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Sectoral Development Multipliers
WP 24-02
Francisco Buera
Washington University in St. Louis
Nicholas Trachter
Federal Reserve Bank of Richmond
Sectoral Development Multipliers∗
Francisco Buera
Washington University in St. Louis
Nicholas Trachter
Federal Reserve Bank of Richmond
March 13, 2024
Abstract
How should industrial policies be directed to reduce distortions and foster economic development? We
study this question in a multi-sector model with technology adoption, where the production of goods and
modern technologies features rich network structures. We provide simple formulas for the sectoral policy
multipliers, and provide insights regarding the power of alternative policy instruments. We devise a simple
procedure to estimate the model parameters and the distribution of technologies across sectors, which we
apply to Indian data. We find that technology adoption greatly amplifies the multipliers’ magnitudes, and
it changes the ranking of priority sectors for industrial policy. Further, we find that adoption subsidies
are the most cost-effective instrument for promoting economic development.
∗
Buera: fjbuera@wustl.edu. Trachter: trachter@gmail.com. We thank participants at many seminars and conferences
for comments and suggestions. We also thank Samira Gholami, Lindsay Li, Filip Milosavljevic, and Matthew Murphy for
outstanding research assistance. Francisco J. Buera acknowledges support from National Science Foundation Grant SES2049674. The views expressed herein are those of the authors and do not necessarily represent the views of the Federal Reserve
Bank of Richmond or the Federal Reserve System.
1
1
Introduction
Economic development hinges on the adoption of modern, complementary, technologies. Distortions can
hinder this process, resulting in underdevelopment. Industrial policies targeting keys sectors are called
for alleviating these distortions and promoting the investment in modern technologies (Hirschman, 1958).
Recent developments formalize the relevance of sectoral distortions in economies with input-output linkages
(Baqaee and Farhi, 2020), and how the propagation of these distortions through the economy motivates
the case for policies at the sector level to enhance production efficiency (Liu, 2019). However, these studies
abstract from technology adoption. We extend the analysis to allow for technology adoption, and develop a
laboratory to answer three key questions: Which sectors should be targeted in order to promote economic
development? What is the role technology adoption, relative to production efficiency, in driving the impact
of industrial policies? Which policy instrument is the most effective?
We propose a model of technology adoption in a multisector economy, where sectors are interconnected
through rich production and technology adoption networks. We provide analytical expressions for sectoral
development multipliers—the elasticity of aggregate consumption to a sectoral subsidy divided by the fiscal
cost of running the policy. We consider three set of sector specific policies: revenue, factor, and adoption
subsidies. Plant level and input-output data for India, together with information on the investment network
of developed economies, is combined with structural estimation and calibration methods to quantify the
multipliers and measure which sectors should be targeted to promote economic development. We find that
the introduction of technology adoption greatly amplify the effect of desirable policies and changes our
understanding of which sectors are the most relevant for industrial policy, and that adoption subsidies are
the most effective policy instruments at promoting economic development.
Monopolistically competitive establishments across different sectors produce differentiated intermediate inputs, charging a markup over marginal cost. Intermediate inputs are combined to produce the final
consumption bundle, bundles used by other establishments as a production input, and bundles to produce
adoption goods. Establishments can operate a traditional technology or pay for the adoption goods and
modernize their production techniques. Sectors are interconnected through heterogeneous networks. Intermediate input bundles used by different sectors combine intermediate inputs in different ways—production
network, and the adoption good used by different sectors also combine intermediate inputs in different
ways—investment network.1 These networks propagate the markup distortion through the economy, altering the production efficiency across sectors and, importantly, the technology adoption incentives. Correcting
these distortions provide the grounds for sectoral industrial policy.
We perform a local theoretical analysis of the effect of alternative sectoral policy instruments. We provide
a decomposition of the sectoral development multipliers into a production efficiency channel, the focus on
earlier studies, and a productivity channel. The productivity channel fully encodes the technology adoption
margin of the economy. This second channel features a double-Leontieff inverse. This double inverse
1
We focus on modern technologies as being capital embedded (Caunedo and Kala, 2021; Jones and Liu, 2022). Thus, we
interpret the investment network as the relevant object for modern technology adoption.
2
captures the fact that in a model with technology adoption there are further rounds of complementarities.
Policies that subsidizes a sector trickles down through production and investment networks as the cost of
intermediate inputs and adoption goods decline. The lower costs promote technology adoption, resulting
in increases in the productivity of individual sectors, triggering additional rounds of cost reductions and
technology adoption.
We apply the theory to quantify sectoral development multipliers for the India economy. We devise a
simple yet powerful estimation procedure to estimate the model parameters. With the estimated model at
hand, we study quantitatively the extent by which technology adoption changes our understanding of which
sectors are most relevant. We find that the adoption margin is highly relevant, greatly amplifying the size
of the multiplier in some sectors—by a factor of two in key sectors, and reducing the size of the multiplier
in other sectors. Further, accounting for technology adoption changes the relevance of sectors as engines of
development. Intuitively, investment goods producing sectors are featured among the top 5-sectors in terms
of the development multipliers when the technology adoption margin is taken into account.
Revenue-based policies are useful as promote economic development by directly affecting inefficiencies
and motivating firms to adopt modern techniques. However, they are relative ineffective vehicle for economic
development as they require a high cost to run. We show that adoption subsides, while more modest in
terms of their direct effects, provide more ’bang for the buck’ in the sense that the adoption-based subsidy
elasticity gains are much larger than the fiscal cost of running the industrial policy. Intermediate input
subsidies, which directly address the distortions, are the second most cost effective instrument, while labor
subsidies are the least effective. The effect of labor subsidies are crowd-out in general equilibrium as labor
is a fixed factor.
Related literature
Understanding the relevance of sectoral shocks and distortions in setting with sectoral
linkages has been the focus of several important recent contributions. Acemoglu et al. (2012) shows how
productivity shocks to particular sectors can have aggregate effects. Baqaee and Farhi (2020), Bigio and
La’O (2020) and Caliendo et al. (2022) study how distortions get amplified through sectoral linkages and
provide measures for the aggregate losses. Baqaee and Farhi (2021) explores the role of the entry margin.
We contribute to this literature by studying technology adoption and the role of alternative sectoral policies.
The analysis of industrial sectoral policies is also the subject of an active recent literature. Liu (2019)
studies how to target labor subsidies as a tool to alleviate sectoral distortions, and Liu and Ma (2021) studies
how to target subsidies for R&D when the process of innovation is also subject to sectoral interactions, with
a natural focus on developed economies. Bartelme et al. (2019) considers the case in which sectors are
subject to external economies of scale, and use trade data to infer economies of scale.2 We contribute to this
literature by adding a technology adoption margin, and studying how distortions interact with technology
choice, quantifying the gains from targeted sectoral policies aimed at generating economic development
through the adotion channel. In addition, we analyze alternative policy instruments.
2
Other recent papers in this literature include Choi and Levchenko (2021) and Farrokhi et al. (2023).
3
Our model builds on the work by Buera et al. (2021), which studies the non-linear effects of distortions
and policies, and the possibility of multiple equilibria, in a roundabout economy with complementarities in
technology adoption. We generalize their theoretical framework by considering a multisector economy with
rich production and investment interactions across sectors and a more general structure of firm heterogeneity
that is more amenable to infer the distribution of technologies at the sector level. More broadly, our paper
relates to a recent literature exploring the quantitative relevance of complementarities and technology choice,
such as Alvarez et al. (2023), Boehm and Oberfield (2023), Crouzet et al. (2023), and Demir et al. (2024).
Finally, a recent literature studies the role of investment networks in the propagation of sectoral productivity shocks. Foerster et al. (2022) and vom Lehn and Winberry (2022) show how investment and
production networks interact to amplify trend sectoral shocks in the postwar US economy, while Casal and
Caunedo (2023) provides novel and harmonized investment network data across countries and show how
they change over the process of development. In our work, the investment network plays a central role in
propagating the effect of sectoral policies through technology adoption, as the investment network affects
the adoption cost of capital embedded technologies.
2
Framework
We consider a multisector economy composed of S sectors. Within each sector s, establishments produce
differentiated varieties, used in the production of intermediate inputs and the final consumption aggregate.
The economy is populated by a measure L of individuals, each of whom supply a unit of labor inelastically,
and a measure 1 of potential entrants in each sector.
At each sector s, an active establishment has the ability to produce a differentiated variety j. Production
of the differentiated varieties yjs is aggregated to the sector level output Ys using a Dixit-Stiglitz aggregator,
Z
Ys =
η−1
η
η
η−1
yjs dj
, η>1,
where η denotes the elasticity of substitution across the differentiated varieties produced within sector s.
The output of all sectors is then combined to create aggregate consumption and the intermediate input
aggregate used by each sector s,
C=
Y
(Cs )γs , Xs =
Y
(Xss′ )ωss′ ,
s′
s
with γs ∈ [0, 1], Γ = [γs ]S×1 denoting a column vector collecting all γs , with 11×S Γ = 1. Likewise, let
Ω = [ωss′ ]S×S denote a matrix collecting all ωss′ .
Potential entrants in sector s are endowed with productivity z, where F (z) denotes the cumulative
distribution of z within each sector and f (z) ≡ ∂F (z)/∂z denotes its density. We further assume that F (z)
is Pareto, with parameter ζ > 0. Establishments can be either inactive or active. Inactive establishments
4
do not operate and remain out of the market. Active establishments face an extra productivity shock eε ,
which is realized after the entry and adoption decisions, but before the input choices. We assume that ε is
normally distributed with mean µ and standard deviation χ. Further, we set µ = −(η −1)χ2 /2, a convenient
normalization.3
An active establishment in sector s, endowed with productivity duple {z, ε} and using technology i ∈
{t, m}, produces output ys by combining labor l and the intermediate aggregate x according to the following
Cobb-Douglas production function
yjs =
zAi eε xν l1−ν
ν ν (1 − ν)(1−ν)
, ν ∈ [0, 1] ,
where ν denotes the intermediate aggregate input elasticity, and where Ai is a technology shifter specific to
i, with At < Am . We refer to i = t as the traditional technology and to i = m as the modern technology.
An active establishment in sector s operating the traditional technology must incur in an entry cost κts .
An active establishment in sector s operating the modern technology must incur in the entry cost κts and
modern technology adoption cost Pms κms , where Pms denotes the price of the intermediate aggregate used to
produce the adoption good used by sector s. The adoption good in sector s is produced by a representative
competitive establishment that combines the output of the different sectors using the technology flow matrix
Λ = [λss′ ]. Likewise, the entry cost in sector s is also produced by a representative establishment, converting
labor one-to-one into the entry cost.
Finally, we consider a variety of different instruments aimed at improving industrial outcomes: sector
revenue subsidies rs , sector labor cost subsidies rsl , sector intermediate input cost subsidies rsx , and sector
technology adoption subsidies rsa . We stack all revenue subsidies in a column vector, r = [rs ]S×1 . We stack
all other subsidies in similar vectors. The government levies lump-sum taxes from consumers and runs a
balanced budget.
Demand and Price Indexes.
The economy has two relevant layers. In the first layer, the inner layer,
the output of differentiated establishments is aggregated to produce sector output Ys . Let Ps denote the
sector price index corresponding to this output. Straightforward calculations provide that the demand for
the output of a establishment producing variety j is given by yjs = (pjs /Ps )−η Ys , and the price index
R
1/(1−η) . In the second layer, the outer layer, the output of the different sectors
satisfies Ps = ( p1−η
js dj)
is aggregated to produce the consumption bundle C and the intermediate aggregate of each sector Xs ,
s = 1, ..., S. As before, it is immediate to obtain that final demand for the output of sector s satisfies
Q
Cs = γs (Pc /Ps ) C, where the price index of aggregate consumption is defined as Pc = s (Ps )γs . Likewise,
we can obtain the intermediate demand by sector s for the output of sector s′ , Xss′ = ωss′ (Pxs /Ps′ ) Xs ,
Q
Q
where Pxs = s′ (Ps′ )ωss′ and Pms = s′ (Ps′ )λss′ .
3
This normalization for the mean of the ex-post distribution implies that ex-post heterogeneity does not affect equilibrium
aggregates in the economy. Under this normalization, Eε [e(η−1)ε ] = 1. Ex-post heterogeneneity will be instrumental for the
model to match the size distribution of establishments observed in the data.
5
The problem of a intermediate input producer.
The operating profits of an active establishment
using technology i in sector s are given by
o
πis
(z, ε)
p
Ps
−η
Pxs
1
x− ll
x
p,x,l
rs
rs
−η
ε
ν
1−ν
p
zAi e x l
≥
Ys ,
subject to
(1−ν)
Ps
ν ν (1 − ν)
= max rs p
Ys −
where we normalized the wage to one, i.e. w = 1. That is, for a given productivity duple {z, ε} and
technology choice i, an active establishment in sector s chooses price p, intermediate inputs x and labor l
demands in order to maximize profits, subject to revenue and input subsidies.
Using the first order conditions with respect to p, x and l we obtain the following expressions for prices
and factor demands,
1−ν
1 η
1
Pxs ν 1
pis (z, ε) =
,
rs η − 1 rsl
rsx
Ai eε z
−η
1 η
(Ai eε z)η−1 Psη Ys
lis (z, ε) =(1 − ν)
rsl
ν η−1 ,
1−ν
rs η − 1
P
1
(2)
xs
rsx
rsl
xis (z, ε) =
(1)
ν rsx lis (z, ε)
.
