Analysing the evolution of firms: discontinuity and growth

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Analysing the evolution of firms discontinuity
and growth

• Enrico DElia (Istat and MEF - DT)
• Leopoldo Nascia (Istat DCSP)
• Alessandro Zeli (Istat DCSP)

Istat, Aula Magna, Rome, November 21 22, 2011
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Introduction

• Typically firms grow (or downsize) through an
  irregular sequence of (large) leaps
• Thus, at firms level, continuous models are
  likely unfit to describe accurately the actual
  dynamics of output, employment, profit,
  investment, etc.
• The discontinuity of firms growth processes
  entails a number of consequences
• The heterogeneity among firms performances is
  expectedly huge
• The frequency distribution of firms by size,
  profit, investment, etc. is not continuous and
  embodies a number of peaks and lows
• The statistical relationships between firms
  level indicators (size, performance, etc.) are
  strongly non linear
• The statistical relationships between firms
  level behavior and the corresponding
  macroeconomic aggregates is expectedly weak
• The size and investment of existing firms are
  very inertial
• This paper focuses mainly on the topic iii

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Summary

• A new database has been built on the Italian
  manufacturing firms with more than 19 employees.
  It is a balanced panel of annual data ranging
  from 1998 to 2007, including about 40,000 firms.
• This database has been exploited to test some
  prediction of a model of firms growth which
  takes into account explicitly lumpy investment,
  and consequently the discontinuity in the
  evolution of firms.
• The model relies on the hypothesis that the
  entrepreneur is unwilling to make profit less
  than a given minimum threshold even during the
  transition between exploiting of current plants
  and installing the new ones
• Both non parametric and parametric estimates
  provide some evidence favorable to a couple of
  very challenging predictions of the model, i.e.
• profitability is higher when the firms size
  reaches some characteristic values (almost stable
  over time), and
• taking into account the firms size, the
  propensity to invest of each firm is larger when
  profitability is lower.

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Why firms change their size discontinuously

• Some traditional explanations
• Discontinuity can be induced by
• the indivisibility of plants
• institutional constraints
• sunk costs of changing (one off non proportional
  charges)
• optimal timing of investment under uncertainty
• A criticism to the traditional approach
• Most traditional explanations are adhockeries or
  truisms.
• In principle, firms can adjust their productive
  capacity almost continuously by leasing plants,
  having recourse to outsourcing, etc.
• Possible effects of institutional constraints can
  be easily set aside from the analysis of firms
  behavior.
• Discontinuity has been observed even when i iv
  do not hold.
• Often assuming the relevance of hardly measured
  sunk costs (including the advantage of waiting
  for better information) is only an ex post
  rationale for whatever discontinuous change

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A very simple model of discontinuous growth the
assumptions

• The assumptions on firms and entrepreneurs
• The firm is price taker on the input, labor and
  output markets
• The firm does not face financial and
  institutional constraints
• Changing the firms size
• is risky (because investment is almost
  irreversible, while the market conditions are
  uncertain) and
• bears sunk costs (search for plants, building,
  etc. hiring or firing workers training
  reshaping firms internal structure etc.)
• In an uncertain framework, the entrepreneur aims
  at achieving at least some (minimum) profit level
  even during an investment (or disinvestment)
  process
• Some (standard) assumptions on the short run
  profit function p(q, qt)
• it is continuous respect to the production level
  (q)
• it is twice differentiable respect to q
• it reaches a unique maximum (at the production
  level qt) within its definition field
• p(q1, q1) p(q0, q0) if q1 q0

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A rationale for the assumptions (a) (h)
Most assumptions (particularly (a), (b), (e),
(h)) serve to exclude from the analysis some
obvious sources of discontinuity. This fact
strengthens the scope for the model. E.g. For a
price taker firm strategic considerations on
supply level are irrelevant. Financial and
institutional constraints to grow are excluded. A
continuous (and first degree homogeneous)
production and profit function sets aside trivial
discontinuities. Other (standard) assumptions
simply make the problem analytically more
tractable (in particular (f) and (g) on the shape
of the profit function) The assumption (c), on
the cost of changing firms size, and (d), on the
minimum profit requirement, are the only sources
of discontinuity in the model. The first one is
very standard, and is not sufficient to generate
discontinuities alone. The hypothesis (d) is
milder than (and encompasses) the usual
assumption on an ever-profit-maximizing
entrepreneur. It is consistent with a managerial
model of firm in which the manager (who is not
necessary the firms owner) aims at expanding the
business and keeping his/her job achieving the
minimum targets set (expected) by the
shareholders.
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The consequences of the minimum profit assumption

