Benjamin David
email: benjamin.david@u-paris10.fr.

On the information and communication technologies -productivity nexus: a long-lasting adjustment period

Keywords: Information and Communication Technologies, General Purpose Technologies, Malmquist productivity index, Data envelopment analysis, regression tree JEL Classification: C14, D24, O33

This paper investigates the relationship between Information and Communication Technologies (ICT) and productivity within 240 industries from 8 OECD countries. Specifically, it aims at providing explanations for the coexistence of this strong technological evolution together with the absence of break in the productivity trend during the last decades. We calculate the total factor productivity changes and their components (technical progress and pure efficiency changes) over the period 1973-2005 using the Malmquist productivity index and we then relate these measures with data on ICT diffusion using regression trees. Our results suggest that ICT diffusion is accompanied by opposite movements that conceal the potential of these technologies. Indeed, we find evidence of a clearly identifiable positive relationship between computerization and technical progress, while ICT diffusion negatively affects pure efficiency changes. Our findings support the existence of an adjustment period and are consistent with the fact that the economies under consideration are still in a phase of adaptation.

Introduction

In 1987, Solow coined his famous paradox: « You can see the computer age everywhere but in the productivity statistics. » This statement referred to the introduction of ICT tools in the economy without any apparent improvement in productivity. Since this observation, a large literature about the impact of computerization on productivity has been produced. After inconclusive or contradictory results, a consensus in favor of a positive impact of ICT on productivity has emerged (see e.g. OCDE (2012) and [START_REF] Cardona | ICT and productivity: conclusions from the empirical literature[END_REF]). However, productivity gains seem to have a limited magnitude although ICT are traditionally accompanied by many promises in terms of performance improvements. Even if we acknowledge the possibility of methodological issues which prevent the accurate measurement of ICT impact [START_REF] David | The Dynamo and the Computer: An Historical Perspective on the Modern Productivity Paradox[END_REF] and [START_REF] Cardona | ICT and productivity: conclusions from the empirical literature[END_REF]), there is no doubt that the pace of productivity growth in the computer era (from 1970s to now) is not « miraculous » (OECD data 1 , [START_REF] Jorgenson | A retrospective look at the US productivity growth resurgence[END_REF], [START_REF] Timmer | Productivity and Economic Growth in Europe: A Comparative Industry Perspective[END_REF]). There is no paradigm shift, and there is to some extent a persistence of the Solow paradox. This fact is illustrated in Figure 1 displaying productivity changes together with the evolution of the share of ICT in the total capital stock for eight industrialized countries.

Figure 1: Average trends of productivity and share of ICT in total capital stock in eight industrialized countries2 1 Indeed, between 1973 and 1995 -a period which corresponds to the beginning of ICT diffusion-labour productivity and Total Factor Productivity (TFP) growth clearly slowed down in the EU and the US, while during the second part of the 1990s, only the US had a limited « resurgence of productivity growth » (Timmer et al. (2008)). According to [START_REF] Jorgenson | A retrospective look at the US productivity growth resurgence[END_REF], this dynamic originates in the industries producing ICT equipment. Companies belonging to this sector experienced a strong growth in productivity contributing to a relative fall in prices, associated with massive investment in ICT facilities and capital deepening. This period ended with the burst of the internet bubble in 2001. From the mid-2000s growth rates in labour productivity and TFP have slowed down [START_REF] Fernald | Productivity and Potential Output before, during, and after the Great Recession[END_REF], [START_REF] Byrne | Is the Information Technology Revolution Over?[END_REF]).

The gap between the expected improvement in productivity and the levels of productivity actually observed has prompted some authors to develop a pessimistic viewpoint about the capacity of ICT to profoundly reshape production performance positively. [START_REF] Gordon | Is US economic growth over? Faltering innovation confronts the six headwinds[END_REF] establishes a chronology where the computer and internet revolution begins in the 1960s and reaches its maximum at the end of the 1990s in the US. According to him, the US is now facing up to « six headwinds »3 which are « the end of "demographic dividend" », rising inequality, factor price equalization, educational problems, consequences of environmental regulations and consumer and government debt. Gordon considers that the technological advance including the field of ICT is too weak to offset these problems. This pessimistic analysis is consistent with the work of [START_REF] Cowen | The Great Stagnation: How America Ate All The Low-Hanging Fruit of Modern History, Got Sick, and Will (Eventually) Feel Better[END_REF] who argues that the US economy has reached a « technological plateau » that severely limits the perspective of productivity growth.

Falling into this strand of the literature, this paper aims at contributing to the analysis of the link between ICT and productivity. Specifically, we argue that such pessimistic analyses may lead to hasty conclusions. Indeed, this representation does not match the data and literature on technological improvement and ICT diffusion, and the idea of a decrease in technological advance is not empirically supported. On the contrary, as shown by [START_REF] Nagy | Statistical Basis for Predicting Technological Progress[END_REF], technological progress is often rapid and can be mostly approximated by exponential laws, while the evolution of ICT performances is possibly faster. By analyzing data on storage,transportation and transformation of information in terms of volume and cost in a long-term perspective, [START_REF] Nagy | Superexponential long-term trends in information technology[END_REF] suggest that they may follow a « superexponential » growth. Moreover, the massive decrease in the cost of ICT is accompanied by the apparition of many tools capable of improving productivity to a large extent in firms: computers and means of communication with increasing capacities, software which has dramatically facilitated information processing. The chronology exposed by Gordon merits comment. He dates the beginning of the computer revolution from the 1960s, although key inventions such as microprocessors or personal computers (PC) did not exist at that time. Similarly, he locates the climax of this era at the end of the 1990s. This hypothesis seems too strong with regard to the US situation in 2000: only 43% of the population used Internet and only 39% had a mobile phone,4 whilst the share of ICT capital in the total capital stock was roughly equal to 9.5% although it has doubled since. 5To explain the absence of a break in the productivity path despite ICT diffusion, we follow a different approach based on the hypothesis that the ICT effects can appear after an extended adjustment period [START_REF] David | The Dynamo and the Computer: An Historical Perspective on the Modern Productivity Paradox[END_REF]). [START_REF] David | The Dynamo and the Computer: An Historical Perspective on the Modern Productivity Paradox[END_REF] draws a parallel with the electrification period and argues that the emergence of this new technology has produced large effects in production activities but after many years of adaptation. The key elements of the « electric age » date from around the 1880s and the positive impact on the US productivity growth is mainly visible towards 1920s6 suggesting that, in 1900, economists could certainly formulate a first productivity paradox. This delay can be attributed to the slow pace of electrification in factories due to the necessity of large organizational changes. Indeed, appropriation of a new technology is a difficult task requiring massive material and human reorganization. It is necessary to reshape production units, firms, sectors, distribution chains and mobilise new abilities. It involves the hiring of people with specific skills and learning-by-doing processes. Transition is complicated by the presence of a previous technology which should be profitable but causes detrimental overlay of new and old technologies. Surprisingly, [START_REF] Gordon | Is US economic growth over? Faltering innovation confronts the six headwinds[END_REF] himself suggests that the two first industrial revolutions (the steam engine and electricity) reached their full potential at the end of, respectively, 150 and 100 years. This approach proposed by [START_REF] David | The Dynamo and the Computer: An Historical Perspective on the Modern Productivity Paradox[END_REF] is connected with the nature of ICT that can be viewed as a General Purpose Technology (GPT) characterized by « binary logic » (Bresnahan and Trajenberg, (1995)). Each GPT is defined by « pervasiveness » that refers to the idea that this technology diffuses in all economic sectors,by increasing capacities and by creating complementarities. This definition suggests a diffusion process with different steps and gradual effects. In the same way, Helpman and Trajtenberg (1998a) propose a GPT-based growth model with a first phase marked by a « productivity slowdown » and a second stage where development of complementary inputs and the diffusion of new technology produce positive effects. This model is summarized by the title of their paper: « A Time to Sow and a Time to Reap ». From this point of view, ICT could have a strong effect on productivity performances but only after the economy has realized a costly transition. This approach is a natural explanation to the actual situation and constitutes our working hypothesis. Some authors, such as [START_REF] Sichel | The Computer Revolution: An Economic Perspective[END_REF] or Hempell (2005b), have suggested that the absence of ICT impact in the 1970s or 1980s on productivity performances arises simply because at this time the size of ICT capital stock is too limited to have significant effects. We do not consider this point of view as a competing explanation but just as a symptom of the adjustment hypothesis. Indeed, the low levels of ICT stock in firms that prevailed over several years can be interpreted as being due to the difficulty of incorporating these technologies.