1 − ν Pxs rsl
(3)
Using the normalization of the distribution of ex-post heterogeneity, µ = −(η − 1)χ2 /2, expected operating
profits for an active establishment in sector s, with ex-ante productivity z, operating technology i = t, m,
are given by
o
πis
(z)
≡
o
Eε [πis
(z, ε)]
1
=rsη
η
η−1
η
η−1
(Ai z)η−1 Psη Ys
ν η−1 .
1−ν
1
rsl
(4)
Pxs
rsx
Operating profits are continuous in establishment productivity z. Also, given that η > 1, operating profits
o (z) are increasing in z. As a result, optimal entry and adoption decisions are given by threshold rules
πis
{zts , zms }∀s . Furthermore, provided that the modern technology is relatively expensive, both technologies
would be used in each sector, zts < zms . A establishment is active and operates the traditional technology in
sector s if and only if zts ≤ z < zms , and is active and operates the modern technology if and only z ≥ zms .
For each sector s, the marginal entrant zts and marginal adopter zms satisfy
o
πts
(zts ) =κts and,
o
o
πms
(zms ) − πts
(zms ) =
6
Pms
κms .
rsa
(5)
(6)
The next lemma summarizes an important, albeit evident, relationship between the thresholds.
Lemma 1 For all s, zms ≥ zts . Further, if 1 + Pms κms /κts > (Ams /Ats )η−1 , zms > zts .
In other words, for establishments that decide to be active in sector s, i.e. those with z ≥ zts , only a fraction
ζ
(zms )
zts
of them operate the modern technology, aess = 1−F
≤ 1, where as and es denote the mass of
=
zms
1−F (zts )
adopters and entrants in sector s, respectively. Notice that there is selection in adoption: adopters are those
with the highest productivity within a sector. This will have important implications for the distribution of
activity across establishments.
We now define a symmetric equilibrium,
Definition 1 Given vector of subsidies r, r l , r x and r a , a symmetric equilibrium consists of thresholds
{zts , zms }s
∈ S,
{Ps , Pxs , Pms }s
demand for labor and intermediate inputs by the different active establishments, prices
∈ S
and Pc , and a level of aggregate consumption C, such that establishments maximize
profits, markets clear,
X
X
Pc
Pxs′
Pms′
C+
ωs ′ s
Xs′ +
λs′ s
as′ κms′ ,
Ps
Ps
Ps
s′
s′
X
X
L=
Ls +
es κts ,
Ys =γs
s
where Xs ≡
R zms
zts
xts (z)df (z) +
R∞
zms
(7)
(8)
s
xms (z)df (z) and Ls ≡
R zms
zts
lts (z)df (z) +
R∞
zms lms (z)df (z),
with xis (z) =
Eε [xis (z, ε)] and lis (z) = Eε [lis (z, ε)].
Adoption and Sectoral Productivity.
Aggregating the output of establishments in sector s we obtain
an expression for the output of sector Ys , as a function of the sector’s inputs Xs and Ls ,
Ys =
ν ν (1
1
Zs Xsν L1−ν
,
s
− ν)1−ν
where Zs is a neutral Total Factor Productivity (TFP) shifter in sector s, defined as
Z
Zs ≡ Aη−1
t
zms
z η−1 f (z)dz + Aη−1
m
zts
Z
∞
z η−1 f (z)dz
1
η−1
1
1− η−1 η−1
1− η−1
η−1
ζ
ζ
η−1
+
A
−
A
a
= Aη−1
e
.
s
s
t
t
m
(9)
zms
We let Z = [Zs ]S×1 .
In turn, changes in sectoral productivity and policies affect sectoral and factor prices as these changes
percolates through the production network. Using (1) on the definition of the price index Ps , we can obtain
a condition relating the sector price Ps , the price of the intermediate aggregate Pxs , and sector productivity
7
Zs ,
1−ν
Ps =
1 η
rs η − 1
1
rsl
Zs
Pxs
rsx
ν
.
(10)
This is the sectoral couterpart of (1). The price index of a sector s is a markup over its marginal cost,
which depends on the price of the sector’s intermediate aggregate Pxs , its productivity Zs , and subsidies.
We combine (10) with the definition of the price index Pxs to obtain,
ln P =1
1
η
ln
− (I − νΩ)−1 ln Z + ln r + (1 − ν) ln r l + ν ln r x .
1−ν η−1
(11)
1
where P = [Ps ]S×1 , and where we used that (I − νΩ)−1 1 = 1 1−ν
. Because of the input-output network,
sector prices compound all sector aggregate TFP levels and markup distortions η/(η − 1) through the
Leontieff inverse (I − νΩ)−1 = I + νΩ + ν 2 Ω2 + ν 3 Ω3 + ... .
From the definition of the consumption price index Pc and technology adoption price index Pms , we
obtain the following relationships,
d ln Pc =Γ′ d ln P , and d ln Pm = Λd ln P .
(12)
These expressions show how changes in sector prices translate into changes in the consumption bundle and
technology adoption price indexes.
A key feature of our economy is that, through adoption, sector productivity is endogenous. Higher
adoption in sector s increases the share of value-added within the sector produced by modern establishments
and, thus, the sector’s productivity. In particular, the elasticity of sector TFP, Zs , to a change in the mass
of adopters, as , is given by
η−1
d ln Zs
1 ζ + 1 − η Aη−1
m − At
Ms ≡ βMs > 0 .
=
d ln as η − 1
ζ
Aη−1
m
(13)
The value Ms represents the share of value-added in sector s attributed to modern establishments; M =
[Ms ]S×1 collects these shares for all sectors. The first expression shows how changes in adoption are converted
into changes in sector TFP. Importantly, the elasticity is higher (i) the larger the productivity wedge among
the two technologies, (ii) the lower the elasticity of substitution and heterogeneity within a sector, as
captured by η and 1/ζ, which are key determinants of complementarities across establishments within a
sector (see Buera et al., 2021), and (iii) the higher is the share of value-added of modern establishments in
the sector Ms .
8
3
Sectoral Development Multipliers
In this section we study the mechanism by which technology adoption affects industrial policy. To keep
things as simple as possible, our benchmark analysis for the characterization follows Baqaee and Farhi
(2020), and only considers revenue subsidies r (that is, ln r l = ln r x = ln r a = 0). However, while we put
special emphasis on revenue subsidies, as we advance through the section we provide insights on how the
different subsidies operate as a tools for industrial policy.
To maintain things as simple as possible in this section, we abstract from the entry margin, thus treating
es as an exogenous object across all s and abstract from the entry costs. We do this to provide a more
transparent analysis of the adoption margin, which is the main focus of the paper. The entry margin is
relevant for matching the data, but it is somewhat orthogonal to endogenous technology adoption. In fact,
in our quantitative analysis, as shown in Table IV, we show that the entry margin accounts for a very small
fraction of the development multipliers, and thus little is lost by ignoring this margin.4 Still, we emphasize
that we do take the entry margin into account when we quantify the multipliers using data.
As in Liu (2019) we define the sectoral revenue development multiplier in sector s as
subsidy elasticity
ϵrs
z }|
d ln C
≡
d ln rs
{
/(Fiscal cost of policy) .
(14)
r=1
Basically, the multiplier accounts for the elasticity of aggregate consumption C to a revenue subsidy in
sector s—–the subsidy elasticity, d ln C/d ln rs , around the equilibrium with no subsidies, ln r = 1, relative
to the fiscal cost of the policy. A multiplier ϵr,s above one implies that an industrial policy in sector s
increases aggregate consumption by more than the fiscal cost, a multiplier equal to one is neutral, a positive
multiplier lower than one implies that the increase in aggregate consumption is lower than the fiscal costs,
and a negative multiplier implies that aggregate consumption falls as result of the policy.5
3.1
Subsidy and Productivity Elasticity of Aggregate Consumption
As a first step, we present a characterization of the elasticity of consumption with respect to a revenue
subsidy and a sectoral productivity.
Let δs =
Pms as κms
Pc C
denote sector s adoption share of GDP, with ∆ = [δs ]S×1 a vector collecting these
shares. We define the Domar and Forward Domar weights in the economy as
−1
η−1
′
Ψ = Γ +∆Λ I −ν
Ω
, Ψ̃ = Γ′ + ∆′ Λ (I − νΩ)−1 .
η
′
′
′
4
(15)
Follows from comparing the columns for ϵ and ϵe of Table Table IV.
While in our setting lump-sum taxation would imply that any positive subsidy elasticity would suffice to motivate the use
of the instrument, one could consider that there are alternative uses for the income taxed away from consumers. In such a
setting, taking into account the fiscal cost of the policy is appropriate.
5
9
The Forward Domar weight is usually referred in the literature as the Cost Domar weight. Our choice of
language will be clear in the discussion of the following proposition.
Proposition 1 Consider independent vectors of changes in revenue subsidies d log r and changes in sector
TFP induced by changes in adoption d log Z = β diag (M ) d ln a, then
TFP channel
production channel
d ln C =
z
}| {
′
Ψ̃ − Ψ′
z
}|
{
η−1 ′
d ln r + Ψ̃ −
Ψ d ln Z .
η
′
(16)
′
The term Ψ̃ − Ψ′ gives the aggregate consumption gains that follow from changes in the allocation of
resources across sectors. Ignoring the resources used, the change in the policy induces changes in the quantity produced by the different sectors. The first round effects of these changes are proportional to the final
demand elasticities, which are measured by the consumption elasticities vector Γ′ and the adoption elasticities ∆′ Λ. The effects from the subsequent rounds are proportional to the intermediate input elasticities,
which are measured by the matrix νΩ, and its subsequent powers, which themselves give the increase in the
output of all sectors as a result of the greater availability of inputs from the promoted industries. The impact of these subsequent rounds are ultimately also proportional to the aggregate demand elasticities, thus
′
(Γ′ + ∆′ Λ) νΩ + (νΩ)2 + ... = (Γ′ + ∆′ Λ) (I − νΩ)−1 = Ψ̃ . The Forward Domar weight Ψ̃ accounts
for all these cumulative effects.6 Increasing production requires reallocating labor across sectors. The cost
of the reallocation is measured by the Domar weight Ψ, which accounts for the labor used by the sector
directly and, indirectly, by all sectors supplying inputs to this sector. This cumulates to the labor or Gross
Output share of the sector. Because of markup distortions, gross labor shares and cumulative final demand
′
elasticities differ, and thus Ψ̃ − Ψ′ can be positive. The larger is the markup, the larger the divergence
between these vectors, and larger the gains. When the markup approaches zero, i.e. η/(η − 1) → 1, forward
and backward Domar weights are equal and thus the production channel of the subsidy elasticity is zero.
Overall, gains are large when a sector has large (forward) cumulative final demand elasticities and small
(backward) cumulative resource shares.
Endogenous technology adoption affects the subsidy elasticity by inducing a change in sector produc′
tivities Z, the TFP channel. The term Ψ̃ −
η−1 ′
η Ψ
regulates this margin. Notice that the intuition that
we discussed for the production channel also applies here, with the only difference being that the vector
of Domar weights, which accounts for the cost of reallocating labor, is now discounted by (η − 1)/η. This
occurs as the total resources required to implement a given increase in TFP combines a direct labor reallocation cost, Ψ′ , with the way this labor is transformed into sector productivity through adoption. The
reciprocal of the adoption elasticity of TFP, (∂ ln Zs /∂ ln as )−1 , accounts for how adoption is transformed
into productivity, and it is proportional to (η−1). Generating this increase in adoption requires changing the
identity of the marginal adopter. Thus, the cost per unit of adoption is given by the profit of the marginal
6
In deriving these expression it is convenient to work with the dual, and trace out the fall in sector and final consumption
prices induced by the change in policies—see (11) and (12).
10
adopter, which are proportional to 1/η. The ratio (η − 1)/η combines the marginal rate of transformation
from adoption to sector productivity with its marginal cost. As the elasticity of substitution declines, a
lower increase in adoption is required to implement a given change in TFP, relative to the cost per unit of
adoption. Therefore, as the elasticity of substitution declines, fewer resources are needed to increase sectoral
productivity.
′
′
′
The entries of the vectors (Ψ̃ −Ψ′ ) and (Ψ̃ − η−1
η Ψ ) in (16) give the differential effects of sector-specific
subsidies and TFP changes, respectively. The relative magnitude of each entry depends critically on the
details of the production network, given by the matrix Ω. Aggregating these entries results in the effect
of a uniform and unitary policy or change in TFP. The following remark gives a simple expression for the
elasticity of aggregate consumption to an uniform unitary change in revenue subsidies and TFP.
Remark 1 Consider a uniform unitary changes in revenue subsidies and TFP in all sectors, d ln r = 1 and
d ln Z = 1, then the expression in (16) reduces to
d ln C = 1 + δ̄
where δ̄ =
ν
1
+
1−ν
1−ν
1
1
,
η 1 − ν η−1
η
P
s δs .
A notable feature about this result is that the expression shows that, under uniform policies, the complexities of the network structure of production do not play any role. Indeed, the gains accrued by the uniform
policy and TFP gains are exactly those obtained in a one-sector economy with roundabout production.
