Omega 41 (2013) 336–344

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Omega

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journal homepage: www.elsevier.com/locate/omega

Learning to learn and productivity growth: Evidence from a newcar-assembly plant

Carlos Saenz-Royo a, Vicente Salas-Fumas b,*

a Centro Universitario de la Defensa, Carretera Huesca s/n, 50090 Zaragoza, Spainb School of Economics and Business, University of Zaragoza, Gran Vıa, 4, 50005 Zaragoza, Spain

a r t i c l e i n f o

Article history:

Received 17 October 2011

Accepted 29 March 2012

Processed by Associate Editor Peschlearned. We compare the extended deterministic learning model with Jovanovic and Nyarkos’ [26]

stochastic learning. The theoretical models are tested with data on the total factor productivity of a

Available online 6 April 2012

Keywords:

Organizational learning

Learning by experience

Total factor productivity

Production function

Car-assembly plant

83/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.omega.2012.03.010

esponding author. Tel.: þ34 976761803.

ail address: vsalas@unizar.es (V. Salas-Fumas

chnical progress is the result of successf

c, technical and organizational discoveries, s

and innovation. Technical progress as an

is connected with the pioneering work

d productivity growth as a ‘‘residual’’ reflec

that are unknown to the researcher. Resea

ivity growth can then be viewed as the sea

a b s t r a c t

This paper models learning by experience beyond the experience curve, including the possibility of

‘‘learning to learn’’: the pace of learning increases over time by building on what has already been

car-assembly plant in its first months of operation. We find that the deterministic ‘‘mixed learning

model’’, where the speed of learning is equal to a constant plus a learning to learn effect, is the one that

best fits the empirical data. The mixed learning model results in a time pattern of total factor

productivity growth, first increasing and later decreasing, different from the always decreasing rate of

growth of the learning curve, opening new perspectives on the study of learning by experience.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Productivity, or the rate at which input quantities are turnedinto outputs, has received much attention at the macro (asdeterminant of differences in the per capita income of countries[43]), at the firm (as explanatory of differences in competitive-ness and profitability of firms [6]), and at the operational level(explaining differences in efficiency and costs across produc-tion units [35]). The bulk of recent productivity research hasconcentrated on explaining the observed differences in produc-tivity levels across firms within and between industries andcountries (Syverson [42], for a review). Much less is known,however, on what determines the time path of productivity foran individual production unit, even though macro productivitygrowth comes from the aggregation of efficiency gains at theproduction unit level.

There are two primary explanations of productivity gains at themicro level. One considers productivity growth as the consequenceof a general time trend of technological progress that continuouslyexpands output at rates faster than the growth in inputs.1 The

ll rights reserved.

).

ul practical applications of

ustained by research, devel-

explanation of productivity

of Solow [40,41] who first

ting the causes behind this

rch on productivity and on

rch for explanations of why

other explanation is rooted in the learning curve, where the rate ofproductivity growth is positive but decreases over time (Zangwilland Kandor [48], for a formal generalization of the learning curve).The first explanation implicitly assumes ‘‘Schumpeterian’’ innova-tion, constantly reinventing the production technology as well asthe products and services sold in the market. The second takes theproduction technology and product attributes as constant, seeingproductivity gains as the result of continuous and gradual improve-ments in the way things are done in the production process.

This paper advances the explanation of productivity growth,proposing a general model of learning by experience. The modelincludes unlimited technical progress and the learning curve asparticular cases, but it covers two additional forms of ‘‘determinis-tic’’ learning. The model is formulated at the level of an operatingunit, i.e. a production plant, and it is applied to the learning processunderlying the observed productivity growth in a car-assemblyplant in the first years of operations. The measure of productivityused in the analysis is invariant to the intensity of capital and laborinputs used in production, i.e. it captures the total factor productiv-ity (TFP) of the plant. We do not observe the specific actions takenby managers and workers to improve the efficiency of the produc-tion process; rather, we postulate a relationship between theunderlying process of discovery and application of better waysof doing things, and the observed track of improvement in termsof measured TFP. In addition, our paper includes a comparison

(footnote continued)

there are differences in the levels of output produced with similar combinations of

inputs, such as labor and capital [42].

C. Saenz-Royo, V. Salas-Fumas / Omega 41 (2013) 336–344 337

between the proposed class of deterministic models with the‘‘stochastic’’ learning model of Jovanovic and Nyarko [26].

We find that the learning model that best fits the empirical datais what we call mixed learning, i.e., workers and managers of theplant combine a fixed rate of learning with a ‘‘learning to learn’’capability as more knowledge is acquired. The mixed learning modelimplies that the results of learning translate into a first period ofaccelerated productivity growth, followed by another period ofdecelerated growth, until the maximum level of operating efficiencyis attained. This is precisely what we observe in the data. Otherlearning models, such as the exponential version of the learningcurve, and Jovanovic and Nyarko’s [26] stochastic learning model, donot capture the S shape in the evolution of TFP over time. Althoughthe evidence is obtained from a single plant, the results of the papersuggest that existing explanations of TFP growth, such as general-ized technical progress and the learning curve, are incomplete, andother forms of deterministic learning such as learning to learn ormixed learning should also be considered.

As for the relation of the paper to the existing literature, thetheory section of the paper is in line with Zangwill and Kandor[48], who model the process of continuous improvement compa-tible with the learning curve [4,47] as evidence of such improve-ment. Our paper is different in that we model the process oflearning without limiting the results of the process to thosecompatible with the evidence of the learning curve. In fact, asmentioned earlier, the pattern of performance improvement ofthe learning curve is one of four possible results in the class of‘‘deterministic’’ learning models.

