Andrew Mearman
email: andrew.mearman@uwe.ac.uk

Don J Webber

Paul Dunne

Paul Downward

Student Participation in Sporting Activities

Keywords: I12 -Health Production: Nutrition, Mortality, Morbidity, Disability, etc. < I1 -Health < I -Health, Education, and Welfare, L83 -Sports|Gambling|Recreation|Tourism sport, participation, social capital, time, students JEL Classification: L83, I12, J22 Sport, Participation, Time, Social capital, Students

come

Introduction

Most universities encourage students to participate in sports and physical recreation activities during their time at university and often provide a wide range of sports and recreation opportunities to cater for all levels of participant needs. Universities often encourage participation in other physical recreational activities by students who are not interested in sporting activities. For example, the University of Birmingham's Student Charter explicitly encourages students who do not already have an active lifestyle to participate in physical recreational activities through the provision of a relaxed, nonintimidating environment and an attractive and relevant programme (University of Birmingham, 2004). Although many universities spend large sums of money in supplying sporting facilities for their students to use, comparatively little is known about the factors that influence the quantity of student participation in sporting activities and whether a broad or narrow range of sporting facilities is necessary. This paper presents an analysis of factors that influence the quantity of participation in sporting activities by students in a British university by drawing on data collected from a survey and employing ordered logistic regression analysis.

Theoretical background

There is now a vast literature on the demand for sports participation. 1 This literature reflects the state of the art of sports studies and economics: it is varied; it comes from various theoretical perspectives, and from outside Economics. Nonetheless, the standard treatment of demand for sport remains the neo-classical theory, which analyses 1 There is also a large literature on the demand for watching professional sport (see, for example, [START_REF] Jones | Our fans in the North: the Demand for British Rugby League[END_REF]. The two demands might be related, given that watching professional sport might inspire emulation. However, the linkages between the two types of demand are not explored here. activity, the prices of other goods, and income. Theoretically, preferences are also included, as they must be for a neo-classical treatment; but empirically, tastes are often omitted.

In [START_REF] Becker | A Theory of the Allocation of Time[END_REF] and [START_REF] Gmt | The Economics of Leisure and Recreation[END_REF], it is acknowledged that sports is a composite good, which involves several derived demands, such as equipment, clothing, membership of organisations, transportation to the place of the activity, and price of the facilities (assuming facilities are available). The composite nature of the good significantly complicates the analysis. For example, neo-classical treatments tend also to include a time element in their analysis. Clearly sport, as with all forms of leisure, involves consumption of time; moreover, time often plays a significant role in affecting a person's demand for sport. Furthermore, the time required for sports varies according to the sport, i.e., some sports are more 'time-intensive' than others, which might be more 'goods-intensive':

mountaineering is considerably more time consuming than table tennis, for example.

Typically neo-classical treatments analyse time allocation via the labour (or income)leisure trade off. That framework applies utility theory, usually indifference analysis, to the choice of taking more or less leisure, usually in response to changes in wage or tax rates, subject to physical limits such as the need for sleep and the absolute limit of hours per time period. The analysis of such changes tends to be decomposed into familiar income and substitution effects. In standard analysis, the substitution effect usually acts to shift demand away from the good (or activity) whose opportunity cost has increased as the result of a price change. With regard to the income effect, it is usually assumed that leisure is a normal good. Thus, in response to an increase in wage rates, the substitution overall effect depends on the relative sizes of the two effects. Thus, according to that analysis, historically, rising wage levels in Western countries have caused leisure levels to increase, as the income effect has dominated (Gratton and Taylor, 2000, Ch. 2).

For a number of reasons (Gratton and Taylor, 2000, pp. 58-9), the standard analysis has been regarded as overly restrictive for analysis of the participation decision. Therefore, sports economics has become multidisciplinary, culminating in a much more complex picture of sports demand than in the neo-classical model; however, many of the additional variables cited as causing sports participation can be reconciled with the neo-classical

model. An obvious one is age, which is shown to be negatively correlated with sports participation (see Gratton and Taylor, 2000, p. 74;[START_REF] Thompson | Leisure Time in Midlife: What are the Odds?[END_REF]. However, [START_REF] Rodgers | Rationalising Sports Policies; Sport in its Social Context: International Comparisons[END_REF] argued that age per se is not related with participation; rather, people who have an established familiarity with sports -what Rodgers calls 'sports literacy' -and have engaged regularly and deeply in sports as younger people ('sports careers'), will tend to carry on with sports later in life. For the rest, who are coerced into exercise at school but otherwise did not participate in sports, this is not the case, and their participation rates will fall. With such concepts, it is clear that the analysis has moved out of the raw economic model and into notions of habits (and their persistence).

