DISCUSSION PAPER SERIESDP15530

A TECHNOLOGY-GAP MODEL OF
PREMATURE DEINDUSTRIALIZATION
Ippei Fujiwara and Kiminori Matsuyama
INTERNATIONAL TRADE AND REGIONAL ECONOMICS
MACROECONOMICS AND GROWTH

ISSN 0265-8003

A TECHNOLOGY-GAP MODEL OF PREMATURE
DEINDUSTRIALIZATION
Ippei Fujiwara and Kiminori Matsuyama
Discussion Paper DP15530
Published 07 December 2020
Submitted 05 December 2020
Centre for Economic Policy Research
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Copyright: Ippei Fujiwara and Kiminori Matsuyama

A TECHNOLOGY-GAP MODEL OF PREMATURE
DEINDUSTRIALIZATION
Abstract
This paper presents a simple model of what Rodrik (2016) called “premature deindustrialization,”
the tendency that late industrializers reach their peaks of industrialization at lower levels of percapita income with the lower peak shares of manufacturing, compared to early industrializers. In
this model, the hump-shaped path of the manufacturing share in each country is driven by thefrontier technology whose productivity growth rate differs across the sectors. The countries areheterogenous in their “technology gaps,” their capacity to adopt the frontier technology, whichmight affect adoption lags across sectors differently. In this setup, we show that prematuredeindustrialization occurs, for example, if adoption takes longer in the service sector, and yet the
productivity growth rate in the service sector is sufficiently smaller such that cross-country
productivity differences are smaller in the service sector.
JEL Classification: O11, O14, O33
Keywords: structural change, premature deindustrialization, sectoral productivity growth rate
differences, adoption lags, technology gap
Ippei Fujiwara - ippei.fujiwara@keio.jp
Keio University and Australian National University
Kiminori Matsuyama - k-matsuyama@northwestern.edu
Northwestern University and CEPR

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A Technology-Gap Model of Premature Deindustrialization*

Ippei Fujiwara
Faculty of Economics, Keio University
ippei.fujiwara@keio.jp;
Crawford School of Public Policy, ANU
ippei.fujiwara@anu.edu.au

Kiminori Matsuyama
Department of Economics
Northwestern University
k-matsuyama@northwestern.edu

December 4, 2020
Abstract
This paper presents a simple model of what Rodrik (2016) called“premature deindustrialization,” the tendency that late industrializers reach
their peaks of industrialization at lower levels of per capita income withthe lower peak shares of manufacturing, compared to early industrializers.
In this model, the hump-shaped path of the manufacturing share in eachcountry is driven by the frontier technology whose productivity growthrate differs across the sectors. The countries are heterogenous in their“technology gaps,” their capacity to adopt the frontier technology, whichmight affect adoption lags across sectors differently. In this setup, weshow that premature deindustrialization occurs, for example, if adoption
takes longer in the service sector, and yet the productivity growth rate in
the service sector is sufficiently smaller such that cross-country
productivity differences are smaller in the service sector.
Keywords: Structural change, premature deindustrialization, sectoral
productivity growth rate differences, adoption lags, technology gap
JEL classifications: O11, O14, O33__________________*We thank Shinnosuke Kikuchi, Martí Mestieri, and Philip Ushchev for their comments on an earlier draft.
This project started in December 2018 during Matsuyama’s visit to Keio University, whose hospitality hegratefully acknowledges.

Page 1 of 13

1.

Introduction
In most countries, the share of manufacturing (measured in employment or value-

added) followed an inverted 𝑈𝑈-shaped or hump-shaped path over the course ofdevelopment, as well-documented by Herrendorf, Rogerson, and Valentinyi (2014).

Recently, Rodrik (2016) presented the finding that recent industrializers entered the stageof deindustrialization at lower income levels with lower peak manufacturing shares,
compared to more advanced economies that had industrialized earlier. See, in particular,
Rodrik (2016, Figure 5). Rodrik called this finding “premature deindustrialization.”In this paper, we present a simple model of premature deindustrialization. The
model has three sectors: agriculture, manufacturing, and services, which produce theconsumption goods that are gross complements. As the frontier technology improves
over time, productivity grows at an exogenously constant rate in each sector, which is thehighest in agriculture, the lowest in services, with the manufacturing in the middle. This
generates a hump-shaped path of the manufacturing in each country, as in Ngai and
Pissarides (2007). The countries differ in their ability to adopt the frontier technology,
which we call “technology gap,” following Krugman (1985). Unlike Krugman, however,
we allow for the possibility that the extent to which the country’s technology gap affects
its adoption lags varies across the sectors. Within this setup, we study under what
conditions the model generates premature deindustrialization.
For example, suppose that the technology gap of a country affects its adoption
lags in all sectors uniformly, so that adoption lags are the same in all sectors in eachcountry. Then, poorer countries, which suffer from larger technology gaps, reach theirpeaks later than richer countries, but they reach the same peak shares of the
manufacturing at the same level of the real per capita income. Instead, suppose that, forany given level of technology gap, technology adoption takes longer in the service sector
than in agriculture and manufacturing, capturing the inherent difficulty of adoption in theservice sector due to the intangible nature of its technology, and a larger technology gap
magnifies the difference in the adoption lags across sectors. Yet, in spite of the longer
adoption lag in the service sector, the productivity growth rate is sufficiently smaller inthe service sector, so that productivity differences are smaller in services across countries,

Page 2 of 13

as observed empirically. 1 Then, we obtain premature deindustrialization, as poorer
countries that suffer from larger technology gaps, reach the peak later and at the lower
level of per capita income with the lower peak share of manufacturing, compared toricher countries with smaller technology gaps, which industrialized earlier.
In Section 2, we set up the three-sector model and solve for the equilibrium
sectoral shares as functions of sectoral productivities. In Section 3, we introduce thefrontier technology and adoption lags in each sector to obtain the hump-shaped pattern of
manufacturing. In Section 4, we finally introduce technology gap differences acrosscountries, and show under what conditions premature deindustrialization occurs. We
conclude in Section 5.
2.