The learning curve, and in general, learning by doing, has beenapplied to units of varying complexity, from single machines (espe-cially scheduling problems, [9,24,45]) to plants [1,5,39] and firms[46,31,32,6]. The performance measures considered in prior researchinclude cost [47,48,39], productivity [1,6,21], and quality [15], as wellas complex measures such as overall equipment effectiveness [46].

This paper is unique in that the learning unit is a start-upassembly plant, enabling us to study learning at the moment intime when it can be expected to be particularly important. Theplant produces a homogeneous output and we have monthly dataon the number of cars assembled. The time interval betweenmeasurements of performance is short and the effects of learningon performance are observed shortly after management decisions,spurred by what has been learned, are implemented. The monthlyfrequency of observations assures sufficient observations toestimate the learning model for a total time period when thecar model assembled in the plant remained unchanged, as well asthe main parameters of the production function different from theTFP parameter. The parameters of the production function areestimated jointly with those that capture the features of thelearning process, using the Error Correction Mechanisms [17],which is another innovation of the paper. TFP has been usedbefore as an indicator of the results of the learning process at thefirm and industry level, but not often, if at all, at the plant level.

The rest of the paper is organized as follows. Section 2 presentsa description of the theories of learning and their respectiveanalytical formulations for empirical estimation purposes. Section3 contains the application of the theory to the case study of theassembly plant. Finally, in Section 4, we present the discussion ofour results and the main conclusions of our paper.

2 See for example: Osterman [36], Ichniowski and Shaw [22], Cappelli et al.

[11].3 See the papers of Adler and Clark [1], Jones and Kato [25], Ichniowski and

Shaw [22], Ichniowski, Shaw and Pernushi [23], Cappelli and Neumark [12], Kato

and Morishima [28], Lieberman and Dhawan [31], Ben-Ner and Lluis [8].

2. Learning theories and proposed models

2.1. A brief review of the literature

There are three main learning mechanisms identified in theextensive literature on this topic [2]. In one, individuals and

groups learn from their own experience, refining the procedurespreviously set as the most effective way to perform the assignedtasks. This mechanism is known as learning-by-experience [4,47,3].In another mechanism, individuals combine repetition with trialand error experimentation, intending to extract informationabout individual or group capabilities and about their absorptivecapacity. This is defined as matching [34,29,26]. Finally, indivi-duals learn from observation of the behavior and performanceof others, or social learning [20,33]. The learning by experiencemechanism is modeled as a deterministic process, while thematching mechanism has stochastic properties. In deterministiclearning, there is an optimal way to perform the tasks, known byall collaborating agents, although internal forces condition thepace at which individuals and groups converge towards theoptimal solution. When learning takes place in a stochasticenvironment, individuals do not exactly know the best way toperform an activity, since the observed outcomes from such a bestway must be progressively inferred from a noisy signal.

Organizational learning and its translation into higher perfor-mance of firms, i.e. cost, productivity, quality, profits, and the like,has been extensively studied in the operations, management andeconomics literature. The research strategies vary. At the highestlevel, researchers see organizational learning as an endogenousresource depending on factors such as the absorptive capacity ofthe organization [14,19], the organization’s culture [38], and thetransfer mechanisms from one part of the organization to theother [13]. At one level below, research on learning looks at theentry to the market of new ways of organizing work and ofmanaging those involved, and investigates the rate at whichinnovations are adopted and diffused among firms, in one orseveral industries.2 In this vein, certain studies go one step furtherand investigate the link between the adoption of new forms ofwork organization and human resource management practices,and the observed operating and financial performance of firms.3

Another line of research focuses on the measurement of firmperformance over time (productivity growth, for example) andexplains performance as a function of some modeled learningprocess. Most of the research on the learning curve referencedabove follows this approach, and it is our methodology in thispaper, with TFP being the performance measure used to trackimprovements.

2.2. Deterministic learning models

Let Q ðtÞ ¼ AðtÞFðK ,L; tÞ be the production function that sum-marizes a state of knowledge, at a moment of time, of theproduction of a good or service, in our case the assembly of cars.Q(t) is the output flow in period t; K is the level of capital inputservices; L is the level of labor input services; t refers to the timeperiod, and A(t) is the total factor productivity parameter mea-suring the level of operating efficiency in period t. The learningmodels considered explain the time evolution of the TFP term A(t)for cases where the other parameters in FðK ,L; tÞ remain constantover time (the time variable t applies only to input quantities).This is a standard assumption in studies that explain TFT as aresult of learning by experience, where it is additionally assumedthat the parameters inFðK ,L; tÞ are the same for all plants andfirms (see Balasubramanian and Lieberman [6] for example). Webelieve that the assumption is appropriate for our empiricalanalysis, since our data come from only one production unit,

C. Saenz-Royo, V. Salas-Fumas / Omega 41 (2013) 336–344338

and during several months when the plant produced the same carmodel with an invariant technology, plant layout and organiza-tional structure.

The actual functional form or time dynamics for A(t) variesacross different studies. For example, the hypothesis of perma-nent and constant growth implies A(t)¼A(0)egt, where g is theconstant growth rate. A popular functional form used to representthe deterministic learning curve in the social sciences [18,37] isdAðtÞ=dt¼ b A*�AðtÞ

� �for A(0) known and where b and A* are

constant. This function captures the scope of learning A*/A(0)(ratio between maximum and minimum TFP) and the pace oflearning b.