Unsurprisingly, psychology has been influential in helping to explain sport participation.

Several authors have highlighted the importance of sport in generating psychological well-being through stimulation [START_REF] Scitovsky | The Joyless Economy[END_REF], so-called 'peak experiences' [START_REF] Lipscombe | The Relevance of the Peak Experience to Continued Skydiving Participation: A Qualitative Approach to Assessing Motivations[END_REF], feelings of control [START_REF] Csikszentmihalyi | Beyond Boredom and Anxiety[END_REF] [START_REF] Alexandris | Demographic Differences in the Perception of Constraints on Recreational Sport Participation: Results from a Study in Greece[END_REF]. Furthermore, sport participation might be affected by gender (see Gratton and Taylor, 2000: 75;[START_REF] Thompson | Leisure Time in Midlife: What are the Odds?[END_REF], ethnicity, and educational attainment [START_REF] Thompson | Leisure Time in Midlife: What are the Odds?[END_REF].

Therefore, there is a large range of possible causal factors for sports participation. The literature does not suggest one simple model which might be estimated and/or tested. The goal of this study is to identify whether there is any evidence to support these theories from students who might have more time to participate in sporting activities than workers.

Knowledge of factors influencing sporting participation by students is important if universities are going to optimally allocate funds to meet the needs of students, especially in the UK where recent evidence suggests a move towards greater proportions of students working long hours to support themselves while studying for university qualifications, which can impinge on the number of hours available for study and socialising.

Data

Data were collected via a questionnaire of self-reported, closed questions designed to gather evidence for and against the theories discussed above and was distributed to students following two modules in two levels in one British university. All respondents in the sample were classified as being full-time students. The questions attempted to capture the diversity of motives for sports found in the literature. A series of questions dealt with the types of sports played; constraints, including cost, on the ability to play sports, which takes into account competing demands on their time; physical, psychological and social motives for sports participation; and details of the sports played. Given the sample size (n = 85), it would clearly not be appropriate to make strong inferences about the population

F o r P e e r R e v i e w 6
of students in general, nor indeed of the whole student body at the university at which the data were gathered. Descriptive statistics are presented in Table 1.

Table 1 here

The sample descriptive statistics show a number of key features. First, the sample comprises active sporting participants, who on average play 3 or 4 sports. Indeed only 6

of the sample played no sports whatever. Sport is defined broadly: first, by allowing the students to define the sports they participate in; second, by allowing those responses to stand. The range of sports cited is extremely broad, including walking, which was a common response, skiing, and yoga; however, otherwise, the most common sports were as might be expected, including football, rugby, netball, (field) hockey, swimming, tennis, and running.2 To some extent, sports played conform to gender stereotypes (no men play netball, for example); however, a number of women play cricket, football and rugby, reflecting the shifting gender profile of those sports. A slight majority of the sports played were competitive, although this was less often in an organised competition, and even less often intensive (in its level of activity and exertion).

Most respondents were around the age of 20; all had access to university sports facilities;

all lived in the same city (term-time), so differential access to local facilities was not relevant. Given the situations of the respondents, none were engaged in high level managerial work. Moreover, given that all of the respondents are full-time students, and are assumedly not the main wage earner in their family (although they might be in their student accommodation), personal income could be less relevant to their sports choices than it might otherwise be. Nowadays, student income in England and Wales is comprised done by the student. However, parental income (if made available to the student) could affect the student's need to work while at university, and therefore their time available for sports participation. Furthermore, parental income can affect the range of sports available to them prior to university, either through their location or the expenditures necessary to pay for those sports, or again by affecting the students' need to work prior to university.