The Model
Consider an economy with three competitive sectors, indexed by 𝑗𝑗 = 1,2,3. Eachsector produces a single consumption good, also indexed by 𝑗𝑗 = 1,2,3. We interpret
sector-1 as agriculture, sector-2 as manufacturing and sector-3 as services.
2.1

Households

The economy is populated by 𝐿𝐿 identical households. Each household supplies

one unit of labor, which is perfectly mobile across sectors, at the wage rate 𝑤𝑤 and 𝜅𝜅𝑗𝑗 unitsof the factor specific to sector-𝑗𝑗 at the rental price, 𝜌𝜌𝑗𝑗 , to earn income, 𝑤𝑤 + ∑3𝑗𝑗=1 𝜌𝜌𝑗𝑗 𝜅𝜅𝑗𝑗 . It

spends its income to consume 𝑐𝑐𝑗𝑗 units of good-𝑗𝑗, purchased at the price, 𝑝𝑝𝑗𝑗 , subject to thebudget constraint,3�to maximize its CES utility

𝑗𝑗=1

3

𝑝𝑝𝑗𝑗 𝑐𝑐𝑗𝑗 ≤ 𝐸𝐸 = 𝑤𝑤 + �

𝑗𝑗=1

(1)

𝜌𝜌𝑗𝑗 𝜅𝜅𝑗𝑗 ,𝜎𝜎
1
11−𝜎𝜎 𝜎𝜎−1
𝑈𝑈 = �� �𝛽𝛽𝑗𝑗 �𝜎𝜎 �𝑐𝑐𝑗𝑗 � �
𝑗𝑗=1

(2)

3

1

For example, Kravis, Heston and Summers (1982, p.8) wrote that “services are much cheaper in therelative price structure of a typical poor country than in that of a rich country.” This finding has been
confirmed by many subsequent studies.
Page 3 of 13where 𝐸𝐸 denotes the per capita income (and expenditure) and 𝛽𝛽𝑗𝑗 > 0 and 0 < 𝜎𝜎 < 1, sothat the three goods are gross complements. Maximizing eq.(2) subject to eq.(1) yields
the household’s expenditure shares,
1−𝜎𝜎

where

𝛽𝛽𝑗𝑗 �𝑝𝑝𝑗𝑗 �
𝑝𝑝𝑗𝑗 1−𝜎𝜎𝐸𝐸 ⁄𝑝𝑝𝑗𝑗
𝑝𝑝𝑗𝑗 𝑐𝑐𝑗𝑗
= 3
=
𝛽𝛽
�
�
=
𝛽𝛽
�
�
𝑗𝑗
𝑗𝑗
∑𝑘𝑘=1 𝛽𝛽𝑘𝑘 (𝑝𝑝𝑘𝑘 )1−𝜎𝜎
𝐸𝐸
𝑃𝑃
𝑈𝑈
3

is the cost-of-living index and
𝑈𝑈 =

𝐸𝐸
=
𝑃𝑃

𝑃𝑃 = ��

𝛽𝛽𝑘𝑘 (𝑝𝑝𝑘𝑘 )1−𝜎𝜎 �𝑘𝑘=1[∑3𝑘𝑘=1

𝐸𝐸

Production

(3)

,

1
1−𝜎𝜎

3

𝐸𝐸

𝛽𝛽𝑘𝑘 � �
1 = ��
𝑝𝑝𝑘𝑘
𝑘𝑘=1
1−𝜎𝜎
1−𝜎𝜎
(𝑝𝑝
)
]
𝛽𝛽𝑘𝑘 𝑘𝑘is the real per capita income (and expenditure).
2.2.

𝜎𝜎−1

1
𝜎𝜎−1 𝜎𝜎−1�Sector-𝑗𝑗 produces good-𝑗𝑗, with its specific factor and labor as the inputs, by using
the following Cobb-Douglas Technology:
𝛼𝛼

1−𝛼𝛼

𝑌𝑌𝑗𝑗 = 𝐴𝐴𝑗𝑗 �𝜅𝜅𝑗𝑗 𝐿𝐿� �𝐿𝐿𝑗𝑗 �(4),where 𝐴𝐴𝑗𝑗 > 0 is the TFP of sector-𝑗𝑗; 𝜅𝜅𝑗𝑗 𝐿𝐿 is the total supply of specific factor-𝑗𝑗; and 𝐿𝐿𝑗𝑗 is

the labor employed in sector-𝑗𝑗, with 𝛼𝛼 ∈ [0,1) being the share of specific factor. Labor
employed in the three sectors are subject to the labor supply constraint:
3

�

𝑗𝑗=1

(5)

𝐿𝐿𝑗𝑗 = 𝐿𝐿.