3.2
Subsidy Elasticity of Adoption and Development Multiplier
To find an expression relating how changes in the subsidy affect sector productivity, we need to find an
expression for d ln a from the marginal adopter’s condition. In this case, it is convenient to consider both
revenue and adoption subsidies, d ln r and d ln r a .
Log-differentiating the vector of conditions for the marginal adopter in (6), and using (10), we obtain
a condition linking adoption, sectoral productivity, gross output, and the price of the adoption good, with
the vector of revenue and adoption subsidies,
d ln r − (η − 1)d ln Z + d ln (P ◦ Y ) −
η−1
d ln a = −d ln r a + d ln Pm ,
ζ
(17)
where the symbol ◦ denotes the Hadamard (element-wise) product. Notice how revenue r and adoption r a
subsidies have the same direct effect on adoption. Revenue subsidies also affect adoption indirectly through
its effect on sector prices, which affect the aggregate demand channel and, importantly, the vector of prices
of the adoption good Pm .
A simple case to study is when an adoption subsidy is applied to a sector where the modern share is
zero, i.e. Ms = 0. The next Remark describes the result.
11
Remark 2 Assume that Mŝ = 0 for some sector ŝ. Then,
d ln aŝ
d ln rŝa
=
ζ
η−1 ,
and
d ln as
d ln rŝa
= 0 for all s ̸= ŝ.
The Remark follows immediately after noticing that (13), given that Mŝ = 0, provides that the elasticity of
the sector’s TFP to adoption is also zero, and thus d ln Z = d ln (P ◦ Y ) = d ln Pm = 0. Adoption increases
in sector ŝ because of the direct effect of the policy: ζ measures the elasticity of the distribution of ex-ante
heterogeneity, which is normalized by the curvature of the profit function with respect to this productivity,
η − 1. But the percentage increase in adoption in this sector does not translate into a percentage increase
in the sector’s TFP. As a result, the sector’s price index does not vary, and the same occurs to the price
of adopting a modern technology in other sectors, and thus adoption in other sectors is unaffected by the
policy. Remark 2 showcases the relevance of the modern share Ms : through its effect on the elasticity of
TFP with respect to adoption, is a crucial ingredient for understanding how subsidies in one sector percolate
to other sectors through adoption.
We now follow to solve for d ln a from (17), together with the price feedbacks in (11) and (12). Combining
the relationships between prices, policies and TFP in (11), (12) and (17), we obtain an expression for the
elasticity of adoption d ln a with respect to an adoption subsidy d log r a . The next Proposition presents this
result.
Proposition 2 The elasticity adoption with respect to a adoption subsidy is given by
direct effect
amplification
z
}|
z }| {
−1{
i
ζ
ζ h
ζ
d ln a =
d ln r a .
I−
Λ (I − νΩ)−1 − (η − 1)I βdiag (M ) −
∇P Y,a
η−1
η−1
η−1
|
{z
}
(18)
≡∇a,ra
Details on the derivation are available in Appendix A.3. The operator diag(·) converts a column vector
into a diagonal matrix, and ∇P Y,a denotes the elasticity of P ◦ Y with respect to a, which is presented
in close-form in the appendix. To keep things simple, we will provide intuition abstracting from this last
channel. The elasticity of adoption with respect to an adoption subsidy combines the direct effect of the
policy, measured by the term ζ/(η −1), with the amplification effect that follows from the sector interactions.
The degree of amplification depends on the structure of the production (Ω) and investment (Λ) networks,
as the effect of policies percolates through them. A subsidy to a sector promotes adoption and TFP in that
sector, which in turns lower the sector price, lowering the cost of adoption in all sectors. This leads to a
further increase in adoption in other sectors, resulting in further feedback rounds. The magnitude of these
multiplier effects depends on parameters governing the elasticity of productivity with respect to adoption
given in (13), which are subsumed in β and the vector of modern shares M and, for given values of these
quantities, the effects are increasing in the elasticity of substitution and heterogeneity within a sector, as
captured by η and 1/ζ, which are key determinants of complementarities across establishments (see Buera
et al., 2021).
12
In virtually all research exploring the relevance of input-output structures for economic aggregates, the
key object is the Leontieff Inverse (I − νΩ)−1 , its backward counterpart, or linear combinations of them.
This occurs as heterogeneity in the way sectors combine sectoral output to use as a intermediate aggregate
in production results in heterogenenity in sector price indices. Because the intermediate aggregate used by
any sector aggregates the output of all other sectors through Ω, so does sector prices P . Absent technology
adoption, aggregate effects of policies are the result of comparisons of objects that directly map to the
Leontieff Inverses. Technology adoption adds a double inverse to the loop. Adoption increases TFP, which
lowers sector prices and thus the cost of adoption. This is the inner inverse. Also, adoption in a sector
reduces the marginal cost of production in all sectors, thus increasing the profits accrued from adoption by
the marginal adopter. This feedback accounts for the outer loop of the Double-Leontieff Inverse.
To further understand the role of individual parameters and of the value-added share of modern establishments in shaping the Double-Leontieff Inverse, the next remark characterizes the impact of a uniform
unitary change in adoption subsidies on the elasticity of adoption, under the additional restriction that
there is no heterogeneity in the modern share of value-added across sectors. This expression is the same one
would obtain in a one-sector economy with roundabout production.
Remark 3 Assume M = mI with m ≥ 0, and d ln (P ◦ Y ) = 0. Consider a uniform unitary change in
adoption subsidies in all sectors, d ln r a = 1, then the expression (18) reduces to
1
d ln a =
1−
h
1
1−ν
ζ
i
.
ζ η−1
− (η − 1) βm η−1
(19)
As briefly discussed earlier, the subsidy elasticity of adoption features two nested multipliers. Subsidies
promote adoption which enhance productivity, mediated by the elasticity βmζ/(η − 1), which lowers sector
prices and, therefore, the price of adoption. These effects get amplified as they percolate through the inputoutput structure. The inner multiplier 1/(1 − ν), which is the standard multiplier of a roundabout economy,
encodes these effects. The increase in the productivity of competing producers, captured by the term η − 1,
partially dampens this effect. The outer multiplier captures the infinite rounds of adoption lowering the
marginal cost of production, further enhancing adoption. Finally, ζ/(η−1) is the direct elasticity of adoption
with respect to the subsidy, which we described earlier.
The expression in (19) also allows us to see ingredients that, through the adoption channel, can greatly
amplify the effect of policies even under uniform policies. While in a simple production economy the
multiplier becomes unbounded as ν approaches one, with adoption the multiplier becomes unbounded when
ν approaches 1 −
βmζ
1
η−1 1+βmζ .
Under our calibration later on, this upper bound is always between zero and
one. This implies that, through the adoption channel, the multiplier becomes unbounded for lower values
of the intermediate input elasticity ν. In other words, through the adoption channel, even the effect of
uniform policies can be greatly amplified, and it is possible to generate larger amplification than through
the production channel for a wider set of parameter values. In our calibrated economy this is not the case,
13
i.e., technology adoption greatly amplifies the multiplier for key sectors, but not the multiplier associated
with a uniform policy, which is consistent with the findings in Buera et al. (2021) for a symmetric economy
with roundabout production.
Putting together the results from Propositions 1 and 2, we obtain the following expression for the
elasticity of aggregate consumption with respect to independent revenue and an adoption subsidies:
h
i
η−1 ′
d ln C = Ψ̃ − Ψ d ln r + Ψ̃ −
Ψ β diag (M ) ∇a,ra I + ∇P Y,r + Λ (I − νΩ)−1 d ln r
η
η−1 ′
′
(20)
Ψ β diag (M ) ∇a,ra d ln r a ,
+ Ψ̃ −
η
′
′
′
where ∇P Y,r denotes the elasticity of P ◦ Y with respect to r, which is provided in close-form in Appendix
A.3.
We note the relevance of the Double-Leontieff inverse, present in the key object ∇a,ra , as characterized in
Proposition 2. The matrix ∇a,ra converts subsidies into adoption, which in turn is converted to TFP through
η−1
d ln Z = βdiag (M ) d ln a. A key determinant of this conversion rate is the technology gap Aη−1
m − Am , as
discussed
in (13).
Then, the change in TFP is converted to aggregate consumption through the elasticity
′
η−1 ′
Ψ̃ − η Ψ . We provided a discussion of this term in Proposition 1.
In settings with exogenous productivity, Acemoglu et al. (2012) noted that Ψ̃ accounts for the Influence
matrix. That is, the entry s of the Influence matrix provides the change in aggregate consumption resulting
from a shock to exogenous sector productivity Zs .7 With endogenous sector TFP, we note that Ψ̃ accounts
for Influence in production, while (∇a,ra )′ βdiag (M ) Ψ̃ accounts for influence in adoption. That is, the
entry s of this object accounts for the resulting change in aggregate consumption following from an exogenous
shock to the adoption technology in sector s.
With elasticities at hand, we need expressions for the fiscal cost of the different policies to be able to
produce the multipliers. Similarly to Liu (2019), the fiscal cost of the revenue subsidy is given by the Domar
weight Ψ, as this weight provides the size of each sector, and thus, around the equilibrium with no subsidies,
the fiscal cost of the subsidy. Therefore, obtaining an expression for ϵrs requires dividing the s entry of (20)
by the s entry of vector of Domar weights Ψ. Similarly, the fiscal cost of the adoption subsidy in sector s is
given by the adoption share of the sector, the s entry of ∆, the fiscal cost of a intermediate input subsidy
in sector s is given by the s entry of νΨ, and the fiscal cost of a labor subsidy in sector s is given by the s
entry of (1 − ν)Ψ.
The following remark provide expressions for the development multipliers associated with uniform and
unitary revenue and adoption subsidies,
Remark 4 Suppose that M = mI and d ln P ◦ Y = 0 in the marginal adopter’s problem in each sector.
7
Exogenous sectoral productivity implies no adoption, and thus ∆ = 0. Then, under the restriction that Γ = 1, we get that
Ψ̃ = 1′ (I − νΩ)−1 which is the influence measure described in Acemoglu et al. (2012).
′
14
Then, under uniform and unitary revenue and adoption subsidies, i.e. d ln r = d ln r a = 1, we have that
ϵur
ν 1 η−1
1
u
=
,
+
βmϵra 1 +
1−νη
η
1−ν
where
ϵura =
1
1
h
η−11−ν1−
1
1
1−ν
ζ
i
.
ζ η−1
− (η − 1) βm η−1
These expressions highlight the counteracting forces driving the relative importance of the development
multiplier associated with revenue and adoption subsidies. On the one hand, revenue subsidies have a direct
effect enhancing production efficiency, as captured by the first term of ϵur . In addition, revenue subsidies
affects the adoption incentive directly and indirectly, through their effect on the price of the adoption good,
as capture by the last bracketed term. On the other hand, adoption subsidies have an advantage given their
lower fiscal cost as the adoption share is smaller than the Domar weight, i.e. δs = ((η − 1)/η)βMs Ψs < Ψs .
3.3
Additional Results
In this section we present two additional analysis. First, we study the sectoral development multipliers
associated with input subsidies, and compare them to the multipliers associated with revenue subsidies.
To simplify the analysis, we study them in the case with no adoption. This section sheds light of the
relevance of different policy instruments as an engine of development, and will aid on our quantitative
analysis contrasting the power of different instruments. Second, we consider a version of the model where
the adoption good is produced with labor only. This section sheds light on the importance of using fixed or
variable inputs in the production of the adoption good.
3.3.1
Alternative Policy Instruments
Consider the case of revenue r, intermediate inputs r x and labor subsidies r l in the economy with no
adoption. The results in this section will shed light on (i) the differential relevance of the adoption margin
depending on whether the adoption good is produced with intermediate inputs or labor, and (ii) the potential
of the different policy instruments as the engine of industrial modernization.
In this case, the subsidy elasticity of aggregate consumption is given by
′
d ln C = Ψ̃ − Ψ
′
η−1 ′
η−1
′
′
x
d ln r + Ψ̃ −
Ψ νd ln r + (1 − ν)Ψ̃ − 1 − ν
Ψ′ d ln r l .
η
η
′
Naturally, the contribution of the revenue subsidy (Ψ̃ − Ψ′ ) is the sum of the contributions of the
intermediate input and labor subsidies. When considering intermediate input subsidies, the Forward Domar
′
weight Ψ̃ is weighted by the elasticity of intermediate inputs in production, ν, and the Backward Domar
15
weight Ψ′ is weighted by the intermediate input share, ν(η − 1)/η. A similar argument applies for labor
subsidies.
To further gain insights about the power of the different policy instruments we restrict to uniform and
unitary subsidies, d ln r = d ln r x = d ln r l = 1, and we contrast the resulting multipliers. Under the uniform
policy,
ϵurl = 0 < ϵur =
ν 1
1 1
< ϵurx =
.
1−νη
1−νη
While a labor subsidy to an individual sector has generically an effect on aggregate consumption, either
positive or negative, these individual effects cancel out when aggregated up, implying that the multiplier
of a uniform labor subsidy is zero. Key to this result is that labor is a fixed input. When aggregating
the Backward Domar weight we obtain the gross output share of consumption, 1/(1 − ν(η − 1)/η), which
−1
results from accumulating infinite rounds of the (variable) intermediate inputs share, I − ν η−1
Ω
1=
η
n
P
∞
η−1
1. This share is then multiplied by the share of the (fixed) labor input, one minus the
n=0 ν η Ω
′
share of the variable input. A similar argument applies to (1 − ν)Ψ̃ 1, resulting in zero consumption effect
of a labor uniform subsidy. Intuitively, in the aggregate, nothing can be gained by promoting uniformly the
use of a fixed input. This is reminiscent of the results in Liu (2019).
