Revisiting the “Missing Middle”:

Productivity Analysis∗

Hien Thu Pham † Shino Takayama‡

June 22, 2015

Abstract

This paper investigates empirically the relationship between firm size and production effi-

ciency and inefficiency associated with the production scale. We study the possible sources of the

missing middle phenomenon, which refers to the fact that most employment in developing coun-

tries is located in either small-sized or large-sized firms. Using Vietnamese data, we show that

middle-sized firms’ production efficiencies tend to be lower than small-sized or large-sized firms

in most of the manufacturing industries, and that the least efficient firm size differs across sectors,

which indicates the necessity of sector-by-sector analyses to study the underlying mechanisms of

the missing middle phenomenon.

Key Words: firm size distribution, missing middle, productivity, efficiency, data envelopment analy-

sis, free disposal hull.

JEL Classification Numbers: D21, D22, L25.

∗We would like to thank Fabrizio Carmignani, Begona Dominguez, Jeff Kline, Do Won Kwak, Hideyuki Mizobuchi,

Andrew McLennan, Flavio Menezes, Thanh Le, Antonio Peyrache, Rodney Strachan, Satoshi Tanaka, Yuichiro Waki,

Haishan Yuan, Valentin Zelenyuk, and other participants at UQ Macro Workshop. We are grateful to the General Statistical

Office in Vietnam for giving us permission to use the micro data of the Enterprise surveys. Any remaining errors are our

own.†Pham; School of Economics, University of Queensland, Level 6 - Colin Clark Building (39), QLD 4072, Australia;

e-mail: thu.pham3@uq.net.au‡Takayama (corresponding author); School of Economics, University of Queensland, St Lucia, QLD 4072, Australia;

email: s.takayama@economics.uq.edu.au; tel: +61-7-3346-7379; fax: +61- 7-3365-7299.

1 Introduction

The missing middle refers to the empirical fact that most employment in developing countries is lo-

cated in either small-sized or large-sized firms. The missing middle was first documented in Liedholm

and Mead (1987). Tybout (2000) also finds that there is a large spike in the size distribution for the

small-sized category, and that it drops off quickly in the middle-sized category in the poorest coun-

tries, and then argues that strong business regulation could be a reason behind the existence of too

many small firms1.

Our objective is to identify the reasons for the missing middle phenomenon, which is more evident

in developing countries than in developed countries. To achieve this objective, we use firm-level data

from Vietnam and examine whether there are any differences in production efficiencies across differ-

ent sized firms. We estimate each firm’s production efficiency as well as the inefficiencies associated

with scale in each manufacturing sector and find that the middle-sized firms generally produce less

efficiently than other size firms in most sectors.

Our analysis shows that in most of the manufacturing industries that we study, middle-sized firms’

production efficiencies tend to be lower than those of small-sized or large-sized firms, and the scale

at which minimum efficiency occurs and the level of this efficiency are different across industries.

We also find that there are many firms whose efficiency is low, less than half of the most efficient

firms, across different firm sizes in all sectors. Almost half a century ago, Leibenstein (1966) intro-

duced the concept of X-efficiency, which is mainly caused by incentive misalignments in a workplace,

unlike allocative efficiency. Furthermore, he provides a review of empirical studies, showing that

allocative inefficiency such as that caused by a tax could cause very small losses in total production,

while improving management or motivational efficiency in a workplace could increase production by

approximately 50%. Although in general it is difficult to separate these two inefficiencies empirically,

our data suggest that there are many inefficient firms of different sizes in each sector, and that this

feature is more prominent in middle-sized firms across different sectors.

Hsieh and Olken (2014) study the possibility that regulatory obstacles generate a missing middle,

but found no evidence of discontinuities in the firm size distribution. On the other hand, Tybout

1See also Little et al. (1987) for South Korea and Steel and Webster (1992) for Ghana. The presence of the missing

middle phenomenon and how it is defined have been extensively discussed in the literature (see Hsieh and Olken, 2014;

Tybout, 2014a,b).

1

(2014b) claims that the missing middle is at least partly policy-induced and states that when policies

are imperfectly enforced, producers tend to stay small in order to avoid taxes and costly regulations.

One difference in our approach from those of the previous literature is that rather than looking at

taxes or regulations themselves, we study production efficiency as a manifestation of the business

environment in developing nations.

Business environments vary across nations. Although it is difficult to measure how good the

business environment is, or how well each firm is managed, the level of production efficiency can

reflect the management of an organization. We provide empirical evidence that available resources

might be less efficiently used in developing nations and that this could be an important factor for the

missing middle phenomenon.

To the best of our knowledge, we do not know any other studies that link the missing middle and

production efficiencies across different firm sizes in various sectors of a developing country, particu-

larly because the analysis of the underlying mechanisms for the observed firm size distribution is less

developed, perhaps because of the unavailability of detailed data.2 The data set that we use to mea-

sure productivity and efficiency is the Enterprise Census conducted by the General Statistics Office

of Vietnam (GSO). The Enterprise Census contains information on all registered firms in Vietnam.

Our data include capital, labor, and intermediate materials as inputs, and goods measured in monetary

terms as output. Table 1 lists the sectors we study in this paper.3

To see if the missing middle phenomenon is observed in our data set and if there are any differ-

ences in the firm size distributions across sectors, we compute the bimodality coefficients for each

sector.4 We then consider the situation where each firm uses capital, labor, and intermediate materials

2Recently, there has been growing interest in the relationship between the firm size distribution and productivity in

developed nations. Leung et al. (2008) investigate the US–Canada difference in the distribution of employment over firm

size and confirm that a larger average size supports higher productivity at both the plant and firm level. Crosato et al.