To investigate the link between ICT and productivity and assess the relevance of the existence of an adjustment period, we rely on the following empirical strategy. Firstly, we study the productivity performances in 240 industries from eight industrialized countries. To this end, we calculate the Malmquist productivity index relying on Data Envelopment Analysis (DEA). This enables us to obtain the changes in TFP and its decomposition into Technical Changes (TC), Efficiency Changes (EC) and Scale Changes (SC). This method had the added advantage of not making strong assumptions in terms of returns to scale or functional form. Secondly, we relate these estimations to data on ICT through regression trees. This nonparametric methodology allows us to take into account the particular form of ICT data characterized by nonlinear trends. Our results suggest that while the effect of ICT on TFP variations is not clear-cut, it impacts its components by positively affecting TC and by reducing EC. These opposing movements corroborate the existence of the adaptation process postulated by [START_REF] David | The Dynamo and the Computer: An Historical Perspective on the Modern Productivity Paradox[END_REF].

The rest of the paper is organized as follows. Section 2 presents arguments in favour of an ongoing adaptation phase. Section 3 contains the productivity analysis and section 4 assesses the link between ICT and productivity performances. Section 5 concludes the paper.

The relevance of an adaptation period

In line with David's hypothesis, we can find in the literature about ICT the probable signs of the adaptation process that may explain the expected delayed effect. Indeed, the electrification episode has shown that changes in organization are a key element for incorporating new technology into the economy. Bresnahan et al. (2002) found complementarities between ICT investments, the hiring of high-skilled workers, new work organization, product and service innovations. In particular, ICT reshape the modality of information diffusion in firms because these technologies substantially increase the amount of available data and their flows (« Information overload »). In response, firms need a more flexible and horizontal organization where high-skilled workers operate. Using data from 379 US firms for the period from 1987 to 1994, the authors obtain results in line with their assumptions. Tambe et al. (2011) enrich this approach by arguing that the positive impact of ICT productivity depends on a combination of decentralization, ICT investments and external focus.7 However, the phase of reorganization is not necessarily easy and some initiatives of this kind may fail. An illustration of this trial and error process is given by [START_REF] Aral | Which came first, IT or productivity? The virtuous cycle of investment and use in enterprise systems[END_REF]. Their work enables the observation of a microeconomic foundation for ICT productivity gains and details about the dynamics of the link between ICT investment and performance growth. Using panel data from 623 US firms over the 1998-2005 period, they show that ICT investments are carefully implemented and their success (productivity growth) provides an incentive to the achievement of other specific expenses. This forms a feedback process that gradually strengthens productivity gains. Inversely, when investment is not conclusive (no additional productivity gains), firms reduce their propensity to make new ICT investment.

Another way to support the assumption that ICT need an adjustment period is to consider that they are deployed in part in the form of network because ICT cover technologies such as mobile phones, the Internet or any computer network. Observations and theory [START_REF] Curien | Economie des réseaux[END_REF]) suggest that their diffu-sion displays specific trajectories mainly characterized by nonlinear dynamics on both the demand and supply sides. Networks are characterized by a positive externality called « club effect » that corresponds to the situation where when the number of network users grows, individual satisfaction increases. On the other hand, on the supply side, networks are characterized by the fact that investments are concentrated in the deployment phase of the physical structure. These high fixed costs are independent of the production volume but the use of the network is virtually cost-free due to very low variable costs. Then, firms that develop networks benefit from increasing returns to scale (IRS) (until a threshold where complexity costs could become important). Gains in profitability constitute an incentive to improve the quality of the service offered. Due to « club effects » and the specific costs structure, networks produce nonlinear dynamics. During the first phase, the size of a network is limited and the costs are important for producers and users (due to learning costs). After achieving a critical mass of users, a new phase begins with a strong growth of the network sustained by retroactions. The increasing number of users attracts new customers while the producer could improve the service offered which attracts new users. This kind of dynamics produces an increase in both the capacity and the quality of the network. On the basis of these theoretical precisions, it seems to be logical that the network goods used in the production process by firms (they constitute a part of capital stock) do not immediately cause productivity gains. For a time, networks generate mainly costs for users and producers. A certain level of maturity is necessary before a potential improvement is realized. For example, at the beginning of the Internet, the first users have no possibility to contact in this way with their suppliers or customers due to the low adhesion and technical level. This argumentation concerns many networks used for economic activities (fax machines, mobile phones and, specifically the Internet) from the middle of the 1990s, and all other computer networks used internally by firms (Intranet).

From an empirical point of view, the hypothesis of the delayed and increasing effects of ICT on productivity performances has been studied by several authors for example [START_REF] Jovanovic | General Purpose Technologies » Handbook of Economic Growth[END_REF]. Considering the Hodrick-Prescott filtered annual growth in output per man-hour data for US over the 1874-2004 period, they show that, at the beginning of the electricity diffusion, the US economy suffered from a productivity slowdown and that this situation was reflected again in the first years of the ICT diffusion.8

Brynjolfsson and Hitt (2003) use panel data from US firms between 1987 and 1994, and regress TFP variations on computer capital growth using varying differences (from one to seven years). Their results suggest that in the short term (one-year difference), the contribution of computers is approximately equal to its factor share but as the time horizon grows, the effect becomes increasingly important. These results are similar when time and industry controls are taken into account. Basu et al. ( 2004) study the differential productivity path in the US and the UK during the 1990s. They estimate the impact of contemporaneous and lagged variations in computer and software capital stock on TFP changes. Their results for the US are in line with the hypothesis of a delayed effect. Indeed, the shareweighted computer and software capital growth over the 1995-2000 period are positively associated with ICT capital growth for the 1980-1990 period while it is negatively associated with contemporaneous ICT capital growth. On the other hand, Basu et al. ( 2004) do not obtain similar results for the UK.

3 Productivity analysis

Malmquist productivity index

We consider the Malmquist Productivity index to grasp productivity variations. It is constructed on the basis of distance functions estimated by DEA (Fare et al., 1994). The starting-point of this approach is the concept of production technology frontier that is a function based on inputs used (x t and outputs (y t ) produced by the Decision Making Unit (DMU) considered.9 For a given period, the most efficient DMU(s) define(s) this frontier and a production possibility set (S t ) that give the possibility to assess the level of efficiency of the other DMUs compared to this benchmark. The concept of distance function (D o )10 can be used for measuring productivity growth through the Malmquist productivity index. This index was introduced by [START_REF] Caves | The Economic Theory of Index Numbers and the Measurement of Inputs, Outputs, and Productivity[END_REF], building upon the work of [START_REF] Malmquist | Index numbers and indifference curves[END_REF]. It is constructed on the basis of four distance functions differentiated by time horizon (see details in Appendix 1):