The results of the uniform labor policy are useful as an input to understand the values of the uniform
revenue and intermediate input multipliers. In fact, the revenue and intermediate input elasticities are the
same,
d ln C
d ln r
=
r=1
d ln C
d ln r x
= (1 + δ̄)
r x =1
ν 1
1
.
1 − ν η 1 − ν η−1
η
They are equal because a uniform subsidy provides no gains in terms of labor reallocation and all gains
follow from the intermediate input channel. This is why ν appears in the numerator. But, although the two
uniform policies share the same value for the elasticity, they differ in the cost of the policy. The revenue
uniform policy is ’paying’ for labor reallocation with no gains, while the intermediate inputs uniform policy
circumnavigates paying for the added cost. Therefore, while both policies have the same subsidy elasticities,
the intermediate input policy cost is only ν for every unit of cost in the revenue policy. As a result,
ϵur /ϵurx = ν < 1.
While the intermediate input subsidy dominates the revenue subsidy in the economy with no adoption,
this is not necessary the case in the model with adoption. This occurs as, while both the revenue and
intermediate input subsidies affect the marginal adopter’s condition indirectly through their effect on sector
prices, the revenue subsidy also adds a direct effect on the marginal adopter’s profit. As a result, we resort
to our quantitative analysis of multipliers to gauge the relative importance of the different subsidies as an
engine of development.
16
3.3.2
Adoption goods produced with labor only
While we see technologies in our model as embedded in intermediate inputs (closer to capital in a dynamic
version of the model), it is useful to evaluate the extent to which sectoral multipliers are sensitive to this
assumption. In this section we study the polar opposite case, where the adoption good is produced solely
with labor instead of solely with goods.
When labor is the sole input of production for the adoption good, the subsidy elasticity expression in
(16) is given by
′
′
d ln C = Ψ̃ − Ψ
1
η−1
′
d ln r +
(1 − ν)Ψ̃ − 1 − ν
Ψ′ d ln Z ,
1−ν
η
−1
′
where, as before, Ψ̃ = Γ′ (I − νΩ)−1 , Ψ′ = Γ′ I − ν η−1
Ω
, and d ln Z = βdiag(M )d ln a.
η
First, notice that when ν = 0 we have that d ln C = 0′ d ln r+0′ d ln Z. While there are markup distortions
in the economy, they do not manifest in creating a wedge between labor shares and final demand elasticities,
thus rendering industrial policy useless.
Second, consider the effect of uniform subsidies in this economy. As before, we set d ln r = d ln Z = 1.
The contribution of the production margin is analogous to the uniform policy in the baseline economy,
1
ν 1
η−1 . However, this analogy does not follow through for the contribution from productivity,
1−ν η
1−ν
η
"
−1 #
′
′
η−1
η−1
η−1
−1
′
Ω
1=0.
(1 − ν)Ψ̃ − 1 − ν
Ψ̃ 1 = Γ (1 − ν) (I − νΩ) − 1 − ν
I −ν
η
η
η
(21)
This result is similar to the case of a labor subsidy in the economy without adoption. While the cumulative
effect of the policy must be zero, aggregate consumption can increase or decrease by reallocating adoption
labor across sectors. In fact, if a subsidy in one sector has a positive effect on aggregate consumption, there
must be another sector with a negative effect on aggregate consumption.
Finally, in a symmetric economy with roundabout production in each sector, i.e. Ω = I, we have that
the vector of TFP contributions satisfy
"
Γ′
η−1
(1 − ν) (1 − ν)−1 I − 1 − ν
η
#
η − 1 −1
1−ν
I =0′ .
η
That is, each sector’s contribution through TFP must be zero, and thus there are no gains from reallocating
adoption. This last result is reminiscent of that one in Atkeson and Burstein (2010). As a result, adoption
reallocation can have relevant effects for aggregate multipliers if there is substantial heterogeneity in the
network structure of production.
17
4
Estimation and Calibration
We begin this section by presenting a simple identification strategy that allow us to structurally estimate
the model and the distribution of technologies across sectors. Our strategy consists of three stages, and
combines structural estimation and calibration methods. After describing our strategy, we introduce the
data that we use and present the estimates. Finally, we perform a validation exercise of the structural
estimates using independent proxies of technology adoption.
4.1
Identification
While the model is able to capture complex sectoral production and adoption complementarities, it is parameterized in a relatively parsimonious fashion. We need to assign values to the following set of parameters
(scalars, vectors and matrices): At , Am , ζ, χ, η, ν, {κts , κms }for all s , Γ, Ω, Λ. To do this, we follow a threestage procedure. In the first stage, we exploit the theory implications for the size of an establishment
within a sector, and estimate a reduced-form representation of the parameters governing the distribution of
technologies in the model, i.e. combination of the deep parameters of the model. In the second stage, we
combine direct measurements from the data and estimates from other sources to calibrate the production
and adoption networks, i.e. Ω and Λ, and the elasticity of substitution across varieties within a sector η.
Finally, in the third stage, we use the equilibrium construct of the model to unbundle the reduced-form
estimates obtained in the first stage and recover the deep parameters of the model governing technology
adoption. We also show that this procedure provides a unique mapping from the reduced-form and deep
parameters estimates. Further, following the working assumptions for multipliers, we assume no subsidies
in the starting equilibirum. That is, ln r = ln r l = ln r x = ln r a = 1.
Brute force estimation of the multisector economy is a challenging task. For example, given that entry
and adoption in a sector depend on entry and adoption in other sectors, it is unclear how to operationalize the
procedure for the estimation of entry and adoption costs. This problem gets exponentially more convoluted
the more sectors are considered. Our insight to address these complications is the following: In order to
explain variation in the size of establishments within a sector, a common estimation target in the literature,
one does not need to understand how sectors are connected, nor the level of aggregate prices, nor entry
and adoption rates in other sectors. This partial information procedure allows us to provide reduced-form
estimates for entry and adoption thresholds for each sector, productivity parameters, and the distribution
of ex-ante and ex-post heterogeneity.
Taking logs to (2), using (10), and setting rs = rsl = rsx = 1, implies that
ln lis (z, ε) = ln Ãi + ln z̃s + ln ε̃ ,
, z̃s ≡ (1−ν) η−1
where Ãi ≡ Aη−1
i
η
Ps Ys η−1
z ,
Zsη−1
(22)
and ε̃ ≡ εη−1 . In this representation, the size of an establishment
within sector s combines (i) a technology-specific component Ãi , (ii) a mix of a sector component and ex-
18
ante heterogeneity of the establishment, z̃s , and (iii) an idiosyncratic ex-post component, ε̃. Notice that the
sector component cannot be identified by observing heterogeneity in size of establishments within a sector.
Unbundling the sector and the ex-ante heterogeneity components is the main objective of the third stage in
our devised procedure.
Within a sector, heterogeneity in technology use manifests through its impact on the employment-size
distribution. Let Hs (l) = Pr (ls (z̃, ε̃) ≤ l), with density hs (l) ≡
∂Hs (l)
∂l ,
be the employment size distribution
of establishments within sector s.
!
2 ζ̃
ln
z̃
−
ln
l
+
µ̃
+
χ̃
ms
−
e
Φ
hs (l) =l−ζ̃−1
−ζ̃
χ̃
z̃ts
!#
"
!#)
2 ζ̃
ln z̃ts − ln l + µ̃ + χ̃2 ζ̃
ln
z̃
−
ln
l
+
µ̃
+
χ̃
ms
Φ
+ Ãζ̃m 1 − Φ
,
χ̃
χ̃
ζ̃
where ζ̃ ≡
ζ
η−1 ,
2
µ̃ζ̃+ χ̃2 ζ̃ 2
(
"
Ãζ̃t
(23)
χ̃ ≡ (η − 1) χ, µ̃ ≡ (η − 1)µ = −χ̃2 /2, and where z̃ts and z̃ms are the entry and adoption
thresholds under the reduced-form representation. Details on the derivation are available in Appendix A.4.
If one shuts down technology heterogeneity and the ex-post shock ε, i.e. Ãt = Ãm and χ̃ → 0, hs (l)
is Pareto. Departures from Pareto are the result of heterogeneity in technology adoption and dispersion
following from ε̃. Similarly, we can define the employment
weighted size distribution across establishments
R
ˆ
ˆ
ˆ
Gs (l), where gs (l) ≡ ∂Gs (l)/∂l = lhs (l) / lhs l dl.
Inference using hs (l) or gs (l) in the reduced-form model provides estimates for Ãt , Ãm , χ̃, ζ̃, and for the
reduced-form entry and adoption thresholds {z̃ts , z̃ms }for all s . The next two remarks show how the shape
of hs (l) and gs (l) is informative about the extent of heterogeneity in technology use within sector s, which
is encoded in the technology parameters Ãt and Ãm , and in the entry and adoption thresholds z̃ts and z̃ms .
We explore the informativeness of the distributions regarding technology adoption by characterizing two
extreme cases. In the first remark we show that if there is no variation in technology use within a sector
the resulting distributions are unimodal. In the second remark we show that with no ex-post heterogeneity,
heterogeneity in technology use results in bimodal distributions.
Remark 5 Suppose only one technology is used in sector s. Then, the distributions Hs (l) and Gs (l) are
unimodal.
The proof is available in Appendix A.5. The proposition states that a combination of Pareto and Log-Normal
shocks, both unimodal distributions, imply equilibrium distributions that are also unimodal. Because the
ex-post shock ε̃ is iid, it shifts mass in such a way that does not generate two modes.
Remark 6 Let χ → 0. Then, the distributions Hs (l) and Gs (l) have two modes if and only if z̃ms > z̃ts .
In this case, the two modes are at l = z̃ts and l = z̃ms . If z̃ms = z̃ts , both distributions have one mode,
located at l = z̃ts .
19
The proof is available in Appendix A.6. The proposition establishes that, by shifting mass from the middle
of the distribution to the right of the distribution, technology adoption generates bimodality.
Under the reduced-form representation the endogenous choices of establishments regarding entry and
technology adoption are summarized by the thresholds z̃ts and z̃ms for all s. In order to compute development
multipliers and perform counterfactual analysis we need to unbundle these thresholds and uncover the entry
and adoption costs, i.e. κts and κms for all s, that rationalizes these thresholds. The next proposition
uses the equilibrium conditions of the model and establishes that there is a unique mapping between the
reduced-form thresholds and the underlying entry and adoption costs.
Proposition 3 Given values for η, ν, Γ, Ω, Λ, and the reduced-form objects ζ̃, Ãt , Ãm , {z̃ts , z̃ms }for all s ,
then the vectors of entry and adoption costs, {κts , κms }for all s , are uniquely identified.
The proof of this proposition is available in Appendix A.7, and follows from the fact that the equilibrium
of the model can be described by a system of linear equations given the reduced-form thresholds, z̃ts and
z̃ms for all s.
4.2
Data and calibration
We focus our quantitative analysis on the India economy and combine three sources of data: (i) Indian
Economic Census, (ii) the World Input-Output Database (WIOD) and, (iii) data on the investment network
from vom Lehn and Winberry (2022).
The Indian Economic Census is a complete count of all establishments/units located within the geographical boundaries, engaged in production or distribution of goods or services other than for the sole purpose of
own consumption, crop production and plantation, public administration and defence, or activities of households as employers of domestic personnel. We further drop establishments in animal production, forestry
and logging, and fishing and aquaculture. The sectoral data is given at the 3-digit level as per the National
Industrial Classification (NIC 2004 and 2008, for the 2005 and 2012-13 Economic Censuses, respectively).
We use the Sixth Indian Economic Census from 2012-13, which is our main source for data on the size
distribution of establishment by sector, which is the data used to estimate the reduced-form representation
of technology adoption by sector. We measure employment in an establishment as the sum of non-hired and
hired workers, including family workers and owner.8 We also partially rely on the Fifth Indian Economic
Census as it provides information about the type of power used by individual establishments, a proxy for
the technology used at the establishment level. This data is used in the validation exercise at the end of
this Section. The final dataset for the Sixth (Fifth) Indian Economic Census includes 45363786 (35171881)
establishments employing 108411367 (84134947) individuals.
8
The question ask for the number of persons found working comprising, hired, non- hired (including family members; unpaid
apprentice and owner), on the last working day in the establishment. Regular wage/salaried workers, owner/other family
workers, who are temporarily absent on the last working day are also counted.
20
The WIOD (Timmer et al., 2015) contains international comparable information on the input-output
linkages and sectoral intermediate and final uses, covering 43 countries, for the period 2000-2014. Data for
56 sectors are classified according to the International Standard Industrial Classification revision 4 (ISIC
Rev. 4).9 We use the table for India in 2010. This data gives us information to directly calibrate Γ and Ω.
There is no data available for India that can be easily mapped to the network of investment Λ. We
overcome this limitation by using the network of investment produced in vom Lehn and Winberry (2022)
for the United States in 2010. In particular, the investment network records the share of new tangible and
intangible investment expenditures of sector s that were purchased from sector s′ for each pair of sectors.
