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Big Push in Distorted Economies
WP 21-07
Francisco Buera
Washington University in St. Louis
Hugo Hopenhayn
UCLA
Yongseok Shin
Washington University in St. Louis
Nicholas Trachter
Federal Reserve Bank of Richmond
Big Push in Distorted Economies∗
Francisco Buera
Hugo Hopenhayn
Washington University in St. Louis
UCLA
Yongseok Shin
Nicholas Trachter
Washington University in St. Louis
Federal Reserve Bank of Richmond
March 26, 2021
Abstract
Why don’t poor countries adopt more productive technologies? Is there
a role for policies that coordinate technology adoption?
To answer these
questions, we develop a quantitative model that features complementarity
in firms’ technology adoption decisions: The gains from adoption are larger
when more firms adopt.
When this complementarity is strong, multiple
equilibria and hence coordination failures are possible.
More importantly,
even without equilibrium multiplicity, the model elements responsible for the
complementarity can substantially amplify the effect of distortions and policies.
In what we call the Big Push region, the impact of idiosyncratic distortions
is over three times larger than in models without such complementarity. This
amplification enables our model to nearly fully account for the income gap
between India and the United States without coordination failures playing a
role.
∗
Buera: fjbuera@wustl.edu. Hopenhayn: hopen@econ.ucla.edu. Shin: yshin@wustl.edu.
Trachter: trachter@gmail.com. We thank Andy Atkeson, Ariel Burstein, Ezra Oberfield, Michael
Peters, and participants at several seminars and conferences for comments and suggestions. We
thank Eric LaRose, Reiko Laski, and James Lee for outstanding research assistance. 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
Many countries have industrialized and grown rapidly by adopting modern
technologies. Why don’t poor countries adopt more productive technologies? What
policies can effectively promote technology adoption? The standard view emphasizes
the role of distortions or barriers to technology adoption (e.g., Parente and Prescott,
1999; Hsieh and Klenow, 2014; Cole et al., 2016; Bento and Restuccia, 2017).
According to this view, eliminating the distortions is the obvious policy response. An
alternative view emphasizes the role of complementarity and coordination failures:
Firms in poor countries use unproductive technologies because other firms use
unproductive technologies, even though the gains from adoption increase with the
number of adopters so that they will all benefit from a coordinated decision to adopt.
According to this view, policies can have large aggregate effects by coordinating
firms’ decisions. This view has a long tradition in policy circles (e.g., RosensteinRodan, 1943; Hirschman, 1958) and is supported by more recent theoretical works
(e.g., Murphy et al., 1989; Matsuyama, 1995; Ciccone, 2002). However, there have
been few quantitative analyses, if any, of the coordination failure view of economic
development. Our paper bridges these two paradigms in a quantitative framework.
This paper makes two contributions. First, our theoretical analysis goes beyond
determining the existence of multiple equilibria to show that, even if they do not
exist, complementarity in technology adoption can amplify the effect of distortions
and policies. Second, our quantitative analysis based on aggregate and micro-level
data examines the empirical relevance of multiple equilibria and coordination failures
and, more importantly, shows the economic significance of the amplification channel
in the absence of multiple equilibria. In what we call the Big Push region, the impact
of idiosyncratic distortions is over three times larger than in models without such
complementarity. This amplification in our model helps account for the vast income
gap between poor and rich countries without coordination failures playing a role.
In our model, firms are ex-ante heterogeneous, produce differentiated goods, are
subject to idiosyncratic distortions, and are connected to one another through inputoutput linkages. Firms first choose whether or not to pay a fixed cost and enter the
market. Active firms can operate a traditional technology or, upon paying adoption
costs, a more productive modern technology.
We first theoretically analyze the conditions under which the complementarity in
2
firms’ technology adoption decisions can amplify the effect of policies and distortions
and even support multiple equilibria. When more firms adopt the productive
technology, the aggregate price falls, which has the direct effect of reducing the
price and the profit of the marginal adopter. However, the lower aggregate price
also means a larger demand for the marginal firm’s output, a lower cost of the
intermediate input for production, and less expensive goods portion of the adoption
costs, raising the marginal adopter’s gains. If the latter forces dominate the direct
competition effect, we have complementarity.
More heterogeneity in firm productivity implies that the density of the firms near
the adoption margin is smaller, implying a weaker feedback effect from the marginal
firm’s adoption to the aggregate economy and hence weaker complementarity.1 On
the other hand, idiosyncratic distortions correlated with firm productivity (i.e., more
productive firms being distorted more) compress the effective heterogeneity across
firms, which strengthens complementarity.
For simple versions of our model, we derive the conditions for amplification and
equilibrium multiplicity. What contributes to amplification are, consistent with
the discussions above: a small elasticity of substitution across differentiated goods,
adoption costs in goods rather than labor, a high intermediate input intensity of the
modern technology, small heterogeneity in firm productivity, and a close correlation
between idiosyncratic distortions and firm productivity. Multiplicity is an extreme
form of amplification, and indeed the conditions for the existence of multiple equilibria
are stronger than those for amplification, a result corroborated by our quantitative
model that exhibits potent amplification even without multiple equilibria.
Next, we use aggregate and micro-level data from the United States and India and
conduct a quantitative analysis. The United States is an undistorted benchmark, and
India is a large developing country for which relevant micro-level data is available.
The full version of our model has several layers, but it is tractable enough that most
of the parameters can be identified transparently from the data. In spite of the
potential presence of multiplicity, under the assumption that the data comes from
an equilibrium where adopters and nonadopters coexist, the values of the key model
parameters are obtained in closed form from the establishment size distribution.
The quantitative analysis has three main results. First, although our calibration
1
This may explain why multiplicity results in this literature invariably come from models with
homogeneous firms. In contrast, firm heterogeneity is essential for our quantitative strategy.
3
targets the moments from the establishment size distribution but not the income
level of either country, the model generates nearly as large an income gap between
the United States and India—a factor of 7—as in the data, due to the higher adoption
costs and degrees of idiosyncratic distortions in India, whose impact is amplified by
the complementarity. Second, the United States is in the unique equilibrium region,
but India is in the multiplicity region. However, India is found to be in the good (or
high adoption/output) equilibrium, and hence coordination failures do not explain
why India is poorer than the United States.
The final result is that the effect of idiosyncratic distortions on aggregate
productivity and GDP can be highly nonlinear. When the complementarity in firms’
adoption decisions is strong enough, multiple equilibria appear above a threshold
degree of idiosyncratic distortions, but eventually only the bad equilibrium survives
with even more distortions. This implies that, for a heavily distorted economy in a
unique bad equilibrium, a reform that reduces the distortions just enough to place
it in the multiplicity region gives it a chance to coordinate to the good equilibrium
featuring more adoption and vastly higher GDP. More important, even when the
model has a unique equilibrium for any degree of distortions, the model elements
responsible for the complementarity still amplify the impact of distortions. In
an empirically plausible range of distortions, a small reduction in distortions can
disproportionately increase adoption and aggregate productivity by several times
more than in the standard models without complementarity (e.g., Hsieh and Klenow,
2009; Hopenhayn, 2014; Restuccia and Rogerson, 2017). These positive nonlinear
effects with or without multiplicity are the Big Push in distorted economies, and we
call this range of distortions the Big Push region.
Our quantitative results offer two broader implications. First, the powerful
amplification of the impact of distortions through complementarity in our model can
help account for the huge income differences across countries. Second, the existence
of the Big Push region, where a small reduction in distortions can unleash massive
improvements, can be an explanation of why some distortion-reducing reforms are
more successful than others.
Related literature The idea that underdevelopment can result from coordination
failures goes back to Rosenstein-Rodan (1943). It has been formalized by Murphy
et al. (1989) in a model with aggregate demand spillovers and by Ciccone (2002) in
4
a model with differences in intermediate input intensities across technologies.2
Some empirical works have applied the idea of multiple equilibria and coordination
failures to historical contexts (Davis and Weinstein, 2002, 2008; Redding et al., 2011;
Kline and Moretti, 2014; Lane, 2019; Crouzet et al., 2020). The evidence so far is
mixed, suggesting that the possibility of multiple equilibria depends on the details of
the economic environment, a theme we emphasize.
Such advances in the theoretical and empirical literature have not been actively
followed by quantitative work with few exceptions. Valentinyi et al. (2000), although
a theoretical work, makes the important point that multiplicity is overstated in
representative agent models. Using a heterogeneous agent version of the two-sector
model of Matsuyama (1991), in which the economies of scale that are external
to individual producers cause multiplicity, they show that sufficient heterogeneity
restores a unique equilibrium. Graham and Temple (2006) study a representative
agent version of a similar two-sector model and find that a quarter of the world’s
economies are stuck in a low output equilibrium. Caucutt and Kumar (2008)
numerically explore a model in the theoretical literature.3
Relative to these papers, our contribution is to quantitatively analyze a richer,
more granular model of coordination failures, bringing together elements emphasized
in the theoretical literature and disciplining the analysis with micro-level data. More
important, we find that, even in the absence of multiplicity, these model elements
amplify the impact of idiosyncratic distortions and policies.
Our model builds on widely used models of heterogeneous firms, including those of
Hopenhayn (1992) and Melitz (2003). We extend the standard model to incorporate
discrete technology adoption choices.4 Our modeling choice is partly motivated by the
evidence in Holmes and Stevens (2014), who show wide variations in the size of plants,
2
Krugman (1992) and Matsuyama (1995) review the earlier papers on this topic and the more
recent theoretical contributions. Additional examples include Okuno-Fujiwara (1988), Rodrı́guezClare (1996), and Rodrik (1996), which analyze open-economy models of coordination failures.
3
Owens et al. (2018) study a quantitative urban model in which residential externalities cause
multiple equilibria at the neighborhood level. Another related literature explores the role of
coordination failures in accounting for the Great Recession (Kaplan and Menzio, 2016; Schaal
and Taschereau-Dumouchel, 2019) in the tradition of Cooper and John (1988), but this literature
abstracts from micro-level heterogeneity.
4
Yeaple (2005) and Bustos (2011) also consider firms’ technology choice decisions, but they either
ignore the possibility of multiplicity or make assumptions that happen to guarantee uniqueness. The
small- vs. large-scale sector choice in Buera et al. (2011) can also be thought of as a technology
choice, but that model also has a unique equilibrium.
5
even within narrowly-defined industries. In our model, small firms producing with
the traditional technology coexist with large firms operating the productive modern
technology, with the technology choice driven by and magnifying the underlying
heterogeneity in firm-level productivity.
Another important element of our model is the input-output linkages in the form of
roundabout production as in Jones (2011), which helps make firms’ adoption decisions
complementary in our model and amplifies the effect of distortions in general.