From eq.(4), output per worker and output per capita in sector-𝑗𝑗 can be expressed as:

𝛼𝛼

𝑌𝑌𝑗𝑗
−𝛼𝛼
= 𝐴𝐴̃𝑗𝑗 �𝑠𝑠𝑗𝑗 � ;
𝐿𝐿𝑗𝑗

𝑌𝑌𝑗𝑗
1−𝛼𝛼
= 𝐴𝐴̃𝑗𝑗 �𝑠𝑠𝑗𝑗 �𝐿𝐿(6)where 𝐴𝐴̃𝑗𝑗 ≡ 𝐴𝐴𝑗𝑗 �𝜅𝜅𝑗𝑗 � and 𝑠𝑠𝑗𝑗 ≡ 𝐿𝐿𝑗𝑗 ⁄𝐿𝐿 is the employment share of sector-𝑗𝑗 with
3

�

𝑗𝑗=1

(7)

𝑠𝑠𝑗𝑗 = 1.

Firms in each sector chooses the inputs to minimize their cost of production,
while taking the wage rate 𝑤𝑤 and the rental price 𝜌𝜌𝑗𝑗 given. Under the Cobb-Douglas

technology, eq.(4), this leads to:Page 4 of 13

𝑤𝑤𝐿𝐿𝑗𝑗 = (1 − 𝛼𝛼)𝑝𝑝𝑗𝑗 𝑌𝑌𝑗𝑗 ;𝜌𝜌𝑗𝑗 𝜅𝜅𝑗𝑗 𝐿𝐿 = 𝛼𝛼𝑝𝑝𝑗𝑗 𝑌𝑌𝑗𝑗

Thus, the shares of labor and of the specific factor in the value-added of each sector are
1 − 𝛼𝛼 and 𝛼𝛼 respectively, which are common across all sectors. Using the household

budget constraint, eq.(1), this implies the aggregate budget constraint, 𝐸𝐸𝐸𝐸 = ∑3𝑗𝑗=1 𝑝𝑝𝑗𝑗 𝑌𝑌𝑗𝑗 .
Furthermore, using the labor supply constraint, eq.(5), the sectoral shares measured inlabor employment are equal to those measured in value-added.
𝑠𝑠𝑗𝑗 ≡
2.3

Equilibrium

𝐿𝐿𝑗𝑗
𝑝𝑝𝑗𝑗 𝑌𝑌𝑗𝑗
𝑝𝑝𝑗𝑗 𝑌𝑌𝑗𝑗= 3=.
𝐿𝐿 ∑𝑘𝑘=1 𝑝𝑝𝑘𝑘 𝑌𝑌𝑘𝑘 𝐸𝐸𝐸𝐸

(8)From eq.(6) and eq.(8), we obtain
𝑌𝑌𝑗𝑗
𝐸𝐸
−𝛼𝛼
= = 𝐴𝐴̃𝑗𝑗 �𝑠𝑠𝑗𝑗 � .
𝑝𝑝𝑗𝑗 𝐿𝐿𝑗𝑗(9)

By inserting this expression in eq.(3), the expenditure shares can now be expressed as:
𝑝𝑝𝑗𝑗 𝑐𝑐𝑗𝑗𝐴𝐴̃𝑗𝑗 �𝑠𝑠𝑗𝑗 �= 𝛽𝛽𝑗𝑗 �𝐸𝐸𝑈𝑈−𝛼𝛼 𝜎𝜎−1�.

Since the expenditure shares are equal to the value-added (and employment) shares,
𝑝𝑝𝑗𝑗 𝑌𝑌𝑗𝑗
𝑝𝑝𝑗𝑗 𝑌𝑌𝑗𝑗 𝑝𝑝𝑗𝑗 𝑐𝑐𝑗𝑗𝐴𝐴̃𝑗𝑗 �𝑠𝑠𝑗𝑗 �
𝑠𝑠𝑗𝑗 = 3=== 𝛽𝛽𝑗𝑗 �∑𝑘𝑘=1 𝑝𝑝𝑘𝑘 𝑌𝑌𝑘𝑘
𝐸𝐸𝐸𝐸
𝐸𝐸
𝑈𝑈

Solving this equation for 𝑠𝑠𝑗𝑗 yields
where𝑠𝑠𝑗𝑗 =𝑎𝑎 ≡1

�𝛽𝛽𝑗𝑗 𝜎𝜎−1 𝐴𝐴̃𝑗𝑗 �−𝑎𝑎

𝑈𝑈 −𝑎𝑎−𝛼𝛼 𝜎𝜎−1�

.

,

1 − 𝜎𝜎> 0.
1 − 𝛼𝛼(1 − 𝜎𝜎)Then, by using the adding up constraint, eq.(7), this implies
𝑠𝑠𝑗𝑗 =

1

�𝛽𝛽𝑗𝑗 𝜎𝜎−1 𝐴𝐴̃𝑗𝑗 �
1−𝑎𝑎∑3𝑘𝑘=1 �𝛽𝛽𝑘𝑘 𝜎𝜎−1 𝐴𝐴̃𝑘𝑘 �

(10)

−𝑎𝑎

Page 5 of 13

3

𝑈𝑈 = ��

𝑘𝑘=1

(11)

1
−𝑎𝑎 −𝑎𝑎1
�𝛽𝛽𝑘𝑘 𝜎𝜎−1 𝐴𝐴̃𝑘𝑘 � �Eq.(10) and eq.(11) show the equilibrium values of the sectoral shares (measured in
employment, value-added, and expenditure) and of the real per capita income, as
3functions of the sectoral productivities, �𝐴𝐴̃𝑗𝑗 �𝑗𝑗=1 .