They construct the investment network using the BEA Fixed Assets and Input-Output databases for a
sample of 37 private non-farm sectors from 1947-2018. We consider this as the best available proxy for the
production network of modern technologies in India.
After harmonizing the sectors across the three datasets, the India economy that we consider is composed
of 30 2-digit sectors. Details on how we armonize the different data sources into these 30 cohesive sectors are
available in Table III in the Appendix. Also, we follow Broda and Weinstein (2006) and Hsieh and Klenow
(2009) and we set η = 3. Given the value of η, we set ν = 0.75 so that the intermediate input share of gross
output equals 0.49, the value in the WIOD for India in 2010.
4.3
Structural Estimation
In this section we structurally estimate the parameters of the model. We use the insights of previous
sections and estimate reduced-form parameters for each sector within the Indian economy. Then, we use
the system of equilibrium equations in the model to back out the fundamental parameters that make the
partial information estimates that we obtained for each sector consistent across sectors.
We assume that all sectors in the economy share the same distribution of (normalized) ex-ante and
ex-post heterogeneity, {ζ̃, ε̃}, and gains from adoption Ãm . Without loss of generality, we set Ãt = 1. Thus,
for each sector we estimate the (normalized) entry and adoption thresholds {z̃ts , z̃ms }s
in S .
While our
approach may seem restrictive, we show that this parsimonious approach is able to provide a good fit of the
data. Operationally, we estimate the parameters of the model by matching the binned employment share
distribution for each sector—we use 21 bins per sector, where for the common parameters we weight equally
all sectors. We want to notice that, unlike for other parameters, we estimate the tail parameter ζ̃ by using
direct information from the tail of the size-distribution of establishments for the whole Indian economy.
Table I presents the implied share of modern establishments, as /es , and the their value-added share, Ms ,
for the 30 sectors of the Indian economy, as well as the estimated entry cost κes and the entry cost relative
to the cost of adoption, κes /(Pms κas ). The table shows large heterogeneity across sectors. For example, the
estimation provides that modern establishments account for 60% of value-added share in the Mining sector,
0.4% in the Manufacturing of Wood sector (M-Wood), and 87% in the Manufacturing of Motor Vehicles
9
The industry codes of the ISIC Rev. 4 and the NIC 2008 classifications coinceide at 2-digit, which is the level of aggregation
that we use to harmonize them with the investment network in vom Lehn and Winberry (2022).
21
Table I: Modern establishments in India
Division
as /es
Ms
κes
κes
Pms κas
1. Mining
2. M-Food
3. M-Textiles
4. M-Wood
5. M-Paper
6. M-Printing and Media
7. M-Petroleum
8. M-Chemicals and Pharma.
9. M-Plastics
10. M-Other Non-Metallic
11. M-Basic Metals
12. M-Metal Products
13. M-Computer and Electronic
14. M-Electrical Equipment
15. M-Machinery and Equipment
16. M-Motor Vehicles
17. M-Other Transport
18. M-Furniture and Other
19. Utilities
20. Construction
21. Trade
22. Transportation
23. Food and Accommodation
24. Information
25. Professional Services
26. Finance and Insurance
27. Real Estate
28. Education
29. Health and Social Work
30. Other Services
0.021
0.002
0.001
0.000
0.534
0.000
0.031
0.149
0.761
0.098
0.054
0.002
0.070
0.074
0.011
0.227
0.100
0.000
0.024
0.001
0.000
0.000
0.000
0.000
0.015
0.617
0.001
0.994
0.000
0.000
0.606
0.358
0.305
0.004
0.948
0.033
0.648
0.824
0.978
0.780
0.711
0.340
0.742
0.748
0.532
0.868
0.782
0.000
0.622
0.277
0.003
0.209
0.005
0.000
0.568
0.961
0.313
0.999
0.000
0.002
1.387
0.235
0.337
0.262
0.366
0.819
0.607
1.745
0.536
0.938
1.078
0.810
1.083
0.990
0.624
8.340
50.892
0.422
0.640
0.369
0.156
0.125
0.391
0.585
0.577
0.185
0.316
0.282
0.614
0.125
0.023
0.005
0.004
0.000
0.179
0.000
0.029
0.080
0.224
0.061
0.042
0.005
0.049
0.051
0.016
0.104
0.062
0.000
0.025
0.003
0.000
0.002
0.000
0.000
0.019
0.196
0.004
0.265
0.000
0.000
sector (M-Motor Vehicles). Heterogeneity in these shares is rationalized by heterogeneity in the relative
cost of adoption: in Mining the entry cost is 2.3% of the added cost of adoption, in M-Wood it is basically
negligible (in other words, the cost of adoption is very large), and in M-Motor Vehicles is 10% of the added
cost.
Figure 1 presents the empirical and the model implied employment share distributions for these three
22
Figure 1: Examples of employment share distributions, data vs. model
1. Mining
4. M-Wood
16. M-Motor Vehicles
sectors.10 The model closely replicates the distributions in each sector, even though the only degrees of
freedom at the sector level are the entry and adoption thresholds. The Mining sector presents an evident
bimodal employment share distribution, implying that both small and large establishments are highly relevant. As a result, the estimation procedure matches this by having 60% of value-added share accounted
for by modern establishments. Consistent with this, Table I shows that the cost of adoption in the sector,
Pm sκms , is large relative to the entry cost, κes , but not extreme. Highly productive establishments end up
adopting and, while accounting for a small share of establishments in the sector, they account for a relatively
large share of value-added. In the M-Wood sector, the empirical distribution resembles a Pareto distribution
and thus the estimation procedure considers the sector to have almost no establishments adopting modern
technologies, with a value-added share of modern establishments close to zero, which, as shown in Table I,
is obtained in the model by setting large adoption costs relative to entry costs.11 For the M-Motor Vehicles
sector, the empirical distribution shows that most of employment is concentrated in large establishments,
which the model matches with a high adoption rate. In fact, as previously discussed, the value-added share
of modern establishments in the sector is almost 0.87. To generate high adoption rates in this sector, the
estimation recovers entry and adoption costs that are relatively similar, with the adoption cost accounting
for around 90% of total effective cost of adoption.
While we estimate parameters to match individual sector employment share distributions, the model
provides a employment share distribution for the whole India economy which we can contrast with the
empirical one. This can be found in Figure 2. The model-implied distribution closely resembles the distribution observed in the data, although over-predicting the employment share of mid-size establishments,
and under-predicting the employment share of the very large ones.
Given that we calibrated the elasticity of substitution to be equal to 3, it is immediate to back out our
estimates for ζ, χ and Am from the reduced-form estimates. We also note that we provide 95% bootstrap
confidence intervals for the different objects of interest. Table II presents the estimates and confidence
10
Figure 11 in the Online Appendix presents the empirical and model implied employment share distributions for all sectors.
It is also relevant that the M-Wood sector has relatively small establishments relative to the other sectors in the economy.
Otherwise the estimation procedure would interpret the sector as having full adoption of modern technologies.
11
23
Figure 2: Employment share distribution in the aggregate Indian economy
intervals for the common parameters.
Table II: Common parameters, estimates
Ex-ante heterogeneity, ζ
Ex-post heterogeneity, χ
Technology, Am
3.16
0.57
2.18
[3.03, 3.19]
[0.49, 0.60]
[2.01, 2.31]
We also provide an off-sample test of the validity of our estimates. We do so by contrasting adoption
measures implied by the estimated model with adoption proxies that can be observed in the data. In particular, the Fifth Indian Economic Census provides the type of power (i.e. electricity, horsepower, etc.) that a
establishment employs. We then classify each power source as either traditional (without power, fire wood,
animal power, non conventional, and others) or modern (electricity, coal/sift coke, petrol/diesel/kerosen, and
liquefied petroleum gas/natural gas). Figure 3 contrasts the employment share of modern establishments
within a sector using this proxy (x-axis) with that one implied by the structural estimation of the model
(y-axis). The figure shows a strong correlation between the two measures. In fact, for many sectors we see
that the values are close to the 45 degree line. We note that sectors that will exhibit high development
multipliers in our quantitative exercises are among those exhibiting good fit in this here. This is reassuring
of the lack of relevance of the bad fit we observe in Figure 3 for some sectors.
24
Figure 3: The employment share of modern establishments, proxy vs. estimated
4.4
Sectoral revenue multipliers
We begin our quantitative analysis of sectoral development multipliers by presenting estimates of revenue
multipliers for all sectors, ϵr . Figure 4 presents our estimates, with 95% bootstrap confidence intervals. As
seen in the figure, estimates are ’tight’, in the sense that confidence intervals are small and they do not
seem to affect the way one should interpret the ranking of the multipliers. Regarding the estimates, the
figure showcases large heterogeneity in multipliers across sectors. Some sectors are cost-effective, i.e. ϵr > 1,
some sectors are cost ineffective, i.e. ϵr < 1, and even some sectors are such that it is counterproductive to
subsidize them, i.e. ϵr < 0. Among cost-effective sectors, there are a few that stand out as key engines for
development, in decreasing order of relevance: 11. M-Basic Metals, 15. M-Machinery and Equipment, 1.
Mining, 13. M-Computer and Electronic, and 14. M-Electrical Equipment.
Multipliers are the result of normalizing the revenue subsidy elasticity of consumption by the cost of the
subsidy, measured by the Domar weight. The left panel of Figure 5 provides a scatter plot with the subsidy
elasticity in the y-axis and the revenue multiplier in the x-axis, while the right panel presents a scatter
plot with the Domar weight in the y-axis and the revenue multiplier in the x-axis. As it is clear from the
figure, there is a strong positive relationship between the subsidy elasticity and the multiplier, and a strong
negative relationship between the Domar weight and the multiplier. The figure shows that sectors with
high multipliers are those with intermediate-to-high values for the subsidy elasticity and low values for the
25
Figure 4: Sectoral revenue multipliers, ϵr
Domar weight. The M-Basic Metals sector falls in this category, exhibiting the highest multiplier, a subsidy
elasticity that is high, but 30% lower than that one in the Trade sector, and a Domar weight that is very
low, 85% lower than in the Trade sector. Overall, sectors with high multipliers exhibit intermediate values
for consumption elasticity and, relative to sectors with large values for the elasticity, are disproportionately
smaller, thus severely reducing the fiscal cost of the policy.
The relevance of adoption margin as a key determinant of sectoral multipliers can be measured by
comparing ϵr with ϵea
r , where this last one is the multiplier that would be obtained after shutting down
entry and adoption margins. The left panel of Figure 6 presents the multipliers for these two cases. The
effect of the adoption margin is highly heterogeneous across sectors. In some key sectors, the adoption
margin greatly amplifies the degree of the sectoral multiplier. In some sectors, the adoption margin turns
the multiplier negative. And in other sectors, the adoption margin barely affects the level of the multiplier.
Heterogeneity of the effect of adoption for multipliers across sectors manifests on the way sectors are
ranked in terms of multipliers with and without technology adoption. The right panel of Figure 6 presents
in the x-axis the ranking of the sector revenue development multiplier in the economy with no entry and
adoption, i.e. the ranking using ϵea , and in the y-axis the ranking of the sector revenue development
multiplier in the economy with entry and adoption, i.e. the ranking using ϵ. While there is little variation
for those sectors considered less relevant in the economy with no adoption—i.e. those sectors with low
26
Figure 5: Components of sectoral revenue multipliers
Subsidy elasticity of C
Domar weight, Ψ̃
Figure 6: The role of adoption
ϵr and ϵea
r
Ranking
multipliers, there is substantial variation in rankings among those exhibiting intermediate and high values
for multipliers. For example, the top 5 sectors abstracting from adoption are, in descending order of
relevance, 11. M-Basic Metals, 1. Mining, 19. Utilities, 12. M-Metal Products, and 10. M-Other NonMetallic, while the top 5 sectors in the economy with adoption are, in descending order of relevance, 11.
M-Basic Metals, 15. M-Machinery and Equipment, 1. Mining, 13. M-Computer and Electronic, and 14.
M-Electrical Equipment. With a finite amount of resources to be allocated to industrial policy, this result
suggests that accounting for the adoption margin is crucial for the policy.