M o x t , y t , x t+1 , y t+1 = D t+1 o x t+1 , y t+1 D t o (x t , y t ) EC D t o x t+1 , y t+1 D t+1 o (x t+1 , y t+1 ) D t o x t , y t D t+1 o (x t , y t ) 1/2 T C (1)
As shown in equation ( 1), the Malmquist index measures productivity changes as the geometric mean of efficiency change (EC) and technical change (TC). A variation in efficiency change refers to the movement of a DMU over the best practice frontier. Therefore, an improvement (or a decrease) in efficiency corresponds to the diminution (or an augmentation) of the distance over the most efficient DMU -this kind of movement is called « catching up » by [START_REF] Färe | Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries[END_REF]-whilst technical change reflects shifts in the frontier of S t . When Variable Returns to Scale (VRS) are considered, EC can be decomposed into « pure efficiency change » (PEC) and scale change (SC) by using the difference between estimates under the two hypotheses. Formally, we have:

11 SC = D t+1 ov x t+1 , y t+1 /D t+1 oc x t+1 , y t+1 D t+1 ov (x t , y t ) /D t+1 oc (x t , y t ) D t ov x t+1 , y t+1 /D t oc x t+1 , y t+1 D t ov (x t , y t ) /D t oc (x t , y t ) 1/2
(2)

P EC = EC SC (3)
Numerical interpretation of the Malmquist index is simple. When its values are greater than unity, this indicates an increase in TFP between t and t+1, while a decrease is observed when values are inferior to unity. The same interpretation holds for its components (TC, EC, SC and PEC). The estimations of the distance functions are realized by DEA analysis. This is a linear programming method for constructing a nonparametric envelopment frontier that contains all data points. Details on the calculation procedure are given in Appendix 1.

In our case, using the Malmquist index and DEA analysis is very convenient because it is a nonparametric approach. It does not assume any specific functional identity for the production frontier described above. The form and the location of the combination output(s)/input(s) are just deduced from the data of the DMUs considered. Moreover, this method allows us to make a minimum of arbitrary hypotheses because it does not require classical and strong assumptions regarding constant returns to scale and perfect market used in growth accounting approach [START_REF] Solow | Technical Change and the Aggregate Production Function[END_REF]. This last restrictive assumption can be problematic to take into account the influence of the technological change since it is located in an equilibrium framework. On the contrary, as stressed by OECD ( 2001), some evolutionist economists [START_REF] Nelson | An Evolutionary Theory of Economic Change[END_REF], [START_REF] Dosi | Sources, Procedures and Micro-economic Effects of Innovation[END_REF]) argue that the notion of « disequilibrium » is fundamental to understanding innovation and productivity changes. Another drawback of the growth accounting approach avoided by the Malmquist index estimated by DEA analysis is to consider only TFP changes. Indeed, the Malmquist index provides a more precise view on productivity changes, since it allows decomposition between TC, EC and SC.

Data

Our analysis of the productivity changes relies on data from the EU Klems database (2009 release). Data are available for 30 sectors12 for 8 countries (Australia, Denmark, Finland, Italy, Japan, the Netherlands, the UK and the USA) during the 1973-2005 period. We therefore have 240 DMUs observed over a 33 year period. We calculate the Malmquist productivity index in a standard way by considering one output, Value Added (VA), and two inputs, total capital stock (K) and labor expressed as the number of hours worked (LAB). To have the same unit of measurement for the variables VA and K, we divide the real values of the variables by the purchasing power parity data provided by OECD. 13 The size of our sample is the result of the tradeoff between the need for a « long » interval, and the maximum of industries in a harmonized framework. This delimitation is convenient because we have, in our balanced panel, a sufficient number of observations for implementing DEA analysis (7920 observations for each variable and 240 observations for each period).14

Results of productivity estimates

Due to the number of DMUs considered and the length of the time interval, we obtain a lot of results,15 and we will only comment the main features here. We present the geometric mean results. 16At the country level, we can compare these results with the TFP estimations realized by the OECD using the growth accounting method ( available. We found that the Malmquist index is well correlated with OECD data because we obtain, on average, a correlation of 0.591. One exception is the US which displays a positive, but low correlation coefficient. Japan has a slightly lower value than the mean, while Australia, Denmark and the UK have values above the average. Finland, the Netherlands and, in particular, Italy have the strongest correlation with the OECD data. This comparison is interesting because it assures the plausibility of our results, and emphasizes that using different methods does not lead to similar results. The productivity changes in the countries studied are clearly heterogeneous (Table 2). Indeed, half of them have experienced on average a decline in TFP (Australia, Denmark, Italy and the UK) while four other countries have witnessed positive variations (Finland, Japan, the Netherlands and the US).

The magnitudes are also distinctly differenced. For example, Italy displays a strong negative variation but the Netherlands has large gains in TFP. Regarding the TFP components, we observe that all countries have, on average, benefited from a positive technical change while the pure efficiency performances are more nuanced. Only three countries (Japan, the Netherlands and the USA) display a growth in efficiency. Low values are obtained in Australia and in the UK, and especially in Italy with a mean of 0.9820.

At the sectoral level, results are also contrasted (Table 3). Some sectors such as agriculture, hunting, forestry and fishing; chemicals and chemical products; basic metals and fabricated metal; electricity, gas and water supply; post and telecommunications; the electrical and optical sectors, all display large gains in productivity. Other sectors like Coke, refined petroleum and nuclear fuel; construction; hotels and restaurants; public administration and defense; compulsory social security sectors, have strong negative evolutions. Furthermore, the coexistence of technical progress accompanied by losses in efficiency observed at the country level is also discernable at the sectoral level. This observation will be commented on further. As suggested by [START_REF] Färe | Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries[END_REF], it may be interesting to identify which DMU(s) shift the frontier over time. Formally, DMU that shift the frontier have the following characteristics:

• T C > 1 • D t o (x t+1 , y t+1 ) > 1 • D t+1 o (x t+1 , y t+1 ) = 1
Table 4 summarizes the evolution of the contributions to the frontier in our analysis. The country whose industries have the best contribution, is the US with construction; renting of machinery and equipment and other business activities, and the electrical and optical equipment sectors. A more surprising contribution is that of Italian industries because the Coke, refined petroleum, nuclear fuel sector and the real estate activities sector successively sharply modify the technological frontier during the period studied. Traditionally, Italy is not seen as a technological leader. However, these results are plausible since there are only two specific industries which contribute to the shifting of the frontier. Indeed, we can suppose that the importance of tourism to the Italian economy may have a positive influence on productivity performances in real estate and the very significant weight of the petroleum company ENI (the largest company in Italy) in the energy sector explains this contribution to the evolution of the efficient frontier. Other countries have industries that significantly move technological frontiers such as Finland, Japan and the UK.

In the Netherlands, only two industries have influence; whilst Australia and Denmark do not have any industry that shifts the frontier. Now let us provide some comment about the sectoral level. We found that from the beginning of our interval until the end of the 1990s, the best frontier practice is defined mainly by Coke, refined petroleum, nuclear fuel; construction and the renting of machinery equipment and other activities sectors.

From the end of the 1990s, we note a contribution of financial intermediation and the real estate activities sectors but especially of the electrical and optical equipment sector. In this category are included many products that constitute ICT.17 Otherwise, the most efficient DMUs at the end of the period studied (last eight years) are DMU producing ICT. This point emphasizes the fact that ICT have an important and increasing role in the productivity path, at least in terms of technical change.

4 On the link between diffusion of ICT and productivity changes

Methodology: regression trees

Our aim is to check if ICT diffusion has an impact on productivity changes and its components. The methodological challenge concerns the unit root property of our series, and the potential nonlinearities in their interaction. Specifically, 240 series are characterized by polynomial trends but their order is difficult to identify (probably degree three or four) and there is no reason to consider that interactions between our variables are linear. These characteristics prevent us to use classical panel data methods. We rely on the regression tree approach, a methodology most used in machine learning but which may be very fruitful in economics as suggested by [START_REF] Varian | Big Data: New Tricks for Econometrics[END_REF]. 18

Regression trees model the response of a continuous dependent variable to explicative variable(s) by partitioning « the space of all joint predictor variable values into disjoint regions . . . as represented by the terminal nodes of the tree [START_REF] Hastie | The Elements of Statistical Learning: Prediction, Inference and Data Mining[END_REF]). » 19 In each region R m for m=1,. . . M, the response is mostly modelled as a constant (c m ). Formally, the model can be expressed as follow:

f (x) = M 1 c m I(x ∈ R m ) (4)
We identify the dependent variable (f(x)) with, successively, the Malmquist index or its components (TC and PEC). We consider only one predictor (x), the share of ICT in the total capital stock (%), that proxies ICT diffusion in the economy. 20 Thus, we expect that there is a link between the Malmquist index (or TC and PEC) and the importance of ICT used by industries. 21

In order to construct our regression tree, we use the popular CART algorithm 22 defined by [START_REF] Breiman | Classification and Regression Trees[END_REF] which proceeds by binary partition.