(2009) investigate the data of a micro-survey in Italy and, using nonparametric estimates, find that firms in the concave

part of the Zipf plot of the Italian firm distribution overwhelmingly experience increasing returns to scale, while firms in

the linear part are mainly characterized by constant returns to scale. Diaz and Sanchez (2008) use a stochastic frontier

model to investigate small-sized and medium-sized manufacturing enterprises in Spain and find that these firms tend to be

less efficient than their large-sized peers.3These sector codes are based on the International Standard Industrial Classification (ISIC) codes. We choose these

sectors because each sector based on the two-digit-level ISIC code has more than 200 observations per year, so that we

can ensure robustness of the nonparametric estimations.4The logic behind the bimodality coefficient is that a bimodal distribution would have very low kurtosis, an asymmetric

2

to produce goods, and study production efficiency in various sectors of the manufacturing industry by

using the nonparametric frontier approach.

Table 1: ISIC Codes and Industry Description

ISIC Industry Description ISIC Industry Description

14 Other mining and quarrying 15 Food products and beverages17 Textiles 18 Wearing apparel etc.19 Tanning and dressing leather 20 Wood and wood products21 Paper and paper products 22 Publishing, printing, etc.24 Chemical products, etc. 25 Rubber and plastic products26 Nonmetallic mineral products 28 Fabricated metal products29 Machinery and equipment, etc. 36 Furniture

In recent years, many methods have been developed to estimate production efficiency under the

frontier approach.5 Comparing average products across different inputs is a complicated task. As

pointed out by Ray (2004), measures of a firm’s productivity relying on a single input disregarding

other inputs may fail to reflect total factor productivity. Taking this into account, we use the frontier

approach.

Our analysis relies on the following theoretical framework. Consider one sector in the manufac-

turing industry. Firm i’s input and output combination is called a productive unit, denoted by (xi, yi)

with xi ∈ IR3+ and yi ∈ IR+. Then, define firm i’s production set by Ψi = {(x, y) : x ≥ xi, y ≤ yi}.

The union of all the firms’s production sets in the sector,⋃

i Ψli, is the production set of this sector.

The surface of this set is the production function. Using these data, we construct a production function

for each sector.6 Then, we compute each firm’s efficiency score, which is the minimal proportional

reduction of all inputs while maintaining the same output level within⋃

i Ψli. Figure (1a) provides

a graphical illustration in the case of a single input and a single output (an explanation for the case

of multiple inputs is provided later). Each dot represents a productive unit. A solid line is the con-

structed production function. The multiple θ in Figure (1a) shows the efficiency score of this white

circle whose inputs are given by x. This is our first method to compute the efficiency scores.

character, or both, all of which increase this coefficient. The formula itself does not assume a particular distribution. On

the other hand, the value for the uniform distribution or the exponential distribution is 5/9. Values greater than 5/9 may

indicate a bimodal or multimodal distribution in the data.5The production frontier can be estimated using parametric estimates (deterministic or stochastic frontier analysis) or

nonparametric ones (data envelopment analysis or free disposal hull). For more details, see Kalirajan and Shand (1994b).6To clarify, we do not impose any functional form on the estimation, and a production function is estimated as a

frontier of a production set.

3

Second, by using the algorithm developed by Soleimani-Damaneh and Reshadi (2007), we study

returns to scale and inefficiencies associated with production scale.7

To further explain our second method, note that the circled dot A has the highest average product.

The other black dots on the production function such as B and C are efficient in terms of efficiency

scores, unlike the white circles, but do not have the highest average product. If the same technology

and management skills of Firm A are available for Firm C and the underlying technology is constant

returns to scale, Firm C could achieve Firm A’s average product by adopting the production practices

of Firm A, which is represented by the point C. Otherwise, this indicates that there is some ineffi-

ciency associated with production scales or decreasing returns to scale technology between A and C.

Similar logic applies to the points A and B, although in this case the technology can be increasing

returns to scale, but not decreasing returns to scale.

Conceptually, this method studies whether there is some management inefficiency related to scales

or decreasing returns to scale technology (or both), although the method does not identify what per-

centage of such a difference between Firm 1’s and Firm 2’s production levels is attributable to man-

agement inefficiency and the returns to scale technology. For instance, suppose that there are two

firms, where Firm 1 uses 10 units of labor to produce 5 units of goods and Firm 2 uses 15 units of

labor to produce 7 units of goods. If the technology is constant returns to scale, then by simply ex-

trapolating Firm 1’s production, Firm 1 should be able to produce 7.5 units by using 15 units of labor.

Then, comparing Firm 2’s production, the technology may be decreasing returns to scale, or there

may be some inefficiency in managing the resources in Firm 2. 8 Merging data from multiple sectors

might mix these two features even further, and we may not be able to identify a relationship between

7Diewert et al. (2011) uses the following definition of returns to scale. If all inputs are multiplied by a positive scaler,

t, and the consequent output can be represented as tγy, the value of γ may be said to indicate the magnitude of returns

to scale. If γ = 1, there are constant returns to scale; if γ > 1, there are increasing returns to scale and if γ < 1,

there are decreasing returns to scale. In the literature on productivity analysis, a production function is constructed from

the data, and we apply this definition to each firm’s production by comparing it to the production of the firm with the

highest average product among the set of firms with similar production scales. Applying this definition to each firm’s

production might be confusing, because in the classical production theory of economics, returns to scale is a property of

the production technology. Thus, we only use the term returns to scale when we talk about production technology itself.8As pointed out by Feng and Zhang (2014), not considering the technological heterogeneity of an individual firm even

within a single sector can cause bias in measuring returns to scale. Attempting to separate the two factors by measuring

returns to scale considering technological heterogeneity for each firm might be an interesting and promising research path

to further investigate to what extent inefficiencies only associated with scale affect the missing middle.