We explore more flexible forms of representing the learningprocess and propose the following equation on the time motion ofthe observed performance of such a process, to be validated bythe empirical evidence:

dAðtÞ

dt¼ bþaAðtÞ

A*

� �ðA*�AðtÞÞ ð1Þ

The left-hand side of the equation is the absolute change inTFP in period t. The right-hand side has two terms. One, (A*

�A(t)),is the difference between the maximum level of operatingefficiency of the process or target of the learning process, A*,and the current level of operating efficiency that results from pastlearning, A(t). Thus, the difference (A*

�A(t)) is the stock ofknowledge still pending incorporation into the production pro-cess, in period t. When A* is known, the learning process belongsto the class of deterministic learning models. Deterministiclearning is realistic in our empirical analysis since the plantbelongs to a leading global car manufacturer with many decadesof experience and therefore with extensive knowledge of effi-ciency standards for assembly plants with a given technology.

The other term on the right-hand side of (1), bþa AðtÞA*

� �, is the

speed of learning, defined as the proportion of the pending stockknowledge that is learned in period t. The parameters b and a, arenon-negative and, depending on their values, the learning processtakes one form or another as follows.

2.2.1. Constant learning

One version of model (1) is what we call the constant learningprocess, which corresponds to the parameter values: b40 anda¼0. The speed of learning will be constant and equal to b. In thissituation, the organizational learning capability, the proportion ofknowledge still pending in period t, is exogenously given andremains constant throughout the whole learning process. Thegeneral model includes other models of learning by experience,referenced above, as a particular case.

4 The constant learning model contains the learning curve model as a special

case. Zangwill and Kandor [48] derive three mathematical formulations of the

learning curve from modeling a continuous improvement process where man-

agers take an action, observe the results and learn how to improve the process in a

new round. Each formulation expresses the observable outcome from continuous

learning as, respectively, a potential, exponential and ‘‘finite’’ function of cumu-

lative production. The potential functional form is the most extended in the

literature (where cost of n-esima unit produced is a log linear function of n-esima

unit of cumulative output), but the exponential function is also used in certain

other papers, for example Wang and Lee [46]. Our model of constant speed of

learning expresses TFP over time as an exponential function of time; if production

per unit of time remains relatively constant over time, then the cumulative output

is the production per period times the number of time periods, so in this case, the

exponential function of time will coincide, except for a constant, with the

exponential learning curve.

2.2.2. Learning to learn

Another version of model (1), unexplored in the literature,corresponds to the parameter values: b¼0 and a40. The speed oflearning is now equal to aðAðtÞ=A*Þ and it varies over time as afunction of the ratio between accumulated knowledge up to t,A(t), and total potential knowledge A*. The speed of learning at agiven moment of time is an increasing function of the learningaccumulated up to t, relative to the total potential cumulativelearning. Since the speed at which the organization learns today isa function of past cumulative learning, we interpret this result asevidence that the learning capacity of the organization increasesover time with the accumulated relative knowledge. We definethis learning model as learning to learn. Notice that what deter-mines the speed of learning is not the absolute value of theknowledge acquired in the past, but this value relative to themaximum possible knowledge that can be acquired.

2.2.3. Learning to learn with unlimited knowledge model

A special case of interest in the learning to learn model is whenA* tends to infinity. Under this additional assumption, togetherwith b¼0, Eq. (1) is written as dAðtÞ

dt ¼ aAðtÞ. The learning to learnprocess, together with the assumption that the stock of knowl-edge is unlimited, implies that TFP will increase over time at aconstant rate equal to a.

2.2.4. Mixed learning model

The general formulation of model (1), where the two para-meters b and a are positive, can then be described as a mixed

learning model, combining an exogenously given, constant, orga-nizational learning capacity, together with acquired learningcapabilities from knowledge already accumulated. The relativeimportance of each component of the process in a particularproduction environment is an empirical question.

Technically, Eq. (1) and the variations under different values ofthe learning parameters are differential equations that can besolved for the function giving the time evolution of the level ofTFP, A(t). Table 1 shows a summary of the functional forms of A(t)obtained by solving the respective differential equations, as wellas the graphical representation of the respective function A(t).

The process of constant learning implies a time profile of thelevel of TFP increasing, but at a decreasing rate (A(t) is an increasingand concave function of time t).4 The case of learning to learnimplies an always-increasing time profile of TFP too, but now therate of increase is first increasing and later decreasing (A(t) is alogistic function of time with the familiar S shape). In the particularcase of learning to learn with unlimited bound of knowledge, thelevel of TFP is an increasing and convex function of time (the slope isincreasing with time). This implies also that the log of A(t) will be alinear function of time. Finally, the mixed learning model gives aprofile of the level of TFP over time that is first convex and, aftersome inflexion point, becomes concave (S-shaped functional form).The profile is then similar to that resulting from learning to learn,but the functional form resulting from the mixed learning model ismore flexible, having two parameters, b and a.

2.3. Stochastic model

In the stochastic learning models, individuals involved in pro-duction (managers, technicians, workersy) do not know the stan-dards of full efficiency of the production process in which theyparticipate, i.e. A* is unknown, and perform experiments to discoverthem. The model of Jovanovic and Nyarko [26] is one of the mostgeneral formulations of learning by experimentation in an environ-ment of uncertainty and noise. The model assumes that managersand workers have an a priori estimate of the most efficient way toperform a task, and update this estimate from the informationcontained in the result observed after each trial. The process oflearning, updating knowledge, is then tied to the number of trials,

Table 1Solutions to the differential equations resulting from combinations of the values of the parameters in Eq. (1).