At each non terminal node, there are necessarily only two leaves where j is the splitting variable and s the split point:

R 1 (j, s) = X|X j ≤ s and R 2 (j, s) = X|X j > s (5)

18 [START_REF] Breiman | Classification and Regression Trees[END_REF] underline that the first work on tree method was co-written by economist James N. Morgan in the context of Automatic Interaction Detection program (AID) (see [START_REF] Morgan | Problems in the analysis of survey data, and a proposal[END_REF]).

19 Our brief presentation is based on [START_REF] Hastie | The Elements of Statistical Learning: Prediction, Inference and Data Mining[END_REF]. 20 ICT capital contains hardware, software and communication equipment.

21 Note that we do not include control variables given the lack of relevant data at the fine level used in our analysis since it would be necessary to have data for the 240 industries. Furthermore, in several studies, authors consider TFP as the dependent variable with only one ICT variable as predictor (for example Brynjolfsson and Hitt (2003);Basu et al. (2003)).

22 CART = Classification And Regression Trees.

In our case, there is just one splitting variable because we have one explicative variable. The search of the best partition consists in solving the following expression:

min   min c1 x i ∈R 1 (s) (y i -c 1 ) 2 + min c2 x i ∈R 2 (s) (y i -c 2 ) 2   (6)
For any s, the best ĉm is equal to ave(y i |x i ∈ R m ). Determination of s is realized by comparing residuals obtained in all partitions computable with input data. In the two regions obtained, we repeat the same procedure to construct the regression tree. The size of the tree (i.e. number of terminal nodes) has crucial implications. The problem consists in arbitrating between grasping informative structure in data and the risk of overfitting. In CART framework, the idea is to construct a large tree and to prune it by using cost-complexity pruning criterion. More detailed explanations regarding this procedure are provided in Appendix 2. We apply CART algorithm at different levels with all sample, by aggregating sectors and by country.23

Regression trees results

Results of regression trees are displayed in Figure 2. The presented results apply only to variables aggregated at the sectoral level because this level corresponds to the format of the data source. Application of the CART algorithm does not provide interesting results when TFP changes are considered as the dependent variable. In fact, any partition is realized in the final tree (i.e. there is only one region after the pruning procedure). This means that we do not identify any relation between the share of ICT in capital stock and the TFP changes. Inversely, when we consider the variables TC and PEC as dependent variables in the regression, we obtain very interesting findings (Figures 2 a,b,c,d). Indeed, in the case of TC, we obtain a tree with three leaves delimited by two split points equal to 3.092% and 5.247%. Constants (ĉ 1 , ĉ2 , ĉ3 ) in each region are clearly different because in the first region (R1), the value is equal to 0.980, in R2 to 1.006 and in R3 to 1.057. This means that in our sample, the higher the share of ICT in capital stock the more important are the gains in technical progress. Regarding to PEC, we have a tree with two terminal nodes delimited by a split point equal to 4.603%. In region R1*, the constant is equal to 1.013 and in region R2*, there is a value of 96.02. These results suggest a negative relation between share of ICT and efficiency growth. We plot respectively on Figure 2 the trees obtained and the observations partitioned. Note that even if we apply the same methodology by country or by considering all observations not pooled by sector, we obtain the same kind of results (Table 6 in Appendix). TC increases and PEC decreases with the share of ICT in capital stock. The number of terminal nodes is ranged from two and four.

The absence of clear relation between our ICT variable and TFP changes and the fact that the share of ICT in total capital stock is statistically associated with different levels of technical progress and efficiency changes highlight that ICT require an adjustment period before expressing their full potential.

From this point of view, the positive link between ICT and the contribution of TC to TFP indicates the capacity of new technologies to strongly improve production capacity. Indeed, regression trees show that, on average, when the level of ICT reaches a threshold value, very large gains in TC are observable (for example, 5.7% by year when data are pooled by sector). This result is in line with promises of paradigm shift due to the introduction of computer tools in the economy.

On the other hand, performances in PEC seem to be negatively affected by ICT diffusion. We view this relation as « the cost of the introduction » of these technologies in the economic area. Behind these statistical results, there are probably obsolescence of machines, of skills, overlapping of technologies, groping in research of efficient organization into firms, lack of anticipation of managers or psychological barriers (resistance to change). . . that affect efficiency performances.

The coexistence of these two contradictory relations can be interpreted as the sign of the adjustment period postulated by [START_REF] David | The Dynamo and the Computer: An Historical Perspective on the Modern Productivity Paradox[END_REF]. This hypothesis explains why any structural change is observable in productivity path in the ICT era. This means that from this perspective, productive systems in the countries studied are submitted to deep and ongoing adaptation which creates temporary losses in efficiency. Logically, this situation should end and large productivity gains should appear in the following years or decades. The existence of important growth in TC and a decrease in efficiency has also been identified by [START_REF] Färe | Productivity Growth and Convergence in the European Union[END_REF] who report a similar evolution in sixteen European countries mainly from the 1990s. They propose that these trends can be explained by the costs, in the short run, of regulatory reforms operated in these economies. This argument seems to be plausible and probably explains a part of these observations but it is also limited by the fact that all the countries studied have not simultaneously carried out this

R1 R2 R3

q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5e-04 2e-03 1e-02 5e-02 2e-01 0.8 0.9

1.0

1.1 1.2 q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5e-04 2e-03 kind of reform. 24 The advantage of our approach is to refer to ICT diffusion as a uniform process (even if there are differences in timing) to explain the broader movement identified by [START_REF] Färe | Productivity Growth and Convergence in the European Union[END_REF] and in our analysis.

Robustness checks

We assess the robustness of our results using other statistical approaches.

We compute the correlation coefficient and the Kendall's tau coefficient (1938) between each variable TFP, TC, PEC and our ICT variable. The first method allows us to identify linear dependence for each pair of series, whereas Kendall's tau coefficient is a measure of the rank correlation. It can detect potential dependence between the ranks of the two series considered. As shown in Table 5, we observe that the positive link between the diffusion of ICT and the growth of technical progress previously identified is visible with the two correlation measures. Results show positive and highly significant correlations. Values for PEC are also consistent with the previous analysis since we observe a negative and strongly significant correlation between the variables. When we consider TFP and ICT, results are less compelling because correlations are very low even if significant in the case of Kendall test. These observations coincide with our previous findings presented in part 4-2.

Conclusion

We study the coexistence of the massive spread of ICT in the economy and the absence of paradigm shift in the productivity path. Although a contribution of ICT to productivity variations has been observed since the 1990s, it does not correspond to the « revolution » expected. We suggest that this situation is linked to the hypothesis formulated by [START_REF] David | The Dynamo and the Computer: An Historical Perspective on the Modern Productivity Paradox[END_REF] stating that new technology needs a relatively long period to adapt a productive system before it achieves its potential. To assess the relevance of this hypothesis, we measure the evolution of productivity performances in 240 industries from eight industrialized countries. We use the Malmquist productivity index estimated by DEA analysis in order to have estimations of TFP changes and its components. Then, we explore the link between these measures and ICT diffusion by using regression trees. Our results are consistent with the adjustment period hypothesis: no clear relation between TFP changes and ICT diffusion can be observed, while as the share of ICT in total capital stock increases, the performances in technical progress improve. Inversely, we find evidence of a negative association between ICT and pure efficiency changes.

