Introduction

1 The emergence of the knowledge society seems a main feature of developed economies at the start of the 21^st century. There is no question that new information technologies (NIT) represent a source of wealth for a society taken as a whole. The question of the impact of these technologies on distribution issues, either at a national level or at an international one remains open. Newspapers, for instance, are full of articles which express the fear of a digital divide between people who are connected and people who are not. The former at the opposite of the latter have access to knowledge which is a source of opportunities and wealth. In the same vein, the idea of an increase of the North-South gap is often mentioned. This paper questions the validity of this fear and investigates the main factors that can influence the evolution of inequality in a given society after the introduction of internet. We organize the discussion around a very simple model which figures out the adoption of internet in a closed economy. The question is so broad that we have to focus on some issues and to ignore some very important ones intentionaly. Among income sources, capital incomes are omitted. Indeed all inequality decomposition studies agree on the definite importance of earnings inequality in industralized countries see for instance Jenkins (1995) or Sastre and Trannoy (2001). Hence we restrict our attention to this income component. As a consequence our model is a growth model without capital. Another limitation of the analysis comes from the consideration of a closed economy. The interaction between international trade and IT adoption cannot be analysed in such a framework. Therefore the question precisely addressed in this paper is the impact of the IT revolution on earnings inequality at a domestic level.

2Two main ideas govern the model. From a qualitative point of view, the digital revolution can be analysed in the same way as two former revolutions in the knowledge technology. The first one is the invention of writing, the second one the invention of printing. What are the main characteristics of the diffusion of knowledge that writing brought to human societies ? According to Goody (1996), "writing overcame the limitations of memory in oral societies by providing for quasi-permanent storage in material form, which permitted precise communication over time and over space. Writing renders knowledge public in that its publication makes it available to all who can read. Restrictions come on the diffusion of knowledge before that particular moment. Afterwards it is open to a speed of circulation and to the accumulation and augmentation by others that change the nature of knowledge systems". Clearly, if we analyse the change operated by printing in occidental societies, it enormously extends the benefits brought by writing. The digital technology like printing has an impact on the two essential components of the costs borne by information providers, see Shapiro and Varian (1998) for developments. It reduces both the reproduction and the distribution costs. This change is captured in the model by a parameter that figures out the proportion of the knowledge stock of a given society that an individual can mobilize on its own. The value of this parameter increased once with the printing revolution and again with the digital one. A question raised here is whether internet will decrease the cost to be literate as printing surely did. Let us recall the importance of the first complete Bible in English published in 1535-1536 for the reading practice in Britain, a fact which is well documented (see Oxford (1997)). The evidence that internet will induce such a similar shock on education technology is not obvious for the moment, but it may be still to come (see Gates (1995)). In the reference model, we adopt a pessimistic view, and we assume that it will not produce any productivity gain in the education technology.

3The second main idea is that the interest to be connected to internet depends on your literacy. If you are illiterate, the interest of a connection is small if any. Since it is costly financially – hardware, software and connection spell – to say nothing of cognitive costs, we can suspect that people with a poor literacy score will not choose to be connected. On the opposite, people with a medium or high literate level will find an advantage to be connected to get a better job or a better life. In view of the asymetry between literacy choice and connection choice, it is useful to modelize the decision as a sequential one, first to decide to be literate or not, then to be connected or not for those who have chosen to be literate. Then at a personal level, it seems that we can establish a link between literacy and connection decisions. We still have to find some empirical evidence of such a link at a more aggregated level. Let us first agree on the meaning of literacy.