3.

Productivity Growth Rates, Adoption Lags and Structural Change
Let us now see how the sectoral shares respond to the sectoral productivities

3change over time. 2 Suppose that �𝐴𝐴̃𝑗𝑗 (𝑡𝑡)�𝑗𝑗=1 change according to:(12)𝐴𝐴̃𝑗𝑗 (𝑡𝑡) = 𝐴𝐴𝑗𝑗̅ �𝑡𝑡 − 𝜆𝜆𝑗𝑗 � = 𝐴𝐴𝑗𝑗̅ (0)𝑒𝑒 𝑔𝑔𝑗𝑗�𝑡𝑡−𝜆𝜆𝑗𝑗� = 𝐴𝐴𝑗𝑗̅ (0)𝑒𝑒 −𝜆𝜆𝑗𝑗𝑔𝑔𝑗𝑗 𝑒𝑒 𝑔𝑔𝑗𝑗𝑡𝑡 ,with 𝑔𝑔𝑗𝑗 > 0. and 𝜆𝜆𝑗𝑗 ≥ 0, while the other parameters stay constant. 3 Here, 𝐴𝐴𝑗𝑗̅ (𝑡𝑡) =𝐴𝐴𝑗𝑗̅ (0)𝑒𝑒 𝑔𝑔𝑗𝑗𝑡𝑡 is the frontier technology in sector-𝑗𝑗 at time 𝑡𝑡, which grows at a constant rate
𝑔𝑔𝑗𝑗 > 0. With 𝐴𝐴̃𝑗𝑗 (𝑡𝑡) = 𝐴𝐴𝑗𝑗̅ �𝑡𝑡 − 𝜆𝜆𝑗𝑗 �, 𝜆𝜆𝑗𝑗 represents the adoption lag in sector-𝑗𝑗. Note thatboth the growth rates and the adoption lags are sector-specific. Note also that the

adoption lag in each sector does not affect the productivity growth rate of that sector.
However, the “level” effect of the adoption lags, 𝑒𝑒 −𝜆𝜆𝑗𝑗𝑔𝑔𝑗𝑗 , depends on the growth rate. A

large adoption lag would not matter much in a sector with slow productivity growth, buteven a small adoption lag would matter a lot in a sector with fast productivity growth, afeature that plays a crucial role later. For the moment, “the base year,” 𝑡𝑡 = 0, is chosen
arbitrarily, but we will later set the calendar time to ease the notation.

Inserting eq.(12) into eq.(11) yields the time path of the real per capita income:3

𝑈𝑈(𝑡𝑡) = ��

𝑘𝑘=1

(13)

1
−𝑎𝑎 −𝑎𝑎1
�(𝛽𝛽𝑘𝑘 )𝜎𝜎−1 𝐴𝐴̅𝑘𝑘 (0)𝑒𝑒 𝑔𝑔𝑘𝑘(𝑡𝑡−𝜆𝜆𝑘𝑘) � �

Clearly, larger adoption lags would shift down the time path of 𝑈𝑈(𝑡𝑡).

To understand the patterns of structural change, let us first take the ratio of theshares of two sectors, 𝑗𝑗 and 𝑘𝑘 ≠ 𝑗𝑗, given in eq.(10) and using eq.(12), to obtain:

Without any means to save, the equilibrium path of this economy can be viewed as a sequence of thestatic equilibrium.
𝛼𝛼3Since 𝐴𝐴̃𝑗𝑗 ≡ 𝐴𝐴𝑗𝑗 �𝜅𝜅𝑗𝑗 � , 𝑔𝑔𝑗𝑗 includes both the growth rate of 𝐴𝐴𝑗𝑗 and the growth rate of 𝜅𝜅𝑗𝑗 .
2Page 6 of 13

1

𝑠𝑠𝑗𝑗 (𝑡𝑡)𝛽𝛽𝑗𝑗 𝜎𝜎−1 𝐴𝐴̃𝑗𝑗 (𝑡𝑡)= �� �
���
𝑠𝑠𝑘𝑘 (𝑡𝑡)𝛽𝛽𝑘𝑘𝐴𝐴̃𝑘𝑘 (𝑡𝑡)
=�

where

−𝑎𝑎

𝛽𝛽𝑗𝑗
= �� �
𝛽𝛽𝑘𝑘

1
𝜎𝜎−1

−𝑎𝑎

(14)

𝐴𝐴𝑗𝑗̅ �𝑡𝑡 − 𝜆𝜆𝑗𝑗 ����𝐴𝐴𝑗𝑗̅ (𝑡𝑡 − 𝜆𝜆𝑘𝑘 )𝛽𝛽�𝑗𝑗 𝑎𝑎(𝜆𝜆 𝑔𝑔 −𝜆𝜆 𝑔𝑔 ) −𝑎𝑎�𝑔𝑔 −𝑔𝑔 �𝑡𝑡
𝑘𝑘𝑗𝑗� 𝑒𝑒 𝑗𝑗 𝑗𝑗 𝑘𝑘 𝑘𝑘 𝑒𝑒𝛽𝛽�𝑘𝑘

1𝛽𝛽�𝑗𝑗 ≡ �𝛽𝛽𝑗𝑗 𝜎𝜎−1 𝐴𝐴𝑗𝑗̅ (0)�

−𝑎𝑎

> 0.