27
4.4.1
Determinants of amplification through adoption
Figure 6 showcases the relevance of the adoption margin for multipliers: through adoption, there is substantial amplification, and this amplification is heterogeneous across sectors. In this section we study the
determinants of amplification through adoption. To do so, we study the case with no entry as it is simpler
and, as shown in Table IV in the Appendix, multipliers barely change when the entry channel is considered. Using (20) together with the definition of a sectoral revenue multiplier, we define the contribution of
adoption to the multiplier as
h
i
η−1 ′
′
−1
A=
Ψ̃ −
⊘ Ψ′ ,
Ψ βdiag (M ) ∇a,ra I + ∇P Y,r + Λ (I − νΩ)
η
where ⊘ denotes element-by-element division. We exploit the following decomposition of A to study the
determinants of amplification,
direct incentive
z
}|
{
η−1
ζ
′
′
A = Ψ̃ ⊘ Ψ −
1 βdiag (M )
η
η−1
price of adoption, Pm
z
}|
{
η−1 ′
ζ
′
+ Ψ̃ −
Ψ βdiag (M )
Λ (I − νΩ)−1 ⊘ Ψ′
η
η−1
feedback through adoption
z
}|
{
h
i
η−1 ′
ζ
′
+
Ψ̃ −
Ψ βdiag (M ) I + Λ (I − νΩ)−1 ∇a,ra −
I
⊘ Ψ′
η
η−1
aggregate demand
}|
{
z
η−1 ′
′
+ Ψ̃ −
Ψ βdiag (M ) ∇a,ra ∇P Y,a ⊘ Ψ′ ,
η
(24)
The direct channel in A relates to Remark 2, and considers the direct effect of a subsidy on adoption, as
showcased in (18), abstracting from heterogeneity that stems from ∇a,ra . To assess the average relevance
of this channel we consider the case where M = m̄I, where m̄ is equal to the average equally-weighted
modern share across sectors. In this case, the direct effect on adoption implies a change in sectoral TFP,
(d ln Z/d ln a) (d ln a/d ln ra ) = β m̄ζ/(η − 1). For a particular sector s,
=0.115
=0.038
z
}|
{
<3 z }| {
z}|{
ζ
ζ
direct incentive = ϵea
+ β m̄
.
rs β m̄
η−1
η(η − 1)
(25)
From Table IV in the Appendix, the multiplier ϵea
rs is lower than 3. Thus, the direct channel of adoption can
generate amplification raging from 0.038 to 0.383. Given that amplification is larger than 2 for many sectors,
28
these simple calculations imply that the average contribution of the direct margin is relatively small.12
The second term in the decomposition measures the importance of the revenue subsidy in lowering
the price adopting modern technologies, Pm . The third term in the decomposition of A accounts for the
relevance of feedback effects in adoption, as captured by ∇a,ra − ζ/(η − 1)I. This object captures the
Double-Leontieff inverse described in detail in Proposition 2. Sectors showcasing large values for this term
are those for which an increase in adoption generates large increases in adoption in other sectors. Finally,
the fourth term in the decomposition of A accounts for the relevance of the feedback effects through the
aggregate demand channel.
Table V in the Appendix shows each term in this decomposition. Consistent with the discussion above,
the direct incentive term is small relative to A. Furthermore, the aggregate demand term is also small. As
a result, we now focus on the main two terms: the price Pm and the feedback through adoption terms.
Figure 7: Determinants of amplification
Figure 7 presents the contributions of the Pm channel and the feedback in adoption channel. We also
ψ̃rs
Notice that the term ϵea
rs = ψrs − 1 in (25) is the same as the key statistic in Liu (2019). Here, because of adoption, the
key statistic is ’transformed’ by the elasticity of TFP with respect to the subsidy.
12
29
add A to the figure to aid in the analysis. The other terms in the decomposition are not included as they
are small relative to the level of amplification observed through adoption. The figure shows that the channel
through the price of adoption Pm accounts for a large part of amplification through adoption A. While
the price of adoption channel is relevant for all sectors, the feedback channel is only substantial for sectors
exhibiting high values of A. For example, for the top 5 sectors in terms of multipliers, the feedback channel
accounts for 20 to 30% of amplification through adoption.
What are the forces driving the relevance of the price of adoption channel? Specifically, what is the
relative importance of centrality of a sector in the production Ω and investment Λ networks? To gain
insights, we further decompose the expression for the Pm term, exploiting the identity
Λ (I − νΩ)−1 =Ω (I − νΩ)−1 + (Λ − Ω) (I − νΩ)−1 .
(26)
This identity allows us to study the relevance of the two different input-output networks in shaping up the
price of adoption term in the decomposition above. We decompose the price of adoption effect into two
terms. The first term accounts for the Pm channel under the counterfactual assumption that the adoption
good is produced with the same input-output structure as the intermediate aggregate, i.e., Ω. The second
term in the decomposition measures the relative importance of a sector in the actual investment network Λ
relative to Ω.13
Figure 8 reproduces the contribution of the price of adoption term to amplification, and the difference
between this term and a counterfactual price of adoption term where we replace the actual investment
network Λ with Ω (labeled Λ − Ω in the figure). A sector exhibiting a high value for this term is one that is
more central in the investment than in the production network. Figure 8 shows that the investment network
is important for explaining amplification for some key sectors. For example, while for sector 11. M-Basic
Metals, this channel is muted, for sector 15. M-Machinery and Equipment centrality in Λ is instrumental
for amplification.
4.5
Alternative policy instruments
Which policy instrument is the most appropriate to promote economic development? The different instruments differ in (i) the way they improve production efficiency and promote the adoption of modern
technologies, and (ii) the fiscal cost of implementation (for example, see Remark 4 in Section 3.2 and results in Section 3.3.1). As there are clear trade-offs among the different policy instruments, we resort to a
quantitative analysis to understand which instrument is the most cost-effective for development.
Figure 9 presents, for each sector, the resulting multiplier under the four alternative policy instruments:
13
In particular, the price of adoption channel in (24) is decompose as follows:
η−1 ′
ζ
η−1 ′
ζ
′
′
Ψ̃ −
Ψ βdiag (M )
Ω (I − νΩ)−1 ⊘ Ψ′ +
Ψ̃ −
Ψ βdiag (M )
(Λ − Ω) (I − νΩ)−1 ⊘ Ψ′ .
η
η−1
η
η−1
30
Figure 8: Investment Λ or production Ω networks?
revenue subsidies r, intermediate input subsidies r x , labor subsidies r l , and adoption subsidies r a . Next
we discuss the main takeaways.
First, among all instruments, labor subsidies rl are the least effective tool to promote development, as
evident from the fact that, for all sectors, labor subsidies provide the lowest multipliers. This is expected,
as labor is in fixed supply, and consistent with the results in Section 3.3.1.
Second, revenue multipliers ϵr are always above labor multipliers ϵrl and below intermediate input
multipliers ϵrx . While consistent with the results in 3.3.1, it is surprising that there is no sector where
mutlipliers under revenue subsidies are above multipliers under intermediate inputs, given that revenue
subsidies have a direct effect on the marginal adopters’ condition and intermediate input subsidies do not.
Key for this result is that the cost of implementing a intermediate input subsidy is a fraction ν of the cost
of implementing a revenue subsidy.
Third, and most importantly, adoption subsidies appear to be the most cost-effective way to promote
economic development, in spite from the fact that they do not have a direct effect on the production
31
Figure 9: Multipliers under alternative policy instruments
efficiency channel, as evident from (20). Overall, among all sectors and among all policy instruments,
the top 5 multipliers are obtained through implementing adoption subsidies in the following sectors (in
descending order of relevance): 11. M-Basic Metals, 1. Mining, 15. M-Machinery and Equipment, 12.
M-Metal Products, and 13. M-Computer and Electronic.
What does it make adoption subsidies the most cost-effective policy instrument? To shed light on this
question, we perform a decomposition along the lines we did in (24). In particular, we decompose the
adoption multiplier into a direct effect and a feedback through the adoption channel,
feedback through adoption
ϵr a
direct incentive
}|
{
z
}|
{ z
′
η
−
1
ζ
η
−
1
η
ζ
′
= Ψ̃ ⊘ Ψ′ −
1
+ Ψ̃ −
Ψ′ βdiag (M ) ∇a,ra −
I
⊘ ∆′ . (27)
η
η−1η−1
η
η−1
As before, simple calculations can be used to gauge an idea of the magnitude of the direct incentive
effect in a particular sector s,
32
=2.37
direct incentive, ra = ϵea
rs
}|
=0.79
}|
{
ζ
ζ
η
η
+
.
η − 1 η − 1 η − 1 η(η − 1)
z
{
z
Different from the case with revenue subsidies in (25), the direct incentive effect is guaranteed to be
large with adoption subsidies: It is over two times the size of the multiplier in a model with no entry and
adoption, ϵea
rs , and it is bounded below by 0.79. The larger magnitude results from the fact that the fiscal
cost of an adoption subsidy δs is substantially lower than the fiscal cost of a revenue subsidy ψs . In fact,
δs = ((η − 1)/η)βMs ψs < ψs . Indeed, the difference in the fiscal cost is the only difference between the
direct incentive effect in (24) relative to (27).
Figure 10: Adoption multipliers
Figure 10 presents the contribution of the direct incentive and feedback through adoption effects for
all sectors. As suggested by the simple calculations for the direct incentive under adoption subsidies, the
contribution of the direct incentive effects is now dominant but, as with revenue subsidies, the feedback
33
through adoption effect is still substantial for the top sectors in terms of the multiplier.
5
Concluding remarks
This paper provides a theoretical and quantitative analysis of the role of sectoral industrial policies as a tool
to foster economic development. We find that the technology adoption margin has important effects on the
magnitude and relative ranking of sectoral development multipliers, i.e., the effect of a sectoral policy on
aggregate consumption, relative to its fiscal cost. In addition, we show that technology adoption subsidies
are the most cost-effective policy instrument.
More generally, the analysis in this paper can be seen as illustrating a framework that can be used to
analyze policies aimed at promoting the adoption of new technologies. While we applied the framework to
study technology adoption for economic development, it can also be applied to alternative questions. For
example, we view the case of the adoption of green technologies as a natural application.
In the current paper, traditional and modern technologies use factors with the same intensity, a counterfactual assumption (Boehm and Oberfield, 2023). This assumption limits the complementarities in technology adoption, which have been shown to be quantitatively important by Buera et al. (2021). While we have
abstracted from this margin in order to be as transparent as possible, we conjecture that the conclusions
would be strengthened by incorporating this heterogeneity. We also abstracted from dynamics. As such, the
analysis should be understood as describing the effect of policies across steady-states. A dynamic version of
the analysis, using the methods in Alvarez et al. (2023), is a natural next step.
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Appendix
A
Proofs and derivations
A.1
Domar weights
Multiplying both sides of (7) by Ps and using that Ps Cs = γs Pc C provides that
γs Pc C +
X
ωs′ s Pxs′ Xs′ +
s′
X
λs′ s Pms′ as′ κms′ =Ps Ys ,
s′
and by using the intermediate input demand in (3), the expression relating Ps and Pxs in (10), and the
definition of sector productivity in (9) we obtain
γs Pc C + ν
X
η−1X
ωs′ s rs′ Ps′ Ys′ +
λs′ s Pms′ as′ κms′ =Ps Ys .
η
′
′
s
(28)
s
When rs = 1 for all s, and by dividing by Pc C we obtain for each s that
X
Ps Ys
η−1X
P s ′ Ys ′
Pms′ as′ κms′
−ν
ωs′ s
=γs +
λs′ s
,
Pc C
η
Pc C
Pc C
′
′
s
s
X
Ps Ys
η−1X
P s ′ Ys ′
−ν
ωs′ s
=γs +
λs′ s δs′ .
Pc C
η
Pc C
′
′
s
s
We can stack these equations and obtain an expression for the Domar weigths Ψ,
η−1 ′
Ψ= I −ν
Ω
η
−1
Γ + Λ′ ∆ .
Or, expressed in row vectors,
−1
η−1
Ψ = Γ +∆Λ I −ν
Ω
.
η
′
A.2
′
′
Derivation of equation (16)
We begin by noting that we can combine labor input demand in (2), teh expression relating Ps and Pxs in
(10), the definition of sector productivity in (9) with the labor market clearing condition in (8) to obtain
the following expression,
L =(1 − ν)
η−1X
rs Ps Ys .
η
s
37
Log-differentiating this expression around r = 1,
X
Ps Ys (d ln rs + d ln Ps Ys ) =0 .
(29)
s
Note how, by dividing by Pc C, this expression provides that Ψ′ [d ln r + d ln (P ◦ Y )] = 0. That is, weighted
by Domar weights, the total change in subsidies is offset by the change in gross product.
Log-differentiating the goods market clearing condition in (28) around r = 1 provides,
γs Pc C (d ln Pc + d ln C) + ν
η−1X
ωs′ s Ps′ Ys′ (d ln rs′ + d ln Ps′ Ys′ )
η
′
s
+
X
λs′ s Pms′ as′ κms′ (d ln Pms′ + d ln as′ ) = Ps Ys d ln Ps Ys .
s′
Adding across all sectors s, using (29), and dividing by Pc C,
d ln C = − d ln Pc −
X Pms′ as′ κms′
s′
Pc C
(d ln Pms′ + d ln as′ ) −
X Ps Ys
s
Pc C
d ln rs ,
or, in matrix form,
d ln C = − d ln Pc − ∆′ (d ln Pm + d ln a) − Ψ′ d ln r
Using that from (11) we obtain that d ln P = − (I − νΩ)−1 (d ln Z + d ln r), and using (12),
d ln C = Γ′ + ∆′ Λ′ (I − νΩ)−1 (d ln Z + d ln r) − ∆′ d ln a − Ψ′ d ln r .
′
Finally, noting that Ψ̃ = [Γ′ + ∆′ Λ′ ] (I − νΩ)−1 ,
′
′
d ln C = Ψ̃ − Ψ′ d ln r + Ψ̃ d ln Z − ∆′ d ln a .
(30)
Combining (4), (6), and (10), and specializing to the case with only revenue subsidies, we obtain
rs
η
− η−1
η−1
Aη−1
as ζ
m − At
Zsη−1
Ps Ys = Pms κms
or
rs
η−1
βMs Ps Ys = Pms as κms .
η
Finally, writing the conditions in vector form, dividing both sides of the expression by Pc C and setting
38
r = 1, there is a tight connection between the GDP share of a sector, or Domar weight, and its adoption
share of GDP,
η−1
β diag (M ) Ψ = ∆ .
η
Applying this expression to (30) completes the derivation.