4According to the International Adult Literacy survey (IALS) (see OECD (2000), [2]) literacy is defined as "the ability to understand and employ printed information in daily activities, at home, at work and in the community, – to achieve one's goals, and to develop one's knowledge and potential". This broad definition encompasses the multiplicity of skills that constitute literacy in advanced countries. This definition is make more precise for the sake of measurement and is fragmented into prose literacy, document literacy and quantitative literacy. The first one covers "the knowledge and skills needed to understand and use information from texts including editorials, new stories, brochures and instruction manuals". The second one embodies "the knowledge and skills required to locate and use information contained in various formats, including job applications, payroll forms, transportations schedules, maps, tables and charts, while the third deals with "the knowledge ans skills required to apply arithmetic operations, either alone or sequentially, to numbers embedded in printed materials, such as balancing a chequebook, figuring out a tip, completing an order form or determining the amount of interest on a loan from an advertisement". The IALS stresses that it no longer defines literacy in terms of an arbitrary standard of reading performance, distinguishing the few who completely fail the test (the "illiterates") from nearly all the remaining in industrialized countries who reach a minimum threshold "those who are literate"). Indeed, it defines five levels of literacy from 1 to 5 according to scores achieved at some tests. Nevertheless it turns out that among the five levels of literacy, the first two, levels 1 and 2 are considered below a reference line [3]. It is this kind of reference line that we try to take into account here. In our model, we consider that there is a threshold between people who are literate and people who aren't.

5In a cross-section analysis made among 20 industrialized countries, it is possible to check roughly the existence of a relation between connection rate and illiteracy rate. The scattered diagram illustrates the relation between the ratio of computers connected to internet (at work and at home) per 1000 inhabitants (source : United Nations (1999) [4]) and the arithmetic mean of the proportion of people who are below level 3 at prose literacy, document literacy and quantitative literacy tests [5]. Indeed we consider that to be connected mobilize the three types of literacy already mentioned to some degree.(See Data Values in Table 1 in Appendix).

6The figure captures a potential log-lin relation. The empirical evidence gives some credit to this kind of relation and the results of the regression are displayed below.

7This result [6] does not infirm the view that there is a significant negative influence of the illiteracy rate on the growth rate of the proportion of people connected to internet. Since this latter variable is linked to an investment in information technology it is a reminiscence of a finding of Romer (1989) which shows that the initial level of literacy does help to predict the subsequent rate of investment.

8A more technical remark is in order. Endogeneous growth theory (see for instance Aghion and Howitt (1998)) has focused on the crucial role played by the accumulation of technological knowledge on the growth process. General interest into questions of how technical change and endogeneous growth affect inequality has been recently revived by new empirical evidence. In particular the possibility of a skill-biased technical progress has been intensively discussed. This bias reveals and enhances new differences in abilities among workers across or within educational cohorts (see Juhn, Murphy and Pierce (1993)). In this burgeoning literature (see for example Aghion, Caroli and Penelosa (1999)) one can detect a somewhat irritating feature for the specialist of the measurement of inequality. Very peculiar income distributions have often been considered. For instance the density is assumed to be concentrated on two values : unskilled and skilled wages. Inequality is then easily encapsulated by the ratio of these two numbers. One can kindly remark that the tremendous work of statisticians and economists to establish rigorous measures of inequality measurement comes from the fact that income distributions cannot be summed up in such a simple parabola. Our aim would be to attack the question of the influence of technological shocks on the acquisition of knowledge on economic inequality in incorporating an individual heterogeneity. Results in terms of social dominance tools like Lorenz quasiorderings (see for instance Atkinson (1970) and Sen (1993)) will be investigated. In view of the generality and robustness of the results it will be worth it to overcome the technical difficulties generated by handling more complex instruments.

9I begin in Section 2 with a model of literacy decision in a world where printing already exists. Section 3 describes an extension of the model which analyses literacy and connection choices in a society where internet has been invented. Section 4 contains policy implications and some conclusive comments. All proofs but one are given in the companion working paper [7].

Literacy and Inequality

10At each period t, t=1,...,ω, a generation, called the generation t, composed of a continuum of agents, lives one unit of time. Individuals are supposed to be identical with respect to their physical ability, w₀>0. They are heterogeneous with respect to cognitive ability ω. This variable is distributed according to a cumulative distribution F(.) which admits a density f(.) over a finite support

11The model outlines an artisan economy with no land and capital. The focus is on the role of knowledge in income distribution. Since we want to explain inequality of lifetime incomes, decisions of labour supply are not modelized. Nickell and Layard (1999) [8] find that the best predictor at a macroeconomic level of earnings inequality in OECD countries is the inequality of scores obtained at quantitative literacy tests. It is such a finding that the model tries to capture. At each generation, individuals have only one choice to make, to be literate or not. Of course education plays a great part to become literate but remaining literate demands an effort during all your life. In that sense, an individual can confirm or infirm a choice made by his parents while a child.