On the whole, our findings are consistent with the existence of a delay between ICT diffusion and an improvement in productivity, and with the fact that the considered economies are still in their adaptation phase.

D t (x t , y t ) = inf θ : (x t , y t /θ) ∈ S t = sup θ : (x t , y t θ) ∈ S t -1 (7)

According to [START_REF] Färe | Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries[END_REF], « the distance function is defined as the reciprocal of the maximum proportional expansion of the output vector y t , given input x t . » If D t o (x t , y t ) = 1, (x t , y t ) are situated on the boundary of the production possibility set. This corresponds to the maximum level of efficiency that can be reached by a DMU at t. Inversely, when D t o (x t , y t ) < 1, this situation reflects inefficiency. Figures 3a and3b show an example with five DMUs and only one input and one output. We consider the CRS case and VRS with Decreasing Returns to Scale (DRS). In Figure 3a, DMU 3 is the most efficient DMU, therefore it defines the frontier of the production possibility set. All other DMUs (1, 2, 4, 5) are situated below this frontier, meaning they are inefficient. When VRS is considered (Figure 3b), the shape of the frontier is piecewise because it is constructed by several DMUs (3, 4, 5). Again, inefficient DMUs are located under the limit of the production possibility set.

Malmquist index

In order to compute the Malmquist productivity index, several distance functions are used as shown in Equation (1). Some distance functions are defined in t, t+1 or with mixed period. 4a represents two production possibility sets in t and t+1 when CRS and DRS are considered. In t, DMU which defines the frontier of this set has a ratio input/output located at point (1; 1). If we consider another DMU which has the ratio (1; 0.5), its distance function in t, D t o (x t , y t ), is equal to (0A)/(0B). Since the ratio input/output of this DMU is under the limit of the production possibility set, it is inefficient and therefore the value of (0A)/(0B) is lower than one. In t+1, the frontier has positively moved because the DMU that defines this frontier has improved its own performance. That means technical progress has occurred. It is possible to have distance function in t+1. If we consider that the inefficient DMU has the ratio input/output equal to (1.25, 1.65) the distance function D t+1 o (x t+1 , y t+1 ) is (0D)/(0F ). Furthermore, we can define mixed period distance functions evaluating (x t , y t ) in relation to technology in t+1 D t+1 o (x t , y t ) = (0A)/(0C) and evaluating (x t+1 , y t+1 ) in relation to technology in t D t o (x t+1 , y t+1 ) = (0D)/(0E). Figure 4b is similar except for the shape of the frontier of the possibility production set because DRS are considered. In the same way, we can define the precedent distance functions. All distance functions combined permit to construct the Malmquist productivity index.

The general formulation is: (2)

M o x t ,
P EC = EC SC (3)
It should be noted that there exist other types of decompositions (see [START_REF] Coelli | An introduction to efficiency and productivity analysis[END_REF]

p293).

Data envelopment analysis

There are many methods for calculating the Malmquist productivity index,26 among which the DEA approach that has been coined and defined by [START_REF] Charnes | Measuring the Inefficiency of Decision Making Units[END_REF]. The distance functions are obtained by linear programming to establish a nonparametric envelopment frontier that contains all data points. These data must be located under or on the frontier. By considering k=1, . . . , K DMUs (here industry), n=1, . . . , N inputs and m=1,. . . , M outputs, λ = λ 1 , ..., λ r a vector of weights, the purpose is to get for each DMU four distance functions.

This corresponds to solving the following maximization programs:

D t o (
+ Y t+1 λ ≥ 0 x t+1 i -X t+1 λ ≥ 0 N 1 λ = 1 λ ≥ 0
The number of problems to solve is equal to N ×(4T -2). Thus, in our application, the calculation of the Malmquist productivity index for all industries needs to solve 240 × (4 × 33 -2) = 31200 linear-programming problems. To achieve this task, we use the DEAP Version 2.1 computer program designed by [START_REF] Coelli | A guide to DEAP version 2.1: a data envelopment analysis (computer) program[END_REF].

Figure

  Figure 2: Regression trees results

  Figure 2b

  Figure 2d

Figure 3 :

 3 Figure 3: production possibility sets

Figure 4 :

 4 Figure 4: production possibility sets

Table 1

 1

	). Our

Table 1 :

 1 Correlation between DEA results and OECD estimations

	Country	Correlation
	Australia	0.628
	Denmark	0.531
	Finland	0.715
	Italy	0.852
	Japan	0.483
	Netherlands	0.726
	UK	0.608
	US	0.190
	Mean	0.591

Table 2 :

 2 Geometric mean by country

	Country	EC	TC	PEC	SC	TFP
	Australia	0.9890 1.0044 0.9884 1.0006 0.9934
	Denmark	0.9924 1.0052 0.9934 0.9989 0.9975
	Finland	0.9995 1.0048 0.9991 1.0004 1.0043
	Italy	0.9773 1.0049 0.9681 1.0094 0.9820
	Japan	1.0029 1.0064 1.0025 1.0003 1.0093
	Netherlands 1.0098 1.0033 1.0077 1.0020 1.0131
	UK	0.9868 1.0040 0.9826 1.0042 0.9908
	USA	1.0014 1.0045 0.9986 1.0028 1.0059

Table 3 :

 3 Geometric mean by industry

	Sector

Table 4 :

 4 Industries contributing to the evolution of the frontier

	Year Industry

Table 5 :

 5 Correlation tests between productivity variables and ICT

	Test	TFP	Dependent Variable TC PEC
	Pearson	0.004	0.278	-0.177
	p-value	(0.7376) (<2.22e-16) (<2.22e-16)
	Tau Kendall 0.0246	0.234	-0.178
	p-value	(0.0012) (<2.22e-16) (<2.22e-16)

  y t , x t+1 , y t+1 = By inserting the distance functions of Figures4a and 4b, the Malmquist index becomes:M o x t , y t , x t+1 , y t+1 =In our application, we prefer to use VRS that is why we decompose EC into PEC and SC. The last two terms can be expressed as follows: , y t ) /D t+1 oc (x t , y t ) D t ov x t+1 , y t+1 /D t oc x t+1 , y t+1 D t ov (x t , y t ) /D t oc (x t , y t )

		D t+1 o	x t+1 , y t+1	D t o x t+1 , y t+1		D t o x t , y t	1/2
		D t o (x t , y t )	D t+1 o	(x t+1 , y t+1 )		D t+1 o	(x t , y t )
			EC				T C
									(1)
		0D 0F	0B 0A	0D 0E	/	0D 0F	0A 0B	/	0A 0C	1/2
			EC			T C
									(8)
	SC =	D t+1 ov D t+1 x t+1 , y t+1 /D t+1 oc ov (x 1/2 x t+1 , y t+1

t

  x t , y t ) -1 = Max θ,λ θ Two additional programs are necessary for taking into account the possibility of VRS. We insert in the program a convexity restriction (N 1 λ = 1) where N1 is a vector of one.

					D t+1 o (x t+1 , y t+1 )	-1 = Max θ,λ θ
	s.t. -θy t i + Y t λ ≥ 0	s.t. -θy t+1 i	+ Y t+1 λ ≥ 0
	x t i -X t λ ≥ 0	x t+1 i	-X t+1 λ ≥ 0
				λ ≥ 0	λ ≥ 0
	D t o (x t+1 , y t+1 )	-1 = Max θ,λ θ	D t+1 o (x t , y t )	-1 = Max θ,λ θ
	s.t. -θy t+1 i	+ Y t λ ≥ 0	s.t. -θy t i + Y t+1 λ ≥ 0
		x t+1 i	-X t λ ≥ 0	x t i -X t+1 λ ≥ 0
				λ ≥ 0	λ ≥ 0
	D t o (x t , y t )	-1 = Max θ,λ θ	D t+1 o (x t+1 , y t+1 )
	s.t. -θy t i + Y t λ ≥ 0
		x t i -X t λ ≥ 0
			N 1 λ = 1
				λ ≥ 0

-1 = Max θ,λ θ s.t. -θy t+1 i

The eight countries considered in this paper are: Australia, Denmark, Finland, Italy, Japan, the Netherlands, the UK and the US. Labor productivity is defined by the ratio value added/number of hours worked. This series and share of ICT data are from EUKlems database. TFP changes data correspond to OECD estimations.

