Growth and the Fragmentation of Production∗
Johannes Boehm

Ezra Oberfield

Sciences Po & CEPR
johannes.boehm@sciencespo.fr

Princeton University & NBER
edo@princeton.edu

September 8, 2023
First draft, preliminary

Abstract
How much do changes in the fragmentation of production contribute to growth? Using detailed plantlevel data on the manufacturing sector in India between 1990 and 2014, we study a version of Smithian
growth, the link between greater fragmentation of supply chains and productivity. We propose a measure of a plant’s vertical span, which corresponds roughly to the number of stages in a supply chain thatthe plant performs in-house; when plants have smaller vertical spans, production is more fragmented.
We find that fragmentation increases with development in both the cross-section and time series. Further, within locations at a point in time, larger plants tend to have smaller vertical spans, and those that
increase sales tend to decrease vertical span. Using changes in demand during the tariff liberalizationin the 1990s, we provide evidence that increased demand causes specialization. We find evidence from
economies of scale in specialization. We construct a general equilibrium model to rationalize these findings and estimate the sources and magnitude of scale economies. Goods are produced in a successionof steps, each combining labor and a set of intermediate inputs, giving rise to a tree-like structure.
Firms exert effort to find suppliers for inputs, and choose the set of production stages (and thereby
inputs) to produce the output at lowest cost. The structure implies that the returns to searching are
more strongly diminishing for inputs that are further upstream. Firms with high productivity draws
are therefore more likely to choose to be more vertically specialized.
KEYWORDS: Growth, Organization, Intermediate Inputs, Productivity, Production Networks, Specialization
JEL: O11, O14, L23, L25∗
We thank Santiago Caicedo, Murat Alp Celik, and Fabiano Schivardi for helpful comments, and to Stephen
O’Connell for sharing data. We are grateful to the Princeton-Sciences Po partnership for financial assistance.

1

Introduction

Since Adam Smith, economists have been postulating that specialization is one of the primary
drivers of improvements in standards of living.1 Specialization can take many forms; perhaps themost salient is the fragmentation of supply chains, with different vertically related production stages
often done by different firms in different locations.2 Whether this increased specialization plays
an important role in growth is ultimately an empirical question: What is the relationship between
increased fragmentation and productivity growth?
In this paper we study the fragmentation of production in the Indian economy, from the timeof reform at the onset of the 1990s, up until 2015. We use detailed data on plants’ inputs andoutputs from annual manufacturing surveys to construct a measure of plants’ vertical spans of production, a measure intended to capture the range of sequential production steps performed within
the plant. The first part of the paper explores the empirical relationship between plants’ verticalspecialization choices—as measured by their vertical spans of production—and productivity andmarket conditions. We first examine the macro-level relationship between specialization and development across time and space. We find that vertical span and income per capita are strongly
related. At the macroeconomic level, plants in richer districts of India are more vertically specialized, and state-level income growth is associated with within-plant reductions in vertical span.
This is consistent with Smith’s famous statement that “the division of labour is limited by theextent of the market” (Stigler, 1951). Second, we assess the micro-level relationship between plantsize and vertical specialization among plants that produce the same product in the same location
at the same time. In contrast to the usual intuition that larger plants perform a wider range ofactivities, we find that those with higher sales tend to have shorter vertical spans, i.e., perform
fewer sequential stages of production. This holds both across plants in the cross-section and also
for within-plant changes over time.
These correlations suggest a link between vertical specialization and growth. To make inference
about the direction of causality between size and vertical span, we exploit India’s tariff liberalization
at the start of the 1990’s. We find that plants respond to negative demand shocks (induced bya reduction in the import tariff to the plant’s output) by increasing their span, and to positive
demand shocks by reducing their span, i.e., specializing in fewer sequential steps of production. In
addition, increases in demand are associated with using fewer inputs.
Finally, inspired by Young (1928), we study the presence of network economies. When tariffs ona good ω are reduced, this negative demand shock propagates upstream to industries that supply ω.
At the same time, we also find that industries that are downstream from these upstream industries
also see a reduction in sales (even excluding ω itself). This suggests industry-level scale economies.
These network economies could be the results of internal economies of scale in production in the1
As virtually all papers on Smithian growth have noted, Adam Smith begins the first chapter of Wealth of Nations
with the statement: “The greatest improvement in the productive powers of labour [...] seem to have been the effectsof the division of labour.” (Wealth of Nations, Chapter 1, 1776, emphasis added). Smith goes on to present the
famous example of the pin factory, and how an increased division of labor leads to higher labor productivity.
2
See, for example, Johnson and Noguera (2017) and Antràs (2020).

1

upstream industries, or the result of external economies of scale operating along the value chain.
We rationalize and explain these findings using a simple model, which we later develop into afull quantitative model. In the simple model, firms produce an output (“shirts”) using one of two(perfectly substitutable) production functions: either from cloth (in which case the firm is verticallyspecialized, and just tailors the cloth into shirts) or from yarn (in which case the firm performs both
the weaving of the cloth from yarn, and the tailoring of the shirts). Both production steps follow
a Cobb-Douglas production function in labor and the respective intermediate input. In the case of
the firm being integrated, it produces using the nested production function. Firms are born with
ex-ante heterogeneous Hicks-neutral productivity, and need to search for suppliers for each of thetwo inputs, paying a cost that is convex in search effort. More search effort leads to more matcheswith potential suppliers, and therefore on average a lower effective cost of that input. Following
realization of these draws, firms choose one of the two production structures (as well as suppliers)
to minimize cost.
The model implies that firms that are born with higher productivity have higher returns tosearching, and will therefore search more and obtain even lower costs, a manifestation of scaleeconomies. The sequential nature of production gives rise to a nonhomotheticity: when productivity
increases, the firm searches more for both inputs, but disproportionally more so for the downstream
input (cloth). This is because searching for cloth lowers the cost of the out-sourcing option by morethan seraching for yarn lowers the cost of the in-house option, as the latter requires labor and th
ecost of labor is invariant to search. Thus as the firm searches more, the expected expenditure
share of cloth rises relative to the expected expenditure share of yarn. The firm responds to thisby shifting it search effort toward suppliers of cloth, increasing the likelihood of a short verticalspan of production. This mechanism, whereby more productive firms select into being specialized,
is consistent with our empirical findings on the positive correlation between specialization and size.
The full quantitative model embeds this mechanism into a setup where value chains can be madeup of an arbitrarily large number of sequential steps, where each step combines primary factors withan arbitrary number of inputs. Firms search for suppliers in all input markets in this tree, and facea make-or-buy decision for each step. Building on Kortum (1997) and Oberfield (2018), we choosefunctional form assumptions such that, conditional on search efforts, the cost of buying each input
from the market follows a Weibull distribution. The make or buy decision at each node is tractable
due to a functional form assumption that is new to the literature for the distribution of task-specificproductivity draws, which yields a Weibull-distributed unit cost for in-house production with thesame shape. Firms make profits from sales to households, and entry is elastic. The search processfeatures network economies through a matching function that is increasing in the mass of potential
suppliers. This elasticity, together with firm-level scale economies in the form of an elastic search
effort, drives the propagation of shocks across industries and speaks to our empirical results.
In the final section, we describe an approach to solve and estimate the model. Estimation isstill in progress.

2

1.1

Related Literature

Our paper builds on a large theoretical literature that studies the relationship between specialization, market size, and economic performance, including classic papers by Young (1928), Stigler(1951), Rosen (1978), and Becker and Murphy (1992), and more recent work by Yang and Borland(1991), Baumgardner (1988a), Rodriguez-Clare (1996), Kelly (1997), Chaney and Ossa (2013),
Legros, Newman and Proto (2014), and Menzio (2020). In contrast to this work, our model iswritten in such a way that it can be taken to the data and used for quantitative work.
There is a good amount of evidence that the specialization among workers is limited by theextent of the market: Baumgardner (1988b) shows that the range of tasks performed by physicians
shrinks as local labor market grows. Garicano and Hubbard (2009) show that the fraction of lawyers
working in firms that specialize in particular fields grows with the size of the market. Duranton
and Jayet (2011) provide evidence that scarce specialist occupations are over-represented in large
cities. Tian (2018) shows that Brazilian manufacturing firms that are located in cities tend to hire
workers in more occupations. Atalay, Sotelo and Tannenbaum (2021) show that workers are more
specialized, relative to other workers in the same firm or the same occupation, when they work inlarger markets. Hansman et al. (2020) studies the vertical specialization and its relationship to thedemand for quality in the Peruvian fishmeal industry. Brown (1992), in a study of the Germancotton textile industry at the beginning of the 20th century, also finds a positive correlation between
scale and the degree of vertical specialization. He attributes the high degree of vertical integration(compared to British cotton textile) to the underdevelopment of input markets.
Our modeling approach builds on existing approaches. We allow firms to choose from a set of
production functions, called “recipes”, similarly to Boehm and Oberfield (2020), and also related
to Acemoglu and Azar (2020), and Ciccone (2002). Our measure of the vertical span builds onwork by Alfaro et al. (2019) and Boehm and Oberfield (2020). In terms of its application to thevertical scope of production, the paper relates to recent work by Chor, Manova and Yu (2021).
In terms of context, we build on work that studies the Indian trade liberalization in the early
1990s (Panagariya (2004), Sivadasan (2009), Topalova and Khandelwal (2011), and Goldberg et al.
(2010), among others) and manufacturing productivity growth in India more generally (Hsieh andKlenow (2014), Bollard, Klenow and Sharma (2013)).
Finally, our work is related to the growing literature on endogenous production networks, e.g.
Oberfield (2018), Eaton, Kortum and Kramarz (2022), Lim (2018), Chaney (2014), Dhyne et al.
(2021), Startz (2021), Grant and Startz (2021), Taschereau-Dumouchel (2017), Huneeus (2018),
Panigrahi (2021), and Miyauchi (2018), among others. Our work also builds on studies that focusedon cross-country differences in input-output structure and its relation to development, Chenery et al.
(1986), Jones (2013), Boehm (2022), Fadinger, Ghiglino and Teteryatnikova (2021) and Bartelme
and Gorodnichenko (2015)

3

2

Reduced-form regressions

2.1

Data and Context

Our main dataset is the Indian Annual Survey of Industries (ASI), a panel of formal manufacturing
plants that the Ministry of Statistics and Programme Implementation constructs and makes available to researchers. Each year the ASI surveys all plants that have more than 100 employees, anda fifth of all plants with more than 20 employees (or more than 10 employees, if the plant is using
power). We use survey rounds from between the years 1989/90 and 2014/15 except 1990 to 1993
and 1995/96, for which no detailed data is available. Most relevant for our paper, the ASI contains detailed information about the plants’ mix of outputs and intermediate inputs in addition tocommonly available factor costs. We concord all product codes to the five-digit classification usedin the 2007/08 round, which contains about 5,500 items that are similar in detail to six-digit HS
codes. Some early ASI rounds have low quality for certain parts of the survey and we exclude thesein our regressions; we discuss these choices along with more detail on concordances and summary
statistics in Appendix A.
Our empirical strategy employs the large changes in India’s barriers to importing from around
India’s vast trade liberalization at the beginning of the 1990’s. Before this liberalization, India’s
economic strategy emphasized self-sufficiency (Panagariya, 2004). Imports were generally subject
to licensing. While the list of items exempted from licensing grew towards the end of the 1980’s,
tariffs remained high. In 1991, India experienced a balance of payment crisis and depreciated therupee. Following the crisis, in July 1991 abolished the import licensing regime on all but a few items,
and started a series of large tariff cuts. Between 1991 and 1997, tariffs fell from an (unweighted)
average of 113 percent to an average of about 35 percent. While there was substantial variationin tariffs before the reforms (the top tariff bracket was 355 percent), post-reform tariffs were much
more even. Since the levels of pre-reform tariffs had been determined much earlier, Topalova andKhandelwal (2011) conclude that the “differential tariff changes across industries between 1991 and1997 were as exogenous to the state of the industries as a researcher might hope for in a real-world
setting.”
We therefore combine the manufacturing survey data with information on import tariff levels.
With the exception of a few years at the start of the 1990’s, tariff data is available from UNCTAD;
for the remaining years we digitize and transcribe commercial publications that contain effective
import tariffs by six-digit HS code. We concord all tariffs to the common product classification
used by the ASI in the 2000’s. See Appendix A.1 for details.
Most of the import competition changes due to import tariff reductions constitute negative
demand shocks. To study whether variation coming from positive demand shocks has a different(i.e. asymmetrical) effect, we construct an export demand shifter similar to Hummels et al. (2014)4

as an alternative source of exogenous variation:
ExDemandωt =Xξ¯ωd Imωdtd∈D

where ξ¯ωd is the average share of country d in India’s export basket, and Imωdt is the log of importsof ω of country d at time t from all countries except India.

2.2

Measuring the vertical span of production

We follow our earlier work (Boehm and Oberfield, 2020) to construct a measure of the verticalspan of production, by which we mean the number of consecutive production steps performed in
the plant. We give the precise definition in Appendix A.2, and convey the intuition here. In thefirst step, we construct a measure of vertical distance between each output-input pair (ω, ω 0 ). If
there is a single path in the IO matrix from input to the output in the IO matrix, then the verticaldistance is the number of links in that path, or the number of plants along the value chain fromthe input to the output. If there are multiple such paths from the input to the output in the IO
matrix, then vertical distance is simply the cost-weighted average of the number of links in eachpath. Suppose the (aggregate) materials expenditure mix of shirts is 70% cloth and 30% yarn, andthat the materials mix of cloth is 100% yarn. The distance between the output shirts and the input
cloth is one (because cloth only shows up directly as an input to shirts), and the distance between
shirts and yarn is 0.3 × 1 + 0.7 × 1.0 × 2 = 1.7. In the second step, we construct the vertical spanof a (single-product) plant, which measures whether a plant tends to use inputs that are distant orclose. A plant’s vertical span is the cost-weighted average distance between its output and each of
its inputs. In this example, a plant that produces shirts and uses cloth as an input would have a
vertical span of 1, whereas one that uses yarn as an input would have a longer vertical span of 1.7.
In our view, the measure of vertical span does not measure a primitive attribute of technology,
as the vertical distance is partly determined by plant decisions in equilibrium. Further, the numberof plants between an input and an output in a value chain is not the same as the number of steps
between the input and the output, however one might define a “step.” Nevertheless, when wecompare plants that produce the same product, the plant with the higher vertical span arguably
performs more steps in-house then the plant with the lower vertical span.

2.3

Motivating facts

Our first fact is a robust relationship between the aggregate level of development and the degree ofvertical fragmentation of production. This relationship holds both in the cross-section and within
locations over time, and hence motivates our study of fragmentation in the context of economic
growth:Fact 1 (Macro-level correlation between development and fragmentation) We have:(a) Plants in richer districts have shorter vertical spans of production (within their industry)5

Cotton
Cloth

100%

CottonYarn

70%
CottonShirts30%
CottonYarn
Figure 1 Example of Vertical Distance between Output (Shirts) and Inputs(b) Plants in states that grew faster have, on average, reduced their vertical span of production
more

-.04

Vertical Span, de-meaned
-.02
0
.02

.04

Figure 2 Plants in richer districts are more vertically specialized

-1

-.5
0
.5
Log District GDP/capita, de-meaned

1

Binscatter with n=50. Y and X variable de-meaned by industry x year. SP plants only.

Part (a) of Fact 1 is corroborated by Figure 2, which shows a binned scatter plot for the relationship between the (residualized) measure of the vertical span of production and the (residualized) log
income per capita of the district where the plant is located in, after projecting on 5-digit industry
dummies. Among plants in the same industry, those in richer districts have shorter vertical spans of
production. Concerning part (b) of Fact 1, Table I shows that plants shortened their vertical spansof production more in states that grew faster.3 In these (and subsequent) regressions we exclude
multi-product plants that, while economically important, are not easy to measure the vertical spanfor.
We next study the relationship between plant-size and vertical span. In many settings, larger
firms do more activities: it is well documented that they sell a wider variety of products and operate
3

We use state-level growth rather than district-level growth because district-level income per capita is only available
for a single cross-section in 2005.

6

Table I Plants in states that grew faster have vertically specialized more
Dependent variable: Vertical Span
(1)
Log GDP/capitast
Year FE
Plant FE
5-digit Industry FE
5-digit Industry × Year FE
Plant × 5-digit Industry FE
R2
Observations

(2)
∗

(3)
∗

-0.0716(0.028)-0.0601(0.026)

Yes
Yes

Yes
Yes
Yes

-0.0551∗(0.026)

Yes
Yes
0.592
2700030.656
2693990.808
163668

Standard errors in parentheses, clustered at the state × 5-dgt industry level. SP plants only.
p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01+

in a wider range of industries, operate more plants, and sell to a wider variety of destinations.4
Nevertheless, we show here that larger and more successful plants tend to have shorter verticalspans, and hence do fewer steps.
Fact 2 (Micro-level correlations between size and vertical span) We have:(a) Plants with higher sales within their industry on average have a shorter vertical span(b) Plants that, within their industry, grow faster on average reduce their vertical span more.
Figure 2.3 shows (a) by plotting the relationship between vertical span and size within 5-digit
industry × year cells. Table II shows that this correlation also holds within narrow industry ×
district × year cells, and also conditional on plant age and employment. Table III corroborates
part (b) of Fact 2 and shows regressions of the change in sales on the change in vertical span, over
all time horizons where firms produce the same single five-digit product. The point estimates are
consistently negative across specifications.
Finally, we mention some other covariates of vertical span: firms with shorter vertical span havehigher materials share of cost, are more likely to import, and have a higher share of relationshipspecific inputs in their materials basket. These correlations hold in the cross-section (within industries) and over time (within plant-industries). Table XVI in Appendix C shows these results.

2.4

Determinants of Vertical Specialization

A plant’s vertical span and size are, of course, jointly determined. How a plant organizes itself may
affect its marginal cost and therefore its size, and a plant’s size might affect incentives to organize
its production in a particular way. In this section we attempt to identify a causal channel: if aplant grows in response to higher demand, how does it change its vertical span?
4

See, e.g., Bernard, Redding and Schott (2010), Cao et al. (2022), and Bernard and Jensen (1999).

7

-1

-.5

Log Sales0.51Figure 3 Larger plants are more vertically specialized

-2

-1

0
Vertical Span

1

2

SP plants only. Regression includes industry x year FE's.

Table II Within industries, more specialized plants are larger
Dependent variable: Log Sales(1)
Vertical Span(2)∗∗-0.719(0.024)(3)∗∗-0.670(0.023)(4)∗∗-0.431(0.034)

Age

(5)
∗∗

-0.432(0.034)-0.193∗∗(0.015)0.00616∗∗(0.0012)-0.00314∗∗(0.00068)
1.067∗∗(0.015)

Log Employment
Year FE
5-digit Industry FE
District FE
Industry × District × Year FE
R2
Observations

Yes
Yes

0.394
353659

Yes
Yes
Yes
0.440
295094

Yes

Yes

Yes

Yes

Yes

Yes

0.700
1406100.701
1368310.859
136608

Standard errors in parentheses, clustered at the 5-dgt industry level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

As discussed in Section 2.1, we use shifts in demand as a source of exogenous variation, as
captured by tariff changes that affect firms through import competition and changes in foreign
demand which affects firms through demand for their exports.
Table IV shows that these shifters indeed affect sales of the firms, showing their relevance as
instruments for firm scale. Column (1) shows that changes in import tariffs are positively correlatedwith changes in the sales of single-product plants that produce this good (the import competition
channel). Note that tariff changes can both alter import competition and act as a supply shock bychanging the cost of materials. In column (2), we also control for the industry’s change in input
tariffs (weighted by the shares in the aggregate industry’s materials basket). The coefficient on thisterm is negative, meaning that plant sales increase as input tariffs decrease. These relationships

8

Table III Plant growth is correlated with increased vertical specialization
Dependent variable: ∆ log Sales(1)∆ Vertical Span
Year FE
Product × Year FE
Plant FE
Plant × Product FE
2

R
Observations

(2)
∗∗

(3)
∗∗

-0.0655(0.0082)(4)∗-0.0445(0.0087)-0.0284(0.013)-0.0259∗(0.011)

Yes

Yes
Yes

Yes

Yes

Yes
0.008191204360.149111244

0.432
83026

0.431
74707

Changes within plant-productsStandard errors in parentheses, clustered at the state-industry level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

mean tariff changes act as industry-level demand shifters and cost shifters. Column (3) shows thatexport demand raises firm sales as well. And column (4) shows the joint significance of importtariff changes and export demand.
Table IV Impact of Import Competition and Export Demand on Plant Sales
Dependent variable: Log Sales(1)
ExDemandωt(2)∗∗

0.0493(0.0086)

outputlog(1 + τit
)

(3)
∗∗

0.0539(0.011)

0.0526∗∗(0.011)0.478∗∗(0.15)

0.722∗∗(0.16)

input
log(1 + τ̄it)Year FE
Plant FE
Plant × Industry FE
2R
Observations-0.604∗∗(0.20)
Yes

Yes

Yes

Yes

Yes

Yes

0.947
1624210.949
1241060.949
124098

Standard errors in parentheses, clustered at the state × industry level.
p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01+

We turn to exploring the direction of causality between scale and vertical span. Table V shows
regressions of the plant’s vertical span on sales (again, for single-product plants only), within plant× product pairs, and controlling for year effects. Columns (1) and (2) report OLS results, andfind a negative correlation between in sales and in span. Columns (3) and (4) instrument sales bylog import tariffs on the plants’ output, and columns (5) and (6) instrument sales by the export
demand shifter. The IV coefficient estimates are larger than the OLS ones, and point to a negative
causal relationship: as plants shrink because of a reduced output tariff, they increase their verticalspan of production. Because output tariff changes may be correlated with input tariff changes,
columns (2), (4), and (6) also control for plant-level input tariffs (where we weigh tariffs by the9plants’ expenditure shares at time of first observation), and whether those input tariff changes arelarger for inputs that are more distant.5
Taken together, the regressions in Table V suggest that Smith’s famous adage about the divisionof labor being limited by the extent of the market applies to our context in the sense that verticalspan captures the division of labor across production units.
Table V Firms vertically specialize when they receive a positive demand shock
Dependent variable: Vertical Span
(1)
Log Sales(2)∗∗-0.0191(0.0020)

input
log(1 + τ̄jωt
)

(3)
∗∗

-0.0197(0.0024)(4)+-0.512(0.28)(5)∗-0.253(0.13)(6)∗-0.164(0.073)-0.268∗∗(0.094)-0.0211(0.050)0.0198(0.065)0.0552(0.10)

input)spanji αi log(1 + τ̄it

-0.0905(0.056)-0.219∗(0.099)-0.372∗∗(0.13)

input)(distanceωi − spanj )i αi log(1 + τ̄it

-0.121(0.095)-0.337∗(0.15)-0.498∗(0.21)

P
P

Year FE
Plant × Product FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Estimator

OLS

OLS

IV 1

IV 1

IV 2

IV 2

0.765

0.731

186628145165-1.0495.566
138204-0.232
10.91
137059-0.081626.53
143428-0.22715.701105782

RFirst stage w. id. robust F
Observations

Standard errors in parentheses, clustered at the state-industry level. + p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

Regression contains single-product plants only. The left-hand side is the plant’s vertical span ofproduction; the right-hand side is log sales. Columns (3) and (4) instrument log sales by the log
tariff on the plant’s output ω. Columns (5) and (6) instrument log sales by the export demand
shifter.

2.5

Network Externalities and Industry Returns to Scale

Next, we turn to searching for network externalities. When firms receive a negative demand shock,they scale down or exit. This means that there may be in turn lower demand for the goods ofproducers further upstream, etc. In neoclassical models with constant returns, demand shocks
propagate upstream. In the presence of scale economies or network externalities, they may also
propagate downstream again.
5

The sample in Table V contains only observations in which the plant produces the same, single product, as these
are the situations in which we can clearly measure vertical span. Nevertheless, one might be concerned that somefirms respond to an increase in demand by changing their product mix, and that by focusing on those that remain
single-product plants we are selecting the sample in a way that somehow biases our estimates. We take two approaches
to examine whether selection might impact our results. First, we expand the sample to include multi-product plants,
treating them as all of its revenue came from its best-selling good. Here there is no selection issue, but the plant’s
vertical span will be measured with error. The results are very similar to our baseline specification. Second, we buildon the observation that plants that are currently single-product, but had been multi-product in the past, are more
likely to become multi-product again following a positive demand shock. Selection is likely to be a larger issue forthese plants than for plants that have always been single-product. Nevertheless, estimates from these two groups are
virtually indistinguishable. See Appendix C.6 for details.

10In Table VI, we search for upstream propagation of demand shocks. We regress log sales ofplants j onto the weighted average log number of producers of j’s inputs in j’s state, where theweights are again j’s materials shares at the time of first observation:
pj yj = β

X
ω̂∈Ω

p x
P j ω̂ j ω̂log(#producers of ω̂ in state d)t
ω 0 pjω 0 xjω 0!
+ αjω + αωt + εjt

The inclusion of plant-product fixed effects means that we again look at changes in the dependentand independent variables. Columns (1) and (2) estimate this relationship using OLS. In columns(3) and (4) we instrument the number of producers in upstream industries ω̂ by the log tariff on
goods ω̃ weighted by the share of ω̂’s sales to ω̃, but excluding j’s industry ω itself:X

pj ω̂ xj ω̂ω 0 pjω 0 xjω 0


X

P
ω̂∈Ω

νω̂ω̃ log(1 + τω̃t )

ω̃∈Ω, ω̃6=ω

where νω̂ω̃ denotes the share of sales of industry ω̂ to industry ω̃ (but ignoring sales to ω). In
other words, we instrument the number of producers in upstream industries by demand shocks toindustries downstream from these industries, but excluding the industry of j itself. In columns(5) and (6) we instrument the number of producers in upstream industries by a demand shifter tothe downstream industries that is analogously constructed, but using the export demand shifter
instead of the tariff 1 + τω̃t . The regressions in Table XIX of Appendix C confirm that tariff shocks
in downstream industries affect entry. TODO update entry regressions w/ demand shifter
The IV regressions in columns (3) to (6) find a positive and significant coefficient: the demand
shocks to industries that are not directly vertically related also change the cost of production in
the industry. These changes to cost may be coming from internal economies of scale in upstream
sectors, or from network externalities as postulated by Young (1928). The model in the next section
will help to understand the role of both channels and serve as a basis for identifying them.