12In case of a negative answer, the lifetime income of an illiterate individual of type ω belonging to generation t denoted y_t(ω,0), is equal to

13He can only sell his brute physical strength on the labor market as for instance a road worker. In case of a positive answer, earnings are given by a C.E.S return function defined by two inputs, the cognitive ability and the stock of knowledge accumulated at the previous period, denoted K_t-1. The lifetime income of a literate individual of type ω belonging to generation t, denoted y_t(ω,1), is equal to

14where θ_t a parameter in (0,1) represents the part of knowledge which an individual can resort to. This formulation tries to capture the key elements which influence the earnings of an intellectual job, let us say for instance, a writer. Her income is generated by the combination of two production factors, a private one, the innate talent, and a public one, the used knowledge of a generation, θ_tK_t-1. Among the literates, the natural ranking is preserved, but the public good effect of knowledge mitigates the inborn difference beween individuals. The portion of knowledge that an individual, θ_t, can mobilize for her benefit is dependent on the technology and can change from one generation to another. The elasticity of substitution between talent and knowledge

15Before pursuing, let us establish a link between this expression and the human capital earnings function. It seems to be easier with the Cobb-Douglas formulation. Just for the exercice of comparison, one adopts a double indices notation : y_it denotes the earnings of an individual i belonging to generation t. Assuming just for the exercice that θ is specific to an individual i, (ii) is more suitably written

16Trying to estimate this expression, it seems clear that the last term of RHS is a residual, since ω_i is not observable. The same remark may be made about θ_i, but we can postulate that education and experience increase it. Let us assume that

17where S represents years of completed education, E represents the working experience and η is a statistical residual.. If we combine the two expressions above, we are back to Mincer's model (1974) for which the log of individual earnings in a given time period can be decomposed into an additive function of a linear education term and a quadratic experience term

18where e is a statistical residual and letting

19which allows to infer the value of α. We can conclude that formulation (2) is compatible with standard human capital earnings function and can offer a plausible interpretation of the constant in Mincer's equation. Endogeneous growth theory puts knowledge in the forefront. The expression above suggests that it may be a good idea to do it as well for labor economics.

20Going back to the model, utility is assumed to be quasi-linear in income and in case of a illiterate person, his lifetime utility is given by his income. A parameter c_t enters in the lifetime utility of a literate individual and it figures out the financial cost to be literate as well as a monetary appraisal of the cognitive effort implied by such a learning. Since we do not want to cope with two parameters of individual heterogeneity, we assume that this learning cost does not vary across individuals. With obvious notations we define

21Innovations in information technologies, educative training or government intervention through for instance free compulsory public education or vouchers, can reduce the learning cost c. An individual decides to become literate iff

22Hence, at each generation, a threshold in terms of cognitive talent, ω_t^*, is implicitly defined between those with a cognitive ability larger or equal who will choose to become literate and those with a strictly smaller cognitive value who find this effort unvaluable. This threshold is defined by

23Quite naturally, this threshold increases in learning cost and decreases in knowledge as well as in the proportion of knowledge absorbed by an individual. The income cumulative distribution of a generation which presents a point mass in ω ₀, can be easily deduced

24When a generation is fully literate, the income distribution is described by one of the last two equations.

25To end the description of the model, we have to specify the law of accumulation of knowledge. We assume that only literate people can extend the knowledge of a society. Moreover we assume that knowlege cannot become obsolete. Knowledge grows at a constant rate β-(0,1) in a fully literate society. Since the literacy rate of generation t is equal to

26The dynamics across generations of such an economy can be easily expressed if we assume that the initial stock of knowledge, K₀, accumulated by the oral tradition is strictly positive. Moreover we suppose that

27The invention of writing by itself proves that in the history of mankind there was an individual who satisfied this inequality [9].