Gordon asserts that these problems are US specific and that other countries are subject to different mix of headwinds.

http://www.itu.int/en/ITU-D/Statistics/Pages/stat/default.aspx

EUKlems database

See also[START_REF] Devine | From Shafts to Wires: Historical Perspectives on Electrification[END_REF].

Authors define external focus « to be a set of practices firms use to detect changes in their external operating environment ».

[START_REF] Jovanovic | General Purpose Technologies » Handbook of Economic Growth[END_REF] point out that it is just a correlation and that a debate towards a possible causality relation is still open.

DMU is a general formulation. It can be firm, sector, country...or any component of a panel.

Subscript "o" indicates the output orientation of the distance function.

Subscripts "v" and "c" refer respectively to VRS and CRS.

The complete list of sectors is given in Table3.

http://stats.oecd.org/Index.aspx?DataSetCode=PPPGDP#

DEA analysis may be biased when the number of observations is insufficient.

All results are available from author upon request.

Indeed, since the Malmquist index is a multiplicative index, the mean is also multiplicative.

These subsectors are office, accounting and computing machinery; electrical engineering; electrical machinery and apparatus, NEC; insulated wire; other electrical machinery and apparatus nec, radio, television and communication equipment; electronic valves and tubes; telecommunication equipment; radio and television receivers; medical, precision and optical instruments; scientific instruments; other instruments.

We use the R software with tree package version 1.0-35 designed by[START_REF] Ripley | tree: Classification and regression trees[END_REF]: http://cran.r-project.org/web/packages/tree/.

The panel considered is heterogeneous, including countries as Sweden, Portugal, the Netherlands, Greece. . .

Distance functions can be constructed with input orientation.

See Färe et al. (1994) p73.

Appendix 1: Malmquist productivity index estimated by DEA analysis

The following presentation is broadly based on [START_REF] Färe | Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries[END_REF] and [START_REF] Coelli | An introduction to efficiency and productivity analysis[END_REF].

Distance function

We consider DMUs that realize the transformation of input(s) (x t ) in outputs (y t ). A production possibility set S t = {(x t , y t ) : x t can produce y t } models, at each period t, this process and the frontier of S t is defined by the fully efficient DMU. The distances of the other DMUs over this limit allow to assessing their level of inefficiency. In a formal way, Shepard (1970) defines the output distance function 25 as follows:

Appendix 2: Pruning algorithm in CART

The way for building a regression tree in CART framework is, in a first step, to construct a large tree T 0 and then apply it «weakest link pruning algorithm». One has to successively collapse the internal node identified as « weakest link » and continue until we produce the root tree. In this sequence, we would like to identify T α , the tree that minimizes the cost complexity criterion C α (T ). This criterion is defined as follows:

is the number of nodes in T , a real number α ≥ 0. α is the parameter that sets the size of the tree. If α = 0, the final tree is T 0 because any penalty is defined for any new node. If T 0 is large enough, each observation is contained in one specific region. Inversely, large values of α produce small trees since growth tree is costly.

We can also define the contribution of node (t) to total cost after pruning:

Pruning becomes interesting at threshold C α ({t}) = C α (T t ) where C α (T t ) is the cost of the branch T t , α is deduced from this equality.

By collapsing successively internal nodes, we obtain a sequence of subtrees of T 0 : T 0 T 1 T 2 T 3 T k where T k is the root node of the tree and a sequence of α as 0 = α α 1 α 2 α 3 α k . For each α there is one subtree which minimizes the squared error. In order to find the best tree in the sequence, it is necessary to identify the best α. Several methods may be used but generally cross-validation is preferred. We consider a learning sample L v = L -L v where v = 1, . . . , V are subsets of the same size from data. Mostly, v = 10 therefore L v contains 90% of observations. The pruning procedure is repeated with the ten learning samples L v . We compute at each time the sum of squared errors on test sample L v that allows finding the best α (in terms of minimization of squared error).

Appendix 3: Additional results

Table 6: Additional regression trees results

Dependent


Éric Langlais

Eric Langlais
email: eric.langlais@u-paris10.fr.

The structure of …nes in the light of political competition

Keywords: timing of sanction, law enforcement, deterrence, political competition, majority rule. JEL classi…cation codes : D72, D73, H1, K14, K23, K32, K4, Q52, Q53, Q58

This paper analyzes the determinants of public law enforcement policies when citizens vote for the timing and level of …nes. We consider situations where citizens and politicians disagree on the value of the expected social harm associated with some activities. We …nd that citizens vote for act-based (harm-based) sanctions when they expect that social harm is low (resp high). On the other hand, we show that the equilibrium …nes may be higher or lower than the optimal one, depending on the di¤erence between politicians and citizens'expectations.

Introduction

Public law enforcement (Becker, 1968; Polinsky and Shavell, 2000) focuses on the control of activities imposing external costs to society, such as criminal acts, environmental pollution, speeding on a highway. A public authority sets a …ne, given general enforcement e¤orts (i.e. a given probability of detecting and convicting any kind of wrongful act), in order to induce optimal deterrence. 1 The issue of the timing of sanction arises when the external cost is uncertain rather than certain. Should the …ne be applied when the activity is detected (act-based) or when the external cost is (harm-based)? Polinsky and Shavell (2000) show that, for a given and equal probability of detection, the social planer should be indi¤erent between act and harm based sanctions, even when citizens and the authority disagree about the probability of harm.2 Furthermore, the level of deterrence is the same for both regimes, and determined only by the authority's estimation of the expected harm.

A crucial aspect is that most of the law enforcement literature issuing from Becker's seminal article assumes a benevolent law enforcer, which chooses according to a social welfare function aggregating both the preferences of law abiding citizens and o¤enders. This important methodological issue has been early debated both by lawyers and economists,3 although no convincing solution has been provided up to now. 4 More astonishing, the debate in the Law & Economics literature regarding the objectives of the law has focused on the issue of e¢ ciency vs fairness,5 in a sense ignoring to a large extent the more general debate in Social Choice theory initiated after the seminal work of Arrow (1951).

From this point of view, our paper is the …rst attempt (as far as we know) to analyze formally the issue of the deterrence of wrongful acts under political in ‡uences. We introduce here a simple model of electoral competition in a public law enforcement setting where the harm is uncertain, in order to characterize the structure of penal sanctions. In the vein of the framework known as Dowsian electoral competition (see Persson and Tabellini, 2000), we assume that citizens vote for an enforcement policy based on a speci…c timing of sanction (act based vs harm based) and a level of …nes. We provide a formal analysis of the structure of sanctions associated with the equilibrium emerging under political competition, and compare its e¤ects to those of the optimal one.

The paper is organized as follows. Section 2 introduces the basic model and notations, and brie ‡y reminds the case of a benevolent enforcer. Section 3 analyzes the timing of sanction (act-based versus harm-based sanction) that emerges at equilibrium under electoral competition. Section 4 concludes.

The basic model

We rely on the usual model of law enforcement à la Becker, assuming a disagreement exists between the population and the public enforcer regarding the assessment of the expected external cost associated with o¤enses.

