ESi ) ∈ RS≥0 . Superscripts j refer to different outputs, and subscripts (s, i) index
inputs s and firms i. We assume that standard production data only contain information on
firm-level aggregate inputs and not how inputs are allocated across product lines j.
We define a firm’s technology as its production possibility set, Pi , where the firm is only
6This problem has a similar structure to the models of scale and scope considered in Ding (2022) andArgente et al. (2020); our key distinction is that we load spillovers across product lines through non-rival
inputs that directly affect production, rather than knowledge or ideas which affect production throughchanges in productivity.
7
See concurrent work by Khmelnitskaya et al. (2023) on identifying scope economies in the beer industry
using the class of technologies proposed in this paper.

5

capable of producing with output and input vectors (Yi , Xi ) ∈ Pi .8 A firm’s production
possibilities frontier summarizes the maximal quantities of output that may be obtained
for a given vector of inputs. As in Caves et al. (1982), we characterize a firm’s production
possibility frontier using an output distance function that maps all possible non-negative
output and input vectors (Yi , Xi ) to a non-negative scalar δ:
Definition 1 A firm’s output distance function, Di (Yi , Xi ) : RJ+S≥0, 6=0 → R+ , solves:
Di (Yi , Xi ) ≡ min δ

s.t.

δ

Yi
, Xi
δ


∈ Pi .

(1)

The solution to this problem tells us by what (minimum) factor a firm must scale an
output vector Yi such that it can produce Yδ i with Xi . If Di (Yi , Xi ) > 1, the firm cannot
produce Yi with Xi ; if Di (Yi , Xi ) < 1, the firm can produce a vector of outputs which isstrictly larger than Yi with Xi . This leads to the following definition:
Definition 2 A firm’s production possibility frontier, PFi ⊂ Pi , is the set of all (Yi , Xi )satisfying
Di (Yi , Xi ) = 1.
(2)
For the rest of the paper, we make the following assumption on the shape of the output
distance function, Di (Yi , Xi ):Assumption 1 Di (Yi , Xi ) is continuous, twice-differentiable, and quasi-convex.
We now define several properties of a firm’s technology that are relevant for resultsthat follow: heterogeneity in marginal rates of transformation, joint production, and nesting
single-product production. We then use these properties to characterize whether there is joint
or non-joint production based on the shape of the output distance function.

2.1

Heterogeneous Marginal Rates of Transformation

Large firms tend to produce multiple products with the firm’s core product often definedas the product that is produced with the highest level of productivity or that which generates
the highest revenue (Eckel and Neary 2010, Mayer et al. 2014, Orr 2022). It is reasonable
to examine technologies that allow the core product to vary across firms. We consider
broad classes of technologies —including joint production technologies where the notion of
product-line specific productivity can be difficult to define. To this end, we capture the idea
8

We assume Pi always includes (0, Xi ) so that a firm can always produce nothing, and that production
possibility sets satisfy a no free lunch property, such that for Yi > 0, (Yi , 0) ∈/ Pi .

6

of core products varying across firms by considering technologies where the marginal ratesof transformation (MRT) for various products can vary across firms.
Marginal rates of transformation can be obtained from a firm’s output distance function
by totally differentiating a firm’s output distance function at Di (Yi , Xi ) = 1 for an arbitrarily small change in the output of good j and k, holding the output of all other goods
fixed:∂Di (Yi , Xi ) j ∂Di (Yi , Xi ) k
dYi = 0.
(3)
dYi +
∂Yik
∂Yij
The marginal rate of transformation, which we represent as ∆kj
i (Yi , Xi ), is defined as
the marginal change in good k needed to increase good j by one unit:∂Di (Yi ,Xi )
dYik
∂Yij
kj
∆i (Yi , Xi ) ≡.
=−∂Di (Yi ,Xi )
dYij
k
∂Y

(4)

i

We define a model as having heterogeneous marginal rates of transformation — or MRT
heterogeneity for short — if ∆kj
i (Yi , Xi ) is indexed by i:Definition 3 A production model allows for MRT heterogeneity, if for any (Yi , Xi ),∆kj
i (Yi , Xi ) can take on any value in R+ and can differ across i.
The simplest version of a production model allowing for MRT heterogeneity is a firm-level
Ricardian model where labor Li is the only factor of production, there is no joint production,and there are constant returns to scale. In this case, the output distance function takes the
Yi1
Y2
+ i2
A1
A
i
i

A2

j
i
form Di (Yi , Xi ) = Li , and ∆12i = − A1 , where Ai are product-line Hicks specific TFPishifters. Here (A1i , A2i ) acts a vector of unobserved heterogeneity at the firm-level.

2.2

Joint Production

Joint production can occur when there are public (i.e. shared) inputs that affect theoutput of all product lines simultaneously. Examples include shared managerial inputs, machines that produce multiple outputs at once, or production processes involving by-products,
such as how the refining of sugar generates molasses.
Non-joint production is a setting where production takes place through a series of productline specific production functions, Yij = Fij (Xji ), where Xji is a vector of input quantities
P
allocated to product j, and j Xji = Xi . Since the inputs are allocated across productlines in a mutually exclusive and exhaustive manner in non-joint production settings, we can
Xjjdefine the input share of input s into production line j as Ssi≡ P siX k , with Sji representingkj
the 1 × S vector of input shares Ssi
.

7

si

We define non-joint and joint production following Hall (1973) as:
Definition 4 A technology is non-joint if there exists a set of functions {Fij (.)}j suchthat for all (Yi , Xi ) ∈ PFi there exist input shares {Sji }j such that Yij = Fij (Sji ◦ Xi ) andP jj Si = 1. Otherwise the technology involves joint production.
In general, whether a particular technology involves joint or non-joint production will
depend on the shape of a firm’s production possibility frontier (Samuelson 1966, Hall 1973).
Section 2.4 formalizes this by characterizing how the shape of a firm’s output distance function reveals whether production is joint or non-joint. We first describe the remaining features
a firm’s technology that we rely on for our results.

2.3

Nesting Single-Product Firms

We focus on production possibility frontiers that allow firms to choose to only produce
a single product line j because we are interested in what features of a firm’s technology canbe estimated from single-product firms as in De Loecker et al. (2016). To start, we define
the concept of nesting single-product production as follows:
Definition 5 A technology nests single-product production if for any Xi and j, the set
PFi contains Yij , Xi where Yij is a 1 × J vector of outputs where Yij ≥ 0 for j, and Yik = 0∀k 6= j.
This definition formalizes the idea that a multi-product firm using Xi can choose to
produce only a single product. This requires that Di Yij , Xi be well defined for all Yij .
This rules out translog distance functions of the form considered in Caves et al. (1982).9 Asufficient condition for a technology to nest single-product production is that Di Yij , Xi =YijjFi (Xi ). In this case, the production possibility frontier is characterized by a standard production function:Yj
Di Yij , Xi = j i= 1 =⇒ Yij = Fij (Xi ) .
Fi (Xi )2.4

A Theorem on Joint vs Non-Joint Production Under Seperability

When specifying technologies using production possibility frontiers, it is common forresearchers to rely on separable functional forms that possess well defined output and input
aggregators (Mundlak 1963, Grieco and McDevitt 2017, Grieco et al. 2018, Maican and Orth
9

Such functional forms imply that firms must produce positive quantities of every product, which rules
out shutting down particular product lines.

8

2021, Dhyne et al. 2022). While separable transformation functions have useful properties
for the purpose of estimation, Hall (1973) emphasizes that separable functional forms caneasily imply joint production. In this subsection, we build on his result to show exactly
when separability implies joint, or non-joint, production. For this purpose, we rely on thefollowing definition of separability:
Definition 6 An output distance function is separable if it can be written as:
Gi (Yi ).
Fi (Xi )Di (Yi , Xi ) =(5)Theorem 1 (below) builds on Hall (1973) to establish a simple functional form test on
a firm’s output distance function to establish whether a firm’s technology involves joint or
non-joint production. For this to hold, we require two further restrictions on the space offeasible distance functions:
Assumption 2 Di (Yi , Xi ) is separable, with Fi (Xi ) continuous, differentiable, strictly increasing in all arguments, quasi-concave, and homogeneous of degree φi > 0.

YjAssumption 3 Di Yij , Xi = Aj F i(X ) for all Yij > 0.
iiiAssumption 2 imposes separability of the output distance function and establishes standard regularity conditions on Fi (Xi ). The most important of these assumptions is that
Fi (Xi ) be homogeneous degree φi > 0. Assumption 3 simply requires that firms can choose
to produce a single product using a standard production function, while still respecting theseparability assumption.10 These two assumptions make characterizing non-joint production
straightforward.
Theorem 1 Suppose Assumptions 2 and 3 hold. Then a firm’s technology is non-joint if
and only if:φP  Yij  φ1i ij

Di (Yi , Xi ) =Aji

Fi (Xi ).

(6)

Proof. See Appendix A.
An important implication of Theorem 1 is that if the firm’s technology is separable andnests single product production, then the firm’s technology involves joint production if theφP  Yij  φ1i i
output aggregator, Gi (Yi ) 6=
. As a result, assuming non-joint productionj Ajiimplies specific restrictions on the shape of firm’s output distance function.

Yj
Notice that Di Yij , Xi = Aj F ji(X ) would violate separability, since the input aggregator would havei i i different functional forms for Di Yij , Xi and Di Yik , Xi , k 6= j, which would make Fi () depend on Yi .
10

9

3

Firm-Product Markups

We now use our output distance function approach to derive what objects must be observed to identify firm-product markups. We also show how past approaches have dealt withthe relevant identification issues. We do this to spell out the restrictions that have been
made in the past, and to compare these restrictions to those made here.
The identification of markups using the production-side approach of De Loecker andWarzynski (2012) requires three key assumptions: i) cost-minimization, ii) the existence ofat least one static input in production where the cost-minimizing firms are price takers, andiii) knowledge of some features of each firm’s technology. We first present a more precise
representation of these three assumptions, and show how they generate a mapping fromstandard production data (Yi , Ri , Xi , Ei ) to unobserved markups, similar to Grieco et al.
(2018) and Dhyne et al. (2022).
We partition the set of inputs into two subvectors, Xi = (Mi , Ki ). Mi represents avector of static inputs, and Ki denotes the vector of dynamic inputs. The set M of static
inputs is the set of inputs purchased each period at exogenous prices Wsi . The set K ofdynamic inputs may be accumulated over time through a dynamic investment process. We
assume that there is at least one static input used by all firms in a given industry (e.g.
materials), and assume that each firm’s conditional cost function Ci (Yi , Ki , Wi ) is given by
the following minimization problem:11
Ci (Yi , Ki , Wi ) = minMiX

Wsi Msis.t. Di (Yi , Xi ) = 1.

(7)

Msi ∈M

The first-order condition for any static input Msi ∈ M is given by:
Wsi + λi

∂Di (Yi , Xi )
=0
∂Msi

(8)

where λi is the relevant Lagrangian multiplier. The Envelope Theorem implies that condii ,Ki ,Wi )can be written as:
tional marginal costs M Cij ≡ ∂Ci (Y∂Yji

M Cij = λi

∂Di (Yi , Xi )
.
∂Yij

11

(9)

Costs related to the accumulation of dynamic inputs will, in general, show up in a firm’s profit function.
Since the quantity of dynamic inputs are being conditioned on in the conditional cost minimization problem,the exact structure of these costs can be ignored when solving for the conditional cost function.

10

Combining equations (8) and (9) yields∂Di (Yi ,Xi )

1
1
∂Msi
,
j = − ∂Di (Yi ,Xi )
Wsi
M Ci
j

(10)

∂Yi

Pj

and the firm-product markup µji ≡ M Ci j is obtained by multiplying equation (10) by Pij :i∂Di (Yi ,Xi )jsiµi = − ∂Di∂M(Yi ,Xi )∂Yij∂ ln Di (Yi ,Xi )

Pij
Rij
Msi
= − ∂ ln∂Dlni (Yi ,Xi ) E
Wsi
si
j

(11)

∂ ln Yi

where the second equality uses the fact that Pij Yij = Rij , and Xsi Wsi = Esi .
Equation (11) extends the De Loecker and Warzynski (2012) firm markup formula to thefirm-product level for a (potentially) multi-product firm. If the firm produces a single productso thatYi = Yij , their formula holds as long as the relevant output distance function satisfies
i (Yi ,Xi )i ,Xi )
Assumption 2, in which case ∂ ln ∂Dln
= 1 and ∂ ln∂Dlni (Y= − ∂ ∂lnlnFiM(Xsii ) ≡ −θsi (Xi ), so
Msi
Yj
i

Rj

µji = θsi (Xi ) Esii . With a single-product firm, the researcher only needs to estimate a single
elasticity, θsi (Xi ). Equation (11) extends the De Loecker and Warzynski (2012) result byshowing that product-level markups at multi-product firms can be recovered from standardi (Yi ,Xi )i ,Xi )and ∂ ln ∂Dln
. We now highlight
production data if the researcher can estimate ∂ ln∂Dlni (YMsi
Yijidentification challenges that a econometrician faces when tackling this problem.

3.1

Identification Challenges

We use equation (11) to illustrate what needs to be known to identify the ratio of markups
across product lines µji /µki :∂ ln Di (Yi ,Xi )∂Di (Yi ,Xi )

µji
Rij
Pij
Pij
∂ ln Yik∂Yikkj= ∂ ln Di (Yi ,Xi ) k = ∂Di (Yi ,Xi ) k = −∆i (Yi , Xi ) k .
µki
Ri
Pi
Pi
j
j
∂ ln Yi(12)∂Yi

The final equality in this expression tells us that it is crucial to identify the marginal
rate of transformation across goods when recovering within-firm markup differences. In particular, conditional on output prices, markup ratios are completely determined by marginal
rates of transformation. Unfortunately, equation (12) also implies that MRT heterogeneity
can generate a serious underidentification problem because equation (12) only provides asingle equation to identify two unknowns; relative markups, µji /µki , and the marginal rate oftransformation between goods k and j. In short, relative output prices within a firm can berationalized in two different ways: differences in market power, or differences in technologies.
11

To see this more clearly, consider the following parameterization of a firm’s output distance function, which we derive in Appendix B based on a model of rival and non-rival
intermediates as in Baumol et al. (1982):12
φβ
P  Yij  φβ1
Ajij

Di (Yi , Xi ) =,F (Xi )

(13)

where F (Xi ) is an arbitrary homogenous of degree φ > 0 function and β ∈ (0, 1] is theshare of rival intermediates in total costs such that 1 − β is the share of non-rival, or public,
intermediates. Applying Theorem 1 to equation (13), the technology is non-joint if and only
if β = 1; otherwise, the technology involves joint production. The Aji terms function as
product-line specific Hicks-neutral productivity shifters. By including these terms, we allow k  φβ1
 j  1−φβ
φβ
Yi
Ai
kj
for MRT heterogeneity since ∆i (Yi , Xi ) = − Y kdepends on i, conditionalAjiion Yi and Xi .
Evaluating equation (12) using the technology in equation (13) delivers:
µji
=
µki

Yij
Yik

! 1−φβ
φβ 

Aki
Aji

 φβ1

Pij
.
Pik

(14)
µj

Ak

i

i

There are two (i, j, k) specific unknowns in this expression: µki and Aij . De Loecker et al.
(2016) deal with this under-identification problem by assuming Aji = Aki ∀j, k, in which case j  1−φβ
φβ
µji
Pij
Y
we obtain µk = Yik. This allows the researcher to identify product-level markups
Pk
i

i

i

because markup-ratios rationalize the observed prices and quantities. However, if Aji 6= Aki ,this can potentially generate a mismeasurement problem.
Valmari (2016), Gong and Sickles (2021), Orr (2022) offer another path. They first model
demand and price setting behavior and recover markups µji /µki . This then allows them touse equation (14) to recover Aki /Aji .13 However, such an approach requires the researcher totake a stand on a particular demand system and pricing setting behavior, which is precisely
what the production-side approach wishes to avoid.
Equation (14) also points to another identification problem: researchers must address
whether there is joint, or non-joint production before they can identify product-level markups.
12

Note that this representation of a firm’s production possibility frontier is similar to that used in Eslava etal. (2023) and De Roux et al. (2021), with the important difference that φ, β, and Aji are purely technology,rather than preference, parameters.
13
These papers only consider non-joint production in which case β = 1. φ can then be recovered byestimating a firm-product specific production function.

12 j  1−φβ
φβ
µj
Pij
Y
Even if Aji = Ai ∀j such that µki = Yikk , a researcher must still know the values ofPiiiβ and φ to identify product-level markup ratios. De Loecker et al. (2016) address this byassuming that production is non-joint (β = 1), in which case β = 1, and φ can identified byestimating returns to scale for single-product firms.14
The severity of the potential identification and mismeasurement problems associatedwith the problems and potential solutions above depends crucially on how the magnitudewith which MRT heterogeneity and joint production manifest in the data. We explore
this further in the next subsection, where we use simple Monte-Carlo simulations to ask how
biased the estimated plant-product markups might be if the researcher’s model ignores MRT
heterogeneity or joint production when they are present.

3.2

Monte Carlo Evidence

We base our Monte Carlo evidence around the output distance function in equation (13),which allows us parameterize MRT heterogeneity with Aji /Aki differences, and the importanceof joint production with β; the share of non-rival intermediates in production. We present
two Monte Carlo exercises to highlight how erroneous assumptions of no MRT heterogeneity
and non-joint production quantitatively affect estimated markups. In the first exercise, we
switch off joint production but keep within-firm productivity dispersion in the true model.
We compare markups from the true model to one derived under the incorrect assumptionof there being no within-firm productivity dispersion. In the second exercise, we switch off
within-firm productivity dispersion but keep joint production in the true model. We thencompare the true markups to those derived under the incorrect assumption of production
being non-joint. In both settings, we assume that there are demand shocks both within andacross firms.15 We now discuss these two exercises in detail.
There are 1000 firms i, and 10 possible products j, a random subset of 5 of which eachfirm produces. Consumers spend a constant share of their income 1/J in each industry j.
Labor is the only factor of production and is paid an exogenous wage w. We assume constantreturns to scale in all industries for all firms such that φ = 1, and F (Xi ) = Li where Li
is the total amount of labor used by the firm. This allows us to derive true markups using
14

Recall that F (Xi ) in equation (13) is an arbitrary homogenous of degree φ > 0 function. In Appendix
B, we show that this function also governs the product-line specific production technologies, and therefore
can also be interpreted as the production function for single-product firms, with φ > 0 governing overall
returns to scale.
15
We do this so that there is within-firm dispersion in output even when there is no within-firm productivity
dispersion.