Finally, following Restuccia and Rogerson (2008) and Hsieh and Klenow (2009),
we introduce idiosyncratic distortions, which stand in for various types of frictions,
including barriers to technology adoption as modeled by Parente and Prescott (1999)
and Cole et al. (2016). The interaction between distortions and technology adoption in
our model is related to the impact of distortions on productivity-enhancing investment
in Bento and Restuccia (2017) and Bhattacharya et al. (2013). Our emphasis is
the amplification of the effect of distortions through the complementarity in firms’
adoption decisions, which results in highly nonlinear effects of distortions with or
without multiple equilibria. Relative to the distortion literature, our model is unique
in its ability to generate large income differences across countries with moderate
degrees of idiosyncratic distortions.
2
Setup
The economy is populated by a mass L = 1 of workers and measure one of potential
firms, each of them producing a differentiated good j.5 Workers supply their labor
inelastically and use their labor income to consume a final good. The differentiated
goods produced by firms are combined to produce an intermediate aggregate,
Z
X=
η−1
η
yj
dj
η
η−1
, η>1,
where η is the elasticity of substitution, governing how complementary are
differentiated goods in the production of the intermediate aggregate.
The
intermediate aggregate can be transformed with a linear technology to produce
the final consumption good and the intermediate input to be used by firms.
Firms are heterogeneous in their productivity z, drawn from a cumulative
5
Thus, j indexes a differentiated good or a particular firm.
6
distribution F (z), with density f (z) = ∂F (z)/∂z. Based on their productivity, firms
choose to be active or inactive. An active firm with idiosyncratic productivity z must
incur κe units of labor to enter the market and operate. An active firm produces
using technology i ∈ {t, m}, labor l, and intermediate aggregate x, according to
y=z
Ai
l1−νi xνi , νi ∈ [0, 1] ,
(1 − νi )1−νi
νiνi
where νi is the intermediate input elasticity. Technology m, the “modern”
technology, is more productive and intermediate input intensive than technology t,
ν
m
the “traditional” technology. Specifically, we assume that Am / νm
(1 − νm )1−νm >
At / νtνt (1 − νt )1−νt and νm ≥ νt .
The modern technology requires κa units of an adoption good. The adoption
good is produced by a competitive fringe in the adoption good sector by combining
labor and the intermediate aggregate using a Cobb-Douglas production function. We
denote by Pa the price of the adoption good.
Finally, firms are subject to idiosyncratic gross output distortions given by τ z −ξ ,
where ξ ∈ [0, 1] is the elasticity of distortions with respect to productivity, and τ is a
budget-balancing scale parameter. With ξ > 0, low productivity firms are subsidized,
and high productivity firms are taxed, which is the empirically relevant range in the
misallocation literature as we show in Section 5.
As is standard, the demand for differentiated good j and the price index of the
intermediate aggregate are
yj =
P
pj
η
Z
X and P =
pj
1−η
1
1−η
.
dj
We assume the labor market is perfectly competitive with equilibrium wage w.
Problem of Intermediate Input Producers An active firm with productivity
z producing with technology i earns operating profits πio (z). The firm chooses the
price p for its differentiated good and the amount of labor l and intermediate input
m required for production. The problem is
πio (z) = max τ z −ξ p
p,l,x
P
p
η
X − wl − P x
Ai
s.t. z νi
l1−νi xνi ≥ y =
νi (1 − νi )1−νi
7
(1)
P
p
η
X.
(2)
From the first order conditions of this problem, we obtain expressions for the
optimal price and input demands:
η w1−νi P νi 1
,
η − 1 Ai τ
z 1−ξ
(1−νi )η+νi
η−1 η
η P
li (z) =
(1 − νi ) τ
XAη−1
z η(1−ξ)−1 ,
i
η
w
νi w
li (z).
xi (z) =
1 − νi P
pi (z) =
(3)
(4)
(5)
From these expressions, we obtain an expression for the maximized operating profit
πio (z)
1
=
η
η−1 1
η w1−νi
η−1
τ η P η(1−νi )+νi XAη−1
z η(1−ξ)−1 ,
i
which is increasing in z provided η − 1/(1 − ξ) > 0.
It is useful to analyze the effect of distortions ξ on a firm’s decisions in two steps.
First, the firm acts as if having a lower productivity—lowered by a factor z −ξ , as
evidenced in equation (3). This effect also translates into an equivalent drop in
output (equation 2). Thus, this effect is captured by defining effective productivity
as z 1−ξ . Second, as seen in equations (4) and (5), distortions imply a lower elasticity
of factor demands and profits with respect to z than in the undistorted case, i.e.,
η − 1/(1 − ξ) ≤ η − 1. To see this, notice that we can rewrite the productivity term
in equation (4) as z (1−ξ)(η−1/(1−ξ)) ; the first term in parentheses corresponds to the
productivity reduction effect, while the second one corresponds to the lower demand
elasticity resulting from the wedge.
Finally, the profits of a firm with productivity z, given the optimal entry and
adoption decisions, are
π (z) =
max
inactive,active
o
{0, max {πto (z) , πm
(z) − Pa κa } − wκe } .
(6)
The entry and adoption decisions are characterized by thresholds ze and za , where
ze ≤ za . That is, a firm with productivity z will be active if and only if z ≥ ze ,
and will adopt the modern technology if and only if z ≥ za . This is because, under
the assumption that η − 1/(1 − ξ) > 0, the operating profit πio (z) is increasing in
productivity z for i ∈ {t, m}.
8
Problem of Adoption Good Producers A representative competitive firm
producing the adoption good takes the adoption good price Pa as given and solves
max
La ,Xa
γγ
1
γ
Pa L1−γ
a Xa − wLa − P Xa ,
(1 − γ)1−γ
(7)
where γ is the intermediate aggregate share in the production of the adoption good.
Equilibrium We consider symmetric equilibria where all firms of a given
productivity make the same decision.
Definition 1. A symmetric equilibrium consists of entry and adoption decisions
by differentiated goods producers, factor demands by producers of differentiated and
adoption goods, and relative factor prices w/P and Pa /P = (w/P )1−γ such that (i)
firms maximize profits and (ii) the markets for labor and intermediate aggregate clear
Z
za
Z
∞
lm (z) dF (z) + (1 − F (ze )) κe
lt (z) dF (z) +
ze
za
P γ
+ (1 − F (za )) (1 − γ)
κa = L ,
w
Z za
Z ∞
w 1−γ
C+
xt (z) dF (z) +
xm (z)dF (z) + (1 − F (za )) γ
κa = X .
P
ze
za
(8)
(9)
Equation (8) is the labor market clearing condition, and the four terms in the lefthand side are the labor used for traditional technology production, modern technology
production, entry costs, and adoption good production. Equation (9) states that
the intermediate aggregate used for consumption, traditional technology production,
modern technology production, and adoption good production must add up to the
quantity produced on the right-hand side.
Manipulating the equilibrium conditions, we can characterize the equilibrium by
three equations in three variables, ze , za , and P/w, which are shown in the appendix.
3
Understanding Amplification and Multiplicity
A firm’s incentives to adopt the modern technology can be affected by other firms’
adoption decisions. We show that the strength of this complementarity has important
implications for the amplification of the impact of distortions and policies and also
for the possibility of multiple equilibria and coordination failures.
9
Before examining the role that different parameters play in our model, we first
consider the complementarity at a more abstract level for the case of a policy that
subsidizes technology adoption.
Let D (z; a, s) denote the net gains from adoption for a firm with productivity z,
when the mass of adopters in the economy is a, and the adoption subsidy is s:
o
D (z; a, s) ≡ πm
(z; a, s) − πto (z; a, s) − (1 − s) P (a) κa ,
(10)
where πio (z; a, s) is the operating profit of a firm with productivity z using technology
i, and where we made explicit the dependence on the aggregate variables a and s.
The aggregate price index when the mass of adopters is a is P (a). We are assuming
that the adoption costs are in units of goods only (γ = 1 in our model).
A firm adopts the modern technology if D ≥ 0 and operates the traditional
technology otherwise. Consistent with our model, we assume that D is increasing in
z, so the adoption decision rule is characterized by a productivity threshold za : that
is, a firm adopts the modern technology if and only if z ≥ za , with the corresponding
mass of adopters being a = 1 − F (za ), where za satisfies
D (za ; a, s) = 0 .
(11)
We now consider a small change in subsidy, ds. We define the direct effect as the
impact on za holding fixed the value of a at the initial equilibrium level:
dza
ds
direct
=−
Ds (za ; a, s)
.
Dz (za ; a, s)
(12)
The total effect of this policy includes the equilibrium response of a and is
dza
ds
total
= −
Ds (za ; a, s)
.
Dz (za ; a, s) − Da (za ; a, s) f (za )
The ratio between the total and the direct effect is
[dza /ds]total
[dza /ds]
direct
=
1
1−
Da (za ;a,s)f (za )
Dz (za ;a,s)
,
(13)
which can be interpreted as a multiplier measuring the degree of the feedback
effect. When this ratio is greater than 1, the feedback is positive, an indication of
complementary in adoption decisions. The intensity of complementarities depends
10
on the feedback ratio:
f (za )
r (za , a, s) ≡ Da (za ; a, s)
.
|
{z
} Dz (za ; a, s)
|
{z
}
incentive
(14)
feedback
The feedback ratio can be interpreted as the product of two terms that determine the
adoption complementarities. The incentive term is the effect of a change in the mass
of adopters on the incentives (net gains) for adoption Da (za ; a, s). The feedback term
shows how the changes in the net gains from adoption feed back into a change in the
adoption threshold, f (za ) /Dz (za ; a, s). The expression 1/Dz (za ; a, s) measures how
the productivity of the marginal adopter relates to changes in the incentives to adopt,
and f (za ) is the density of firms at the adoption threshold. A positive feedback ratio
r(za , a, s) implies a multiplier exceeding one and hence amplification. In addition, the
closer the feedback ratio is to one, the higher the multiplier will be.
The feedback ratio also plays an important role in the analysis of equilibrium
determination and the possibility of multiplicity. An equilibrium for a given s consists
of a threshold za and a = 1 − F (za ) satisfying equation (11). In examining the
equilibrium, it is useful to consider the following mapping:
T (a) = a0 |a0 = 1 − F (za ) and D (za , a, s) = 0
.