General assumptions

Let us consider the case where the illegal activity allows the (risk neutral) criminal to obtain a bene…t equal to b, while an honest citizen earns an income normalized to 0; b will be called the type of the criminal. Public authorities do not observe the type b, and only know that b follows a uniform distribution function on [0; B]. The (external) loss/harm to the rest of the society is h in case of crime, whatever the private bene…t for the criminal.

The citizens and the public authority do not agree on the probability of occurrence of h; for the former, the likelihood of h is denoted as e 2 (0; 1), while g 2 (0; 1) is for the government. We require the following assumption:

Assumption 1: B > max f g ; e g h.
One of the main consequence of this assumption is that it does not matter (see Kaplow (1992) for this discussion) whether acts are de…nitively undesirable (h > B) or not (h < B).

Monitoring the criminal activity entails a cost for public authorities, equal to m(p), where for the sake of simplicity p is the probability of control (encompassing arrest, conviction and punishment for an illegal behavior). As is usual in the literature, we assume that this cost is …nanced through a lump sum tax t plus the expected …ne levied on the fraction of the population which is seen as criminal (either for whom the harmful activity entails the harm, or is not deterred from committing the crime). However, we assume general enforcement expenditures, such that p is exogenously set throughout in the text.

We focus here on punishment, i.e. the choice of a monetary sanction (penalty or …ne) f > 0. We assume that the management costs associated with the monetary penalty are negligible. We also assume that the maximal …ne is the legal wealth of the population, i.e. f 2 [0; w], assuming w > maxf e ; g g h p . We assume that the cost a crime imposes to the society is de…ned as a externality term a¤ecting individuals'utility level, with a very simple formulation: E = e qh, where q 2 (0; 1) is the probability of crime. As usual in the literature on crime enforcement, we will show that q = Pr(b b), with b denoting the deterrence threshold (to be more speci…cally de…ned therein). Hence

q = B Z b 1 B db = 1 b B .

Act-based sanctions

For simplicity, let us denote the individual who abides the law the "honest" people, and those who violates the law the "criminals". Under the act-based regime, let us denote the utility level of a criminal as:

u a c = w + b t pf 1 ba B ! e h
where ba is for the level of deterrence, while for an honest individual, we have:

u a h = w t 1 ba B ! e h
Hence as usual, ba is de…ned by u a c = u a h ; under the act-based regime, a potential criminal decides to undertake the activity if the bene…t he retrieves from doing so is higher that the expected punishment, i.e. if b pf = ba .

Regarding the behavior of the public enforcer, the public budget constraint writes in this case:

m(p) = t + 1 ba B ! pf (1)
We consider here only balanced-budget policies.

Harm-based sanctions

Under the harm-based regime, the utility level of a criminal is:

u h c = w + b t e pf 1 bh B ! e h
where bh is the level of deterrence, while for an honest individual, we have:

u h h = w t 1 bh B ! e h
Using the condition u h c = u h h , we obtain a deterrence level equal to bh = p e f . The budget constraint of the enforcer writes now as:

m(p) = t + 1 bh B ! g pf (2)

Optimal timing of sanctions

When the policy maker acts as a benevolent planner, the social welfare function under an act-based sanction writes as:6

S a = w m(p) + 1 B Z B pf (b g h)db (3)
The integral term in S a corresponds to the expected private bene…t net of the external cost associated with the illegal activity. The other terms (exogenous here) are the citizens wealth net of the cost of monitoring for public authorities.

It comes that the optimal sanction is:

f a u = g p h. The rate of crime is q a u = 1 pf a u B = 1 g B h.
Under a harm-based sanction, the enforcer has to choose a …ne f in order to maximize:

S h = w m(p) + 1 B Z B epf (b g h)db (4)
Then solving for the optimal sanction yields f h u = g e h p , and the rate of crime is q h u = 1 e pf h u B = q a u . Thus, despite a higher level of the …ne in the harm-based regime, both achieve the same level of deterrence determined by the expected sanction g h, which only re ‡ects the beliefs of the authority on the occurrence of harm. Thus it makes no di¤erence in terms of social welfare that law enforcement policies rely on act-based rather than harm-based sanctions: S a = S h (Polinsky and Shavell, 2000; Garoupa and Obidzinski, 2010).

Remarks. It is obvious that f a u < f h u .
The comparative statics of the optimal sanctions are worth to notice. On the one hand, the act-based sanction f a u is increasing in g and does not depend on e ; on the other hand, the harmbased sanction f h u is increasing in g = e .

Electoral competition

Assume now that citizens vote for the law enforcement policy. To simplify, assume there exist two candidates i = 1; 2, representative of two political parties, competing for national (presidential or legislative) or local (municipal) elections. 7 A programme or electoral platform (R; f ) consists in a regime of sanction R and a …ne f . We consider for obvious reasons that their exist only two regimes of sanction -the harm-based regime, R h , or the act-based regime, R a .

Regarding the monetary sanction, we still assume that the maximal …ne is the legal wealth of the population w. A candidate proposes to electors an electoral platform that will indeed maximize his chance to win the elections. We consider the (simple) majority rule for voting. All citizens are electors and do participate: each voter simply votes for the candidate whose platform allows him to reach the highest utility level, and in case where he is indi¤erent, he tosses a coin to decide for whom he votes. 8 The timing of the electoral competition game between the candidates and the citizens/voters is as follows: after that Nature moves at stage 0 (choosing the type of citizens, not observable for politicians), the electoral competition begins at stage 1, which is a simultaneous move (non cooperative) game between the candidates, where they both choose and announce their platforms, both satisfying the associated balanced budget constraint; at stage 2, elections take place, and citizens simultaneously choose between the two candidates; at stage 3, the elected candidate implements his policy9 -it becomes a law; at stage 4, citizens choose to abide or not the law; at stage 5, the law is enforced.

Equilibrium timing

We denote as f i h (f i c ) the choice of sanction by the honest (respectively, criminal) population in regime R i , i = a; h.

Proposition 1 When e 6 = g and e h 6 = B 2 , the unique symmetric equilibrium has the following features: 10 i) Assume e h > B 2 ; then: -If e g > 0, both candidates announce (R h ; f h h ) where

f h h = 1 2 h e g h p + B p e i ;
and the rate of crime is

q h h = 1 2 h 1 2 e g B h i .
-If e g < 0, both candidates announce (R a ; f a h ) where

f a h = 1 2 h e p h + B p i ;
and the rate of crime is

q a h = 1 2 1 e B h . ii) Assume e h < B 2 ; then: -If e g > 0, both candidates announce (R h ; f h c ) where f h c = 1 2 h e g h p + g e g B p e i ;
and the rate of crime is

q h c = 1 2 h 1 + e g 1 e B h
i .

-If e g < 0, both candidates announce (R a ; f a c ) where f a c = 1 2 e p h; and the rate of crime is

q a c = 1 1 2 e B h.
The proof is available on request 11 .

Remarks. The comparative statics are in a sense the dual of those found for the optimal sanctions. On the one hand, a act-based sanction is increasing in e and does not depend on g ; on the other hand, a harm-based sanction is increasing in e = g . It is also easy to see that

f a h > f a c and f h h > f h c , meaning that if e h > B
2 , a strong enforcement equilibrium emerges with a high sanction (f a h or f h h ); while if e h < B 2 , a weak enforcement equilibrium emerges with a low sanction (f a c or f h c ).

Under political competition, the characteristic features of the equilibrium depend on the sign of ( g e ) and e h B 2 . Let us consider …rst the case where citizens expect that crime yields a large external loss, e h > B 2 . In this case, the majority is composed of citizens who will abide the law, and vote for a large sanction. However, two alternative equilibria may emerge. If citizens' assessment of the probability of harm is larger than politicians' one, e > g , the prevailing equilibrium is associated with harm-based sanction; in contrast, if e < g , then the act-based sanction arises at equilibrium. Now when citizens expect that crime yields a small external loss, e h < B 2 , the majority is composed of citizens who will not abide the law, and vote for a low sanction. However the circumstances under which the ham-based (act-based) sanction arises at equilibrium are similar to those found before.