However, the information on the financial background of the breadwinner in the household was not sought: it was felt that any information received on this question would be inaccurate.

Most respondents live with other students, as is typical of the population. There was a small majority of men in the sample, and a larger majority of white respondents; however the data did not suggest that ethnicity is much of a factor affecting sports participation.

Indeed, all those in the sample (men and women) who did not participate in sports were white. At least in our sample, some preconceptions about ethnicity (often through religion) and its impact on gender roles and hence participation in sports are challenged.

Results

Initially, a series of bivariate analyses and pivot tables were estimated. A selection of these is presented in Tables 3-5 in the Appendix. The bivariate analysis suggested that respondents do sport because of the feeling it gives them; and to augment their social capital. This suggests that for universities, therefore, arranging opportunities for sports is a sensible strategy, one that might also have positive social spillovers, further enhancing the student experience. Further, sports participation was part of an investment in health [START_REF] Grossman | On the Concept of Health Capital and the Demand for Health[END_REF]; however, interestingly, citing fitness as a motive for sports suggest a strong sports literacy effect was present. Cost of sports participation did not affect participation rates. However, time did have an effect. We note that students adopted strategies for dealing with the perceived time constraint, which included doing less sports, but also involved changing their chosen sports. With all of these categories, gender did not seem to play an important role in affecting participation.

Multivariate Analysis

To obtain a better understanding of the factors that influence the quantity of sports that students participate in, an ordered logistic regression was employed to identify the determinants of the quantity of sporting participation. The corresponding results are presented in Table 2.

The literature suggests a wide range of plausible causal factors for sports participation.

Consequently, socioeconomic variables, preferences for types of sports, reasons for not participating more, motives, partner's sporting activities and work hours were all employed as explanatory variables. We have employed the 'general-to-specific' modelling strategy (see originally, [START_REF] Davidson | Econometric Modelling of the Relationship between Consumers' Expenditure and Income in the United Kingdom[END_REF] of two distinct types: first, we eliminate variables from the model on the basis of theoretical reductions; second, the elimination process is purely statistical. One process acts as a check on the other; and both processes generate similar results. The general model is presented in column 1 in Table 2. In line with the discussion above of sports literacy [START_REF] Rodgers | Rationalising Sports Policies; Sport in its Social Context: International Comparisons[END_REF], if the student participated in sports before attending university 'Sportsb4uni', then this had a positive and significant effect on the quantity of sports participation (measured by the number of sports participated in). Similarly the evidence that a lack of time is a reason for not sports; the partner might encourage the person to participate in the sports even when he/she does not necessarily feel like participating in sports at that time. The quantity of time that the student devotes to work has a negative effect on the quantity of sports participation; the greater the time spent on work then the greater the effect on reducing the quantity of sports participation. This is also borne out in the squared term of work hours. This result largely supports the findings from the bivariate analysis.

Column 1 might be biased as most people have other interests and being not interested in some sports does not preclude an individual participating in a different type of sports (perhaps they just haven't found the sports yet in which they are interested). Also, if the respondent's partner does a different sport then this is again not necessarily a direct reason why the person does not participate in other sports. These corresponding variables are then excluded to simplify the model and the results are presented in column 2 in Table 2. The magnitude and significance of the coefficients of the explanatory variables remain 2 suggests that, in line with the earlier results, sports literacy accounts for an important part of sports participation: those individuals who did sports before coming to university were statistically significantly more likely to participate in a greater quantity of sports. Interestingly, respondents whose motive was to meet new friends also participated in more sports; in this way, sporting participation could be seen as a fit way of dating or of enlarging the individual's peer group. The results from the theoretical reduction of the general model yield a specific model, which consistently suggests a strong and statistically significant effect of greater working hours impacting on sports participation. We then employ the log-likelihood ratio test for variable deletion to reduce statistically the model to only the most statistically significant (and stably so) variables; these are presented in column 5 and empirically support the finding above that sports literacy, meeting new friends and work hours all influence the quantity of sports participation. However, once some variables have been omitted it also indicates that being male increases the quantity of sports.