This function is analogous to a best response map, giving the fraction of adopters
T (a) when firms behave optimally in response to the equilibrium prices that prevail
with an adoption rate a. An equilibrium is given by a fixed point of this mapping
a = T (a) . The slope of this mapping is
T 0 (a) =
Da (za ; a, s) f (za )
,
Dz (za ; a, s)
which is precisely the feedback ratio r(za , a, s). The equilibrium is unique if T 0 (a) < 1
for all a. The possibility of multiple equilibria arises if there is a range where T 0 (a) > 1
followed by a range where T 0 (a) < 1, as shown in the left panel of Figure 1. In the
right panel, we zoom in on the right-most equilibria of the left panel. We show the
effect of a change in the subsidy rate s, breaking it into the direct and the total
effects. Note that the condition for multiplicity is stronger than the condition for
amplification, r(za , a, s) > 0.
In the remainder of this section, we illustrate the role of various model elements
11
The equilibrium mapping T (a)
Direct and total effects
! "
! "
&'&()
!"#$%&
"
"
Figure 1: Equilibrium Mapping and Amplification: An Example
in generating amplification and multiplicity using simple examples.
3.1
Simple Model I: Heterogeneity
Consider the case with no intermediate inputs in the production of differentiated
goods, νi = 0, and an exogenous population of active firms, i.e., no entry margin.
Adoption costs are in units of goods only (γ = 1). We further assume that the
differentiated goods producers’ productivity z follows a Pareto distribution, z ∈
[1, ∞) ∼ 1 − z −ζ . There is no idiosyncratic distortion.
In this case, the net gains from adoption presented in equation (10) are
1
D(z; a, s) =
η
η−1
η
η−1
η−1
z η−1 − (1 − s)P (a)κa ,
P (a)η X(a) Aη−1
−
A
m
t
and the feedback ratio presented in equation (14) becomes
r (za , a) = [(1 − η)P a + Xa ]
ζ
,
η−1
(15)
where P a = −(∂P/∂a)(a/P ) and Xa = (∂X/∂a)(a/X) are the elasticities of the
price index and the aggregate demand with respect to the mass of adopters. We
dropped the dependence of the feedback ratio on s, as the subsidy rate does not show
up in this particular formulation.
As in the general case, the feedback ratio is the product of the incentive and the
feedback terms. The incentive term measures the impact of the mass of adopters on
the profitability of the marginal adopter. It is a function of the adoption elasticities of
12
the price index and the aggregate demand with respect to the adoption rate, P a and
Xa , and the demand elasticity, η. More adoption affects a firm’s profit through two
channels. First, it shifts the demand in proportion to the change in the intermediate
aggregate X. This is captured by the elasticity Xa . The second channel is through
the impact on the price index. This is a negative competitive effect, whose strength
is governed by the demand elasticity η. In addition, the decline in the price index has
a direct effect on the net gains from adoption by lowering the cost of adoption. The
net effect of the price index on the feedback ratio is captured by the term (1 − η)P a
in equation (15).
The feedback term in equation (15) captures the response of the mass of adopters
to the increase in the net gains from adoption for the marginal adopter and depends
on the demand elasticity η and the elasticity of the density of the productivity
distribution to the mass of adopters ζ. In the feedback term, η − 1 measures the
elasticity of the net gains from adoption with respect to productivity z, and its inverse
measures the response of the adoption threshold to the changes in the net gains from
adoption. The impact of changes in this threshold on the mass of adopters depends
on the density of firms at this point, captured by the parameter ζ.
To further understand the feedback ratio, we now analyze the adoption elasticities
of the price index P a and the aggregate demand Xa :
P a = Xa =
1
1
−
η−1 ζ
η−1 1−
a
Am
η−1
ζ
1−
h
Aη−1
t
Aη−1
m
Aη−1
t
!
M(a) ,
1− η−1
ζ
1−
Aη−1
m a
η−1
ζ
i
where M (a) =
+
∈ (0, 1) is the
/
1−a
modern sector’s share of total value added and employment. The demand elasticity
η has a negative effect on the adoption elasticities P a and Xa . A higher elasticity
of substitution η implies that the additional contribution of the marginal, less
productive, adopter will be smaller relative to the contribution of the infra marginal,
and more productive, adopters. On the other hand, a more elastic density of the
productivity distribution (higher ζ) implies that the productivity of the marginal
adopter declines less as the mass of adopters rises. This naturally results in larger
elasticities with respect to the mass of adopters.6
In making these arguments, we are holding fixed the values of Aη−1
/Aη−1
and M(a), which also
t
m
depend on η and ζ. One interpretation is that we are implicitly adjusting At /Am and the aggregate
adoption rate a by changing the adoption cost κa .
6
13
Using the expressions for the adoption elasticities, we can express the feedback
ratio in terms of the model parameters:
r (za , a) =
2−η
η−1
ζ +1−η
η−1
1−
Aη−1
t
Aη−1
m
!
M (a) .
(16)
For the effect of policies to be amplified, the multiplier in equation (13) must be
greater than one. This requires a positive feedback ratio. In this case, the feedback
ratio is positive if and only if
2−η >0 ,
(17)
since the last three terms in the right-hand side of equation (16) are strictly positive.7
When η is low, the negative competition effect is small, and thus the positive effect
on the gains from adoption of a larger mass of adopters dominates, generating a
positive feedback ratio and hence amplification.8 While the existence of amplification
depends solely on η, the degree of amplification does depend on ζ, the ratio Am /At ,
and M(a). Intuitively, a higher density of marginal firms (high ζ) and a larger
difference in technology Am /At give larger amplification.
Finally, we can obtain a necessary condition for multiplicity, i.e., the condition
T 0 (a) > 1. Using the fact that the last two terms in the right-hand side of equation
(16) are strictly less than 1, a necessary condition for multiplicity is
2−η >
1
.
1+ζ
(18)
This implies that the necessary condition for multiplicity is stricter than the condition
for amplification—that is, amplification is possible even in the absence of multiple
equilibria. Multiplicity requires that complementarity is strong enough to countervail
the negative effect of heterogeneity on the density of marginal adopters, as well as
that on the elasticities P a and Xa . This may explain why most multiplicity results
in the literature come from models with homogeneous producers (ζ → ∞).
7
For aggregates to be finite, ζ + 1 − η must be positive.
With limit pricing, as in Murphy et al. (1989), the negative competition effect is inoperative.
As a result, amplification and multiplicity do not require restrictions on η.
8
14
3.2
Simple Model II: Idiosyncratic Distortions
We now introduce idiosyncratic output distortions (controlled by ξ) into the simple
example with heterogeneity above and explore how they affect complementarity and
amplification. We retain all other simplifying assumptions: no entry margin, νi = 0,
and γ = 1. As we show, distortions have two effects. First, distortions compress
the distribution of the effective productivity of firms. In particular, it is as if the
tail parameter of the productivity distribution were ζ/(1 − ξ) instead of ζ. Second,
distortions create a wedge between the value of output produced by firms and their
profitability and hence their factor demand, reducing the elasticity of profits and
labor demand with respect to firms’ productivity.
With distortions, the feedback ratio is now
r (za , a) = [(1 − η)P a + Xa ]
ζ/(1 − ξ)
.
η − 1/(1 − ξ)
(19)
Comparing the feedback ratio with the one with no distortions in equation (15), we see
that the feedback term is adjusted by the distortion parameter ξ. This adjustment
can be understood as a more concentrated firm productivity distribution, i.e., one
with a tail parameter ζ/(1 − ξ) and a lower elasticity of the net gains from adoption
with respect to firms’ productivity. Both adjustments make the feedback term larger,
holding fixed the adoption elasticities P a and Xa .
We can also derive expressions for the two adoption elasticities:
P a =
1−ξ
1
−
η−1
ζ
1−
1
!
Aη−1
t
Mv (a) ,
1−ξ
1
η − 1−ξ
ζ
− 1−ξ
ζ (η − 1)
1−
Xa = ηεP a − (η − 1)
Aη−1
t
Ml (a)
εP a ,
Mv (a)
where Mv (a) and Ml (a) are the modern firms’ shares of total value added and
employment, respectively.9
The effect of distortions on the elasticity Pa is analogous to what we obtained
9
In particular, Ml (a) = Mv (a1+ξ/ζ ) and
v
M (a) =
Aη−1
a1−
t
(1−ξ)(η−1)
ζ
.
(1−ξ)(η−1)
(1−ξ)(η−1)
1−
ζ
ζ
Aη−1
1 − a1−
+ Aη−1
m a
t
15
in the model with no distortions, adjusted for the compression of the effective
productivity distribution with the tail parameter ζ/(1 − ξ). Accordingly, with more
distortions, and for a given level of Mv (a), the elasticity of the price index with
respect to the mass of adopters is higher.
The effect of distortions on the elasticity of the aggregate demand Xa involves
two effects. The first effect is the same as the one for P a , since the aggregate
demand is a function of the price index. This implies that the compression of the
productivity distribution increases both elasticities. The second is the effect that
distortions have on the aggregate demand, creating a wedge between firms’ input
demand and revenues. Here, holding fixed the modern firms’ shares and P a , an
increase in distortions reduces Xa . However, one can see that the net effect of ξ on
Xa is positive: The larger the degree of distortions, the more responsive the aggregate
demand is to an increase in the mass of adopters.
The relative effect of distortions on the two elasticities, P a and Xa , depends
crucially on the mass of adopters, a, through its effect on the modern firms’ share of
value added and employment, Mv (a) and Ml (a).
The modern firms’ share of employment is smaller than that of value added, i.e.,
l
M (a) ≤ Mv (a), with strict inequality when a < 1. This is because distortions
introduce a wedge between the value of the output produced by the modern firms
and their profitability and hence factor demands. This effect is clearest in the
limit with an arbitrarily small mass of modern firms, where the employment share
of the modern firms is arbitrarily small relative to their value added share, i.e.,
lima→0 Ml (a) /Mv (a) → 0, as the modern firms are infinitely productive and hence
face arbitrarily large distortions. Here, the elasticity of the price index is smaller than
that of the aggregate demand, P a < Xa . At the other extreme where the mass of
adopters is close to one, we have Ml (a) /Mv (a) ≈ 1, and the elasticity of the price
index is larger than that of the aggregate demand, P a > Xa .
Using the expressions for the adoption elasticities, we rewrite the feedback ratio:
r (za , a) =
l
l
ξ
M (a)
−
η v
−1
η−1
ζ +1−η
M (a)
!
ζ +1−η
Aη−1
t
×
1 − η−1 Mv (a) .
η(1 − ξ) − 1
Am
M (a)
1 − (η − 1) M
v (a)
(20)
Again, a multiplier exceeding one (i.e., amplification) requires r (za , a) > 0, which
16
is equivalent to
1 − (η − 1)
Ml (a)
ξ(η − 1)
>
Mv (a)
ζ +1−η
Ml (a)
η v
−1 .