2.6

Spiders, Snakes, and Trees

In this section we provide evidence on the structure of a firm’s production, studying how the numberof inputs a plant uses relates to vertical specialization. Baldwin and Venables (2013) coined namesof two commonly used models of the structure of production, spiders and snakes. In a spider,
input use has only a horizontal dimension. There is only one production stage which combines
multiple intermediate inputs to produce output.6 In a snake, input use is vertical. There areseveral sequential stages of production, and at each stage a single intermediate is combined with6

Spider production functions are standard in the macro literature with input-output structures such as Long
and Plosser (1983), Acemoglu et al. (2012), and Baqaee and Farhi (2019). Spider production structures with an
endogenous number/measure of intermediate inputs include Acemoglu, Antràs and Helpman (2007), Eaton, Kortum
and Kramarz (2022), Taschereau-Dumouchel (2017), Lim (2018), Huneeus (2018), Chan (2017), Blaum, Lelarge andPeters (2015), and Tintelnot et al. (2018).

11

Table VI Demand shocks also propagate downstream
Dependent variable: log Sales(1)
Avg. log #Producers in Upstream Ind.

(2)∗∗

0.0383(0.0050)

0.0375(0.0060)

input
log(1 + τ̄jωt)

input)(distanceωi − spanj )i αi log(1 + τ̄it

P

(3)
∗∗

(4)
∗∗

0.0611(0.017)(5)∗∗

0.0735(0.018)

(6)
∗∗

0.254
(0.036)0.121∗∗(0.018)-0.0242(0.096)-0.0207(0.097)-0.0162(0.097)-0.331∗∗(0.11)-0.334∗∗(0.11)-0.337∗∗(0.11)

Industry × Year FE
Plant × Industry FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Estimator

OLS

OLS

IV 1

IV 1

IV 2

IV 2

0.952

0.954

1990391420060.000636
615.6
1990390.000292
517.1
142006-0.0301
88.48
199039-0.00391
437.5
142006

2

R
First-stage w. id. F
Observations

Standard errors in parentheses, clustered at the industry-year level. + p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

Regression contains single-product plants only. The left-hand side is the plant’s sales; the right-hand side is thechange in log sales. Changes are taken across all time horizons within the same plant-product pairs. Columns (4)to (6) instrument ∆ log sales by the change in the log output tariff.

labor and the output of that stage is passed on to the next stage.7 The length of a snake has aclear connection to a plant’s vertical span, but no relation to the number of inputs. In contrast,
the width of a spider has a clear connection to number of inputs, but an ambiguous connection
to vertical span. A production tree combines the vertical dimension of a snake with the horizontal
dimension of a spider. There are multiple sequentially related stages, and each stage may require
more than one intermediate input.8 Figure 4 depicts these three types of production structures.
The different production structures hard-wire different relationships between the number oftasks that are outsourced and the number of inputs. With a spider production structure, outsourcing more tasks would correspond to purchasing more intermediate inputs. With a snake production
structure, the number of inputs used is invariant to outsourcing. With a tree production structure,outsourcing more tasks would correspond to purchasing fewer intermediate inputs. For example,
the firm pictured in panel (c) can choose to purchase ω̃1 from a supplier or produce it in-house,
in which case it would purchase ω̃4 and ω̃5 from suppliers. The former corresponds to a shorter
vertical span and fewer inputs, whereas the latter corresponds to a longer vertical span and more
inputs.
We therefore study the relationship between the number of inputs, the vertical span, and thescale of production of plants. In particular, we present evidence that that is consistent with animportant role for a vertical dimension of input use, consistent with a snake or a tree structure.
Table VII shows correlations between the vertical span of plants and the number of five-digit
materials inputs that the plant consumes in the production process (alternatively, the inverse
7

Snake production structures have been used Dixit and Grossman (1982), Yi (2003), Levine (2010), Costinot,
Vogel and Wang (2013), Chaney and Ossa (2013), Fally and Hillberry (2018), and Antràs and de Gortari (2017).
8
In graph theory, this is a rooted, directed tree.

12Herfindahl of expenditure shares on 5-digit inputs9 ). In both the cross-section (within industries)
and in the time dimension, plants that have a long vertical span on average use more inputs. We
find that plants with larger share of intermediate inputs in total cost tend to use, if anything, fewer
inputs (Appendix C.2), consistent with a tree but not with a spider.
Table VIII shows regressions of the plant’s log number of inputs (columns (1) to (4)) or inverse
HHI of cost shares of the plant’s materials basket (columns (5) to (8)) on sales, within plant-productpairs. In the IV specifications, columns (2), (3), (7), and (8), we again instrument sales using thelevel of output tariffs. While the OLS estimates of the sales coefficient are perhaps positive, theIV estimates show a clear picture: decreases in scale coming from negative demand shocks areassociated with an increase in the number of inputs, and vice versa.
Table VII Production Structures are Trees
Dependent variable: Vertical Span
(1)
Log Number of Inputs(2)∗∗0.0169(0.0031)(3)(4)

0.0269∗∗(0.0021)0.0337∗∗(0.0042)

Yes
Yes

Yes

∗∗

0.0533(0.0045)

Inverse HHI of Materials Cost SharesYear FE
Industry FE
Plant × Industry FE

Yes
Yes

Estimator

OLS

OLS

OLS

OLS

0.302
3536940.766
1866410.304
3534320.767
1864862

R
Observations

Yes
Yes

Yes

Standard errors in parentheses, clustered at the state-industry level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01
Regression contains single-product plants only. The left-hand side is vertical span; the right-hand side variables
are the plant’s number of 5-digit materials inputs (columns (1) and (2)) and the inverse of the HHI of materials
shares (columns (3) and (4)).

These patterns are consistent with a tree production structure in which reductions in verticalspan and reductions in number of inputs go hand-in-hand. We found that reductions in verticalspan correspond to using fewer intermediate inputs. Further, the same increases in demand thatcause plants to reduce vertical span also cause them to reduce the number of inputs. This finding
will guide the theoretical framework described below.

9

The inverse of the Herfindahl index is a common, ad hoc measure of the diversity of inputs. Simply counting thenumber of inputs would not differentiate between a situation in which a plant splits its intermediate input spending
equally on two inputs, and one in which the plant uses two intermediate inputs but 99.9% of its expenditure on
intermediates is spent on one of them. In the latter case, we would argue that the plant “effectively” uses one
input. Formally, the inverse Herfindahl can be interpreted as the number n of inputs such that, if cost shares were
uniform across those n inputs, would deliver the same level of concentration of input spending as the actual input
bundle. To see this, note that with firm used n inputs each with a share 1/n, the inverse Herfindahl would be
P
 −1
n
1 2
= n. Thus we interpret the inverse Herfindahl index as capturing the “effective” number of inputs.
i=1 n

13

Table VIII Firms reduce the number of inputs when they receive a positive demand shock
Dependent variable: # Inputs(1)
Log Sales(2)∗∗0.0359(0.0030)

input
log(1 + τ̄jωt
)

(3)
∗∗

0.0374(0.0035)(4)∗-1.136(0.54)(5)+-0.468(0.26)(6)+-0.383(0.20)-0.542∗(0.25)-0.441∗∗(0.16)-0.366∗(0.14)-0.207(0.19)

input)spanji αi log(1 + τ̄it

0.125
(0.10)-0.126(0.14)-0.337(0.21)

input)(distanceωi − spanj )i αi log(1 + τ̄it

0.162
(0.10)-0.202(0.25)-0.722∗(0.30)

P

P

Year FE
Plant × Product FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Estimator

OLS

OLS

IV 1

IV 1

IV 2

IV 2

0.856

0.847

186628145165-4.7885.566
138204-0.87410.91
137059-0.57426.53
143428-1.05915.701105782

RFirst stage w. id. robust F
Observations

Changes within plant-productsStandard errors in parentheses, clustered at the state-industry level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01
Regression contains single-product plants only. The left-hand side is the plant’s log number of 5-digit materials
inputs; the right-hand side is log sales and controls. Columns (3) and (4) instrument log sales by the log output
tariff, columns (5) and (6) instrument log sales by the export demand shifter.

Table IX Firms reduce the effective number of inputs when they receive a positive demand shock
Dependent variable: Inverse Input HHI(1)
Log Sales(2)0.0101(0.0072)

input
log(1 + τ̄jωt)

0.00930(0.0082)

(3)

(4)
+

-1.804(1.08)(5)+-1.100(0.63)(6)+-1.705(0.99)-2.135+(1.18)-0.745+(0.42)-0.574+(0.30)-0.354(0.50)

P

input)spanji αi log(1 + τ̄it

0.428
(0.34)-0.0487(0.29)-1.362+(0.78)

P

input)(distanceωi − spanj )i αi log(1 + τ̄it

0.630
(0.39)-0.138(0.55)-1.906+(1.16)

Year FE
Plant × Product FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Estimator

OLS

OLS

IV 1

IV 1

IV 2

IV 2

0.807

0.799

192795145189-3.1374.732
142265-1.132
10.81
137084-2.46428.71
147328-3.56015.73110594

2

R
First stage w. id. robust F
Observations

Changes within plant-productsStandard errors in parentheses, clustered at the state-industry level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01
Regression contains single-product plants only. The left-hand side is the plant’s inverse of the HHI of materials
shares (columns (5) to (8)); the right-hand side is log sales and controls. Columns (3) and (4) instrument log
sales by the log output tariff, columns (5) and (6) instrument log sales by the export demand shifter.

14

ω

ω

ω̃1

ω̃2

(a) Snake

ω̃1

ω

ω̃1

ω̃2

ω̃3

ω̃4

ω̃2

ω̃5

(b) Spider

ω̃3

ω̃6

(c) TreeFigure 4 Production Structures
The figure shows three different ways inputs (denoted by ω̃i ) can be combined to form
an output ω. In a snake, each stage combines one input with primary factors to produce
the stage output, and several of these stages are performed sequentially to produce thefinal output. In the spider, there is only one production stage which combines multiple
intermediate inputs to produce output. In general the production structures form a tree,
which combines multiple vertically related stages with the fact that more than one input
may be required to produce the stage output.

15

3

A Simple Model of Vertical Span

In this section, we describe a simple model of a firm’s vertical span of production. There are a large
number of industries. Each industry is comprised of firms that produce imperfectly substitutablevarieties. Firm j in industry ω produces a variety of ω, and is born with ex-ante productivity qj .
Each firm can, in principle, produce using different recipes that have different vertical spans. Eachfirm pays a convex cost to search for suppliers. Firms encounter potential suppliers according to amatching function.
The key mechanism operates through the decision to search for suppliers of different inputs.
Firms born with higher productivity or that expect higher demand will tend to search more, which
will be a source of increasing returns. As we will show, the vertical structure of the technology menu
makes the returns to searching across inputs asymmetric, and search decisions respond asymmetrically to changes in productivity and demand. This non-homotheticity implies that ex-ante heterogeneous firms select differentially into different vertical spans, generating the size/specialization
relationship emphasized in Section 2.1.
To build intuition, this section focuses on a simple version of the model. We focus mostly on
the problem of a single firm that can produce performing either one task or two vertically related
tasks arranged as a snake. In Section 5 we present a full quantitative model that allows for richer
production structures including trees and that can be taken to the data.

3.1

Environment: Simple Model

There are a large number of industries arranged in a circle. Each industry is indexed by ω. In
each industry there is a finite measure of firms that produce differentiated varieties. Each firm inindustry ω can produce in two ways, either using a variety in industry ω − 1 as an intermediate
input or using a variety in industry ω − 2 as an intermediate input. That is, a firm can perform one
task in production, purchasing a variety of ω − 1 and using labor to transform it into its variety oftype ω; or perform two tasks, purchasing a variety of ω − 2, using labor to transform it into ω − 1,and then using labor again to transform the product into its variety of type ω.
Firm j is born with an idiosyncratic Hicks-neutral productivity draw qj . Before producing, eachfirm j in industry ω exerts search effort hj1 and hj2 to find potential suppliers in industries ω − 1and ω − 2 respectively. Search yields a set of matches with potential suppliers. For each match,firm j draws a match-specific, input-augmenting productivity z that is specific to using supplier s’s
variety as an input. After searching, firm j also gets an idiosyncratic productivity shock specific to
the more upstream task, Bj . If firm j produces using supplier s in industry ω − 1 as an input, its
production function is1−αyj = qj alj1
[zj1s xj1s ]α ,whereas if it uses supplier s in industry ω − 2 its production function is
h
iα
1−α
1−α
yj = qj alj1
Bj alj2(zj2s xj2s )α16

where a ≡ (αα (1 − α)1−α )−1 is a normalizing constant. The nested structure of the production
functions captures the sequential nature of the two tasks in the production process.

3.2

Market Structure

There is a representative household with nested CES preferences. It consumes an aggregate of

 η
η−1
P η1 η−1ηindustry bundles, u ≡δu, where the industry bundle for industry ω is uω =
ω
ω
ω
ε


ε−1
ε−1
R
ε
where Jω is the set of firms producing varieties in industry ω and ε > 1. δωJω uωj dj
is a preference parameter that shifts the household’s demand for industry ω.
Firms sell their goods to the household and to other firms that are further downstream. We
assume in this section that firms price at marginal cost when selling to other firms.10 Firms engage
in monopolistic competition when selling to the household.
The mass of firms in each industry is endogenous. Upon entry, each firm draws its ex-ante
productivity q from a fixed distribution with CDF Q(·). We model entry by having a representative
entrepreneur that takes as given the distribution of profit within each industry, and chooses a massof entrants for each industry subject to a constant elasticity of transformation function
1max
Jω π̄ω − w
1 + 1/χ
{Jω }
ω
X

X

1+β
β

! β (1+1/χ)
1+β

Jω

ω

where π̄ω is expected profit for an entrants into industry ω, or, equivalently, average profit among
all entrants into the industry. Note that this nests free entry (β, χ → ∞) and inelastic entry(β = χ = 0) as special cases. We assume β < ∞, so that an increase in demand is not fully
absorbed by new entrants.

3.3

SearchWe assume that the search happens according to a matching function. In particular, if a firm j inindustry ω exerts search effort hj1 to find suppliers in industry ω −1, the arrival of matches depends
on hj1 M (Jω−1 ), where the matching function M is weakly increasing in the mass of firms |Jω−1 |.
Following the findings of Miyauchi (2018), we assume that the arrival rate increases with the massof potential suppliers, but congestion from others buyers that are also searching for suppliers does
not reduce the matching rate.11 If M is strictly increasing, then the matching function exhibits
increasing returns.
10With a notion of bargaining power as in Oberfield (2018), the assumption that each firm sets a price of marginal
cost in firm-to-firm trade is equivalent to the equilibrium in which buyers have complete bargaining power.
11
Miyauchi (2018) found that when a firm lost its supplier due to an unexpected supplier bankruptcy, the speedwith which the firm found a new supplier increased with density of suppliers, but was uncorrelated with buyer density.
In our view, this is the most credible evidence to date on matching functions in the context of firm-to-firm trade.

17

3.4

TimingFirms born with qj

Supplier set & productivity
draws realized

Search effort choice {hj ω̂ }ω̂

Entry choices

Production,
Consumption

Make/buy & supplier choices

All entering firms draw their Hicks-neutral productivity shifter qj . Then they simultaneously
decide how much to search, choosing hj1 and hj2 . Nature then reveals each firm’s set of potential
suppliers along with match-specific productivities and each firm’s task productivity Bj . Then all
firms set prices and make production choices (i.e. choosing production function and suppliers tominimize cost, and choosing quantities to maximize profit) simultaneously. Finally, production andconsumption occur.
gross
For firm j in industry ω born with productivity qj , let πωjbe the realized gross profit (grossof search costs). If that firm exerts search effort hj1 and hj2 , then E [πωj |qj , hj1 , hj2 ] is its expected
profit taking as given the search decision of all other firms, and expectations are taken over realizations of all firms’ matches and task productivities. We assume that the cost of search effort isisoelastic. Then the firm’s choice of search effort maximizes expected profit net of search costs:
h
i
gross
max E πωj|qj , hj1 , hj2 − w

hj1 ,hj2

3.5

k
k
h1+γ
h1+γ
.
j1 − w
1+γ
1 + γ j2

Functional Form Assumptions

We make some functional form assumptions that will help in the characterization of the equilibrium.
First, we assume that if firm j chooses search effort (hj1 , hj2 ), then the number of potential suppliers in industry ω − 1 with match-specific productivity greater than z is Poisson withmean hj1 M (Jω−1 )z −ζ , and the number of potential suppliers in industry ω − 2 with match-specificproductivity greater than z is Poisson with mean hj2 M (Jω−2 )z −ζ . We will use the shorthandm1 = M (Jω−1 ) and m2 = M (Jω−2 ) when there is no risk of ambiguity. In our numerical implementations we parameterize M as M (J) = J µ .
Second, we assume that the task productivities are independent and identically distributed
across firms and take the form Bj = ebj /ζ , where bj is a random variable with characteristic
Γ(1−it), or equivalently that ebj /α is an α-stable random variable. We discuss the role offunction Γ(1−αit)this functional form assumption in footnote 12 below.
Third, we assume that the distribution of productivity with which firms are born has a sufficiently thin tail: limq→∞ q

ζ 1+γ
γ

[1 − Q(q)] = 0. This assumption ensures that average profits remain

finite.

18

3.6

Characterizing the Equilibrium

We characterize the equilibrium by working backward. Each firm’s production cost exhibits constant returns to scale. Given its search effort and that of other firms, the realization of matches, andall firms’ production choices, let cj denote the realization of j’s unit cost. Let coj1 = mins∈Sj1 zpjss

be the effective cost delivered by the best supplier of input ω − 1, and coj2 = mins∈Sj2 zpjss be theeffective cost delivered by the best supplier of input ω − 2. The cost of producing intermediate ω − 1
in-house is cij1 = B1j w1−α (coj2 )α . Thus the effective cost of intermediate ω − 1 is cj1 = min{coj1 , cij1 }.
Finally, j’s unit cost is cj = q1j w1−α cαj1 . Let Fω (c) denote the cumulative distribution of unit costs
in industry ω.
We first characterize the possible realizations of firm j’s unit cost, given its choices of search
effort. We then use this to solve for the optimal search effort.
Proposition 1 For a firm j in industry ω with productivity qj that chooses search effort hj1 , hj2 ,(a) The effective cost of outsourcing intermediate input ω − k follows a Weibull distribution withcounter-cumulative distribution−ζζ
Pr cojk > c|qj , hj1 , hj2 = e−hjk mk c̄ω−k cwith c̄ω−k ≡Rc−ζ dFω−k (c)

− 1
ζ

.

(b) The effective cost of producing ω − 1 in-house follows a Weibull distribution with countercumulative distributionα α
1−α −ζ ζαPr cij1 > c|qj , hj1 , hj2 = e−hj2 m2 (c̄ω−2 w ) c .
(c) The conditional probability that the firm uses a supplier in recipe using ω − 1 is chosen is
hj1 m1 c̄−ζ
ω−1
α
α
α
1−α
hj1 m1 c̄−ζω−1 + hj2 m2 c̄ω−2 w

−ζ(d) The firm’s expected profit ishhi i− αζ 1−α 1−ε
gross
−ζ
ε−1
α
α
α
1−α −ζ
w
E πωj |qj , hj1 , hj2 = Aω δω qj
hj1 m1 c̄ω−1 + hj2 m2 c̄ω−2 w(ε−1)
where Aω ≡ upη pε−η
ω
εε

ε−1



Γ 1 − α ε−1ζParts (a) and (b) of Proposition 5 characterize the distribution of costs of different choices
conditional on qj and on search choices. Both costs, in-house production and outsourcing, follow a Weibull distribution with the same shape parameter, and the realizations are conditionally19independent.12 The probability that the firm chooses to outsource the input hence follows thewell-known expression that depends on the relative scale parameters (part (c)). Part (d) gives the
expected profit conditional on qj and search choices, which allows us to restate the search effort
choice problem as:
max Aω δω qjε−1
hj1 ,hj2h
 i− αζ 1−α 1−ε−ζααα1−α −ζ
w−w
hj1 m1 c̄ω−1 + hj2 m2 c̄ω−2 w

k
k
h1+γ
h1+γ
j1 − w
1+γ
1 + γ j2
(1)

The optimal choices of search intensity imply a non-homotheticity in input use:
Proposition 2 In industry ω, the probability of using ω − 1 is increasing in qj . Sales are positively
correlated with the decision to use ω − 1.
The empirical counterpart to Proposition 2 is a negative correlation between vertical span and size
within industries: firms are born with varying Hicks-neutral productivity qj , and choose search
efforts based on this draw. Firms born with higher productivity choose higher search intensities
and but also increasingly tilt their search toward suppliers in industry ω − 1, resulting in a higher
probability of choosing the production function in ω − 1. At the same time, those firms born withhigher productivity are likely to have more customers and sell more to the household, and so will
end up being larger on average.
What is the source of this nonhomotheticity? Mathematically, given the curvature in the
expected profit as a function of the search intensities, the returns to search diminish more quickly
for suppliers in ω − 2 than for suppliers in ω − 1 (see equation (1)). Hence, those firms that searchmore increasingly tilt their searches toward suppliers in ω − 1.
The intuition behind this result is subtle. ω − 2 is more complementary with labor than is ω − 1(because labor used for the upstream task has a unitary elasticity with ω − 2 but is a substitute ofω − 1). Labor is special because, unlike for intermediate inputs, the firm cannot reduce the wage.
A firm that increases search effort will gradually expand its expected expenditure share on ω − 1
12
The functional form for the distribution of the task-specific productivity Bj is chosen to solve the following
problem. The effective cost of purchasing input ω − 1 follows a Weibull distribution with shape ζ. The effective costof purchasing input ω − 2 is also Weibull with shape ζ. Without Bj , the effective cost of producing ω − 1 in-house
would be Weibull-distributed with shape αζ because ω −2 has an output elasticity of α in the production of ω −1. This
would be unfortunate because discrete choice problems with extreme-value random variables only work well when the
random variables have the same shape parameter. To make the discrete choice problem tractable, the distributionof Bj is chosen so that the cost of producing ω − 1 in-house (incorporating Bj ) follows a Weibull distribution withshape ζ.
Why does this particular functional form work? We build on the observation of Shanbhag and Sreehari (1977)
that if Z is a standard
exponential randomvariable and X is an α-stable random variable defined by the Laplace


α
Z α
transform E e−uX = e−u , then X
is also a standard exponential random variable. The proof is remarkably
simple:

 Z ∞
h
i
1/α
1/α
Pr ((Z/X)α > u) = Pr Z > u1/α X =e−u x d Pr(X ≤ x) = E e−u X = e−u .
0
α/ζ

1

Z
ζ
If we raise each side of the equation to the power 1/ζ, we obtain that X
α/ζ has the same distribution of Z . Inα/ζother words, the ratio of a Weibull-distributed random variable with shape ζ/α and the random variable X
is a
Weibull-distributed random variable with shape ζ. Bj in the model plays the role of X α/ζ .

20relative to ω − 2. This raises the return to searching for ω − 1 relative to the return to searchingfor ω − 2.13
An example might help to illustrate the mechanism. Consider a producer of polished diamonds
who faces the choice of using cut diamonds or rough diamonds as inputs.14 These two recipes
are substitutable, but each production step requires her to combine labor and the intermediate
input in an imperfectly substitutable manner. When faced with a higher qj , the producer would
exert a higher search effort for both cut and rough diamonds. She would, however, increase her
search effort for rough diamonds (the upstream input) proportionally less than for cut diamonds,
knowing that when she searches more for rough diamonds the cost of self-produced cut diamonds
falls proportionally by less, because the rough diamonds would still need to be combined with labor,
whose cost is not decreasing.

3.7

A Shift in Demand

We now consider a shift in household preferences that raises demand for the industry ω. The
implications are summarized by Proposition 3
Proposition 3 Suppose that entry is not completely elastic or inelastic (β ∈ (0, ∞)). An increasein δω causes• more entry in industry ω: Jω %• the price levels in industry ω falls: pω , c̄ω &13

To see this, consider a related simpler problem:

A firm produces with the production function y =
h

−ε

1+γ

h

1+γ

2
1
and 1+γ. An inqf (l, h1 x1 , h2 x2 ), faces isoelastic demand y = δp , and can choose h1 and h2 at costs 1+γ
crease in q raises incentives to search. Changing either h1 or h2 leads to a change in the marginal product of h1
and of h2 . To determine whether a firm born with higher productivity would tilt their search toward h1 , one must
compare (i) how raising h1 would change the marginal product of h1 relative to the marginal product of h2 , to (ii)how raising h2 would change the marginal product of h2 relative to the marginal product of h1 . These are encodedin the Morishima elasticities of substitution between h1 and h2 : A firm born with higher productivity would tilt its
search relatively more toward h1 (i.e. ddlnlnhq1 > ddlnlnhq2 ) if the direct Morishima elasticity of substitution of input 1 for
input 2 is higher than the elasticity of input 2 for input 1.
More concretely, suppose that f takes the nested CES form,(
y=q(A1 h1 x1 )

η−1
η

h
i φ η−1φ−1
φ−1 φ−1 η+ (A0 l) φ + (A2 h2 x2 ) φ

) η

η−1

.