13

Figure 1: Monte Carlo(A) Within-Firm TFP Heterogeneity(B) Joint Production

Notes: Each triangle in panel (A) presents the correlation of true markups from equation (16) and markups
assuming no within-firm productivity heterogeneity from equation (17). The horizontal axis shows the ratioof within- to across-firm productivity heterogeneity for that correlation. There is no joint production in thispanel. Each triangle is a separate correlation coefficient. Each triangle in panel (B) presents the correlationof true markups from equation (18) and markups assuming no within-firm productivity heterogeneity fromequation (19). The horizontal axis shows the cost share of public inputs 1 − β. Each triangle is a separate
correlation coefficient. There is no within-firm productivity heterogeneity in this panel.

equation (11) and the parameterization in equation (13):P  Yik  β1
µji =

Aki

k

 j  β1
Yi
Aji

Rij
.
wLi

(15)

P
Q
σ j −1αjPreferences of a representative consumer are given by U = j (C j ) where C j = i∈Ωj (cji νij ) σj ,
νij is a demand shock, cji is consumption of a variety of product j produced by firm i andΩj is the set of firms i that produce varieties of product j. We assume that the number offirms in each industry is sufficiently large that markups are constant within each industryjand equal to the standard CES formula: µji = σjσ−1 .16 Elasticities of substitution for the J
industries σ j run from 1.1 to 10.1 in even increments. Appendix C describes the setting in
more detail.
For the first exercise, we impose no joint production (β = 1) in which case equation (15)16

It is easy to confirm that this will also equal the estimated markup using equation (15).

14becomes

P  Yik 
Ak

k

µji =

 j i
Yi
Aji

Rij
.
wLi

(16)

If one (falsely) assumes that there is no within-firm productivity heterogeneity (Aji = Ai ),this expression becomes
P k j
Y Ri
j
.
(17)
µ̂i = k j i
Yi wLiWe now explore whether and how quickly markups derived from equations (16) and (17)diverge from one another as we allow for increasing amounts of within-firm productivity
dispersion. If there is no within-firm productivity dispersion, and one correlates markups
obtained from equations (16) and (17), the correlation is one because the assumption of
Aji = Ai is accurate. This is the first point on the far upper left of the panel (A) of Figure 1.
We then increase the within-firm variance of (log) productivity draws relative to theacross-firm variance of (log) productivity draws as we move along the horizontal axis withthe vertical axis reflecting the correlation between measures obtained between equations(16) and (17). When the variance of within-firm draws is small (10% of the across firm
variance) relative to the across-firm variance, the assumption of no within-firm productivity
heterogeneity is innocuous as reflected by a high correlation of 0.98. However, this changes
quickly. When the ratio of the within- to across-firm variance is 0.3, the correlation falls to
0.22. We interpret this as evidence that markup estimation is sensitive to the precise degreeof within-firm MRT heterogeneity.
In the second exercise (panel B), instead of steadily increasing the within-firm varianceof productivity draws, we steadily increase the cost share of joint inputs (1 − β) and compare
true markups with markups obtained under the false assumption that production is nonjoint. We assume there is no within-firm productivity dispersion in this second exercise.
More formally, true markups in this case are given by
µji =

1 j
k β
Y
Ri
i
k
.
1

j β
wL
i
Yi

P

(18)

where β ∈ (0, 1]. If one falsely assumes non-joint production, then one will believe that thefollowing would accurately measure markupsµ̂ji =jkk Yi Ri.
Yij wLi

P

15

(19)

Analogous to panel (A) of Figure 1, panel (B) of Figure 1 presents the relevant correlations.
When 1 − β = 0, the share of joint inputs is zero, and there is no difference between
restricted firm-product markups (from equation 19) and the true markups (from equation
18). Again, this is the dot at one in the upper left corner of panel (B). However, unlikewith productivity dispersion, we find that the increasing presence of joint production (β ↓)does not materially affect one’s ability to recover markups: the correlations remain well
above 0.99 for cost shares of public inputs (1-β) up to 0.95. Appendix C shows plots of thevariance of estimated markups instead of their correlation with true markups. As one might
expect, the variance of estimates increases dramatically near the same parameter values
when the correlation coefficients diminish. As a whole, these results suggest that within-firm
productivity heterogeneity is more of the threat to identification of firm-product markups
than is joint production.

4

Identification of Firm Markups

We now develop our firm markup and show that its identification does not require an
econometrician to know whether or not there is joint production, nor the magnitude of
MRT heterogeneity. Our object is a weighted average of firm-product markups and can be
identified using only information on total firm revenues, expenditure on a static input, and aset of output distance function elasticities that can be estimated using techniques that havealready been established in the literature subject to minor restrictions.
We obtain our object by multiplying the expression for a firm-product markup (equation
11) by∂ ln Di (Yi ,Xi )j∂ ln YiP∂ ln Di (Yi ,Xi )k∈Yi∂ ln Y ki, and then summing over all j ∈ Yi , where Yi is the set of all j suchthat Yij > 0. This yields

µi ≡

X

ρji µjiρji ≡ P

where

∂ ln Di (Yi ,Xi )∂ ln Yij∂ ln Di (Yi ,Xi )k∈Yi∂ ln Yik

j∈Yi

.

(20)

Equation (20) shows that the firm markup defined here is simply a weighted average of firmproduct level markups, µji , where the weights ρji sum to 1 by construction. These weights
depend on the shape of a firm’s output distance function, as well as equilibrium choices of
product-level outputs and prices.

16

Slight manipulation of equation (20) delivers∂ ln Di (Yi ,Xi )Ri∂ ln Misµi = − P∂ ln Di (Yi ,Xi ) E
sik∈Yi∂ ln Yik(21)

P
where Ri ≡ j Rij . This expression shows that an econometrician can identify µi using total
firm revenue Ri , expenditure on a static input Esi , and output distance function elasticitiesP∂ ln Di (Yi ,Xi )∂ ln Di (Yi ,Xi ). Section 5 shows that these elasticities can be estimatedandk∈Yi∂ ln Mis∂ ln Yikusing standard production data under minor restrictions. Importantly, estimation of theseelasticities does not require a researcher rule out MRT heterogeneity or to take a stand onwhether there is joint production.
Before we show how to estimate these elasticities, we first discuss interpretation of ourfirm-level markup. First, we provide an interpretation of the weights. Second, we show thatthe firm markup can be interpreted as the markup of a composite good produced by the firmin a world where consumers have general preferences over these firm-level composite outputs
and the output distance function is separable. Third, we show that the firm markup is ameaningful object for a social planner who wants to eliminate across-firm misallocation but
who takes within-firm misallocation as given.

4.1

Interpreting the Weights ρji

The weight ρji defined in equation (20) can be interpreted as the share of the percentagechange in firm total cost when scaling up all products that can be attributed to scaling up
product j. This makes it analogous to a cost share. To see this start by multiplying and(Yi ,Xi ) 1:
dividing equation (21) by ∂Di∂M
Wsisi∂Di (Yi ,Xi )∂Yijρji = ∂Di (Yi ,Xi ) 1 Yij
∂Msi
Wsi

∂Ci (Yi ,Ki ,Wi ) jYi
M Cij Yij
∂Yij
k
P
Yi =
=P
k k
∂Ci (Yi ,Ki ,Wi ) k∂Di (Yi ,Xi ) 1
Yik∈Yi M Ci Yi
k∈Yi
k∈Yi
∂Msi
Wsi
∂Yik

,

X

∂Di (Yi ,Xi )∂Yiki ,Ki ,Wi )where the second equality uses the definition of firm-product marginal cost M Cij = ∂Ci (Y∂Yji
defined in equation (10) derived from the firm’s cost minimization problem. This can be
rewritten as:∂ ln Ci (Yi ,Ki ,Wi )∂ ln Yijjρi = P.
∂ ln Ci (Yi ,Ki ,Wi )kk∈Yi∂ ln Yi

17(22)

The numerator in equation (22) is the elasticity of the firm’s cost function with respect to
output j. While the denominator is simply the sum of the cost elasticities for all outputs
within the firm, it is also equivalent to the elasticity of the cost function with respect to thescale of the firm. The denominator is the percentage change in the cost of the firm when it
scales up all outputs by one percent.17 The weight ρji therefore represents how costly it is toincrease the output of product j within a multi-product firm.
The firm-level markup (equation 21) is therefore a weighted average of firm-product levelmarkups where the weights correspond to the share of the product in total cost elasticity of
the firm. This is analogous to the aggregate markup defined in Edmond et al. (2022), where
single-product firm-level markups are weighted by the firm’s cost share in the aggregate costof production. Under their assumption of a common cost elasticity across firms, a firm’s
relative cost elasticity with respect to the cost elasticity of scaling up the entire economy issimply equal to the firm’s cost share in total production. Our result generalizes the idea ofa cost share to joint production settings where input shares are not well defined, due to thenon-existence of product-line specific production technologies (i.e. joint production).

4.2

Single-Product Firm Equivalence Result

The firm markup acts as a firm-level wedge that scales down uniformly the productionof all products within a firm. In order to develop this intuition, we examine a version of
our model that is isomorphic to the single-product firm model of Edmond et al. (2022).
They consider a world populated with single-product firms producing a product over which
consumers have preferences. We first impose restrictions on consumer preferences and firms’technologies under which our multi-product firm model aggregates to their single-productfirm model. We show that our aggregate firm-level markup is equivalent to their singleproduct firm markup under these restrictions.
Suppose that consumer preferences are given by U ({Yij }j,i ) = U ({Yi }i ) where the function Yi = Yi ({Yij }j ) is homogenous of degree 1 and Yi (.) aggregates all products produced
by a firm i. We refer to this as the firm’s composite good. Further, under the assumption of
separability, a firm’s output distance function can be written as Di (Yi , Xi ) = Gi (Yi )/Fi (Xi ).
If Gi (.) is homogenous of degree one, we can collapse our multi-product firm to look like a
17

To see this, start from the cost minimization problem defined in equation (7), and scale up the output
vector by a factor λ. The cost elasticity with respect to the scale of the firm is given by the elasticity of thescaled cost function with respect to λ evaluated at λ = 1:
X ∂ ln C(Yi , Ki , Wi )X ∂ ln Ci (Yi , Ki , Wi )∂ ln C(λYi , Ki , Wi )
=
Yik =
.
k
∂λ
∂Yi
∂ ln Yik
λ=1
k∈Yi

k∈Yi

18

single-product firm that produces the composite good Yi . Assuming that firms operate on
their production possibilities frontier, we can show the following:
Yi (Yi ) =Yi (Yi )Fi (Xi )Gi (Yi )

=⇒

(23a)

Yi (Yi ) = Ai (λi )Fi (Xi )
Y1

where

Ai (λi ) ≡Yi (λi )Gi (λi )

(23b)

Y2

and λi = ( Yir , Yir , ...) is the vector of firm outputs Yij relative to some reference productiiwithin the firm Yir > 0.18
Equations (23) relate any vector of firm outputs Yi to a consumption aggregator Yi (Yi ).
This relationship depends on two separate components of firm production: i) within-firm
output allocations, which are captured by the relative output ratios λi , and ii) firm scale,
which is determined by its aggregate input use Xi .
The production technology given in Edmond et al. (2022) is Yi = Ai Fi (Xi ) where Ai is asingle-product-firm Hicks neutral productivity term that is independent of firm output. In
our setting, firm productivity, Ai is determined by the within-firm output allocations, λi .
To link the two, we consider the case where within-firm output allocations are fixed suchthat relative output levels λi are taken at some initial reference level, λ0i .19 This makes
Ai (λ0i ) = A0i invariant to the scale of the firm. The firm production function can then be
written as:
Yi = A0i Fi (Xi ).

(24)

In this environment, the De Loecker and Warzynski (2012) formula for the markup of
the single-product firm defined in equation (24), µSP
i , with total revenue Ri and expenditure
Esi on some static input Xsi is then given by:∂ ln(Di (Yi ,Xi ))Ri∂ ln (A0i Fi (Xi )) Ri
Ri∂ ln(Xis )SPµi = θsi
=
= −P
= µi
∂
ln(D
(Y
,X
))
i
i
i
Esi
∂ ln (Xsi ) Esi
E
si
k
k∈Yi
∂ ln(Yi )where the final equality follows from applying equation (21) to a separable output distancefunction. This implies that–in this setting–the markup of a single-product firm producing
Yi is equivalent to the weighted average of firm-product markups defined in equation (21).
In this sense, our firm markup, µi , can be interpreted as the markup on a composite good18

We make use of the homogeneity of functions Yi (·) and Gi (·) in going from the first to the second lineof equations (23).
19More formally, we assume that firms only produce output vectors such that {Yi : λi = x × λ0i , x ∈ R+ }19

produced by its single-product counterpart.

4.3

Firm Markups and Across-Firm Misallocation

We now ask what the social planner’s allocation looks like in this economy by multiproduct firms under the conditions set out in section 4.2. From Edmond et al. (2015, 2022),
we know that the social planner would allocate inputs across all of the single-product producers so as to equalize markups, i.e. µi = µ ∀ i, so as to eliminate misallocation of resources
across firms. This leads to the well-known result that variation in markups across firms is a
symptom of misallocation.20
This result holds only if Ai is invariant to firm scale; otherwise, our model is not isomorphic to the single-product firm setting of Edmond et al. (2015, 2022). This requires that
we hold within-firm output allocations fixed, i.e. λi remains unchanged when we consider
counterfactual changes to firm scale through Xi . We interpret this setting as one in whicha social planner targets across-firm misallocation, taking within-firm output allocations asgiven. In this setting, we can then interpret our firm markup as equivalent to that in Edmondet al. (2015, 2022) and, therefore, as a summary statistic for the extent of across firm misallocation, even though our generalization allowed for multi-product firms with heterogeneous
MRT and joint production.
We now extend the same result after relaxing the restrictions placed on consumer preferences and firm technology in the previous subsection. Suppose instead that the representative
consumer has preferences over all products produced in the economy U ({Yij }), and firm’s
technology is characterized by the output distance function Di (Yi , Xi ). A social planner
who wants to eliminate across-firm misallocation taking within-firm output allocations asconstant solves the following problem:
max U ({Yi λi }){Yi }is.t.

Di (Yi , λi , Xi ) ≤ 1

andX

Xsi ≤ Xs

∀ i, s(25)iwhere λi is the vector of relative output levels within the firm, and Yi is the scale of the firm.
The problem outlined in equation (25) states that the social planner chooses the scale of thefirm, Yi , by allocating resources across firms while keeping the ratio of outputs within eachfirm unchanged. The first order conditions from this problem gives the following condition
20

If one extends this setting to have an endogenous supply of resources in the economy (e.g. by assuming
labour is costly for a representative consumer as in Edmond et al. 2022), then you can further show thatmarkups should all be equalized to 1.

20

that has to be met in order to eliminate across-firm misallocation:21∂ ln Di (Yi ,Xi )∂ ln Xsi−P∂ ln Di (Yi ,Xi )k∈Yi∂ ln YikPk∈Yi∂U ({Yi }) k
Yi
∂Yik

Xsi

= γs

∀i

where γs is the shadow price of the input Xsi . The left-hand-side of the above equation in
the decentralized equilibrium is proportional to the product of the price of input Xsi thatthe firm faces and the firm markup, wsi µi (see Appendix D). Under the assumption that all
firms face the same input prices, firms markups must be equalized, i.e. µi = µ ∀ i for an
efficient allocation of resources to hold. Therefore, the constrained social planner would liketo allocate inputs across firms such that firm markups are equalized. The firm markup canthen be interpreted as a output wedge that decreases firm scale so that variation in the firm
markup provides evidence of across firm misallocation, even under more general utility andproduction structures.

5

Estimating Firm Markups

Following De Loecker and Syverson (2021), we now discuss how two econometric techniques standard in the literature can be used to estimate the relevant output distance function
elasticities needed to identify the firm markup given in equation (21). First, we can follow
De Loecker et al. (2016) and use a sample of single-product firms to estimate an output
elasticity for a multi-product firm that counterfactually chooses to be single-product. As
long as the output distance function is separable, this single elasticity is sufficient to recover
the firm markup. Second, we can use a cost-share estimator applied to the full sample ofmulti-product firms following Syverson (2004), Foster et al. (2008), Backus (2020), and Raval(2022).22 We refer to these as techniques 1 and 2, respectively.

5.1

Technique 1: Relying on Single-Product Firms for Estimation

De Loecker et al. (2016) use single-product firms to estimate product-level output elasticities with non-joint production. While identification of firm-product markups in theircontext depends critically on MRT heterogeneity (see figure 1), an almost identical estimation approach can be used to identify the firm markup while not making any assumptions
21

See derivation in Appendix D.
See the excellent discussion in De Loecker and Syverson (2021) for more details of the merits of these
two approaches.
22

21

about MRT heterogeneity even if there is joint production.23
Our extension requires two key restrictions. First, as previously discussed, each firm’s
technology must nest the possibility of a well-defined single-product production function (Assumption 3). This means that a firm uses the single-product production function wheneverit chooses to produce a single product. This allows us to tie output elasticities of singleproduct firms to those of multi-product firms and does not preclude the possibility of joint
production for multi-product firms whenever Yi 6= Yij (Theorem 1). The second restriction
is a set of restrictions on the firm’s output distance function:
Assumption 4 Di (Yi , Xi ) is separable, Gi (Yi ) is homogeneous of degree 1, and Fi (Xi ) =Fg(i) (Xi ) for some mapping for firms i to industries, g.
Combining Assumptions 3 and 4, the shape of the production function only differs at theindustry level for single product firms.24 This is a standard restriction imposed by nearly theentirety of the literature on production function estimation because estimation of production
function parameters requires some sort of averaging across a set of firms operating in thesame industry g.25
Assumption 4 does not restrict firms from having varying degrees of returns to scale,and does not require that single-product production be homogeneous of degree φ > 0. The
output aggregator Gi (Yi ) for for multi-product firms must be homogeneous of degree 1
but can otherwise vary flexibly across firms. This flexibility is key to allowing for MRT
heterogeneity as well as joint/non-joint production.26
Assumptions 3 and 4 combined also imply that the firm markup can be recovered using
an estimate of a single-product firm’s output elasticity for a static input Ms . This is a key
result in this paper which we now state and prove as a proposition:
Proposition 1 Suppose Assumptions 3 and 4 hold. Then,
j
j Ri

P
µi = θsg(i) (Xi )where θsg(i) (Xi ) ≡∂ ln Fg(i) (Xi )∂ ln Msi

Esi

(26)

is the output elasticity for static input s in firms that choose

23This is because our approach sidesteps the need to solve an input allocation problem as in De Loecker
et al. (2016).
24
By “shape” we exclude TFP heterogeneity which we naturally allow to vary across firms and products.
25
See for example Olley and Pakes (1996), Levinsohn and Petrin (2003), Ackerberg et al. (2015), Gandhi
et al. (2020).
26
For example, equation (13) provides an example of an output distance function consistent with Assumption 4. As we previously demonstrated, this class of technology allows for MRT heterogeneity, as well as
joint or non-joint production. Since we are allowed to index the output aggregator, Gi , with i, we can allow
for even greater flexibility than the output distance function in equation (13).