M (a)
(21)
This condition is automatically satisfied when the mass of adopters is small, a ≈ 0,
because then Ml (a) /Mv (a) ≈ 0 and ζ +1−η is assumed to be positive for aggregates
to be finite. Thus, with idiosyncratic distortions, an economy with high adoption costs
(i.e., a small mass of adopters) will feature a multiplier greater than one for adoption
subsidies. To the contrary, distortions tend to make amplification less likely when the
mass of adopters is large, a ≈ 1, since condition (21) is stronger than condition (17)
when Ml (a) /Mv (a) ≈ 1.
How do distortions affect the necessary condition for multiple equilibria? As
before, because multiplicity requires T 0 (a) > 1, the condition for multiplicity is
stronger than that for amplification. From equation (20), holding fixed Mv (a),
more distortions relax the condition for multiplicity. In the limiting case of
lima→1 Ml (a)/Mv (a) = 1, the necessary condition for multiplicity is
2−η >
1
1+
ζ
1−ξ
,
which is weaker than condition (18) for ξ > 0. How ξ reduces the denominator of
the right-hand side of equation (19) is the dominant force. When a < 1, we have
Ml (a)/Mv (a) < 1, which further relaxes the necessary condition for multiplicity.
3.3
Discussion of the Role of Additional Elements
In this section, we briefly describe two additional elements of the model. We first
discuss the role of intermediate input in production. The use of intermediate input
strengthens the role of complementarities in adoption, making both amplification and
multiple equilibria more likely. We then describe the role of the entry margin and
discuss its relevance for the size of firms in the data.
Intermediate Input in Production. One important difference between our
benchmark quantitative model and the simple models explored in Sections 3.1 and
3.2 is the use of differentiated goods. While in the simple models differentiated
goods are used only as intermediate input for the production of the final good (to be
17
used for consumption and adoption costs), in the quantitative model an aggregated
quantity of the differentiated goods is also used as an intermediate input in the
production of the differentiated goods themselves (i.e., roundabout production).
The importance of intermediate input in production is governed by the intermediate
input intensity in the production of the adoption good, γ, and the intermediate input
intensity of the traditional and the modern technologies in the production of the
differentiated goods, νt and νm , respectively.
In the simple examples, we set the intensity of the intermediate aggregate in the
production of the adoption good to be equal to one, i.e., γ = 1. If γ = 0 in the
simple examples, the effect of policies would be dampened by a negative equilibrium
feedback, and neither amplification nor multiplicity would arise. A higher γ leads
to stronger complementarity: More adoption reduces the adoption costs by more
when a larger fraction of the costs is in units of goods rather than labor. The same
logic applies to the intensity of the intermediate aggregate in the production of the
differentiated goods themselves.
Consider first the case where the intermediate input intensity in the production
of the differentiated goods is the same for both the traditional and the modern
technologies: νt = νm = ν. The use of the intermediate aggregate as an input in
the production of the differentiated goods strengthens complementarities and the
equilibrium feedback for two reasons. First, more adoption now reduces the price
of firms’ intermediate input and hence their production costs, partly offsetting the
negative competition effect on firms’ profits, which in turn relaxes the restrictions on
the elasticity of substitution across differentiated goods η. For instance, when ν = 1
in the simple examples, the necessary condition for amplification and multiplicity
is always satisfied for any γ ∈ [0, 1], independently of η. Second, as in the simple
examples, the strength of the net effect of more adoption on the gains from adoption
depends on η. When the intermediate aggregate is itself an input in the production
of differentiated goods, the strength of the net effect also depends on ν, through a
standard intermediate input multiplier 1/ (1 − ν), as in Jones (2011).
When the modern technology uses intermediate input more intensively, that is,
νm > νt as in our quantitative model, there is an additional positive effect of more
adoption on firms’ incentive to adopt the modern technology. More adoption reduces
the price of the intermediate input relative to labor, raising the relative profitability
of the technology that uses the intermediate input more intensively. In this case,
18
amplification and multiplicity can arise even when the production of the adoption
goods requires only labor (γ = 0), which shares some similarity with the result in
Ciccone (2002).
In summary, the goods share of the adoption costs and the intermediate input
intensity in the differentiated goods production both contribute to amplification and
multiplicity, but neither is absolutely necessary for such outcomes.
Entry Margin. Relative to the simple models explored above, our quantitative
model also has the firm entry margin: A firm must pay a fixed labor cost κe > 0 to
become active. If firms’ productivity follows a Pareto distribution, the entry margin
is summarized by the productivity of the marginal entrant ze , which is given by
"
"
(η − 1) ζ
1+
ze = max 1, 1 +
ζ − (η − 1)
Am
At
η−1
!
−1 a
1− η−1
ζ
## 1
ζ
1
ζ
κe
.
(22)
The productivity of the marginal entrant ze and the average size of firms zeζ are an
increasing function of the fraction of adopters a, the cost of entry κe , and the ratio
of the productivity shifter of the modern to tradition technology, Am /At .
4
Identification of Model Parameters
Section 3 shows that several features of the model can generate multiple equilibria.
In general, parameter identification is not granted when a model features multiple
equilibria (see, for example, Jovanovic, 1989), because the mapping from the data to
the model parameters may not be unique. In this section, we construct an argument
that allows us to uniquely identify the parameters of the model despite potential
multiplicity. The key assumption for our identification strategy is that traditional
and modern firms coexist in the data. However, the strategy does not presuppose
multiplicity. Rather, once the parameters are uniquely identified from the data, we
check whether or not the model has any other equilibrium for those parameter values.
To keep things as simple as possible, we provide our constructive argument for
the case where the intermediate input elasticity is the same for the modern and the
traditional technology, i.e., νt = νm = ν, and the adoption good production only
uses intermediate input (γ = 1). In addition, following the common practice in
the literature, we maintain the assumption in Section 3 that firm-level productivity
19
distribution F is Pareto with a tail parameter ζ. We set the elasticity of substitution
across differentiated goods η from outside of the model using estimates common in
the literature on demand or profit share estimation. We also assume that there is
no distortion ξ = 0. However, our identification strategy also holds without these
parameter restrictions.
Our goal here is to identify the following six parameters: The technology
parameters At and Am , the entry and adoption costs κe and κa , the parameter of the
productivity distribution ζ, and the intermediate input elasticity ν. We normalize
the productivity of the modern technology to one, Am = 1. This leaves us with five
parameters. For identification, we rely on the size distribution of establishments
in the data G(l), where size is defined as the number of employees as well as the
intermediate input share in the data.
We begin by showing that, for a given η, we can identify the intermediate input
elasticity ν directly from the intermediate input share in the data:
ν=
η
× intermediate input share.
η−1
(23)
The other four parameters are identified from the establishment size distribution
in the data. In particular, we rely on the implications of the theory for the relationship
between the log of employment (log l) and the log of the fraction of establishments
with size larger than l as illustrated in Figure 2. We hereafter refer to this relationship
simply as the log-log relationship. Our identification strategy relies on the three
thresholds in the figure, which must exist under the assumption that both traditional
and modern technologies are used in the economy: (i) The size of the smallest entrant
lt ≡ lt (ze ), (ii) The size of the largest establishment using the traditional technology
¯lt ≡ lt (za ), and (iii) The size of the smallest establishment operating the modern
technology lm ≡ lm (za ).
From the slope in the right tail of the log-log relationship, i.e., l > lm , we identify
the tail parameter of the productivity distribution ζ for a given η:
ζ
d log (1 − G(l))
d log (1 − G(l))
=−
⇒ ζ = −η
.
η
d log l
d log l
(24)
Given the value of the intermediate input elasticity ν, the size of the smallest
20
log lt = log [(η − 1)(1 − ν)κe ]
log ¯lt
log lm
log l
−ζ log zzae
log fraction w/ emp. > l
(η − 1) log AAmt
− ηζ
Figure 2: Identification from the Establishment Size Distribution
establishment is a simple function of the entry cost, pinning down κe :
lt = [(η − 1)(1 − ν)κe ] ⇒ κe =
lt
.
(η − 1)(1 − ν)
(25)
The theory implies that there should be a gap in the size distribution of
establishments, if both the modern and the traditional technologies are operated
in the economy.10 In particular, there should be no establishment larger than the
largest establishment using the traditional technology ¯lt but smaller than the smallest
establishment operating the modern technology lm ; i.e., G(l) = G (lm ) = G ¯lt for
l ∈ ¯lt , lm . The difference between these two employment levels is a function of the
relative productivity of the two technologies, Am /At , which, given the knowledge of
η and the normalization of Am = 1, identifies At :
log lm − log ¯lt = (η − 1) log
Am
At
1
¯ η−1
lt
⇒ At =
.
lm
Finally, to identify the adoption cost κa , we use the equilibrium condition relating
the ratio of the two thresholds to the ratio of the adoption and the entry costs:
κe
10
za
ze
η−1
=
G ¯lt
P
w
Aη−1
m
Aη−1
t
−1
κa ⇒ κa =
− η−1 Aη−1
m
ζ
Aη−1
t
−1
P
w
This is akin to the concept of missing middle in Tybout (2000).
21
κe .
5
Calibration
We use data from the United States and India to calibrate the parameters of the
model. We think of the United States as an economy that is distortion-free (ξ = 0) and
therefore informative about the tail parameter of the firm productivity distribution
ζ, among others. India is a large developing economy with relatively good data
availability and is informative about the degree of idiosyncratic distortions ξ in
particular. This choice is consistent with the evidence on the relationship between
productivity and idiosyncratic distortions in the United States and India in Hsieh
and Klenow (2014, p.1059).
For our benchmark model without the simplifying assumptions of Section 3 or 4,
the following eleven parameters need to be calibrated: the elasticity of substitution
among differentiated goods η; the share of intermediate input in the production of the
adoption good γ; the intermediate input elasticity of the modern and the traditional
technology νm and νt ; the productivity levels of the modern and the traditional
technology Am and At ; the Pareto tail parameter of the firm productivity distribution
ζ; the entry and the adoption costs κe and κa ; and finally the degree of idiosyncratic
distortions ξ and the budget-balancing scale parameter τ .