4

Figure 1: Illustrative Example

(a) Efficiency

◦

θx x

◦◦◦◦◦◦

•B

�B ••••A ◦◦◦◦◦◦•C�

C• •

• •

◦◦◦◦◦◦◦

◦◦• • • • •

Output

Input

(b) Intuition

◦

◦◦◦◦◦◦

Small-Sized•••

••◦

◦◦◦◦◦

◦◦•• •

• •

Middle-Sized

◦◦◦◦◦◦◦

◦◦• • • • •

Large-Sized

Output

Input

management inefficiency and firm size. It is particularly important to study this measure at the sec-

toral level, as the underlying technology in each sector would not be that different in comparison with

different sectors.9

Figure (1b) describes the intuition of our empirical findings in the case of one input and one output,

although our analysis with multiple inputs is more complicated. Large-sized firms produce closer to

the production function, although compared with the highest average product firms, they tend to have

lower average products (except for the sectors with lower bimodality coefficients). In the case of

multiple inputs, these lines can be thought of as a cone. In contrast, the middle-sized firms tend to be

further away from efficient production, and also have lower average products.

Our first contribution is to show that the middle-sized firms tend to have lower production ef-

ficiency in the sectors whose firm size distribution does not reject bimodality. Together with our

companion paper, Pham and Takayama (2015), which provides evidence that as firm size increases,

the likelihood of paying bribes increases, this paper provides evidence that the middle-sized firms are

disadvantaged.

Furthermore, our work indicates that the large-sized firms may be unable to fully utilize their in-

puts. Hsieh and Olken (2014) claim that “the problem of economic development in low-income and

9Another way to approach this problem is by assuming that the basic natures of the technologies are not so different

between developed and developing nations, one may try to study this measure at the sectoral level in developed nations

and compare the outcomes with ours.

5

middle-income countries is how to relieve the differential constraints faced by large firms.” Our anal-

ysis of inefficiency associated with scale finds that the large-sized category is quite mixed, including

few firms with high average product and many firms with low average product. Our analysis suggests

that one of the problems, in Vietnam at least, is how to activate certain resources that these large-sized

firms hold but tend not to fully utilize in production.10

As our second method was developed for the purpose of measuring returns to scale (see Kalirajan

and Shand, 1994a), our work is also related to the literature on measuring returns to scale. Basu and

Fernald (1997) use US data to estimate returns to scale and find that a typical industry appears to

have constant or slightly decreasing returns to scale. Diewert et al. (2011) also find a similar result

using Japanese data for the period 1964 to 1988, and provide explanations of their findings of the time

variation in returns to scale in the light of economic occurrences and government policies for each

period of time. By using French data from the 19th century, Doraszelski (2004) show that returns

to scale vary across industries and time. These works estimate returns to scale for the production

function of the whole industry. By applying data envelopment analysis to the data of large US banks

in the late 1980s, Miller and Noulas (1996) show that larger and more profitable banks have higher

levels of technical efficiency, but at the same time larger banks tend to be too big in terms of cost

considerations. Feng and Zhang (2014) investigate the returns to scale of large banks in the US over

the period 1997 to 2010 by considering technological heterogeneity and showed that most firms’

production exhibits constant returns to scale. Using Vietnamese data including small-sized firms, our

results show that most industrial manufacturing sectors have a similar feature, and we demonstrate

how the results differ across different sectors.

This paper also contributes to the literature on firm size and productivity in developing coun-

tries by showing that productivity is different across different firm sizes, and that middle-sized firms

have lower efficiencies of production. Johannes (2005) provides evidence on the differences in the

evolution of firm size and productivity distributions across nine sub-Saharan African countries and

developed countries including the US. Banerjee and Duflo (2005) provide a survey and analyze what

causes lower total factor productivity (TFP), such as lack of access to technology or human capital, in

poor countries (which they call “macro puzzles”).

10For example, in Vietnam, sector 17 (Textiles) and sector 21 (Paper and paper products) include many large-sized firms

that were previously or are currently state-owned. As we will see later in Figure 4, these industries exhibit this feature

more visibly.

6

The rest of the paper is organized as follows. The second section describes our data set and

provides a summary of Vietnam’s firm size distribution and average products. The third section de-

scribes our methodology for measuring productivity and efficiency, and then presents an analysis of

production efficiency and inefficiency associated with scale. The last section concludes.

2 Firm Size Distribution at a Glance in Vietnam

2.1 Date Description

In Vietnam, a census has been conducted annually since 2000. In the survey, an enterprise is defined

as “an economic unit that independently keeps a business account and acquires its own legal status.”

In this paper, we focus on the nine-year period from 2000 to 2008. We exclude recent years (from

2009 to present) because the Vietnamese economy has suffered from the effects of the global financial

crisis, and a 30% reduction in the corporate income tax rate for qualifying entities was implemented in

the fourth quarter of 2008 and all of 200911. In addition, small-sized and middle-sized firms involved

in labor-intensive production and processing activities will also benefit from the subsequent tax re-

duction under Decree 60/2011. We exclude inconsistent data from our sample, such as observations

that are recorded twice for the same firm in the same year, those with negative or zero revenue values,

or those with an implausibly large number of employees. We include 14 sectors of the manufacturing

industry that have at least 200 observations for each year, and the number of observations ranges from

approximately 3,350 to 29,700 per industry.

In the data, labor is measured by the total income of employees in a firm. This includes total wages

and other employee-related costs such as social security, insurance, and other benefits. Intermediate

materials include costs such as fuel and the value of other materials. Capital is measured as assets

to be used in production (in terms of present value).12 All input and output values are adjusted to

11See Circular No. 03/2009/TT-BTC published in January 2009 by Vietnam’s Ministry of Finance.12At the firm level, prices and quantities may not be well measured, and therefore revenue, instead of gross output or

cost, is normally used. In the literature, there are some arguments that the elasticities of labor and capital, in a revenue

estimate, may be downward biased (Basu and Fernald, 1997). Kristiaan and Philippe (1999) report that changes in sector

prices are substantially diversified and correlated with changes in labor and capital. However, according to Jacques

and Jordi (2005), the introduction of individual output prices into the production function does not markedly affect the

estimate. In addition, the estimation of a production function in terms of “physical quantities” is, in fact, meaningless,

unless we confine the analysis to a very precisely defined industry where goods are so homogeneous that firm outputs can

be well measured and compared across firms. Accordingly, we use this measure in the analysis.

7

account for inflation to obtain real values. Inflation rates are obtained from the World Bank.