22to produce a single product, i.e. if Yi = Yij , then all cost minimizing firms operate using thesingle-product production technology Yij = Aji Fg(i) (Xi ).
Proof. See Appendix E.
The right-hand side of equation (26) is identical to the standard De Loecker and Warzynski (2012) markup formula, with the caveat that θsg(i) (Xi ) for multi-product firms is understood as the output elasticity for the firm if it counterfactually chose to be a single productproducer. This shows that the De Loecker et al. (2016) estimation approach–using single
product firms to estimate this output elasticity–can in principal be used to identify the firm
markup.
An important concern is that firms may self-select into whether they are single productor multi-product producers depending on their productivity differences, potentially leading
to selection bias. De Loecker et al. (2016) show that it is possible to correct for this bias intheir setting. In Appendix Q, we show that their approach can be extended to environmentsthat involve joint production and MRT heterogeneity based on a core competence model asin Eckel and Neary (2010), Mayer et al. (2014) and Arkolakis et al. (2021). However, one
should not disregard that a key disadvantage to relying on single-product firms for estimationof the relevant elasticities in (21) is that there may be biases introduced by modelling theselection into single-product status incorrectly. This partially motivates Technique 2.

5.2

Technique 2: Cost SharesThis approach does not rely on using single-product firms as a counterfactual for multiproduct firms and therefore is not subject to concerns about selection bias in estimation. This
technique develops a cost-share approach to identify the unknown elasticities in equation (21)following Syverson (2004), Foster et al. (2008), Backus (2020), and Raval (2022). However,the validity of this approach requires restrictions not needed for Technique 1: i) static
first-order conditions must hold for all inputs on average, and ii) each firm’s (separable)
input aggregator must be constant returns to scale Cobb-Douglas varying by industry. We
formalize these as Assumptions 5 and 6 below:
Assumption 5 For any input s, each firm’s static cost minimizing first-order condition,
expressed in terms of total expenditure on each input,27 holds on average:∂Di (Yi , Xi )Xsi .
E (Wsi Xsi ) = −E λi
∂Xsi


27

(27)

i.e, each input’s first order condition is multiplied by Xsi , so the right hand side of equation (8) is
expressed in terms of expenditures, rather than quantities.

23

Assumption 5 requires that firms adjust their inputs consistent with cost minimization
on average. This may accommodate adjustment costs in capital (for example), which may
prevent equation (8) from holding exactly each period or for each firm, even though a firm
facing no adjustment costs will make equation (8) hold since current period costs would be(Yi ,Xi )Xsi .
lower for the firm if they set total capital costs to λi ∂Di∂X
si
P
Assumption 6 Di (Yi , Xi ) is separable, with Fi (Xi ) = Πs (Xsi )βs,g(i) , where s βs,g(i) = 1,
for some mapping of firms to industries g(i).
Assumption 6 requires the output aggreagtor of the distance function to simplify to a
standard constant returns to scale Cobb-Douglas production function whenever firms only
produce a single product j. Since single-product production functions are industry specific,
industry is a property of a firm, rather than a firm-product pair.28 Together, Assumptions
5 and 6 lead to the following result:
Proposition 2 Suppose Assumptions 5 and 6 hold. Then as long as at least one purely
static input, Xsi , exists and firms minimize their conditional costs as in equation (7), then:µi =E (Wsi Msi |g = g(i)) Ri
P.
E ( s0 Ws0 i Xs0 i |g = g(i)) Esi(28)

Proof. See Appendix G.
E(Wsi Msi |g=g(i))
A consistent estimate of E P
is then easily constructed using industry by( s0 Ws0 i Xs0 i |g=g(i))
industry aggregate cost shares, as in Syverson (2004), Foster et al. (2008), Backus (2020)
and Raval (2022).

5.3

Comparing Techniques 1 and 2

While Technique 1 relies on single-product firms to estimate the relevant elasticities for
identifying markups, Technique 2 uses observable cost shares and does not require that we
condition on single-product firms. Therefore Proposition 2 from Technique 2 provides auseful alternative to Proposition 1 in Technique 1 for obtaining firm markups. However,Technique 2 relies on functional form restrictions not needed in Technique 1, most notably
the requirement that output distance function be separable with a constant returns to scale
Cobb-Douglas input aggregator.
28

This restriction can be relaxed is one is comfortable assuming that all inputs are fully static, so that
equation (8) holds firm-by-firm for all inputs. In Appendix F we show that if: i) the output distance function
satisfies constant returns to scale, and ii) all inputs are fully static; then we can identify the firm markup using
the ratio of revenues to costs. Importantly, this result allows for non-separable output distance functions,which can capture situations where firms operate non-joint production technologies that have different factor
intensities (Hall 1973).

24

Because of this, Technique 1 and 2 require non-overlapping restrictions on a firm’s technology and economic environment. While Technique 1 allows for rich models of dynamic
input use and varying returns to scale, Technique 2 requires constant returns to scale CobbDouglas technologies, and that static first-order conditions hold across firms on average.
This latter requirement limits dynamics. Technique 2 directly uses multi-product firms in
its estimation, and does not require disciplining how firms choose their output sets as inTechnique 1. Therefore determining which one is proper is difficult and case-specific. We
therefore pursue both strategies in our estimation.

6

Data and Implementation

This section first briefly describes our data. We then describe how we operationalize theestimation of our output elasticities following the two techniques described above. Finally,
we discuss how to estimate plant-product markups for comparison to our plant-level objects.

6.1

Data

We implement our methodology using the Indian Annual Survey of Industries (ASI) data
set for the years 2001-2008. The ASI is a panel data set of manufacturing establishmentswith 10 or more permanent employees that use electricity, and of establishments 20 or more
employees that do not use electricity. Large plants are always included and therefore ASI isa census of these firms. In addition, ASI contains a 5% random sample of small plants.29 An
observation is a plant-product-year triad. Each plant i reports a single 2-digit NIC industry
g(i) in which it operates. We take g(i) to correspond to a particular production technology
as in De Loecker et al. (2016).30 Each plant reports up to ten products that it produces,with each entry corresponding to a single j in our framework.31
The primary appeal of the ASI is its inclusion of detailed product-line data, including
revenues (measured in rupees) and physical outputs. Product-level output is the physicalquantity of the good manufactured by the plant in a given year.32 We observe capital,
29

The cutoff for a plant to be considered “large” is based on the number of permanent employees, withthe exact threshold varying from state to state.
30
Besides major revisions in 1999 and 2009, the NIC system underwent a minor revision in 2005 from
ISIC-3 to ISIC-3.1. While identical at the 2-digit level, ISIC-3.1 revised some 4-digit industry codes. In
order to have a consistent set of industry codes, we drop plant-years from our pre-2005 data with a 4-digit
industry code that does not have an exact match in the revised NIC. This results in the loss of about 11,000
plant-product-years.
31
Associated with each product entry is a 5-digit ASICC code. Note that some plants report more than
one entry for a given 5-digit ASICC code, in which case we treat each entry as a separate product.
32
Each product level quantity is associated with a 5-digit product code as well as a particular unit of25

Table 1: Summary statistics
Plant-product outcome

Mean

Log Revenue: ln(Ritj )Log Quantity Produced: ln(Qjit )Plant-product-years:

15.9260Plant outcome

Mean

Log Plant Revenue: ln(Rit )Log Plant Capital: ln(Kit )Log Plant Labour: ln(Lit )Log Plant Materials: ln(Mit )Log Plant Wage: ln(Wit )Average products per plant-year:
Plant-years:
Distinct plants:

17.051
9.809
9.455
11.3284.912

Std. dev.

Min.

Max.

2.681 1.386
2.482 -14.28

27.557
15.631355,624

Std. dev.

Min.

Max.

2.235 6.774
2.257 -6.697
1.566 1.792
1.959 2.313.622
027.60319.81316.51220.816
11.832
1.77
200,886
96,047

Notes: See main text and Appendix H for more detail on variable definitions. Log quantities demeaned
within 5-digit product code.

labor, and intermediates input expenditures at the plant level, not the plant-product level.
The plant capital stock is constructed using the perpetual inventory method as described inAppendix H. Labor input is in person-days. Person-days and the wage bill are plant totals
for all employees including supervisors and managers. Intermediate inputs are the sum of
expenditure on domestic and foreign goods. We deflate capital and intermediates using theIndian Wholesale Price Index (WPI) for all manufacturing industries although we offer afurther input price correction described in section 6.2.33 See Appendix H for more details.
Table 1 presents summary statistics. Our data comprise 355,624 plant-product-years,
and 200,866 plant-years. The average number of products per plant-year is less than 2suggesting the prominence of single-product plants. We now discuss how we use the plantdata to estimate physical output elasticities and firm-level markups. We perform estimation
using plants, not firms.34
measure (e.g. kilograms, number of, etc...). We demean these quantities within product code when presenting
summary statistics for quantities.
33
This follows De Loecker et al. (2016). For the years of our dataset, the WPI records price indices forsome 2-digit NIC industries but not others. We opt to use the WPI for all manufacturing industries todeflate expenditures for all plants in our data.
34
Since our data is plant (not firm) level, from now on we primarily let i index plants, rather than firms.
We will compare our plant-product markups to those estimated in De Loecker et al. (2016). The latterare firm-product markups, and are estimated using the Prowess data set covering the years 1989-2003. See
De Loecker et al. (2016) for more details on the Prowess sample frame.

26

6.2

Plant Markups

As discussed in Section 5, we consider two techniques to recover plant markups: theestimation of the production function for single product plants using a control function
approach, as well as a cost-share estimator.
Estimation of the production function for single product plants requires specifying afunctional form. We rely on a translog specification of the production function which imposes
the constraint that the production function is globally homogenous of degree φ > 0:35
yitj = βL lit + βK kit + βM mit + βLK lit kit − 0.5 lit2 + kit2
+ βLM lit mit − 0.5 lit2 + m2it + βKM kit mit − 0.5 kit2 + m2it + ajit (29)where lower case variables denote natural logs, and global returns to scale are given by
φ = βL + βK + βM . In our baseline specification based on Technique 1, we focus on results
where a single translog is estimated for the entire Indian economy, although we also reportresults where we allow production function parameters to vary by two-digit industry inAppendixes N and O.
We follow De Loecker et al. (2016) closely by implementing both a selection correction
for single-product firms, and a control function approach to deal with the endogeneity of
inputs.36 To deal with measurement error in capital, our baseline results identify the coefficient on capital using lagged investment, as recommended by Collard-Wexler and De Loecker(2021). One minor point of departure is how we deal with input price bias. Since we observe
plant-specific wages, we use this information directly to correct for input price bias in a waythat does not require that the input price control function be estimated simultaneously withproduction function parameters- see Appendix I for more details. 37
After having estimated each firm’s (potentially counterfactual) single-product production
function, we then construct plant markups. Under the assumptions underlying Propositions
1 and 2, we assume that materials is a static input (s = M ), and write:
35
See De Ridder et al. (2022) and Orr and Tabari (2022) for similar specifications of the translog with theadded restriction that φ = 1.
36
We also consider dynamic panel based approaches to estimation in Appendixes J, N, and O to verify that
our results are not driven by some of the concerns articulated by Bond et al. (2021) concerning proxy-variableapproaches.
37
To recover input allocations for multi-product firms, we will need the input price control function torecover output-specific input prices. However, since we observe wages at the plant level, we can obtain
more precise estimates of the relationship between input prices and quality by directly regressing observed
wages for single-product producers onto prices and market shares to obtain the relevant input price function.
As a result, we can obtain the input price control function outside of the production function estimation
algorithm. See Appendix I for more details.

27

Pµit = θM g(i) (Xit )

j
j Rit

EM it

(30)

where θM g(i) is either the output elasticity for single product firms in industry g (as in
Proposition 1) or the industry cost share for materials (as in Proposition 2).38
The key decision that needs to be made to implement the cost-share estimator (Technique2) is how to allocate plants to industries, where the input aggregator Fg(i) (.) is assumed tohave the same Cobb-Douglas form for all plants i belonging to industry g.39 We consider two
mappings of plants to industries g(i). First, we simply take a plant’s stated two-digit NIC
code as their industry. Second, we follow Raval (2023) and for each two-digit NIC code, we
split plants into five different quintiles, based on the ratio of labour to material expenditures,and treat each two-digit code × quintile as a separate “industry” g. Raval (2023) proposes
this approach to account for unobserved factor-specific productivity differences across firms,
and documents that this method alleviates some of the issues with the production-side
approach to markup estimation described in Raval (2022). For each mapping of plants toindustries, we then construct output elasticities using the industry-specific cost shares.

6.3

Plant-Product Markups

For comparison to our plant markups, we also estimate plant-product markups adhering
as closely to De Loecker et al. (2016) which relies on a system of equations to solve forthe unobserved input allocations.40 The most basic variant of this approach imposes nonjoint production and that Aji = Ai ∀j. In this case unobserved input allocations ρjit can beobtained by solving the following system of equations for each plant-year (i, t):Xρjit = 1,

(31)

j∈Yi

and
Yitj = Ait F

Eit
ρjit j
c
W
it

38

!
∀j

(32)

We always restrict our sample to plant-years where the estimated θM g(i) (Xit ) is strictly positive. While
this is always true for our baseline sample, we occasionally obtain θM g(i) (Xit ) < 0 for some industries andplants when we consider alternative production function estimators in the Appendices. We drop those plants
whenever this happens.
39
Although another important decision is how to obtain the rental rate of capital for each plant. For thispurpose, we largely follow Raval (2022)— see Appendix H.
40
We calculate plant-product markups instead of simply using the measures from the replication files to
De Loecker et al. (2016) (which uses the Prowess data set) so that we can compare ASI-based plant markups
to ASI-based plant-product markups.

28

c j denotes the product line specific input-price control function.41 The system ofwhere Witequations defined by equations (31) and (32) provide J + 1 equations to obtain down J + 1
unknowns; the J expenditure shares ρjit and Ait . De Loecker et al. (2016) solve this systemof equations numerically.42
Since the translog specification of the production function is homogeneous of degree φ,(31) and (32) provide a simple closed form expression for the input shares:43
Yitjjρit = P

 φ1

cj
W
it
.
1

k φ ck
Y
W
it
it
k

(33)

Since production is non-joint in De Loecker et al. (2016), our estimates of ρjit are sufficient
to recover estimates of product-level markups, which using the standard De Loecker andWarzynski (2012) formula adapted to multi-product firms as in De Loecker et al. (2016):µjit = θM g(i)Eitρjit j
c
W
it

!

Rij
Rij
= θM g(i) (Eit ) jρjit EM i
ρit EM i(34)

Xs
where the second equality uses the fact that θs = ∂F∂X(X)is homogeneous of degree 0 ins F (X)
X if the production function is homogeneous of degree φ > 0.
We also consider specifications where ρjit is given by the within-plant revenue share ofP
product j, Rij / j Rij . This specification is of interest for two reasons; first, allocating
inputs based on on revenue shares is common in the literature (Foster et al. 2008, Atalay
2014, Collard-Wexler and De Loecker 2015, Blum et al. 2018). Second, this input allocation
rule rationalizes within-plant heterogeneity entirely through heterogeneity in marginal ratesof transformation, rather than markup differences. Orr (2022) shows that revenue sharesreflect within-plant productivity differences as long as production functions are identicaland homogeneous of degree φ > 0, and there is no within-firm markup dispersion. As aresult, this allocation rule effectively imposes a constant within-plant markup, µjit = µit ,to allow for flexible Ajit differences, rather than imposing constant within firm productivity,
41
We obtain the input price control function outside of the production function estimation routine byregressing plant wages on various observables, such as prices and market shares, for single-product firms.
See Appendix I for more details. If the input-price control function is correctly specified and estimated,
Eit
Xjit = cj .
Wit42b j , from a variant of their first stage estimating
De Loecker et al. (2016) rely on predicted output, Φitequation to generate (32), rather than output directly. We rely on output in our baseline specification sinceb j or Y j makes any difference for
it is more tightly linked to our theory, but also consider whether using Φitit
the recovered product-level markups as a robustness check.
43
See Appendix K for derivation.

29

Ajit = Ait to identify flexible product-level markup differences µjit .44

7

Results

We now examine plant-product markups using methods popular in the literature, andthen compare them to the plant markups that are the focus of this paper.
Section 7.1 starts by showing that common estimators of plant-product markups delivera large mass of markups close to zero. This implies nearly infinite marginal costs or zero
prices. We show that such markups are nearly always found at multi-product firms suggesting
potential misspecification of within-plant across-product input allocation rules. We thenpresent data-driven exercises suggesting that omission of MRT heterogeneity may be behind
this issue consistent with the Monte Carlo results of 3.2.45
Section 7.2 displays our plant-level markups where output elasticities are derived using
Techniques 1 and 2 as discussed in section 5. Using both techniques, our plant-level markups
display much tighter dispersion than those presented in section 7.1 even without trimming
outliers. Further, we do not see a mass of markups at zero.