A.3
The elasticity of adoption to revenue and adoption subsidies
We begin by computing an expression for d ln (P ◦ Y ). To this end, we first rewrite (28) in matrix form and
log-differentiate to get,
η−1 ′
Ω diag (Ψ) d ln (P ◦ Y ) =Γ (d ln Pc + d ln C)
I −ν
η
d ln Pm
′
+ Λ diag (∆)
+ I d ln a
d ln a
d ln Pm
η−1 ′
′
+ Λ diag (∆)
+ν
Ω diag (Ψ) d ln r .
d ln r
η
Also, we follow the same steps as in Appendix A.2 to obtain an expression for d ln Pc + d ln C,
′
d ln Pc + d ln C = −∆
d ln Pm
′ d ln Pm
+ I d ln a − ∆
+ Ψ d ln r .
d ln a
d ln r
Combining these two expressions provide that
d ln (P ◦ Y ) =∇P Y,a d ln a + ∇P Y,r d ln r ,
h
i−1
′
where ∇P Y,a = I − ν η−1
Ω
diag
(Ψ)
Λ′ diag (∆) ddlnlnPam + I − Γ∆′ ddlnlnPam + I , and ∇P Y,r =
η
h
i−1 h
i
η−1 ′
d ln Pm
′
′
′ d ln Pm
I − ν η−1
Ω
diag
(Ψ)
Λ
diag
(∆)
+
ν
Ω
diag
(Ψ)
−
Γ
∆
+
Ψ
, and where, usη
d ln r
η
d ln r
ing (11) and (12),
d ln Pm
d ln a
= −Λ (I − νΩ)−1
d ln Z
d ln a ,
and
d ln Pm
d ln r
= −Λ (I − νΩ)−1 .
Having obtained an expression for d ln (P ◦ Y ), we can now solve for d ln a using the marginal adopters’
conditions,
d ln r − (η − 1)d ln Z + d ln (P ◦ Y ) −
η−1
d ln a = −d ln r a + d ln Pm .
ζ
Substituting the expression for d ln (P ◦ Y ) produced in (37) and rearranging,
η−1
d ln Pm d ln Z
d ln Pm
I + (η − 1)I −
− ∇P Y,a d ln a = I + ∇P Y,r +
d ln r + d ln r a ,
ζ
d ln Z d ln a
d ln r
39
so that
h
i
d ln a =∇a,ra I + ∇P Y,r + Λ (I − νΩ)−1 d ln r + ∇a,ra d ln r a ,
where
−1
h
i
η−1
−1 d ln Z
=
,
I + (η − 1)I − Λ (I − νΩ)
− ∇P Y,a
ζ
d ln a
(
"
)−1
#
Λ (I − νΩ)−1
ζ
ζ
I−
=
− I ζβdiag (M ) −
∇P Y,a
.
η−1
η−1
η−1
∇a,ra
A.4
The density hs (l)
We are interested in Hs (l) = Pr (lis (z̃, ε̃) < l) = Pr (ln lis (z̃, ε̃) < ln l). Using (22) this reduces to
Hs (l) = Pr ln Ãi + ln z̃s + ε̃ < ln l = Pr ε̃ < ln l − ln Ãi − ln z̃s ,
!
Z z̃ms
Z ∞
ln l − ln Ãt − ln z̃ − µ̃
1
1
−ζ̃−1
Φ
Φ
ζ̃z
dz̃ +
=
−ζ̃
−ζ̃
χ̃
z̃ts
z̃ms
z̃ts
z̃ts
ln l − ln Ãm − ln z̃ − µ̃
χ̃
where Φ(·) denotes the CDF of a standard normal. To compute hs (l) recall that hs (l) =
hs (l) =
Z
1
ln l − ln Ãt − ln z̃ − µ̃
χ̃
ϕ
−ζ̃
χ̃lz̃ts
+
z̃ms
z̃ts
Z
1
−ζ̃
χ̃lz̃ts
∞
ϕ
z̃ms
!
ζ̃z −ζ̃−1 dz̃ ,
∂Hs (l)
∂l .
Then,
!
ζ̃z −ζ̃−1 dz̃
ln l − ln Ãm − ln z̃ − µ̃
χ̃
!
ζ̃z −ζ̃−1 dz̃ .
Integrating provides the expression in the text provided in (23).
A.5
Proof of Remark 5
Without loss of generality let Ãt = 0. Because all entrants are adopters of the modern technology, z̃ts = z̃ms .
Then (23) reduces to
hs (l) =l
−ζ̃−1
ζ̃
−ζ̃
z̃ms
2
µ̃ζ̃+ χ̃2
e
"
Ãζ̃m
1−Φ
ln z̃ms − ln l + µ̃ + χ̃2 ζ̃
χ̃
!#
.
Notice that
"
∂hs (l) hs (l) 1
=
haz
∂l
l
χ̃
ln z̃ms − ln l + µ̃ + χ̃2 ζ̃
χ̃
40
!
− ζ̃ + 1
#
,
ϕ
where haz (·) ≡
ln z̃ms −ln l+µ̃+χ̃2 ζ̃
χ̃
2
1−Φ ln z̃ms −lnχ̃l+µ̃+χ̃ ζ̃
is the hazard rate of the normal distribution, where haz ′ > 0 and
haz ′′ > 0.
Because hs (l) ≥ 0 and the hazard rate decreasing and convex in l, the distribution Hs (l) has one mode.
χ̃2 ζ̃
≤
χ̃
ζ̃
+
1
then the mode is at l = z̃ms . Otherwise, the mode is at some l > z̃ms .
If haz ln z̃ms +µ̃+
χ̃
For the employment share distribution, recall that gs (l) =
R
lhs (l)
,
l̂hs (l̂)dl̂
so that
∂gs (l)
∂l
∝ hs (l) + l ∂h∂ls (l) .
Then,
"
∂gs (l) gs (l) 1
=
haz
∂l
l
χ̃
As a result, again we have that if haz
ln z̃ms − ln l + µ̃ + χ̃2 ζ̃
χ̃
ln z̃ms +µ̃+χ̃2 ζ̃
χ̃
!
#
− ζ̃
.
≤ χ̃ζ̃ then the mode is at l = z̃ms . Otherwise, the
mode is at some l > z̃ms .
A.6
Proof of Remark 6
When χ → 0 we also have that χ̃ → 0. Then, when χ → 0, the expression for hs (l) in (23) reduces to
ζ̃
l−ζ̃−1
z̃ −ζ̃ ζ̃ Ãt
ts
hs (l) = 0
l−ζ̃−1 ζ̃ Ãζ̃m
−ζ̃
z̃ts
if z̃ts ≤ l ≤ z̃ms ,
if z̃ms ≤ l < Ãm z̃ms ,
(31)
if l ≥ z̃ms .
Because ζ̃ > 0 and Ãm > Ãt , Hs (l) has two modes: one at l = z̃ts , and one at l = z̃ms . Likewise, because
gs (l) ∝ lhs (l), Gs (l) has the same two modes. Further, if Ãm = Ãt both distributions have a single mode,
and the mode is at l = z̃ts .
A.7
Proof of Proposition 3
The data is given by values of η, ν, Γ, Ω, and Λ, the normalization Ats = 1, and the reduced form estimates
1
ζ̃ = ζ η−1 , Ãms = Aη−1
ms , and the following reduce-form thresholds
z̃ts = (1 − ν)
η − 1 Ps Ys η−1
z
,
η Zsη−1 ts
(32)
z̃ms = (1 − ν)
η − 1 Ps Ys η−1
z
.
η Zsη−1 ms
(33)
41
zts
zms
The ratio of (32) and (33) provides
of two equations relating
−ζ
zts
,
−ζ
zms
=
z̃ts
z̃ms
1
η−1
. Combining this expression with (33) provides a system
and a sector’s gross value added Ps Ys ,
(1 − ν)
η−1
ζ
−ζ η−1
Ps Ys = z̃ms
zms
Ats
η
ζ − (η − 1)
and
−ζ
zts
=
z̃ts
z̃ms
−
z̃ts
z̃ms
ζ
η−1
− ζ−(η−1)
η−1
η−1
+ Aη−1
,
ms − Ats
−ζ
zms
.
We next derive a system of equations determining the vector of gross output, given values for η, ν, Ω, Λ, the
η−1
η−1
productivity parameters Ams
and Ats
, and the value of the reduced form thresholds. The good market
clearing condition in sector s is given by
γs
X
X
Pxs′
Pms′ −ζ
Pc
C+
ωs′ ,s
Xs′ +
λs′ ,s
z ′ κms′ = Ys ,
Ps
Ps
Ps ms
′
′
s
where Xs =
ν
η−1
η
s
η
ν w 1−ν )(η−1)
Pxs (Pxs
Ps
Psη Ys (Zs )η−1 = ν η−1
η Pxs Ys , where we used that Ps =
ν w 1−ν
η Pxs
.
η−1
Zs
Then, the
goods market clearing condition in sector s can be expressed as
η
X
X
Ps′
Pms′ −ζ
Ys′ +
λs′ ,s
z ′ κms′ =Ys .
Ps
Ps ms
′
ωs′ ,s
s′
(34)
s
Similarly, we can express the labor market clearing condition as follows,
(1 − ν)
X
η−1X
−ζ
Ps Ys = L −
κts zts
.
η
s
s
(35)
We now turn back to the goods market clearing condition in (34). Summing over all sectors s,
Pc C +
X
−ζ
Pms zms
κms
=
s
η−1
1−ν
η
X
Ps Ys ,
s
and using (35),
Pc C =
1 − ν η−1
η
(1 − ν)
η−1
η
"
#
L−
X
−ζ
κts zts
−
X
s
−ζ
Pms zms
κms .
s
Substituting back into (34)
(
γs
1 − ν η−1
η
(1 − ν)
η−1
η
"
L−
#
X
s
−ζ
zts
κts −
)
X
−ζ
zms
Pms κms
+ν
s
X
η−1X
−ζ
ωs′ ,s Ps′ Ys′ +
λs′ ,s zms
Pms′ κms′ = Ps Ys .
η
′
′
s
42
s
The values of κts and Pms κms can be obtained as function of the reduced form thresholds and productivity,
given values for η and ν. In particular, using the marginal entrant’s condition,
κts =
η−1
1
1 η−1 Ps Ys zts
1
Ats
Aη−1
z̃ts ,
=
η−1
η
1 − ν η − 1 ts
Zs
where we used (32). Similarly, using the marginal adopter’s condition,
1 η−1
1
Ams − Aη−1
z̃ms .
ts
1−νη−1
Pms κms =
Thus, given values for η, ν, Γ, Ω, Λ, and reduced form estimates ζ̃, Ãms , z̃ts , and z̃ms , the vector of the
value of sector gross output,Ps Ys , solves
(
γs
1 − ν η−1
η
(1 − ν) η−1
η
"
L−
#
X
−ζ
zts
κts
s
)
−
X
−ζ
zms
Pms κms
+ν
s
X
η−1X
−ζ
ωs′ ,s Ps′ Ys′ +
λs′ ,s zms
Pms′ κms′ = Ps Ys ,
η
′
′
s
s
where
1
1
Aη−1 z̃ts ,
1 − ν η − 1 ts
1 η−1
1
z̃ms ,
Ams − Aη−1
=
ts
1−νη−1
κts =
Pms κms
−ζ
zms
=
(1 − ν) η−1
η Ps Ys
# ,
ζ−(η−1)
−
η−1
η−1 z̃ts
η−1
η−1
Ats
+ Ams − Ats
z̃ms
"
ζ
z̃ms ζ−(η−1)
and
−ζ
zts
=
z̃ts
z̃ms
−
ζ
η−1
−ζ
zms
.
Thus, given the structural thresholds, the price of the adoption good in sector s Pms and the adoption cost
κms can be calculated.
B
Additional tables and figures
In this section we collect additional results.
B.1
Detailed description of sectors and cross-walk
In this section we present a table showing how we harmonize the different dataset sources into the 30
sectors we use to describe the Indian economy. The sectors in the Indian Economic Census of 2013 uses the
43
National Industrial Classification NIC 2008, which is based in the ISIC Revision 4. The WIOD data uses
International Standard Industrial Classification ISIC revision 4. These two industry classification coincide
at 2-digit, which is the lowest level of aggregation in our analysis. Finally, the investment network data from
vom Lehn and Winberry (2022) is provided using a 37-sector partition, which is based on NAICS industry
codes. We harmonize the sectors at the 2-digit. The cross-walk used is shown in the last two columns of
Table III.
44
Table III: Detailed Description of Sectors, and Cross-Walk across Datasets
Short Name
Long Description
1.
Mining
2.
M-Food
3.
M-Textiles
4.
M-Wood
5.
6.
M-Paper
M-Printing and media
7.
M-Petroleum
8.
M-Chemicals and pharma
9.
M-Plastics
10.
M-Other Non-Metallic
11.
M-Basic Metals
12.
M-Metal Products
13.
M-Computer and Electronic
14.
M-Electrical Equipment
15.
M-Machinery and Equipment
16.
M-Motor Vehicles
17.
18.
M-Other Transport
M-Other
19.
Utilities
20.
Construction
21.
Trade
22.
Transportation
23.
Food and accomodation
24.