Under parameter restrictions which guarantee the choices of search effort are interior, whether an increase in q leads
to a larger proportional increase in h1 or h2 depend on whether η ≷ φ. That is, η > φ implies ddlnlnhq1 > ddlnlnhq2 > 0
whereas η < φ implies ddlnlnhq2 > ddlnlnhq1 > 0. Importantly, the cost shares of each input play no role (i.e., the relative
magnitudes of A0 , A1 , A2 play no role).
In the model, expected profit takes a form similar to this nested CES production function, where the elasticity of
substitution of the inner nest translates to one (the elasticity of substitution between labor and input ω − 2 in the
upstream production function), while the elasticity across outer nests translates to 1 + ζ (the effective elasticity of
substitution across production recipes induced by discrete choice across Weibull distributed random variables withshape parameter ζ). Since 1 + ζ > 1, ddlnlnhq1 > ddlnlnhq2 > 0.
14
In the former case, the producer would just polish the cut diamonds, whereas in the latter case she would haveto do both cutting and polishing.

21

• the fraction of firms in industry ω using ω − 1 increases.
The proof shows that the positive demand shock sets in motion a sequence of events. First,
the demand shock raises profit. This increases entry, but since entry is not fully elastic, expecteddemand for each firm rises. Firms increase their search intensity. Both the increased entry andincreased search intensity lower the industry price index, due both to increased variety and to lower
cost for each q. In addition, with increased search intensity, firms tilt their search toward suppliers
in industry ω − 1, for the same reason as above.15
We think of Proposition 3 as speaking to the empirical results in Table ??, that demand shocks
affect the vertical span.

3.8

A Shift in Demand for an Upstream Industry

We next consider the impact of a shift in household demand for industry ω − 1.
Proposition 4 If δω−1 increases, then if γ is sufficiently large (search effort not too elastic):• more entry in industry ω − 1: Jω−1 %, cω−1 &• the fraction of firms in industry ω using ω − 1 increases• total sales in industry ω increase
The first result—that the increase in δω−1 raises entry in ω−1 and increase the value of searchingfor suppliers in ω − 1—is simply an application of Proposition 3. The increase in vω−1 would leadto a reduction in the distribution of cost even if those in ω did not change their search effort. Thisimplies that the price index pω declines. The fact that vω−1 increases raises incentives to search,
but the fact that pω declines reduces incentives to search. If γ is sufficiently large, we can show thatthe former dominates and all firms search more.16 When firms search more, search effort increases
more for suppliers in ω − 1 than for suppliers in ω − 2. This, along with the reduction in c̄ω−1increases the probability that firms use suppliers in ω − 1. Finally, the fact that c̄ω falls and pω
decreases means that total industry sales rise.
15

If entry were completely elastic (β → ∞), profit per firm would be independent of δω , as the number of firms
would increase until profit per firm matched the opportunity cost of entry. With the same profit per firm, incentives
to search would not change, and the fraction of firms in ω using suppliers in ω − 1 would not change. However, Jω
would increase, and gains from variety would imply that pω would fall. If m(Jω ) is strictly increasing in Jω , then vω
would rise as well. If entry were completely inelastic, (β = 0), the number of firms would be fixed by construction.
However, the increased demand would increase incentives to search, leading to lower price index pω and higher vω .
16
For any value of γ, the largest firms will become more likely to use ω − 1 as a supplier. But if γ is low, thesmallest firms may not. In particular, if the firms born with lowest productivity q are extremely unlikely to use a
supplier in ω − 1, then the decline in c̄ω will have less impact on profitability than the decline in pω . If search effort issufficiently elastic (i.e., γ small enough), then the large firms would increase their search enough so that the decline
in pω is large. Then the low-q firms may reduce search effort, tilting their search more toward suppliers in ω − 2.

22

d log u

d log Jd log h1 d log h2,
d log L d log L
0.25

d log L
1.0

d log L
0.30
0.25

0.8

0.20
0.20

0.6

0.15
0.15

0.4

0.10

d log h10.10

d log L

0.2

0.05

0.0

0.2

0.4

0.6

Share
1.0 Outsourcing

0.8(a) Response of Income/capita

d log h20.05

d log L

0.0

0.2

0.4

0.6

0.8

Share
1.0 Outsourcing

0.0(b) Response of Search Effort

0.2

0.4

0.6

0.8

Share
1.0 Outsourcing(c) Response of Entry

Intermediates
Share
0.8
Share of Production Costd log Share Outsourcing
d log L
0.4

0.7

Share of Prod + Search Cost

0.3
0.6
0.2

0.5

0.1

0.0

0.4

0.2

0.4

0.6

0.8

Share
1.0 Outsourcing

0.0(d) Response of Share Outsourcing

0.2

0.4

0.6

0.8

Share
1.0 Outsourcing(e) Cost Share of Intermediates

Figure 5 Elasticity to an Increase in Scale
Note: This figure plots the respective elasticities of income per capita, search effort, entry, andthe share of firms outsourcing to an increase in the labor force. The final plot shows the shareof expenditures on intermediate inputs, both as a fraction of total production cost and as afraction of total production and search cost. The values of parameters are α = 0.75, ε = 2.5,
ζ = 1.5, χ = 0.3, γ = 3, and µ = 0.6.

3.9

An Economy-wide Increase in Scale

We now illustrate some key mechanisms in the model using a numerical example. We study the
impact of an increase in the labor force, L. This could be interpreted as population growth or
growth in efficiency units of labor per person. In particular, we study the elasticity of output per
worker, entry, search effort, and vertical span to L. For this exercise, we assume that all industries
are symmetric, and that the distribution of productivity within each industry is degenerate, so that
all entrants have the same ex-ante productivity q. In this simplified environment, it turns out thatthese elasticities can be expressed in terms of parameters and a single endogenous statistic, O, theshare of firms that choose a vertical span of 1 (or, alternatively,the total spending on intermediate
inputs as a share of production costs which is equal to Oα + (1 − O)α2 ).
First, in the special case in which there is a fixed set of firms (χ = 0) and fixed search effort
log u(γ → ∞), income per capita is invariant to scale: dd log
L = 0. Second, suppose that search effort

remains fixed (γ → ∞) and allow for entry (χ > 0), and allow gains from variety to consumption,
except in returns to search (µ = 0). In this case, there is only the usual returns to scale that would
log u
χ
1
be present in a standard one-sector model with increasing varieties, dd log
L = χ+1 ε−1 .

Consider now the case of interest, with either endogenous search effort (γ < ∞) and/or entry
and returns to scale in search (µ > 0, χ > 0). The economy exhibits increasing returns to scale:panel (a) of Figure 5 shows that the elasticity of income/capita to the size of the labor force ispositive. When the labor force grows, there is more entry and firms search more, as shown in panels

23

(b) and (c). Both mechanisms raise income per capita, and both reduce firms’ vertical spans, as inpanel (d).
These effects are stronger when firms are more specialized, indicating the presence of an inputoutput multiplier effect. From the perspective of canonical input-output models (e.g. Long andPlosser, 1983, Acemoglu et al., 2012 or Jones, 2011, 2013) this multiplier effect may appear puzzling,
since input-output multiplier effects are usually only present with increases in neutral productivity,
not labor-augmenting productivity.17 In our model, there is a sense in which the expansion of labor
supply is similar to an increase in labor-augmenting productivity, for which there is no multipliereffect. But, differently from canonical models, some of that labor gets channeled into search effort
and into the creation of new firms. Both the search effort and the gains from additional potential
suppliers act like neutral shifters of those suppliers’ costs. That is, they both reduce the effective
cost of suppliers in a neutral way, not just augmenting the labor of those suppliers.
− 1
R −ζ
More formally, recall that average effective cost of an intermediate input is c̄ ≡c dF (c) ζ .

−α/ζIn this simple economy, this cost index satisfies c̄ ∝ w1−α h1 M (J)c̄−ζ + hα2 M (J)α (w1−α c̄α )−ζ.
From this, we obtain that the expected effective cost of intermediate inputs falls with entry andsearch effort:

d log c̄/w
s 1=−
d log L1−sζd log Js1 d log h1 s2 d log h2µ++
d log Ls d log L
s d log L

where s is the expected share of production costs spent on intermediate inputs, while s1 and s2 are
the expected shares of production costs spent on inputs from ω − 1 and ω − 2 respectively, so thats = s1 + s2 . The reduction in expected cost is coming fully from search efficiency and search effort.
Panel (d) shows that firms specialize more as the labor force expands. To see why, recall that
O is the fraction of firms that choose to outsource input ω − 1 rather than produce in-house, which
can be expressed as
h1 M (J)c̄−ζ
O
= α
.
1−O
h2 M (J)α (w1−α c̄α )−ζFirst, as firms exert more search effort, they tilt their search effort toward less distant inputs: panelh1
d log h2(b) shows that ddlog
log L > d log L . Second, since α < 1, firms tend to become more specialized as

search effort rises, when the matching rate increases, or when the cost of intermediates falls relative
to the wage. In fact, either of the first two forces would reduce the cost of intermediates relative
to the wage, amplifying the direct impact.18
17

Consider an economy with roundabout production so that output can either be consumed (U ) or used as anintermediate input (X): the resource constraint is Y ≥ X + U . Suppose that the production function is Y =A(BL)1−α X α , so that B is labor-augmenting productivity and A is neutral productivity. In a competitive equilibrium,d log UU1
= 1 and dd log= 1−α.
d log B
log A
18Interestingly, while endogenous search effort is needed for larger firms to be more specialized in the cross-section,
it is not needed for specialization to rise with development; endogenous entry (χ > 0) and a matching rate that rises
with entry (µ > 0) are sufficient.

24

4

Alternative Mechanisms

4.1

Cost of Coordination

Becker and Murphy (1992) emphasized the cost of coordinating tasks as an important delimiter ofthe range of activities done in a single team. An implication they draw is that higher demand (orproductivity) larger firms find it in their interest to pay the coordination cost to do more activities.
In principle there are many types of coordinating costs. Here we discuss a few specific versionsof a coordinating cost, and discuss why we think they do not explain our results about size andvertical specialization.
One such cost of coordination is a fixed cost that is increasing in the number of suppliers (aspecial case, of course, is a separate fixed cost for each supplier).19 In such a setting, an increase indemand would make a firm more willing to pay the cost to engage with more suppliers. However,as discussed in Section 2.6, an increase in demand tends to cause firms to reduce the number ofsuppliers.
Alternatively, it is possible that there is a cost of coordinating many activities within a firm.
For example, suppose a firm needed to do a range of horizontal tasks; it could perform each task inhouse or outsource it, but there was a fixed cost that increased with the number of tasks performedin house (with, again, a special case of there being a fixed cost for each task).20 The predictionof such a model is that an increase in demand would tend to make the firm do more tasks inhouse. This would reduce the share of intermediate inputs. However, as discussed in Table XVI,
an increase in demand tends to raise a firm’s cost share of intermediate inputs.
Finally, if a firm faced a fixed cost that increased with the number of tasks performed in-house,but the tasks were arranged vertically as a snake, an increase in demand would cause the firm to
do more tasks in house, increasing vertical span.

4.2

Limited Span of Control

Another factor that might affect vertical specialization is a limited span of control, as in Lucas(1978). For example, Bloom et al. (2013) found a correlation between the number of male family
members and the number of plants operated by the firm.21 While we are unaware of similar evidence
indicating that span of control considerations limit plant size (or the size of single-plant firms), it
is plausible that some plants are constrained in the quantity of labor that can be hired.
If a firm’s employment is constrained, an increase in demand could induce a decline in vertical span. If demand rises, the firm might choose to meet that demand by shifting employment
19

Such a production structure has been adopted by a large number of papers, including Acemoglu, Antràs andHelpman (2007), Lim (2018), Huneeus (2018), Tintelnot et al. (2018). In general, such models do not have clear
implications about how vertical span would change with demand.
20
For example, in Chan (2017), a firm faces a fixed cost of performing each task in house and a different fixed costof outsourcing a task.
21
Bloom et al. (2013) found that the number of “family members [that] could currently work as directors in thefirm” accounts for 10% of the variation in the number of plants operated by a firm.

25

to a downstream task and outsourcing the upstream task. Indeed, the vertical structure of 3.1
augmented with a constraint on labor would lead to such a prediction.22
One difference between span of control limitations and endogenous search for suppliers is thepredicted response of marginal cost to changes in demand. A limited span of control would suggestthat marginal cost would rise with an increase in demand, whereas search for suppliers would imply
that marginal cost would fall. In Appendix C.5 we study whether an increase in foreign demandleads to a change in the plants’ domestic sales. We find that—in the short subsample of the datain which we can separately see domestic and foreign sales—that the response of domestic sales topositive demand shocks is either positive or statistically insignificant. These results suggest thatplants do not operate under decreasing returns, and that, if anything, unit costs fall with size.23
Our findings are also consistent with those of other papers that study the sales response of firms
in one destination to demand shocks in another destination. Alfaro-Urena, Manelici and Vasquez(2022) find that firms in Costa Rica increase their sales to other firms after starting to supply
multinational firms. Albornoz, Brambilla and Ornelas (2021), who study Argentine firms after thesudden removal of preferential tariffs by the US following an intellectual property dispute. Theyfind that affected firms reduced sales to the US but also reduced sales to other export markets,consistent with marginal cost declining with size.24

4.3

Other Mechanisms

Our empirical findings suggest that larger firms pay lower prices, on average, than smaller firms. We
have posited that such a relationship arises naturally if firms can exert effort to such for suppliers.
An alternative mechanism that yields similar cross-sectional results is that sellers practice second
degree price discrimination and offer quantity discounts, as in Meleshchuk (2019).25

5

A Quantitative ModelIn this section we extend the baseline model to so that production modules use multiple inputs.
Household preferences, market structure, and timing are the same as in the simple model.
22
Both a quantity constraint on labor and increase in search effort would lead to a decline in the cost of intermediates
relative to the shadow cost of labor, leading to substitution toward the downstream intermediate which is a better
substitute for labor (a higher Morishima elasticity of substitution).
23
In principle, a more direct way to gauge this mechanism would be to assess whether unit costs fall in response
to a demand shock. We can construct unit costs from total costs and reported output quantities for some of theplants. Unfortunately the quantities reported in the ASI are so noisy that we cannot detect a response of quantities
to demand or supply shocks. Part of the issue could be that the units in which quantities are denominated changeover time.
24
Our approach here is similar to studies that estimate industry-level returns to scale using trade data, e.g. Bartelme
et al. (2019) and Costinot et al. (2019).
25
Monopsonistic behavior in intermediate input markets, on the other hand, is not consistent with marginal costdeclining with size. Monopsonistic behavior requires an upward sloping supply curve for each intermediate input.
While this could be consistent with larger firms attaining larger markdowns on intermediate input purchases, it alsomeans that when the firm is larger and chooses to buy larger quantities of intermediate inputs, it must pay more forthe marginal unit.

26

5.1

Recipes and Production
Output

Make

Make

Make

or

or

or

or

Buy

Buy

Buy

Buy

Buy

Make

Make

Make

Make

Make

Make

or

or

or

or

or

Buy

Make
or

Buy

Buy

Buy

Buy

Buy

Buy

A firm’s production possibilities can be described by a tree of production modules. A production
module to make ω is a production function that uses a particular set of inputs Ω̂ω and labor. The
firm can either buy each input ω̂ ∈ Ω̂ω from a supplier or it can produce it in-house with anotherproduction module to produce ω̂. We say that Ω̂∞ω is the set of all inputs that are nodes in the tree
formed by the production modules.
The firm searches for suppliers. For each input ω̂ ∈ Ω̂∞ω , its search effort hω̂ delivers a set of
potential suppliers Sj ω̂ from whom it may purchase input ω̂. For each potential supplier s ∈ Sj ω̂ ,the firm draws a match-specific productivity zjs ; with unit price ps , the effective cost of purchasingthat input from that supplier is zpjss . Since the buyer will choose the supplier that delivers the lowest
effective cost, the cost of outsourcing input ω̂ is cojω̂ = mins∈Sj ω̂ zpjss .

The firm can also produce some inputs in-house using a production module. A production
module to produce ω̂ in-house delivers an effective cost cij ω̂ (which we characterize below). The
firm’s effective cost of input ω̂ is thus
cj ω̂ = min cij ω̂ , cojω̂
We assume that each production module is Cobb Douglas. The production module to produce
an input ω delivers an in-house unit cost ofcijω =

1 αω Y αωω̂
w l
cj ω̂
Bjω
ω̂∈Ω̂ω

Bj ω̃ is a random firm-module specific productivity shifter and αω̂ω are the output elasticities of amodule input ω̂.
A firm born in industry ω also has a core production module from which it produces ω. It
cannot outsource the production module, and there is no productivity shifter B. This production

27

module takes the form

1 ω Y αωω̂cj ω̂cjω = wαlqω̂∈Ω̂ω

where q is a firm-specific productivity shifter. Every module has constant returns to scale, αlω +Pωωω̂∈Ωω αω̂ = 1, and labor is essential, αl > 0.

5.2

Entry

The mass of firms in each industry is endogenous. We model entry by having a representative
entrepreneur that takes as given the distribution of profit within each industry, and chooses a massof entrants for each industry subject to a constant elasticity of transformation functionkEJω π̄ω − w
max
1 + 1/χ
{Jω }
ω
X

X

 1
E −β

δω

1+β
β

! β (1+1/χ)
1+β

Jω

ω

where π̄ω is average profit among all entrants into industry ω, and δωE is a shifter of the supply ofentrants for industry ω. The solution isJω = δωE π̄ωβ π̄ χ−β (wk E )−χ
where π̄ ≡

P

E 1+β
ω δω π̄ω 11+β

is an index of average profit. β indexes the ease of shifting entry

between industries, and χ indexes the response of total entry to labor devoted to entry. Note that
this specification nests free entry with β, χ → ∞, in which case equilibrium requires π̄ω = wk E , ∀ω,and exogenous entry with β, χ → 0, in which case Jω = δωE . We assume that β < ∞, so that anincrease in industry demand is not fully absorbed by new entrants.
Each new entrant draws the productivity of its core module, q, from an industry-specific distribution with distribution function Qω (q).

5.3

Functional Form Assumptions

In this section, we make several functional form assumptions that will be useful in facilitating acharacterization of the equilibrium.
First, we specify the distribution of match-specific productivities. We assume that if firm jchooses search effort hj ω̂ for some input ω̂ for use in a module to produce ω, then the number ofpotential suppliers of varieties of ω̂ with match-specific productivity greater than z is Poisson withmean hj ω̂ mωω̂ z −ζ . mωω̂ is a measure of matches, which we will later endogenize.
Second, we specify the distribution of module-specific productivities, which are independent
across firms and modules. For a module to produce ω with output elasticities αlω , {αω̂ω }ω̂∈Ω̂ω , theproductivity shifter Bjω takes the form eb/ζ with b drawn from a distribution with characteristic

28

function26QΓ(1 − it)ωω̂∈Ωω Γ 1 − αω̂ it

Third, we assume that the number of production modules in the tree is finite.
Finally, we assume that the distribution of productivity with which firms are born has a sufficiently thin tail: limq→∞ q

5.4

ζ 1+γ
γ

[1 − Qω (q)] = 0.

Characterizing the Equilibrium

We characterize the equilibrium by working backward. We normalize the wage to 1.
We first characterize possible realizations of firm j’s unit cost, given its choices of search effort.
We then use this to restate the firm’s problem in order to solve for its choice of search effort.
Proposition 5 For firm j in industry ω with productivity qj that chooses search effort {hj ω̃ }ω̃∈Ω̂∞ ,ω

• The effective unit cost of outsourcing input ω̂ ∈ Ω̂ω̃ follows a Weibull distribution with countercumulative distribution−ζ ζPr(cojω̂ > c) = e−hj ω̂ mω̃ω̂ c̄ω̂ cwith c̄ω̂ ≡R

− 1c−ζ dFω̂ (c) ζ and Fω̂ (·) is the distribution function of unit cost among firms inindustry ω̂.
• For any input ω̂ ∈ Ω̂ω̃ that can be produced in-house, the effective unit cost of producing it
in-house follows a Weibull distribution with counter-cumulative distribution−ζ ζ
P cij ω̂ > c = e−Tj ω̂ cwhere {Tj ω̃ }ω̃∈Ω∞are defined iteratively asωω̃

Tj ω̃ = wαω̃li− αω̂Y hζ−ζ−ζhj ω̂ mω̃ω̂ c̄ω̂ + Tj ω̂ω̂∈Ω̂ω̃

where we use the convention that Tj ω̂ = ∞ if the input is a leaf of the production tree (i.e., if
in-house production is infeasible).
• The probability of outsourcing input ω̂ ∈ Ω̂ω̃ conditional on using it in production ishj ω̂ mω̃ω̂ c̄−ζω̂−ζhj ω̂ mω̃ω̂ c̄−ζω̂ + Tj ω̂
26(2)

There are two special cases in which such a random variable has a well-known name. If the recipe uses nointermediate inputs, b follows a Gumbel distribution. If the recipe uses exactly one intermediate input with output
ωω
elasticity αω̂, e−b follows a one-sided Levy-stable distribution with characteristic exponent αω̂, which has the Laplaceihρα−ub̃ji−u ω̂transform E e=e. In Appendix E we show such a random variable exists, building on Shanbhag, Pestanaand Sreehari (1977) who show the existence of a closely related random variable.

29This is independent of the firm’s cost or of the probability of outsourcing other inputs.
• The firm’s expected profit is1−εAω δω qjε−1 Tjω
−

X

wk

ω̂∈Ω̂∞
ω(ε−1)
where Aω ≡ upη pε−η
ω
εε

ε−1

Q

h1+γ
j ω̂
1+γ



ω ε−1 .
Γ
1
−
α
ω̂ ζ
ω̂∈Ω̂ω

An implication of the functional form assumptions is that the effective unit cost of outsourcing
each input ω̂ ∈ Ω̂ω̃ follows a Weibull distribution with scale that depends on search effort and the
27 In addition, conditioning on search effort
availability of good suppliers, hj ω̂ mω̃ω̂ c̄−ζω̂ and shape ζ.

across all inputs, the unit cost for in-house production of each input follows a Weibull distribution
with shape parameter ζ and scale Tjζω̂ .28 Because the two Weibull distributions share the same
shape parameter, the usual discrete-choice logic implies that a simple expression for the probabilityof outsourcing a particular input, (2).
Corollary 1 Given search choices, the industry average cost index satisfies
c̄−ζ
ω =

Y

Z

Γ (1 − αω̂ω )q ζ Tω (q)−ζ dQω (q)ω̂∈Ω̂ω

and the industry price index satisfies
p1−ε
=
ω

Z
Y 
ε
ωε−1
Jω
Γ 1 − αω̂q ε−1 Tω (q)1−ε dQω (q)ε−1ζω̂∈Ω̂ω

5.5

Optimal Search Effort

With this, we can characterize the optimal choice of search effort. The firm’s problem can be
expressed as a recursive cost-minimization problem. In particular, for the module to produce ω,define the cumulative search cost function
Kω (Tω ) =Xmin{hω̂ }ω̂∈Ω̂∞ω27ω̂∈Ω̂∞
ω

k

h1+γ
ω̂
.
1+γ

The logic closely follows Oberfield (2018).
Why does the cost of in-house production follow a Weibull distribution? We show this by induction, starting
upstream. For any input that is a leaf, the effective cost of the input follows a Weibull distribution with shapeparameter ζ. Consider module to produce ω̃, with inputs Ω̂ω̃ . The cost of purchasing any input from a supplierfollows a Weibull distribution with shape ζ. Suppose that unit cost of in-house production of input ω̂ follows a
Weibull distribution with shape ζ. Since the firm will procure ω̂ in the way that delivers the lowest cost, and thefamily of Weibull distributions with the same shape is closed, the cost to the firm of procuring the input will be
Weibull with shape ζ. The cost for the module is the product of the cost of each input, raised to the respectiveω̃
output elasticities {αω̂}ω̂∈Ω̂ω̃ , multiplied by the random variable Bj ω̃ . While the product of independent Weibulls
is not Weibull, the distribution of Bj ω̃ is reverse engineered so that the resulting unit cost of ω̃ follows a Weibull
distribution with shape ζ.
2830This is the minimal search cost across all inputs upstream from the module required to deliver a
distribution of unit cost for the module with scale parameter Tω . With this notation, the firm’s
problem can be expressed as
1−εmax Aω δω qjε−1 Tjω
− wKω (Tjω ) .
Tjω

Further, the cumulative search cost function Kω has an iterative representation

X  k
1+γ
h
+ Kω̂ (Tω̂ )Kω (Tω ) =min
1 + γ ω̂{hω̂ ,Tω̂ }ω̂∈Ω̂ωω̂∈Ω̂ω

subject toTω ≥ w

αω
l

Y h

−ζ
hω̂ mωω̂ c̄−ζω̂ + Tω̂

i− αωω̂ζω̂∈Ω̂ω

The next lemma derives some useful properties of the cumulative search cost functions, Kω .
Lemma 1 Each Kω (·) is such that Kω (0) = 0, is strictly decreasing, strictly convex, twice continuously differentiable, and the policy functions {hω̂ (·), Tω̂ (·)}ω̂∈Ωω are each differentiable and respectively strictly decreasing and strictly increasing in Tω .
This Lemma is helpful in characterizing several features of firms’ choices.
Proposition 6 Among firms in ω, those born with higher productivity choose higher search effort
for all nodes. Further, for any input for which it is feasible to produce in-house, the probability ofoutsourcing the input is strictly increasing in q.
The proposition follows from Lemma 1 and the fact that a firm born with higher productivity
would choose to invest more in lowering its cost by choosing lower Tω . The proof of Lemma 1 notes
first that, for any module, cost minimization implies that lowering its unit cost involves searching
more for direct suppliers and for suppliers further upstream. That is, lower Tω corresponds to both
higher hω̂ and lower Tω̂ for each of the module’s inputs ω̂ ∈ Ω̂ω . Second, for each of input for
which in-house production is feasible, hω̂ rises proportionally more than Tω̂−ζ ,d ln hω̃d ln Tω−ζ>d ln Tω̂−ζd ln Tω−ζ.