Eq.(14) shows that, with the two sectors producing gross complements (𝑎𝑎 > 0 because

𝜎𝜎 < 1), 𝑠𝑠𝑗𝑗 (𝑡𝑡)⁄𝑠𝑠𝑘𝑘 (𝑡𝑡) is decreasing over time if 𝑔𝑔𝑗𝑗 > 𝑔𝑔𝑘𝑘 , and increasing over time if 𝑔𝑔𝑗𝑗 <𝑔𝑔𝑘𝑘 . That is, the sectoral shares shift from sectors with faster productivity growth to thosewith slower productivity growth over time. In contrast, the adoption lags have no effecton the direction of the sector changes over time, but they shifts the time path, with ahigher 𝜆𝜆𝑗𝑗 𝑔𝑔𝑗𝑗 −𝜆𝜆𝑘𝑘 𝑔𝑔𝑘𝑘 raising 𝑠𝑠𝑗𝑗 (𝑡𝑡)⁄𝑠𝑠𝑘𝑘 (𝑡𝑡) at any point in time.

Likewise, using eq.(9) and eq.(14), the relative price can be expressed as:
𝛼𝛼

𝑎𝑎
1−𝜎𝜎

𝑝𝑝𝑗𝑗 (𝑡𝑡)𝛽𝛽𝑗𝑗𝐴𝐴̅𝑘𝑘 (0)= �� � ���𝑝𝑝𝑘𝑘 (𝑡𝑡)𝛽𝛽𝑘𝑘
𝐴𝐴𝑗𝑗̅ (0)(15)

𝑎𝑎(𝑔𝑔𝑘𝑘 −𝑔𝑔𝑗𝑗 )𝑡𝑡 𝑎𝑎(𝜆𝜆𝑗𝑗 𝑔𝑔𝑗𝑗 −𝜆𝜆𝑘𝑘 𝑔𝑔𝑘𝑘 )1−𝜎𝜎𝑒𝑒 1−𝜎𝜎 𝑒𝑒

Eq.(15) shows that 𝑝𝑝𝑗𝑗 (𝑡𝑡)⁄𝑝𝑝𝑘𝑘 (𝑡𝑡) is decreasing over time if 𝑔𝑔𝑗𝑗 > 𝑔𝑔𝑘𝑘 , and increasing overtime if 𝑔𝑔𝑗𝑗 < 𝑔𝑔𝑘𝑘 , so that slower productivity growth causes its relative price to go up overtime. In contrast, the adoption lags shift the time path, with a higher 𝜆𝜆𝑗𝑗 𝑔𝑔𝑗𝑗 −𝜆𝜆𝑘𝑘 𝑔𝑔𝑘𝑘 raising

𝑝𝑝𝑗𝑗 (𝑡𝑡)⁄𝑝𝑝𝑘𝑘 (𝑡𝑡) at any point in time.

In what follows, we restrict ourselves to the case of 𝑔𝑔1 > 𝑔𝑔2 > 𝑔𝑔3 > 0, togenerate the patterns of structural change, well documented, for example, by Herrendorf,
Rogerson, Valentinyi (2014), based on the mechanism put forward by Ngai and

Pissarides (2007). That is, the share of agriculture, 𝑠𝑠1 (𝑡𝑡), is decreasing over time because1
𝑠𝑠1 (𝑡𝑡) + 𝑠𝑠2 (𝑡𝑡) + 𝑠𝑠3 (𝑡𝑡)𝑠𝑠2 (𝑡𝑡) 𝑠𝑠3 (𝑡𝑡)
=
=1+
+
,
𝑠𝑠1 (𝑡𝑡)
𝑠𝑠1 (𝑡𝑡)
𝑠𝑠1 (𝑡𝑡) 𝑠𝑠1 (𝑡𝑡)with both the 2nd and 3rd terms increasing; the share of services, 𝑠𝑠3 (𝑡𝑡), is increasing overtime, because

Page 7 of 13

1
𝑠𝑠1 (𝑡𝑡) + 𝑠𝑠2 (𝑡𝑡) + 𝑠𝑠3 (𝑡𝑡) 𝑠𝑠1 (𝑡𝑡) 𝑠𝑠2 (𝑡𝑡)
=
=
+
+ 1,
𝑠𝑠3 (𝑡𝑡)
𝑠𝑠3 (𝑡𝑡)
𝑠𝑠3 (𝑡𝑡) 𝑠𝑠3 (𝑡𝑡)with both the 1st and 2nd terms decreasing; and the share of manufacturing; 𝑠𝑠2 (𝑡𝑡), is
hump-shaped, because=�1
𝑠𝑠1 (𝑡𝑡) + 𝑠𝑠2 (𝑡𝑡) + 𝑠𝑠3 (𝑡𝑡) 𝑠𝑠1 (𝑡𝑡)𝑠𝑠3 (𝑡𝑡)==+1+
𝑠𝑠2 (𝑡𝑡)
𝑠𝑠2 (𝑡𝑡)
𝑠𝑠2 (𝑡𝑡)
𝑠𝑠2 (𝑡𝑡)(16)