7.1

Product-level Markups and a Zeros Puzzle

7.1.1

Documenting the Problem of Outlier Markups

Figure 2 presents histograms of markups at the firm/plant-product level under the assumption that production is non-joint with no MRT heterogeneity (equations 33 and 34).
Panel (A) uses the control function estimation approach as described in section 6.3 applied
to our plant-level ASI data. Panel (B) plots firm-product markups obtained directly fromDe Loecker et al. (2016) replication files.46
Both panels display a large mass of markups near zero, another above 1, and a long righttail. While the second peak is expected, the mass of estimates near zero is striking, and can
only be rationalized by very high marginal costs, or prices of almost zero. Since this is themultiplicative markup, µ ≡ MPC , perfect competition corresponds to µ = 1. Therefore, µ = 0
Rj

P

Rk

To see this, substitute ρjit = P itRk into equation (34) to obtain µjit = θM g(i) (Eit ) EkM i i , which iskitconstant across j.
45
These are data-driven in that they use actual data as opposed to the Monte Carlo simulations of section
3.2.
46
Both sets of markups are for Indian producers. However, panel (A) uses the ASI and therefore only
considers plant-product levels markups for the years 2002 to 2008, while panel (B) uses Prowess data toconstruct firm-product level markups for the years 1989 to 2003. In both figures we truncate the distribution
at 5 to focus on the lowest set of markups we estimate.
44

30

Figure 2: Product-level Markups: All Plant-products(A) ASI Data

(B) Prowess Data

Notes: This figure displays histograms of firm/plant-product markups. Panel (A) presents plant-productmarkups calculated according to equations (33) and (34) using the ASI data. Panel (B) shows firm-productmarkups taken from replication files for De Loecker et al. (2016) that uses the Prowess data. We trim
markups ≥ 5 for scale.

is unexpected in standard profit maximizing models of firm behavior and likely reflects somedegree of model misspecification.
We quantify this possible misspecification by comparing the ratio of the number of estimated product-level markups that are close to zero (µjit < 0.5), to the number of markups
above 0.5 but below 5.47 We refer to the latter set as reasonable markups and to this ratio
as the zeros outlier ratio. We can also use this metric to examine the long right tail in
estimated markups.48 We quantify the importance of the right tail by calculating the ratioof the number of markups greater than 5 to the number of reasonable markups.49 We referto this latter ratio as the tail outlier ratio.
To explore whether model misspecification is a plausible culprit, we systematically vary
the assumptions used to estimate markups with the goal of pinning down the precise mechanisms that are generating these results. As a benchmark, the two far-left vertical bars in
Figure 3 display the zeros outlier ratio and the tail outlier ratio (respectively) for the firmWhile one might take µjit ≥ 1 to be “reasonable”, i) product-level markups can be less than 1 withcomplementary goods, and ii) when the correct model is perfect competition and there is some (classical)
measurement error in revenues or expenditures, we may find estimated markups distributed above and below
1.
48
While not directly plotted in Figure 2, estimated markups here have a fairly fat, right tail, with the 95th
percentile of product level markups being 57.83, and the 99th percentile involving a markup of 546.
49
We compare our outlier counts to the reasonable markup counts rather than simply examining proportions of markups of each type so that one outlier ratio becoming more likely does not mechanically decrease
the severity of the other outlier ratio.
47

31

product markups from Figure 2, panel (A). The number of zero markup estimates is about
20% the number of reasonable markups estimates. Combined with the right tail outliers(slightly less than 20%), the sample of total outlier markups is roughly 40% the size of reasonable markup sample. The problem gets worse if we focus only on markups estimated at
multi-product plants as seen in the second set of bars (“MP only”). Zero markup observations are over 40% of the size of the reasonable markups sample, and accounting for right tailmarkups generates a total outlier sample 70% the size of the sample reasonable markups,
suggesting that to 40% of the plant-product markups are outliers (0.7/(0.7 + 1) = 0.41).
This suggests that mispecification of the input allocation rule may be partly responsible forthe large number of markups with undesirable properties.
The next three sets of bars (“No Wj ”,“Phi ”, “Unit Adj ”) show that adjusting the input allocation rule by i) removing the input price controls function, ii) purging output of
measurement error, and iii) carrying out additional units adjustments do not substantively
change these results. Appendix L describes these changes in detail. In the final two bars,
we allocate inputs by revenue shares. This rule is interesting because it allows for MRT
heterogeneity as in Orr (2022) but rules out within-plant markup differences. This simple allocation rule almost completely resolves our zero markup puzzle; the number of zero
markup observations is only 2% the size of the reasonable sample and tail outliers are only
0.05%. Figure 4, panel (A) plots the histogram of product-level markups for our baseline
multi-product sample, and compares it to panel (B) which shows the revenue share multiproduct markup sample. Comparing panel B to panel A, both types of outliers become farless common.
While allocating inputs based on revenue shares almost completely solves the zero markup
puzzle, it is unsatisfying because this requires that we abstract from within-plant markup
differences, defeating the point of estimating product-level markups. However, this suggests that establishing ways to allow for MRT heterogeneity that do not restrict within-firm
markups may be fruitful.
7.1.2

Do Parameterized Productivity Ladders Help?

Can parameterized models allowing for MRT heterogeneity help in eliminating the massof markups at zero? If the answer is “yes”, then these fixes may allow researchers to use
input allocation rules for existing models with minor changes. While we find that suchparametrizations may help, results are too sensitive to provide conclusive support for whether
productivity ladders can eliminate the outlier problems.
To see this, consider a simple parameterized input allocation rule that allows for MRT

32

Figure 3: Plant-Product Markup Outlier Ratios Across Methods

Notes: The above bar chart plots two outlier ratios for different models of product-level markups. Letting
Z denote the set of product-level markups where µ ≤ 0.5, T denote the set of product-level markups whereµ ≥ 5, R denote product-level markups between 0.5 and 5, and |.| denote the number of elements in a given|Z||T|set, the Zeros outlier ratio is defined as |R|and the Tail outlier ratio is defined as |R|. Baseline refers to the
product-level markups in Panel A of Figure 2. MP Only only considers the subset of markups produced bydmulti-product plants. No Wj removes the input price control function adjustment Witj in equation (33). Phi
b j in place of realized output Y j in equaiton (33). Units Adj uses the modified input
uses predicted output Φit it  1j
P  Yitk  φ1 c k
φ
j
Yit
j
j
c
Wit /
Wit , where A is average TFPQ obtained from single
allocation rule ρit =
j
k
k
A

A

product-firms by 5 digit product code j. Rev Sh uses revenue shares to allocate inputs in equation (34). All
models except for Baseline look only at markups generated by multi-product plants. See Appendix L for
further details.

heterogeneity (see Appendix K for derivation):


Yitj
Ajit

 φ1

cj
W
it
ρjit =.
1
P Yitk φ c k
Wit
k Ak

(35)

it

If the econometrician knows {Ajit }j , then they can derive the allocation rules necessaryfor estimating firm-product markups. However, {Ajit }j is generally unknown making thisunfeasible without further restrictions. To provide such structure, we consider a variant of
the Mayer et al. (2014) model of within-firm heterogeneity based on core competencies. Eachfirm faces a productivity ladder over all potential products j, with j = 0 indexing a “core”
product that has the highest productivity within the firm. We assume that for all j 6= 0

33

Figure 4: Plant-Product Markups: Multi-Product-Plant-Product(A) Baseline(B) Revenue Share Allocations

Notes: The above two panels display a histogram of product level markups, for the subsample of multiproduct plants. Panel (A) shows our baseline plant-product level markups calculated according to equations(33) and (34). Panel (B) product-level markups where we use within-plant revenue shares to allocate inputs.
We trim markups ≥ 5 for scale.

Aji = A0i × (δ)j(36)

where δ ∈ (0, 1) is a free parameter that implies greater within-plant productivity dispersion
as δ → 0.50 We consider two different rankings.51 First, we assume that the product withthe highest revenue is the firm’s core product, j = 0, their second highest revenue product is
j = 1, and so on. Alternately, we assume that a firm’s lowest revenue product is the firm’s
core product, j = 1 is their second lowest revenue product, and so on. While this secondcase may seem perverse, previous work including Jaumandreu and Yin (2016), Forlani etal. (2016), Atkin et al. (2017), and Orr (2022) documents that TFPQ can be negatively
correlated with demand shifters (e.g. quality), and that these demand shifters often explain
revenues better than productivity differences (Hottman et al. 2016, Eslava et al. 2023). In
short, it may be that the lowest revenue products within a firm are the highest Ajit values.52
Figure 5 plots the zero and tail outlier ratios for various assumed values of 1/δ underthese two cases of rankings. Panel A assumes that top revenue products are j = 0 and panel
B assumes that bottom revenue products are j = 0. As 1/δ increases, MRT heterogeneity
50

With slight abuse of notation, j here indexes a plant-specific ranking of product-specific TFPQ.
Unfortunately, we have no way of knowing what each plant’s TFPQ ranking across products without
first disciplining the input shares across product lines. We also report results based on quantity rankings inAppendix M. These results turn out to be similar qualitatively, although we find that revenue rankings wereport in the main text perform better quantitatively in terms of solving the zeros puzzle.
52
In this case, these are low value “cheap goods”.
51

34

Figure 5: Product-level Markup Outlier Ratios with Productivity Ladders(A) Core Product - Highest Revenue(B) Core Product - Lowest Revenue

Notes: This figure reports outlier ratios for different models of product-level markups as a function of within
firm productivity dispersion, parameterized by 1/δ. Letting Z denote the set of product-level markups whereµ ≤ 0.5, T denote the set of product-level markups where µ ≥ 5, R denote product-level markups between 0.5
and 5, and |.| denote the number of elements in a given set, the Zeros outlier ratio (represented by circles) is|Z||T|defined as |R|, while the Tail outlier ratio (represented by triangles) is defined as |R|. Product-level markups




1
1



j
k
φ
φ
Yit
Yit
cj / Pc k . Panel (A) orders products jWW
calculated using equation (34) with ρjit =ititk (δ)k(δ)jwithin a plant from highest to lowest revenue, with j = 0 having the highest revenue, j = 1 having thesecond highest revenue, and so on. Panel (B) orders products within a plant from lowest to highest revenue,
with j = 0 denoting the lowest revenue product, j = 1 denoting the second lowest revenue product, and soon.

also increases as the steps along the productivity ladder become larger. The lines withtriangles represent the tail outlier ratio while the lines with circles represent zeros outlier
ratios. Assuming that products with small revenue shares have the highest productivity
within the firm as in panel B helps reduce the share of zero ratio but increases the numberof tail outlier ratios, diminishing its usefulness. When 1/δ = 10, the number of outliers
actually exceeds the number of reasonable markups.53 Assuming that high revenue products
have the highest productivity as in panel A makes both outlier ratios worse.
Figure 6 plots the plant-product markup distribution for 1/δ = 20 for the case where the
top-revenue product is a plant’s core product (panel A), as well as the case where a firm’s
bottom revenue product is their core product (panel B). Again, when we treat high revenue
products as having the highest TFPQ, the zero markups puzzle becomes much worse. If weassume low-revenue products have the highest TFPQ, the zero markups puzzle diminishes
greatly but this proposed solution (again) generates a thicker right tail in markups consistent
with continued model misspecification.
53

More precisely, the ratio of the sum of zero and tail outliers to reasonable outliers is greater than 1.

35

Figure 6: Product-level Markups with Productivity Ladders(A) Core product - Highest Revenue(B) Core product - Lowest Revenue

Notes: The above two panels display a histogram of product-level markups
subsampleof multi-product forthe 
1 k  φ1
j
P
φ
Yit
Yit
j
j
k
c
c
plants. Both panels calculate markups using equation (34) and ρit =Wit /Wit tok (δ)k(δ)jcalculate input shares, where δ = 0.05. Panel (A) orders products within a plant from highest to lowest
revenue, with j = 0 having the highest revenue, j = 1 having the second highest revenue, and so on. Panel(B) orders products within a plant from lowest to highest revenue, with j = 0 denoting the lowest revenue
product, j = 1 denoting the second lowest revenue product, and so on. We trim markups ≥ 5 for scale.

MRT heterogeneity is limited in two important ways in the above exercises. First, in these
exercises, within-plant heterogeneity takes on a very specific parametric form that rules out
other forms of heterogeneity (see equation 36). Second, we force TFPQ to rise and fallacross products within a firm using the coarse indicator of product performance. However,revenue is not a sufficient statistic for within-firm productivity dispersion as documented byOrr (2022).54 Knowledge of the form of competition and within-firm markup dispersion isnecessary to extract the required information from quantity and revenue data. The fact that
this approach provides only a partial solution to the zero markup puzzle suggests that thissimple model of productivity ladders is not sufficient to capture the true variation in theunderlying data. However, this does suggest that allowing for more flexible patterns of MRT
heterogeneity may be key to pinning down a reasonable estimate of product-level markups.55
54

See also Appendix M for similar result where quantity rankings are used in place of revenue rankings.
In Appendix O we consider two further sources of potential misspecification; assuming non-joint production when production may be joint, and misspecification of the production technology. We find that thesecannot help resolve the zeros puzzle.
55

36

Figure 7: Baseline Plant-level Markups

Notes: The above histogram displays plant-level level markups according to equation (30) based on ourbaseline production function estimation routine described in the main text (technique 1). We trim markups≥ 5 for scale.

7.2

Plant Markups

Figure 7 displays a histogram of plant markups using technique 1: a control function
estimator with a selection correction. Appendix N presents the estimated elasticities. Plant
markups are well behaved, with a mean of around 1.5 without trimming outliers. Zero andright-tail outliers are now quite rate rare; the ratio of zero to reasonable markup estimatesis 0.03, while the tail outlier ratio below 0.01.
Figures 8 and 9 compare outlier ratios and the distribution of plant markups using
Technique 1 (the control function estimator for single-product firms) with two variants ofTechnique 2 (the cost-share approach): i) a simple cost-share approach where the production
function parameters differ by two-digit NIC code; ii) the cost-share approach advocated in
Raval (2023) which allows cost-share parameters to vary by labour-to-materials cost quintiles
within each two-digit NIC code. Figure 8 reports outlier ratios for each method and Figure
9 reports average markups without any trimming with 90-10 percentiles. We also report theoutlier ratios for our baseline plant-product markups in the first column of Figure 8 for scale.
Figures 8 and 9 show that results using Technique 1 and the first version of Technique 2
are similar. Outlier ratios appear to be almost identical (Figure 8) and the average markup
only differs by 0.02 (Figure 9). It is also reassuring to see that the 90-10 ratios are remarkably
similar across these two approaches, differing by less than 0.04 (Figure 9). For researchers
unsure as to which approach to use in their own work— a cost share or an estimator based37

Figure 8: Markup Outlier Ratios Across Methods

Notes: The above bar chart plots two outlier ratios for different models of product and plant-level markups.
Letting Z denote the set of product-level markups where µ ≤ 0.5, T denote the set of product-level markups
where µ ≥ 5, R denote product-level markups between 0.5 and 5, and |.| denote the number of elements in a|T||Z|
and the Tail outlier ratio is defined as |R|. Plant-Productgiven set, the Zeros outlier ratio is defined as |R|refers to the product-level markups from panel (A) of Figure 2. Baseline refers to the plant-level markups
in Figure 7. Cost Share (NIC2) obtains plant-level markups after estimating the materials output elasticity
using industry level cost shares for each two-digit NIC industry code. Cost Share (NIC2*L/M Quintiles)
obtains plant-level markups using a seperate cost-share estimator for each labour-to-materials quintile ×
two-digit NIC industry.

on single-product firms — it is useful to know that both approaches find consistent evidence
for some degree of across-plant misallocation due to market power.56
The more flexible cost-share approach recommended by Raval (2023) to deal with factor
augmenting productivity differences implies slightly lower markups on average. In particular,
once we allow the Cobb-Douglas parameters to vary by labour-to-materials quintiles withineach industry, the average markup falls from 1.5 to 1.34. Similarly, dispersion in markups also
falls using this approach, with the 90th to 10th percentile difference falling to around 0.80,
which is about 25% smaller than the baseline dispersion. These differences are important
56

However, it is worth emphasizing that our baseline approach using single-product firms estimates asingle set of production-function parameters for the entire Indian economy, while the cost-share approach
estimates a separate Cobb-Douglas technology by two-digit NIC code. Note, however, that since the translog
specification allows output elasticities to vary across all plants (since output elasticities are functions of inputuse), while this cost-share approach forces all output elasticities to be identical for all firms in the same 2digit NIC code, it’s not clear which approach should be regarded as “more” flexible. In Appendix P, we also
report results for single-product firm translog production functions for a subset of India’s largest industries,
but find that occasionally our approach leads to some unreasonable (e.g. negative) output elasticities formany plants (See Appendix N), and as a result there are more zero and outlier issues, which we believe are
driven by insufficient sample size concerns.

38

Figure 9: Plant-level Markup Dispersion Across Methods

Notes: The above plots the mean plant-level markup, as well as the 90th to 10th percentile of markups, forfour different ways of measuring plant-level markups. Baseline refers to the plant-level markups in Figure 7(technique 1). Cost Share (NIC2) obtains plant-level markups after estimating the materials output elasticity
using industry level cost shares for each two-digit NIC industry code (technique 2i). Cost Share (NIC2*L/M
Quintiles) obtains plant-level markups using a separate cost-share estimator for each labour-to-materials
quintile × two-digit NIC industry (technique 2ii).

and potentially point to the importance of allowing for factor-specific productivity differences
as noted by Raval (2022, 2023). On the other hand, note that all methods clearly imply
sizeable differences in plant markups across plants, which indicates that output market power
is responsible for some degree of misallocation in the Indian economy.

8

Conclusion

In this paper, we considered two important problems. First, we asked how to identify
plant-product markups using standard production data. We found that disciplining themagnitude of MRT heterogeneity is of first-order importance, while determining whether
production was joint or non-joint was less empirically relevant. We see further work on new
and novels ways deal with MRT heterogeneity as a very valuable path for future research.
In particular, in this paper we only considered some simple, stylized approaches to deal withthis problem, but expect that further progress can be made beyond our modest first-pass at
this problem.
Another key problem we tackled in this paper was how to interpret the firm markups
uncovered by the De Loecker and Warzynski (2012) approach. We showed that the firm39markup can be interpreted as a cost-weighted average of firm-product markups under fairly
general conditions, and that this object is a welfare relevant statistic that can help uncover
the magnitude of across-firm misallocation due to market power. We also showed that a
number of popular estimators, including the cost-share approach, as well as a control function
approach developed in De Loecker et al. (2016), can be applied to uncover this object, evenin joint production settings. Empirically, we found that both cost-share and control function
approaches generated similar plant markup estimates in the Indian manufacturing sector,
although we also found that adjusting for factor-specific productivity differences can lead tolower markups levels as well as dispersion. We hope these identification results shall prove
useful for uncovering market power in other settings.

40

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44

Appendix
A

Proof of Theorem 1
The Proof of Theorem 1 relies on the following intermediate results:

Lemma 1 Suppose a firm operates a non-joint technology with product-line production functions Yij = Aji Fi (Sji ◦ Xi ), where Fi (Xi ) continuous, differentiable, strictly increasing in
all arguments, quasi-concave, and homogeneous of degree φi > 0. Then the firm’s output
distance function is given by:P  Yij  φ1
Ajij

DiN J (Yi , Xi ; Fi (.), {Aji }j ) =

φ
(37)

Fi (Xi )Proof. Given the proposed product-line production functions, the firm’s output distancefunction will satisfy:
DiN J (Yi , Xi ; Fi (.), {Aji }j ) ≡ minδ
jδ,{Si }j
Yj
= Aji Fi Sji ◦ Xi ∀j
δ
X
j
Sni
= 1, ∀n

s.t.:

(38)

j

The Lagrangian for the optimization problem in (38) is given by:
Li = δ +

X

λji



j

j

Y
− Aji Fi Sji ◦ Xi
δ




!
+

X

µni

n

X

j
Sni
−1

(39)

j


Since Fi (.) is quasi-concave, any solution Γi ≡ {Sji }j , δ, {λji }j , {µni }n to (38) will satisfy
the following first order conditions:F OCS j (Γi ) ≡ −λji Aji
ni


∂Fi j
Si ◦ Xi Xi + µni = 0 ∀(j, n)
∂Xi

(40)

j
j Yi
λi 2 = 0(41)

F OCδ (Γi ) ≡ 1 −

X
j

δ


Yj
F OC (Γi ) ≡− Aji Fi Sji ◦ Xi = 0 ∀jδX j
F OCµni (Γi ) ≡Sni − 1 = 0 ∀n
λji

j

45

(42)
(43)


j∗
∗
∗
It is easily verified that the following Γ∗i ≡ {Sj∗i }j , δ , {λi }j , {µni }n satisfies equations(40) through (43), and therefore solves (38): j  φ1
j∗
Sni
=

Yi
Aji

(44)

P  Yik  φ1
Aki

k

φ

P  Yij  φ1
Aji

j

δ∗ =



λj∗
i =

X

Yij

j

Aji F (Xi )! φ1 2φ j  φ1

P  k  φ1

Yi
Aji

k


µ∗ni =(45)

Fi (Xi )∂F (Xi ) Xi 
∂Xi F (Xi )
j

X

Yi
Aki

(46)

! φ1 φ

Yij


P


(47)



Aji F (Xi )It immediately follows that DiN J (Yi , Xi ; Fi (.), {Aji }j ) =

1
Yij

j

j
Y
i
j
A
i

! 1 φFi (Xi )φby equation (45).