Table 2 here

Conclusions

Given that many universities spend large sums of money in supplying sporting facilities for their students to use, comparatively little is known about the factors that influence the participation rates of students in sporting activities. This paper presents an analysis of factors that influence the quantity of participation in sporting activities by students in a British university by drawing on data collected from a survey and employing ordered logistic regression analysis.

The results from multivariate analysis suggest that the number of hours in work has a Participation in sporting activities is seen as a way of increasing social capital: students participate in sports in order to create new or develop existing social relationships. In addition, the study does support the theory that agents invest in physical capital, i.e., their health by participating in sports, which they perceive as increasing their fitness. In addition to these rationalistic explanations, there is considerable evidence in the data for a strong effect of habit persistence in sports participation, or 'sports literacy'. In short, the paper provides empirical support for a number of theories of participation. However, in contrast to much of the literature, cost of participation and preferences for competitive, organised or intensive sports do not appear to influence the overall quantity of participation.

Universities need to know whether the demand for their supply of sporting facilities is likely to be high. With the increasing number of students being in employment to increase their income to pay for living expenses while at university, working longer hours is a reality for contemporary students but will also impact on the demand for sports facilities. 3 This paper has identified a need for universities to use several strategies to encourage students to participate in sports. These strategies might include organizing more sporting activities, and by attempting to change the perception of sports by students. More specifically, our analysis suggests that universities should focus on providing organised, often competitive, social sports and that they should target these sports in their marketing of sports participation. Our data suggests that the types of students who would engage in intensive sports are those who would be willing sports participants anyway, and thus for whom institutional encouragement is unnecessary. Furthermore, students feel time-3 On the issue of student labour market participation, see, for example, [START_REF] Bailey | The Labour Market Participation of Northern Ireland University Students[END_REF].

  maximisation and a demand function. In said function, demand for sport (measured in various different ways) is determined by the price of the sports

  to work more, whilst the income effect makes them work less. The

  of parental donation, LEA support, student loans and paid employment

  associated with the least amount of sporting participation. Our results

  sporting activities is supported in the multivariate regression analysis. Having other interests reduces the quantity of sports participation. If the student's partner participates in the same sports then they are likely to participate in more

  stable. Column 3 is a reduced model of column 2. In column 3, two variables are excluded: 'partnersame' and 'mot: friends'. Their exclusion is justified on the grounds of the direction of causation: the respondent might have found their partner and their friends doing the sporting activity. Once these two variables are excluded, the only important change in the results is that 'mot: new friends' now becomes important. The numbers of variables in column 3 is now reduced to form column 4. In this final column, 'Not: cost' and 'Not: time', 'Not: TV', and 'Not: Bed' are all removed as they might be simultaneous to the number of hours worked: the more a person works then the more money the respondent might have, the less spare time, the less time to watch TV and the less time in bed. Column 4 in Table

  strong and negative effect on sporting participation, suggesting support for a trade off leisure. Throughout the results, time constraints negatively affect sports participation -this also applies to students who previously rejected sports.

Page 11 of 15 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  to participate; thus universities might be wise to change the work culture of the university and the nature of the students' working week, to give them more opportunity to participate in sports. Our bivariate results in particular show that such changes might lead to improved sporting activities and, moreover, higher general levels of activity.

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Table 1 : Descriptive Statistics