Learning models Parameters Flow of learningdAðtÞ

dt

Cumulative learningAðtÞ

Graphical representation ofAðtÞ

Constant learningb40

dAðtÞ

dt¼ bðAn

�AðtÞÞ AðtÞ ¼ An�ðAn�Að0ÞÞe�bta¼0

An finite

Learning to learn

b¼0dAðtÞ

dt¼ aAðtÞ

AnðAn�AðtÞÞ AðtÞ ¼

An

1þbe�atwhere b¼ An

�A 0ð Þ�

=A 0ð Þa40

A* finite

Unlimited learningb¼0

dAðtÞ

dt¼ aAðtÞ AðtÞ ¼ Að0Þeata40

An infinite

Mixed learning

b40dAðtÞ

dt¼ bþaAðtÞ

An

� �ðAn�AðtÞÞ AðtÞ ¼ An 1�ðb=aÞYe�ðbþaÞt

1þYe�ðbþaÞtwhere Y¼

An�Að0Þ

ðb=aÞAnþAð0Þ

a40

A* finite

C. Saenz-Royo, V. Salas-Fumas / Omega 41 (2013) 336–344 339

repeated experience. The number of trials can be converted intotime-related experience by modifying the model with a parameterthat converts trials to number of time periods. This parameter willbe process-specific, since the number of times a task is repeated perunit of time will vary across products and technologies.

The model is formalized as follows. Consider a process oractivity with N tasks where there is a maximum efficiency levelfor each task a*

j , for j¼1,..,N. The current level of efficiency for taskj is given by aðtÞj and the total level of efficiency for the processwith N tasks is equal to

AðtÞ ¼ APð1�ða*j�aðtÞjÞ

2Þ,j¼ 1,:::,N ð2Þ

When N¼1 the activity or process is assumed to be simple andfor higher values of N the process is described as complex

(complexity increases with N).The level of maximum efficiency for activity j is not known for

certain; what is known is that the value is drawn from a jointprobability distribution with unknown mean yj and known variances2yj. In each time period, there is a realization of the process in terms

of observed level of efficiency, a(t)j resulting from the application toproduction of existing knowledge. These observed values will be therealizations of the independent and identically distributed randomvariables aðtÞj ¼ yjþojt where ojt are independent normally distrib-uted random variables with mean zero and variance s2

oj. The variableojt is a noise that distorts the information of the signal of the truevalue of the mean of the distribution, yj.

The application of a Bayesian revision mechanism updatingthe current knowledge of the efficiency of the system from the

information content of the observed realizations on the perfor-mance variable of the process, gives the following result [26]:

AðtÞ ¼ Að1�XðtÞ�s2oÞ

Nð3Þ

where XðtÞ ¼ ð1=s2yþmtÞ�1 and m is the parameter converting the

number of trials to the equivalent number of time periods.When time t tends to infinity, a large number of trials, then

X(t) tends to zero and the limit of the maximum efficiency will bereached:

A*¼ Að1�s2oÞ

Nð4Þ

Dividing (3) by (4), the relative efficiency of the process inperiod t is equal to

AðtÞ=A*¼ 1�ð1=s2

yþmtÞ�1

1�s2$

" #N

ð5Þ

The Jovanovic and Nyarko [26] theory of learning implies that thelevel of relative efficiency will increase over time with the number oftrials, which give noisy information signals about the unknownparameter of the system. The observed value of the relative efficiencyat a given moment of time will decrease with the noise of the signals(s2

o), with the uncertainty around the mean best practice of theprocesses (s2

y), and with the complexity of the activity-process (N).However, it increases with the parameter that converts trials intotime period effects (m). Fig. 1 represents the shape of the function in(5) for possible scenarios obtained combining high and low values ofthe parameters capturing the environmental noise of the process, andthe uncertainty about the parameter determining the efficiency of the

Fig. 1. Learning rate with environmental noise s2o¼0.2, system complexity N¼10

and changing values of process uncertainty (s2y).

C. Saenz-Royo, V. Salas-Fumas / Omega 41 (2013) 336–344340

process. In all cases, it is assumed there is one trial per unit of time(m¼1).

In theory, the shape of the process performance variable fromthis learning model can vary; for example, an increasing andconcave function (when environmental noise is low), or a firstincreasing convex and later decreasing concave function (whenenvironmental noise is low and complexity N is high).

3. Empirical application: operational learning in anautomobile assembly plant in the early months of operation

In this section, we estimate the time evolution of the TFP of acar-assembly plant in the first five years of operation, and weexplain this evolution as the result of the learning processesmodeled above. The plant belongs to a multi-national car manu-facturer and began assembling cars regularly in 1984. Our interestis in responding to the question of what is the model of learning –deterministic or stochastic, exogenous learning, learning to learn,or mixed learning – that best explains the empirical evidence fromthis particular plant. We respond to the question by comparing theestimated values of the plant’s TFP over time with the predictedoutcomes from each of the proposed learning models.

3.1. Functional form of the production function and data

The TFP parameter of the production technology in period t,A(t), is calculated as the ratio between the observed number ofautomobiles assembled in period t, Q(t), and the number pre-dicted by the function FðK ,L; tÞ expressing the number of auto-mobiles expected to be assembled in period t, given theparameters of the technology of the assembly plant when built,and given the quantity of capital, K, and labor, L, used inproduction in period t. Therefore A(t)¼Q(t)/FðK ,L; tÞ . The actualfunctional form of F( ) proposed in the paper is of the Cobb–Douglas type, FðK ,L; tÞ ¼ KðtÞgLðtÞ1�g. The parameter g, elasticity ofoutput to capital, will be empirically estimated with data oninputs and output of the plant. We estimate the parameters of therespective learning model in the functional form for A(t).