This happens because the cost of finding suppliers rises less steeply than the cost of reducing thein-house production cost by searching for suppliers further upstream. The critical step is that
producing in-house requires labor, and searching for suppliers further upstream cannot reduce thecost of labor used in-house, whereas outsourcing an input replaces labor. The labor required to
produce in-house ultimately limits the effectiveness of reducing cost by searching for upstream
suppliers relative to doing so by searching for suppliers further downstream.29
With this, we can derive some cross sectional predictions.
29

Another way to see the role of labor is to consider the following thought experiment. For a module to produce
d ln T−ζd ln hω̂ω̂ω, no labor was required to produce in house, so that αlω̂ → 0 for all ω̂ ∈ Ω̂∞ω . In that limit,−ζ →−ζ , i.e.,d ln Tωd ln Tω
production becomes homothetic.

31

Proposition 7 In industry ω, the probability of outsourcing any intermediate input (conditionalon using it) is increasing in sales.
The empirical counterpart to Proposition 7 is a negative correlation between vertical span andsize within industries, as discussed in Section 2.3. The leap from the previous proposition is movingfrom q to sales. Firms with higher q search more and tend to have lower cost, enabling them to sellmore, on average, to both the household and to other firms. To prove the result, we simply notethat conditioning on q, a firm’s sourcing decisions are independent of its sales.

6

Estimation and Quantitative Evaluation (In Progress)

6.1

Solving the Model

Before describing the algorithm, we collect the model’s key equations in one place.
For a firm born with productivity q, the firm’s problem gives its profit and the first orderconditions give its choices:πω (q) = max δω Aω q ε−1 Tω1−ε − wKω (Tω )
Tω

where

(ε − 1)ε−1
Aω = upη pε−η
ω
ε

Y

ε



ωε−1
Γ 1 − αω̂,ζω̂∈Ω̂ω

the industry price index satisfies
p1−ε
=
ω

 Z
Y 
ε
ωε−1
Jω q ε−1 Tω (q)1−ε dQω (q),
Γ 1 − αω̂ε−1ζω̂∈Ω̂ω

and the cumulative search cost function Kω for module to produce ω can be expressed iteratively
as


X  k
1+γ
+ Kω̂ (Tω̂ )Kω (Tω ) =minh
1 + γ ω̂{hω̂ ,Tω̂ }ω̂∈Ω̂ωω̂∈Ω̂ω

subject toTω ≤ w

αω
l

Y h

−ζ
hω̂ mωω̂ c̄−ζω̂ + Tω̂

i− αωω̂ζω̂∈Ωωwhere the average cost index satisfies
c̄−ζ
ω =

Y

Γ (1 − αω̂ω )Z

q ζ Tω (q)−ζ dQω (q).

ω̂∈Ω̂ω

Equilibrium entry decisions satisfy
Jω = δωE π̄ωβ π̄ χ−β (wk E )−χ32with π̄ =

P

E 1+β
ω δω π̄ω 11+βand π̄ω =Rδω Aω q ε−1 Tω (q)1−ε − wKω (Tω (q)) dQω (q).

Total spending up equals total revenue, which, in turn, equals the total wage bill for variableεproduction times the markup, or up = ε−1
wLP roduction .

The labor market clearing condition is
Lproduction + LEntry + LSearch = L
Total labor used for search and for entry across all industries are
SearchL=

X=
Jω L̄Searchωω

LEntry =k E  π̄1 + 1/χ wk E

X

ω
1+χ

Z
Jω

Kω (Tω (q)) dQω (q).

The Algorithm:
In the model, the structure of production is a directed acyclic graph. Our approach to solving
the model will be to begin upstream to solve the decisions of firms and then iteratively work ourway downstream. As we describe below, we can do this conditional on a single general equilibrium
variable.
π̄
1. Guess u(p/w)η and w. These variables summarize general equilibrium considerations, respec-tively shifting industry demand curves and industry entry schedules.
2. Proceed iteratively, starting with the most upstream industry to solve for industry variables
and cost functions and then proceed downstream. For industry ω, the information neededfrom any upstream industry ω̂ ∈ Ω̂ω is the cost index c̄ω̂ , the matching efficiency, mωω̂ , and, if
it possible to produce ω̂ in-house, the cost function Kω̂ (·). We then solve for the key objects
for industry ω:(a) First, solve for the cost function Kω (·). We first convert the cost to units of labor.
Given the wage, w, let K̃ω (T̃ ) ≡ Kω (wT̃ ), so that T̃ is in units of labor. Then K̃ω canbe defined recursively asK̃ω (T̃ω ) =

 
X  k
1+γ
h
+ K̃ω̂ T̃ω̂
1 + γ ω̂min{hω̂ ,T̃ω̂ }ω̂∈Ω̂ω ω̂∈Ω̂ωsubject to

αω

− ω̂
 c̄ −ζ
Y 
ζ
ω̂
−ζ
T̃ω ≤
hω̂ mωω̂+ T̃ω̂wω̂∈ΩωThe cost of any primary input is exogenous, and for any input ω̂ that is further upstream,
we have already solved for K̃ω̂ (·) and c̄wω̂ from the previous step or, if a primary input,
from technology.
33

(b) We next solve for Awω . Given Awω and K̃ω , we can solve for T̃ω (q) = Tωw(q) for each
and Tωw(·) are sufficient for expected profit for entrants, π̄wω . Note that Awω can
− ε−η
ε−1 R ε−1  Tω (q) 1−εEδωAωπ̄ χ−β π̄ω βηdQ(q)be expressed as w = u(p/w) (kE )χ w
q
Xω ,
ω
w
w
η−1
ε−η
1−ε+
nQ

o

1−ε 1
ε−1
ε
ω ε−1
.
Γ
1
−
α
where Xω ≡ ε−1ω̂ ζεω̂∈Ω̂ωq.

Aωw(c) Tωw(·) is also sufficient to recover the average cost index relative to labor, c̄wω .
β π̄ χ−β E −χ(k ) .
(d) The measure of firms in industry ω is Jω = δωE π̄wωw(e) With this, we can compute the industry price index pwω , and total labor used for searchacross all firms in the industry.
3. Solve for remaining aggregates(a) Once we have π̄wω and pwω for all industries, we can compute the average profit index
P
 1
P
 1

pω 1−η 1−ηπ̄
E π̄ω 1+β 1+β and the aggregate price index p =
=
δ
δ
.
ω ω
ω ω w
w
w
w

ε
(b) We can recover u from up = ε−1
w L − LSearch − LEntry , where LSearch was computed
1+χ
kE
π̄
separately for each industry and LEntry = 1+1/χ.
wkEπ̄
and u(p/w)η match the initial guesses.
4. Check that w

6.2

Estimation of Model Parameters

6.2.1

Constructing the Production Tree

We take the model to the manufacturing data by assuming that each product ω corresponds to a
five-digit product code. We proceed in the following steps. The details are in Appendix A.4.
1. We set some of the industries ω to be leaf industries, i.e. industries which produce only
using labor. In the data, this corresponds to product codes that fall outside of manufacturing(i.e. agriculture, mining) as well as some goods where consumption vastly exceeds domestic
production. These industries are assumed to have a final demand share of zero, and a fixedprice index of pω = 1.
2. We then construct a directed acyclical graph of input-output relationships between products.
Starting with leaf industries, we find all goods that produced only using leaf industry inputs,
then all goods that are produced from those goods and leaf goods, etc. In order to allow
the construction of such a graph without any cycles, we treat some less important observed
inputs as primary inputs (cf. Tintelnot et al. (2018)).30
3. Starting again upstream, we recursively calibrate the production function parameters αω̂ω foreach module ω and each direct input ω̂ to match the aggregate expenditure on direct inputs
and inputs upstream from them. We calibrate αlω to match the residual expenditure on
30

Some large firms span many products, making it more likely that cycles arise. In contrast, the graph we construct
is based on the production functions of plants that produce single products. As a result, fewer links need to be severed.

34

primary inputs, netting out primary factor expenditure associated with intermediate input
expenditures from non-direct inputs (which are determined by the α’s of modules further
upstream).
6.2.2

Calibration of Shifters

In this section, we take as given the production trees and the elasticities γ, β, χ, ζ, ε, η, φ and show
how the we pin down all of the shifters in the economy.
We have data on shares of each industry in final consumption, HHω , the measure of firms ineach industry, Jω , and for each of an industry’s direct inputs, the fraction of firms in an industry
that purchase the input from a supplier, Ōω̂ . We also can compute each firm’s value added andwage bill.31
1. Normalizations and choice of units(a) We choose units of labor so that L = 1.
(b) We normalize each qω to unity.32(c) We normalize the cost of search parameter k to unity.33(d) For any leaf input, normalize c̄ω and mωω̂ to equal the wage.34(e) We normalize the sum of the preference shifters and entry shifters to unity:
P Eω δω = 1.

Pω δω =

2. Next, we infer LP roduction , LSearch , and LEntry . First, the sales to the household is equalεto value added. Since firms charge the usual markup of ε−1, the share of labor used in
Value Addedε
production satisfies wL
P roduction = ε−1 . Second, we assume that labor used for entry does

not appear in our data, so that the remaining labor in the wage bill represents search forsuppliers: wLSearch = Wage Bill − wLP roduction . Third, note also that the P
ratio of profit to
J π̄

ω ω
expenditure on entry depends on the rate of diminishing returns to entry: wLωEntry= 1 + χ1 ,and that total profit is the difference between aggregate Value Added and the Wage Bill.
Since we have chosen units so that L = 1, these together implyχχ+1 (Value Added − Wage Bill)χ1
χ+1 Value Added + χ+1 Wage Billε−1
ε Value Added
LP roduction = χ1χ+1 Value Added + χ+1 Wage Bill
Wage Bill − ε−1
ε Value Added
LSearch = χ1χ+1 Value Added + χ+1 Wage Bill

LEntry =31

Note that herethe wage bill corresponds to total expenditure on primary inputs.
A model with qω , δω , {mωω̂ } is equivalent to one with qω0 = 1, δω0 = δω qωε−1 , and m0ωω̂ = mωω̂ qω̂ζ .
33
A model with k and {{mωω̂ }} is equivalent to one with k0 = 1 and m0ωω̂ = mω1ω̂ .

32

k 1+γ

34

Since production of any good is ultimately Cobb-Douglas across primary inputs, changing the cost of any primary
input would simply be absorbed by changes in the demand shifters {δω }.

35

3. Calibrate {mωω̂ } proceeding iteratively, starting with the most upstream industries.
(a) Consider industry ω. For each direct input ω̂ ∈ Ω̂ω , we already have the cumulative
search cost function, Kω̂ .
(b) For each non-leaf, direct input, ω̂, guess mωω̂ . We already haveRq ζ Tω̂ζ dQω̂ (q) from theprevious step. Thus we have vωω̂ .
(c) For each leaf direct input, ω̂, we have normalized c̄ω̂ = 1 and mωω̂ .
(d) Given mωω̂ c̄−ζω̂ for each direct input and Kω̂ for each non-leaf direct input, we can solve
for Kω .
1−ηpω(e) We next solve for pwω . We guess a value for pwω . Then, noting that HHω = δωp1−η
andεup = ε−1
wLproduction , imply thatεp1−η η ε−ηwLproduction pε−1
= HHωδω upη pε−η=HH
ω 1−η up pω
ω
ω
ε
−
1
pω
With this, we can solve for Aω , which satisfies(ε − 1)ε−1 Yωε−1δω Aω =
Γ 1 − αω̂δω upη pε−ηωεεζω̂∈Ω̂ωε(ε − 1)ε−1 Y
ωε−1Γ 1 − αω̂
HHωwLproduction pε−1
=
ω
ε
ε
ζ
ε−1

(3)

(4)

ω̂∈Ω̂ω

With this, one can solve for Tωw(q) for each q. Finally, we check that
 p 1−ε
ω

w

 Z

1−ε
Y 
ε
ε−1 Tω (q)
ωε−1
Jω q
dQω (q) .
=Γ 1 − αω̂ε−1ζwω̂∈Ω̂ω

(f) Compute the fraction of firms in industry ω that outsource each direct input. Check
that these match the data. If not, adjust the guess of {mωω̂ }.
1−ηpω
4. After we have solved for each pω /w, we next solve for δω . Using HHω = δωp1−η
from householdP
optimization and the normalization ω δω = 1 givesδω = P

HHω pη−1ωη−1
ω 0 HHω 0 pω 0

5. Solve for δωE using Jω = δωE π̄ωβ π̄ χ−β (wk E )−χ and the normalizationyieldδωE = PJω π̄ω−β−β
ω 0 Jω 0 π̄ω 036Eω δω = 1, which togetherP6. k E is set to match labor used for entry
LEntry =6.2.3

k E  π̄ 1+χ.
1 + 1/χ wk E

Calibration of Elasticities

We have seven elasticities to pin down. We set ε − 1 = ζ = 1.5. These parameters characterize theelasticity of substitution across varieties in the same industry of the household and of downstream
firms. We assume that these elasticities are the same. We set the elasticity of substitution acrossbaskets equal to the within-basket elasticity, η = ε. We set the overall entry elasticity χ = 0.3,
and β = 1. Together, these calibration choices imply that seven percent of labor is used on search,and 20 percent of labor is used on entry. For the distribution of firm productivities q we choose aPareto distribution with lower bound of unity and tail exponent of 8. For the matching function,
we follow Miyauchi (2018) and set M (Jω ) = Jω0.6 . In the baseline calibration, we set γ = 3.

6.3

A counterfactual increase in L

Similarly to Section 3.9, we explore numerically the economy’s response in utility, the price level,
and allocation of labor to a change in the labor force L. Figure 6 shows the proportional responseof a simulated economy that contains all industries up to 10 production steps away the exogenous
industries (which corresponds to 80% of all industries, and 84% of sales to households). Following
a 100 percent increase in L, the price level in this simulated economy falls by 27 percent. The shareof labor allocated to entry and to search increases by about one half.

.5

1

1.5

2

2.5

3

Figure 6 Counterfactual increase in size of the economy L

1

1.2

1.4

1.6

1.8

2

L
U
L_search7

p/w
L_entry

Conclusion

How do firms organize the vertical extent of their activities, and what are the implications for the
relationship between specialization and growth? We present data on the vertical span of production37

of Indian manufacturing plants. At the macro level, specialization—plants having short verticalspans—is positively correlated with the level of development. At the micro level, among plantsthat produce the same output, plants that produce at larger scales have shorter vertical spans.
We present a theory where firms are born with ex-ante heterogeneous Hicks-neutral productivity
and search for suppliers of inputs at different stages in their value chain, eventually choosing
the cost-minimizing bundle of inputs and suppliers. When intermediate inputs are imperfectly
substitutable with primary production factors, a non-homotheticity emerges: firms that are exante more productive search disproportionately more for inputs further downstream, and end up
more likely to have a short vertical span. A full quantitative model points to the presence ofconsiderable economies of scale in production, creating a strong link between specialization andgrowth at the aggregate level.

38

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43

A

Data Appendix

A.1

Data Sources and Variable Definitions• Plant-level data: Our plant data is India’s Annual Survey of Industries (ASI), published by
the Central Statistics Office, Ministry of Statistics and Program Implementation (MOSPI).
The data is at annual frequency, each reporting year starts on April 1st and ends on March
31st. Our data covers the years 1989/90, 1993/94, 1994/95, and 1996/97 to 2014/15. In thefollowing, we denote ASI rounds by the year in which they end (e.g. ‘2004” for “2003/04”).
Product codes for inputs and outputs vary across the years. We map all product codes
to the ones used between 1998 and 2008 (“ASIC 2008”). We map the pre-1997 productcodes to ASIC 2008 using a concordance that we create manually from the input and output
descriptions published by MOSPI. We concord input and output product codes for 2008/09
and 2009/10 to ASIC 2008 using the concordance used in Boehm and Oberfield (2020). The
years 2010/11 to 2014/15 use the NPCMS product classification, which we convert to ASIC
2008 codes using the concordance published by the Ministry. Codes for 1997 largely follow
the 2008 classification, but unlike the other rounds, the 1997 ASI does not list packaging
materials and auxiliary consumables (nuts, bolts, etc) as separate input categories, making
the breakdown of the plants’ materials basket inconsistent with the other years. We therefore
exclude observations from that round in all regressions where variables that rely on thecomposition of plant inputs are used (most notably, regressions with the vertical span).
• Total cost: Sum of the user cost of capital, the total wage bill, energy, services, and materials inputs. Total cost is set to be missing if and only if the user cost of capital, the wage
bill, or total materials are missing. The user cost of capital is constructed using the perpetual inventory method as in the Appendix of Greenstreet (2007), using depreciation rates
of 0%, 5%, 10%, 20%, and 40% for land, buildings, machinery, transportation equipment,
and computers & software, respectively. Capital deflators are from the Ministry’s wholesale
price index (except buildings, for which we use the CPWD building cost index up to 1994,
and the Construction Cost Index CIDC Average after 1994), and the nominal interest rate is
the India Bank Lending Rate, from the IMF’s International Financial Statistics (on average
about 11%). Services input expenditures include payments for “work done by others on materials provided by the factory”, operating expenditures (inward freight and transportation
charges, local taxes and licence fees), and non-operating expenditures (payments for communication, accounting services, financial and insurance services, legal services, contractor fees,and others).
• Materials expenditure in total cost: and subsequent tables) Total expenditure on 5-digit
intermediate inputs (which excludes energy and services) divided by total cost (see above).
• Rauch classification of goods: From James Rauch’s website, for 5-digit SITC codes. Con-44

corded from SITC codes to ASIC via the SITC-CPC concordance from UNSTATS, and the
NPCMS-ASIC concordance from the Indian Ministry of Statistics (NPCMS is based on CPC
codes).
• Dependence on relationship-specific inputs, by industry: Total expenditure of single-productplants in an industry on relationship-specific inputs (according to the concorded Rauch classification), by 3-digit industry, divided by total expenditures on intermediate inputs that areassociated with a 5-digit product code (which excludes services and most energy intermediate
inputs).
• Gross domestic product per capita, by district: Nominal district domestic product was assembled from various state government reports, for the year 2005 (to maximize coverage).
Missing for Goa and Gujarat and some union territories, and for some individual districts inthe other states. Population data from the 2001 and 2011 Census of India, interpolated to2005 assuming a constant population growth rate in each district. Whenever district domestic product per capita was unavailable, we used gross state domestic product per capita, asreported by the Ministry of Statistics and Program Implementation.
• Gross state domestic product per capita: Nominal gross state domestic product per capita,
by state and year, from CMIE’s “States of India” database.
• Vertical Span: See Appendix A.2.
• Import Tariffs: Effective applied tariffs, whenever available, otherwise MFN applied tariffs.
Tariffs for the tariff years 1996 to 2014 are from TRAINS. For the tariff years 1989, 1993,
and 1994, we digitized and coded the six-digit harmonized system codes from the respective
edition of Arun Goyal’s Customs Tariff books. This excludes tariffs for headings 84 (“nuclear
reactors, boilers, machinery and mechanical applications, and parts thereof”) and 85 (“electrical machinery and equipment, and parts thereof”), which cannot be reasonably mapped tothe six-digit level because of long lists of exceptions (140 pages in Goyal’s 1989 book). We
concord all codes to the 2007 revision of the harmonized system, and then via CPC/NPCMS
to ASIC 2008, as in Boehm and Oberfield (2020). We impute tariffs for years where they are
missing by linearly interpolating within the ASIC code, and assign each tariff year to the ASI
year that maximizes the overlap (e.g. tariff year 1989 to the ASI year 1989/90).

A.2

Definition of Vertical Span

See Appendix B of Boehm and Oberfield (2020) for definition and examples of vertical distanceand vertical span. We follow these steps:
1. For a given product ω, construct the materials cost shares of industry ω on each input
2. Recursively construct the cost shares of the input industries (and inputs’ inputs, etc...),
excluding all products that are further downstream.
45

3. Vertical distance between ω and ω 0 is the average number of steps between ω and ω 0 , weighted
by the product of the cost shares.
Tables X and XI give examples of vertical distance.
Table X Vertical distance examples for 63428: Cotton Shirts
Mean Vertical DistanceFabrics/Cloths
Yarns
Ginned & pressed cotton
Raw cotton

1.66
2.58
3.44
4.09

Table XI Vertical distance examples for 73107: Aluminium Ingots
Vertical DistanceAnodes, copper
Aluminium scrap
Aluminium oxide
Bauxite, calcined
Caustic soda (sodium hydroxide)
Bauxite, raw
Coal

1.00
1.19
1.25
2.18
2.39
3.03
3.43

The vertical span is the cost-weighted distance of a plant’s inputs from its output. We definevertical span for single-product plants only.

A.3

Sample definition and summary statistics

For the stylized facts and linear regressions in Section 2, our sample consists of all plant-year
observations with valid panel identifiers and comparable intermediate input data (see A.1 above)
between 1990 and 2015 that report to be operating in the given year and produce a single 5-digit
product good (except in regressions that explicitly include multi-product plants). We also drop allobservations from Jammu and Kashmir, Andaman and Nicobar, and from the Seven Sister Statesin northeast India (Assam, Nagaland, Manipur, Tripura, Meghalaya, Mizoram).
Table XII and XIII show the number of 5-digit inputs and outputs respectively, for single- andmulti-product plants. Table XIV shows summary statistics for the single-product plants in our
sample.
Table XII Number of 5-digit Inputs, by single-product status
Multi-productSingle-product
Total

A.4

min

p25

p50

p75

max

count

1
1
1

1
1
1

3
2
3

5
4
5

117
69
117

235,566
359,894
595,460

Mapping the data to the model

46

Table XIII Number of 5-digit Products
Multi-productSingle-product
Total

min

p25

p50

p75

max

count

2
1
1

2
1
1

2
1
1

3
1
2

26
1
26

242,440
373,889
616,329

Table XIV Summary statistics for single-product plant-year observations
Age
Log Sales
Total Employment
Labor Share of Cost
Capital Share of Cost
Materials Share of Cost
Share of Materials relationship-specific
Vertical Span

A.4.1

min

p25

p50

p75

max

count

0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00

6.00
15.14
13.00
0.05
0.02
0.73
0.02
1.08

13.00
16.62
30.00
0.09
0.05
0.84
0.46
1.29

23.00
18.48
106.00
0.16
0.09
0.91
0.96
1.70

70.00
26.81
2185508.00
1.00
1.00
1.00
1.00
7.13

396,104
402,601
402,368
377,412377,412377,412
347,989
354,128

Defining the sample for the quantitative exercise

Interpreting the data through the lens of our model imposes some requirements on the data. We
therefore restrict the sample to observations that we can use for the estimation:• We remove plant-year observations that have missing total cost (which arises mainly becauseof missing labor or capital expenditures).
• We remove plant-year observations that do not report at least one 5-digit materials input
with positive nonmissing associated expenditures• We remove observations from years 1996/97, 1998/99, and 1999/2000, because plants seemto underreport intermediate inputs in those rounds.
Furthermore, for the calibration and estimation we only use single-product plants.
A.4.2

Constructing the graph of input-output relationships

In general we assume that each five-digit ASIC code corresponds to an industry ω, and model theirposition in the production tree as determined by the inputs that plants use to produce them. Thatsaid, we model some industries ω as leaf nodes, i.e. we do not model their inputs explicitly:• Non-manufacturing industries: Industries corresponding to 5-digit codes where total
consumption exceeds total production by more than 30%, and some additional codes thatare part of agriculture, mining, utilities, and services. Also industries where the aggregatematerials share is below 5%.