Lemma 2 Suppose Assumptions 1 and 2 hold. If there exists Feij (.) such that for all (Yi , Xi ) ∈PFi there exist input shares {Sji }j such that Yij = Feij (Sji ◦ Xi ), the only Feij (.) satisfying thiscondition are Feij (.) = Aji Fi (.), where Fi (.) is defined in Assumptions 1 and 2; otherwise, nosuch Feij (.) exist.
Proof. We establish this by contradiction. In particular, suppose not, so that there exists
Feij (.) such that for all (Yi , Xi ) ∈ PFi there exist input shares {Sji }j such that Yij = Fij (Sji ◦Xi ),b i such that F j (Xb i ) 6= Aj Fi (Xb i ). By Assumption 3, for any
and there existsat least one Xiib j , Xi ∈ PF . Denote this value of Y j , given Xi , as Yb j (Xi ). Assumption
Xi we can find Y
i
i
i
i
j
j
b
2 implies that Yi (Xi ) = Ai Fi (Xi ). Since Fi (Xi ) is strictly increasing in all it’s arguments,
bj
this implies
 that Yi (Xi) is also strictly increasing in all it’s arguments.
Let Sj Yb j (Xi ) , Xi denote the optimal input shares for the proposed production func-

i
i
j
bj
tion Fi (.). The fact
 that Yi (Xi ) is strictlyincreasing inall it’s arguments implies that for
any Ybij (Xi ) , Xi ∈ PFi , we must have Sji Ybij (Xi ) , Xi = 1. To prove this, suppose not,
so that Yij (Xi ) = Fij (Sji Ybij (Xi ) , Xi ◦ Xi ), where Sji Ybij (Xi ) , Xi is strictly less than 1
e i = Sj Yb j (Xi ) , Xi ◦ Xi , where X
e i is strictly less thanfor at least one input. Denote Xii
46

Xi in at least one dimension. SinceYbij (X i ) is strictly increasing in all it’s arguments, thise i) > Y j X
e i ,which further implies
means that Yij (Xi ) = Fij (X
i
 
ei
Yij X
e i)
Fij (X

<1

  

ei ,X
e i ∈ PF . Since the production possibility frontier provides
Next, consider Ybij Ximaximal quantities of output, given inputs, it must be that:e i) ≤ F j
Fij (Sji ◦ X
i   
j bj eee
Si Yi Xi , Xi ◦ Xi

for any feasible Sji . Note that this implies, when combined with the above inequality:
 
 
ei
ei
Yij X
Yij X
   

≤ j
<1
e i)
ei ,X
ei ◦ Xei
Fi (XFij Sji Ybij X
  

j e
ei ,X
e i ∈ PF we must have j j Yji (Xi )Which is a contradiction since if Ybij Xie i ) =1.
e i )◦Xe i ),X
Fi (Si (Ybi (XTherefore, Sji Ybij (Xi ) , Xi = 1 for all Ybij (Xi ) , Xi ∈ PFi .
  b i ) 6= Aj Fi (Xb i ), and consider Yb j X
bi ,X
b i ∈ PF .
b i such that F j (XNext, choose any X
i
i
i
i
j
j
j b
j
j
b
b
The previous argument has established Yi = Fi (Si ◦ Xi ) = Fi (Xi ), i.e. Si = 1 , whileb i ), which together imply F j (Xb i ) = Aj F (Xb i ), generating
Assumption 2 implies Yb j = Aj F (X
i

i

i

i

a contradiction. Therefore, the only Feij (.) that may support non-joint production are Feij (.) =Aji Fi (.).

We now turn to the statement and proof of Theorem 1, which will make use of these twolemmas.
Theorem (1). Suppose Assumptions 2 and 3 hold. Then a firm’s technology is non-joint if
and only if:φ
P  Yij  φ1i ij

Di (Yi , Xi ) =Aji

Fi (Xi )(48)

Proof. The proof proceeds in two steps. First, in Step 1 we show that if the firm’s input
distance function satisfies (48), then the technology is non-joint. Then in Step 2 we showthat if the technology is non-joint, it must have an output distance function given by (48).

47

Step 1: Suppose a firm’s input distance function satisfies (48). To prove that thisproduction possibilities set involves non-joint production, we must show that there exist Feij (.)
such that for all (Yi , Xi ) ∈ PFi , there exist input shares {Sji }j such that Yij = Feij (Sji ◦ Xi ).
Take Feij (.) = Aji Fi (.). Lemma 1 tells us that the output distance function for this
proposed class of non-joint technologies (Fi (.), {Aji }j ) has an output distance function equal



P

j

to DiN J (Yi , Xi ; Fi (.), {Aji }j ) =

j
Y
i
j
A
i

! 1  φi
φi
.
Since the firm’s output distance function satisfies (48), (Yi , Xi ) ∈ PFi is the same as theFi (Xi )P

set of all (Yi , Xi ) such that



P

j

j
Y
i
j
A
i

j

j
Y
i
j
A
i

! 1  φiφiFi (Xi )= 1. However, since DiN J (Yi , Xi ; Fi (.), {Aji }j ) =! 1  φiφiFi (Xi )was generated under the assumption that Yij = Aji Fi (Sji ◦ Xi ), then if we



P

j

j
Y
i
j
A
i

! 1  φi
φi
consider the set of (Yi , Xi ) such that DiN J (Yi , Xi ; Fi (.), {Aji }j ) =Fi (Xi )immediately implies that for all (Yi , Xi ) ∈ PFi , there exist input shares {Sji }jYij = Feij (Sji ◦ Xi )= 1, thissuch that

Step 2: Suppose the technology is non-joint. By Lemma 2, Yij = Aji Fi (.) is the only setof non-joint technologies consistent with Assumptions 1 and 2. Lemma 1 then immediatelyimplies Di (Yi , Xi ) =

B

P

j

j
Y
i
j
A
i

! 1 φφ

Fi (Xi ).

A model of input sharing in production

In this Appendix, we show that the output distance function (13) can be obtained froma model of production where inputs are shared across product lines, similar to the CES costfunction described in Chapter 15 of Baumol et al. (1982).
Suppose that each firm assembles final output using two classes of intermediate inputs;
private intermediates, Iirj , and a public intermediate, Iip . Private intermediates are specialized inputs, such as product-line specific equipment or machinery, that can only be used in
the production of product line j. Public intermediates, on the other hand, are non-rival andcan be used in the production of all product lines at once— these might be thought of as
managerial inputs, or buildings and structures that are useful for producing many things at
once.
Each of these intermediate inputs are built entirely in-house, using the firm’s vector of48

inputs Xi . We let Xrj
i denote the vector of inputs allocated to producing the specialized
private intermediate Iirj , and Xpi denote a vector of inputs allocated to producing the public,
non-rival intermediate, Iip . Because Xpi contains public inputs across product lines, it does
not have an index j. The production of the public and private intermediates within the firm
are given by:
Iip = G(Xpi );Iirj = G(Xjri )Note that the aggregation of inputs to make the public and private intermediates are determined by the same function G(·). We then assume that the production function for productj at a given firm i is
Yij =Aji rj βφ p (1−β)φ Aji
p (1−β)φβφ(I ) (Ii )
=
G(Xrj
.
i ) G(Xi )
κ i
κ

(49)

where β ∈ [0, 1] governs the intensity of private intermediate use relative to the public
intermediate use, βφ and (1 − β)φ are the private and public returns to scale respectively, Aji
is a Hicks-neutral productivity shifter specific to product line j, and κ = (β)βφ (1 − β)(1−β)φis a constant. Note that even though there exist product-specific production functions inside
the firm, the above defines a joint production technology because of the presence of public
inputs. Therefore, product line production functions within the firm vary only due to the
productivity shifters Aji .57 We now make two assumptions that will facilitate the derivationof the output distance function Di (Yi , Xi ) for this firm.
Assumption 7 G(·) : RN → R continuous and twice differentiable, equal to zero if anyof its arguments are equal to zero, strictly increasing in all arguments, quasi-concave, andhomogeneous of degree 1.
Assumption 8 Both static and dynamic inputs can be costlessly transferred between public
and private use.
The firm’s output distance function specified in definition 2 can be characterised as57

We can also derive a closed form expression for a firm’s output distance function that has varying total
returns to scale by product line, i.e. φj . We show the derivation for the simple case here.

49

follows:
Di (Yi , Xi )≡minδ,Xpi ,{Xrj
i }j

δ

Yij
Aji
p (1−β)φ
βφ
s.t.
≤
G(Xrj
i ) G(Xi )
δ
κ
X
p
Xsi +
Xsirj ≤ Xsi ∀ s

∀j

(50)

j

The above optimization problem has the following Lagrangian:
L=δ+

X
j

λji

Yij
Aj
p (1−β)φβφ− i G(Xrji ) G(Xi )
δ
κ

!

!
+

X
s

µsi

Xsip +

X

Xsirj − Xsi

(51)

j

Since the production functions are increasing in all inputs, all constraints will bind withequality. The first order necessary conditions where all constraints bind are given by:
FOCδ (δ, Xri , Xpi , λi , µi )≡1−

X λj Y ji

δ2j

FOCX rj (δ, Xri , Xpi , λi , µi )si≡

FOCXsip (δ, Xri , Xpi , λi , µi )≡FOCλj (δ, Xri , Xpi , λi , µi )≡i

FOCµsi (δ, Xri , Xpi , λi , µi )

i

=0

j j
G(s) (Xrji ) λi Yi− µsi = 0 ∀ {s, j}
βφ
G(Xsr
δ
i )
G(s) (Xpi ) X λji Yij(1 − β)φ− µs = 0 ∀ {s}G(Xpi ) jδAjYijp (1−β)φβφ=0− i G(Xrji ) G(Xi )δκ& λji > 0 ∀ {j}X rj≡ Xsip +Xsi − Xsi = 0

(52)
(53)
(54)

(55)

j

& µsi > 0 ∀ {s}(56)

where G(s) (·) = ∂G(·)/∂Xsi , and λi and µi denote the vectors of Lagrangian multipliers.
Stacking equations (52) - (56) allows us to represent the solution to the cost minimization problem as a vector (δ, Xri , Xpi , λi , µi ) that satisfies FOC (δ, Xri , Xpi , λi , µi ) = 0 with
(λi , µi ) >> 0.

50

Consider the following candidate solution for the input allocation rule:1 j  βφXsirj∗ = β

Yi
Aji

1
P  Yik  βφXsi

∀ {s, j}

(57)

∀ {s}

(58)

Aki

k

Xsip∗ = (1 − β)Xsi

Substituting the above into the product line production function that is binding (equation55), and using the homogeneity of G(·):
   1 βφ
Yij βφjjAi βφ YiAji
 G(Xi )βφ (1 − β)(1−β) G(Xi )(1−β)φ=β
1  P  Y k  βφ
δ∗
κ
i
Aki

k

1
 j  βφ(δ ∗ )

1
βφ

Yi
Aji

1
P  Yik  βφ

Yij
Aji G(Xi )φ

=

Aki

k

1
 j  βφ(δ ∗ )

1
βφ

Yi
Aji

X

1
P  Yik  βφ

j

X

Yij

j

Aji F (Xi )YijAji G(Xi )φ

X

=

j

Aki

k


=⇒ δ ∗ = = (δ ∗ )

1
βφ

1
! βφ

1
! βφ

βφ
1
! βφ
where F (X) = G(X)φ



(59)

Further, also consider the following solutions for the Lagrangian multipliers:
λji = XYijjAji F (Xi )1 j  βφ

2βφ
1
! βφ


Yi
Aji

1
P  k  βφkXµsi = φ j

YijAji F (Xi )

βφ
1
! βφ


Yi
Aki

G(s) (Xi )G(Xi )

1
Yij

∀ {s}

∀ {j}

(60)

(61)

p∗
∗
∗
It is easily verified that the candidate solution (δ ∗ , Xr∗
i , Xi , λi , µi ) defined in equations (57)
- (61) satisfies the first order conditions stated in equations (52) - (56). Therefore, the output

51

distance function for a firm that shares public intermediates across product lines is given by:
Di (Yi , Xi ) = 

X

Yij

j

Aji F (Xi )

βφ
1
! βφ


(62)

where F (X) is homogeneous of degree φ. An important takeaway from the above expressionis that β governs whether there is joint-production or not.

C

Monte Carlo Details

This appendix discusses in more detail the Monte Carlo exercises presented in section
3.2. There are J product-specific CES aggregates C j each comprised of firm-specific varieties
cji as described in the text. Because the ASI data lists a maximum of ten products a firm
can list, we set J=10. We also assume that each firm produces a random subset of 5 of these10 products. Production of other products is exogenously set to zero. For the Monte Carlo
exercises, we assume that the output distance function takes the form of equation (13), withF (Xi ) = Li and therefore φ = 1. Labour, Li , is paid an exogenous wage w=1 in both uses.
β is the cost share of rival (non-public) inputs in production at the firm as in Appendix
B. The constant elasticity of substitution demand structure combined with goods market
clearing implies thatj(νij )σ−1 (Pij )−σ j
j
γ E
(63)
ci = P j j
1−σi (Pi /νi )where E is aggregate expenditure, γ j is a Cobb-Douglas expenditure share 1/J, and νij areidiosyncratic demand shifters whose properties are discussed below. The true CES markup
is given byσj
Pij = jM Cij .
(64)
σ −1
where
∂Di (Yi ,Li )∂Y jj
M Ci = − ∂Di (Yii ,Li ) w.
∂Li

(65)

Using equations (13), (64), and (65), and substantially manipulating, we obtain:" #α−1
Yij
w
σ
Pij =
.
σ − 1 Li
(Aji )α

52

(66)

1
where α = φβ
. If returns to scale are constant and there is no joint production, thenφβ = 1, and this simplifies to the familiar CES constant markup of price over marginal cost:
σ w.
Pij = σ−1
AjiParameters
Lower case letters represent log transformations such that x = ln X. We assume thatfirm-product productivity can be parameterized by aji = ai + ãji . Firms draw ai from arandom normal distribution N (0, σF2 ) where we set σF2 = 5. ãji is drawn from a different
random normal distribution N (0, σF2 J ). The ratio σF2 J /σF2 is the key parameter that we vary
to assess the importance of within-firm productivity heterogeneity. We assume that quality
can be written as νij = exp dji where dji = di + d˜ji − aji . di and d˜ji are random variables drawn
22
22= 0.5 in all cases. The= 5 and σDJ). We set σD) and N (0, σDJfrom distributions N (0, σD
jfinal term (−ai ) captures the negative correlation of productivity and quality documented
in Orr (2022). We assume that the cost of one unit of labor is exogenous and equal to 1, andthat aggregate expenditure is also exogenous, E = 100. We assume that there are 10,000
firms such that N = 10, 000. We assume constant returns to scale in all specifications φ = 1.
Unless otherwise stated, we assume that β = 1 such that production is non-joint.

Fixed Point
We start by nominating a N × J matrix of prices Pij . Using equation (63) and goods
market clearing, we obtain N × J matrices of revenue and output where typical elements are
Rij and Yij , respectively. Using equation (13) with F (Xi ) = Li , we obtain a N × 1 vector of
total labor demands which we then use to generate a new vector of firm-product prices Pbij
using (66). We then update Pij until Pij and Pbij converge. When there is no joint production(β = 1), this convergence occurs immediately. This delivers values of Yij , Rij , and Li that
we combine with exogenous parameters Aji and β to calculate true and measured markups
using equations (16)-(19) as we discuss below.

Within-Firm Productivity Heterogeneity (Figure 1, panel a)
All parameters are as described above except σF J which we initially set to zero imposing
no productivity heterogeneity across product lines within a firm. We then calculate thecorrelation between markups implied by equations (16) and (17). This correlation is 1 whichis the first triangle seen in the upper left of panel a. We then increase σF J by 0.1, take therelevant draws solve the model again, and calculate the correlation between markups implied

53

Figure 10: Monte Carlo: Log Variance of Estimates(A) Within-Firm Productivity Heterogeneity(B) Joint Production

Notes: Each triangle in panel a presents the log of the variance of estimated markups. The horizontal axis
shows the ratio of within- to across-firm productivity heterogeneity for that correlation. There is no joint
production in this panel. Each triangle is a separate variance term. Each triangle in panel b presents thelog of the variance of estimated markups. The horizontal axis shows the cost share of public inputs 1 − β.
Each triangle is a separate log of variance term. There is no within-firm productivity heterogeneity in thispanel. The log of the variance of markups when there is no within-firm productivity heterogeneity and no
joint production is 6.78=ln(885.4).

by equations (16) and (17), plotting it as the second triangle in this figure. Seeds are set
such that the distribution of across firm draws is the same for each point in the figure. We
continue increasing σF J , and calculating/plotting the relevant correlation until σF J /σF = 1.

Joint Production (Figure 1, panel b)
All parameters are as described above except β which we initially set to 1 imposing nojoint production. We then calculate the correlation between markups implied by equations(18) and (19). This correlation is 1 which is the first triangle seen in the upper left of panel b.
We then decrease β by 0.1, take the relevant draws, solve the model again, and calculate thecorrelation between markups implied by equations (18) and (19). Again, seeds are set suchthat draws of the relevant exogenous variables are held constant. We continue decreasing β,
and calculating/plotting the relevant correlation until β = 0.2. At this point, we begin toencounter scaling issues in the shares.

Variance of Estimated Markups
Figure 10 presents the log of the variance of estimated markups analogous to figure 1.
The horizontal axis of each panel is exactly the same in each figure. We present the y-axis
54

in logs because of the scale of panel (a). Panel (a) shows that the variance starts to increase
dramatically when within-firm productivity heterogeneity is roughly 20 percent of acrossfirm heterogeneity. Panel (b) again shows that the omission of joint production produces
less dramatic results although the log of the variance of estimated results increases slightly
to 7.41 inputs for joint production are rougly 80 percent of total costs.