Information
Mining and quarrying
Manuf. of food products,
beverages, and tobacco products
Manuf. of textiles, apparel,
leather and related products
Manuf. of wood and products
of wood, cork and straw, except furniture
Manuf. of paper and paper products
Printing and reproduction of recorded media
Manufacture of coke and refined
petroleum products
Manuf. of chemicals and chemica,
pharmaceuticals, and botanical products
Manuf. of rubber and plastics products
Manuf. of other non-metallic mineral
products
Manuf. of basic metals
Manuf. of fabricated metal products, except
machinery and equipment
Manuf. of computer, electronic and optical
products
Manuf. of electrical equipment
Manuf., repair and instalation of machinery
and equipment n.e.c.,
Manuf. of motor vehicles, trailers and
semi-trailers
Manuf. of other transport equipment
Manuf. of furniture and other manufacturing
Electricity, gas, and water supply; sewerage,
wast collection and other waste management
services
Construction of buildings, civil engineering,
and specialized construction activities
Wholesale and retail trade
Land, water and air transportation, including
transport via pipelines; warehousing and
support activities for transportation; post
and courier activities
Foor and accommodation service activities
Publishing activities; motion picture, video
and television programme production, sound
recording and45music publishing activities;
broadcasting and programming activities
Codes
NIC 2008/
ISIC Rev. 4
Vom Lehn
& Winberry
05-09
1
10-12
15
13-15
16-17
16
4
17
18
18
19
19
20
20-21
21
22
22
23
5
24
6
25
7
26
9
27
10
28-33
8
29
11
30
31-32
12
13-14
35-39
2
41-43
3
45-47
23-24
49-53
25
55-56
35-36
58-61
26
Short Name
25.
Professional Services
26.
Finance and insurance
27.
Real estate
28.
Education
29.
Health and social work
30.
Other services
Long Description
Computer programming and consultancy;
accounting, advertising, architecture,
engineering,legal, management consultancy,
and market research activities; activities of
head offices; technical testing and analysis;
scientific research and development;
veterinary activities
Financial service, insurance, reinsurance
and pension funding, except compulsory
social security
Real estate, including rental and leasing,
services to building and landscaping;
employment activities, security and
investigation, office administrative,
office support and other business
support activities
Education
Health, residential care, social work
activities without accommodation
Other services
46
Codes
NIC 2008/
ISIC Rev. 4
Vom Lehn
& Winberry
62-63,
69-75
29
64-66
27, 30
68, 77-82
28
85
32
86-89
33
90-96
37
B.2
Sectoral revenue multipliers
Table IV produces the revenue development multipliers for all sectors. The first column provides the
(backward) Domar weight or influence measure Ψ̃ and the second column provides the (forward) Domar
weight Ψ—the size of the sector. In the spirit of Baqaee and Farhi (2020), the third column provides the
development multipliers fixing entry and adoption, ϵea . The fourth column presents the multiplier fixing
only entry, ϵe , and the fifth column presents the multiplier with both entry and adoption being active. The
fact that there are no substantial differences between column 4 and 5 provides reassurances of our analysis
abstracting from the entry margin in Section 3.
B.3
Decomposition of amplification under revenue multipliers
In this section we present the results of the decomposition of the amplification term, as described in Section
4.4.1. We perform the decomposition abstracting from the entry channel, as accounting for this channel is
barely relevant for multipliers as evident in Table IV. Each column of Table V accounts for a term in the
decomposition of A.
47
Table IV: Sectoral (revenue) development multipliers
Division
1. Mining
2. M-Food
3. M-Textiles
4. M-Wood
5. M-Paper
6. M-Printing and Media
7. M-Petroleum
8. M-Chemicals and Pharma.
9. M-Plastics
10. M-Other Non-Metallic
11. M-Basic Metals
12. M-Metal Products
13. M-Computer and Electronic
14. M-Electrical Equipment
15. M-Machinery and Equipment
16. M-Motor Vehicles
17. M-Other Transport
18. M-Furniture and Other
19. Utilities
20. Construction
21. Trade
22. Transportation
23. Food and Accommodation
24. Information
25. Professional Services
26. Finance and Insurance
27. Real Estate
28. Education
29. Health and Social Work
30. Other Services
48
Ψ̃
Ψ
ϵea
r
ϵer
ϵr
0.16
0.30
0.16
0.02
0.05
0.04
0.29
0.17
0.06
0.04
0.15
0.08
0.07
0.05
0.07
0.08
0.02
0.08
0.18
0.23
0.47
0.47
0.21
0.09
0.07
0.37
0.14
0.10
0.05
0.08
0.04
0.20
0.11
0.01
0.02
0.02
0.12
0.07
0.03
0.01
0.04
0.03
0.03
0.02
0.03
0.05
0.01
0.04
0.06
0.11
0.25
0.24
0.11
0.04
0.05
0.15
0.13
0.09
0.05
0.04
2.67
0.48
0.43
1.27
1.76
1.22
1.48
1.50
1.43
1.85
2.68
1.91
1.59
1.79
1.16
0.76
1.25
1.11
1.91
1.12
0.90
0.97
0.94
1.42
0.50
1.43
0.06
0.09
0.10
0.76
4.54
0.54
0.48
1.77
2.83
1.68
2.43
2.39
2.42
3.55
5.70
3.87
4.27
4.11
4.63
2.20
4.07
1.76
3.35
2.28
1.36
1.45
1.23
2.64
3.95
2.44
0.04
0.17
0.13
0.99
4.57
0.45
0.38
1.91
2.77
1.78
2.44
2.33
2.33
3.52
5.71
4.03
4.22
4.06
4.60
2.13
3.95
1.90
3.36
2.36
1.43
1.46
1.26
2.88
3.80
2.39
-0.06
0.10
-0.00
1.02
Table V: Decomposition of A
Division
Direct
Pm
Feedback
Demand
0.45
0.07
0.06
0.00
0.47
0.01
0.29
0.37
0.40
0.42
0.53
0.20
0.35
0.39
0.20
0.23
0.31
0.00
0.35
0.10
0.00
0.07
0.00
0.00
0.12
0.40
0.03
0.10
0.00
0.00
1.18
0.12
0.12
0.53
0.57
0.44
0.51
0.49
0.54
1.07
1.95
1.35
1.73
1.47
2.48
0.93
1.91
0.65
0.89
0.88
0.47
0.40
0.40
1.04
2.50
0.59
0.04
0.09
0.12
0.31
0.39
0.05
0.05
0.15
0.15
0.13
0.17
0.15
0.15
0.38
0.72
0.48
0.74
0.55
0.98
0.34
0.78
0.18
0.32
0.30
0.13
0.13
0.12
0.28
1.17
0.18
0.02
0.04
0.04
0.09
-0.16
-0.18
-0.17
-0.19
-0.13
-0.12
-0.03
-0.11
-0.11
-0.16
-0.17
-0.06
-0.14
-0.09
-0.17
-0.06
-0.12
-0.18
-0.12
-0.13
-0.14
-0.13
-0.22
-0.09
-0.32
-0.16
-0.11
-0.14
-0.13
-0.16
1. Mining
2. M-Food
3. M-Textiles
4. M-Wood
5. M-Paper
6. M-Printing and Media
7. M-Petroleum
8. M-Chemicals and Pharma.
9. M-Plastics
10. M-Other Non-Metallic
11. M-Basic Metals
12. M-Metal Products
13. M-Computer and Electronic
14. M-Electrical Equipment
15. M-Machinery and Equipment
16. M-Motor Vehicles
17. M-Other Transport
18. M-Furniture and Other
19. Utilities
20. Construction
21. Trade
22. Transportation
23. Food and Accommodation
24. Information
25. Professional Services
26. Finance and Insurance
27. Real Estate
28. Education
29. Health and Social Work
30. Other Services
49
C
Online Appendix - Not for publication
C.1
The elasticity of aggregate consumption to revenue and adoption subsidies
In this Appendix we derive an expression for the elasticity of aggregate consumption with respect to revenue
and adoption subsidies where both entry and technology adoption decisions are considered.
We begin by expanding the expression in (16),
′
′
= βdiag (M ),
d ln Z
d ln e
d ln C = Ψ̃ − Ψ
where
d ln Z
d ln a
η−1 ′
d ln r + Ψ̃ −
Ψ
η
′
d ln Z
d ln Z
d ln e +
d ln a ,
d ln e
d ln a
= βe I − βdiag (M ) and βe ≡
1 ζ−(η−1)
.
η−1
ζ
(36)
Obtaining an expression for
the elasticity of aggregate consumption to revenue and adoption subsidies requires us to solve for d ln e and
d ln a.
We begin by computing an expression for d ln (P ◦ Y ). To this end, we first rewrite (28) in matrix form
and log-differente to get,
η−1 ′
I −ν
Ω diag (Ψ) d ln (P ◦ Y ) =Γ (d ln Pc + d ln C)
η
d ln Pm
′
+ Λ diag (∆)
+ I d ln a
d ln a
d ln Pm
′
+ Λ diag (∆)
d ln e
d ln e
d ln Pm
η−1 ′
′
+ Λ diag (∆)
+ν
Ω diag (Ψ) d ln r .
d ln r
η
Also, we follow the same steps as in Appendix A.2 to obtain an expression for d ln Pc + d ln C,
d ln Pc + d ln C = −∆
"
′
d ln Pm
+ I d ln a
d ln a
′ d ln Pm
− ∆
− ∆′
d ln e
d ln Pm
d ln r
where ∆e ≡ κt ◦e/(Pc C). Also, using (11) and (12),
and
d ln Pm
d ln r
+
1 − ν η−1
η
(1 − ν) η−1
η
+ Ψ d ln r ,
d ln Pm
d ln e
#
∆e
′
d ln e
= −Λ (I − νΩ)−1
d ln Z d ln Pm
d ln e , d ln a
= −Λ (I − νΩ)−1
= −Λ (I − νΩ)−1 .
Combining these expressions provide
d ln (P ◦ Y ) =∇P Y,a d ln a + ∇P Y,e d ln e + ∇P Y,r d ln r ,
50
(37)
d ln Z
d ln a ,
where
−1
′
d ln Pm
η−1 ′
I −ν
Ω diag (Ψ)
Λ diag (∆) − Λ∆′
+I ,
η
d ln a
#
−1 "
η−1
1
−
ν
η−1 ′
d
ln
P
η
m
= I −ν
Λ′ diag (∆) − Γ ∆′
∆e ′ ,
Ω diag (Ψ)
−Γ
η
d ln e
(1 − ν) η−1
η
−1
η−1 ′
η−1 ′
′
′ d ln Pm
′
= I −ν
Ω diag (Ψ)
Λ diag (∆) − Γ∆
+ν
Ω diag (Ψ) − ΓΨ .
η
d ln r
η
∇P Y,a =
∇P Y,e
∇P Y,r
Having obtained an expression for d ln (P ◦ Y ), we can now solve for d ln e and d ln a using the marginal
entrant and marginal adopters’ conditions,
η−1
d ln e = 0 ,
ζ
η−1
d ln r − (η − 1)d ln Z + d ln (P ◦ Y ) −
d ln a = −d ln r a + d ln Pm .
ζ
d ln r − (η − 1)d ln Z + d ln (P ◦ Y ) −
Substituting the expression for d ln (P ◦ Y ) produced in (37) and rearranging,
≡∇E,e
z
≡∇E,a
≡∇E,r
}|
}|
z
{
{
z
}|
{
η−1
d ln Z
d ln Z
I + (η − 1)
− ∇P Y,e d ln e + (η − 1)
− ∇P Y,a d ln a = (I + ∇P Y,r ) d ln r ,
ζ
d ln e
d ln a
≡∇A,a
≡∇A,e
z
}|
z
}|
A,r
{
{ z ≡∇
}|
{
d ln Z
η−1
d ln Z
(η − 1)
− ∇P Y,e d ln e +
I + (η − 1)
− ∇P Y,a = (I + ∇P Y,r ) d ln r + d ln r a .
d ln e
ζ
d ln a
Notice that these equations are linear in the vectors d ln e and d ln a, and so there is a unique solution to
the system. Stacking these equations in matrix form and solving for these two vectors provide,
d ln e
!
d ln a
=
∇E,e ∇E,a
!−1 "
∇A,e ∇A,a
∇E,r
∇A,r
!
d ln r +
0
I
!
#
d ln r a
.
Using these expressions in (C.1) imply that we fully specified the elasticity of aggregate consumption with
respect to revenue, r, and adoption subsidies, r a .
C.2
Model fit, by sector
51
Figure 11: Employment share distributions, data vs. model
1 - Mining
2 - Food, beverages & Tabacco
3 - Textiles, wearing & Leather
4 - Wood and Cork
5 - Paper & Paper Products
6 - Printing & Media
7 - Coke & Petroleum
8 - Chemicals & Pharma. Products
9 - Rubber & Plastic Products
10 - Other Non-metallic Mineral
11 - Basic Metals
12 - Metal Products, except M&E
52
13 - Computer, Electronic & Optical
14 - Electrical Equip.
15 - Machinery & Equipment
16- Motor Vehicles
17 - Other Transport Equip.
18 - Furniture & Other
19 - Utilities
20 - Construction
21 - Trade
22 - Transportation
23 - Accomodation & Food Serv.
24 - Information
53
25 - Professional Services
26 - Finance & Insurance
27 - Real Estate
28- Education
29 - Human Heatlh & Social Work
30 - Repair & Other Serv.
54
@FedFRASER