𝛽𝛽�1 𝑎𝑎(𝜆𝜆 𝑔𝑔 −𝜆𝜆 𝑔𝑔 ) −𝑎𝑎(𝑔𝑔 −𝑔𝑔 )𝑡𝑡
𝛽𝛽�312
𝑒𝑒 1 1 2 2 � 𝑒𝑒
+ 1 + � 𝑒𝑒 𝑎𝑎(𝜆𝜆3𝑔𝑔3 −𝜆𝜆2𝑔𝑔2 ) � 𝑒𝑒 𝑎𝑎(𝑔𝑔2−𝑔𝑔3)𝑡𝑡 ,
𝛽𝛽�2
𝛽𝛽�2

with the 1st term exponentially decreasing and the 3rd term exponentially increasing.
Differentiating eq.(16) with respect to 𝑡𝑡 yields the peak time of the manufacturing

share as:

𝑡𝑡̂ =

where

Using 𝛽𝛽�𝑗𝑗 ≡ �𝛽𝛽𝑗𝑗

1
𝜎𝜎−1𝐴𝐴𝑗𝑗̅ (0)�𝑡𝑡̂0 ≡−𝑎𝑎

𝜆𝜆1 𝑔𝑔1 −𝜆𝜆3 𝑔𝑔3+ 𝑡𝑡0̂ ,𝑔𝑔1 − 𝑔𝑔3

1
𝑔𝑔1 − 𝑔𝑔2 𝛽𝛽�1ln �� � �.
𝑎𝑎(𝑔𝑔1 − 𝑔𝑔3 )𝑔𝑔2 − 𝑔𝑔3 𝛽𝛽�3> 0, this can be rewritten as11𝛽𝛽1 𝜎𝜎−1 𝐴𝐴1̅ (𝑡𝑡̂0 )𝑔𝑔1 − 𝑔𝑔2 𝑎𝑎
=�
� �
� .
̅
𝛽𝛽3
𝑔𝑔2 − 𝑔𝑔3
𝐴𝐴3 (𝑡𝑡0̂ )In what follows, we reset the calendar time to simplify the notation, such that 𝑡𝑡̂0 = 0 and𝑡𝑡̂ =𝜆𝜆1 𝑔𝑔1 −𝜆𝜆3 𝑔𝑔3.
𝑔𝑔1 − 𝑔𝑔3

(17)This choice of the calendar time also implies that, if an economy could use the frontier
technology without any adoption lag, the share of the manufacturing sector would peak at
𝑡𝑡̂ = 𝑡𝑡̂0 = 0, so that eq.(17) shows how adoption lags affect the peak time. Notice that alarger 𝜆𝜆1 , which makes the relative productivity of the agriculture sector lower and hence
agriculture relatively more expensive, causes a further delay in the manufacturing peak,

while a larger 𝜆𝜆3 , which makes the relative productivity of the service sector lower andhence services relatively more expensive, reduces a delay in the manufacturing peak.
From eq.(17), simple algebra can verify:

Page 8 of 13

𝑔𝑔1 (𝑡𝑡̂ − 𝜆𝜆1 ) = 𝑔𝑔3 (𝑡𝑡̂ − 𝜆𝜆3 ) = �

𝑔𝑔2 (𝑡𝑡̂ − 𝜆𝜆2 ) =𝜆𝜆1 − 𝜆𝜆3
� 𝑔𝑔 𝑔𝑔 ;𝑔𝑔1 − 𝑔𝑔3 1 3(𝜆𝜆1 − 𝜆𝜆2 )𝑔𝑔1 𝑔𝑔2 +(𝜆𝜆2 − 𝜆𝜆3 )𝑔𝑔2 𝑔𝑔3.
𝑔𝑔1 − 𝑔𝑔3

Using these expressions, the peak manufacturing share, 𝑠𝑠�2 ≡ 𝑠𝑠2 (𝑡𝑡̂), can be derived as:3 )𝑔𝑔2 𝑔𝑔3 +(𝜆𝜆3 −𝜆𝜆1 )𝑔𝑔3 𝑔𝑔1 ]𝛽𝛽�1 + 𝛽𝛽�3 𝑎𝑎[(𝜆𝜆1−𝜆𝜆2 )𝑔𝑔1𝑔𝑔2+(𝜆𝜆2𝑔𝑔−𝜆𝜆−𝑔𝑔
1
1
3
=1+�
� 𝑒𝑒
𝑠𝑠�2
𝛽𝛽�2

(18)

1
(𝜆𝜆 −𝜆𝜆 )𝑔𝑔 𝑔𝑔 +(𝜆𝜆 −𝜆𝜆 )𝑔𝑔 𝑔𝑔 −𝑎𝑎
𝜆𝜆 −𝜆𝜆−𝑎𝑎�𝑔𝑔1 −𝑔𝑔3 �𝑔𝑔1 𝑔𝑔3−𝑎𝑎 1 2 1 𝑔𝑔2 −𝑔𝑔 2 3 2 3� = ��𝛽𝛽�1 + 𝛽𝛽�3 �𝑒𝑒
1
3
1
3
𝑈𝑈
+ 𝛽𝛽�2 𝑒𝑒
� .

(19)

� ≡ 𝑈𝑈(𝑡𝑡̂), as:and the real per capita income at the peak, 𝑈𝑈

4.

Technology Gaps and Premature Deindustrialization
Now imagine that there are many countries, whose adoption lags are given by(𝜆𝜆1 , 𝜆𝜆2 , 𝜆𝜆3 ) = (𝜃𝜃1 , 𝜃𝜃2 , 1)𝜆𝜆, with 𝜆𝜆 varying across countries. The countries are otherwise
identical, including 𝜃𝜃1 > 0 and 𝜃𝜃2 > 0. Thus, the countries differ in one dimension, 𝜆𝜆.