D

Constrained Social Planner and Decentralized Economy

We will show that the object that the constrained social planner outlined in section
4.3 cares about is the aggregate firm-level markup µi . We first re-state the social planners
problem here and derive the first order conditions. Then we characterize the decentralised
economy and show that firm-level aggregate markups are the object that the social planner
wants to equalize across firms.
Constrained Social Planner: A social planner that wants to eliminate across-firm misallocation while holding within-firm output allocations constant faces the following constrained
optimization problem:
max U ({Yi λi }){Yi }is.t. Di (Yi λi , Xi ) = 1
X
Xsi ≤ Xs∀i∀siwhere λi is the vector of relative output levels within the firm, and Yi is scale of the firm.
Suppose δi and γs are the lagrangian multipliers associated with firm i’s technology constraint
and input s’s resource constraint respectively, then the first order conditions for the above
problem are given by:
FOC wrt Yi :X ∂U ({Yi })k

FOC wrt Xsi :

δi

∂Yik

λki = δi

X ∂Di (Yi , Xi )j∂Di (Yi , Xi )
= −γs
∂Xsi

55

∂Yik

λki

∀i

(67)

∀ {s, i}(68)

Substituting equation 68 into 67, and substituting for λki = Yik /Yir for some reference productr, we can rewrite the above conditions to be:
X ∂U ({Yi })X ∂Di (Yi , Xi )
γs
k
=
−
Y
Yik
i
k
k
∂Di (Yi ,Xi )
∂Y
∂Y
i
i
k∈Yi
k∈Yi
∂Xsi
P
∂U
({Y
∂Di (Yi ,Xi )
i })
Xsi
Yik
k∈Yi
∂Yik
∂Xsi
Di (Yi ,Xi )
−P
= γs
Yik
∂Di (Yi ,Xi )
X
k∈Yi

=⇒

∂Yik

si

Di (Yi ,Xi )∂ ln Di (Yi ,Xi )∂ ln Xsi−P∂ ln Di (Yi ,Xi )k∈Yi∂ ln YikPk∈Yi∂U ({Yi }) k
Yi
∂Yik

Xsi

= γs

(69)

Therefore the social planner will allocate resources across firms such that the object on theleft hand side of equation (69) is equalized across all firms.
Decentralized Economy: Consumers maximize their utility given prices, and firms will
minimize costs. We do not require disciplining how firms set prices for deriving markups,
but rather use prices we observe in the data as the set of prices that maximize firms’ profits.
The consumer problem is given by:max
U ({Yi })
j

XX

s.t.

{Yi }i,j=⇒i

Pij Yij ≤ I

j∂U ({Yi })= θu Pij
j
∂Yi

(70)

where θu is the Lagrangian multiplier associated with the consumer budget constraint. From
the firm’s cost minimization problem specified in section 3 and the firm-level aggregate
markup defined in section 4, we have:
P k
P k k
∂ ln Di (Yi ,Xi )∂ ln Di (Yi ,Xi )∂ ln Xis∂ ln Xis
k Ri
k Pi Y iµi = − P= −P∂ ln Di (Yi ,Xi )
∂
ln
D
(Y
,X
)
i
i
i
Esi
wsi Xsik∈Yik∈Yi∂ ln Yik∂ ln Yik
Substituting for price Pik from the consumer FOC, we have:∂ ln Di (Yi ,Xi )∂ ln Xisµi = − P∂ ln Di (Yi ,Xi )k∈Yi∂ ln Yik=⇒∂ ln Di (Yi ,Xi )∂ ln Xis−P∂ ln Di (Yi ,Xi )k∈Yi∂ ln Yik

P

k∈Yi

Xsi

P

56

∂U ({Yi }) k
Yi
∂Yij

k∈Yi

∂U ({Yi }) k
Yi
∂Yij

Xsi

1
θu wis= θu wis µi(71)The left hand side of equation 71 is exactly equal to the object that the constrained social
planner wants to equalize across all firms. Therefore, under the scenario where all firms face
the same input costs, wsi = ws , the social planner wants to equalize firm-level markups µi
across all firms in order to eliminate across firm misallocation while holding the within-firm(mis)allocation fixed.

E

Proof of Proposition 1

Proof. By Assumption 4, we have
Di (Yi , Xi ) =Gi (Yi ),Fg(i) (Xi )(72)

where Gi (Yi ) is homogenous of degree 1. This implies
X ∂ ln Di (Yi , Xi )k∈Yi∂ ln Yik=X ∂ ln Gi (Yi )k∈Yi∂ ln Yik=

1 X ∂Gi k
Y =1
Gi k∈Y ∂Yik ii

Substituting this into (21) yields:
P j∂ ln Di (Yi , Xi ) j Ri
µi =
∂Mis
Esi

(73)

P j
P j
∂ ln Fg(i) (Xi ) j Ri
j Ri
µi =
= θsg(i) (Xi ).
∂ ln Msi
Esi
Esi

(74)

While (72) implies:

The last part of the Theorem then follows from Assumptions 3 and 4, where if Yi = Yij ,Yjthen Di Yij , Xi = Aj F i (X ) . Since cost minimizing firms will operate on their produci
i g(i)
tion possibilities frontier, and therefore Di Yij , Xi = 1, this immediately implies Yij =∂ ln F(Xi )Aji Fg(i) (Xi ), and therefore θsg(i) (Xi ) ≡ ∂ lng(i)can be interpreted as the output elasticMsi
ity for static input s for firms that choose to produce a single product.

F

Firm-level Cost Shares

In this Appendix, we consider an alternative set of restrictions that will allow researchers
to identify the firm-markup for any output distance function satisfying constant returns toscale. In a multi-output context, constant returns to scale is defined as follows:57

Assumption 9 Each firm’s production possibility frontier involves constant returns toscale, i.e. for any λ > 0 and (Yi , Xi ) ∈ Pfi
Di (λYi , λXi ) = 1(75)

Equation (75) essentially states that if we scale inputs by λ from any previously feasible
production vector (Yi , Xi ), then we can also produce an output vector that is scaled by λ.
To identify the firm-markup for all output distance functions satisfying Assumption 9,
we require the following further restriction on the firm’s operating environment.
Assumption 10 All inputs are static, i.e. Xi = Mi
Assumption 10 is a very strong restriction, requiring that all inputs, including capital,
be available for purchase for some (potentially firm varying) input price WiX . A key benefit of this restriction is that with it, we can prove that the firm markup is identified forany technology that satisfies constant returns to scale; importantly, this includes technologies with non-separable output distance functions. This is important, since non-separable
specifications of a firm’s technology are needed to capture situations where firms operate
non-joint production technologies that have different factor intensities (Hall 1973)— for example, situations where a high quality good may be more capital intensive than a low quality
good.
Proposition 3 If firms minimize total costs and Assumptions 9 and 10 hold, then :∂ ln Di (Yi ,Xi )∂ ln Xsi∂ ln Di (Yi ,Xi )k∈Yi∂ ln Yik−P

Wsi Xsi=P
s Wsi Xsi

(76)

Proof. Since firms cost minimize, Di (Yi , Xi ) = 1. Totally differentiating this expression
yields:X ∂Di (Yi , Xi )
j∈Yi

∂Yij

dYij +X ∂Di (Yi , Xi )∂XsisdXsi = 0or:
X ∂Di (Yi , Xi )s

∂Xsi

dXsi = −X ∂Di (Yi , Xi )
j∈Yi

∂Yij

dYij

si
Divide by Di and multiply the left hand side by X
Xsi

X ∂Di (Yi , Xi ) Xsi dXsis∂Xsi

Di Xsi=−X ∂Di (Yi , Xi ) Y j dY j
i

j∈Yi

58

∂Yij

i

Di YijX ∂ ln Di (Yi , Xi ) dXsi
∂ ln Xsi

s

Xsi

=−

X ∂ ln Di (Yi , Xi ) dY j
∂ ln Yij

j∈Yi

i
j
Yi

dY j

Assumption 9 implies that if dX= 1 + δ ∀X, then Y ji = 1 + δ ∀j, which implies:Xi

X ∂ ln Di (Yi , Xi )∂ ln Xsis=−

X ∂ ln Di (Yi , Xi )j∈Yi∂ ln Yij(77)

Next, since firms cost minimize and all inputs are static by Assumption 10, the static
cost minimization condition (8) holds for all inputs s, which implies:
Wis = −λi

Wsi Xsi∂Di Xsi
∂Di
→
= −λi
∂Xsi
Di
∂Xsi Di

Since Di = 1
Wsi Xsi = −λi∂ ln Di (Yi , Xi )∂ ln Xsi(78)

Sum over all sX

Wsi Xsi = −λisX ∂ ln Di (Yi , Xi )∂ ln Xsis(79)

Divide (78) by (79), yielding:∂ ln Di (Yi ,Xi )W X
si
P si si = P ∂∂lnlnDX(Yii ,Xi )
W
X
si si
s
s
∂ ln Xsi

Substituting using (77) to substitute out

P ∂ ln Di (Yi ,Xi )s∂ ln Xsiin this expression then yields:∂ ln Di (Yi ,Xi )W X∂ ln Xsi
P si si = − P∂ ln Di (Yi ,Xi )s Wsi Xsi
k
k∈Y
i

(80)

∂ ln Yi

Proposition 3 tells us that the series of output distance function elasticities that arenecessary to recover the firm markup in (21) can be point identified, firm-by-firm, by simply
examining the cost shares of one of the inputs. This is a powerful result, as the only restriction
on the shape of the output distance functions is that it be constant returns to scale, which
incorporates an extremely broad class of joint and non-joint technologies, including non-joint
technologies with different product-line production functions. However, it requires the strong
restriction that all inputs be static, which may not be appropriate in many settings. Since59the requirement that all inputs be static means that the static first-order conditions hold for
all inputs, Proposition 3 also implies to following corrolary:
Corollary 1 If firms minimize total costs and Assumptions 9 and 10 hold, then:
P j
j Riµi = Ps Wsi Xsi

(81)

Proof. Implied by substituting (76) into (21) and cancelling out WEsisiXsi = 1
This result is reminiscent of a well known result that the ratio of revenues to costs
measures the markup when there are constant returns to scale and all inputs are static (e.g.
De Loecker et al. 2020). However, a key distinction is that previous variants of this result
focused on settings where revenue and cost data were interpreted as if they applied to asingle product. Here, we allow firms to produce arbitrary numbers of products, and allow
firms to operate using an extremely wide class of joint and non-joint production technologies.
In this more general setting, the “markup” cannot always be interpreted as the price overmarginal cost of a single product (this is only true if a firm produces a single product), butPrather a weighted average of a series of production level markups, µi = j ρji µji .58

G

Proof of Proposition 2

Proof. Since Di (Yi , Xi ) = 1, then∂Di (Yi , Xi )Xsi
∂Di (Yi , Xi )Xsi == −βs,g(i) ,
∂Xsi
∂Xsi
Di (Yi , Xi )where the last equality follows from Assumption 6. Combining this with Assumption 5 thenimpliesE (Wsi Xsi |g = g(i)) = βs,g(i) E (λi |g = g(i)) .
Since this holds for all inputs:
!
E

X
s

Wsi Xsi |g = g(i)=Xβs,g(i) E (λi |g = g(i)) = E (λi |g = g(i)) .

s

58
De Loecker and Warzynski (2012) emphasize a similar result when addressing the issue of multiple
products shipped to different markets. Specifically, they show that the firm level markup derived in their
setting is an input share weighted average of product-level markups. However, their results apply only to
non-joint production settings with product line production function.

60

Takign the ratio of the previous two expression then yields:
E (Wsi Msi |g = g(i))
P= βs,g(i)E ( s0 Ws0 i Xs0 i |g = g(i))
Since s is assumed to be a static input, (21) holds for this input, and the result immediately
follows.

H

Data Appendix• Labour (Lit ) : Labour input: We use labour input in person-days, as recorded in
ASI Block E. We use the total labour input for all classes of employees, including
managers. Wage bill, wL: The wage bill is the total expenditure on wages for allclasses of employees. Dividing the wage bill by labour input gives the implied wage,w.
• Capital (Kit ) : Stock of Capital : We rely on the perpetual inventory method, use theheterogeneous depreciation rates of 0%, 5%, 10%, 20%, and 40% for land, buildings,
machinery, transportation equipment, and computers & software, respectively, whenconstructing the capital stock over time as in Boehm and Oberfield (2018). The starting
value for each component of a plant’s capital stock is given by its book net opening
value as recorded in block C of the ASI, and investment is taken as the value of grossadditions plus gross revaluations minus gross deductions. The perpetual inventory
method is only used to construct the capital stock for plants operating in multiple
sequential years. We use the book net opening value as an estimate of the capital stock
for all plants that are missing past values. Value of Capital, rK: For the cost shareapproach, we need to calculate the total user cost of capital for each firm. We largely
follow Raval (2022) for this purpose. In particular, we obtain the real interest rate byt −πt, where it is the private sector lending rate and πt is the inflation
year as Rt = i1+πtrate.59 The rental rate of capital is then defined as the sum of the real interest rate plus
depreciation for each type of capital (buildings, machinery, transportation equipment,
and computers software). Then value of the capital stock is then the rental rates times
the value of the capital stock, summed over each type of capital.
• Materials (Mit ) : Domestic and directly imported inputs are recorded in ASI Blocks H
and I, respectively. For domestic inputs, we keep the total value of intermediates used,

59

We obtain the private sector lending rate from IMF Financial Statistics, and the inflation rate for India
from World Development Indicators.

61

including electricity. For directly-imported intermediates we also keep the total, andimpute zero imports for firms that do not fill out the block. Total materials expenditureis the sum of expenditure on domestic and foreign intermediates.

I

Baseline Estimation Details

In this Appendix, we outline in detail the construction of our control function estimator
for the production function using single product firms. In particular, we describe how we deal
with input price bias, selection into single product firm status, and finally how we construct
out estimating equation based on innovations to productivity.

I.1

Input prices

Although we observe output quantities in the ASI, we do not observe input prices for all
inputs. As De Loecker and Goldberg (2014) explain, using deflated input expenditures as
a proxy for physical inputs can bias elasticity estimates when physical output is observed.
De Loecker et al. (2016) deal with unobserved input prices using a control function approach.
Modeling input prices as a function of output prices, market shares, and other state variables, they estimate the parameters of this function simultaneously with translog parameters.
Recovering input price control function parameters allows them to predict input “prices,”
which are used to deflate expenditures and construct output elasticities.
We exploit a unique feature of the ASI to correct for input price bias during the production
function estimation routine, without having to use a control function. The ASI includes dataon both labor expenditure and physical input in man-days, which allows us to recover labor
unit values (wages). Following De Loecker et al. (2016) in assuming common deviations fromthe national price index for each of capital, labor and intermediates, we use these unit values
to deflate plant expenditures for capital and intermediates, in addition to using labor input
in man-days.60 More formally, we assume that prices for input s can be decomposed into
time- and time-plant varying components, orWsit = Wst × Wit .

(82)

Recall that we deflated nominal expenditures using the Indian WPI, which is time-varying
only. By subsequently deflating using the observed wage, we capture components of input
60

De Loecker et al. (2016) require that input “prices” be the same across inputs within each product line
in order to identify input price control function parameters.

62prices that vary cross-sectionally. We refer to the plant-level wage as an input “price,” it isproperly interpreted as the plant’s idiosyncratic input price deviation from the WPI.
Since we only observe plant-level wages, not plant-product wages, this approach does notallow for plant-product input prices as in De Loecker et al. (2016). Since we only use singleproduct firms for estimation, this is without a loss of generality at the estimation stage.
However, this distinction matters once we turn to identifying markups in multi-product producers. In particular, firm-product input price control functions can incorporate informationon unobserved quality differences by product line. This is a key piece of identifying information that can, and should, be incorporated in the algorithm for identifying the unobserved
input allocations. As a result, when using their approach to uncover input allocations, wefollow De Loecker et al. (2016) and allow for firm-product input prices. In practice, we
do this by directly estimating a firm-product input price controls function using observed
plant-level wages for single-product producers. See Section K below for details.
On the other hand, when we consider our measure of firm-level markups, we do not allow
for firm-product specific input prices. The reason for this is that our approach allows for joint
production, i.e. situations where inputs are not directly allocated to particular product lines,
but rather enter the production of all goods simultaneously. In these cases, a firm-productinput price does not have a clear meaning. Therefore, to truly have our approach nest bothjoint and non-joint production, we do not allow for within-firm input price variation.

I.2

Selection correction

A concern with using single-product plants is that this may introduce selection bias dueto the non-random addition of product lines. As De Loecker et al. (2016) note, plants may
choose to add product lines if their realized productivity exceeds an idiosyncratic threshold.
Moreover, this threshold may be negatively correlated with past levels of dynamic inputs(for example, floor space in a factory) because firms with large stocks of these inputs at t − 1
would rather add a product at t, even if their productivity is low, rather than leave inputs
underutilized. This may induce a negative correlation between the firm’s time-t productivity
shock and levels of dynamic inputs at t, which are positively correlated with levels at t − 1.
This violates the exclusion restrictions described below.
We correct for potential selection bias of this form using a similar procedure to De Loecker
et al. (2016). An important component of this approach is that selection into multi-productfirm status at time t be based on the productivity of a single-product firm through a simple
cutoff rule based on their productivity at time t − 1. For this purpose, we assume that thedecision to add a product at time t is made at time t − 1. We then require that this decision63be based on a simple threshold productivity rule, where, letting Jit denote the number of
products by i at time t:61
Jit > 1 if ai,t−1 > ai,t−1 = at−1 (li,t−1 , ki,t−1 , ii,t−1 , Γi,t−1 ),otherwise Jit = 1(83)

where ii,t−1 is investment, ai,t is log TFPQ for the single product produced by firm i, andΓi,t−1 is a vector of firm-level state variables at time t − 1, excluding dynamic inputs andait .62
If this is the case, we can then write:
P r(Jit = 1|Ji,t−1 = 1) = P r (ai,t−1 ≤ ai,t−1 )= P r (ai,t−1 ≤ at−1 (li,t−1 , ki,t−1 , ii,t−1 , Γi,t−1 )= κt−1 (li,t−1 , ki,t−1 , ii,t−1 , Γi,t−1 )≡ SPit(84)

Importantly, this expression can be estimated using a standard binary outcome model (e.g.
a probit) on a dummy for whether a firm changes their status from single product to multiproduct in the subsequent period. This allows us to then obtain probabilities that a firmexits the sample, which can then be used to correct for selection. We provide the details of
this procedure I.3 below.
An important consideration is whether the more general setting of MRT heterogeneityand joint production considered in this paper is likely to generate the required threshold
selection rules as in (83). In particular, De Loecker et al. (2016) appeal to the model inMayer et al. (2014), which is explicitly non-joint. In Appendix Q, we consider a particularparametric example that extends the Mayer et al. (2014) environment to our setting withjoint production, and show that this model generates exactly this class of selection rule.
Note, however, that this particular parametric structure is not strictly necessary; rather,
any model that generates a selection rule as in (83) will be valid.63
61

De Loecker et al. (2016) consider a slightly different formulation of the this problem, based on firm’s
productivity at time t, rather than t − 1. Note however, that since the decision to add products is made attime t − 1, it is natural to state the selection rule in terms of realized productivity at time t − 1 (when thefirm makes their decision to add a product), rather than time t. Note, however, that both approaches to the
problem lead to basically identical selection corrections.
62
We set Γit ≡ (wit , pit , msit , Dit ) where wit is the firm wage, pit is a log output price, msit is the firm’s
quantity market share for product j, Dit is a vector of year and 5-digit ASICC product code dummies forthe product produced by the single product firm.
63
An important condition for this type of selection rule to be valid is that the degree of joint-production
not vary across firms; otherwise, the selection cutoff will depend on other unobservables that would need tobe estimated.