We perform a preliminary statistical analysis of the variablesmonthly number of assembled cars, nd capital and labor inputs,before deciding the econometric estimation procedure used toobtain the values of the parameters of model. The results showthat these variables are unit root non-stationary and co-inte-grated, so the Error Correction Mechanism (ECM), Engle andGranger [17], is adequate for the estimation. The ECM impliesestimating both long-term and short-term models of the eco-nomic process, in our case the production function of thecar-assembly plant. The basic model is completed with other

explanatory control variables capturing observed alterations inthe production process over time, including strikes, holidays,vacations and additional shifts. The complete formulation of themodel for one-step ECM estimation (the long-term and short-term relationships are estimated simultaneously) is as follows:

Dqt ¼C0þC1 qt�1�aðt�1Þ�gkt�1�ð1�gÞlt�1

�þdiþ1Dqt�i

þX12

i ¼ 0

diþ14Dkt�iþX12

i ¼ 0

diþ27Dlt�iþX3

i ¼ 1

xiDHitþX7

i ¼ 1

xiþ3DVit

þx11DTtþUt ð6Þ

The low case letters, q, k and l are, respectively, number ofautomobiles assembled Q, stock of capital, K, and number ofemployees, L, all expressed in logs, and t is the time period inmonths. D is a difference operator so Dqt indicates the differencebetween qt and qt�1. Since qt is the number of cars (output)assembled in month t, then Dqt is the rate of output growth inmonth t. The parameter g is the elasticity of output to capitalinput, to be estimated. The term a(t) is the log of TFP value inmonth t, a(t)¼LnA(t). The variables DH, DV and DT are dummyvariables controlling for alterations of the normal activity of theassembly plant: strikes (DH), different working days of the month(DV), and changes in the number of production shifts (DT). Finally,Ut is the error term, which is assumed to be white noise.

We now explain each of the main terms in Eq. (6). The terma(t�1)þg kt�1þ(1�g) lt�1 is the long-term relationship betweenthe inputs and output of production, with the variables in logs. Thedifference {qt�1�a(t�1)�g kt�1�(1�g) lt�1} is the deviationbetween the current level of production in logs, qt�1, and the valuepredicted by the long-term model. The parameter c1is the speed atwhich observed and predicted output converge and must have anegative sign for co-integrated variables. The term

P12i ¼ 0 diþ1Dqt�iþP12

i ¼ 0 diþ14Dkt�iþP12

i ¼ 0 diþ27Dlt�I captures the short-term rela-tionship among input and output values obtained from non-antici-pated changes in the respective variables, changes in productionplans, changes in labor and capital input quantities, and so on. Theestimation allows for a lag structure of 12 months and diþT areparameters to be estimated. Finally

P3i ¼ 1 xiDHitþ

P7i ¼ 1 xiþ3DVitþ

x11 DTt are perturbations of the production process anticipated by theplant managers and include: strikes (DH), different working days ofthe month (DV), and changes in the number of production shifts (DT).The coefficients xiþM are parameters to be estimated.

We use monthly data on output and inputs for the period fromSeptember 1984 to March 1992 (91 observations). During thisperiod, the plant was assembling the same basic automobilemodel so the time series finishes when the plant lay-out,machinery and organization of work were adapted to a newmodel to be assembled. Our analysis intends to measure thelearning of the plant in the time period when it produced the firstcar model of its operating life. The output data are number of carshomogenized, so that the output of each and all time periodsin the sample is in physical units and homogeneous over time.The inputs are equivalent full-time employees working during themonth, and the stock of capital is equal to net plant andequipment at constant 1983 prices at the beginning of month t.

Table 2 shows certain descriptive statistics of the data. Thecumulative annual growth rate in number of equivalent full-timeemployees and stock of capital at constant prices is very similar,around 1.5%, and smaller than the cumulative annual growthrate in output (number of cars assembled), 5.9%. The differencebetween output and inputs growth rate, 4.4%, gives a first estimateof the magnitude of productivity growth resulting from learning.Notice that annual growth rates show relatively high volatility(standard deviations of growth rates between 4.3% and 7.8%). Ourinterest in the detailed empirical analysis is to model the pattern ofgrowth rates over time from the different learning theories.

Table 2Descriptive statistics of inputs and outputs of the assembly plant.

Source: Data collected from company files.

Labor Capital Output

Initial value 8,383 86,249 21,655

Final value 9,255 95,225 31,761

Cumulative annual growth rate (%) 1.52% 1.51% 5.85%

Standard deviation annual growth rate (%) 4.30% 7.80% 7.75%

Table 3Results from the estimation of Eq. (6) under different assumptions on the dynamics of TFP foreach particular learning modelþ .

Unlimited learning (I) Mixed learning (II)1 Jovanovic–Nyarko (III)a,b

Speed of adjustment (c1) �0.17536**(0.07086) �0.18432**(0.07134) �0.14510***(0.06477)

g(kt�1) 0.39316***(0,02472) 0.38665***(0.01067) 0.38772***(0.01375)

y 0.00454***(0.00108)

a 0.00685***(0.00128)

b 0.03021***(0.00620)

so2 0.17270***(0.06471)

sy2 0.04117***(0.00543)

RA2 0.84082 0.8639 0.84001

Observations 81 81 81

þValues of the estimated coefficients of the lagged and dummy variables are not reported (see appendix).a Estimated values A*

¼1.53025 A0¼1.04558.b Default parameters N¼10 m¼1.