The idea is that each country tries to adopt the frontier technologies, which keep

improving at exogenously constant growth rates, but the countries differ in their ability toadopt, indexed by the country-specific parameter, 𝜆𝜆, which we shall call “technology

gap,” following Krugman (1985). Unlike Krugman (1985), who assumed 𝜆𝜆𝑗𝑗 = 𝜆𝜆 in all
sectors, we allow for the possibility that the extent to which technology gap affects the
adoption lag varies across sectors. That is, 𝜃𝜃1 and 𝜃𝜃2 are sector-specific parameters,common across countries, capturing the inherent difficulty of adoption in the agriculture
and manufacturing sectors, relative to the service sector.

First, consider the case where 𝜃𝜃1 = 𝜃𝜃2 = 1, so that 𝜆𝜆1 = 𝜆𝜆2 = 𝜆𝜆3 = 𝜆𝜆 > 0, as in

Krugman (1985). Then, from eqs.(17)-(19),𝑡𝑡̂ = 𝜆𝜆;1
𝛽𝛽�1 + 𝛽𝛽�3
=1+�
�;
𝑠𝑠�2
𝛽𝛽�2

1

� = �𝛽𝛽�1 + 𝛽𝛽�2 + 𝛽𝛽�3 �−𝑎𝑎 .
𝑈𝑈

Thus, if technology gaps affect the adoption lags in all the sectors uniformly, they cause adelay in the peak time by 𝜆𝜆 > 0, but they do not affect the peak manufacturing share andthe real per capita income at the time of the peak. This means that poorer countries that
suffer from larger technology gaps are late industrializers, with a larger 𝑡𝑡̂ = 𝜆𝜆, but they

Page 9 of 13

reach the same peak manufacturing share at the same level of the real per capita income
as earlier industrializers with a smaller 𝑡𝑡̂ = 𝜆𝜆.

Now, consider the case where 𝜃𝜃1 = 𝜃𝜃2 = 𝜃𝜃 with 𝑔𝑔3 ⁄𝑔𝑔2 < 𝜃𝜃 < 1. The condition,𝜃𝜃 < 1, implies

𝜆𝜆1 = 𝜆𝜆2 < 𝜆𝜆3 .

This means that technology adoption takes longer in the service sector than in the
agriculture and manufacturing sector, capturing the inherent difficulty of adoption in theservice sector due to the intangible nature of its technology. The condition, 𝜃𝜃 > 𝑔𝑔3 ⁄𝑔𝑔2 ,ensures 𝜃𝜃𝑔𝑔1 , 𝜃𝜃𝑔𝑔2 > 𝑔𝑔3 and hence,

𝜆𝜆1 𝑔𝑔1 , 𝜆𝜆2 𝑔𝑔2 > 𝜆𝜆3 𝑔𝑔3 .

From eq.(15), this implies that, in spite of the longer adoption lag in the service sector,
the productivity growth rate is sufficiently smaller in the service sector, so that crosscountry productivity differences are smaller in services, and equivalently that the services
are relatively cheaper in poorer countries, as observed empirically by Kravis, Heston andSummers (1982) and many others.
Then, from eqs.(17)-(19),𝑡𝑡̂ = �𝜃𝜃𝜃𝜃1 −𝑔𝑔3
� 𝜆𝜆.
𝑔𝑔1 − 𝑔𝑔3

2 )𝑔𝑔3 (1−𝜃𝜃)𝜆𝜆𝛽𝛽�1 + 𝛽𝛽�3 𝑎𝑎(𝑔𝑔𝑔𝑔1 −𝑔𝑔
1
−𝑔𝑔
1
3
=1+�
� 𝑒𝑒
𝑠𝑠�2
𝛽𝛽�2

� = ��𝛽𝛽�1 + 𝛽𝛽�3 �𝑒𝑒
𝑈𝑈

𝑎𝑎𝑔𝑔1 𝑔𝑔3(1−𝜃𝜃)𝜆𝜆𝑔𝑔1 −𝑔𝑔3+ 𝛽𝛽�2 𝑒𝑒𝑎𝑎𝑔𝑔2 𝑔𝑔3(1−𝜃𝜃)𝜆𝜆𝑔𝑔1 −𝑔𝑔3

1

−𝑎𝑎

� .

This case thus generates premature deindustrialization. Poorer countries that suffer frombigger technology gaps, 𝜆𝜆, are late industrializers in the sense that they reach their peaks

later (𝑡𝑡̂ is larger), have smaller peak manufacturing shares, 𝑠𝑠�2 , and reach their peaks at the�, compared to richer, early industrializers withlower levels of the per capita income, 𝑈𝑈
smaller technology gaps, 𝜆𝜆.

Figure 1 illustrates this case with 𝑔𝑔1 = .10 > 𝑔𝑔2 = .06 > 𝑔𝑔3 = .02, 𝜃𝜃 = .35, 𝛼𝛼 =�=
1⁄3, and 𝜎𝜎 = 0.6 (hence 𝑎𝑎 = 6⁄13). The other parameters are chosen such that ln 𝑈𝑈

0 and 𝑡𝑡̂ = 0 for 𝜆𝜆 = 0. 4 The hump-shaped curves, each capturing the rise and fall of the
4

They are 𝛽𝛽𝑗𝑗 = 1, 𝐴𝐴𝑗𝑗̅ (0) = 31⁄𝑎𝑎 = 313⁄6 = 10.8..., so that 𝛽𝛽�𝑗𝑗 = 1⁄3 for 𝑗𝑗 = 1,2,3.