 1

	Variables	Definition	Mean	Standard Deviation	Min	Max	Skew	Kurt
		= sum of the number of sports in which the student
	SportNumber	participates: 0 = 0 sports; 1 = 1 or 2 sports; 2 = 3 sports; 3 = 4 sports; 4 = 5 sports; 5 = 6 or more	2.635	0.153	0	5	0.288	-1.002
		sports
	Age	= age of student	20.412	0.165	19	28	2.630	9.994
	Male	= 1 if Male; = 0 else	0.576	0.054	0	1	-0.315	-1.947
	Ethnicity	= 0 if White; = 1 else	0.282	0.049	0	1	0.984	-1.056
	LiveFriends	= 1 if the student lives with friends; = 0 else	0.859	0.038	0	1	-2.098	2.460
	SportsB4Uni	= 1 if the student participated in sports before university	0.940	0.026	0	1	-3.791	12.676
	Competitive	= 1 if they participate in competitive sports; = 0 else	0.553	0.054	0	1	-0.217	-2.001
	Organised	= 1 if they participate in organised sports; = 0 else	0.424	0.054	0	1	0.315	-1.947
	Intensive	= 1 if they participate in intensive sports; = 0 else	0.271	0.048	0	1	1.051	-0.917
	PartnerSame	= 1 if their partner participates in the same sport; = 0 else	0.072	0.029	0	1	3.364	9.548
	PartnerOther	= 1 if their partner participates in other sports; = 0 else	0.207	0.045	0	1	1.471	0.167
	No-Cost	= 1 if cost stops them participating in more sports; = 0 else	0.213	0.045	0	1	1.436	0.064
		= 1 if they don't have time to participate in more
	No-Time	sporting activities;	0.741	0.051	0	1	-0.756	-0.492
		= 0 else
		= 1 if watching TV stops them participating more in
	No-TV	sporting activities;	0.118	0.035	0	1	2.416	3.931
		= 0 else
	No-OtherInterests	= 1 if they have non-sporting interests; = 0 else	0.435	0.054	0	1	0.266	-1.976
	NotInterested	= 1 if they're not interested in sports; = 0 else	0.059	0.026	0	1	3.818	12.877
	No-Bed	= 1 if they'd prefer to stay in bed; = 0 else	0.224	0.045	0	1	1.351	-0.179
		= 1 if they have family commitments that restrict
	No-Family	participation in sporting activities;	0.012	0.012	0	1	9.220	85.000
		= 0 else
	Motive-Fitness	= 1 if they do sports to keep fit; = 0 else	0.824	0.042	0	1	-1.728	1.009
	Motive-Friends	= 1 if they meet friends doing sports; = 0 else	0.482	0.055	0	1	0.072	-2.043
	Motive-NewFriends	= 1 if they meet new friends doing sports; = 0 else	0.247	0.047	0	1	1.194	-0.588
	WorkHours: 0	= 1 if does no work; = 0 else	0.024	0.017	0	1	6.400	39.903
	WorkHours: 1-5	= 1 if works 1-5 hours per week; = 0 else	0.059	0.026	0	1	3.818	12.877
	WorkHours: 6-12	= 1 if works 6-12 hours per week; = 0 else	0.282	0.049	0	1	0.984	-1.056
	WorkHours: 13-20	= 1 if works 13-20 hours per week; = 0 else	0.329	0.051	0	1	0.739	-1.489
	WorkHours: 21-34	= 1 if works 21-34 hours per week; = 0 else	0.247	0.047	0	1	1.194	-0.589
	WorkHours: 35+	= 1 if works over 35 hours per week; = 0 else	0.059	0.026	0	1	3.818	12.877
		= 0 if no hours work; = 1 if 1-5 hours work;
	WorkHoursSqd	= 4 if 6-12 hours work; = 9 if 13-20 hours work	16.212	0.955	1	36	0.549	-0.190
		= 16 if 21-24 hours work; = 25 if 35+ hours work
	Note: Column D indicates the expected direction of effect of variables on the dependent variable: SportNumber. C
		implies control variable.

Table 2 : What Influences the Number of Sports a Student Participates In?

 2 Dependent variable in each case is 'activorder'. Standard errors are in parentheses. ***, ** and * indicate significance at the 1%, 5% and 10% levels respectively. Normalised observations used throughout. The sample size differs between columns: in (1) it is 80, in (2) it is 82, in (3) and (4) it is 84 and in (5) it is 85. As the results are stable across variable regressions we feel that the differences in sample size are not seriously affecting the results.