C. Saenz-Royo, V. Salas-Fumas / Omega 41 (2013) 336–344 341

3.2. Estimation of the parameters of the learning models

3.2.1. Deterministic learning

We estimate the parameters of Eq. (6) for the case when A(t) isthe result of the more general mixed learning model. FromTable 1, mixed learning implies a functional form for A(t):

AðtÞ ¼ A*1�

baYe�ðbþaÞt

1þYe�ðbþaÞtð7Þ

Y¼A*�Að0Þ

ba A*þAð0Þ

ð8Þ

Since the function Ln A(t) in (6) is highly non-linear, theestimation of the parameters will follow an iterative process.The first iteration considers the particular case of constant growthrate in total factor productivity, Ln A(t)¼a(t)¼yt, where y is theparameter of the unlimited learning model. Since the models ofconstant productivity growth are common in the literature, wepresent the estimations of the first iteration as the results of aparticular learning model. The results of estimating model (6)with a(t�1)¼y(t�1) appear in the first column of Table 3. Theestimated values of the main parameters of the technology, theelasticity of output to capital, and the estimate of averageproductivity growth, are, respectively, g¼0.393 (1�g¼**0.603)and y¼0.00454, both statistically different from zero. The esti-mated value of parameter c1 is negative and statistically sig-nificant, as expected for co-integrated variables; the estimatedvalue of the speed of adjustment is 0.17536, which implies that,on average each month, the system adjusts for 17.5% of thedifference in its long-term production value; and thus the gapwould close in less than 6 months.

Fig. 2I shows the graphical representation of the predicted andobserved values of TFP for each month in the estimation period.

The estimated value of TFP in month t is At ¼Qt

Kgt L1�g

t ebD, where Qt is

the actual number of cars assembled in month t, and the denomi-nator is the number of cars that would be produced as predicted

from the model, excluding the constant time trend, yt. Notice that

the linear time trend in productivity implicit in the constant

growth rate does not match well with the S-shape of the actual

values of At obtained from the empirical data. This justifies moving

to step two in the estimation of the mixed learning model.In step two, the residual values of At obtained from the assump-

tion of constant productivity growth are the dependent variable ofEq. (7) that, together with (8), summarize the TFP time-trendaccording to the mixed learning model. We use non-linear regres-sion estimation procedures to fit function (7) to the data, and obtain

estimates of the parameters b and a. In the estimation, we take asvalues of A0 and A* the lowest and the highest values of At,respectively. Having estimated the parameters, we use (7) and (8)to predict the values of At. These values, in logs, are substituted fora(t�1) in Eq. (6) and the equation is estimated again for new valuesof g and the rest of the parameters. The process is repeated: use thenew estimated parameters of the production function to obtain new

values of At in At ¼Qt

Kgt L1�g

t ebD; re-estimate the values of b and a fitting

(7) with (8) to these values of At, predict new values of At andsubstitute them in (6) for a new estimation of g and the rest of theparametersy The estimation process ends when we reach conver-gence in the values of the parameters.

The values of the estimated parameters from this iterativeprocess, and other information about their statistical significance,are shown in the second column of Table 3, estimation II. Theestimated value of the speed of adjustment parameter is 0.18432and the estimated elasticity of output with respect to capital(labor) is g¼0.387 (1�g¼0.613). These values are very similar tothose obtained in estimation I. The estimated parameters of thelearning function are b¼0.00686 and a¼0.0302. All the para-meters’ estimates are statistically different from zero.

Since the two parameters, b and a, are different from zero, thehypothesis that learning in the assembly plant was driven by adeterministic mixed learning process is not rejected by the data.The speed of learning is then given by the function 0:00686þ

0:0302 At

A* .Thus, the speed of learning goes from a low value of

0.00686 in the first time period, to a high value of 0.0371 in thelast, when the system reaches maximum productive efficiency.

Fig. 2. Estimated and predicted values of At for the three learning models in

Table 3. I—Unlimited learning, II—mixed learning and III—Jovanovic and Nyarko.

C. Saenz-Royo, V. Salas-Fumas / Omega 41 (2013) 336–344342

The second term of the speed of learning, the component oflearning to learn, contributes substantially more (three timesmore on average) than the constant learning parameter.

Fig. 2II shows the values of At in the last iteration, and thepredicted values of A(t) from Eqs. (7) and (8) for the parameterestimates of b and a in estimation II of Table 3. The figuredescribes the S shape of the data points and of the fitted function.The inflexion point of this function occurs at the moment in timewhen the derivative of the function (7) is maximized. Performingthe calculation, we obtain that the derivative is maximized inDecember 1987, after 40 months of operation, and the value ofthe derivative at t¼40 gives a maximum growth rate of 0.45%.

3.2.2. Jovanovic and Nyarko’s model

The progress in TFP is now represented by Eq. (5). Thisfunction gives the evolution of operating efficiency of the plantover time with four parameters: the noise of the signal, s2

o, theuncertainty around the mean best practice of the processes,s2y , the complexity of the activity-process N, and the parameter

that converts trials into time period effects, m. The functional formof the TFP is highly non-linear and the estimation strategy

requires that certain parameters take values fixed in advance.We then fix the value of the parameter m¼1, indicating one trialper month; that is to say, the plant managers and workersobserve the outcome of the production process and learn fromit once a month, which is a realistic hypothesis. Additionally, theestimation procedure will include two iterative processes. One isthe same as was used earlier to estimate the parameters of theTFP function in the case of deterministic mixed learning, and thesecond involves the estimation of the parameters s2

o and s2y for

exogenously given values of the complexity parameter N.