Page 10 of 13

manufacturing sector, are plotted for 𝜆𝜆 = 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, usingeq.(13) and eq.(16). The left panel shows the time paths of 𝑠𝑠2 (𝑡𝑡), which shows that ahigher 𝜆𝜆 causes a delay in the timing of the peak. The right panel traces the movement of
�ln 𝑈𝑈(𝑡𝑡) , 𝑠𝑠2 (𝑡𝑡)�, which shows that a bigger technology gap 𝜆𝜆 shifts the curve down andto the left, with the upward-sloping line connecting the peaks, which captures prematuredeindustrialization.
More generally, what are the conditions for premature deindustrialization? Fromeq.(17), the condition for 𝑡𝑡̂ increasing in 𝜆𝜆 is given by 𝜃𝜃1 > 𝑔𝑔3 ⁄𝑔𝑔1 , whose boundary is

depicted by the vertical line, 𝜃𝜃1 = 𝑔𝑔3 ⁄𝑔𝑔1 , in Figure 2. From eq.(18), the condition for 𝑠𝑠�2
decreasing in 𝜆𝜆 is:

�

𝑔𝑔2
𝑔𝑔1
𝑔𝑔1
𝑔𝑔2
− 1� �𝜃𝜃1 − 1� > � − 1� �𝜃𝜃2 − 1�,
𝑔𝑔3
𝑔𝑔3
𝑔𝑔3
𝑔𝑔3

whose boundary is depicted by the line connecting (𝜃𝜃1 , 𝜃𝜃2 ) = (𝑔𝑔3 ⁄𝑔𝑔1 , 𝑔𝑔3 ⁄𝑔𝑔2 ) and� decreasing in 𝜆𝜆 is:(𝜃𝜃1 , 𝜃𝜃2 ) = (1,1), in Figure 2. From eq.(19), the condition for 𝑈𝑈
�

𝑔𝑔1
𝑔𝑔1
− 1� 𝜃𝜃2 > �𝜃𝜃1 − 1�,
𝑔𝑔3
𝑔𝑔3

whose boundary is depicted by the line connecting (𝜃𝜃1 , 𝜃𝜃2 ) = (𝑔𝑔3 ⁄𝑔𝑔1 , 0) and (𝜃𝜃1 , 𝜃𝜃2 ) =(1,1), in Figure 2. Thus, when (𝜃𝜃1 , 𝜃𝜃2 ) belongs to the triangle area in Figure 2, poorer

countries with bigger technology gaps 𝜆𝜆 reach their peaks of the manufacturing sharelater at lower levels of the real per capita income with lower peak shares of the
manufacturing sectors.
5.

Concluding Remarks
In this paper, we developed a simple model of Rodrik’s (2016) prematuredeindustrialization. We have done so by combining the mechanism of structural change
due to the exogenous productivity growth rate differences across sectors, as in Ngai and
Pissarides (2007), with heterogeneity across countries due to the technology gap, as in
Krugman (1985), with an additional element, which allows the technology gap of acountry to have disproportionate impacts on its adoption lags across sectors.
Needless to say, we are not claiming that the countries differ only in their capacity
to adopt the frontier technology. Nor do we argue that our mechanism is the sole cause

Page 11 of 13

for premature deindustrialization. After all, differential productivity growth rate acrosssectors is not the only mechanism capable of generating a hump-shaped path of the
manufacturing sector. As pointed out by Herrendorf, Rogerson and Valentinyi (2014) intheir survey article, another well-known mechanism is income elasticity differentials
across sectors. See also Comin, Lashkari, and Mestieri (forthcoming) and Matsuyama(2019) for recent examples. It would be interesting to see what types of country
heterogeneity could explain premature deindustrialization in models of structural change
based on income elasticity differentials.

6.

References

Comin D., D. Lashkari, and M. Mestieri, “Structural Transformation with Long-run
Income and Price Effects,” forthcoming, Econometrica.
Herrendorf, B., R. Rogerson, and A. Valentinyi, 2014, “Growth and Structural
Transformation,” in Handbook of Economic Growth, ed. by P. Aghion and S. N.
Durlauf, 855–941.
Kravis, I., Heston, A. and Summers, R., 1982, “The share of services in economic
growth” In Global Econometrics: Essays in Honor of Lawrence R. Klein, ed. F.
G. Adams and Bert Hickman, Cambridge: MIT Press.
Krugman, P. 1985, A ‘Technology-Gap’ Model of International Trade. In Structural
Adjustment in Developed Open Economies, ed. K. Jungenfelt and D. Hague.
London, Macmillan.
Matsuyama, K., 2019, “Engel's Law in the Global Economy: Demand-Induced Patternsof Structural Change, Innovation, and Trade” Econometrica, 497-528.
Ngai, L.R., and C. Pissarides (2007), “Structural Change in a Multisector Model of
Growth,” American Economic Review, 97, 429–443.
Rodrik, D., 2016, “Premature Deindustrialization,” Journal of Economic Growth, 21, 133.

Page 12 of 13

Figure 1: Premature Deindustrialization

Figure 2: Conditions for Premature Deindustrialization

Page 13 of 13