2.2 Summary

To examine the firm size distribution of the manufacturing sector in Vietnam, we start with the figures

for the whole manufacturing sector. In the next sections, we analyze each sector. For the whole

manufacturing sector, the number of observations ranges from 9,143 (year 2000) to 27,918 (year

2008). The number of sectors is 27.

Insert Figure 1

To make the patterns more visible, we present figures similar to those in Hsieh and Olken (2014).

In the first and second rows, we classify firm size into subintervals of 0 to 200 employees (column 1),

10 to 200 employees (column 2), 20 to 200 employees (column 3), 50 to 200 employees (column 4)

and 200 to 3000 employees (column 5). These two sets of histograms present the distribution of firm

size in bins of 10 workers. In the last row, we take the maximums and the minimums of log(Value addedCapital

)

(the first row), log(Value addedCapital

) (the second row) and log(RevenueMaterial

) (the third row), for the entire period of

2000 to 2008, and truncate the intervals into 50 bins.

The key observations in the figures are as follows:

• There is a large number of small firms compared with other firm sizes.

• The distribution of employment in each column is “flatter” compared with that in Figure 1.

• The employment share for very small firms increases over time (from 2000 to 2004 and from

2004 to 2008).

• The distributions of average products do not look bimodal.

We compute the bimodality coefficients for each sector.13 This method examines the relationship

between the bimodality, skewness, and kurtosis of the distribution. The bimodality coefficient (BC)

13We have also tried the Dip tests for each sector as well as for each year of the whole sample.(for details, see Hartigan

and Hartigan, 1985). The detailed results are available upon request. The Dip statistics we obtained are all smaller than

0.05, which indicates the presence of bimodality or multimodality. A more detailed interpretation of this statistic is found

at Freeman and Dale (2013).

8

ranges from 0 to 1, with those exceeding 5/9 ≈ 0.56 suggesting bimodality14. The BCs for each

sector are presented in Table 2.

Table 2: The Bimodality Coefficients in Each Sector

Sector 14 15 17 18 19 20 21 22 24 25 26 28 29 36

BC 0.57 0.68 0.74 0.64 0.68 0.68 0.62 0.61 0.72 0.71 0.36 0.66 0.58 0.69

The BCs in sectors 14, 26, and 29 indicate that these sectors’ firm size distributions are not bi-

modal, unlike other sectors.15 Table 2 tells us that the features of the firm size distributions vary

substantially across sectors.16 We analyze productivity and efficiency for each sector. In what fol-

lows, we will study productivity and efficiency across different firm sizes in each sector. The results

presented in the next section indicate that the sectors with lower BCs in Table 2, particularly sectors

14, 26, and 29, show different features in terms of productivity and efficiency from other sectors.

3 Efficiency Scores and Inefficiency Associated with Scale

This section consists of two parts. In the first part, we measure the efficiency of production, using

the efficiency score17, and in the second part, study the inefficiency associated with scale at the firm

14For more details, see page 1258 in SAS Institute Inc. (2008). The bimodality coefficient ism2

0+1

m1+3(n−1)2

(n−2)(n−3)

, where m0

is skewness and m1 is the excess kurtosis. However, the BC is criticized because of its sensitivity to the skewness of the

distribution. We have also conducted the Dip test, which is considered to be more robust (see Freeman and Dale, 2013).

Unimodality is rejected for all sectors, according to the Dip test. The detailed results are available upon request.15 We have also computed the BCs for each sector by year. Because the number of observations in each year for each

sector decreases, the results fluctuate. In the yearly results, the BCs tend to increase, and even for sectors 14, 26, and 29,

the BCs are higher than 0.56 (except for sector 14 in 2008). Perhaps the interesting case is sector 26. Although it shows the

lowest score for any year, sector 26’s score for each year is higher than 0.56. We further analyzed sector 26’s histograms

of firm size distribution for each year and found that small firms grow noticeably in this sector year by year. According

to Anh et al. (2014), the industry of non-metallic mineral products is one of the “sunrise” industries in Vietnam. This

industry’s ranking in the share of total output has increased from the eighth (in 2001) to the second in 2007, and continues

to be the second highest after Food products.16We have also checked the distributions against Zipf’s law, and the hypothesis that they follow Zipf’s law was rejected

in our data set. The details of the analysis are available upon request.17To compute the efficiency scores, we use the R-package FEAR (see Wilson, 2008). We are grateful to Wilson, who

provided us with a license to use the software.

9

level using nonparametric methods, which impose no or very limited assumptions on the data. The

following summarizes our findings.

• In all the sectors, there are many firms whose efficiency scores are low, smaller than 0.5, across

different firm sizes.

• Except in some sectors whose bimodality coefficients are low, the relationship between firm

size and efficiency score tends to be U-shaped, which indicates that smaller-sized firms and

larger-sized firms produce efficiently compared with middle-sized firms.

• Firm size at the bottom of the U-shape differs across sectors.

• The large-sized firms are likely to exhibit inefficiencies associated with scale.

The next subsections detail our methodologies and the above-mentioned points. We consider the

situation where a firm uses capital, labor, and intermediate materials to produce goods, which are

measured in monetary terms. Real revenue is used as a proxy for output. Three inputs are included

in our estimation: intermediate inputs, labor, and capital. Then, we construct a production function

for each sector without imposing any restrictions on the parametric relationship between inputs and

outputs.

3.1 Efficiency Score and Results

Suppose that there are L industrial sectors in the economy. Take a sector l ∈ {1, · · · , L} (we repeat

the same procedure for each sector) where inputs x ∈ IR3+ are used to produce an output y ∈ IR+.

Let N l denote the number of firms in sector l of the data set. Firm i’s input and output combination

is called a productive unit, denoted by (xi, yi) with xi ∈ IR3+ and yi ∈ IR+.18 In the same way as

discussed in the Introduction, for firm i, we formally define:

Ψli = {(x, y) : x ≥ xi, y ≤ yi}. (1)

Furthermore, we define ΨlFDH =

⋃i Ψ

li, and the convex hull of Ψl

FDH is defined as ΨlDEA. The

efficiency score for a productive unit (xi, yi) is defined by:

E(xi, yi) = infθ{θ : (θxi, yi) ∈ Ψl}, (2)

18To keep the notation simple, we do not explicitly denote a period and an industry for each productive unit.