47• MP-only industries: 5-digit codes that are not produced by single-product plants withvalid input codes• Residual industries: Some industries defined as residuals (“not elsewhere classified”, “others” etc), or govern broad types of inputs (“paint”, “aromatic chemicals”). Plants in theseindustries usually have different input mixes because they produce different goods, not necessarily because the plants differ in organizational form.
The model assumes that the production tree (the input-output relationships) for each goodtakes the shape of a directed acyclical graph, i.e. no circular input structures. We clean the data
manually to remove cycles:(i) We merge a small number of goods that are chemically identical but differ in their state ofmatter (e.g. liquid and gaseous nitrogen), or where the product descriptions are virtuallyindistinguishable (e.g. “aluminium oxide” and “alumina (aluminium oxide)”).
(ii) We remove a small number of input-output relationships that would otherwise give rise tocycles. Some of those cases seem to have arisen from survey respondents confusing inputs andoutputs. Some of them arise because respondents seem to have included packing materials
or capital goods in their reported intermediate inputs (e.g. tin plates in the production ofmushrooms, hospital furniture in amoxycilin). These removed inputs account for less than
TODO% of total intermediate input expenditure.
(iii) Finally, we remove all inputs that account for less than 5% of materials expenditures in anindustry (unless they are reported by more than 15% of all plant-years, and at least 5), andall inputs that are identical to the outputs.
We treat materials expenditures “removed” in steps (ii) and (iii) as primary input expenditures.
The resulting input-output relationships form a directed acyclical graph, which we construct
recursively: we starting from the leaf goods and assign them a downstreamness of 0, then calculate the set of goods ω that only rely on inputs with a downstreamness of 0 and assign them adownstreamness of 1, then calculate the set of goods ω that only rely on inputs with a downstreamness of 0 or 1 and assign them a downstreamness of 2, etc. The resulting DAG has a maximum
downstreamness of 40.
A.4.3

ρ(ω̂)

Calibrating the αω̂We again start with goods that have a downstreamness of 0, and proceed by moving furtherρ(ω)

downstream. For goods ω with a downstreamness of 0, we set αlequal to the share of expenditureρ(ω)
on primary inputs in total cost, and we set each αω̂ to the share of expenditure on ω̂ in total

cost. For goods ω with a downstreamness greater than 2:
ρ(ω)

1. We first set αω̂= 0 for all inputs ω̂ that have a downstreamness of more than one less thanthe downstreamness of ω. In other words, we set the input requirements for non-direct inputs
to zero.
48

ρ(ω)

2. We calculate the αω̂ρ(ω)

for direct inputs ω̂ as well as αl

as expenditure shares of direct

inputs and primary factors in the aggregate input basket. If a good ω̃ is showing up multiple
times upstream from ω, we split expenditures equally between all direct inputs. For each
input observed in the data, we also account a share of primary input expenditures (as givenby the α’s to that input). The remainder of the observed primary factor expenditure is usedρ(ω)

to calculate the direct primary factor coefficient, αl. Whenever this coefficient is below

5%, we assume that primary factors have been underreported and set the share to 5%.

B

Robustness

C

Further Results

C.1

Further correlates of vertical span
Table XV Cross-sectional correlates of Vertical Span
Dependent variable: Vertical Span
(1)

Log Employment

(2)

(3)

(4)

(5)

(6)

∗∗

-0.0661∗∗(0.0028)-0.0589(0.0031)-0.0540∗∗(0.0051)

Sales/Cost Ratio(7)

0.00316(0.0062)-0.250∗∗(0.018)Materials Share of Cost

-0.306∗∗(0.023)-0.163∗∗(0.0094)

Importer Dummy-0.0723∗∗(0.0084)-0.0903∗∗(0.0068)

Exporter Dummy
Share of R-Inputs in Materials Cost

-0.0168∗(0.0075)-0.260∗∗(0.021)-0.242∗∗(0.022)

Year FE
Industry FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

R2
Observations

0.317
3531700.304
3316850.310
3323560.309
3536940.322
1504160.322
3475480.371
147649Standard errors in parentheses, clustered at the 5-dgt industry level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

49

Table XVI Other Plant-level Correlates of Vertical Span
Dependent variable: Vertical Span
(1)
Materials Share of Cost

(2)

R
Observations(5)-0.163∗∗(0.0094)-0.0143∗∗(0.0055)-0.260∗∗(0.021)

Yes
Yes
0.310
332356(6)-0.119(0.015)

Share of R-Inputs in Materials Cost

2

(4)
∗∗

-0.250(0.018)

Importer Dummy

Year FE
Industry FE
Plant x Industry FE

(3)

∗∗

Yes
Yes
0.309
353694

Yes
Yes
0.322
347548

Standard errors in parentheses, clustered at the 5-dgt industry level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

50

-0.181∗∗(0.021)
Yes

Yes

Yes

Yes

Yes

Yes

0.774
173141

0.765
186641

0.773
181958

C.2

Materials shares and number of inputs
Table XVII Materials Shares and Number of Inputs
Dependent variable: Materials Share in Total Cost
(1)

(2)
∗∗

Inverse HHI of Materials Cost Shares-0.00727(0.00063)

R2
Observations(4)-0.00575∗∗(0.00050)-0.00174∗(0.00076)-0.000160(0.00097)

Number of Inputs
Year × 5-digit Industry × District FE
Plant × Product FE

(3)

Yes

Yes
Yes

Yes

Yes
Yes

0.722
137013

0.928
63325

0.722
137013

0.928
63325

SP plants only. Number of inputs exclude those with small cost shares (<5%).
+
p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

Table XVIII Input Adoption
Dependent variable: Input Used Dummy 1(Xj ω̂t > 0)(1)
log(1 + τit )Year FE
Plant × Input FE
Plant × Product FE
R2
Observations(2)∗∗-0.0506(0.0067)-0.0373∗∗(0.0071)

Yes
Yes

Yes
Yes
Yes

0.337
24608310.361
2454899

Standard errors in parentheses.
+
p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01
Regression at the level of plant × input × year, on single-product plants only. The left-hand side
is a dummy for whether expenditure on input ω̂ is positive; the right-hand side is the log tariff on
input ω̂ at time t.

51

Table XIX Supply and demand shifters determine entry
Dependent variable: log Producers |J|dωt
(1)

(2)

input
log(1 + τ̄it)∗∗

-0.108(0.025)-0.0496∗∗(0.015)

outputlog(1 + τit)

0.186∗∗(0.021)0.251∗∗(0.013)

Year FE
State FE
Industry FE
State × Year FE
State × Industry FE
R2
Observations

Yes
Yes
Yes
Yes
Yes
0.481
5481800.844
537013Standard errors in parentheses.
+
p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01
The left-hand side is the log number of producers of a good ω at time t in state d. The right-handside are log import tariffs on ω’s input industries, weighted by I-O table cost shares (“log upstream
tariff”) and log import tariffs on downstream industries, weighted by ω’s sales shares. Weights areinvariant across time and space.

52

C.3

Pre- to post-liberalization
Table XX Tariff changes act as demand and supply shocks: pre/post liberalization
Dep. var.: ∆t1990 log Sales(1)
output∆t1990 log(1 + τωt
)

(2)
+

1.533+(0.79)

1.302(0.75)

input∆t1990 log(1 + τ̄ωt)Year FE
2R
Observations-1.188(0.77)
Yes

Yes

0.08522376

0.0903
2376

Standard errors in parentheses, clustered at the state × industry level.
p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01
The sample consists of all census plants that are observed in 1990 and post-1997 as single-productplants producing the same product. Changes are within plant-product.
+

Table XXI Vertical span and market size: pre/post liberalization
Dependent variable: ∆t1990 Vertical Span(1)
∆t1990 log Sales(2)+-0.147(0.084)

input∆t1990 log(1 + τ̄it
)

(3)
+

-0.166(0.086)-0.237+(0.12)

0.194(0.24)

1.421+(0.77)

input)(distanceωi − spanj )i αi log(1 + τ̄it

-0.747(0.75)

input)spanji αi log(1 + τ̄it

-1.031+(0.62)∆

P∆

P

Year FE
2R
Observations

Yes

Yes

Yes

-0.1942179-0.2552179-0.4982128

Standard errors in parentheses, clustered at the state × industry level.
+ p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

The sample consists of all census plants that are observed in 1990 and post-1997 as single-productplants producing the same product. Changes are within plant-product. The regressions in bothcolumns instrument ∆t1990 log sales by the corresponding change in the log output tariff over thesame horizon.

C.4

Results with Hummels export demand shifter

The following tables show the main results of the paper where, instead of the level or change ofIndia’s import tariffs, an export demand shifter is being used as an instrument:
ExDemandωt =

X
d

53

X,Indiaπωdlog X dωt

(5)

Table XXII Vertical span and market size: pre/post liberalization
Dependent variable: log Sales(1)
Avg. log #Producers in Upstream Ind.

(2)∗∗

0.0655(0.013)(3)∗∗

0.0560(0.018)input
log(1 + τ̄jωt
)

(4)
∗∗

0.0551(0.018)0.0201(0.043)

(5)

(6)
∗∗

0.119
(0.044)0.540∗(0.26)

Year FE
Industry × Year FE
Plant × Industry FE

Yes

Estimator
2

R
Observations

0.115∗∗(0.044)0.519∗(0.26)

Yes

Yes

Yes
Yes

Yes
Yes

Yes

Yes
Yes

Yes
Yes

OLS

OLS

OLS

IV

IV

IV

0.916
13683

0.943
9768

0.943
9757

0.0026213683-0.000638
9768

0.000690
9757

Standard errors in parentheses, clustered at the industry-year level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01
Analogously to Tables XX and XXI, we limit the sample to plants that are observed as singleproduct plants in 1990 and post-1997. The dependent variable is log sales of plant j; the right-handside variable is the average log number of producers in j’s state in industries upstream from j(with weights taken at the time of j’s first observation). Columns (4) to (6) instrument the averagelog number of producers in upstream industries by the tariff on industries downstream of these
upstream industries, excluding the industry of the left-hand side plant.

where X dωt =Po∈Õ Xd,o,ωt is the sum of exports of ω from all countries except India to countryd, andX,India=πωd

Xd,India,ωt
1X
P
T t XIndia,India,ωt + d Xd,India,ωt

is the average weight of exports of ω to d in India’s production of ω.
Table XXIII shows that an increase in the export demand shifter leads to an increase in plantsales, above and beyond any effects of tariff changes.
Table XXIII Hummels export instrument as demand shifter
Dependent variable: Log Sales(1)
ExDemandωt(2)∗∗

0.0493(0.0086)

outputlog(1 + τit
)

(3)
∗∗

0.0539(0.011)

0.0526∗∗(0.011)0.478∗∗(0.15)

0.722∗∗(0.16)

input
log(1 + τ̄it)Year FE
Plant FE
Plant × Industry FE
2R
Observations-0.604∗∗(0.20)
Yes

Yes

Yes

Yes

Yes

Yes

0.947
1624210.949
1241060.949
124098

Standard errors in parentheses, clustered at the state × industry level.
p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01+

54

Table XXIV shows that demand affects the vertical span of firms. In contrast to the importcompetition instrument, this instrument features many more instances where demand increases,
alleviating concerns that effects of demand changes may be asymmetric.
Table XXIV Firms vertically specialize when they receive a positive demand shock – Export
demand instrument
Dependent variable: Vertical Span
(1)

(2)
∗∗

Log Sales-0.0191(0.0020)

input
log(1 + τ̄jωt
)

(3)

(4)

∗∗

∗∗

-0.0190(0.0020)-0.0197(0.0024)-0.0877∗∗(0.026)-0.0211(0.050)(5)∗-0.154(0.075)(6)∗-0.157(0.074)-0.257∗∗(0.095)-0.156∗∗(0.046)

0.0511(0.099)P

input)spanji αi log(1 + τ̄it

-0.0905(0.056)-0.362∗∗(0.13)

P

input)(distanceωi − spanj )i αi log(1 + τ̄it

-0.121(0.095)-0.484∗(0.21)

Year FE
Plant × Product FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Estimator

OLS

OLS

OLS

IV

IV

IV

0.765
1866280.765
1866280.731
145165-0.0703
143428-0.0739
143428-0.2071105782

R
Observations

Standard errors in parentheses, clustered at the state-industry level. + p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

Regression contains single-product plants only. The left-hand side is the plant’s vertical span ofproduction; the right-hand side is log sales. Columns (3) and (4) instrument log sales by the export
demand instrument ExDemandωt .

Table XXV shows the specifications from Table VIII with the export demand instrument.
Results are very similar.
Table XXV Firms reduce the number of inputs when they receive a positive demand shock
Dependent variable: log Number Of Inputs(1)
Log Sales(2)∗∗0.0359(0.0030)

input
log(1 + τ̄jωt
)

(3)
∗∗

0.0361(0.0029)

(4)
∗

(5)
∗

-0.412(0.20)-0.418(0.20)-0.252∗∗(0.096)

Dependent variable: Inverse Input HHI

0.0101(0.0072)-0.258∗(0.10)(6)0.0104(0.0069)

(7)

(8)
+

-1.749(0.99)-1.764+(0.99)-0.591∗(0.27)-0.407+(0.24)

Year FE
Plant × Product FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Estimator

OLS

OLS

IV

IV

OLS

OLS

IV

IV

0.856
1866280.856
186628-0.657143428-0.6741434280.807
1927950.808
192795-2.593147328-2.6391473282

R
Observations

Standard errors in parentheses, clustered at the state-industry level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01
Regression contains single-product plants only. The left-hand side is the plant’s number of 5-digit materials inputs(columns (1) to (4)) and the inverse of the HHI of materials shares (columns (5) to (8)); the right-hand side islog sales. Columns (3), (4), (7), and (8) instrument log sales by the export demand instrument ExDemandωt .

Table XXVI shows regressions of log Sales of j on a measure of the number of plants in the55upstream sectors of j in the state of j, instrumented by demand shocks downstream from those
upstream industries (as in Table VI / Section 2.4). The difference here is that the demand shocks
are coming from the Hummels export demand shifter instead of from the import tariff changes.
Results are stronger than in the baseline results.
Table XXVI Demand shocks also propagate downstream
Dependent variable: log Sales

Avg. log #Producers in Upstream Ind.

(1)

(2)

(3)

(4)

(5)

(6)

0.0466∗∗(0.0041)0.0383∗∗(0.0050)

0.0375∗∗(0.0060)

0.247∗∗(0.030)0.255∗∗(0.036)0.121∗∗(0.019)input
log(1 + τ̄jωt)

input)(distanceωi − spanj )i αi log(1 + τ̄itP

Year FE
Industry × Year FE
Plant × Industry FE

Yes

Estimator
2

R
Observations-0.0242(0.096)-0.0162(0.097)-0.331∗∗(0.11)-0.337∗∗(0.11)
Yes

Yes

Yes
Yes

Yes
Yes

Yes

Yes
Yes

Yes
Yes

OLS

OLS

OLS

IV

IV

IV

0.942
2158050.952
1990390.954
142006-0.0329215805-0.0304
199039-0.00399
142006

Standard errors in parentheses, clustered at the industry-year level. + p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

Regression contains single-product plants only. The left-hand side is the plant’s sales; the right-hand side is thechange in log sales. Changes are taken across all time horizons within the same plant-product pairs. Columns (4)to (6) instrument ∆ log sales by the export demand instrument ExDemandωt .

C.5

Plant-level returns to scale

Table XXVII shows regressions of domestic sales on the export demand shock for the few years
where we have data on foreign and domestic sales (2010-2014). We find that export demand has
either a positive or a zero effect on domestic sales. We interpret this as evidence against the viewthat the plants are operating under decreasing returns to scale, such as what would be implied bya model where firms are facing constraints to the hiring of workers (span of control).

C.6

Selection into single-product status

In order to investigate whether the negative relationship between vertical span and plant size in thebaseline results are the result of selection into single-plant status, we show two sets of additional
regressions. The first set of results, shown in Table XXVIII, shows the baseline results on the(instrumented) size-span relationship from Table ?? on the pooled sample of single- and multiproduct plants. We define the vertical span of a multi-product plant as the weighted average
distance of the plant’s inputs from the 5-digit product with accounts for the largest fraction ofsales of the plant. While in this regression both the dependent and the independent variables are
mismeasured (for the span, some inputs will go into the production of the non-leading output,
and the scale of production relevant for economies may not be total plant scale), the negative
56

Table XXVII Response of domestic sales to export demand shifter
Dependent variable: Log Domestic Sales(1)
ExDemandωt(2)∗0.0214(0.0092)

(3)

(4)
∗∗

0.0208(0.0066)(5)(6)

0.0160(0.013)0.0222(0.014)-0.000381(0.017)0.00264(0.018)outputlog(1 + τit)-0.765(0.52)-0.648(0.52)-0.609(0.41)-0.562(0.41)

input
log(1 + τ̄it)

-0.397(1.19)-0.614(1.55)-1.343+(0.74)-1.771+(0.95)

input)spanji αi log(1 + τ̄it

0.433
(0.83)

0.400(0.44)

input)(distanceωi − spanj )i αi log(1 + τ̄it

0.968
(1.59)-0.0418(0.86)

P

P

Year FE
Plant × Product FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Sample

SP

SP

SP

All

All

All

0.953
64375

0.956
40279

0.956
38335

0.957
109802

0.959
65959

0.959
63433

2

R
Observations

Changes within plant-productsStandard errors in parentheses, clustered at the state-industry level.
+p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01
Regression contains single-product plants in columns (1) to (3) and all plants in columns (4) to (6), using datafrom year 2010 to 2014. The left-hand side is the plant’s log domestic sales; the right-hand side is the export
demand shifter as well as tariff controls.

relationship in this much larger sample prevails (the tariff instrument becomes very weak though,so columns (3) and (4) are hard to interpret).
The second set of results, in Table XXIX, shows results on single-product plants only, but
includes an interaction between sales and a dummy for whether a plant had been a multi-productplant in a previous period (“MP before”). This dummy is predictive of the plant becoming a multiproduct plant again. We find that plants that differ along this characteristic (which is related to
the likelihood of selecting out of the sample) have a very similar elasticity of span with respect tosize, indicating that selection is unlikely to drive this negative relationship.

57

Table XXVIII Size-span relationship on single-and multi-product plants
Dependent variable: Vertical Span
(1)

(2)
∗∗

Log Sales(3)∗∗-0.0116(0.0016)-0.0116(0.0020)

input
log(1 + τ̄jωt
)

(4)

9.814
(196.5)

(5)

(6)
∗

-0.213(0.38)-0.110(0.054)-0.103(0.066)-0.0205(0.042)-0.0378(0.047)-0.0677(0.072)

input)spanji αi log(1 + τ̄it-0.0109(0.044)-0.0922(0.15)-0.169+(0.092)

input)(distanceωi − spanj )i αi log(1 + τ̄it

0.138∗(0.067)0.0155(0.19)-0.133(0.12)

P

P

Year FE
Plant × Product FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Estimator

OLS

OLS

IV 1

IV 1

IV 2

IV 2

0.764

0.732

338887245188-280.40.00261
232632-0.1171.162231254-0.026336.16
260217-0.0186
43.04
186983

2

R
First stage w. id. robust F
Observations

Standard errors in parentheses, clustered at the state-industry level. + p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

The table shows the baseline results on the relationship between size and span, Table ??, on the pooled sampleof single- and multi-product plants. For multi-product plants the vertical span is defined as the weighted average
distance of the plant’s inputs from the 5-digit product with accounts for the largest fraction of sales of the plant.
“IV 1” uses the import competition demand shifter as instrument, “IV 2” uses the Hummels export demand
shifter as instrument for sales.

Table XXIX Size-span relationship for single-product plants that have been multi-product vs not
Dependent variable: Vertical Span
(1)

(2)

(3)
∗∗

(4)

(5)

(6)

Log Sales∗∗-0.0184(0.0020)-0.0188(0.0024)-0.542(0.31)-0.278(0.15)-0.164(0.074)-0.269∗∗(0.095)Log Sales × MP before-0.00147∗∗(0.00041)-0.00187∗∗(0.00047)

0.00784+(0.0048)0.00404+(0.0024)0.0000738(0.00088)

0.000932(0.0013)

input
log(1 + τ̄jωt
)

P

+

+

∗

-0.0191(0.050)0.0254(0.068)0.0545(0.100)input)spanji αi log(1 + τ̄it

-0.0918+(0.055)-0.227∗(0.10)-0.370∗∗(0.13)

input)(distanceωi − spanj )i αi log(1 + τ̄it

-0.117(0.095)-0.361∗(0.17)-0.499∗(0.21)P

Year FE
Plant × Product FE

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Yes
Yes

Estimator

OLS

OLS

IV 1

IV 1

IV 2

IV 2

0.765

0.731

186628145165-1.1742.592
138204-0.2834.762
137059-0.081713.16
143428-0.2277.909
1105782

RFirst stage w. id. robust F
Observations

Standard errors in parentheses, clustered at the state-industry level. + p < 0.10, ∗ p < 0.05, ∗∗ p < 0.01

The table shows the baseline results on the relationship between size and span, Table ??, but also including aninteraction between sales and a dummy for whether a plant had been a multi-product plant in a previous period(“MP before”). This dummy is predictive of the plant becoming a multi-product plant again. “IV 1” uses theimport competition demand shifter as instrument, “IV 2” uses the Hummels export demand shifter as instrument
for sales.

58

D

Proofs and Additional Theoretical Results

D.1

Proofs for Simple Model of Section 3

We first derive an expression for the effective cost of outsourcing each input. Let Fω (c) be the
distribution of unit cost among those in industry ω.
Lemma 2 If firm j in industry ω exerts search effort hjk to search for suppliers of ω − k then


− 1
R −ζ
−ζ
ζ
Pr cojk > c|qj , hj1 , hj2 = ehjk mk c̄ω−k c with c̄ω−k =c dFω−k (c) ζ .
Proof. If firm j exerts search effort hjk , the arrival rate of potential suppliers of ω − k with matchspecific productivity better than z is hjk mk z −ζ . Thus the probability that the firm’s best outside
supplier delivers cost greater than c isRRcs−ζ−1 dz
Pr cojk > c = e−hjk mk 1{ z ≤c}dFω−k (cs )ζzRRζ1{ cus ≤1}dFω−k (cs )ζu−ζ−1 du= e−hjk mk c−ζ= e−hjk mk c̄ω−k cζ

where c̄ω−k is defined to satisfy
c̄−ζ
ω−k

Z Z
≡

1
Z Z

=
Z
=

nc

s

u

o
≤ 1 dFω−k (cs )ζu−ζ−1 duζ−1
1 {t ≤ 1} c−ζdt
s dFω−k (cs )ζtc−ζ
s dFω−k (cs )Claim 1 If firm j in industry ω exerts search effort h2 , its cost of producing ω − 1 in-house will
follow a Weibull distribution
−ζ ζ

α
α
(1−α) c̄αcω−2 )Pr cij1 > c|hj2 = e−hj2 m2 (wProof. If firm j in industry ω uses supplier s in industry ω − 2, its effective cost of ω − 1 willbe B1j w1−α

ps
zs

α

. Conditional on task productivity Bj and match-specific productivity zs , theprobability that the supplier’s price ps is low enough so that the the supplier delivers effective costof ω − 1 weakly less than c–that is, B1j w1−α

ps
zs

α

59

1/α

< c– is Fω−2 Bj

1−αzs w− α c1/α . Integrating

across possible match-specific draws, the arrival rate of such a supplier isZ
hj2 m2 Fω−2



1−α
1/α
Bj zw− α c1/α



ζz

−ζ−1

dz = hj2 m2

= hj2 m2 w

ζ
1−α
1/α
Bj w− α c1/α

− 1−α
ζ
α

ζ
α

ζ
α

ζ

ζ

Z

Fω−2 (u) ζu−ζ−1 dz

u−ζ dFω−2 (u) du

c Bj

1−α

Z

= hj2 m2 w− α ζ c α Bjα c̄−ζω−2Given Bj , the probability that no such supplier arrives is thene−hj2 m2 w

− 1−α
α ζ

ζ

(Bj c) α c̄−ζ
ω−2

h
i
ζ/α
α
Finally, integrating across realizations of Bj and using the functional form E e−uB
= e−u
gives
"
Pr cij1 > c|hj2 = E e


ζ
ζ
1−α
α
−hj2 m2 w− α ζ c α c̄−ζω−2 Bj#αα= e−hj2 m2 (w−ζ ζ(1−α) c̄αcω−2)Claim 2 If firm j chooses search effort hj1 and hj2 , the probability that the firm uses a supplierin ω − 1 is

hj1 m1 c̄−ζω−1αα(1−α) c̄αhj1 m1 c̄−ζω−1 + hj2 m2 w
ω−2

−ζ

Proof. Given h1 and h2 , what is the probability that the firm chooses to use a supplier from ω − 1?
Pr

cij1 ≤ coj1 |hj1 , hj2



Z



Pr cij2 > c d Pr coj1 < cZe−hj2 m2 (w

=
=
=

α

α

−ζ ζ
(1−α) c̄α
c
ω−2

)

−ζ

ζ

e−hj1 m1 c̄ω−1 c h1j m1 ζcζ−1 dc

hj1 m1 c̄−ζω−1αα(1−α) c̄αhj1 m1 c̄−ζω−1 + hj2 m2 w
ω−2

−ζ

Claim 3 If firm j chooses search effort hj1 and hj2 , its expected gross profit isE [πj |qj , hj1 , hj2 ] = Aω δω qjε−1
ε−1

(ε−1)
where Aω ≡ upη δω pε−η
ω
εε

(

α
α
hj1 m1 c̄−ζω−1 + hj2 m2
Γ 1 − α ε−1
ζ

60



w(1−α) c̄αω−2−ζ − αζ

)1−ε
w1−α

Proof. Given h1 , and h2 , the probability that the firm’s effective cost of ω − 1 is greater than c is
Pr min ci1 , co1 > c|h1 , h2 = Pr (co1 > c|h1 ) Pr ci1 > c|h2
−ζ

α

α

= e−hj1 m1 c̄ω−1 c e−hj2 m2 (wζ−ζ ζ(1−α) c̄α
c
ω−2

)

−ζ ζ

= e−T1 ch−ζ i− ζ1αα(1−α) c̄αwhere T1 ≡ hj1 m1 c̄−ζω−2
ω−1 + hj2 m2 w

Given q, h1 , and h2 , the probability that the firm’s cost is greater than is c

Pr


α
1 1−α
w
min ci1 , co1> c q, h1 , h2
q




= Pr

 qc 1/αmin ci1 , co1 >q, h1 , h2
w1−α

−T1−ζ

= e


−

= e



qc
w1−α

qc
T1α w1−α



ζ/α

ζ/α

The firms gross profit comes only from sales to the household. Given isoelastic
demand, uj ≤
h

upη δ

ε−η −εεω pω pj , the firm chooses price pj = ε−1 cj , so its expected gross profit is E
ε−1

i
c1−ε
=
j

h
i
. The final term is
E c1−εj(ε−1)
upη δω pε−η
ω
εε



E c1−ε |q, h1 , h2 =

=
=
=

D.1.1

ε−1

(ε−1)
upη δω pε−η
ω
εε



ζ/α 


qc
−
α 1−α
c1−ε d 1 − e T1 w0ε−1 Z ∞
n
o
q
1−ε
−uζ/α
u
d
1
−
e
T1α w1−α0ε−1 Z ∞q−α ε−1
ζ d 1 − e−v
v
α
1−α
T1 w
0

ε−1 

q
ε−1
Γ 1−α
T1α w1−αζZ ∞Optimal Search Behavior

In this section, we normalize the wage to unity. Total expected profit is then for a firm withproductivity q is
max Aω δω q

h1 ,h2ε−1h

α α −αζh1 m1 c̄−ζω−1 + h2 m2 c̄ω−261iα ε−1ζ−kk
h1+γ −
h1+γ
1+γ 1
1+γ 2

The first order conditions are
khγ1
khγ2

h
iα ε−1 −1 ε − 1ζ
α α −αζα= Aω δω q ε−1 h1 m1 c̄−ζ+hmc̄m1 c̄−ζ
2 2 ω−2ω−1
ω−1ζhiα ε−1 −1 ε − 1ζ
α α −αζα2= Aω δω q ε−1 h1 m1 c̄−ζhα−1mα2 c̄−αζ
ω−1 + h2 m2 c̄ω−22
ω−2ζNote that 1 + γ > α ε−1ζ gurantees that the second order condition hold.
hi− 1Aω δω α ε−1
ζ
−ζ
−αζ
ζ
α
α
amd r =
Let T ≡ h1 m1 c̄ω−1 + h2 m2 c̄ω−2. Then these FOCs can be expressed
k
as
hγ1
hγ2
Let O =

h1 m1 c̄−ζω−1α α −αζh1 m1 c̄−ζω−1 +h2 m2 c̄ω−2α ε−1 −1ζ
m1 c̄−ζ= rq ε−1 T −ζω−1α ε−1 −1ζ= rq ε−1 T −ζαhα−1
mα2 c̄−αζ2
ω−2be the probability of outsourcing the more upstream task.
d ln T −ζd ln h1
d ln h2=O+ (1 − O) αd ln q ε−1
d ln q ε−1
d ln q ε−1

d ln h1γd ln q ε−1
d ln h2γd ln q ε−1


ε − 1 d ln T −ζ= 1− 1−αζd ln q ε−1
ε − 1 d ln T −ζd ln h2= 1− 1−α− (1 − α)ε−1ζd ln qd ln q ε−1Together, these imply that




1
ε − 1 d ln T −ζ= O
1− 1−αγζd ln q ε−1




1
ε − 1 d ln T −ζ
+ (1 − O) α1− 1−αγ + (1 − α)ζd ln q ε−1

d ln T −ζd ln q ε−1or

d ln T −ζ=d ln q ε−1

1>01+ 1 − α ε−1αζO γ1 +(1−O) γ+(1−α)

Plugging this in givesγ

d ln h1
d ln q ε−1=(γ + 1 − α)d ln h2d ln q ε−1

=

1
h

α
1 + O γ1 + (1 − O) γ+(1−α)

1
1+

h

α
O γ1 + (1 − O) γ+(1−α)

62

i
 >0
1 − α ε−1ζ
 >0i1 − α ε−1ζPr(use ω−1)
Claim 4 Pr(useω−2) is strictly increasing in q.