28Proposition 2.1 Let c_t and θ_t be constant over time. Under the assumptions, there is a period t^* from which the society is fully literate,

29Hence in case of a stability of the parameters of the economy, each generation becomes more literate than its precursor until a generation becomes fully literate. From this generation, knowledge grows at a constant rate. Then two periods can be distinguished, a period of fully literate generations,a mature period, and an initial period where illiterate and literate people coexist, a transition period.

30First, we begin with the analysis of the evolution of income inequality for the mature period. More specifically, it is instructive to learn the consequences of choosing a particular value for the elasticity of substitution between talent and knowledge on the shape of the evolution of income inequality.

31Inequality is measured by an index of inequality consistent with the Lorenz criterion. The Lorenz ordering of distributions involves the comparison of the income shares accruing to different fractions of the population. Given a cumulative distribution function G defined on a support

32With x=G^-1(p), the Lorenz curve of a distribution G is given by

33Actually, L_G(p) represents the proportion of total income possessed by the px100 % poorest income units in configuration G.

34Definition 2.1 Given F and G two distribution functions, we say that F weakly dominates G in the (relative) Lorenz sense, which we write F≥_LG if L_F(p)≥ L_G(p) for all p-[0,1].

35We denote as >_L the asymetric component of ≥_L. Sometimes, it is more suitable to present the results in terms of inequality indices.

36Definition 2.2 Let F be the set of distribution function on X. A relative inequality index is a real valued function I defined on F which is Lorenz consistent, i-e, I(F)≥ I(G)< = > F≥_LG and which is equal to zero in case of a point mass.

37The set of relative inequality indices will be denoted I.

38Our second proposition gathers some results about the earnings inequality in fully literate societies. In this particular case, the income distribution G_F is derived from the distribution of talents F through the following relation

39When we make comparisons of inequality, we would like their domain of validity to be as extensive as possible, namely, that they do not depend on the talent distribution. Here we stick to this requirement, which is justified by our ignorance of the true distribution of skills. But we have to recognize that this care about robustness has a cost. In some circumstances, it can be impossible to conclude to an increase (or a decrease) in inequality whatever the distribution of talents. From a formal point of view, this investigation relies on results obtained about the progressivity of taxation schemes, see Jakobsson (1976), Eichhorn Funke and Richter (1984), Le Breton Moyes and Trannoy (1996).

40Proposition 2.2 Let t≥ t^*. It is composed of five statements valid for all I- I and for all F- F.

41(i) If K_t-1=0, the income inequality is null for all ρ≤0 and the income inequality is equal to the natural inequality, namely, I(G_F)=I(F) for the case 0<ρ≤1.

42(ii) Whatever the values of K_t-1 and ρ,

43(iii) In the Cobb-Douglas case, inequality is invariant to the stock of knowledge, provided it is positive.

44(iv) Let θ_t=θ. Then,

45Figure 2 illustrates the evolution of inequality according to the value of the elasticity of substitution which is the key parameter. If talent and knowledge are rather substitute, a fully literate society will converge toward a fully equal society. If talent and knowledge are rather complementary, the inequality of talents will become the dominant factor in the long run for a fully literate society. The Cobb-Douglas case provides a unique evolution, the steady state is reached immediately. A gain in knowledge increases the income of each literate person in the same proportion. In the following, we will refer to the inequality in a Cobb-Douglas economy as the Cobb-Douglas inequality.

46In view of these results, the plausibility of all scenarii does not appear to be the same. It seems clear that the case for the substitution is rather weak. Let us now examine the Cobb-Douglas and complementary cases. For obvious reasons the data available on earnings inequality on the long run, since for example the invention of printing, are scarce. A noticeable exception is Britain for which we have access to statistical elements from the late eighteenth century (Williamson (1985)). Lindert (2000) [10] estimates on the basis of the more recent articles that "It is hard to say there was any rise-fall pattern in pay gaps within the non-farm sector across the nineteenth century". The beginning of the twentieth century corresponds surely to a configuration where almost all Britons received a compulsory education. Piketty (2001) finds a similar empirical evidence of a more or less constant earnings inequality over the twentieth century for France. Hence there is no strong empirical evidence against the Cobb-Douglas case and for this reason it will occupy a proeminent place in the following. To simplify the notations, from now on G_F(y_t ;ρ)≡ G_F(y_t).