10

for each Ψl = ΨlFDH under free disposal hull (hereafter, FDH), or Ψl

DEA under data envelopment

analysis (hereafter DEA), respectively. Furthermore, the PS Ψl is assumed to satisfy the regularity

conditions; namely, boundedness, closedness, no free-lunch19, and free disposability20. We say that Ψl

exhibits convexity if for every (x1, y1), (x2, y2) ∈ Ψl, and any α ∈ [0, 1], α(x1, y1)+(1−α)(x2, y2) ∈Ψl holds.

The efficiency score21 lies between 0 and 1, and represents the minimal proportional reduction of

all inputs while maintaining the same output level within the production set (hereafter, PS). Solving

Problem (2) requires constructing the PS, using linear programming. In this analysis, we use two

different methods, namely the FDH and DEA methods as defined above. The PS for the FDH method

is different from the PS for the DEA method, and the difference between the two scores indicates the

existence of nonconvexity in the production technologies.

Figure 2 is a scatter-plot of efficiency scores for the FDH method in our sample industries for

the entire nine-year period, and the curve represents the smooth trend line of the efficiency scores for

every 0.1 interval of log(size).22 The x-axis represents the logarithm of firm size. In our analysis, we

will also present the ratios of the two efficiency scores obtained from the DEA and the FDH methods

in Figure 3. The ratio of the two scores indicates the existence of nonconvexity in the production

technology.

Insert Figure 2, and Figure 3

According to Figure 2, except in sectors 14 (Other mining and quarrying), 26 (Nonmetallic min-

eral products), and 29 (Machinery and equipment, etc.), the relationship between firm size and ef-

ficiency score is clearly U-shaped, which indicates that smaller firms and larger firms produce effi-

ciently compared with middle-sized firms. On the other hand, Figure 2 for sectors 14, 26, and 29,

which show the low BCs in Table 2, shows that the smooth trend lines of the efficiency scores are

mostly decreasing as the size increases. In sector 29, there is a spike in the trend line for very large-

sized firms, although this seems to be caused by few efficient large-sized firms. Moreover, firm size

19A positive amount of production cannot occur without a positive amount of inputs.20The increase in inputs must lead to increased or constant outputs, and a smaller output vector than a feasible vector is

also feasible.21This is called the “input-oriented” efficiency score. There is another definition called the “output-oriented” efficiency

score. In this analysis, we use the input-oriented efficiency score because we have computed output-oriented scores for

some sectors, which did not change substantially.22 We use Matlab’s command smooth. This inserts a smooth trend line by using the local regression method.

11

at the bottom of the U-shape—that is, the lowest efficiency score—differs across sectors. The het-

erogeneity of firm size at the bottom of the U-shape across industries indicates the importance of

sector-level analysis in investigating the missing middle.

Finally, the ratios between the DEA scores and the FDH scores are less than one for most of the

firms, which indicates that there is nonconvexity in the production technologies, and the shapes of the

trend lines are quite different across sectors. This indicates that the nonconvexity of the technologies

is also significantly different across sectors.

3.2 Inefficiency Associated with Scale and Results

In this section, we study the relationship between production scale and average product at the firm

level in each industry. Figure 3 in the previous section indicates the existence of some nonconvexities

in production technologies.23 Thus, we use an FDH estimate, which does not require the assumption

of convex technology.

Similar to the previous section, we consider manufacturing sector l. Let Dl denote a set of all the

observations in sector l. Then, we consider the following problem for each productive unit (xi, yi) ∈Dl:

Minimize θ

subject to∑

j∈N l λjxj ≤ θxi,∑

j∈N l λjyj ≥ yi, λj = δwj,

wj ∈ {0, 1} for every j ∈ N l, δ ≥ 0,∑

j∈N l wj = 1.

(3)

The idea is to first compute each firm’s hypothetically best productive unit by projecting it onto

the line of the highest average products. To illustrate, let us consider the inefficiency associated with

scale C in Figure (1a). In Figure (1a), B and C correspond to the projected points of B and C,

respectively. Note that here, only one λj is nonzero, but it does not equal one. The algorithm finds

some nonzero values of (λjxj, λjyj), which are associated with (θixi, yi) for each i. In Figure (1a), A

has the highest average product. To obtain (λCxA, λCyA) = (θCxC , yC), which is represented by C, it

must be the case that λC > 1. This implies that if the same technology and management skills of Firm

A are available for Firm C and the underlying technology is constant returns to scale, Firm C could

23Farrell (1957) emphasizes indivisibilities and economies of scale as important sources of nonconvexity and empha-

sizes that convexity can only be justified in terms of time divisibility (ignoring not only any setup times but also indivisi-

bilities, increasing returns to scale, positive or negative production externalities, etc., that can lead to nonconvexity). For

some recent theoretical or empirical studies, see Brown (1991) or (Ramey, 1991).

12

achieve Firm A’s production efficiency by adopting the production processes of Firm A. Otherwise,

it indicates that there is some inefficiency associated with production scales or decreasing returns to

scale technology between A and C. A similar argument applies to points A and B.

Now, let θi denote the solution θ associated with observation i, and λ+i and λ−

i denote the maximal

and the minimal of the solutions∑

j∈N l λj to Problem 3 for each i, respectively.24 When λ+i is smaller

than 1, it indicates that there is a firm with higher average products whose scale is larger than firm i.

Figure 4 presents the bar graphs indicating how many percentages of firms have λ+i < 1 (dark

color), λ−i > 1 (light color), and else (white) in each group of firm sizes. To grasp this easily, we

group the samples by number of employees into five groups, where groups 1 to 5 include firms with

1−−10, 11−−50, 51−−100, 101−−200, and 201 or more employees, respectively.25 The white

portions are the groups of firms with the highest average products, while in Figure 1 they are the

circled dots. According to the categorization defined above, we calculate how many firms belong to

each group and then divide these numbers by the total number of firms in each group, so that we

can obtain the percentages of firms in each category of each group. We repeat this procedure for all

sectors.