• If the entrepreneur expects the demand flow D,
  exceeding the current optimal (most profitable)
  output level qt, he/she may either
• continue to exploit the current plant, even
  though at less efficient conditions, or
• increase the firms size from qt to qt1
• According to the thumbs rule (d), the
  entrepreneur prefers the strategy ii if
• p(q, qt) p(q, qt1) p 1
• where p(q, qs) is the profit achieved by
  producing q with a plant designed to be most
  profitable at qs p is the minimum profit
  threshold.
• Let us consider only a growth process. Expanding
  1 in a 2nd order Taylor series p(q, qs) near
  qs, and elaborating on the results, it reads
• qt1 - qt (q - qt) 2
• where ps , since p(qt, qt)
  p(qt1, qt1) and 0
• for the assumptions on p(q, qs).

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The limiting growth of firm
The condition 2 sets a ceiling to the maximum
size increase of firm when the entrepreneur is
not disposed to reduce the profit below p even
during the investment process (in compliance to
the assumption (d)). The following figure
provides a graphical illustration of this result
if the concave curve p(q, qt1) must cross p(q,
qt) at q lt qt1, then qt1 cannot be
indefinitely far from q.
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How much general is the limit 2?

• The condition 2 derives from very general (and
  standard) assumptions on p(q, qs) and a decision
  rule for investing which encompasses the usual
  intertemporal profit-maximizing model.
• Provided that the assumptions (e) - (g) hold, the
  function p(q, qs) may take into account of, among
  the other
• capital amortization, financial charges and the
  normal capital remuneration
• price elasticity of to q,
• tax and incentives,
• possible scale economies (but not scale
  diseconomies)
• the discounted value of expected future gains and
  losses (possibly related to the dynamics of
  demand and prices)
• A (very similar) variant of 2 holds even
  including in 1 the cost of investment (say c
  (qt1 - qt ))
• The profit threshold p may be time varying and
  may depend on qt

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Some consequences of the ceiling 2
The first consequence of 2 is that, even under
the most favorable conditions, a single firm
cannot grow too fast. The limit 2 could be
binding during a very strong expansive phase of
the business cycle. If this is the case, the
overall productive capacity may increase enough
only if newborn firms (are let free to) enter the
market. It follows that, in case, the market
share of existing firms is intended to fall,
otherwise the national output would grow less
than its potential. For instance, the
demography of Italian firms contributed by 15
to overall employment growth during the 1998-2007
decade. Apparently, the inequality 1, which
2 is based on, implies that the firm invests
much more when its profit is relatively low.
Nevertheless, this prediction holds only ceteris
paribus, and particularly comparing profit to the
maximum p(qt, qt) attainable by using the current
plants. As far as overall profit raises with
output level (because of the assumption (h)), in
the course of a growth process investment likely
increases with output (and consequently with
profit).
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Combining the ceiling 2 and sunk costs
If the firm faces sunk costs for each single
investment project, the ceiling 2 gives the
actual optimal size adjustment as well, since
the firm saves from amortizing such costs on the
largest feasible amount of fixed capital. In this
event, the actual growth process follows the
sequence of leaps provided by 2. Nevertheless,
the subscripts t and t1 in 2 are not meant as
periods of time measured along the ordinary scale
(e.g. the sequence of quarters, years, etc.),
but only as markers of the sequence of size
adjustments, possibly made erratically over time,
on the basis of demand and price expectations.
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A very special case
The firm adjusts its size at a regular pace over
time only under very special conditions e.g. if
the demand and the minimum profit p grow at a
constant rate over time, as in the graph
below () For the assumed
dynamics of demand
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The relationship between profit and output
Even in the very special (and simplified)
framework assumed in the previous slide, the
statistical relationship between output and
profit of the same firm is strongly non
linear On the one hand, overall
profit tends to raise with output (as expected),
on the other, the peaks and lows predicted by the
model are apparent in the graph
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The relationship between profit and investment
Also the statistical relationship between profit
and investment is very challenging In
general, investment raises with profit (as
expected), but non null investment concentrates
only at special levels of profit (corresponding
to the threshold assumed in the model). Thus the
relationship is highly non linear
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Adding the business cycle
The statistical relationships between output,
profit and investment come to be more and more
complicated simply by taking into consideration a
deterministic business cycle
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A new database to test a new model