Proof. Let Oω (q) be the probability that a firm in ω born with productivity q ends up using ω − 1
as a input. We haveh1 m1 c̄−ζOω (q)
ω−1=
1 − Oω (q)hα2 mα2 c̄−αζ
ω−2Differentiating with respect to q ε−1 and using the expressions forOω (q)
d ln 1−Oω (q)d ln q ε−1
The conclusion that=d ln h1
d ln h1
d ln h2d ln h1αγ−α=−d ln q ε−1
d ln q ε−1
d ln q ε−1 γ + (1 − α) d ln q ε−1

O (q)ωd ln q ε−1ωd ln 1−O
(q)



αγ
d ln h1> 0 follows from 1 − γ+(1−α)= (1−α)(1+γ)γ+(1−α) > 0 and d ln q ε−1 .

Lemma 3 Among those born with productivity q, sales is independent of input choice.
Proof. Among those with productivity q, the realization of unit cost is independent of input choice:Pr

co1 < ci1 |c1 = c, q, h1 , h2=Pr co1 = c, ci1 > c|q, h1 , h2
Pr (c1 = c|q, h1 , h2 )=−ζα(1−α) c̄αcζζ−1 e−h1 m1 c̄ω−1 cζ e−hα2 m2 (wω−2 )h1 m1 c̄−ζω−1 ζc
i
h
h
−ζ ζ
α α
−ζ i
(1−α) c̄αc− h1 m1 c̄−ζ−ζ
ω−2 )
ω−1 +h2 m2 (w
α
α
α
(1−α)
ζ−1
h1 m1 c̄ω−1 + h2 m2 w
c̄ω−2
ζc e


−ζ

=

h1 m1 c̄−ζω−1α α(1−α) c̄αh1 m1 c̄−ζω−1 + h2 m2 w
ω−2

−ζ

Since this probability of producing in-house is the same for all values of c, it must be independentof sales, which is a random variable hat is measurable with respect to the firms unit cost but notwith respect to its choice of input.
Claim 5 Among those in industry ω, those with higher sales are more likely to use a supplier inindustry ω − 1
Proof. Using the law of toal covariance, we can express this as
Cov (Sales, use ω − 1) = E [Cov (Sales, use ω − 1) |q] + cov [E [Sales|q] , E [use ω − 1|q]]
The first term is zero and the second term is positive.
Since using a supplier in industry 1 maps directly into a shorter vertical span and higher
intermediate input share, we immediately have the following corollary
Corollary 2 Among those in industry ω, those with higher sales are more likely to have a shorter
vertical span and higher intermediate input share63

D.1.2

Demand Shocks

We consider in this section the impact of an increase in the demand for industry ω, i.e., and increasein δω .
Proposition 8 If δω increases, Jω increases, pω and c̄ω fall, and fraction of firms using ω − 1rises.
Proof. A shock to δω does not change profit in ω − 1 or ω − 2, so it must be that c̄ω−1 and c̄ω−2A ω δω
are unchanged. We first establish that d ln
d ln δω > 0. Let π̄ω (q) be the expected profit of a firmborn with productivity q.
hiα ε−1ζα α −αζ+hmc̄π̄ω Aω δω q ε−1 = max Aω δω q ε−1 h1 m1 c̄−ζ−
2 2 ω−2ω−1
h1 ,h2

k
k
h1+γ
−
h1+γ
1
1+γ
1+γ 2

Using the envelope theroem, we haveiα ε−1hdπ̄ω Aω δω q ε−1ζ−ζ
α α −αζε−1
h1 m1 c̄ω−1 + h2 m2 c̄ω−2= Aω δ ω q>0d ln Aω δω q ε−1dπ̄ω (Aω δω q ε−1 )dπ̄ω (Aω δω q ε−1 )A ω δω
The elasticity with respect to δω is simply= d ln Aω δω qε−1 d ln
dln
δd ln δω . With this, weωR
can see how average profit π̄ω = π̄ω (q) dQ (q) changes with δω :d ln π̄ω=d ln δωR dπ̄ω (Aω δω qε−1 )d ln δωRdQ (q)π̄ω (q) dQ (q)We next study the impact of δω onR

R dπ̄ω (Aω δω qε−1 )=d ln Aω δω q ε−1

R

dQ (q) d ln Aω δωπ̄ω (q) dQ (q)d ln δω


h





ε−1 h m c̄−ζ + hq
E c1−ε |q dQ (q). Note that E c1−ε |q = Γ 1 − α ε−1
j1 1 ω−1ζDifferentiating yields
d ln E c1−ε |qε−1d ln h1
d ln h2=αOω (q)+ (1 − Oω (q)) αd ln δωζd ln δωd ln δω
Similarly, the first order conditions for h1 and h2 imply that these are functions of Aω δω q ε−1 .
Following the logic about how serach effort changes with q ε−1 in the introduction to the section,
we haved ln h1 Aω δω q ε−1γd ln Aω δωd ln h2 Aω δω q ε−1(γ + 1 − α)d ln Aω δω



=

1
h
i

α
1 + O γ1 + (1 − O) γ+(1−α)
1 − α ε−1
ζ

=

1
h
i

α
1 + O γ1 + (1 − O) γ+(1−α)
1 − α ε−1ζ64Plugging these in yieldshi 1−ε 
ε−1
1
α
α
O
(q)
+
(1
−
O
(q))
ω
ω
d ln E c |q
ζ
γ
γ+1−α
h
i
 >0
=
d ln Aω δω
1 + O (q) 1 + (1 − O (q)) α
1 − α ε−1
ω

ω

γ

γ+1−α

ζ

With these, we can derive an expression for the change in the household’s price index for industry
ω. This price index satisfies
p1−ε
ω

Z
=
Jω


=

where we used Jω = π̄χ− β1d ln p1−εω(1 − ε)d ln δωp1−ε
ωj dj = Jω

ε
ε−1

1−ε
π̄

"

Z
E

χ− β1

π̄ωβ

Z

1−ε #εc|q dQ (q)ε−1E c1−ε |q dQ (q)π̄ωβ . Differentiating yields
R d ln E c1−ε |q dQ (q)d ln π̄ω= β+d ln δωd ln δω R

R dE[c1−ε |q]dπ̄ω (Aω δω q ε−1 )
dQ
(q)
dQ
(q)
A ω δω
 d ln Aω δω
R d ln Aω δω= β+ R d ln1−εd ln δωπ̄ω (q) dQ (q)E [c |q] dQ (q)Let X denote the term in brackets, with X > 0. Sinced ln Aω δωd ln Aω δωd ln pωε−ηX= 1 + (ε − η)=1−d ln δωd ln δωε−1d ln δωA ω δω
Solving for d ln
d ln δω gives

d ln Aω δω
1
>0
=
ε−η
d ln δω1 + ε−1 X

Some simple consequences ared ln pωd ln δωd ln Jωd ln δωd ln c̄ωd ln δωX d ln Aω δω
<0
ε − 1 d ln δωd ln π̄ω d ln Aω δω= β>0d ln Aω δω d ln δωR  −ζ  d ln E[c−ζ |q]
E c |q d ln Aω δω dQ (q) d ln AωR=>0d ln δωE [c−ζ |q] dQ (q)= −E[c1−ε |q ]dQ(q)> 0. Finally, the fraction
d ln Aω δωh1 m1 c̄−ζOω (q)
ω−1of firms in ω born with productivity q is that use a supplier in industry ω −1 is 1−O
=
.
α −αζ
ω (q)
hα
2 m2 c̄ω−2where the last line follows the same logic as showing that

65

d ln

R

Differentiating yields
Oω (q)
d ln 1−Oω (q)d ln δω1
αd ln h1 d ln h2
d ln Aω δω
γ − γ+1−α
h
i

=
−
=
>0
1
α
ε−1
d ln δωd ln δωd ln δω
1 + Oω (q) γ + (1 − Oω (q)) γ+1−α 1 − α ζSince this is true for each q, it must be true for the industry as a whole.
D.1.3

Upstream Demand ShockIn this section we study the impact on industry ω of a shock to δω−1 . We know from above thatan increase in δω−1 raises reduces c̄ω−1 . It has no impact on c̄ω−2 . It will, however have an impact
on Aω because firms in ω respond to the change in expected input cost.
Using the envelope theorem, we haveiα ε−1hζα α −αζ+hmc̄π̄ω (q) = max Aω δω q ε−1 h1 m1 c̄−ζ−
2 2 ω−2ω−1
h1 ,h2

k
k
−
h1+γ
h1+γ
1
1+γ
1+γ 2

iα ε−1 d ln Aiα ε−1 −1 ε −
h
h
ζ
ζ
ω
−ζ
α α −αζα α −αζ
ε−1
+
h
m
c̄
h
m
c̄
= Aω δω q ε−1 h1 m1 c̄−ζ
+
A
δ
q
α
1
1
ω
ω
2
2
ω−2
ω−1 + h2 m2 c̄ω−2
ω−1−ζζd ln c̄ω−1()
hiα ε−1d ln Aωε−1ζα α −αζ= Aω δω q ε−1 h1 m1 c̄−ζ
+
h
m
c̄
+α
Oω (q)
2 2 ω−2ω−1−ζζd ln c̄ω−1dπ̄ω (q)d ln c̄−ζ
ω−1

gross
(q)dQ(q)Rπω> 1 be the ratio of gross profit (without subtracting search costs) to netπ̄ω (q)dQ(q)R grossπ(q)x(q)dQ(q)profit (subtracting search costs), and for any quantity x (q), let let x̄ = Rωπgross (q)dQ(q) be theωLet λ =R

gross-profit-weighted average. With this we can derive an expression for how profit changes with
upstream characteristics.
Rd ln π̄ωd ln c̄−ζω−1=dπ̄ω (q)dQω (q)d ln c̄−ζω−1π̄ωNext, we derive expressions for how

Y (q) ≡

h

1+



!

d ln Aωchanges. Defined ln c̄−ζω−1α α −αζh1 m1 c̄−ζω−1 + h2 m2 c̄ω−2

Σ0 (q) ≡Σ1 (q) ≡

=λ

ε−1
+α
Ōω
−ζ
ζ
d ln c̄ω−1d ln Aω

1 − α ε−1ζiα ε−1ζ1hiαOω (q) γ1 + (1 − Oω (q)) γ+1−αOω (q)
hi.
1
α
1 + 1 − α ε−1
O
(q)
+
(1
−
O
(q))
ω
ω
ζ
γ
γ+1−α


The FOCs for h1 and h2 can be expressed as

66

khγ1

= Aω δω q ε−1 Y

khγ2= Aω δω q ε−1 Y−−

1−α ε−1ζα ε−1ζ
1−α ε−1
ζ
α ε−1
ζ

α

ε−1
m1 c̄−ζω−1ζ

α2

ε − 1 α−1 α −αζh2 m2 c̄ω−2ζDifferentiating yields

γ(γ + 1 − α)d ln h1 (q)d ln c̄−ζω−1d ln h2 (q)d ln c̄−ζω−1==1 − α ε−1d ln Y
ζ
−
+1
ε−1
−ζ
α ζ
d ln c̄ω−1d ln c̄−ζω−1d ln Aωd ln Aωd ln c̄−ζω−1−1 − α ε−1d ln Yζε−1
α ζd ln c̄−ζ
ω−1

and
"
!
#
d ln h1 (q)ε−1d ln h2 (q)Oω (q)= α+ 1 + (1 − Oω (q))ζ
d ln c̄−ζd ln c̄−ζω−1 ω−1 
 
ε−1
1−α
Y
ω
− α ε−1ζ d ln−ζ
+1 +1Oω (q) γ1 d ln Aε−1d ln c̄−ζd ln c̄ω−1
ζ
ω−1 

 
= α
ε−1


1−α
ζ
d ln Aω1
Y+ (1 − Oω (q)) γ+1−α− α ε−1ζ d ln−ζ−ζd ln Y (q)d ln c̄−ζω−1d ln c̄ω−1Solving forζd ln c̄ω−1d ln Ygives
d ln c̄−ζω−1hd ln Y (q)d ln c̄−ζ
ω−1


i

d ln Aω
1Oω (q) γ1 + (1 − Oω (q)) γ+1−α+ γ1 + 1 Oω (q)−ζd ln c̄ω−1
ε−1

h
i
= α
1
1
ζ
1 + 1 − α ε−1
O
(q)
+
(1
−
O
(q))
ω
ω
ζ
γ
γ+1−α
"

 #
ε − 1 1 − Σ0 d ln Aω
1++ 1 Σ1
= α
ε−1
−ζ
ζ
γ
1 − α ζ d ln c̄ω−1

We next study the impact of c̄ω−1 on

R







E c1−ε |q dQ (q). Note that E c1−ε |q = Γ 1 − α ε−1
q ε−1 Y (q).
ζ

Differentiating yields
d ln E c1−ε |qd ln c̄−ζ
ω−1

"

 #
ε − 1 1 − Σ0 d ln Aω1==α++ 1 Σ1−ζζγ
1 − α ε−1d ln c̄−ζζ d ln c̄ω−1
ω−1d ln Y (q)As a result, integrating yields
R d ln E c1−ε |q dQ (q)d ln c̄−ζ
ω−1

"

 #
ε − 1 1 − Σ̄0 d ln Aω1=α++ 1 Σ̄1−ζζγ
1 − α ε−1
ζ d ln c̄ω−167With these, we can derive an expression for the change in the household’s price index for industry
ω. This price index satisfies
Z

p1−ε
ω

=
Jω


=

where we used Jω = π̄
(1 − ε)d ln p1−ε
ω

χ− β1

= β

d ln c̄−ζ
ω−1

ε
ε−1

1−ε
π̄

"

Z
E

χ− β1

π̄ωβ

Z

1−ε #εc|q dQ (q)ε−1E c1−ε |q dQ (q)π̄ωβ . Differentiating yields

d ln π̄ωd ln c̄−ζ
ω−1

= βλ

p1−ε
ωj dj = Jω

+


R 
d ln E c1−ε |q dQ (q)d ln c̄−ζ
ω−1
!
"
 #

ε − 1 1 − Σ̄0 d ln Aωd ln Aωε−11Ōω + α+ 1 Σ̄1
+α
+
−ζ
ζ
ζ
γ
1 − α ε−1d ln c̄−ζζ d ln c̄ω−1
ω−1d ln pωωω= (ε − η) d ln p−ζ= − ε−ηFinally, since d ln A−ζ−ζ , we haveε−1 (1 − ε)d ln c̄ω

ε−η
=−
−ζ
ε−1
d ln c̄ωd ln Aωd ln c̄ω(βλd ln c̄ω

ε−1
+α
Ōω
−ζ
ζ
d ln c̄ω−1d ln Aω

!

"

 #)
ε − 1 1 − Σ̄0 d ln Aω1+α++ 1 Σ̄1−ζζγ
1 − α ε−1
ζ d ln c̄ω−1ωSolving for d ln A−ζgives

d ln c̄ω



 
ε−1
1
βλ
Ō
+
+
1
Σ̄1
α
ω
ζ
γ
d ln Aω
=
−
ε−1
 <0
α ζ
ε−1
d ln c̄−ζ
ω
+
βλ
+
1
−
Σ̄
0
ε−1
ε−η
1−α
ζ

d ln Aωd ln π̄ωd ln Jω1
ε−1
Claim 6 If η−1ε−η ≥ γ then d ln vω−1 ≥ −Ōω α ζ and d ln vω−1 > 0 and d ln vω−1 > 0Jω= βλProof. First, we have that ddlnlnvω−1in

d ln Aω+ α ε−1
ζ Ōω
d ln c̄−ζ
ω−1
. Under what conditions is the term

d ln Aω+ α ε−1ζ Ōω positive? We haved ln c̄−ζω−1d ln Aωd ln c̄−ζ
ω−1
+1
α ε−1
ζ Ōω

= −

ε−1 γ+1
βλα ε−1ζ Ōω + α ζ
γ Σ̄1
ε−1
ε−η + βλ +

= −

α ε−1
ζ
1−α ε−1ζ

1 − Σ̄0Σ̄1
βλ + γ+1
γ Ō

ω

α ε−1
ζ
ε−1
+
βλ
+
ε−η
1−α ε−1

1 − Σ̄01
 α ε−1 Ōω + 1
ζ

 +1

ζ

=

α ε−1
ζ
ε−1
+
ε−η
1−α ε−1ζε−1ε−η + βλ +

68


Σ̄1
1 − Σ̄0 − γ+1
γ Ō

ω

α ε−1
ζ
1−α ε−1ζ

1 − Σ̄0We are looking for conditions under which the numerator is positive. Since Oω (q) is increasing in
q, and Σ0 (q) =1hi is decreasing in q, we have that Σ̄0 Ōω > Σ̄1 , so1α1+ 1−α ε−1O(q)+(1−Oω (q)) γ+1−αωζγthat the numerator is bounded below by
α ε−1
 γ+1
ε−1
ζ
+
1 − Σ̄0 −Σ̄0ε−1
ε−η 1−α ζγThe RHS is decreasing in Σ̄0 , so since since Σ̄0 <1 
, the following condition guarα
1+ 1−α ε−1ζγ+1−αantees the numerator is positive

0 <

α ε−1
ζ



ε−1
1
1 −


+
ε−1
ε−η 1−α ζ
1 + 1 − α ε−1
ζ

=

γ+1−α

η−1


+1+
ε−η
1 + 1 − α ε−1

α
γ+1−α

α
−1
η−1
 γ+1−α γ+ε − η 1 + 1 − α ε−1

α
γ+1−α

1
− α
η−1
 γ γ+1−α−
ε − η 1 + 1 − α ε−1

α
γ+1−α

ζ

≤

1
− γ+1γ 1 + 1 − α ε−1

γ+1
α
α ε−1
ζ γ+1−α − γ
ζ

=

α
γ+1−α

α
− γ+1
α ε−1
ε−1
ζ γ+1−α  γ+
αε − η 1 + 1 − α ε−1
ζ

=



ζ

or

1
α
η−1
γ − γ+1−α≥ε−η
1 + 1 − α ε−1
ζ

α
γ+1−α

A simpler sufficient condition that guarantees this condition holds is
η−1
1
≥
ε−η
γ

Claim 7O (q)ωd ln c̄−ζω−1ω
d ln 1−O
(q)

> 0.

69

ζ

α
γ+1−α

Oω (q)
Proof. Differentiating each side of 1−O=
ω (q)Oω (q)
d ln 1−Oω (q)d ln c̄−ζω−1=d ln h1 (q)d ln c̄−ζ
ω−1

+1−α

h1 m1 c̄−ζ
ω−1
α −αζ
hα
2 m2 c̄ω−2gives

d ln h2 (q)d ln c̄−ζω−1!
!
1 − α ε−11 − α ε−1d ln Aωd ln Yd ln Aωd ln Y1αζζ
=−+1 +1−−
γ d ln c̄−ζγ + 1 − α d ln c̄−ζ
α ε−1
α ε−1
d ln c̄−ζd ln c̄−ζ
ζ
ζ
ω−1
ω−1
ω−1
ω−1
"





 #! 
1d ln Aωε−1
1 − Σ0 d ln Aω
1
1
α
=
− 1−α
+
+
−
+ 1 Σ1+1−ζ
γ γ+1−αζγγ
1 − α ε−1d ln c̄−ζζ d ln c̄ω−1
ω−1




 ! 

d ln Aω
ε−1
1
1
1
α
Σ0
− 1−α
=
−
+ 1 Σ1 ++1
γ γ+1−αζγγd ln c̄−ζω−1d ln Aω< 0 and bounded below by −α ε−1ζ , Σ0 (q) is decreasing in q and bounded above
d ln c̄−ζ
ω−1
1 
by 
, and Σ1 (q) is increasing in q and bounded above by  1 ε−1  1 . Together,α1+ 1−α ε−1
1+ 1−α ζ
ζ
γ+1−α
γ

Since

these give a lower bound of



 



ε−1
1
ε−1
1
ε−1
− 1−α
+ 1 Σ1 ≥ Σ0 −α− 1−α+ 1 Σ1
Σ0ζγ
ζζγ
d ln c̄−ζ
ω−1



ε−1
1
α ε−1
+
1
−
α
+
1
Oω (q)
ζ
ζ
γ
h
i

= −
1
α
O
(q)
+
(1
−
O
(q))
1 + 1 − α ε−1
ω
ω
ζ
γ
γ+1−α



ε−1
1
α ε−1
ζ + 1−α ζ
γ + 1 Oω (q)





= −
ε−1
α
α
1
1 + 1 − α ε−1
+
1
−
α
−
ζ
γ+1−α
ζ
γ
γ+1−α Oω (q)



ε−1
1
α ε−1
ζ + 1−α ζ
γ + 1 Oω (q)




= −
1−α
ε−1 γ+1α
1 + 1 − α ε−1
+
1
−
α
ζ
γ+1−α
γ+1−α
ζ
γ Oω (q)



ε−1
1
α ε−1
+
1
−
α
+
1
Oω (q)
ζ
ζ
γ

i
h


= −h
α
ε−1 1−α
1−α
ε−1
−
α
+
α
+
1−α
1 + 1 − α ε−1ζ
γ+1−αζ γ+1−αγ+1−αζd ln Aω

Since the first term of the denominator is positive, this is minimized when the fraction is maximized,or Oω (q) → 1. This gives the boundΣ0d ln Aωd ln c̄−ζ
ω−1


− 1−α

ε−1
ζ





1
+ 1 Σ1
γ




ε−1
1
α ε−1
+
1
−
α
+
1
ζ
ζ
γ


≥ −
1
1 + 1 − α ε−1
ζ
γ


1
1 + 1 − α ε−1
ζ
γ


= −
1
1 + 1 − α ε−1
ζ
γ
= −1

70

Plugging this back in yields
Oω (q)
d ln 1−Oω (q)d ln c̄−ζ
ω−1


≥

1
α
−
γ γ+1−α(−1) +

1
+1
γ



γ+1
>0
γ+1−α

=

c̄ω
Claim 8 If γ is large enough, d dlnlnδω−1
< 0.
R


d ln E[c1−ε |q ]dQ(q)Proof. Following the same logic as for the expression for, we have E c−ζ |q =d ln c̄−ζ
ω−1
h
iα
−ζ
−αζ
ζ
α
α
Γ (1 − α) q h1 m1 c̄ω−1 + h2 m2 c̄ω−2 , so that differentiating yields

d ln E c−ζ |qd ln c̄−ζω−1"1 − Σ0 d ln Aω=α+−ζ
1 − α ε−1
ζ d ln c̄ω−1



 #
1
+ 1 Σ1γ

Integrating with across firms gives
d ln c̄−ζωd ln c̄−ζ
ω−1

=


R 
d ln E c−ζ |q dQ (q)d ln c̄−ζω−1"1 − Σ̄ζ0 d ln Aω=α+−ζ
1 − α ε−1
ζ d ln c̄ω−1



 #
1
+ 1 Σ̄ζ1γ

E[c−ζ x]where we define the weighted average x̄ζ ≡ E c−ζ . Since limγ→∞ Σ̄ζ1 = Ōωζ and limγ→∞ Σ̄ζ0 = 1,
[ ]taking the limit as γ → ∞ givesR
1 d ln Cω (q)−ζ dQ (q)lim≥ lim Ōωζ
γ→∞ α
γ→∞
d ln vω
Finally, we show that limγ→∞ Ōωζ > 0. The FOCs for h1 and h2 can be rearranged as
h1 m1 c̄−ζω−1

11+γ−α ε−1
ζ

1−α ε−1ε−11ζAω δω q ε−1 Oω
α
k
ζ



1
ε−1
1−α ε−1ζ α2Aω δω q ε−1 (1 − Oω )kζ=hα2 mα2 c̄−αζ
ω−2 =







m1 c̄−ζω−1



1+γ1+γ−α ε−1ζ 1+γ 1 ε−1 
α −α ζ

1+γ

mα2 c̄−αζ
ω−2

 1+γ α ε−1
α −α ζ

Taking the limit of the ratio gives
h1 m1 c̄−ζOω (q)
ω−1
= lim
lim
= lim
γ→∞ 1 − Oω (q)γ→∞ hα mα c̄−αζ
γ→∞ n
2 2 ω−2n1
ε−1 O 1−αk Aω δ ω q

ε−1
ζ

α ε−1
ζ

o

1
1+γ−α ε−1ζ

ε−11
ε−1 (1 − O)1−α ζ α2 ε−1
k Aω δ ω qζSince this is the same for all q, we have limγ→∞ Ōωζ =m1 c̄−ζω−1α −αζm1 c̄−ζω−1 +m2 c̄ω−2

m1 c̄−ζω−1

o 1+γ 1 ε−1 
α −α ζ



1+γ1+γ−α ε−1ζmα2 c̄−αζ
ω−2



> 0.