47Now we study the evolution of income inequality in the transition period. On the one hand, we would like to compare inequality of income distribution within the generation t^* and within a generation t<t^* and on the other hand we would like to compare the inequality between two transition generations. The first question raised is about the comparison of a fully literate society and a partially literate society, while the second question addressed is : Does the extension of literacy bring inequality in uncomplete literate societies ? As stated by the next proposition a conclusion independent of the distribution of talents is impossible to achieve.

48Proposition 2.3 Let c_t and θ_t be constant over time. (i) It is impossible to obtain a ranking of the Lorenz curves associated to the income distribution of generation t^* and to the income distribution of a generation t with t<t^* valid for all F- F.

49(ii) It is impossible to obtain a ranking of the Lorenz curves associated with the income distribution of generation t and to the income distribution of a generation t' with t<t'<t^* valid for all F-F.

50The proof of the above proposition teaches us that the trouble comes from the discontinuity of the income function at ω_t^* which jumps from ω₀ to ω₀+c. Hence the discontinuity introduced by the literacy cost produces such an impossibility to rank income distributions from an inequality point of view [11]. Unfortunately the obtention of positive ones implies the restriction of the domain of talent distributions. The next proposition follows this route. Hence we can expect that a condition requiring the discontinuity to be not too large will help to obtain explicit comparisons. Indeed one of the conditions which emerges bounds the ratio

51Let us denote

52Since α is the elasticity of the return function to the talent, we term ω^α the "dollar-talent" and μ_F(α) the "dollar-talent" average. The dollar-talent average up to ω is equal to

53The dollar-talent ratio up to ω is defined as

54This ratio is bounded by 1 and

55Proposition 2.4 Let c_t and θ_t be constant over time. Let F- F be such that T(ω) is decreasing. Then for any such F- F, there exists a period t_F with

56 Proof. See appendix A

57Hence the lower the literacy cost is, the more complete the ranking of income distributions is. The most plausible dynamics is that starting from a complete equal income distribution, the invention of writing or printing introduces inequality, albeit many partially literate generations experiment a level of inequality strictly smaller than the level characterizing a fully literate society. It may be the case that inequality is higher in final transition periods than in the steady state. It is still possible that beyond t_F not definite conclusion is obtained. Let us recall that these findings concern the Cobb-Douglas case.

The Connection Decision and Inequality

58We provide an extension of the model [12] which captures the invention of internet. Individuals have the possibility to be connected to internet at a cost c_t'. It figures out the financial cost to be connected (personal computer, connection costs) augmented by cognitive costs associated to the learning period. Albeit individuals can choose to be connected whatever their literacy mastery is, we assume that the benefits to do so are substantial only if they are fully literate. In this version of the model we capture these benefits through a parameter

59Hence the lifetime income of a connected literate individual of type ω belonging to generation t, denoted y_t(ω,1,1), is equal to

60Since a rational illiterate person has clearly no interest to connect, the choice of an individual is between three options ; to be illiterate and unconnected, to be literate and unconnected and to be literate and connected. The utility associated to the first option is defined by

61the utility of the second by

62and the utility of the third by

63An individual decides to become literate and connected iff

64The first inequality defines a threshold

65as well as the second inequality

66An individual becomes literate and connected iff

67while an individual chooses to become literate and unconnected iff

68Finally an individual remains illiterate iff

69Two regimes can be distinguished according to the respective values of this three thresholds.

70Proposition 3.1 (i) First Regime. If the following condition holds,

71then, in any transition period, there only exists two kinds of individuals, the literate and connected ones for which

72(ii) Second Regime. Otherwise, in any transition period, there exists three groups of individuals, the literate and connected for which

73The condition stated in this proposition means that the connection benefit is larger than the connection cost relatively to their respective values associated to literacy. If this condition holds, we are going back to the configuration studied in the second section, except that the threshold value is different. If this condition does not stand, there are three groups, a regime reflecting the present configuration in many countries. We start by studying inequality evolution in the simplest case of a fully literate society.