Six of the eleven parameters are assumed to be the same for the United States and
India: η, γ, νm , νt , Am , and ζ. Four of these six are fixed outside of the model. We
maintain the normalization of Am = 1. We set η = 3, which is considered to be on the
lower side, as discussed in Hsieh and Klenow (2009).11 As in Section 4, we set γ = 1,
i.e., the adoption good production uses intermediate goods but not labor. In addition,
we set νt = 0, so that labor is the only input of the traditional technology. This last
assumption maximizes the difference between the two technologies’ intermediate input
elasticity, νm and νt . Our choices of η, γ, and νt make amplification and multiplicity
more probable as explained in Section 3, but the conclusions from our quantitative
analysis do not rest on these assumptions. In Section 6.3, we show the result with
γ = 0—i.e., labor is the only input of the adoption good production—and with
νt = νm . Appendix B reports a sensitivity analysis with different values of η and νt .
One of the remaining common parameters, νm , is then calibrated to match the
11
In our simple example of Section 3.1, the necessary condition for amplification and multiplicity
was η < 2. However, as explained in Section 3.3, with intermediate input in the production of the
differentiated goods, the restriction on η is relaxed.
22
intermediate input share in the United States data, yielding νm = 0.70.12 The other,
the tail parameter of the productivity distribution, is calibrated to match the tail
of the establishment size distribution for the United States in the Census Bureau’s
2007 Business Dynamics Statistics (BDS), giving ζ = 2.42 under the assumption of
no distortion (ξ = 0 and τ = 1).
We now describe how we calibrate the five parameters that differ between the
United States and India: two distortion parameters (ξ, τ ) and three technology
parameters (At , κe , κa ). Following the procedure in Section 4, we calibrate these
country-specific parameters from their establishment size distribution only. In
particular, we do not use any information on the level of income or productivity
in either country. Since there is a priori no tight relationship between a country’s
establishment size distribution and its income level, it is an open question how the
model-predicted income gap between the United States and India will measure up to
the data.
Since we assume that the United States has no distortions, the United States
calibration has ξ = 0 and τ = 1. For India, given the common tail parameter of
the productivity distribution ζ calibrated to the United States data, we calibrate
its degree of distortions ξ to match the tail of the establishment size distribution,
utilizing
∂ log(1 − G(l))
ζ
=−
,
η (1 − ξ) − 1
∂ log l
(26)
which is the distortion-augmented version of equation (24).
The calibration of the other three country-specific parameters, At , κe , and κa ,
closely follows the procedure in Section 4: They are chosen to match many features
of the empirical establishment size distribution (the log-log relationship) for each
country. The empirical moments are chosen to capture the nonlinearities of the loglog relationship in Figure 2. For the United States we use eight points from the
empirical log-log relationship and 26 points for India.
Figure 3 is the log-log plots of the establishment size distribution from the
calibrated model for the United States and India, together with the empirical log12
The intermediate input share in the United States in 2007 was 0.46, calculated from the BEA
input-output tables. Because we assume that the traditional firms use no intermediate input (νt = 0),
the intermediate input share of the modern firms has to be 0.47 in order for the share in the entire
η
as in (23), we obtain νm = 0.70.
economy to be 0.46. Multiplying 0.47 by η−1
23
United States
India
Note: The source of the U.S. data is the 2007 Business Dynamics Statistics. For the data
for India, we combine the 2003 National Sample Survey and the 2005 Economic Census.
See Buera et al. (2020) for details.
Figure 3: Establishment Size Distribution: Model and Data
log relationship. The United States data comes from the 2007 BDS. To produce
the figure for India, we use the size distribution of establishments for the entire
economy in India constructed by Buera et al. (2020), who combine data from the
Fifth Economic Census in 2005 and the 2003 Survey of Land and Livestock Holdings
carried out in the fifty-ninth round of the National Sample Survey (NSS). The Census
has comprehensive information for all entrepreneurial units, excluding agriculture.
The NSS provides information on employment by productive units in agriculture. In
order to obtain an accurate establishment size distribution for the entire economy
in India, it is crucial to account for its agricultural sector, which accounted for 57
percent of the total employment in 2004.13
As shown in the left panel, the model calibrated to the United States (solid
line) generates a flat region that is the size gap between firms using the traditional
technology and those using the modern technology, in order to match the concavity
of the log-log relationship in the data (circles) for small establishments. The
vertical location of the flat region shows that roughly half of all establishments use
the modern technology. The calibration for India in the right panel also shows
both the traditional and the modern technology in use in equilibrium, separated
by a flat region (solid line). However, a much larger fraction of firms uses the
13
By comparison, the agricultural employment share was only 1.4 percent in 2007 in the United
States.
24
Parameter
U.S.
Elasticity of substitution, η
Intermediate aggregate share in adoption good production, γ
Productivity distribution Pareto tail parameter, ζ
Modern technology productivity, Am
Modern technology intermediate input elasticity νm
Traditional technology intermediate input elasticity, νt
Entry cost, κe
Traditional technology, At
Adoption cost, κa
Degree of distortions, ξ
Distortion scale parameter, τ
India
3
1
2.42
1
0.70
0
0.50
0.43
16
0
1
0.50
0.07
272
0.19
2.14
Table I: Calibrated Parameters
traditional technology in India, as evidenced by the fact that less than 1 percent
(0.25 percent, to be precise) of establishments are to the right of the flat region in
the calibrated economy. The calibrated model captures the conspicuously flat region
over intermediate establishment sizes in the data for India (dots): a missing middle.14
In Table I we report the calibrated parameters. Some remarks are in order. First,
the United States and India have the same entry cost κe , which is identified from
the size of the smallest establishment using (25). Because the smallest establishment
is of the same size in both countries (one employee) and we assume that η and νt
are the same for both countries, so is κe . However, this does not mean that the
entry rate of firms is the same in the two countries. In fact, as shown in Table II,
the fraction of firms that enter in India is three times that in the United States.
Second, the traditional technology parameter At for the United States is six times
that for India. Since both countries have the same productivity level of the modern
technology Am by assumption, the technology gap between the modern and the
traditional technology is six times as high in India. Third, the cost of adoption
is seventeen times higher in India. The cost of adoption in India must be higher in
order to rationalize the minuscule fraction of firms adopting the modern technology
in spite of the enormous productivity gains from doing so.15 When measured in units
14
This does not contradict Hsieh and Olken (2014), who do not find a missing middle in the
manufacturing sector in India. Buera et al. (2020) find a missing middle when studying the broad
cross-section of establishments, and within narrow sectors, e.g., the agricultural sector.
15
The high adoption cost can be viewed as standing in for other inhibitors of technology adoption
that are not explicitly modeled in our theory, such as the shortage of skilled labor necessary for
using the modern technology, financial constraints, and bureaucratic or anti-competitive barriers to
25
Gross domestic product per capita
Consumption per capita
Average establishment size
Fraction of entrants
Fraction of entrants that adopt Am
Employment share of Am firms
Value added share of Am firms
U.S.
India
4.34
3.92
19.0
0.05
0.50
0.96
0.98
0.66
0.54
5.7
0.17
0.003
0.58
0.81
Table II: Statistics from the Calibrated Economy
of labor (P κa , since γ = 1), the cost of adoption in India is over eighty times higher,
because the (endogenous) price of the intermediate aggregate is five times higher in
India.
Finally, the calibration for India exhibits significant idiosyncratic distortions, as
given by ξ = 0.19. The tail of the establishment size distribution in India is thinner,
and hence the log-log relationship in the right tail is steeper. Equation (26) pins down
ξ.16 The implied firm-level revenue taxes and subsidies are not extreme. As we show
in Figure 12 in the appendix, the firm at the top 10−4 percentile of the active firm
productivity distribution is taxed at about 40 percent, while the maximum subsidy
is 80 percent for the least productive firm with z = ze .
At face value, some of the calibrated parameters in Table I seem contradictory to
the fraction of entrants reported in Table II: The United States exhibits less firm entry
than India in spite of the substantially lower cost of adopting the modern technology
and the significantly better traditional technology. However, as explained in Section
3, when more firms adopt the modern technology, they crowd out less productive
firms through the negative competition effect as well as through higher wages. That
is, the general equilibrium effects from more firms adopting the modern technology
are responsible for the lower entry rate in the United States. In spite of the lower
entry rate in the United States, the significantly higher rates of modern technology
adoption means that the United States GDP per capita is nearly seven times that of
India.
This last result is a success for the model. Even though the calibration is based
on the difference in the establishment size distribution between the United States and
adoption.
16
A back-of-the-envelope calculation of the ξ for India from Hsieh and Klenow (2014, p.1059) gives
0.4. One possible interpretation is that our model elements amplify the impact of distortions so that
a lower degree of idiosyncratic distortions can match the data for India.
26
India and does not use any information on the income or productivity gap between
the two, the model generates a huge income gap. In the Penn World Tables, the GDP
per worker of India is 6 percent of the United States level in 2005 while in the model
it is 15 percent. That is, the model accounts for 73 percent of the United States-India
income gap.17
6
Quantitative Exploration
In this section, we explore quantitatively the role of the various model elements—
mechanisms and parameter values—in amplifying the impact of distortions,
generating coordination failures, and ultimately accounting for the United StatesIndia income gap. We also discuss what a Big Push in distorted economies is.
6.1
Multiple Equilibria and the United States-India Income
Difference
The calibrated United States economy has a unique equilibrium, but the economy
in India features multiple equilibria. We show this by examining the net gains from
adopting the modern technology for the marginal adopter with productivity za
o
πm
(za ) − πto (za ) − P κa ,
which must be 0 in an equilibrium. Figure 4 shows this object for the United States
and India. As we vary za , the price of the intermediate aggregate and the wage adjust
to clear markets.
For the United States (solid line), the net gains are monotonically decreasing in the
fraction of adopters (and hence increasing in the productivity of the marginal adopter
za ) and intersect the zero line once. This intersection is the unique, stable equilibrium.
For India (dashed line), the net gains cross the zero line three times, twice from above
and once from below. The leftmost intersection is the stable, low adoption or “bad”
equilibrium, and the rightmost one is the stable, high adoption or “good” equilibrium.
The one in the middle is unstable. Our calibration selects the good equilibrium to
17
We use the output-side real GDP at chained PPPs and the number of persons engaged from the
Penn World Tables 9.0. The income gap in the data is a factor of 16.7 and in our model 6.7. Since
16.7 ≈ 6.7 × 2.5, the model explains 6.7/(6.7 + 2.5) × 100 ≈ 73 percent of the actual income gap.
27
Note: The figure shows, for the U.S. and India, the gains for the marginal adopter from
operating the modern technology, net of adoption costs.
Figure 4: Net Gain of the Marginal Adopter, United States and India
match the data for India. That is, despite equilibrium multiplicity, coordination
failures do not explain the observed income difference between the United States and
India. As we show in Section 6.2, if firms in India were to fail to coordinate and get
trapped in the bad equilibrium, India’s GDP would further shrink by a factor of 4.