64

I.3

Proxy-variable estimator

We rely on a proxy-variables approach as in De Loecker et al. (2016), where we assume
that the physical materials demand function can be written as
mit = mg(i) (ait , kit , lit , Γit ) ,(85)

Under the assumption that this function is monotonic in ait for single product plants, we
can invert this expression, yielding ajit as a function of observables:
ait = hg(i) (mit , kit , lit , Γit ) = hg(i) (xit , Γit ) .

(86)

Estimation relies of a two step procedure as in De Loecker et al. (2016), In the first step
we estimate the following model for all single product firms:

yit = ln Fg(i) (Xit ; β) + ait + it = ln Fg(i) (Xit ; β) + hg(i) (Xit , Γit ) + it = Φg(i) [Xit , Γit ] + it(87)
where Fg(i) (Xit ; β) has the translog form in (29), it is an error term capturing unanticipatedshocks when inputs are chosen at time t, or measurement error in yit , and Φg(i) [Xit , Γit ] ≡ln Fg(i) (Xit ) + hg(i) (Xit , Γit )We use (87) to estimate Φg(i) [Xit , Γit ], which allows us to then construct an estimate ofb g(i) [Xit , Γit ] − ln Fg(i) (Xit ; β) .64
ai,t for single product firms as b
ai,t = ΦThe production function parameters are then estimated in the second step. We assume
that productivity follows a Markov process, so E (ai,t |Ii,t−1 ) = E (ai,t |ai,t−1 ), where Ii,t−1 isfirm i’s information set at time t. Since in this step we need to condition of firms thatwere single product in both time periods, and we assume that selection in multi-productfirm status, conditional on being single product at time t − 1, follows selection rule (83) andprobability (84), we work with the following moment:
ai,t = E (ai,t |ai,t−1 , ai,t−1 ≤ ai,t−1 ) + ξit
ai,t = E ai,t |ai,t−1 , ai,t−1 ≤ fg(i) (SPit ) + ξitai,t = mg(i) (ai,t−1 , SPit ; γ) + ξit(88)

where γ denotes a vector of parameters governing the shape of the unknown function mg(i) (, ),and ξit is a productivity shocks unknown to firms at time t − 1.65 Note that the second
64

In practice, we approximate the unknown function with a second order polynomial in all continuous
variables, plus the various fixed effects.
65
We approximate g(.) with a second order polynomial in all it’s arguments.

65equality uses the fact that according to (84), SPit = P r (ai,t−1 ≤ ai,t−1 ), and the fact thatP r (ai,t−1 ≤ ai,t−1 ) will be a monotonic function of ai,t−1 , and therefore we can invert thisexpression to obtain ai,t−1 = fg(i) (SPit ).
We simultaneously estimate γ, and the production function parameters β, by substitutingb g(i) [Xit , Γit ] − ln Fg(i) (Xit ; β) into (88), which yields the following expression for ξitbai,t = Φas a function of unknown parameters (β, γ):ξit (β, γ) = Φ̂it − ln[Fg(i) (Xit ; β)] − mg(i)c it ; γΦ̂i,t−1 − ln[Fg(i) (Xi,t−1 ; β)], SP



(89)

c it is the predicted values obtained by estimating a binary choice model for remaining
where SP
single product based on (84). In practice, we do this by estimating the following probit model
DSPit = κt−1 (li,t−1 , ki,t−1 , ii,t−1 , Γi,t−1 )(90)where DSPit is a dummy for whether a firm remains single product at time t, and κt−1 (.) isapproximated with a second degree polynomial in all continuous variables.
We rely on equation (89) to construct sample equivalents of moments based on
E[ξit × Zit ](91)

where Zit is a vector of instruments.66 We instrument labour and materials with their lagged
values, and to deal with measurement error in kit , we instrument kit with lagged investment,
as suggested by Collard-Wexler and De Loecker (2015). Together (li,t−1 , mi,t−1 , ii,t−1 ) identify
the linear terms in (29), while we generate further instruments for identifying the nonlinear
translog terms of the form x1 x2 − 0.5 ((x1 )2 + (x2 )2 ) for each (x1 , x2 ) ∈ (l, m, k) by replacing
xs with the instrument chosen for that particular input.67
66

b as in Levinsohn and Petrin (2003), whereb in an “inner-loop”, given β,
In practice, we concentrate out γbconditional on a guess β, we estimate mg(i) () using OLS, and then use the error terms for from this toconstruct the GMM criterion function. To account for differences in units across product codes, we demean
all variables within 5-digit ASICC code when constructing the GMM criterion function.

67
So, for example, we
generate an instrument for lit mit − 0.5 (lit )2 + (mit )2 using li,t−1 mi,t−1 −
0.5 (li,t−1 )2 + (mi,t−1 )2 , and an instrument for kit mit − 0.5 (kit )2 + (mit )2 using ii,t−1 mi,t−1 −
0.5 (ii,t−1 )2 + (mi,t−1 )2 .

66

J

Linear dynamic panel estimator

As an alternative to the proxy variable approach with a selection-correction for single
product plants, we also apply a linear dynamic panel style estimator to the sample of singleproduct firms. While this approach does not deal with selection into single-product plantstatus, it does address some concerns articulated in Bond et al. (2021) concerning internalconsistency of proxy-variable methods when there are variable markups.
The estimation approach relies on the assumption that the productivity of single productplant follows a linear AR(1) process:ait = a0 + ρai,t−1 + ξit(92)

We can use the above to ρ-difference yit = ln[Fg(i) (Xit ; β)] + ait yielding:yit = ρyi,t−1 + ln[Fg(i) (Xit ; β)] − ρ ln[Fg(i) (Xit ; β)] + ξit(93)

We estimate the above model using a non-linear GMM routine, using the same set of
instruments as used in the control function estimator, plus yi,t−1 to identify ρ. To accountfor differences in units across product codes, we demean all variables within 5-digit ASICC
code when estimating this expression.

K

Recovering Shares in Closed FormIn our baseline procedure for recovering markups by product line, we follow De Loecker
et al. (2016) as allow for product-line specific input prices, to potentially account for quality
differences across products. First, we assume as in De Loecker et al. (2016) that plantproduct input prices can be decomposed into time-varying and time- and plant-productvarying components, orc j = Wst × W
cj .
W
(94)
sit
it
where Wst is simply an industry-level deflator for input s. We then assuming that the crosssection varying component can be modeled as a function of firm state variables, including
output prices, market shares, vectors of state and product dummies, and a trade statusindicator, or
ln Witj = w(pjit , msjit , IM Pit , Dit , Git ).
(95)
Since we actually observe input prices (plant wages) in the data, we can use single-product67

plants to predict input prices for multi-product plants. In particular, we estimate the modelln Witj = δ · Cjit + it(96)

on the sample of single-product plants, where Cjit contains a low-order polynomial in log
output prices and market shares, plus product, state, import status, and year dummies.
Indicators are permitted to enter log-additively, identically to De Loecker et al. δ is a vectorc j as product-levelof parameters. We estimate (96) by OLS, and use the fitted values W
itdeviations from the price index, for all firms.
With the predicted product-line specific input prices in hand, we can then derive ourclosed for expression for input shares. We start with generalization of (32) that allows for
MRT heterogeneity, and then obtain our baseline by imposing Ajit = Ait .
Since F (.) is homogenous of degree φ > 0, we can write:
Eit
ρjit jc
W

Yitj = Ajit F!
= Ajit

it

ρjit
cj
W

!φ
F (Eit )(97)

itWhich implies
Yitj
Ajit

! φ1
1

c j = ρj (F (Eit )) φ
W
it
it

(98)

Sum over jXjwhere we use the fact that

P

Yitj
Ajit

! φ1
1

c j = (F (Eit )) φ
W
it

(99)

j
j ρit = 1. Divide (100) by (99):



Yitj
Ajit

 φ1

cj
W
it
= ρjit1
P Yitk φ c k
Wit
k Ak

(100)

it

Equation (33) in the main text is then obtained by imposing Ajit = Ait .

L

Alternative Share Procedures for Figure 3

In this Appendix, we describe in detail three alternative to our baseline input allocation
rule that we report in Figure 3 of the main text.
68

1: We consider modifying the input allocation rule by dropping the input price control
1(Yitj ) φdjjfunction, Wit from the problem, generating the modified input share rule ρit = P
1 (“Nok φk (Yit )Wj”). This case corresponds to the standard setting considered in this paper where inputs
are the same when used across various product lines. We find that dropping the input-pricecontrol function does little to change the proportion of right tail outliers, while it slightly
increases the proportion of zero outliers.
2: We replace realized output in (33) with “predicted” output, as suggested by De Loecker
et al. (2016) to adjust for measurement error. More precisely, in this specification (“Phi”),
we re-rerun the first step of our control function estimation routine for the full sample ofsingle- and multi-product plants. We then use this regression to generated predicted output
b j , which we use in place of Y j in (33)— this specification appears to increase both theΦititproportion of tail, as well as zero, outliers.
3: We consider whether the Ajit = Ait restriction is generating issues due to differences inquantity units across product lines. In particular, Ajit is a quantity-based TFP measure, andtherefore will partly vary with the units product j is measured in. To correct for this, we
j
Y
it
j
A

consider a units adjusted input allocation rule


P

j

k

!1

Yk
it
k
A

φ

cj
W
it

1

φ

, (“Units Adjustment”), whereck
W
it

A is average TFPQ for single-product plants producing product code j. Unfortunately, we
find that this approach does little to alleviate the zero markup puzzle.

69

M

Product Markups with Quantity Ranked TFPQ
Figure 11: Product-level Markup Outlier Ratios with Productivity Ladders(A) Core Product - Highest Quantity(B) Core Product - Lowest Quantity

Notes: The above reports outlier ratios for different models of product-level markups. Letting Z denotethe set of product-level markups where µ ≤ 0.5, T denote the set of product-level markups where µ ≥ 5,
and R denote product-level markups between 0.5 and 5, the Zeros outlier ratio (represented by circles)|Z||T|
is defined as |R|, while the Tail outlier ratio (represented by triangles) is defined as |R|, where |.| denotes the number of elements in a given set. These statistics are report as a function of within firm
productivity dispersion, parameterized by 1δ . Product-level markups calculated using equation (34) with



1
j φ
P  Yitk  φ1 c k
Yit
j
j
c
Wit / k (δ)kWit . Panel (A) orders products j within a plant from highest to lowest
ρit =
(δ)j
quantity sold, with j = 0 having the highest quantity sold, j = 1 having the second highest quantity, andso on. Panel (B) orders products within a plant from lowest to highest quantity, with j = 0 denoting thelowest quantity product, j = 1 denoting the second lowest quantity product, and so on.

N

Production Function Estimates

In this Appendix, we report output elasticities and overall returns to scale for the various
approaches used to estimate plant and plant-product markups. The approaches consideredin the paper are:
1. Baseline: We rely on a translog specification of the technology as in equation (29), anduse the control function estimation approach outlined in Appendix I. In particular, weonly use single product plants for estimation and correct for this with a De Loecker et al.
(2016)-style selection correction, and also correct for input price bias using plant-levelwages. We include lagged investment as an instrument for capital in this specification,
as recomended by Collard-Wexler and De Loecker (2021). We only estimate a single set70

of production function parameters for the entire Indian economy in this specification;
however, this still allows for heterogeneity in output elasticities across firms due to thetranslog form. Estimates reported in Table 2.
2. Linear dynamic panel: We rely on a translog specification of the technology as inequation (29), and use the dynamic panel approach outlined in Appendix J. We only
use single product plants for estimation but do not correct for this with a De Loecker
et al. (2016)-style selection correction. However, we do correct for input price bias
using plant-level wages. We include lagged investment as an instrument for capital inthis specification, as recomended by Collard-Wexler and De Loecker (2021). We only
estimate a single set of production function parameters for the entire Indian economy
in this specification. Estimates reported in Table 3.
3. Baseline by industry: Same as Baseline, except we estimate production function
parameters by 2-digit NIC industry. Estimates reported in Table 4.
4. Linear dynamic panel by industry: Same as Linear dynamic panel, except weestimate production function parameters by 2-digit NIC industry. Estimates reported
in Table 5.
5. Baseline alternative instruments: Same as Baseline, except we use kit as an
instrument, rather than lagged investment. Estimates reported in Table 6.
6. Baseline Cost Share: Cost-share approach described in the main text, where production function parameters only differ across 2-digit NIC industries. Estimates reported
in Table 7.
7. Heterogeneous Cost Share: Cost-share approach described in the main text, where
production function parameters differ across 2-digit NIC industries × labour-to-material
quintiles, as suggested by Raval (2023). Estimates reported in Table 8.
For each specification, we report the mean and standard deviation for each output elasticity, as well as the mean and standard deviation of returns to scale.68 In the first column,
we report these statistics for all available industries. In the subsequent columns, we reportstatistics by industry, for a subset of India’s largest industries. The output elasticities are calculated over single-product plants when we rely on approaches that only use single-productplants for estimation (Tables 2 through 6). In the cost-share approaches (Tables 7 and 8, werely on all plant-years, including multi-product plants.
68

Sometimes the standard deviation is zero, if that particular specification holds the statistic constantwithin a particular industry-group.

71

Table 2: Output Elasticities: Baseline estimation using single product plants

θk
θl
θm
RTS

All

15

17

0.187
(0.020)0.057(0.022)0.900(0.037)
1.144
(0)

0.186
(0.021)
0.049(0.020)0.909(0.037)
1.144
(0)

0.188
(0.020)0.055(0.021)
0.901(0.037)
1.144
(0)

18

21

0.189
0.188
(0.019) (0.015)
0.064
0.051
(0.020) (0.014)
0.890
0.905
(0.036) (0.026)
1.144
1.144
(0)
(0)

24
0.184
(0.021)
0.054(0.020)0.905(0.037)
1.144
(0)

25

26

0.188
0.194
(0.016) (0.017)
0.050
0.071
(0.017) (0.018)
0.906
0.879
(0.030) (0.030)
1.144
1.144
(0)
(0)

27

28

29

31

0.179
(0.018)0.040(0.021)
0.925(0.035)
1.144
(0)

0.185
(0.019)0.056(0.019)0.902(0.035)
1.144
(0)

0.187
(0.018)0.060(0.019)0.897(0.034)
1.144
(0)

0.183
(0.020)0.051(0.020)0.909(0.037)1.144(0)Notes: The above table reports the estimated average output elasticities and their standard deviation across single-productplants. Output elasticities are based on a single translog estimated for the entire Indian economy. The first column reports thestatistics for all sectors, while the remaining columns report statistics for India’s largest manufacturing sectors (organized by
2-digit NIC code). The bottom row reports estimated returns to scale.

Table 3: Output Elasticities: Linear dynamic panel estimation using single product plants

θk
θl
θm
RTS

All

15

17

0.219
(0.051)0.030(0.042)0.860(0.065)
1.109
(0)

0.223
(0.052)0.012(0.035)0.873(0.065)
1.109
(0)

0.224
(0.051)
0.025(0.040)0.860(0.064)
1.109
(0)

18

21

0.219
0.226
(0.046) (0.039)
0.046
0.017
(0.035) (0.028)
0.844
0.866
(0.062) (0.046)
1.109
1.109
(0)
(0)

24
0.213
(0.057)0.028(0.041)
0.869(0.064)
1.109
(0)

25

26

0.227
0.228
(0.041) (0.048)
0.015
0.057
(0.033) (0.040)
0.868
0.825
(0.053) (0.052)
1.109
1.109
(0)
(0)

27

28

29

31

0.210
(0.045)0-.001(0.039)0.901(0.061)
1.109
(0)

0.215
(0.048)0.030(0.035)0.864(0.061)
1.109
(0)

0.218
(0.046)0.037(0.034)0.855(0.060)
1.109
(0)

0.211
(0.049)0.022(0.036)0.876(0.065)1.109(0)Notes: The above table reports the estimated average output elasticities and their standard deviation across single-productplants. Output elasticities are based on a single translog estimated for the entire Indian economy. The first column reports thestatistics for all sectors, while the remaining columns report statistics for India’s largest manufacturing sectors (organized by
2-digit NIC code). The bottom row reports estimated returns to scale.

72

Table 4: Output Elasticities: Multi-industry baseline estimation using single product plants

θk
θl
θm
RTS

All

15

17

0.219
(0.765)0.162(0.602)0.828(0.658)
1.209(.114)

0.170(0.062)0.069(0.025)0.897(0.043)
1.135
(0)

0.129
(0.095)–0.459(0.341)1.497(0.246)
1.167
(0)

18

21

0.056
0.425
(0.400) (0.384)
0.049
0.087
(0.447) (0.373)
0.914
0.686
(0.326) (0.666)
1.019
1.198
(0)
(0)

24
–0.049(0.663)–0.006(0.156)1.064(0.508)
1.008
(0)

25

26

0.122
0.262
(2.308) (0.670)
0.624
0.853
(1.131) (0.274)
0.659
0.181
(1.247) (0.686)
1.404
1.296
(0)
(0)

27

28

29

31

0.257
(0.220)0.052(0.251)0.888(0.261)
1.197
(0)

0.298
(0.597)0.073(0.504)0.867(0.106)
1.238
(0)

0.311
(0.508)–0.161(0.168)1.217(0.365)
1.367
(0)

0.706
(0.606)–0.182(0.811)0.740(0.312)
1.264(0)Notes: The above table reports the estimated average output elasticities and their standard deviation across single-productplants. Output elasticities come from seperate translog production functions estimated for each of the listed 2-digit NIC sectors.
The first column reports the statistics for all sectors, while the remaining columns report statistics for by sector. The bottom
row reports estimated returns to scale.

Table 5: Output Elasticities: Multi-industry linear dynamic panel estimation using single
product plants

θk
θl
θm
RTS

All

15

17

0.284
(0.462)0.140(0.704)0.710(0.668)1.134(.099)

0.170(0.211)
0.096(0.108)0.853(0.198)
1.12
(0)

0.104
(0.213)–0.134(0.570)1.109(0.676)
1.079
(0)

18

21

0.235
0.561
(0.760) (0.695)
0.161
0.122
(1.305) (0.177)
0.462
0.482
(0.585) (0.856)
.857
1.164
(0)
(0)

24
0.292
(0.217)0.066(0.059)0.692(0.233)
1.05
(0)

25

26

0.550
0.275
(0.292) (0.560)–0.0540.704(0.460) (0.166)
0.640
0.247
(0.479) (0.534)
1.136
1.225
(0)
(0)

27

28

29

31

0.153
(0.230)0.048(0.183)0.918(0.409)
1.119
(0)

0.186
(0.486)0.233(1.779)0.902(1.439)
1.321
(0)

0.366
(0.338)–0.313(0.123)1.103(0.215)
1.156
(0)

0.692
(0.557)–0.266(0.435)0.702(0.180)1.128(0)Notes: The above table reports the estimated average output elasticities and their standard deviation (in parentheses) acrosssingle-product plants. Output elasticities come from seperate translog production functions estimated for each of the listed
2-digit NIC sectors. The first column reports the statistics for all sectors, while the remaining columns report statistics for by
sector. The bottom row reports estimated returns to scale.