Insert Figure 4

Figure 4 shows that in most of the sectors, the proportion of firms with the property λ+i < 1

is highest in Group 1. This is understandable, because their size is small and thus their production

perhaps tends to be small compared with the optimal scale. Somewhat more strikingly perhaps, we

observe that in many sectors, the large-sized firm category includes the most firms with the property

of λ−i > 1, particularly in sectors 22 (Publishing, printing, etc.) through 25 (Rubber and plastic

products), unlike in sectors 14 (Other mining and quarrying), 26 (Nonmetallic mineral products), and

29 (Machinery and equipment, etc.), which are the sectors with the low BCs in Table 2. The images

of the production functions that the analysis shows are typical of firms using large quantities of inputs

because they show decreasing returns to scale, and many large-sized firms are located in this region.

As Figure 3 shows, at larger scales, the production sets in most of the sectors are more convex. This

24A more detailed description of the numerical algorithm to calculate λ+ and λ− is found in the online Appendix. In

the productivity analysis literature, a firm with λ+i < 1 is said to be operating under Increasing Returns to Scale and a

firm with λ−i > 1 is said to be operating under Decreasing Returns to Scale.

25This set of cutoff sizes ensures that we have enough firms in each group.

13

implies that the production functions in the regions of larger quantities of inputs show concavity.

From the analysis of Figure 2, we can see that many large-sized firms operate close to the function.

This may suggest that even though the large-sized firms can produce in a relatively efficient manner,

they may not be able to take full advantage of scale.

Finally, we should note that we have also computed the annual efficiency scores, λ+i and λ−

i , for

each sector. In this annual exercise, because the number of observations for each sector decreases

dramatically, it becomes more difficult to obtain a general trend, and as described in Chapter 3.3 of

Daraio and Simar (2007), under both the FDH and DEA method, larger quantities of data provide

more sensible estimates.26 Still, similar to the analysis including all the years for each sector, the

annual analysis also shows that the middle-sized firms tend to have low efficiency scores. On the

other hand, as also mentioned in footnote 15, the BCs of sectors 14, 26, and 29 tend to increase and

the annual bar graphs of these three sectors are similar to those of other sectors. Finally, we also

observe that there are still many low-efficiency firms in Figure 2.

4 Conclusion

In this paper, we have focused on the relationship between firm size and production efficiency using

Vietnamese data. We have presented results indicating that middle-sized firms tend to produce less

efficiently relative to small-sized and large-sized firms. Our study analyzes firms in a developing

country in which small-sized firms dominate the economy and exhibit higher efficiency levels com-

pared with middle-sized firms.27 Moreover, the heterogeneity of firm size at the bottom of the U-shape

across industries indicates the importance of sector-level analysis in investigating the missing middle.

Our analysis also contributes to this point.

Needless to say, one might question whether our findings from a transition economy—Vietnam—

provide a sufficient basis for generalization, particularly because much of the missing middle literature

concerns Africa. The business environment and its features vary substantially from country to country.

It would be interesting to see if we observe a similar feature in a different country.

26The detailed results are available from the authors.27In the literature, there has been increasing interest in the relationship between firm size and characteristics such as

innovation and market structure (see Acs and Audretsch, 1987), growth and productivity (see Bentzen et al., 2011), and

job creation (see Dalton et al., 2011). However, there are few empirical studies on the relationship between firm size and

efficiency level, and those examining developing countries are rare. Furthermore, in previous studies, large-sized firms

are found to be the most efficient (see Angelini and Generate, 2008; Leung et al., 2008).

14

Several interesting future research paths follow from our analysis. A next step is to conduct a

similar analysis with data from developed countries on production efficiency and to compare the

results with those for Vietnam. If we still observe lower productivity for middle-sized firms, we could

then consider the causes. If we only observe this U-shaped pattern in developing countries, we could

examine the reasons why this pattern exists in developing countries but not in developed countries.

Finally, it would be very interesting and important to study the causes of these differences in

productivity across different firm sizes in developing countries. OECD Publishing (2013) reports

two main reasons among OECD countries for the differences in productivity across different firm

sizes: (i) firm size matters for productivity, and (ii) structural differences in the industrial composition

of economies impact on the relative performance of large- and small-sized firms across countries.

First, in most countries, it is shown that there is the possibility of improving production efficiency by

increasing firm size. Larger-sized firms are on average more productive than smaller-sized firms, and

this generally holds for all industries. Second, in countries with large industrial sectors and relatively

low income per capita, large-sized firms are, on average, 2–3 times more productive than small-sized

firms. However, in countries with large services sectors and relatively high income per capita, small-

sized firms are often more productive than large-sized firms. It would be interesting to investigate

whether these factors are also important in developing nations.

15

Lis

tofT

able

sand

Figu

res

Tab

le3:

Sum

mar

yof

Ente

rpri

seC

ensu

sD

ata

Sta

tist

ics

(Fir

mS

ize)

Siz

eC

apit

alM

ater

ial

Sec

tor

#O

bse

rvat

ion

sM

ean

Med

ian

St.

Dev

.M

ean

Med

ian

St.

Dev

.M

ean

Med

ian

St.

Dev

.