That is, the United States-India income gap could have been a factor of 28 rather
than 7 in Table II.
If coordination failures do not account for the income difference between the
United States and India, then what does? The two countries in our model have
different productivity of the traditional technology At , adoption costs κa , and
distortions ξ, all identified only from their establishment size distribution.18 In Table
III, we calculate the contribution of each of these elements to the United States-India
gap in per capita consumption. To do so, we compute the hypothetical aggregate
consumption of the United States by replacing one of the parameters with its value
in the calibration for India, holding all others constant. This result is in the first
column, where we replace κa , ξ, and At one by one. In the second column, we do the
reverse: Starting from the calibration for India, we replace one of the parameters with
18
The entry cost is also country-specific, but the calibrated κe ’s for the United States and India
coincide. As ξ changes, τ adjusts to balance the budget.
28
U.S. w/ India Parameters
India w/ U.S. Parameters
Benchmark
1.0
0.14
Adoption cost, κa
Degree of distortions, ξ
Traditional technology, At
0.37
0.41
1.03
0.71
0.34
0.19
Table III: Explaining Consumption Difference
its value in the United States calibration. All per capita consumption is normalized
by the United States level in the benchmark calibration.
The first row shows that the model generates a factor of 7 difference between
the United States and India consumption (=1/0.14).19 Starting from the United
States calibration, we see that the adoption cost difference has the largest impact:
Giving the United States the high adoption cost of India shrinks the United States
consumption by a factor of 2.7 (=1/0.37). The role of idiosyncratic distortions is of a
similar magnitude: Introducing idiosyncratic distortions of the proportions of India
(ξ = 0.19) to the United States economy reduces the consumption by a factor of 2.5
(=1/0.41). The last row of the first column shows that, if we replace the traditional
technology productivity At of the United States with the lower value from India, the
United States consumption actually rises modestly. This is because the very low At
leads to more adoption of the modern technology.20
The same set of counterfactual exercises for India in the second column leads to
similar conclusions, although now adoption costs play a more important role. Giving
India the much lower adoption cost of the United States while holding all other
parameters constant results in a five-fold increase in consumption, which is much
larger than the factor of 2.7 in the first column: The rise in adoption caused by the
lower adoption costs represents a larger increase in productivity when the traditional
technology is less productive as in India. Eliminating idiosyncratic distortions in
India raises consumption by a factor of 2.4 (=0.34/0.14), which is nearly identical to
the result in the first column (2.5). Finally, replacing the traditional technology At
with the higher United States value has a modest positive effect on consumption in
India. The higher At nearly doubles the number of firms but further discourages the
adoption of the modern technology.
19
The gap in consumption is slightly larger than the gap in output, because entry costs and
adoption costs are a larger fraction of the output in India than in the United States.
20
In this case, the number of entrants is nearly halved, but all the entrants adopt the modern
technology.
29
Note: The figure shows, for the U.S. (cross) and India (dot), the combination of ξ and At
for which multiple equilibria exist. The larger cross and dot correspond to the calibrated
U.S. and India, respectively.
Figure 5: Region of Multiple Equilibria, (ξ, At ) Space
To summarize, the model nearly replicates the large income gap between the
United States and India in the data, without directly targeting the income or
productivity level of either country. Coordination failures turn out not to be part
of the story despite the existence of multiple equilibria, and adoption costs and
distortions explain most of the income gap.
6.2
Multiplicity, Amplification and Distortions
In this section, we further explore the role of adoption costs κa , relative technology
productivity Am /At , and idiosyncratic distortions ξ. We first identify the set of these
three parameter values that generates multiple equilibria, holding fixed the other
parameters as calibrated. Second, we show how per capita income and the average
size of firms change with the idiosyncratic distortion ξ. This exercise showcases the
potentially huge effect of distortions with or without equilibrium multiplicity.
Region of Multiplicity Figure 5 shows, for a low adoption cost economy (the
United States, cross) and a high adoption cost economy (India, dot), the combination
30
of the distortion parameter ξ and the traditional productivity At that generates
multiple equilibria. (Recall that Am for both is normalized to 1.) We hold all other
parameters fixed at their respective calibrated values, except that we adjust τ so that
the budget balances. The larger cross and dot represent the calibrated United States
and India economies, respectively. We see that India is in the region of multiplicity
but the United States is not.
Multiple equilibria arise for economies with both high degrees of distortions and
unproductive traditional technology, toward the lower right corner of the figure. One
interesting result is that, holding fixed the productivity of the traditional technology
At , as we increase idiosyncratic distortions ξ (moving horizontally), we enter and
then exit the region of multiplicity. To the right of the region, the only equilibrium is
the one with nearly no adoption. Similarly, holding fixed the degree of idiosyncratic
distortions, as we lower the productivity of the traditional technology At (moving
downward), we enter and then exit the region of multiplicity, although it is hard
to see this for the high adoption cost case (India, dot). The unique equilibrium
with At close to 0 has a small number of entrants, nearly all of whom adopt the
modern technology: With a useless traditional technology, entry also implies adopting
the modern technology, which effectively raises the cost of entry and results in few
entrants. Finally, the region of multiple equilibria is smaller for the high adoption
cost economy. In this case, the model features a unique equilibrium with few adopters
in most of the (ξ, At ) space. However, multiplicity can occur for smaller degrees of
idiosyncratic distortions (as low as ξ = 0.17) than in the low adoption cost economy.
Nonlinear Impact of Distortions We now explore the role of distortions in
generating large differences in income levels with or without multiple equilibria. We
start with the United States and the calibration for India, and vary the degree of
distortions ξ, holding fixed the other parameters at their respective calibrated values,
except that τ adjusts to balance the budget. In addition, since there is no multiplicity
for any ξ in the United States calibration, we also consider a modified United States
case that has the lower traditional technology productivity At of India.
In Figure 6 we show the equilibrium consumption per capita as we vary the
degree of distortions ξ for the United States (left panel) and India (right panel).
The consumption per capita in the vertical axis (log scale) is normalized by the per
capita consumption in the United States calibration.
31
United States
India
Note: Equilibrium consumption per capita as ξ goes from 0 to 0.5. Consumption is
normalized by its level in the U.S. calibration and in log scale. The dotted line in the left
panel is the no-multiplicity result of the United States. For the modified U.S. and the
India cases, the solid lines are the high adoption equilibrium (low adoption threshold za )
and the dashed lines are the low adoption equilibrium (high za ).
Figure 6: Distortions and Consumption per Capita
For the United States, the equilibrium is unique for any value of ξ (dotted line).
There are two notable features. First, the impact of distortions is large, reducing
consumption by nearly 90 percent for large values of ξ. Second, for intermediate
values of ξ, small changes in the degree of distortions have a highly nonlinear effect
on consumption. That is, even without multiplicity, distortions can have an amplified
impact. We discuss these features more rigorously in Section 6.3.
We now turn to the modified United States case (i.e., with India’s At ) in the left
panel and the India case in the right panel, both of which exhibit multiplicity. When
the distortions are small, the equilibrium is unique in both countries. The solid line
is the per capita consumption in the equilibrium with a higher fraction of adopters,
or the good equilibrium. As we increase distortions, a second equilibrium, one with
a small fraction of adopters emerges (dashed line). This is the bad equilibrium. For
India, both equilibria exist over a short interval of intermediate values of ξ, above
which only the bad equilibrium survives.
Tracing either the good (solid line) or the bad (dashed line) equilibrium,
idiosyncratic distortions have moderate to large effects on consumption. The effect is
even larger, however, since distortions can make the economy jump between the two
lines. Near the boundaries of the region of multiplicity, the effect of distortions are
extremely disproportionate. Once the economy enters the multiplicity region from
32
United States
India
Note: Average firm size (the number of employees, in log scale) as ξ goes from 0 to 0.5.
The dotted line in the left panel is the no-multiplicity result of the U.S. For the modified
U.S. and the India cases, the solid lines are the high adoption equilibrium (low adoption
threshold za ), and the dashed lines are the low adoption equilibrium (high za ).
Figure 7: Distortions and Average Firm Size
left, coordination failures can send the economy to the bad equilibrium. On the other
hand, even without better coordination, a small reduction in distortions can push
the economy from the bad equilibrium region to the unique good equilibrium region,
which discontinuously increases consumption. This happens around ξ = 0.3 in the
modified United States case and around ξ = 0.2 in the India case. Our calibrated
India economy is in the good equilibrium with ξ = 0.19 near the end of the solid line.
If ξ were to rise past the narrow interval of multiplicity, its per capita consumption
will shrink by a factor of 5, without coordination failures playing any role.
These results highlight the potentially disproportionate gains from reducing
idiosyncratic distortions. Multiplicity is an extreme form of amplification. However,
as the no-multiplicity United States case shows (dotted line), even without
multiplicity, our model elements that are responsible for complementarity in
adoption decisions amplify the impact of distortions to a magnitude unattainable in
conventional models, a statement we make precise in Sections 6.3 and 6.5.
Figure 7 illustrates the corresponding effect of distortions on the average size of
firms, measured by the number of employees. The vertical axis is in log scale. For
the modified United States and the India cases, the solid lines trace the average size
of firms in the good equilibrium. In this equilibrium, there are fewer entrants but
many of them adopt the modern technology, resulting in large firms. As distortions
get bigger, there is more entry but less adoption, bringing down the average firm
33
size, which is more pronounced in the high adoption cost economy (right panel). The
dashed lines, which appear when distortions are high enough, are the average firm
size in the bad equilibrium. In this equilibrium, the number of entrants is large, but
very few adopt the modern technology, which implies that firms are small on average,
in line with equation (22).21
6.3
Unpacking the Mechanisms
Our model combines several elements whose importance in explaining cross-country
income differences has been studied in isolation in the literature. In this section, we
illustrate the role of each element, in comparison with the findings in the literature. In
Section 6.3.1, we examine how the model elements amplify the impact of idiosyncratic
distortions. In Section 6.3.2, we analyze which elements are responsible for the large
income gap between the United States and India that our calibrated model generates.
6.3.1
Nonlinear Impact of Idiosyncratic Distortions
We start with the basic model in the distortion literature that abstracts from
technology choices and input-output linkages, as in Restuccia and Rogerson (2008)
for example. Next, we consider a case with a technology choice but without
intermediate input or roundabout production, which is our rendition of Bento and
Restuccia (2017). We then introduce input-output linkages in the form of roundabout
production but remove the technology choice. This is our adaptation of Jones (2011).
The fourth one has a technology choice and roundabout production as in our
benchmark model, but with the modification that the adoption costs are in units
of labor only instead of goods. Finally, we consider a case with technology choices,
roundabout production, adoption costs in goods, except that both the modern and
the traditional technologies have the same intermediate input intensity (νt = νm ),
unlike in our benchmark with νt νm .