Claim 9 If η > 1 and γ is large enough, an increase in δω−1 leads to an increase in industry ω
sales
71

1+γ
α
1+γ
ε−1
α −α ζ

=m1

mα2

Proof. Since η > 1 and pω declines, so sales to household increases. Further, since c̄ω declines,
sales for intermediate use rise as well.

D.2

Proofs for Quantitative Model of Section 5

In this section, we prove the propositions in Section 5. Several of te results parallel proofs for thesimple model of Section 3. Nevertheless, we include the results here both for completeness andbecause the notation is not always the same.
We first derive an expression for the effective cost of outsourcing each input. Let Fω (c) be the
distribution of unit cost among those in industry ω.
Lemma 4 If firm j exerts search effort hj ω̂ to search for suppliers of input ω̂ ∈ Ω̂ω̃ , then Pr cojω̂ > c =
− 1
R −ζ
−ζ ζ
e−hj ω̂ mω̃ω̂ c̄ω̂ c with c̄ω̂ =c dFω̂ (c) ζ .
Proof. If firm j exerts search effort hj ω̂ , the arrival rate of potential suppliers of ω̂ with matchspecific productivity better than z is hω̂ mω̃ω̂ z −ζ . Thus the probability that the firm’s best outside
supplier delivers cost greater than c is
Pr (coω̃ > c) = e−hω̂ mω̃ω̂RR= e−hω̂ mω̃ω̂ cζ

1{ czs ≤c}dFω̂ (cs )ζz −ζ−1 dz

RR

1{ cus ≤1}dFω̂ (cs )ζu−ζ−1 du−ζ ζ= e−hω̂ mω̃ω̂ c̄ω̂ cwhere c̄ω̂ is defined to satisfy
c̄−ζ
ω̂

Z Z
≡

1
Z Z

=
Z
=

nc

s

u

o
≤ 1 dFω̂ (cs ) ζu−ζ−1 duζ−1
1 {t ≤ 1} c−ζdt
s dFω̂ (cs ) ζtc−ζ
s dFω̂ (cs )Lemma 5 Suppose that there are independent exponential random variables X1 ,...,Xn that with 
Pr (Xi > x) = e−κi x , and that b is a random variable with characteristic function E ebit =Γ(1−it)Γ(1−α1 it)...Γ(1−αn it) . Suppose that α1 , ..., αn are non-negative numbers such that α1 + ... + αn ≤ 1.
Then Y = e−b X1α1 ...Xnαn is an exponential random variable with countercumalutive distributionα1αnPr (Y > y) = e−κ1 ...κn y
h
i
R∞
α it
−α it R ∞−α it
Proof. Note that E Xj j
= 0 xαj it κj e−κj x dx = κj j 0 uαj it e−u du = κj j Γ (1 + αj it).
With this, we have that

72


h
i h
it 
i
 it 

αn−b α1= E e−bit E X1α1 it ...E Xnαn it
E Y= E e X1 ...Xn==Γ (1 + it)
n itκ−α1 it Γ (1 + α1 it) ...κ−αΓ (1 + αn it)1Γ (1 + α1 it) ...Γ (1 + αn it) 1
1Γ (1 + it)
α1(κ1 ...καnn )itα1αnSuppose that Z is an exponential random variable with Pr (Z > z) = e−κ1 ...κn z . Then the char acteristic function of log Z = E Z it = α1 1 αn it Γ (1 + it). Since there is a one-to one mapping(κ1 ...κn )between distribution functions and characteristic functions, Y and Z have the same distribution.

Lemma 6 Suppose that there are independent Weibull variables M1 , ..., Mn so that Pr (Mi > m) = ζe−κi m , and that b is a random variable with characteristic function E bit = Γ(1−α1Γ(1−it)it)...Γ(1−αn it) .

Suppose that α1 , ..., αn are non-negative numbers such that α1 +...+αn ≤ 1. Then Y = e−b/ζ M1α1 ...Mnαn

is an Weibull random variable with .
α1

αn ζ

Pr (Y > y) = e−κ1 ...κn y
Proof. If M is Weibull, X = M ζ is exponential, as Pr (X > x) = Pr M ζ > x = Pr M > x1/ζ =α 1 α n1/ζ ζe−κ1 (x ) = e−κ1 x . Since Y ζ = e−b M1ζ
= e−b X1α1 ...Xnαn , it follows that from the... Mnζα1αnprevious lemma that Y ζ follows an exponential distribution with Pr Y ζ > t = e−κ1 ...κn t , which
implies that


α1
αn ζ
Pr (Y > y) = Pr Y ζ > y ζ = e−κ1 ...κn y



−ζ ζ
Lemma 7 Given search effort {hj ω̃ }ω̃∈Ω̂∞ , for any input ω̃, Pr cij ω̃ > c = e−Tj ω̃ c where {Tj ω̂ }ω̂∈Ω∞ωωsatisfy the relationships

Tj ω̃ = w

−αω̃
l ζ

Y 

−ζ
hj ω̂ mω̃ω̂ c̄−ζω̂ + Tj ω̂− αω̃ω̂ζω̂∈Ω̂ω̃

where we use the convention that Tj ω̂ = ∞ if in-house production of ω̂ is infeasible.
Proof. We proceed by induction. Consider first a terminal module. The cost of producing inhouse isciω̃ =

1 αω̃ Y αω̃ω̂
w l
cω̂
Bω̃
ω̂∈Ω̂ω̃orciω̃
1 Y  cω̂ αω̃ω̂
=
w
Bω̃
w
ω̂∈Ω̂ω̃73

Since the distribution of each cwω̃ is Weibull with Pr
ci

cω̂
w >x



−ζ ζ

ζ

= e−w hω̂ mω̃ω̂ cω̂ x , the previous lemma

implies that wω̃ is Weibull with
 i


αω̃
Q
ω̂ ζ
cω̃
− ω̂∈Ω̂ wζ mω̃ω̂ c−ζ
x
ω̂
ω̃
Pr
>x =e
w
or
Pr

ciω̃ > c




 i

αω̃

αω̃
ω̃ Q
Q
ω̂ ζ
ω̂ c ζ−ζ ζcω̃cc− ω̂∈Ω̂ wζ hω̂ mω̃ω̂ c̄−ζ−w−αl ζ ω̂∈Ω̂ hω̂ mω̃ω̂ c̄−ζ
(
)
ω̂
ω̂
w
ω̃
ω̃
= e−Tω̃ c
=e
=e
= Pr
>
w
w


−ζ ζ
Suppose that for all modules with depth of D − 1, the Pr ciω̂ > c = e−Tω̂ c . Then since the cost−ζ ζof the outsourcing the input is also Weibull, Pr (coω̂ > c) = e−hω̂ mω̃ω̂ c̄ω̂ c , we have that the effective
cost of input ω̂ follows a Weibull distribution
Pr (cω̂ > c) = Prciω̂ > chi− Tω̂−ζ +hω̂ mω̃ω̂ c̄−ζcζ−Tω̂−ζ cζ −hω̂ mω̃ω̂ c̄−ζ
cζ
o
ω̂
ω̂
e
=e
Pr (cω̂ > c) = e

Again, since the distribution of each cwω̃ is Weibull with Prciprevious lemma implies that wω̃ is Weibull with

cω̂
w >x



=e

h
i
−wζ Tω̂−ζ +hω̂ mω̃ω̂ c̄−ζ
xζ
ω̂

, the


 i
 h
iαω̃
Q
ω̂ ζ
cω̃
x
− ω̂∈Ω̂ wζ Tω̂−ζ +hω̂ mω̃ω̂ c̄−ζ
ω̂
ω̃
>x =e
Pr
w
or
Pr

ciω̃ > c



h
iαω̃
 i

 h
iαω̃
ω̃ Qαω̃Qω̂ ζω̂ c ζω̂−ζ ζ−w−αl ζTω̂−ζ +hω̂ mω̃ω̂ c̄−ζccω̃c− ω̂∈Ω̂ wζ Tω̂−ζ +hω̂ mω̃ω̂ c̄−ζ
)
(
ω̂
ω̂
w
ω̂∈
Ω̂
ω̃
ω̃
= Pr
=e
>
=e
= e−Tω̃ cw
w

Claim 10 Given search effort {hω }, the probability of outsourcing input i conditional on using itin production ishω̃ mω̃ω̂ c̄−ζω̂−ζhω̃ mω̃ω̂ c̄−ζω̂ + Tω̃This is independent of the firm’s cost or of the probability of outsourcing other inputs.

74

Proof. This follows from usual math with Weibulls
Pr coω̃ ≤ ciω̃ , cω̃ = coiPr cω̃ ≤ cω̃ |cω̃ = c =Pr coω̃ = c
Pr coω̃ ≤ ciω̃ , coω̃ = c
=Pr coω̃ = c=


Pr ciω̃ ≥ c, coω̃ = cPr ciω̃ ≥ c, coω̃ = c + Pr coω̃ ≥ c, ciω̃ = c
−ζ ζ

=

=
=

−ζ ζ

ζ−1 e−hω̃ mω̃ω̂ c̄ω̂ c e−Tω̃ chω̃ mω̃ω̂ c̄−ζω̂ ζcPr coω̃ = c
h
i
−ζ ζ
−ζ ζ
−ζ
−ζ
−ζ
h
m
c̄
+
T
ζcζ−1 e−hω̃ mω̃ω̂ c̄ω̂ c e−Tω̃ c
ω̃
ω̃
ω̂
ω̃
ω̂
hω̃ mω̃ω̂ c̄ω̂

o =c
−ζ
Pr
c
hω̃ mω̃ω̂ c̄−ζ
+
T
ω̃
ω̃
ω̂

hω̃ mω̃ω̂ c̄−ζω̂−ζhω̃ mω̃ω̂ c̄−ζω̂ + Tω̃

Since this is true for all c, it follows that the choice of in-house or outsourcing is independent ofdecision downstream from the input ω̃, and the same argument used for inputs upstream fromω̃ implies that outsourcing decisions are independent. Finally, integrating gives Pr coω̃ ≤ ciω̃ =hω̃ mω̃ω̂ c̄−ζω̂E Pr coω̃ ≤ ciω̃ |cω̃ = c =−ζ−ζ .
hω̃ mω̃ω̂ c̄ω̂ +Tω̃

We next turn to the firm’s expected profit
Claim 11 Given its productivity q and its search effort {hω̃ }ω̃∈Ωoω , a firm’s expected profit is
Aω q

ε−1

X

Tω1−ε −

ω̂∈Ω̂oω

where
Tω̃ =

Y 

−ζ
hω̂ mω̃ω̂ c̄−ζω̂ + Tω̂

h1+γ
wk ω̂
1+γ

− αωω̂ζ,∀ω̃ ∈ Ωiωω̂∈Ω̂ω̃with the convention that Tω̂ = ∞ for inputs that for which in-house production is infeasible, i.e.,ω̂ 6∈ Ωiω .

1−ε
ε
1−ε .
upη pε−η
Proof. For a firm in industry ω with cost c, profit gross of search costs is 1ε ε−1
ω c
h
i
ω Q
αω
It is thus sufficient to show that E c1−ε |q, {hω̃ }ω̃∈Ω̂o = κq ε−1 Tω1−ε . Since c = 1q wαl ω̂∈Ω̂ω cω̂ω̂ωand the cost of each input is independent conditional on search effort, we have

1−ε

 

Y  cω̂ αωω̂
c 1−ε
1

|q, {hω̃ }ω̃∈Ω̂o= E 
|q, {hω̃ }ω̃∈Ω̂o 
E
ω
ω
w
q
w
ω̂∈Ω̂ω  ωYcω̂ αω̂ (1−ε)
ε−1
= q
E
|q, {hω̃ }ω̃∈Ω̂oωwω̂∈Ω̂ω

75h  iζ−rNote that if X follows a Weibull distribution with Pr (X > x) = e−κx , Then E wx
=

 r
R
r R
r
ζ
−
wr x−r κζxζ−1 e−κx dx = wr κ ζ u ζ e−u du = wr Γ 1 − ζr κ ζ . Thus we can express this as αω ε−1
Y 
ε−1 
c 1−ε
ω̂ ζ
−ζ
(ε−1)
ε−1 (1−αω
ω
)
l
E
= q w
Γ 1 − αω̂|q, {hω̃ }ω̃∈Ω̂ohω̂ mωω̂ c̄−ζ
+
T
ω̂
ω̂
ω
w
ζ
ω̂∈Ω̂ω




Y 
ω
ε
−
1
w(1−αl )(ε−1)Γ 1 − αω̂ω=
q ε−1 Tω1−εζω̂∈Ω̂ω

Therefore Aω = 1ε
D.2.1



ε
ε−1

1−ε

n

o
Q
ω
upη pε−ηw(1−αl )(ε−1) ω̂∈Ω̂ω Γ 1 − αω̂ω ε−1.
ωζOptimal Search Effort

The firms problem can be expressed as a cost-minimization problem. let Kω (Tω ) be the minimum
cost of delivering a cost distribution indexed by Tω . Then the firm’s problem can be expressed ashi1−ε
E Π|q, {hω̂ }ω̂∈Ω̂o = max Aω q ε−1 Tω ζ − wKω (Tω )ωTω

where each cost function Kω can be expressed recursively

X  k
1+γ
Kω (Tω ) =minh+ Kω̂ (Tω̂ )1 + γ ω̂{hω̂ ,Tω̂ }ω̂∈Ω̂ωω̂∈Ω̂ω

subject to

iαω
Y h
−ζ ω̂
hω̂ mωω̂ c̄−ζω̂ + Tω̂ωTω ≤ w1−αlω̂∈ΩωWe now derive some properties of Kω . In doing so, it will be useful to define {κω , βω } iteratively
as follows.
βω =Xω̂∈Ω̂leafωXαω̂ω +

non−leafω̂∈Ω̂ωαωκω =Yω̂∈Ω̂leafωβω̂ αω̂ω! ω̂ c̄ −ζ 1+γ αω βω
ω̂
ω̂
mωω̂
w
βω


 βω̂ αωω̂βω̂ αω̂ω βωκω̂βωnon−leafYω̂∈Ω̂ω

Claim 12 Each Kω (·) is such that Kω (0) = 0, strictly decreasing, strictly convex, twice continuously differentiable, and the policy functions {hω̂ (·) , Tω̂ (·)}ω̂∈Ωω are differentiable and strictly
ln hω̂
monotonic, with dd ln
Tω >d ln Tω̂−ζd ln TωOω̂0> 0, and dd ln
ln Tω > 0. The elasticity of Kω (·) is bounded by76

00
ζ(1+γ)
ω (Tω )Tω
≤ K−K
− 1 ≤ ζ(1+γ). Finally0βω1−αωω (Tω )lζ(1+γ)1−αωllim Kω (Tω ) Tω

Tω →0

=

1

k
1+γ
Q

ζ(1+γ)

lim Kω (Tω ) Tω βω

=

Tω →∞


ω̂∈Ω̂ω

mωω̂ c̄−ζ
ω̂

1+γ

αω
ω̂
1−αρl

 αωω̂ω1−αlk 1
1 + γ κω

Proof. By induction. If there is no in-house option (i.e., a module of depth 1) then we haveKω (Tω ) = min{hω̂ }kh1+γ
1 + γ ω̂X

subject to Tω ≥ w

Y 

αω
l

ω̂∈Ω̂ω

hω̂ mωω̂ c̄−ζω̂− αωω̂ζω̂∈Ω̂ω

Letting λ be the multiplier, we have, the FOCs areωkhγω̂ = wαl λorαω̂ω 1
Tω
ζ hω̂
ω

h1+γ
=
ω̂

αω̂ω wαl λTω
ζ
k

1+γ

Multiplying each side by mωω̂ c̄−ζand combining across ω̂ gives
ω̂
ω

αω

Y 
ω̂∈Ωω

hω̂ mωω̂ c̄−ζω̂1+γ  ζω̂


=

ω
wαl λ

k

Tω

l
 1−α
ζ
Y 

αω

mωω̂ c̄−ζω̂ω̂∈Ω̂ω

Using the constraint and rearranging, we haveωω −(1+γ)
/wαl

Tω
=
αω

1+γ ω  ζω̂Qαω̂mωω̂ c̄−ζω̂ζω̂∈Ω̂ω

77ω
w αl λ

k

l
 1−α
ζ

Tω

1+γ αω  ζω̂ω̂ζWe can then use these to derive an explicit expression for the cost function:ωKω (Tω ) =ω̂∈Ω̂ω

ω̂∈Ω̂ω

(
=

=

1−αω
l

)k
ζ(1+γ)Qω

X k αω wαl λTω
(1 − αlω ) k wαl λTω
k
ω̂
=
h1+γ
=
ω̂
1+γ
1+γ ζ
k
ζ (1 + γ)kXω̂∈Ω̂ω

mωω̂ c̄−ζ
ω̂

k
1+γ



1+γ

αω
ω̂
ζ



αω
ω̂
1−αω
l

Tω
ω
w αl

− ζ(1+γ)
ω
1−α

l



1
Q


ω̂∈Ω̂ω

mωω̂ (c̄ω̂ /w)−ζ

1+γ

αω
ω̂
1−αω
l



αω
ω̂
1−αω
l

Tω
w

− ζ(1+γ)ω
1−αlSuppose that the hypothesis is true for all depths less than D. The problem for a module at depth
D is


X  k
1+γ
+ Kω̂ (Tω̂ )hKω (Tω ) = min
1 + γ ω̂{hω̂ ,Tω̂ }ω̂∈Ωωsubject toTω ≥ w

αω
l

Y 

−ζ
hω̂ mωω̂ c̄−ζω̂ + Tω̂− αωω̂ζω̂∈Ω̂ω

Let λ be the multplier, and define Oω̂ =produced in-house, then FOC ishω̂ mωω̂ c̄−ζω̂−ζhω̂ mωω̂ c̄−ζω̂ +Tω̂ωkhγω̂ = wαl λTω

. If input ω̂ is a leaf so that it cannot be

αω̂ω
ζhω̂

Otherwise, the FOCs areωhω̂ : khγω̂ = wαl λOω̂ Tω

αω̂ω
ζhω̂

ω

Tω̂ : Kω̂0 (Tω̂ ) = −wαl λ (1 − Oω̂ ) Tω

αω̂ω
Tω̂

These along with the constraint define the policy functions and the Lagrange multiplier. The
implicit function theorem implies that the Lagrange multiplier is differentiable with repect toTω . Differentiating the FOCs with respect to Tω−ζ and noting that d ln Oω̂ = (1 − Oω̂ ) d ln hω̂ −(1 − Oω̂ ) d ln Tω̂−ζ gives(1 + γ)1−ζKω̂00 (Tω̂ ) Tω̂
+1
Kω̂0 (Tω̂ )d ln hω̂d ln Tω−ζd ln Tω̂−ζd ln Tω−ζ==d ln λTωd ln hω̂d ln Tω̂−ζd ln Tωd ln Tωd ln Tω−ζ+ (1 − Oω̂ )−ζ

d ln λTωd ln Tω−ζ− Oω̂

ln hω̂
We can use these two expressions to solve for dd ln
Tω and78− (1 − Oω̂ )−ζ

d ln hω̂d ln Tω−ζd ln Tω̂−ζd ln Tω−ζ+ Oω̂

d ln Tω̂−ζd ln Tω−ζ(6)(7)

in terms of d ln λT−ζω for the nonleaf
d ln Tωnodes:d ln hω̂d ln Tω−ζd ln Tω̂−ζd ln Tω−ζ=1d ln λTω
γ + Oω̂ + rω̂ (1 − Oω̂ ) d ln Tω−ζ=rω̂d ln λTω
γ + Oω̂ + rω̂ (1 − Oω̂ ) d ln Tω−ζ

where we defined rω̂ ≡ −  K 00 (T γ)T.