Sec

tor

14

93

69

48

0.0

03

08

.00

66

9.4

03

40

.63

12

3.5

07

02

.11

26

2.5

01

7.0

07

02

.11

Sec

tor

15

29

71

32

68

.63

27

.50

69

9.8

02

92

.63

35

.50

68

3.1

35

05

.75

30

9.5

06

83

.13

Sec

tor

17

62

51

38

5.0

07

6.0

06

97

.90

29

4.1

32

4.0

02

71

6.6

92

30

3.0

01

75

3.5

02

71

6.6

9

Sec

tor

18

10

17

74

27

.00

22

4.5

06

63

.52

27

5.5

01

9.5

06

89

.05

35

8.8

87

3.5

06

89

.05

Sec

tor

19

33

47

89

18

.63

49

25

.00

12

17

2.4

73

62

.63

31

.50

25

26

.35

22

63

.63

20

15

.50

25

26

.35

Sec

tor

20

11

09

62

83

.75

24

.50

69

4.9

87

12

.88

23

0.0

06

84

.00

31

6.8

88

0.0

06

84

.00

Sec

tor

21

69

86

30

9.7

56

1.0

06

86

.64

33

7.8

81

49

.50

10

87

.30

64

2.7

58

4.0

01

08

7.3

0

Sec

tor

22

84

35

28

7.8

81

3.0

06

96

.97

30

8.1

35

7.0

06

91

.28

30

5.5

03

3.5

06

91

.28

Sec

tor

24

69

27

33

0.6

39

4.0

06

80

.31

29

8.7

55

7.0

06

93

.97

36

0.1

37

2.0

06

93

.97

Sec

tor

25

94

57

30

0.5

04

2.0

06

89

.49

32

9.6

35

1.0

06

99

.46

27

0.5

01

6.0

06

99

.46

Sec

tor

26

11

81

32

76

.63

22

.00

69

7.4

12

83

.50

40

.50

69

8.7

83

88

.63

73

.50

69

8.7

8

Sec

tor

28

15

85

33

54

.50

11

0.0

06

76

.98

29

7.8

85

3.5

07

05

.04

41

6.1

31

06

.00

70

5.0

4

Sec

tor

29

42

03

52

6.0

03

9.0

09

26

.41

51

14

.63

11

29

.50

69

0.3

53

21

.75

33

.50

69

0.3

5

Sec

tor

36

10

38

73

01

.25

16

.50

69

4.9

22

86

.88

54

.50

11

56

.72

10

87

.25

86

2.0

01

15

6.7

2

1S

ize

ism

easu

red

by

the

nu

mb

ero

fem

plo

yee

s.2

Cap

ital

and

Mat

eria

lar

em

easu

red

inh

om

ecu

rren

cy(n

ot

infl

atio

nad

just

ed).

3S

t.D

ev.

isst

and

ard

dev

iati

on

.

16

Tab

le4:

Sum

mar

yof

Ente

rpri

seC

ensu

sD

ata

Sta

tist

ics

(Num

ber

of

Obse

rvat

ions)

Yea

rT

ota

lS

ec1

4S

ec1

5S

ec1

7S

ec1

8S

ec1

9S

ec2

0S

ec2

1S

ec2

2S

ec2

4S

ec2

5S

ec2

6S

ec2

8S

ec2

9S

ec3

6

20

00

80

20

47

62

27

63

36

47

42

21

73

53

55

23

93

69

42

09

88

51

42

01

41

6

20

01

92

82

57

72

46

23

81

58

52

53

66

24

83

34

34

42

52

31

04

66

91

26

75

67

20

02

11

28

77

55

27

24

50

07

26

28

88

45

56

04

85

55

66

93

11

06

10

13

32

97

07

20

03

12

98

98

33

28

33

54

98

72

31

81

08

66

16

62

06

33

80

51

18

51

30

23

95

94

2

20

04

15

83

49

65

33

16

66

21

16

83

98

11

75

77

98

94

75

61

00

81

36

71

69

64

82

11

68

20

05

17

91

59

76

36

32

79

41

27

84

44

13

09

91

41

06

88

57

12

21

14

65

20

23

55

41

38

0

20

06

19

46

19

26

38

59

93

41

35

13

92

15

97

92

01

47

69

90

13

36

14

21

22

52

56

91

43

8

20

07

24

14

91

14

04

58

71

08

61

84

94

92

19

07

11

58

17

29

11

39

17

26

17

05

30

66

73

31

83

2

20

08

25

07

72

72

14

02

41

00

91

87

45

41

17

80

12

01

15

81

11

85

17

25

15

30

32

96

67

31

93

7

Not

e:E

ach

nu

mb

ersh

ow

sth

en

um

ber

of

ob

serv

atio

ns

inea

chse

cto

ro

fea

chin

du

stry

.

17

Fig

ure

1:

Illu

stra

tive

Fig

ure

of

the

2008

Vie

tnam

ese

Man

ufa

cturi

ng

Indust

ry

•Fir

mS

ize

Dis

trib

uti

on

Mea

sure

dby

Num

ber

of

Em

plo

yee

sPercentage

0<

Siz

e≤

200

Fir

msi

ze

10<

Siz

e≤

200

Fir

msi

ze

20<

Siz

e≤

200

Fir

msi

ze

50<

Siz

e≤

200

Fir

msi

ze

200<

Siz

e≤

3000

Fir

msi

ze

•Dis

trib

uti

on

of

Em

plo

ym

ent

Shar

eby

Fir

mS

ize

Percentage

0<

Siz

e≤

200

Fir

msi

ze

10<

Siz

e≤

200

Fir

msi

ze

20<

Siz

e≤

200

Fir

msi

ze

50<

Siz

e≤

200

Fir

msi

ze

200<

Siz

e≤

3000

Fir

msi

ze

Percentage

Val

ue

Ad

ded

toC

apit

al

log

(Val

ue

add

edC

apit

al)

Val

ue

Ad

ded

toW

ork

er

log

(Val

ue

add

edW

ork

er)

Rev

enu

eto

Raw

Mat

eria

ls

log

(Rev

enu

eM

ater

ial)

Not

e:F

or

the

firs

ttw

oro

ws

of

the

fig

ure

s,th

eb

insi

zeis

10

emp

loy

ees.

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hb

inco

nta

ins

the

up

per

bo

un

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ot

the

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ou

nd

.F

or

the

last

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eta

ke

the

max

imu

ms

and

min

imu

ms

oflog(

Val

ue

add

edC

apit

al),log(

Val

ue

add

edW

ork

er),log(

Rev

enu

eM

ater

ial),

resp

ecti

vel

y,an

dtr

un

cate

the

inte

rval

sin

to5

0b

ins.