• The main purpose of the panel is to represent the
  Italian firms with 20 persons employed and over
  for all industrial and services sectors
  (excluding monetary and financial intermediation)
  
• The proposed panel is a catch-up panel, not a
  perspective one.
• A catch-up panel involves the selection of a
  cross-sectional data-set from an archival source
  at some time in the past
• Then locates the units of analysis in the present
  by subsequent observation.
• The catch-up panel is a particularly attractive
  design when the researcher manages to isolate a
  source of baseline archival data that is
  especially rich in information
• On these basis the panel has drawn all the links
  between answering firms in 1998 survey with 2007
  survey respondents.
• The panel is linked with TRADE database and with
  innovation and RD databases in order to make
  available exports and innovation and RD
  variables for further researches

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The database as a subsample of available data
Ts
T
T2
T1
Balanced panel
Panel Firms (starting year)
S cross-sections available
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The primary sources of data
The panel is mainly based on cross-sectional
enterprises surveys micro-data with the
integration of administrative micro-data for
ensuring the matching of items over time and of
possible wave non respondents.
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The main characteristics of the panel
Criterion Description
Target population all firms with 20 person employed and over firms involved in MA events by enterprise included in the panel
Starting criterion for the starting year all Istat surveys respondent enterprises are considered with at least 20 persons employed all enterprises with 100 persons employed and over for which BIL source is available
Continuity criterion the key in order to identify the firm over time is the Firm Identification Code (FIC), assigned by the Istat Business Register to all enterprises
Persistence criterion an enterprises is considered present in a year other then starting year if it is an Istat surveys respondent enterprises or if it is included in the BIL database. If the number of presences is greater then 4 the enterprise is included in the panel
Integrity criterion all variables considered in the panel have to be present for all firms in all panel period
Classification variables the classification variables are enterprise size and economic activity
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Panel coverage with respect to target population
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Testing some predictions of the model

• As noticed above, the model has a number of
  testable consequences at firms level. A couple
  of them are particularly challenging
• the statistical relationship between profit and
  size is highly non linear and shows a sequence of
  peaks and lows for some (few) particular firms
  sizes, regardless to the dynamics of demand,
  prices, etc.
• coeteris paribus, the propensity to invest is
  higher when the profitability is decreasing, and
  the other way round
• Notably, the predictions (i) and (ii) hold at
  firms level, but not necessary for the
  corresponding macroeconomic aggregates
• In this paper, the hypotheses (i) and (ii) were
  tested twice by running both parametric and non
  parametric regressions on the firms level data
  of a balanced panel of Italian manufacturing
  firms with more than 20 employees, described
  above.

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Profit and employment a non parametric
estimation

• Let total employment (E) be a proxy for the size
  of the firm, and the ratio of after tax net
  profit to overall production cost (P) an
  indicator of profitability.
• By fitting local linear regressions of P on E at
  firms level, the estimates seem to confirm some
  predictions of the model, namely
• the relationship between P and E is highly non
  linear (even by running a substantial smoother,
  with a 20 bandwidth around each estimation point
  , instead of 10 suggested by the usual normal
  approximation)
• it shows a number of peaks and lows located at
  some special values of E which are quite stable
  over time, regardless to the phase of the
  business cycle (with few exceptions, mainly for
  the largest firms, and manufacturing of
  non-metallic minerals, machinery and n.e.c.)
• this evidence holds both for the manufacturing
  industry as a whole, and for every ATECO
  sub-section, as shown in the graph 3 in the
  paper.