For each model, we recalibrate the parameters to the same set of target moments
as in our United States benchmark (Section 5) and calculate the effect of idiosyncratic
21
The modified United States case uses India’s At , which is smaller than its value in the United
States calibration. As a result, the modified United States with no distortion (ξ = 0) has more
adopters and a larger average firm size than the benchmark calibration (dotted line). The direct
negative effect of the traditional technology productivity on the average firm size can be seen in
equation (22).
34
Consumption
Adoption
Note: The figure presents the equilibrium consumption per capita and fraction of adopters
in the U.S. as ξ goes from 0 to 0.5. Consumption is normalized by the consumption in the
recalibrated, no-distortion U.S. economy and is in log scale.
Figure 8: Unpacking the Mechanisms
distortions. Below we present the results for the United States calibration of each
model, which does not feature multiple equilibria for any degree of distortions ξ. The
left panel of Figure 8 shows the effect of distortions (ξ) on consumption per capita,
and the right panel shows the effect on the fraction of adopters.
The solid line reproduces the effect of distortions in our benchmark economy (the
dotted line in the left panel of Figure 6). It confirms that idiosyncratic distortions have
a large negative effect on aggregate consumption, which is particularly pronounced
around ξ = 0.2, close to its value in the calibration for India (ξ = 0.19). Consumption
is down by 60 percent, and the fraction of adopters collapses to nearly 0 at ξ = 0.2.
The first comparison model we consider is the basic model in the distortion
literature that abstracts from intermediate inputs and technology adoption (νt =
νm = 0 and At = Am = 0.69, with recalibration), shown by the dashed line. This
specification should be considered the polar opposite of our benchmark model. For
this model, consumption falls almost linearly with the distortion parameter ξ in the
semi-log scale and by much less than in the baseline model. At ξ = 0.2, consumption
goes down by less than 20 percent from its no-distortion level. Even with ξ = 0.5,
the loss in consumption is only 30 percent.
Next, the dotted line introduces technology adoption to the basic model but
without intermediate input. Consistent with the literature, for example, Bento
and Restuccia (2017), introducing the technology adoption by itself makes the
35
effect of distortions on consumption only marginally bigger and only at extreme
degrees of distortions (ξ near 0.5): The dotted line and the dashed line are nearly
indistinguishable in the left panel, and the reduction in consumption at ξ = 0.2 is
again less than 20 percent.
The dashed line with triangles instead adds roundabout production (intermediate
input) to the basic model, but with no technology choice. Roundabout production
more than doubles the effect of distortions on consumption, which decreases by almost
40 percent as ξ goes to 0.2. However, the effect here is nearly linear in the semi-log
scale with respect to ξ and are still significantly smaller than those in our benchmark.
The solid line with squares is a modified benchmark. It has the technology choice
and roundabout production, but the adoption costs are in units of labor only, i.e.,
γ = 0 instead of γ = 1. We see that the effect of distortions on consumption is smaller
than, yet comparable to, that in the benchmark, except for intermediate values of ξ
between 0.2 and 0.3. The same is true for the impact of distortions on the fraction
of adopters in the right panel. At ξ = 0.2, consumption is about 50 percent lower
than in the no-distortion case (compared to 60 percent in the benchmark), and about
20 percent of active firms adopt the modern technology (compared to nearly 0 in
the benchmark). The difference between the γ = 0 and the γ = 1 cases shows the
quantitative relevance of the feedback effect of adoption on the price of the adoption
goods, as discussed in Section 3.1.
Finally, the solid line with circles is the modified benchmark case where the two
technologies have the same intermediate input intensities, νt = νm = 0.69, instead
of νt νm . (We reinstate γ = 1.) The effect of distortions on consumption and
technology adoption is more measured than in the benchmark until ξ becomes large
enough (ξ > 0.35). This highlights another feedback mechanism operating in the
benchmark model: As more firms adopt, the lower is the price of the intermediate
goods relative to labor, and therefore the higher the profitability of the modern
technology that uses the intermediate input more intensively. Because this feedback
mechanism is absent in this modified model, the negative effect of distortions on
adoption and consumption is smaller than in the benchmark. At ξ = 0.2, consumption
is 40 percent lower than in the no-distortion case, and nearly a quarter of active
firms adopt the modern technology. On the other hand, when distortions are large
enough that the fraction of adopters approaches 0 (ξ > 0.35), the negative effect on
consumption is considerably larger than in the benchmark that has νt = 0. This
36
At
κa
Pa κa
GDP p.c.
Ratio
Case 1: ν = 0, no adoption
U.S.
India
1
1
0
0
0
0
1.89
1.62
1
0.86
Case 2: νt = νm = 0, γ = 0
U.S.
India
0.54
0.28
2.17
321
2.17
321
1.64
0.83
1
0.51
Case 3: ν = 0.69, no adoption
U.S.
India
1
1
0
0
0
0
5.33
3.37
1
0.63
Case 4: Benchmark
U.S.
India
0.43
0.07
15.9
271
9.35
810
4.34
0.66
1
0.15
Case 5: γ = 0
U.S.
India
0.43
0.06
8.74
1123
8.74
1123
3.33
0.34
1
0.11
Case 6: νt = νm = 0.69
U.S.
India
0.54
0.29
12.1
272
7.00
1034
4.15
0.49
1
0.12
Note: The price of the adoption good is Pa = P γ w1−γ . With γ = 0, the price of the
adoption good is Pa = w = 1 while with γ = 1 it is Pa = P .
Table IV: Alternative Model Results
is because the dearth of adopters makes the intermediate input expensive, but the
traditional technology in this modified model is still dependent on the intermediate
input (νt = 0.69), reducing its effective productivity.
Overall, the analysis in this section emphasizes the interaction among our model
elements that is more than simply additive. Technology adoption, roundabout
production, and the nature of the adoption cost (i.e., labor or goods) jointly
explain the large, nonlinear effect of idiosyncratic distortions even in the absence of
multiplicity or coordination failures.
6.3.2
Income Gap between the United States and India
Our benchmark model calibration generated a large income gap between the United
States and India, a factor of nearly 7, accounted for by the high adoption cost κa and
the high degree of idiosyncratic distortions ξ in India (Table III). Since other models
in the literature rarely generate such a large income gap, it is natural to ask which
model elements of ours are responsible for this result.22
In Section 6.3.1, we calibrated various comparison models to the United States
data with ξ = 0 and calculated the impact of higher degrees of distortions ξ. In this
22
Francesco Caselli suggested the exercise in this section, for which we are grateful.
37
section, we calibrate each of the comparison models separately to the United States
and the India data, following the procedure in Section 5. We report the resulting
GDP per capita of the two economies in Table IV, which also shows the calibrated
productivity of the traditional technology At , the adoption cost parameter κa , and
the adoption cost in units of labor Pa κa . One thing to note is that the degree of
distortions ξ is assumed to be 0 for the United States and is identified from the right
tail of the establishment size distribution for India. Since the model elements that
vary across the comparison models do not affect the right tail of the size distribution,
across all these calibrations the ξ for India remains at 0.19.
We start with the basic model in the distortion literature that has no intermediate
input nor technology adoption (ν = 0, At = Am and κa = 0; case 1 in Table IV).
For this model, the only parameters we can use to match the size distribution in
either economy is the entry cost κe , the Pareto tail parameter ζ, and the distortion
parameter ξ. The United States right tail pins down ζ and the India right tail ξ, given
the assumptions of a common ζ and ξ = 0 for the United States. Unsurprisingly, this
simple model has a hard time matching the size distribution of either economy. The
result is that the GDP per capita of India is only 14 percent lower than that of the
United States, a magnitude comparable to the corresponding model result in Figure
8 (dashed line, ξ = 0.19).
The second comparison model has a technology choice with labor-only adoption
costs but still has no intermediate input (case 2 in Table IV). To match the
establishment size distribution, the adoption costs become vastly different between
the two economies. However, the gap in the traditional technology productivity At
becomes smaller than in the benchmark model (case 4 in the table). The resulting
GDP per capita gap is now larger: India’s is half that of the United States. This gap
is larger than what we see in Figure 8 (dotted line) but much smaller than what our
benchmark model generates.
The third comparison model has intermediate input but no technology adoption
(case 3 in the table). Like the first comparison model, without the technology choice
and the related parameters, this model cannot closely match the establishment size
distributions either. However, the linkage in the form of the roundabout production
generates a meaningful GDP per capita gap between the two economies: India’s is
nearly 40 percent smaller than that of the United States, which is nearly identical to
the effect of ξ shown in Figure 8 (dashed line with triangles).
38
When the technology choice and the linkages through intermediate input uses are
both incorporated, as in our benchmark, the model generates a much larger GDP per
capita gap between the two countries. The exact size of the gap can vary depending
on the nature of the adoption costs (goods vs. labor, case 4 vs. 5 in the table) and
the intermediate input intensity of the two technologies (case 6), but what does not
change is the insight that these model elements interact and generate a cross-country
GDP per capita gap larger than the sum of their respective individual effects.
6.4
Big Push
“Big Push” is the name Rosenstein-Rodan (1961) gave to the idea that a minimum
scale of investment is necessary for economic development. The rationales are
indivisibilities in the production function, especially social overhead capital and
complementarities across sectors. Under these conditions, individual firms may
not find it profitable to industrialize alone, even though all firms are better off
industrializing together.
Murphy et al. (1989) note that government investments in infrastructure do
not automatically solve the coordination problem. In fact, if unaccompanied by
firms’ coordinated decision to industrialize and utilize the infrastructure, the modern
infrastructure becomes a classic “white elephant.” In this regard, the role of the
government is to promote a coordinated, collective decision.
If all we need is the coordination of firms’ decisions so that they all become
better off, why do so many countries still remain unindustrialized and poor? Our
framework helps address this question in two ways. First, the heterogeneity across
firms implies that not all firms are better off in the good equilibrium. As discussed in
our explanation of Figure 7, many firms that would be active (and make profits) in the
bad equilibrium are inactive (and make no profit) in the good equilibrium. Although
we have not specified preferences or welfare criteria, it is easy to see that the presence
of losers as well as winners can validate the explanations that vested interests block
the adoption of better technologies (Olson, 1982; Parente and Prescott, 1999).