ω̂ ω̂ ω̂ +1
K 0 (Tω̂ )
ω̂

1
ζ

+1

To solve for d ln λT−ζω , we differentiate the constraint to get
d ln Tω1=αω̂ω̂∈Ω̂leafω

Using the expressions ford ln λTωd ln Tω−ζ(ω d ln hω̂X=d ln Tω−ζd ln hω̂d ln Tω−ζ


 X

X

+

αω̂ω

Oω̂

ω̂∈Ω̂non−leaf
ω

and

αω̂ω



ω̂∈Ω̂leafωd ln Tω̂−ζd ln Tω−ζ1+γ+1d ln hω̂d ln Tω−ζ+ (1 − Oω̂ )d ln Tω̂−ζ)d ln Tω−ζand solving for d ln λT−ζω gives
d ln TωXαω̂ωω̂∈Ω̂non−leaf
ω

−1


Oω̂ + rω̂ (1 − Oω̂ )γ + Oω̂ + rω̂ (1 − Oω̂ ) K 00 (T )T

ω̂ ω̂
ω̂
The bounds on −K
0 (T ) imply that rω is bounded inside the [0, 1] interval
ω̂
ω̂

γ
γ
0 < 1+γ≤ rω̂ ≤ 1+γ
<1
−
1
ω̂
βω̂ − 1
1−α

(8)

l

In particular, since rω̂ is strictly smaller than 1 and larger than 1+γγβω̂and below:
d ln λTωd ln Tω−ζd ln λTωd ln Tω−ζ


X

≤

αω̂ω


ω̂∈Ω̂ω X≥ω̂∈Ω̂leaf
ω

−1


1
γ + 1

αω̂ω

1
+
γ+1

=

αω̂ω

non−leafω̂∈Ω̂ωβω̂ω̂where the last line used γ+r= 1+γ
and the definition of βω .
ω̂79

, we can bound d ln λT−ζω above
d ln Tω1+γ
1 − αlω
X

r

−1

−1


rω̂
γ + rω̂ 

=

1+γ
βω

Three consquences are that the changes in the policy functions are bounded away from 0:d ln hω̂d ln Tω−ζd ln Tω̂−ζd ln Tω−ζOω̂d ln 1−Oω̂−ζd ln Tω

=

=


X

−1


1
1
rω̂
d ln λTω≥αω̂ω>0γ + Oω̂ + rω̂ (1 − Oω̂ ) d ln Tω−ζ
γ + rω̂ 
γ + rω̂ ω̂∈Ω̂ω

−1


X
rω̂
d ln λTωrω̂ω r ω̂
≥α>0ω̂γ + Oω̂ + rω̂ (1 − Oω̂ ) d ln Tω−ζ
γ + rω̂ 
γ + rω̂ ω̂∈Ω̂ω

=d ln Tω̂−ζd ln hω̂
1 − rω̂d ln λTω−=−ζ−ζ
γ + Oω̂ + rω̂ (1 − Oω̂ ) d ln Tω−ζd ln Tωd ln Tω
−1≥

1 − r̄ω̂  X ω rω̂ αω̂
γ + rω̂ 
γ + rω̂ >0ω̂∈Ω̂ω

In addition, since all of these derivatives are positive and bounded away from 0, we have thefollowing asymptotic properties.
lim hω̂ (Tω ) = ∞Tω →0lim Tω̂ (Tω ) = 0

Tω →0lim Oω̂ (Tω ) = 1

Tω →0

lim hω̂ (Tω ) = 0

Tω →∞lim Tω̂ (Tω ) = ∞Tω →∞lim Oω̂ (Tω ) = 0

Tω →∞
00

ω )Tω
0
We next derive bounds on KKω (T
0 (T ) . The envelope theorem implies that Kω (Tω ) = −λω < 0,ωω

so taking logs and differentiating gives
Kω00 (Tω ) Tωd ln λd ln λTωd ln λTω==− 1 = −ζ
−10Kω (Tω )d ln Tωd ln Tωd ln Tω−ζThis can be rearranged as

Kω00 (Tω ) Tωd ln λTω
−1=ζ0−Kω (Tω )d ln Tω−ζThe bounds on d ln λT−ζω gived ln Tωζ (1 + γ)
K 00 (Tω ) Tω
ζ (1 + γ)
≤ ω 0
−1≤
βω
−Kω (Tω )1 − αlωFinally, we turn to the asymptotic behavior of Kω . We first solve for limiting behavior of λTω

80

by rearranging the constraint
Tω = w

Y 

αω
l

−ζ
hω̂ mωω̂ c̄−ζω̂ + Tω̂− αωω̂ζω̂∈Ω̂ω

= w

Y 

αω
l

hω̂ mωω̂ c̄−ζω̂− αωω̂ω̂∈Ω̂leafω= w αlhω̂ mωω̂ c̄−ζω̂− αωω̂Yζ1+ω̂∈Ω̂non−leafωω̂∈Ω̂ω= w−ζhω̂ mωω̂ c̄−ζω̂ + Tω̂− αωω̂ζ

non−leafω̂∈Ω̂ω

Y 
ω

Y 

αω
l



Y

ζ

hω̂ mωω̂ c̄−ζω̂− αωω̂Yζω̂∈Ω̂non−leafωω̂∈Ω̂ω1Oω̂!− αωω̂Tω̂−ζζhω̂ mωω̂ c̄−ζω̂− αωω̂ζThe FOCs for hω̂ implyωY hω̂ mωω̂ c̄−ζω̂− αωω̂ζ=ω̂∈Ω̂ω

Yωwαl λT!− αζω̂1

ω  1+γω αω̂kmωω̂ c̄−ζ
ω̂

ζ

=

ω

wαl λTωk(Oω̂ )

11+γ

− αωω̂ζω̂∈Ω̂non−leafωω̂∈Ω̂ω





Y

1−αω
l
− (1+γ)ζ

Y  αω 



ω̂

ζ

αω

mωω̂ c̄−ζ
ω̂

αω

ω̂
1+γ − (1+γ)ζ

ω̂−Oω̂ ζ(1+γ)

Yω̂∈Ω̂non−leafωω̂∈Ω̂ω

Plugging this in gives
ω


Tω =

ω

wαl λTωk1−αl
− (1+γ)ζ

w

Y  αω 

αω
l

ω̂

ζ

αω

mωω̂ c̄−ζω̂ω̂1+γ − (1+γ)ζ

ω

Tω =

ω
wαl λTω

1−αl
− (1+γ)ζ

w

k


αω̂ω

Y

mωω̂

ζ

Oω̂

non−leafω̂∈Ω̂ω

ω̂∈Ω̂ω



ω
γ αω̂
1+γ ζ

Y

 c̄ −ζ 1+γ!− αωω̂(1+γ)ζ

ω̂

Y

w

ω
γ αω̂
1+γ ζ

Oω̂

ω̂∈Ω̂non−leafωω̂∈Ω̂ω

Rearranging and taking the limit and using limTω→0 Oω̂ = 1 gives

lim

Tω →0

w

αω
l

λTω
k



Tω
w

 ζ(1+γ)
ω
1−α

l



1  Y
= lim αωTω →0 w l 


ω 

αω̂
ζ

mωω̂

 c̄ −ζ 1+γ!− αωω̂ω̂=ω̂∈Ω̂

ω 

αω̂
ζ

mωω̂

ω

81

Y

w

ω̂∈Ω̂ω

 Y

ζ(1+γ)

ω̂∈Ω̂ω

 c̄

ω̂

w


! αωω̂ω −1

1+γ
1−α

−ζ
l




αω
ω̂ γ
ζ 1+γ

Oω̂

 ζ(1+γ)
1−αω
l





We next use the objective function
k
h1+γ
+
ω̂
1
+
γ
leaf



X

X

ω̂∈Ω̂ω

ω̂∈Ω̂non−leafωKω (Tω ) =

k
h1+γ
+
ω̂
1
+
γ
leaf

k
h1+γ
ω̂
1
+
γ
non−leafX=ω̂∈Ω̂ω

kh1+γ + Kω̂ (Tω̂ )1 + γ ω̂X

!

Kω̂ (Tω̂ )

1+

1+γ
k
1+γ hω̂ω̂∈Ω̂ω

Using the FOCs for hω and rearranging gives
αω

ω

X

αω

ω

wαl λTω ζω̂

wαl λOω̂ Tω ζω̂XKω̂ (Tω̂ )!

1 + k 1+γ+
1+γ
1+γ
non−leaf
1+γ hω̂ω̂∈Ω̂ω

!
ω
X
αω̂ω
Kω̂ (Tω̂ ) k wαl λTω  X αω̂ω+Oω̂ 1 + k 1+γ


1+γ
k
ζ
ζ
h
non−leaf
leaf

Kω (Tω ) =ω̂∈Ω̂leafω=ζ(1+γ)1−αωl(Tω̂ )
Note that since Kω̂1+γ= Kω̂ (Tω̂ ) Tω −ζ 1+γhω̂lim Kω (Tω )

Tω →0

Tω
w

 ζ(1+γ)
ω
1−α

l

=

Tω
hω̂

 ζ(1+γ)1−αωT0. As result, multiplying through by

ω

l

w

1+γ ω̂

ω̂∈Ω̂ω

ω̂∈Ω̂ω1
ζ(1+γ) 1− 1−αωlTω(Tω̂ ), taking the limit gives limTω →0 Kω̂1+γ=hω̂

and taking the limit yields  ζ(1+γ)
ω
ω
α
X αω
1−α
l
w λTω   X αω̂ω
k
Tω
l
ω̂

lim
+
lim Oω̂
Tω →0
1 + γ Tω →0
w
k
ζ
ζ
leaf
ω̂∈Ω̂ω

ω̂∈Ω̂ω

=

=

P
αω
ω̂
k
ω̂∈Ω̂ω ζ  αωω̂ω 
1+γ

1−α
−ζ 1+γQlαωω̂mωω̂ c̄wω̂
ζ
ω̂∈
Ω̂
ω


1

k
1+γ


Q

ω̂∈Ω̂ω

αω
ω̂

1−αω
l



mωω̂

 1+γc̄ω̂ −ζ αωω̂ω1−αlw

We next turn to the limit as Tω → 0 and take a similar approach
ζ(1+γ)βω̂Suppose that limTω̂→∞ Kω̂ (Tω̂ ) Tω̂ζ(1+γ)

lim

Tω̂→∞

β
Kω̂0 (Tω̂ ) Tω̂ ω̂

+1

k 1
→ 1+γ
κω̂ . Then limTω̂ →0ζ(1+γ)
Kω̂0 (Tω̂ ) Tω̂βKω̂ (Tω̂ ) Tω̂ ω̂ == lim
Tω̂→∞ Kω̂ (Tω̂ )82



0 (T )TKω̂ζ(1+γ)
ω̂ ω̂
andKω̂ (Tω̂ ) = − βω̂ζ (1 + γ)
−
βω̂



k 1
1 + γ κω̂


=−

ζ 1
k
βω̂ κω̂

1+Kω̂ (Tω̂1+k1+γ hω̂

The FOCs giveTω = w

Y 

αω
l

−ζ
hω̂ mωω̂ c̄−ζω̂ + Tω̂− αωω̂ζω̂∈Ω̂ω

= w

Y 

αω
l

hω̂ mωω̂ c̄−ζω̂− αωω̂ζω̂∈Ω̂leaf
ω

Y 

ω

= w αlY ω−ζhω̂ mωω̂ c̄−ζω̂ + Tω̂− αωω̂ζω̂∈Ω̂non−leafωhω̂ mωω̂ c̄−ζω̂− αωω̂
ζ

αω
ω̂

Y

hω̂ mωω̂ c̄−ζ
ω̂



αω
− ζω̂

ω̂∈Ω̂leafω

non−leafω̂∈Ω̂ωαωYhω̂ mωω̂ c̄−ζ
ω̂
+1
Tω̂−ζ

Y

Tω̂

ω̂∈Ω̂non−leafωω̂∈Ω̂leafω= w αlYω̂∈Ω̂non−leaf
ω

ζ

αω
ω̂

Y

Tω̂ ω̂!− αωω̂(1 − Oω̂ ) ζnon−leafω̂∈Ω̂ωPlug in the FOCs to getωTω = wαωlYω̂∈Ω̂leaf
ω

Tω
w


=

ω

wαl λTωkωwαl λTω αω̂ωkζβω− ζ(1+γ)

1
1+γ!− αζω̂− βω̂ αωω̂ζ(1+γ)αωωY w l λ (1 − Oω̂ ) Tω αω̂ (1 −ζ(1+γ)+1βω̂non−leaf
non−leaf0ω̂∈Ω̂ω
ω̂∈Ω̂ω
−Kω̂ (Tω̂ ) Tω̂ωα! βω̂ αωω̂
!− ω̂


ζβ
+1
ζ(1+γ)
ζ(1+γ)
1+γ


0
ω̂
ω
Y
Y
−Kω̂ (Tω̂ ) Tω̂αω̂c̄ω̂ −ζ
mωω̂
ω
ζ
w
kαω̂
non−leafnonYω̂∈Ω̂leafωmωω̂ c̄−ζω̂Yω̂∈Ω̂ω

ω̂∈Ω̂ω

Rearranging and taking the limit givesω

wαl λTω
lim
Tω →∞
k



Tω
w

Q

 ζ(1+γ)
βω

=

=

lim (
Q

Tω →∞



 Y



ω̂∈Ω̂leaf
ω

= ζ



 Y


ω̂∈Ω̂leaf
ω

αω
ω̂
ζ



mωω̂

(1 − Oω̂ )non−leafω̂∈Ω̂ω ω 1+γ αω̂ Qc̄ω̂ −ζ
w

αω
ω̂
ζ


  ζ(1+γ)βω̂βω
1− 1+γ
non−leafω̂∈Ω̂ωαω! ω̂ c̄ −ζ 1+γ βω
αω̂ω
ω̂
mωω̂
ζ
w

Y

αω̂ω

ω̂∈Ω̂non−leafωζ 1βω̂ κω̂kαωω̂0 (T )T ζβω̂ +1
−Kω̂
ω̂ ω̂


! βω̂ αωω̂ −1
βω 



αω



αω̂ω mωω̂ω̂∈Ω̂leaf
ω

83

 c̄

ω̂

w

! ω̂
−ζ 1+γ βωYnon−leafω̂∈Ω̂ω

(κω̂ βω̂ αω̂ω )βω̂ αω
ω̂
βω

−1





βω̂ αω )ω̂We rearrange the value of the objective function Kω
k
h1+γ
+
ω̂
1
+
γ
leaf

X

Kω (Tω ) =ω̂∈Ω̂ω

kh1+γ+
1 + γ ω̂leafXω̂∈Ω̂ω

kh1+γ + Kω̂ (Tω̂ )1 + γ ω̂




k
Kω̂
h1+γ
+
−Kω̂0 (Tω̂ ) Tω̂ω̂0
1+γ−Kω̂ (Tω̂ ) Tω̂

non−leafω̂∈Ω̂ω

XX=



non−leafω̂∈Ω̂ω



Usign the FOCs, this is


ω
ω
X

αω
k wαl λTω αω̂ω
k wαl λTωKω̂ωKω (Tω ) =wαl λ (1 − Oω̂ ) Tω αω̂ω+Oω̂ ω̂ +
0
1+γ
k
ζ
1+γ
k
ζ
−Kω̂ (Tω̂ ) Tω̂ω̂∈Ω̂leafω̂∈Ω̂non−leaf
ω
ω




ω
X
αω̂ω
k wαl λTω  X αω̂ωKω̂ (Tω̂ )(1 − Oω̂ )=Oω̂ + ζ (1 + γ)
+


1+γ
k
ζ
ζ
−Kω̂0 (Tω̂ ) Tω̂
leaf
non−leafXω̂∈Ω̂ω

Multiplying through by (Tω /w)
lim Kω (Tω )

Tω →∞

Tω
w

 ζ(1+γ)βωω̂∈Ω̂ω

ζ(1+γ)βω=klim
1 + γ Tω →∞

=

k
1+γ
Q

=

E

and taking the limit givesω
wαl λTωω̂∈Ω̂leaf
ω



k

Tω
w
P





 ζ(1+γ)
βω

lim


 X

αω̂ω
+
ζ
leaf

αω̂ω
ζ
non−leafXTω →∞ ω̂∈Ω̂ω
ω̂∈Ω̂ω
Pωω
non−leaf αω̂ βω̂leaf αω̂ +ω̂∈Ω̂ω
ω̂∈Ω̂ω

αω̂ω mωω̂ 1+γc̄ω̂ −ζw

 αωω̂βωQω̂∈Ω̂non−leafωκω̂ βω̂ αω̂ω βω̂ αωω̂

k 1
1 + γ κω

Notes about Characteristic Function

In this section we prove the existence of an infinitely divisible random with characteristic functionP
Q Γ(1−it)j αj ≤ 1. The proof builds on Shanbhag,
Γ(1−αj it) for any α1 , ..., αJ such that αj > 0 andj

Pestana and Sreehari (1977) who showed the existence of a closely related random variable.35
35

Shanbhag, Pestana and Sreehari (1977) show the existence of a random variable with characteristic function VβΓ(1+θ α)αsuch that E e−θV = Γ(1+θ)Γ(1+θβ)
for β ≥ 1−αand Re θ > −α/β. One can use this result to show the existence ofour random variable. To see this, their result implies the existence of a random variable with characteristic
P functionΓ(1−it)p
for any p ∈ (0, 1) by using θ = −i(1 − p), α = p, and β = 1−p. Next, letting ᾱj = jj̃=1 αj̃ , weΓ(1−pit)Γ(1−(1−p)it)
can decompose the characteristic function into a product.
Γ (1 − it)Γ (1 − ᾱ2 it)Γ (1 − ᾱ3 it)Γ (1 − it)=...
Γ (1 − α1 it) ...Γ (1 − αn it)Γ (1 − α1 it) Γ (1 − α2 it) Γ (1 − ᾱ2 it) Γ (1 − α3 it) Γ (1 − ᾱn it)84βωOω̂ + ζ (1Lemma 8 (Malmsten’s formula): If Re (z) > 0, thenZ ∞elog Γ (z) =−t0e−tz − e−t(z − 1) +1 − e−t



dt
t
0

(z)
Proof. Start with Gauss’s expression for the digamma function ψ (z) ≡ ΓΓ(z)when Re (z) > 0:no
R ∞ e−te−tz
ψ (z) = 0t − 1−e−t dt and integrate from 1 to z

Z z
log Γ (z) =====ψ (x) dxZ1 z Z ∞  −t
e
e−tx
dtdx
−
t
1 − e−t10

Z ∞ Z z  −t
e
e−tx
dxdt
−
t
1 − e−t01Z ∞  −te
1 e−tz1 e−t(z − 1) +−dttt 1 − e−tt 1 − e−t0Z ∞
e−tz − e−t dt−te (z − 1) +1 − e−tt0Lemma 9 If α positive and real and Re(θ) > −α, then the following two identities hold:Γ(α + θ)
log
Γ(α)
log

Γ(α + θ)
Γ(α)

Z 0 


eαt
eθt − 1 − θtdt|t| (1 − et )
−∞

Z 0 
θt
eαt dtθt= [ψ (α) + bα ] θ +e −1−
1 + t2 (1 − et ) |t|−∞= ψ (α) θ +where ψ (·) is the digamma function and bα ≡

(9)
(10)

R ∞ e−αs0s2ds is positive and bounded.
1−e−s (1+s2 )Proof. Malmsten’s formula can be rearranged, changing variables, to deliverZ ∞  −zu
e
− e−u
−u du
+ (z − 1) elog Γ (z) =1 − e−uu0Z −∞  zt
t
e −e
dt
=
+ (z − 1) et
t
1−e
t
0

Z 0  zt
t
e −e
dt
=
+ (z − 1) et
t
1−e
|t|
−∞
Since Re (α + θ) > 0 and Re (α) > 0, we can apply Malmsten’s formula to both Γ(α + θ) and Γ(α)
Let Xj be a random variable with characteristic functionΓ(1−it)

 
.
ᾱj−1
α
it Γ 1− ᾱj it
Γ 1− ᾱjSimilarly, let S be the randomjΓ(1−it)variable with characteristic function Γ(1−(this is the log of a stable random variable). Then S +ᾱn it)the characteristic functionΓ(1−it).
j Γ(1−αj it)

Q

85

Pn

j=2 ᾱj Xj has

and simplify to get)
Z 0 ( (α+θ)tZ 0  αtdte
− ete − ettt dt+ (α + θ − 1) e
+ (α − 1) e=−
1 − et|t|
1 − et
|t|
−∞
−∞
(
)
Z 0
e(α+θ)t
eαt
dt
=
−
+ θet
t
t
1−e
1−e
|t|
−∞
Z 0 h
Z
i eαt dt0dtθet=eθt − 1+(11)t
(1 − e ) |t|
|t|
−∞
−∞

Γα + θ)log
Γ(α)

To get (9) we subtract and add θt from the inner brackets of the first term and simplify to getΓ (α + θ)
log
Γ (α)

Z 0 h

Z 0
eαt dt
dt
=
θt
+
θet
t
(1 − e ) |t|
|t|
−∞
−∞
−∞

Z 0 
Z 0 h
i eαt dt
αt
t
t e
e
+θ
+
dt
=
eθt − 1 − θtt ) |t|
t)
(1
−
e
|t|
(1
−
e
|t|
−∞
−∞

Z 0 h
Z ∞  −si eαt dte−αseeθt − 1 − θt
=
−
ds
+
θ
s
1 − e−s(1 − et ) |t|
−∞
0
Z 0 h
i eαt dteθt − 1 − θt+ θψ (α)(1 − et ) |t|−∞ie − 1 − θtθteαt dt+(1 − et ) |t|

Z 0

where the last line uses Gauss’s expression for the Digamma function.
θt
To get (10),we subtract and add 1+t
2 from the inner brackets of the first term in 11 and simplify

to getΓ (α + θ)
log
Γ (α)

Z 0 


Z 0
Z 0
eαt dtθtθteαt dt
dt
=
e −1−
+
+
θet
2
t
2
t
1 + t (1 − e ) |t||t|−∞−∞ 1 + t (1 − e ) |t|
−∞


Z 0 
Z 0 
αt
αt
t
θt
e dt
t
e
e
=
eθt − 1 −
+θ
+
dt
2
t
2
t
1 + t (1 − e ) |t||t|−∞−∞ 1 + t (1 − e ) |t|θt

The second integral can be expressed as
Z 0 
−∞

t
eαt
et
+
2
t
1 + t (1 − e ) |t| |t|bα is positive.

Z ∞  −s
e
1
e−αs
−
ds
dt =
s
1 + s2 1 − e−s0
Z ∞  −s
Z ∞  −αs
e
e−αs
e
e−αs
=
−
ds +
−
ds
s
1 − e−s1 − e−s (1 − e−s ) (1 + s2 )00
= ψ (α) + bα

1s
To show that bα is bounded, we use 1+s
2 ≤ 1 and 1−e−s ≤ 1 + s (the latter

follows from adding ses to each side of 0 ≤ es − (1 + s) to get ses ≤ (es − 1) + s (es − 1), ors ≤ (1 + s) (1 − e−s )). Together, these give
Z ∞
bα =
0

e−αs
s2
ds =
1 − e−s 1 + s2

Z ∞
e
0

−αs

s
1
s
ds ≤
−s
1 − e 1 + s2
86

Z ∞
0

e−αs s (1 + s) sds < ∞rLemma 10 Let φr (x) = 1−x
1−x , 0 < x < 1, r > 0. Then for fixed r ≤ 1, φr (x) is decreasing in x;
for r ≥ 1, φr (x) is increasing in x.
Proof. Consider first the case where r ≤ 1. We will use a Taylor expansion of f (t) = 1 − (1 − t)rQk−1(j − r),around t = 0 for t ∈ (0, 1). Noting that f 0 (t) = r(1 − t)r−1 and f (k) (t) = (1 − t)r−k r j=1
k ≥ 2, the Taylor expansion is thusf (t) = f (0) + f 0 (0)t +∞X
f (k) (0)tk

k!

k=2

= 0 + rt +

h Q
i
k−1
∞ r
(j
−
r)
tk
X
j=1
k!

k=2

Then, we have, for x ∈ (0, 1),

φr (x) =1 − xr1−x=

f (1 − x)
=r+
1−x

∞
X

i
h Q
(j
−
r)
(1 − x)k−1
r k−1
j=1
k!

k=2
r

Second, for the case of r ≥ 1, we have φr (x) = 1−x
1−x =

1

= φ

1−(xr )1/r
1−xr

>0
1

r
1/r (x ). φ1/r (xr ) is

decreasing because 1r ≤ 1. Therefore φr (x) = φ1/r (xr )−1 is increasing.
Lemma 11 For every fixed α1 ,...,αJ such that αj > 0 andPj αj ≤ 1 and x ∈ (0, 1). Then1
X x αi
1 x
≥0
−
1
x 1−x
1 − x αi
j



1/αj
P
Proof. Consider the case in which j αj = 1. The previous lemma implies that log 1−x
is
1−x


1/αj
P
is increasing, for x ∈ (0, 1).
increasing, and hence j αj log 1−x
1−xX

αj logj

1 − x1/αj
1−x

!




i Z x 1
1/α
Xh
X
j
u
 u −
 du
= [− log (1 − x)]−−αj log 1 − x1/αj =
u
1
−
u
1
−
u1/αj
0
j

j

i R w−1hw]
x u(because − log[1−x= 0 1−uw dx). Taking the derivative of each side gives
w

d X
0≤
αj logdx jNext, if1 − x1/αj
1−x

!


1
=
 x

P

X x1/αjx−
1−x
1 − x1/αjij αj < 1, then let α̃i = αi for i = 1, ..., J and α̃J+1 = 1 −from the previous case to α̃1 , ..., α̃J+1 .

87

!

P

j αj , and apply the result

Lemma 12 For any real and positive number α,
Z ∞
log Γ (1 − αit) = −itcα +

itu

e
0

uit
−1−
1 + u2e−u/αduu (1 − e−u )with cα finite.
θx + 1 in the case where x is real and positive Re (θ) > −x.
Using (9) and the change of variables s = xt , and then using (9) again, we haveProof. We begin by rearranging log Γ


log Γ

θ
+1
x


=
=
=
=

θ
x +1




Z 0 
θ
θ
et
t
log
= ψ (1) θ +ex − 1 − tdtΓ (1)x |t| (1 − et )
−∞
Z 0 

exs
ds
ψ (1) θ +eθs − 1 − θs|s| (1 − es )
−∞
Z 0 

exs
eθs − 1 − θs
ψ (1) θ − ψ (x) θ + ψ (x) θ +ds|s| (1 − es )−∞
Γ (θ + x)[ψ (1) − ψ (x)] θ +Γ (x)ΓThen using (10) and then a change of variables u = −s, we have

log Γ

θ
+1
x




Z 0 
s
exs
θs
= [ψ (1) − ψ (x)] θ + [ψ (x) + bx ] θ +e −1−θ
ds
1 + s2 |s| (1 − es )
−∞

Z ∞
u
e−xu
−θu
= [ψ (1) + bx ] θ +
e
−1+θ
du
1 + u2 u (1 − e−u )0

The result follows from evaluating this expression for θ = −i and x = α1 , and defining cα ≡ψ(1) + b1/α .
Pj αj ≤ 1. There exists an infinitely divisibleΓ(1−it)random variable with characteristic function Q Γ(1−αj it) .
j

Proposition 9 Fix α1 , ..., αJ such that αj > 0 and

Proof. We show this by showing that such a characteristic function has a Levy-Canonical Representation (see, for example, Lukacs (1960), Thm 5.5.2). Specifically, we can rearrange the functionasΓ (1 − it)log Qj Γ (1 − αj it)it= log Γ (1 − it) −log Γ 1 −
1/αj
j


Z ∞
X


= −it c1 −c1/αj +
eitu − 1 − it
X

0

j


= −it c1 −


X
j

Z ∞

c1/αj  +

itu

e
0

88

u
1 + u2u− 1 − it
1 + u2





e−u

1
−
u 1 − e−udN (u)

1X e− αj uj

1 − e−u duwhere N (x) is defined for x ≥ 0 as
Z ∞N (x) ≡ −x
1
X e− αj u
1  e−u du .
−
u 1 − e−u
1 − e−uj

To verify that this conforms to a Levy Canonical representation, we must verify that N satisfies
several conditions. First, N (x) is non-decreasing on (0, ∞) because Lemma 11 guarantees thatRεN 0 (x) ≥ 0. Second, limx→∞ N (x) = 0. Finally, 0 u2 dN (u) < ∞ for all ε > 0:Z ε0

u2 dN (u) ≤Z ∞u2 dN (u)0Z ∞− α1 u
−u
X
j
e
e
 du
=
u
−
−u
1
−
e
1
− e−u
0
j
Z ∞
−u
e
du
≤
u
1 − e−uZ0 ∞≤(1 + u)e−u du0= 2
where the last inequality uses 1−eu−u ≤ 1 + u as discussed in the proof of Lemma 989