Thus whatever the regime of sanctions, it may be associated with a high level of punishment (strong enforcement), or a low one (weak enforcement). Nevertheless, it can be veri…ed that when e > (<) g , we have e pf h h > (<)pf a h as well as e pf h c > (<)pf a c : in words, political competition allows that the regime of sanctions for which citizens vote yields the highest level of deterrence at equilibrium.

Equilibria and optimum

The next propositions discuss more accurately the ine¢ ciencies associated with an electoral equilibrium. First, we compare the properties of a strong enforcement equilibrium to the optimum: Proposition 2 Consider the strong enforcement equilibria.

A/ i) If B 2 + e 2 g h > 0, the act-based sanction is larger than the social welfare maximizing one, and the rate of crime is smaller (i.e. f a h > f a u and q a h < q a u ); ii) if B 2 + e 2 g h < 0, the act-based sanction is smaller than the 1 1 See also the extended draft, Langlais and Obidzinski (2014).

social welfare maximizing one, and the rate of crime is larger (i.e. f a h < f a u and q a h > q a u ). B/ The harm-based sanction is larger than the social welfare maximizing one, and the rate of crime is smaller (i.e. f h h > f h u and q h h < q h u .

Proof. A/ The conditions under which f a h is enforced imply that g h > e h > B 2 ; on the other hand, we have:

f a h f a u = 1 p B 2 + e 2 g h = 1 2p ((B g h) + ( e g ) h)
which has an ambiguous sign, since the …rst term is positive (by assumption 1), and e g < 0. Note also that the rate of crime is de…ned as q = 1 pf B ; thus q a u q a h has a sign opposite to f a u f a h . Hence the results i) and ii). B/ The conditions under which f h h is enforced imply that g < e and e h > B 2 ; we have now:

f h h f h u = 1 p e B 2 + 2 e 2 2 g h g = 1 2p e (B g h) + 2 e 2 g h g > 0
given assumption 1. Note once more that the rate of crime is de…ned as q = 1 p ef B ; hence q h u q h h has a sign opposite to f h u f h h . Hence the result.

In the harm-based regime, a strong enforcement equilibrium is associated with a sanction larger than the optimal one. In contrast, under the act-based regime it depends roughly speaking, on whether the di¤erence between citizens and politicians assessment of the external cost of crime is large or small enough. When g is close enough to e , then f a h f a u tends to (B g h) which is positive -thus f a h f a u > 0. However, when g increases, the likelihood that f a h f a u < 0 also increases.

Then, we compare the properties of a weak enforcement equilibrium to the utilitarian contract:

Proposition 3 Consider the weak enforcement equilibria.

A/ The act-based sanction is smaller than the welfare maximizing one, and the rate of crime is higher (i.e. f a c < f a u and q a c > q a u ).

B/ i) If B 2 ( g e ) +
2 e 2 2 g h > 0, the harm-based sanction is larger than the social welfare maximizing one, and the rate of crime is smaller (f

h c > f h u and q h c < q h u ); ii) If B 2 ( g e ) +
2 e 2 2 g h < 0, the harm-based sanction is smaller than the social welfare maximizing one, and the rate of crime is larger (f h c < f h u and q h c > q h u ).

Proof. A/ Given the conditions under which f a c is enforced (since g > e ), it comes:

f a c f a u = h p e 2 g < 0 B/
The conditions under which f h c is enforced imply g h < e h < B 2 ; we obtain:

f h c f h u = 1 p e g B 2 ( g e ) + 2 e 2 2 g h = 1 2p e g B ( g e ) 2 g h + 2 e 2 g h
since under the condition g e < 0, the …rst term is negative whereas the second is positive. Hence the results. Thus, in a weak enforcement equilibrium, the act-based sanction is always smaller than the optimal one, whereas the harm-based sanction may be lower as well as larger. Once more, it depends roughly speaking, on whether the di¤erence between citizens and politicians assessment of the external cost of crime is large or small enough. When e is close enough to g , then

f h c f h u tends to 2 g h which is negative -thus f h c f h u < 0.
However, when e increases, the likelihood that f h c f h u > 0 also increases.

Concluding remarks

This paper is a …rst attempt to introduce some political inputs in a law enforcement analysis. Although a utilitarian enforcer is indi¤erent to the timing of sanctions, we show that this does not hold in a model of downsian electoral competition. We analyze the circumstances under which citizens vote between act based and harm based sanctions, and characterize the associated ine¢ ciencies. Interesting enough, our point rejoins a discussion found in classical textbooks touching to the timing of sanctions. The informal argument is related to the quality of information at the disposal of the public agent: the higher the quality of information held by the state relative to the knowledge of citizens themselves, the more attractive will be legal sanctions focused at the early stages of private decisions, and vice versa. 12 To explain when act versus harm based sanctions emerge, this paper takes the slightly di¤erent view that citizens and public agents may hold di¤erent information and assess di¤erent beliefs regarding the consequences of dangerous acts. This is relevant both in a regulatory context (think of new although unknown risks, such as related to innovation and R&D) and a criminal one (think of the depenalization of some activities). We show how both citizens'expectation about social harm, on the one hand, and the divergence between public agent's belief and citizens'one on the other hand, shape the timing and level of sanctions at equilibrium, under electoral competition.

Example are numerous. This authority can be an agency, the State, a local government. A mayor decides of the level of parking …nes in her town. An environmental agency checks …rms' compliance to environmental regulation. The legislative process determines the maximal …nes an o¤ender faces, according to the o¤ense's gravity.

[START_REF] Polinsky | The economic theory of public enforcement of law[END_REF] have highlighted factors such as limited assets, risk aversion, risk information which are relevant to the choice between act based and harm based sanctions. In the simple framework however, optimal deterrence can be achieved under both regimes. See also[START_REF] Garoupa | The scope of punishment: an economic theory[END_REF].

Since Stigler (1970), the introduction of illegal gains in the social value function is a controversial issue. Both the signi…ance and the objective of the penal code are still in debate among scholars; see[START_REF] Dau-Schmidt | An economic analysis of the criminal law as a preference-shaping policy[END_REF] and[START_REF] Lewin | The social value of crime?[END_REF].

See Langlais (2011). Some authors suggest simply that criminals'utility must be weighted; see[START_REF] Garoupa | The theory of optimal law enforcement[END_REF].

See[START_REF] Dari-Mattiacci | The unsolvable dilemmas of a paretian policymaker[END_REF],[START_REF] Fleurbaey | Any non-welfarist method of policy assessment violates the Pareto Principle: a comment[END_REF] and[START_REF] Kaplow | Any non-welfarist method of policy assessment violates the Pareto Principle[END_REF].

Our approach is in line with[START_REF] Garoupa | The scope of punishment: an economic theory[END_REF], and the ex post view of the utilitarian approach, meaning that the planner uses its belief g rather than citizens' belief e in order to assess social welfare.

Competing for elections here is like a rent seeking contest, an exogenous rent being obtained in case of victory is attached to holding o¢ ces, ministries and so on.

8 When both candidates obtain half of the voters, the winner is chosen tossing a

coin. 9 i.e., we assume that candidates commit to their own electoral platforms. 1 0 When g = e or eh = B 2 , a multiplicity of equilibria (symmetric and asymmetric) arises. We let this aside.

2 See for exampleBowles, Faure and Garoupa,

;[START_REF] Shavell | Foundations of Economic Analysis of Law, chapter 25 ‹ ‹ Economic Analysis of the General Structure of the Law and its optimality ››[END_REF] Van der Bergh and Visscher, 2008. We remind that in contrast, the formal literature proves that act and harm based sanctions are equivalent in terms of optimal deterrence.

We thank for their comments the participants to the Conference 2014 of the Association Française de Science Economique.