Second, our framework introduces another dimension to the notion of Big Push,
beyond the coordination over multiple equilibria. Reforms that reduce idiosyncratic
distortions are necessary elements of successful development policies. As Section 6.2
showed, the effects of reducing distortions get amplified in our framework with or
39
United States
India
Note: The elasticity of aggregate consumption to adoption subsidy for the U.S. and India
calibration, as the distortion parameter ξ goes from 0 to 0.5. For the case for India, the
solid line is the high adoption (low adoption threshold za ) equilibrium, and the dashed
line is the low adoption (high za ) equilibrium. The dotted lines are for the simple model
without intermediate input in production and with labor only adoption costs.
Figure 9: Elasticity of Aggregate Consumption to Adoption Subsidy
without multiplicity, which we call Big Push in distorted economies. In this view,
the role of the government is to reduce distortions, identifying and exploiting the Big
Push region where the returns to economic reforms are discontinuously high.
6.5
Big Push and Industrial Policy
The view of economists and policymakers on industrial policy has evolved over time.
In the early years, government planning, public investment, and protectionist trade
policies were the dominant development strategy, but the results were more often than
not disappointing (Krueger, 1997). The mounting evidence of “government failures”
ushered in the wave of market-fundamentalist liberalization of the 1990s, with equally,
if not more, disappointing results (World Bank, 2005). Renewed thinking on industrial
policy emphasizes governments’ coordination of innovation and technology adoption
(e.g., Rodrik, 2004), the very elements central to our framework.
Here we calculate the effect of such a policy: subsidies for technology adoption.
In addition, we solve the problem of a constrained planner who chooses the optimal
technology adoption taking as given the distortions and the set of active firms.
The industrial policy we implement subsidizes a fraction s of the cost of adopting
the modern technology, financed by a lump-sum tax on consumers.
40
United States
India
Note: The aggregate consumption in the constrained planner allocation for the U.S. and
India (dashed line). Consumption is normalized by the equilibrium consumption in the
undistorted U.S. economy and is in log scale.
Figure 10: Constrained Planner Allocation
We show the elasticity of aggregate consumption to the subsidy for both the
United States and the India calibrations, as we vary the degree of distortions ξ.
For comparison, we also compute the elasticity from a simpler model without the
roundabout production (νm = νt = 0) and with labor only adoption costs (γ = 0),
which corresponds to the dotted lines in Figure 8 or case 2 in Table IV. Figure 9
presents two noteworthy results. First, in our benchmark model with the feedback
effects, the elasticity of aggregate consumption to the subsidy is high in the Big Push
region, especially in the multiplicity region for India, but relatively low outside of it.
This can potentially explain why some industrial policies are more successful than
others: It is a matter of whether the economy is in or outside the Big Push region.
Second, the elasticity is uniformly low in the version with no feedback effect and there
is no Big Push region, implying that complementarity and amplification are necessary
conditions for successful industrial policy.
We now calculate the optimal adoption subsidy s, by solving the problem of
a constrained planner, who chooses the adoption threshold za taking as given the
distortions and the set of active firms.23 The aggregate consumption from this
constrained planner allocation is shown with dashed lines in Figure 10, while the
solid lines reproduce the equilibrium outcomes from Figure 6. Consistent with the
23
We also worked out the case in which the planner chooses both the entry and the adoption
thresholds. The entry margin is found to play a minor role, except at high degrees of distortions.
41
elasticity results above, the largest gains from influencing firms’ adoption decisions
are in the Big Push region, where the dashed and the solid lines are farthest apart.
Furthermore, the India case attests to the power of coordination achieved by
the policy: Although the adoption subsidy has small effects on consumption for ξ
greater than 0.2 in Figure 9, the planner can generate massive consumption gains
by coordinating the economy away from the low adoption equilibrium to the high
adoption equilibrium, as long as the degree of distortions is not too high.
Nevertheless, the figure shows that the gains from reducing distortions tend to be
larger than the gains from optimal adoption subsidies, and that adoption subsidies are
not very effective at the low and the high end of the degree of distortions. In other
words, not only do the idiosyncratic distortions determine the effectiveness of the
industrial policy subsidizing technology adoption, but are also an important source
of underdevelopment themselves.
7
Concluding Remarks
This paper provides answers to the following three questions: (i) Can economic
development be explained solely by coordination failures? (ii) Can economic
development be a story of coordination failures and distortions? and (iii) Are there
large nonlinear effects of distortions and policies even without multiplicity, which
can explain the huge income differences across countries? We find that the United
States calibration gives a unique equilibrium and that the calibrated India economy
is in the multiplicity region but is in the good equilibrium, which says no to the
first question. We find that small changes in idiosyncratic distortions can move the
economy in and out of the region of multiplicity, resulting in discontinuously large
changes in the aggregate output. More important, even without multiplicity, the
feedback channels in our model creates Big Push regions, where small changes in
distortions and policies have disproportionate effects. The answer to the second and
the third questions is in the affirmative.
A promising avenue for future research is the exploration of an asymmetric inputoutput structure of production—for example, a multisector extension, in which sectors
differ in adoption costs and forward/backward linkages. We conjecture that this
extension will feature clusters of amplification and multiplicity. Another is a dynamic
extension of the model where only a subset of firms make entry and adoption decisions
42
each period. In this extension, coordination failures may show up as multiple steady
states and history dependence. Whether policies that subsidize adoption or reforms
that reduce idiosyncratic distortions can move the economy from bad to good steady
states is an open question.
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Online Appendix
A
Equilibrium Conditions
In Section 2, we note that the equilibrium conditions can be represented by three
equations in three unknowns. They are
κe =
Aη−1
m
Aη−1
t
w (νm −νt )(η−1)
P
−1
κa
ze
za
η(1−ξ)−1
,
η−1
Z
w νt (η−1) η−1 za (η−1)(1−ξ)
z
dF (z)
At
P
ze
Z ∞
w νm (η−1)
η−1
(η−1)(1−ξ)
+
Am
z
dF (z) ,
P
za
γ
P 1−γ
L
−
(1
−
F
(z
))
κ
−
(1
−
F
(z
))
κa P νt (1−η)
e
e
a
w γ
1
zeη(1−ξ)−1 ,
Aη−1
κe =
t
η(1−ξ)−1
η−1
w
Z
w η−1
P
( Pw )γ
=
(η − 1) τ
η
where
Z
B
η(1−ξ)−1
Z
P νt (1−η) za η(1−ξ)−1
z
dF (z)
≡ (1 −
w
ze
νm (1−η) Z ∞
η−1 P
+ (1 − νm ) Am
z η(1−ξ)−1 dF (z) .
w
za
νt ) Aη−1
t
Sensitivity Analysis
Because the literature lacks precise estimates of the elasticity of substitution across
intermediate goods η and the intermediate input elasticity for the traditional
technology νt , we fixed η = 3 and νt = 0 outside of the model. While η = 3 falls
within the standard range in the literature, there is no available estimate of νt ,
beyond the fact that it is smaller than the modern technology elasticity νm (Chenery
et al., 1986; Blaum et al., 2018; Kim et al., 2020). In this section, we explore the role
of η and νt . As we vary η or νt , we recalibrate the model and re-do the exercises that
produced Figure 6. We consider two values of η, 2.5 and 4, one on either side of the
benchmark value η = 3. We also consider two values for νt : one where νt = 0.35 is
larger than the benchmark value of zero but smaller than νm , and the other where νt
and νm are both 0.69, which is the highest value possible for νt given the restriction
47
νt ≤ νm and the overall intermediate input share. Figure 11 shows consumption per
capita as we vary the idiosyncratic distortion parameter ξ. The two top panels are
the cases with different η’s for the United States and India. The two bottom panels
are for alternative νt ’s. In all cases, consumption is normalized by the United States
consumption in the equilibrium with no distortions.
United States, alternative η
India, alternative η
United States, alternative νt
India, alternative νt
Note: Equilibrium consumption per capita of the United States and India as the
distortion parameter ξ goes from 0 to 0.5. Consumption per capita is normalized
by the U.S. consumption (no distortion) and in log scale.
Figure 11: Sensitivity of Consumption and Equilibrium Multiplicity to η and νt
The top panels of Figure 11 show that a smaller elasticity of substitution η
increases the consumption difference between good and bad equilibria and widens the
set of the distortion parameter ξ that generates multiple equilibria. This is consistent
with the analysis in Section 3.1: A lower η makes goods less substitutable and firms’
48
adoption decisions more complementary. In addition, the income gap between the
United States and India is larger with a smaller η. We draw the conclusion that a
small elasticity of substitution across goods are conducive to explaining the income
differences across countries, with or without multiplicity.
The bottom panels show that a lower intermediate input elasticity of the
traditional technology νt has two effects. On the one hand, a lower νt enlarges the
set of ξ’s generating multiple equilibria. This is consistent with the discussion in
Section 3.3. On the other hand, a low νt compresses the consumption gap between
good and bad equilibria. Intuitively, when νt is small, it is less costly to use the
traditional technology in a world where few firms adopt the modern technology and
the intermediate aggregate is expensive. The two effects run in opposite directions
when it comes to explaining cross-country income differences.
Another robust result in the plots for India is that the model features a very
nonlinear effect of distortions, either through or independently of equilibrium
multiplicity.
C
Idiosyncratic Distortion
We show the magnitude of idiosyncratic distortions in the calibration fro India with
ξ = 0.19. We model distortions as effective taxes and subsidies on firms’ revenue,
and the tax rate is 1 − τ z −ξ . The solid line is the cumulative distribution of tax
rates across firms in the calibration for India, while the dashed line is the same object
but for the re-computed United States economy using India’s ξ. For a given tax
rate on the vertical axis, we show the fraction of firms with higher tax rates on the
horizontal axis in log scale. In both cases, a small fraction of firms pays taxes and
subsidizes the rest of the economy. In India, the most productive firms (those at
the top 10−5 percentile of the active firm productivity distribution) face tax rates of
about 45 percent, and nearly half the firms receive subsidies of at least 60 percent of
their revenue. For the United States case with India’s ξ, the whole tax schedule shifts
up for two reasons. First, the active firms in the United States are more productive
because the entry threshold ze is higher than in India. As a result, holding constant
the τ̄ of India, the United States firms face a higher tax schedule. Second, in the
United States equilibrium with India’s distortions, τ̄ is higher. The top United States
firms face higher taxes than top firms in India—a tax rate of almost 60 percent—and
49
Note: The figure shows the cumulative distribution of taxes across firms. The
solid line is for the calibration for India. The dashed line corresponds to the
recomputed U.S. economy using India’s ξ.
Figure 12: Idiosyncratic Distortions across Firms
half the firms receive subsidies of at least 30 percent of their revenue. Overall, we
conclude that the magnitude of taxes to large firms and subsidies to small firms is
not implausible, relative to the numbers in the misallocation literature.
50
@FedFRASER