The Advanced Country Case

74 The period at which internet appears is assumed to be posterior to t^*. W.l.o.g, we will suppose that internet is discovered in t^*. Therefore an individual is connected if

75and unconnected otherwise. Even if no society can be considered as fully literate in the sense given in the introduction, this case proves to be instructive as a benchmark. We assume that all parameters are constant through time and that internet does not speed up the growth rate of knowledge. Admitting that it represents a pessimistic view, the law of accumulation of knowledge is still given by

76Proposition 3.2 Let

be constant over time. Generations become more and more connected and there is a period t^** from which society is fully connected,

.

77The evolution of inequality in this case is described in the next proposition. The first statement compares the dynamics of inequality with internet and without internet. A superscript equal to 1 refers to the situation "without", a superscript 2 to the situation "with". The evolution of the knowledge stock is the same in the two configurations. In the second one, we already know that inequality will remain constant beyond t^**. The second statement compares the inequality for two generations living in the period of transition between a fully literate society and a fully literate connected society.

78Proposition 3.3 (i) For all t^*<t<t^** and for all F- F

79(ii) The Lorenz curves associated to G_F(y_t) and to G_F(y_t') with t^*<t<t^** and t^*<t'<t^** intersect.

80In a fully literate society, the introduction of internet generates inequality for the transition period but it is impossible to rank income distributions of the period of transition. Indeed both the poorest and the richest individuals experiment a decrease of their income shares with the diffusion of internet.

Developing Country Case

81 We assume that the internet invention is anterior to t^* and occurs in period t_I. We start by the analysis of the first regime.

82Proposition 3.4 Let

be constant over time. Under assumption prevailing in first regime, there is a period t^***from which the society is fully literate and connected, i-e,

. Moreover t^***≤ t^*.

83 Proof. The proof of the first statement is similar to that of proposition 2.1. The second statement derives from the fact that

84The transition period is shorter with internet. It speeds up the convergence process to a fully literate society. Since with a Cobb-Douglas return function a fully literate is more unequal than any partial literate society, we can expect a greater inequality for the transition period. Indeed the next proposition shows that this intuition proves to be true provided the connecting cost is sufficiently large. With the same notations than with the advanced country case we state the following result.

85Proposition 3.5 Let

be constant over time and assume that the first regime holds. Let F- F satisfying the following condition

86Then for any t with t_I≤ t<t^***,

87In this first scenario, the two costs boil down to a generalized literacy cost. If the ratio of the relative literacy cost – the literacy cost relative to the minimum wage – is larger than the ratio of illiteracy rates weighted by their respective thresholds, then the comparison is unambiguous. This condition means that the configurations have to be sufficiently distinct in order to be able to rank the respective income distributions.

88We now turn to the second regime.

89Proposition 3.6 Let

be constant over time. Under assumption prevailing in the second regime, there is a period t^* from which the society is fully literate and a period t^** from which the society is fully connected. Moreover t^**≥ t^*.

90 Proof. The first statement is a consequence of propositions 2.1 and 3.2. The second statement derives from the fact that

91The inequality evolution in this second regime is more in tune with the common wisdom. Internet will generate more inequality at each transition period up to the first fully literate and connected generation.

92Proposition 3.7 Let

be constant over time and assume that the second regime holds. Then

Policy implications

93 The teachings of the model are the following. They concern the Cobb-Douglas case, a case where the elasticity of substitution between talent and knowledge is equal to one. This case is attractive since in a fully literate society inequality remains constant over time as knowledge increases. We show that starting from a totally illiterate society, earnings inequality will increase gradually as the illiterate rate diminishes and at some point can itself exceed its stationary value.

94In a fully literate society the internet revolution produces a temporary upsurge of the earnings inequality like any innovation technology. Inequality will follow an inverted-U curve, a Kuznets curve, as per capita income rises. But in the long run, inequality will return to its stationary path.

95When we move to the case where internet is introduced in an incomplete literate society, a case which can surely describe the situation of developing countries, two configurations must be distinguished. In the first one the relative benefit of internet, in this model a larger access to knowledge, is so high to its relative cost that every literate individual connects. In this case internet rises the interest in being literate and the illiteracy rate decreases at a faster speed than the one which will be observed without internet. Thanks to internet such a society will converge to the inequality stationary state, experimenting a shorter transition period. The most impressive rise of inequality during this period will largely be a by product of this reduction of the transition period. In this case the impact of internet is ambiguous. On the one hand in the short run inequality increases. On the other hand, internet speeds up the convergence process of developing countries toward a fully literate society.