73

Table 6: Output Elasticities: Baseline estimation using single product plants— Alternative
Instruments

θk
θl
θm
RTS

All

15

17

18

–0.13
(0.081)0.160(0.089)1.067(0.031)
1.097
(0)

–0.10
(0.067)0.136(0.081)
1.062(0.031)1.097(0)–0.117(0.078)0.147(0.087)1.067(0.030)
1.097
(0)

21

–0.152 –0.103(0.063) (0.059)
0.175
0.136
(0.072) (0.068)
1.074
1.065
(0.029) (0.022)
1.097
1.097
(0)
(0)

24
–0.133(0.087)0.167(0.100)1.063(0.031)
1.097
(0)

25

26

–0.10
–0.159(0.063) (0.085)
0.133
0.174
(0.071) (0.093)
1.064
1.083
(0.025) (0.025)
1.097
1.097
(0)
(0)

27

28

29

31

–0.093(0.070)0.142(0.076)1.049(0.029)1.097(0)–0.134(0.067)0.167(0.078)1.065(0.029)1.097(0)–0.141(0.062)0.169(0.072)
1.069(0.028)1.097(0)–0.125(0.066)0.163(0.077)1.06(0.031)
1.097(0)Notes: The above table reports the estimated average output elasticities and their standard deviation (in parentheses) acrosssingle-product plants. Output elasticities are based on a single translog estimated for the entire Indian economy. The firstcolumn reports the statistics for all sectors, while the remaining columns report statistics for India’s largest manufacturingsectors (organized by 2-digit NIC code). The bottom row reports estimated returns to scale.

Table 7: Output Elasticities: Cost Shares by 2-digit NIC

θk
θl
θm
RTS

All

15

17

18

21

24

25

26

27

28

29

31

0.064
(0.033)0.068(0.029)0.868(0.043)
1
(0)

0.039
(0)
0.047
(0)
0.914
(0)
1
(0)

0.080
(0)
0.079
(0)
0.842
(0)
1
(0)

0.039
(0)
0.138
(0)
0.823
(0)
1
(0)

0.104 0.081 0.062 0.133 0.075 0.035
(0)
(0)
(0)
(0)
(0)
(0)
0.068 0.040 0.057 0.071 0.048 0.080
(0)
(0)
(0)
(0)
(0)
(0)
0.828 0.878 0.881 0.796 0.877 0.884
(0)
(0)
(0)
(0)
(0)
(0)
1
1
1
1
1
1
(0)
(0)
(0)
(0)
(0)
(0)

0.039
(0)
0.090
(0)
0.871
(0)
1
(0)

0.037
(0)
0.070
(0)
0.892
(0)
1
(0)

Notes: The above table reports the estimated average output elasticities and their standard deviation (in parentheses) across allplants. Output elasticities obtained by the industry cost shares for each NIC2 industry. The first column reports the statistics
for all sectors, while the remaining columns report statistics for India’s largest manufacturing sectors (organized by 2-digit NICcode). The bottom row reports estimated returns to scale.

74

Table 8: Output Elasticities: Cost Shares by 2-digit NIC× Labour-to-materials quintiles

θk
θl
θm
RTS

All

15

17

0.066
(0.031)0.116(0.112)0.817(0.122)
1
(0)

0.044
(0.017)0.063(0.051)
0.892(0.066)
1
(0)

0.079
(0.011)
0.108(0.091)
0.813(0.085)
1
(0)

18

21

0.045
0.105
(0.020) (0.026)
0.168
0.074
(0.110) (0.043)
0.787
0.821
(0.130) (0.058)
1
1
(0)
(0)

24
0.096
(0.016)0.101(0.083)0.803(0.097)
1
(0)

25

26

0.066
0.103
(0.015) (0.027)
0.080
0.218
(0.058) (0.155)
0.854
0.680
(0.072) (0.131)
1
1
(0)
(0)

27

28

29

31

0.078
(0.024)0.052(0.048)0.870(0.070)
1
(0)

0.042
(0.014)0.120(0.092)0.838(0.106)
1
(0)

0.050
(0.021)
0.138(0.098)0.812(0.119)
1
(0)

0.044
(0.014)0.100(0.077)0.856(0.088)1(0)Notes: The above table reports the estimated average output elasticities and their standard deviation (in parentheses) across allplants. Output elasticities obtained by the industry cost shares for each materials-to-labour quintile in each NIC2 industry. The
first column reports the statistics for all sectors, while the remaining columns report statistics for India’s largest manufacturingsectors (organized by 2-digit NIC code). The bottom row reports estimated returns to scale.

75

O

Allowing for Scope and Varying Estimators

In this Appendix, we consider whether two further sources of potential misspecification
can explain the zeros puzzle; assuming non-joint production when production may be joint,
and misspecification of the production technology. Figure 12 considers the case of joint
production by relying on output distance function (13), in which case 1 − β captures theshare of public, non-rival intermediates in production. While we find a slight decrease in
the degree of zeros as we increase the public share, this is more than counterbalanced by asignificant increase the the proportion of tail outliers, suggesting that joint production, on
its own, is unlikely to solve the zero puzzle.
In Figure 13, we examine whether alternative methods of estimating material’s output
elasticity can resolve the zeros puzzle. We report outlier ratios derived from: (1) our baseline estimates, (2) a linear dynamic panel estimator, (3) estimating a separate production
function for each of the largest 2-digit NIC industries, (4) a linear dynamic panel estimatorapplied by industry to the same set of industries as in (3), (5) our baseline approach but
using kit as an instrument to identify βK rather than lagged investment, (6) a cost share estimator by 2-digit NIC industry], and (7), a cost share estimator applied to each 2-digit NIC
industry × five separate labour to material expenditure quintiles, as recommended by Raval(2023). Each set of bars in Figure 13 reports outlier ratios where input shares are calculatedusing (33) as well as using revenue shares. Each set of bars tells a similar story: there is asizeable number of zero and tail outliers when assume that there is no MRT heterogeneity,
but many fewer when we allocate inputs using revenue shares.

76Figure 12: Product-level Markup Outlier Ratios with Scope

Notes: The above reports outlier ratios for different models of product-level markups. Letting Z denote theset of product-level markups where µ ≤ 0.5, T denote the set of product-level markups where µ ≥ 5, and R
denote product-level markups between 0.5 and 5, the Zeros outlier ratio (represented by circles) is defined|Z||T|
as |R|, while the Tail outlier ratio (represented by triangles) is defined as |R|, where |.| denotes the numberof elements in a given set. These statistics as a function of the public input share, 1 − β, where markups are1(Y j ) βφ
calculated using equation (34) with ρjit = P it 1 ..
k βφk (Yit )77

Figure 13: Product-level Markup outlier Ratios Across Methods(A) Baseline(B) Revenue Share

Notes: The above bar chart plots two outlier ratios for different models of product-level markups for the
sub sample of multi-product plants. Letting Z denote the set of product-level markups where µ ≤ 0.5, T
denote the set of product-level markups where µ ≥ 5, and R denote product-level markups between 0.5 and|Z||T|
5, the Zeros outlier ratio is defined as |R|, while the Tail outlier ratio is defined as |R|, where |.| denotes thenumber of elements in a given set. Panel (A) reports outlier statistics when markups are calculated using(33) and (34), as in our baseline. Panel (B) reports outlier statistics when markups are calculated using (34)with ρjit equalling a product’s revenue share (Rshare). The numbers of the x axis denote different waysof estimating relevant output elasticities to recover product-level markups. Specification (1) refers to ourbaseline estimates, where a single production function is estimated for all of India, using a control function
approach with a selection correction. Specification (2) recovers output elasticities using a linear dynamic
panel estimator. Specification (3) applies a control function with a selection correction by 2-digit NIC
industry (industries 15, 17, 18, 21, 24, 25, 26, 27, 28, 29, and 31). Specification (4) applies the linear
dynamic panel estimator by 2-digit NIC for the same set of industries. Specification (5) applies the control
function estimator with the selection correction using, using the current capital kit as instrument to identify
βK , rather than lagged investment as in our baseline. Specification (6) estimates the materials output
elasticity by 2-digit NIC code using the cost share approach. Specification (7) estimates the materials
output elasticity by 2-digit NIC code × labour to material expenditure quintiles using the cost share approach.
Specifications (1) through (5) are rely on a homogeneous translog specification of the production function,
while Specification (6) and (7) rely on Cobb-Douglas.

78

P

Plant-Markups: Alternative Estimation Routines
Figure 14: Plant-level Markup Dispersion Across Methods

Notes: The above plots the mean plant-level markup, as well as the 90th to 10th percentile of markups,
across various methods for calculating the plant-level materials elasticities. Baseline uses the estimatesfrom Table 2 . Dynamic Panel obtains plant-level markups using the elasticities in Table 3. Baseline
by Industry obtains plant-level markups using the elasticities in Table 4. Dynamic Panel by Industry
obtains plant-level markups using the elasticities in Table 5. Baseline, Alternative Z obtains plant-levelmarkups using the elasticities in Table 6.

Q

De Loecker et al. (2016) selection correction when
there is joint production and markup heterogeneity

De Loecker et al. (2016) justify their selection correction with reference to a model similar
to Mayer et al. (2014), which rules out joint production. In this appendix, we show that
their approach to correcting for selection can still be valid even if firms produce using joint
production technologies. In particular, we work through a simple parametric example ofdemand which generates a selection rule as in (83).

79Q.0.1Market Structure, Preferences, and Technologies

We consider a model of monopolistic competition with variable markups, based on Kimball (1995). Specifically, we consider a representative consumer whose overall welfare, V , isimplicitly defined by the following equation:


Z
Υ
ω∈Ω

Y (ω)
V


dω = 1,(101)

where Y (ω) is the total quantity consumed of variety ω, Ω is the set of all available varieties,
00and Υ (ψ) for ψ ≡ Y V(ω) is a function satisfying Υ (ψ) > 0 and Υ (ψ) < 0. For our purposes,
it will be convenient to work with the following specification of Υ (ψ):
Υ (ψ) = η ln(ψ) + β

σ−1σ(ψ) σ ,
σ−1(102)

where η, β and σ are utility parameters.69
We assume that the representative consumer minimizes their expenditure on all goodsin Ω, subject to the preference relation (101). This leads to the following inverse demand
function for variety ω:
P (Y (ω)) =

λ 0
Υ
V



Y (ω)
V


=

λ
V

ηV
+β
Yi



Y (ω)
V

!
 −1
σ
(103)

where λ is the consumer’s Lagrangian multiplier from their expenditure minimization problem. We assume that there are infinitely many firms in the market, so Ω is infinitely large,
and therefore all firms take λ and V as given It is then straightforward to show that these
preferences lead to the following variable markup rule:P (ω)σµ(ω) ==M C(ω)
σ−1

η
1+
β



Y (ω)
V

!
− σ−1σ(104)

While we assume there infinitely many firms in the market, we assume that each firm
produces at most N + 1 products, where N is a finite integer. We then assume that eachfirm produces using the following production possibility frontier.

X

j

Yij
Aji

! α1 α = (Ki )1−βL (Li )βL(105)

Here α 6= 1, which allows for joint production. We assume, similar to Olley and Pakes
69

It is straightforward to verify that if η = 0, then (101) and (102) together imply that the representative
consumer has standard CES preferences.

80

(1996), that capital predetermined at time t, while labour is static.70
Following Mayer et al. (2014), we assume that each firm has a core variety, which is simply
the product that highest the highest single-product productivity index Aji . For simplicity,
we index this product by j = 0 for each firm. Products j = 1, 2, ... decline in productivity
according to the following deterministic productivity ladder:
Aji
= δj
A0i

(106)

Where δ ∈ (0, 1).
Since capital is pre-determined, a firm’s total costs for producing some vector of outputs Yi ≡ (Yi0 , Yi1 , ..., YiN ), conditional on Ki , firm specific wages wi , and their vector of
productivity shocks Ai ≡ (A0i , A1i , ..., ANi ) is given by:C(Yi , Ai , Ki , wi ) =

wi
(Ki )

1−βL
βL


Ji
X

j=0

Yij
Aji

! α1  βαL
=



wi
Ki1−βL A0i


Ji
X

1
βL

j=0

Yij
δj

! α1  βαL
(107)

where the second equality uses (106).
Q.0.2

Characterizing the Variable Profit Function

Each firm can produce their core variety j = 0, at no additional costs beyond the labourand capital necessary to produce each unit quantity. However, each marginal variety beyond
their core variety, j = 1, 2, ... requires that the firm pay a fixed cost f one period before they
can begin selling it. Before we determine the product selection rule, consider a exogenously
firm producing J varieties. Since productivity declines in the index j, a firm producing J
varieties will produces varieties j = 0, 1, ..., J − 1. Given this set of varieties, they choose
their quantities to maximize their profits, i.e.:

max
Yi

J−1
X
j=0


λη + βλ

Yi
V



 σ−1
σ
−

wi
Ki1−βL A0i

 β1 
L

J−1
X
j=0

Yij
δj

! α1  βαL


(108)

After evaluating each of the firm’s J first order conditions, we obtain the following closed
for expression for variable profits, as a function of J, the firm’s predetermined capital stock
Ki , their core product productivity level A0i , and firm specific wages wi .
70
We simply limit out attention to one dynamic input and one static input here to save on notation, but
the approach easily generalizes to multiple static inputs inlucding materials.

81



Π(J, Ki , A0i , wi ) = γ0 J + γ1 

Ki1−βL A0i(σ−1)βL
 β1  σ+β
L −σβL
L




wi

J−1
X

δ

 σ−1
j σ+α−ασσ+α−ασ
! σ+β
−σβ
L

L

(109)

j=0

where:
γ0 ≡ ληγ1 ≡ (βλ)δσσ+βL −σβLV

1−σσ+βL −σβL



σ−1
βL
σ

(σ−1)βL
 σ+β

−σβ
L

L

βL + σ − σβL
σ



σ−1
Note that α + σ − ασ > 1 since σ > 1. Therefore α+σ−ασ> 0 and since δ ∈ (0, 1), then
∈ (0, 1). From the standard formula for geometric series:

σ−1α+σ−ασΠ(J, Ki , A0i , wi ) = γ0 J + γ1 

Ki1−βL A0i(σ−1)βL
 β1  σ+β
L −σβLL

σ−1α+σ−ασσ+α−ασJ  σ+β
L −σβL1 − δσ−11 − δ α+σ−ασ




wi






or, by defining:
γ2 ≡ γ1

σ+α−ασ
 σ+β
−σβ

1
1−δ



C0 (Ki , A0i , wi ) ≡ 

L

L

σ−1
α+σ−ασKi1−βL A0i(σ−1)βL
 β1  σ+β
L −σβL
L




wi




Π(J, Ki , A0i , wi ) = γ0 J + γ2 C0 (Ki , A0i , wi ) 1 − δ

σ−1α+σ−ασσ+α−ασJ  σ+β
L −σβL∂Π(J,A0 ,K ,w )i ii
To guarantee that each firm produces a finite number of products J, we need>∂J20
∂ Π(J,Ai ,Ki ,wi )0 and< 0. Note that it is straightforward to verify that the first condition∂J 2
always holds, since:

82

σ−1
∂Π(J, A0i , Ki , wi )= γ0 −ln δγ2 C0 (Ki , A0i , wi )∂Jσ + βL − σβLσ+α−ασ−1
 σ−1 J  σ+βJ(σ−1)
L −σβLα+σ−ασ1− δδ α+σ−ασ > 0(110)
To obtain the second condition, we need to assume that:δ<

σ + βL − σβL
α + σ − ασ

 α+σ−ασσ−1(111)

In which case it is easily verified that:∂ 2 Π(J, A0i , Ki , wi )(σ − 1)σ−1= (ln ω)2 γ2 C02∂J
σ + βL − σβL α + σ − ασJ(σ−1)
 σ−1 J −1α+σ−ασδ
× 1 − δ α+σ−ασJ(σ−1)(σ + βL − σβL ) 1 − δ α+σ−ασ J(σ−1)
J(σ−1)
α+σ−ασα+σ−ασ<0× δ(α − ασ + σβL − βL ) − (σ + βL − σβL ) 1 − δWhich establishes that the variable profit function is concave in J.
Q.0.3

Product Selection Rule

Recall that we previously made the assumption that a firm will choose to add products
at time t − 1. To account for this, we know index all variables with t. Since a firm needs
to decide to add products one period before selling that product, and Π(.) is concave in Jit ,then a single product firm will only choose to add a new product in the following period if:71
E Π(2, A0i,t , Ki,t , wi,t ) − Π(1, A0i,t , Ki,t , wi,t )|Ii,t−1 ≥ f,(112)

i.e. the marginal gain in profit from adding a new product is less than the marginal fixed
cost of adding a new product. Since it is easily verified that:∂ 2 Π(J, A0it , Ki , wi )> 0,
∂A0it ∂J

(113)

then Π(2, A0it , Kit , wit ) − Πi (1, A0it , Kit , wit ) is strictly increasing in A0it . As long as Ait follows
a Markov process


(114)
A0it = E A0it |Ii,t−1 + ξi,t = g A0i,t−1 + ξi,t
71

For simplicity, we consider myopic firms that one look forward one period when deciding to add products;
however, it is straightforward to extend the logic here to the case of infinitely lived, forward looking firms,
at the cost of some extra notation.

83



such that g A0i,t−1 is strictly increasing in A0i,t−1 , then Π(2, g A0i,t−1 + ξi,t , Kit , wit ) −Πi (1, g (Ai,t−1 )+ξi,t , Kit , wit ) is strictly increasing in A0i,t−1 for every realization of ξit , Kit , wit ,and therefore, as long as the distribution of wit conditional on Ii,t−1 does not depend on A0i,t−1 ,then E Π(2, A0i,t , Ki,t , wi,t ) − Π(1, A0i,t , Ki,t , wi,t )|Ii,t−1 is strictly increasing in A0i,t−1 . Thismeans that the firm will remain single product in the subsequent period as long as A0i,t−1 ≤ A,
where A solvesE Π(2, g A + ξi,t , Ki,t , wi,t ) − Π(1, g A + ξi,t , Ki,t , wi,t )|Ii,t−1 = f,

(115)

Note that the relevant variable in the firm’s information set at time t − 1 for predicting
future period profits are lagged wages, the lagged capital stock, and previous period’s investment (which allows them to perfectly forecast Kit , leading to a cutoff productivity levelthat will be a function of the lagged state variable of the firms, and investment, i.e.
Jit > 1 if Ai,r−1 > Ai,t−1 = At−1 (ki,t−1 , ii,t−1 , wi,t−1 ),which has the same structure as the cutoff rule (83).

84otherwise Jit = 1(116)