The estimation then proceeds as follows: we first fix a value ofN and estimate the parameters s2

o and s2y , together with the rest

of the parameters of the production function, substituting a(t�1)in (6) for the Eq. (5) in logs. The estimation at this stage isperformed in a similar manner as in the case of mixed learning.Next, we change the value of N and repeat the estimation of theparameters of the production and learning functions. The processends when convergence is reached in the R2 of the regression andin the parameter values.

The results of the estimation are shown in column III ofTable 3. The convergence and bet fit is obtained for N¼10. Forthis value, the estimated elasticity of output with respect tocapital is g¼0.388, statistically different from zero, and very closeto the value estimated in the other two learning models. Theestimated speed of adjustment is now slightly less than in the twodeterministic models. All the estimated coefficients for thedummy variables that capture exogenous shocks are also verysimilar to those obtained in the other estimations. The estimatedvalues of the other two parameters of the learning model ares2o¼0.173 ands2

y¼0.041, both statistically different from zero.According to these estimates, the uncertainty around the optimalvalue of the parameter that drives the efficiency of the processs2y is relatively lower than the noise of the signals s2

o.The R2 of the fitted econometric model continues to be above

80%, but a closer look at the plot of the estimated values of theTFP parameter over time, and the plot of the predicted valuesfrom the learning model, reveals that the latter does not capturethe S shape of the former (Fig. 2III). The best estimate of theJovanovic and Nyarko learning model from the data implies atime evolution of the value of TFP increasing, but at a decreasingrate. However, the actual time pattern of the TFP parameter At

shows a time trend with first increasing and later decreasinggrowth rates (S-shaped pattern).

The comparison of the results from the estimations of the threelearning models, shown in Table 3, confirms the similarities in theestimated values of the parameters of the production technology, thevalue of elasticity g around 0.387, and the estimation of the coeffi-cients of the time dummy variables (see Appendix 1). The overall fitof the model, R2, is also similar. The main differences appear in theestimation of the time pattern in the evolution of operating efficiencymeasured by the value of the TFP parameter at each moment of time.The model that captures the S shape of the actual data is the mixedlearning model, combining exogenous learning and capabilities forlearning to learn. Under mixed learning, the speed of learningincreases over time until it reaches its maximum value, whenoperating efficiency also reaches its maximum value, and nothingis left to be learned. Even though the speed of learning at this pointof the learning process is the maximum, the total contribution oflearning to TFP growth tends to zero, since when the target ofefficiency A* is reached, there is nothing left to learn.

4. Discussion and conclusion

Productivity growth (increasing output produced, at a higherrate than the rate of increase in the quantity of inputs used in

C. Saenz-Royo, V. Salas-Fumas / Omega 41 (2013) 336–344 343

production) is viewed as the main driver of economic prosperity,so there is much ongoing research seeking a better understandingof the sources of productivity growth over time. Solow’s [41]ground-breaking paper, pointing to evidence that the aggre-gate output of an economy tends to grow at a higher rate thangrowth in aggregated inputs, identified the difference betweenthe two growth rates as a ‘‘residual’’. Gaining a better under-standing of this residual, and of the factors that lead to sustain-able growth, has been the concern of a great deal of research sincethe 1950s.

This paper studies productivity and productivity growth in amore limited context than that of the whole economy, namely ina car-assembly plant in its first years of operation. During thetime period of the analysis, September 1984 to March 1992,the production technology remained invariant and the plantassembled the same basic model. In this highly controlled andstable production environment, it is most likely that any progressin operating efficiency and productivity will be the result oflearning by employees, and plant managers, the best ways ofdoing things within the constraint imposed by the fixed macrotechnology. Therefore, by studying the time pattern of operatingefficiency in terms of the TFP observed in the plant each month,we are able to discover which of the learning models proposed inthe literature is best represented by the empirical evidence.

Our results indicate that a deterministic learning model, wherethe pace of learning combines an exogenous constant term and atime-increasing term as a function of relative cumulative pastlearning, is the model best represented by the empirical data. Thismixed learning implies that TFP will increase over time, first at anincreasing rate and, after a certain point in time, at a decreasingrate, so the cumulative value of the TFP parameter is S-shaped overtime. The estimated stochastic learning model of Jovanovic andNyarko, fitted to the same data, did not capture the S-shaped timeprofile of TFP observed in the data. One tentative explanation ofwhy deterministic learning is closer to the empirical evidence thanthe stochastic mode is that the assembly plant in this studybelongs to a large and experienced world-wide car manufacturer,and the engineers and managers who designed the plant probablywere quite conversant with the high standards of performance thatwould result from operational learning. Notice also that theprocess of deterministic learning does not mean that there willbe no noise around the TFP trend over time; what it does imply isthat such noise will not be part of the learning process itself, as it isin the case of the stochastic learning models. Thus, the results ofour study do have certain practical implications.

