Analysing the evolution of firms: discontinuity and growth

Analysing the evolution of firms: discontinuity and growth

Enrico D’Elia (Istat and MEF - DT)Leopoldo Nascia (Istat – DCSP)Alessandro Zeli (Istat – DCSP)

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

IntroductionTypically firms grow (or downsize) through an irregular sequence of (large) «leaps»

Thus, at firm’s 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:i.The heterogeneity among firms’ performances is expectedly hugeii.The frequency distribution of firms by size, profit, investment, etc. is not continuous and embodies a number of peaks and lows iii.The statistical relationships between firms’ level indicators (size, performance, etc.) are strongly non lineariv.The statistical relationships between firms’ level behavior and the corresponding macroeconomic aggregates is expectedly weakv.The size and investment of existing firms are very inertial

This paper focuses mainly on the topic iii

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SummaryA 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 firm’s 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 firm’s size reaches some characteristic values (almost stable over time), and•taking into account the firm’s size, the propensity to invest of each firm is larger when profitability is lower.

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Why firms change their size discontinuouslySome traditional explanations

Discontinuity can be induced by:i.the indivisibility of plantsii.institutional constraintsiii.sunk costs of changing (one off non proportional charges) iv.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 firm’s 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 entrepreneursa)The firm is price taker on the input, labor and output marketsb)The firm does not face financial and institutional constraintsc)Changing the firm’s size

o is risky (because investment is almost irreversible, while the market conditions are uncertain) and

o bears sunk costs (search for plants, building, etc.; hiring or firing workers; training; reshaping firm’s internal structure; etc.)

d)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 (q, qt) e)it is continuous respect to the production level (q)f)it is twice differentiable respect to qg)it reaches a unique maximum (at the production level qt) within its definition fieldh) (q1, q1) ≥ (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 firm’s 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 firm’s owner) aims at expanding the business and “keeping his/her job” achieving the minimum targets set (expected) by the shareholders. 6

The consequences of the minimum profit assumptionIf the entrepreneur expects the demand flow D, exceeding the current optimal (most profitable) output level qt, he/she may eitheri.continue to exploit the current plant, even though at less efficient conditions, orii.increase the firm’s size from qt to qt+1

According to the thumb’s rule (d), the entrepreneur prefers the strategy ii if(q*, qt) ≤ (q*, qt+1) = * [1]where (q*, qs) is the profit achieved by producing q* with a plant designed to be most profitable at qs; * is the minimum profit threshold.Let us consider only a growth process. Expanding [1] in a 2nd order Taylor series (q*, qs) near qs, and elaborating on the results, it reads

qt+1 - qt ≤ (q* - qt) [2]

where ”s = , since (qt, qt) ≤ (qt+1, qt+1) and = 0

for the assumptions on (q, qs).

"1

"

1t

t

sqqdqd

(.)2

2 ),(dq

qqd s

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The limiting growth of firmThe condition [2] sets a ceiling to the maximum size increase of firm when the entrepreneur is not disposed to reduce the profit below * 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 (q, qt+1) must cross (q, qt) at q* < qt+1, then qt+1 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 (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 (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 (qt+1 - qt ))

The profit threshold * 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 (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 t+1 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 caseThe firm adjusts its size at a regular pace over time only under very special conditions: e.g. if the demand and the minimum profit * 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 outputEven 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 investmentAlso 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 cycleThe 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 R&D databases in order to make available exports and innovation and R&D variables for further researches

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The database as a subsample of available data

T T+1 T+2 T+s

S cross-sections available

Panel Firms (starting year)

Balanced panel

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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.

The primary sources of data

SCI

PMI20+PMI20+

FINANCIAL REPORTS

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Criterion DescriptionTarget population:

all firms with 20 person employed and over + firms involved in M&A 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

The main characteristics of the panel

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Panel coverage with respect to target population

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

50.0

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

Fatturato

Valore aggiunto

Costo del lav oro

Addetti

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Testing some predictions of the modelAs noticed above, the model has a number of testable consequences at firm’s level. A couple of them are particularly challenging:

i.the statistical relationship between profit and size is highly non linear and shows a sequence of peaks and lows for some (few) particular firm’s sizes, regardless to the dynamics of demand, prices, etc.ii. 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 firm’s 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 firm’s 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 firm’s 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 employmentLet 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 firm’s 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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Food products, beverages and tobacco Textiles and textile products

Leather and leather products Wood and wood products

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Pulp, paper and paper products; publishing and printing Chemicals, chemical products and man-made fibres

Rubber and plastic products Other non-metallic mineral products

0.0%

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Basic metals and fabricated metal products Machinery and equipment n.e.c.

Electrical and optical equipment Transport equipment

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Manufacturing n.e.c. Total manufacturing

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Investment Profit

Sectors b Student's t R2

Food products, beverages and tobacco -0.060 2.47 0.028Textiles and textile products -0.539 11.52 0.462Leather and leather products -0.213 2.53 0.052Wood and wood products -0.312 3.23 0.052Pulp, paper and paper products; publishing and printing -0.196 8.60 0.140Chemicals, chemical products and man-made fibres 0.280 3.60 0.140Rubber and plastic products -0.214 4.50 0.131Other non-metallic mineral products -0.455 9.65 0.418Basic metals and fabricated metal products -1.098 7.81 0.437Machinery and equipment n.e.c. -0.253 7.81 0.368Electrical and optical equipment -0.400 3.54 0.143Transport equipment -0.353 3.99 0.144Manufacturing n.e.c. -0.194 5.92 0.158Total manufacturing -0.764 12.42 0.493

Regression of smoothed investment on profit (*)

(*) Both filtered by a local linear smoothing on employment

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 + j1 it jadd + j2 it jadd 2 + jit [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 2 < 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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Size class 1 2 R2 n20 - 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

<.0001 <.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

The main results of panel regressions

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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 2 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 robustIndeed 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 remarksThe 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 firm’s size, and show a number of peaks for few characteristic numbers of employees (quite stable over time)•controlling for firm’s size, the propensity to invest decreases with profitabilityOf 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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Documents

Analysing the evolution of firms: discontinuity and growth