96A more pessimistic case has also been investigated where internet has only bad effects on the inequality dynamics. This time the relative cost of internet is higher than its relative benefit in comparison with literacy ; by way of consequence only a fraction of the literate population decides to connect. For a long time – the transition period which lasts until everyone is literate and connected- inequality will rise comparatively to a reference situation without internet.

97As we move into the information age, policy-makers are increasingly concerned about the role played by knowledge in enhancing productivity growth and innovation. In view of the results they should also be concerned by its role in shaping inequalities. A public policy can prevent the occurence of the worrying scenario. On the one hand, providing free training public programs to internet and organizing the competition on the market of providers to internet can decrease the generalized connection costs. On the other hand, supplying the ADSL network on the whole territory like in Sweden can improve the benefits brought by internet. Such a policy acknowledges the public good effect played by knowledge which mitigates the effect of talent if both factors are not too complementary in the return function. The less literate a society is, the less favorable the impact of internet will be on the inequality dynamics. The efficiency of the education system to innoculate the basic knowledge and know-how proves to be more crucial than it has been at every prior period. In this respect the scores obtained for instance in France at the entrance of junior high school are rather worrying. Only 68 % (respectively 64 %)of pupils in average pass a prose literacy (resp. quantitative) literacy test (Le Monde (2001)). It is difficult to accept a vision in which 35 % of the population will be left over the cognitive progress. Obviously, internet can provide an improvement of the educational methods, an aspect which is not modelized here but as Bill Gates admits (Gates (1995)), we are still on the sides of the road ahead to this respect. For sure educational sofware will increase earnings inequality through a rise of the gains associated with intellectual property before they maybe contribute to a reduction of the illiteracy rate.

98The model built is certainly a prototype and can be supplemented in several directions. Apart from considering the potential impact of internet on educative technology, the direct impact of internet on the speed of accumulation of the knowledge stock can also be incorporated in the model. An increase of this speed can be viewed as plausible. For instance, Lyman and Varian (2000) estimate that the growth rate of the worldwide production of books of original content is about 2 percent [13]. It will be interesting to see whether this rate grows in the near future. More immediate extensions would be to investigate other cases than the Cobb-Douglas one and to try to make a calibration of the model. We have modelized the literacy and the connection decision as a deterministic discrete choice. Introducing uncertainty will smooth the earnings distributions and make them closer to those observed. All these directions are matters for further research but we think that the main message is already provided by the model. Two forces drive the earnings inequality with internet. On the one hand, the gap between literate and non literate people will increase. On the other hand, the incentive to become literate increases. The first one will surely be dominant for a preliminary period. It is a matter of hope that the second one will prevail in the future.

APPENDIX

99Proof of Proposition 2.4

100 Proof. To save notations G_F(y_t(ω))≡ G_t. We recall that the slope of a Lorenz curve of a distribution G which admits a density over the support

101with

102where at end points the slope must be interpreted as the left or right slope, see Lambert (1993). The Lorenz curves of G_F(y_t(ω)) for t<t^* are not differentiable at

103while the right slope at that point is equal to

104Fact. For any t, the income functions defined by expressions (ii) are rank preserving, namely, they are weakly increasing in ω. Then the proportion of the population which is poorer than or equal to an individual of type ω is always equal to F(ω) for any t. Therefore the slope of the Lorenz curve of G_F(y_t(ω)) evaluated at p =F(ω ) is equal to :

105Step1. We prove that if

106then [14]

107and

108

109Indeed, using the fact

110we obtain

111while

112Then

113which gives the condition expressed in (52). Moreover

114and

115

116Then

117again the condition expressed in (52).

118 Step 2. We now prove that

119Suppose for a contradiction that

120Combined with (54) we obtain

121which contradicts

122Step 3. We now prove that

123Suppose for a contradiction that

124We already know that the slope of L_{G_t} is constant and equal to

125which contradicts

126Step 4. Let t_F be the first period such