First, our findings are consistent with the hypothesis thatknowledge acquisition is a gradual process, so that even whenworkers and managers know the standard of maximum efficiency,that standard may take some time to be attained. Moreover,under production conditions of stable technology and standardproduct characteristics it can be expected that the potentialknowledge acquired by some form of learning by experience willbe finite, with counterpart value of efficiency level A* in ouranalysis. In the particular case of this car-assembly plant, it tookbetween six and seven years to reach the value of A*, which wasaround 50% higher than the level of operating efficiency at thebeginning of the period. In fact, in January of 1992, coincidingwith the end of the complete learning cycle, the management ofthe company began to introduce major changes in the assemblyplant, including the use of more robots in place of workers, as wellas changes in the model being assembled. Therefore, managersappear to have been aware of the stagnation of productivitygrowth from operational learning and, once the limit wasreached, they introduced major changes in the technology, andin the product itself, among other things by beginning a new cycleof learning and productivity growth.

Second, the 50% increase in cumulative TFP does not occur atuniform growth rates over time. Rather, the growth rates varysubstantially, first accelerating and later decelerating. In ourassembly plant, the maximum monthly growth rate in TFP isestimated to occur in month 40, just in the middle of the period ofobservation, and its value was 0.45%. We reject perpetual orunlimited learning (constant growth rate in TFP for unlimitedtime) as a good descriptor of the variable that tracks the results oflearning and improving; since most studies of productivitygrowth with macro data assume constant growth rates in TFPover time, the micro results cast doubt on the macro analysisbeing able to capture what happens at the micro level. Theempirical results also reject the hypothesis of a constant learningrate as the best descriptor of the time evolution of TFP. Since theconstant learning rate is a special case of the widely-used learningcurve, our results suggest that mixed learning should be consid-ered as a plausible alternative to the learning curve.

Third, the econometric methodology used in the estimationof the parameters of the model (ECM) distinguishes between‘‘the speed of learning’’ and ‘‘the speed of adjustment’’ in theworking of the assembly plant. The former is a measure of thepace at which knowledge is acquired, while the speed of adjust-ment is a measure of how fast what is learned is transformedinto higher productive efficiency. The empirical results indicatethat, in this plant, it took between five and six months to adjustfrom current to target production results (speed of adjustmentof 0.183).

Future research should replicate the empirical test of thelearning models derived in this paper with data from otherproduction plants or production processes, including learning inmachine-scheduling activities and any possible process-improvingenvironment. In this respect, it will be important to verify to whatextent the mixed learning model (implying an S-shaped time paceof the performance indicator resulting from learning and improv-ing) is limited to start-up operations, or not. Future researchwith data from multiple plants and other industries should alsoexamine whether there is any plant, firm and/or industry char-acteristic that determines which of the learning models consid-ered in this paper, including the stochastic one, is more likely tooccur. An extension closely related to the research done so farwould be to examine the cross-model learning spillover within thesame assembly plant, as in Levitt et al. [30]; the de-learning orforgetting process, i.e. the rate at which accumulated knowledgedepreciates [7,44]; and the joint learning of the plant togetherwith the learning of a close external supplier [27].

Finally, another limitation to overcome is to incorporate intothe analysis the observation of the actions managers take whenthey learn in contexts of continuous improvement. We havemodeled the presumed cycle of action–performance–action, butthe only information used in the modeling is the observedperformance. As in many manufacturing plants around the world,during the period of study our assembly plant introduced themethods of continuous improvement of TQM and related techni-ques that will underlie the observed improvement in operatingefficiency. It would be of interest for managers who want to knowthe effect of their decisions to trace the measurement of improve-ments in TFP after the introduction of a particular innovation inwork organization, human resources management, and so on.Research aimed at explaining differences in productivity (andeconomic performance in general) across firms from differentindustries and countries, has grown in recent years [10,16].Differences in productivity are evaluated at a given moment oftime, but it would be of interest to investigate the time dynamicsof TFP growth rates to infer the underlying learning model, andhow this model is related to specific management techniquesimplemented.

Table A1

Variable I—Unlimited

learning

II—Mixed

learning

III—Jovanovic and

Nyarko

DTt �0.063789 �0.0640667 �0.066547

DH1t �0.263477 �0.259767 �0.274161

DH2t �0.329069 �0.327435 �0.335293

DH3t �0.161912 �0.161541 �0.162991

DV1t 0.0724797 0.0717038 0.0742428

DV2t 0.146759 0.145737 0.149136

DV3t �0.207215 �0.206192 �0.211352

DV4t 0.0799451 0.0796149 0.0809149

DV5t �0.136466 �0.136303 �0.13683

DV6t �0.184293 �0.183855 �0.187109

DV7t 0.265892 0.264484 0.270482

Dqt�2 0.158715 0.159859 0.157306

Dqt�6 0.185359 0.182958 0.181904

Dkt�3 1.04404 1.06162 1.08947

Dkt�8 0.520535 0.500589 0.517711

Dlt�1 0.777493 0.755712 0.775237

Dlt�5 0.856972 0.848173 0.849558

Dlt�9 �0.756896 �0.778291 �0.726362

C. Saenz-Royo, V. Salas-Fumas / Omega 41 (2013) 336–344344

Acknowledgment

The authors thank two anonymous referees and ProfessorAdenso Dıaz for their comments on an early version of the paper.They also thank the Spanish Ministry of Science and Innovation,Project ECO2009-13158, for financial support.

Appendix A

Error Correction Method estimates of parameters of model (6)unreported in the main text (Table 3). All statistically differentfrom zero (po10%).

Notice that the estimated parameters are all quite similar inthe three models, so the differences in results appear mainly inthe technology and learning function parameters.

(See Table A1 below).

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