Eac

hb

inco

nta

ins

the

up

per

bo

un

dan

d

no

tth

elo

wer

bo

un

d.

18

Fig

ure

2:

Effi

cien

cyS

core

sin

Eac

hIn

dust

ry

Efficiencyscores

log

(Siz

e)

(a)

Sec

tor

14

log

(Siz

e)

(b)

Sec

tor

15

log

(Siz

e)

(c)

Sec

tor

17

log

(Siz

e)

(d)

Sec

tor

18

log

(Siz

e)

(e)

Sec

tor

19

Efficiencyscores

log

(Siz

e)

(f)

Sec

tor

20

log

(Siz

e)

(g)

Sec

tor

21

log

(Siz

e)

(h)

Sec

tor

22

log

(Siz

e)

(i)

Sec

tor

24

log

(Siz

e)

(j)

Sec

tor

25

Efficiencyscores

log

(Siz

e)

(k)

Sec

tor

26

log

(Siz

e)

(l)

Sec

tor

28

log

(Siz

e)

(m)

Sec

tor

29

log

(Siz

e)

(n)

Sec

tor

36

19

Fig

ure

3:

Rat

ioof

the

Tw

oE

ffici

ency

Sco

res

inE

ach

Indust

ry

Ratio(DEA/FDH)

log

(Siz

e)

(a)

Sec

tor

14

log

(Siz

e)

(b)

Sec

tor

15

log

(Siz

e)

(c)

Sec

tor

17

log

(Siz

e)

(d)

Sec

tor

18

log

(Siz

e)

(e)

Sec

tor

19

Ratio(DEA/FDH)

log

(Siz

e)

(f)

Sec

tor

20

log

(Siz

e)

(g)

Sec

tor

21

log

(Siz

e)

(h)

Sec

tor

22

log

(Siz

e)

(i)

Sec

tor

24

log

(Siz

e)

(j)

Sec

tor

25

Ratio(DEA/FDH)

log

(Siz

e)

(k)

Sec

tor

26

log

(Siz

e)

(l)

Sec

tor

28

log

(Siz

e)

(m)

Sec

tor

29

log

(Siz

e)

(n)

Sec

tor

36

20

Fig

ure

4:

Inef

fici

ency

Ass

oci

ated

wit

hS

cale

inE

ach

Indust

ry

Percentageoffirms

Siz

eg

rou

p

(a)

Sec

tor

14

Siz

eg

rou

p

(b)

Sec

tor

15

Siz

eg

rou

p

(c)

Sec

tor

17

Siz

eg

rou

p

(d)

Sec

tor

18

Siz

eg

rou

p

(e)

Sec

tor

19

Percentageoffirms

Siz

eg

rou

p

(f)

Sec

tor

20

Siz

eg

rou

p

(g)

Sec

tor

21

Siz

eg

rou

p

(h)

Sec

tor

22

Siz

eg

rou

p

(i)

Sec

tor

24

Siz

eg

rou

p

(j)

Sec

tor

25

Percentageoffirms

Siz

eg

rou

p

(k)

Sec

tor

26

Siz

eg

rou

p

(l)

Sec

tor

28

Siz

eg

rou

p

(m)

Sec

tor

29

Siz

eg

rou

p

(n)

Sec

tor

36

Not

e:A

lig

ht

colo

rre

pre

sen

tsth

ep

ort

ion

of

firm

sfo

rw

hic

hλ− i>

1,a

dar

kco

lor

rep

rese

nts

the

po

rtio

no

ffi

rms

for

wh

ichλ+ i<

1an

da

wh

ite

colo

rre

pre

sen

tsth

e

rest

of

the

firm

s.T

hex

-ax

isre

pre

sen

tsea

chg

rou

p,

nam

elyn

isG

rou

pn

for

each

n=

1,···,

5.

21

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Online Appendix: Illustration of the Algorithm

We use the FDH algorithm developed by Soleimani-Damaneh and Reshadi (2007) (see also Soleimani-

Damaneh and Mostafaee, 2009). The algorithm is as follows:

Step 1. Compute λij for all i, j ∈ N l by λij = yiyj

where yi, yj ∈ Dl.

Step 2. Compute θji for all i, j ∈ N l by θji = maxs∈S{xjs·λij

xis: (xi, yi) ∈ Dl}.

Step 3. Compute θi and Ai for all i ∈ N l by θi = minj∈N l θji and Ai = {j ∈ N l : θji = θi}.

Step 4. Compute λ+i and λ−

i for all i ∈ N l by λ+i = maxj∈Ai

λij and λ−i = minj∈Ai

λij .

Soleimani-Damaneh and Mostafaee (2009) proves that λ+i and λ−

i are equivalent to the ones de-

fined in Problem 3. The next simple example demonstrates the algorithm. The original version of the

following example is found in Soleimani-Damaneh and Mostafaee (2009).

Table 5: Example for the Algorithm

Firm i Labor Capital Output OutputLabor

OutputCapital θi3

Average Productsλi1

(Labor) (Capital)Firm 1 1 1 4 4 4 1/2 2/3 1/2 -Firm 2 2 1 3 3/2 3 3/8 16/9 2/3 3/4Firm 3 3 4 8 8/3 2 - - - 2

Suppose that labor and capital are used to produce one unit of output and our data include three

firms’ production. We consider firm 3’s production. By Table 5, we can compute θ3 = θ13 = 2/3, and

A3 = {1}. Firm 1 is the most “efficient”. Because λ31 = 2, λ+3 = λ−

3 = 2. This implies that if the

same technology and management skills used by firm 1 are available to firm 3, firm 3 could achieve

firm 1’s production efficiency by reducing its production scale. Otherwise, it indicates that there is

some difficulty in firm 3 achieving firm 1’s efficiency. The above algorithm extends the idea of this

simple example to the case of multiple inputs.

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