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Investment and profit a non parametric
estimation controlling for employment

• Let the ratio of gross investment to output (I)
  be a proxy for the investment propensity.
• By fitting local linear regressions of P on I at
  firms level, the estimates seem to confirm some
  predictions of the model, namely
• the relationship between P and E is highly non
  linear (even by running a substantial smoother,
  with a 20 bandwidth around each estimation
  point, instead of 10 suggested by the usual
  normal approximation)
• generally, the of peaks P match the lows of I
  (with few exceptions, mainly for the largest
  firms, rubber and plastic, metal products,
  machinery and electrical equipment industries)
• regressing P against I (both preliminarily
  smoothed on E), shows a negative relationship
  between the two (with the noticeable exception of
  chemical industry)
• This evidence holds both for the manufacturing
  industry as a whole, and for every ATECO
  sub-section, as shown in the graph 4 in the paper
  (and the following slides)

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Profit and investment (1)
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Profit and investment (2)
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Profit and investment (3)
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Profit and investment (4)
Regression of smoothed investment on profit ()
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() Both filtered by a local linear smoothing on
employment
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A parametric approach
We tested the inverted U-shaped relationship
between profitability and the firms dimension
and growth also by using standard panel
regression techniques Here, the non-linearity is
modeled by including in the regressions also the
squared number of persons employed (add),
yielding the equation jutiit jait jb1 it
jadd jb2 it jadd 2 jeit 3 where
j denotes the range of sizes considered, i refers
to the firm, and t to the period of time If the
relationship between profitability and size can
be represented locally by an inverted U, we
should find b2 lt 0 in every size class.
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The estimation strategy
The model 3 has been estimated over the entire
period 1998-2007 and for 7 size classes, related
to 1998 firm dimension. The classes have been
identified adjusting by a search procedure those
preliminary detected by non parametric
methods. In the estimating 3 we took in
account the panel structure of the data, and run
a fixed effects (within) regression The main
results of regressions are presented in the
following table
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The main results of panel regressions
Size class b1 b2 R2 n
20 - 55 0.071499 -0.01328 0.467 1509
0.0323 0.0029
55 - 92 0.081326 -0.00958 0.352 698
0.0374 0.0534
92 -131 0.237746 -0.02306 0.300 799
0.0006 0.002
131 - 172 0.26715 -0.02567 0.515 542
lt.0001 lt.0001
172 - 629 0.126091 -0.00796 0.391 1104
0.0101 0.078
629 - 1050 0.601275 -0.0483 0.460 108
0.0958 0.0795
more than 1050 -0.1175 0.011172 0.377 110
  0.2467 0.1333    
p values are under each estimate
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The main evidence from the regressions (1)
The model fits quite well for all the size
classes, and the inverted U hypothesis cannot be
rejected for any class on a sound statistical
basis. However, for the largest firm the b2
estimate is positive, although statistically
significant. The inverted U hypothesis seems
quite robust for the firms below 172 employees,
and particularly for the classes between 92 and
172 employees. According to the estimates, the
so called middle size enterprises are grouped
in 3 clusters which are separated by 2
thresholds at 92 and 131 persons employed. This
group is acknowledged as the most dynamic part of
Italian firms. The closer thresholds detected
for this group of firm strengthens this
interpretation. In fact, according to the model,
the closer are thresholds, the larger is the
propensity to increment the productive capacity
(and possibly investment timing is more frequent).
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The main evidence from the regressions (2)
For the firms included in the class between 172
and 1050 persons employed the results of the
model appear less robust Indeed these enterprises
have to compete on broader markets and so
probably follow a different growth pattern, in
which transitory profit reduction is tolerated,
and required investment can be very large (well
beyond the ceiling 2 set by the model) The same
facts may explain why the model seems no longer
valid for the largest enterprises. In addition,
for the management of largest enterprises,
probably, it does not worth the profitability
level but it matters other specific aspects such
as, for instance, to reach dominant positions on
a market or to be included in a corporate group
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Some conclusive remarks

• The database set up for this research will be
  useful for further analyses on Italian firms.
• The model of discontinuous growth presented in
  this paper has revealed quite powerful in
  predicting a number of challenging facts
  apparently not contradicted by available
  empirical evidence on the Italian firms.
• In particular, both the non parametric and the
  parametric analysis tend to confirm that
• profitability varies non linearly with firms
  size, and show a number of peaks for few
  characteristic numbers of employees (quite stable
  over time)
• controlling for firms size, the propensity to
  invest decreases with profitability
• Of course, much more evidence is needed to test
  the model thoroughly. Nevertheless, if the
  assumed rule for investment decisions holds, a
  number of consequences follow
• firms performances expectedly diverge
  systematically from macroeconomic results
• investment could be almost insensitive to
  profitability and demand
• newborn firms play a crucial role in fulfilling
  market demand

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