				1		2		3		4		5
	Age		0.103 (0.180)	0.052 (0.165)	0.097 (0.147)	0.103 (0.146)	1.127 (0.428)***
	Male		0.940 (0.583)	0.807 (0.559)	0.833 (0.515)	0.728 (0.503)	-	-
	Ethnicity		-0.621 (0.658)	-0.739 (0.630)	-0.457 (0.600)	-0.590 (0.563)	-	-
	Live with Friends	1.038 (0.822)	1.008 (0.783)	0.647 (0.726)	0.545 (0.707)	-	-
	Sportsb4uni		4.798 (1.569)***	4.560 (1.517)***	4.050 (1.479)***	4.024 (1.427)***	3.801 (1.246)***
	Competitive		-0.506 (0.520)	-0.232 (0.486)	-0.168 (0.458)	-0.055 (0.454)	-	-
	Organised		-0.454 (0.552)	-0.434 (0.508)	-0.432 (0.474)	-0.303 (0.461)	-	-
	Intensive		1.046 (0.651)	0.863 (0.633)	0.870 (0.573)	0.838 (0.556)	-	-
	Not: Family		1.214 (2.071)	1.552 (2.037)	0.936 (2.002)	-0.180 (1.866)	-	-
	Not: Cost		0.046 (0.671)	-0.070 (0.629)	0.087 (0.577)	-	-	-	-
	Not: Time		1.292 (0.659)**	1.425 (0.635)**	1.204 (0.609)**	-	-	-	-
	Not: TV		0.693 (0.791)	0.630 (0.736)	0.570 (0.732)	-	-	-	-
	Not: Bed		-0.110 (0.650)	-0.167 (0.615)	0.080 (0.602)	-	-	-	-
	Not: Other Interests	-0.942 (0.514)*	-	-	-	-	-	-	-	-
	Not: Not Interested	0.602 (1.037)	-	-	-	-	-	-	-	-
	Mot: Friends	0.808 (0.604)	0.469 (0.549)	-	-	-	-	-	-
	Mot: New Friends	0.635 (0.753)	1.103 (0.682)	1.519 (0.598)**	1.529 (0.595)***	1.482	(0.515)***
	Mot: Fitness	-0.539 (0.649)	-0.320 (0.636)	-0.488 (0.627)	-0.276 (0.618)	-	-
	Partner Same Sports	1.885 (1.045)*	1.933 (1.051)*	-	-	-	-	-	-
	Partner Other Sports	-0.382 (0.673)	-	-	-	-	-	-	-	-
	Work hrs: 0		-	-	-	-	-	-	-	-	-	-
	Work hrs: 1-5	1.318 (1.897)	1.107 (1.834)	1.454 (1.839)	0.953 (1.782)	-	-
	Work hrs: 6-12	-2.448 (1.934)	-2.814 (1.809)	-2.049 (1.769)	-1.941 (1.762)	-	-
	Work hrs: 13-20	-4.807 (2.353)**	-5.085 (2.203)**	-4.458 (2.150)**	-4.238 (2.149)**	-2.010 (0.753)***
	Work hrs: 21-34	-6.392 (3.063)**	-7.076 (2.854)**	-6.200 (2.728)**	-5.793 (2.712)**	-2.791 (1.233)**
	Work hrs: 35+	-8.200 (4.333)**	-9.647 (4.048)**	-8.406 (3.846)**	-8.310 (3.790)**	-4.230 (2.145)**
	Work hours Sqd	0.254 (0.108)**	0.268 (0.101)***	0.256 (0.096)***	0.241 (0.097)**	0.163	(0.068)**
	Ancillary	parameters	Cut 1 Cut 2 Cut 3 Cut 4 Cut 5	0.018 (2.051) 4.810 (2.250) 6.761 (2.298) 7.798 (2.323) 9.036 (2.358)	0.532 (1.788) 4.946 (2.087) 6.787 (2.127) 7.837 (2.151) 8.974 (2.184)	0.318 (1.825) 4.321 (2.056) 6.173 (2.091) 7.196 (2.113) 8.385 (2.142)	-0.488 (1.746) 3.387 (1.953) 5.113 (1.972) 6.115 (1.991) 7.283 (2.017)	-2.125 (1.152) 1.657 (1.351) 3.275 (1.366) 4.227 (1.370) 5.355 (1.388)
	Pseudo R 2		0.206		0.184		0.171		0.151		0.136
	Likelihood Ratio	54.41***	49.88***	47.46***	41.87***	38.23***
	Log likelihood	-104.683		-110.623		-114.988		-117.782		-121.124
	Likelihood Ratio Test									9.32
			Notes:

Table 3 : Of those who cite a motive, how many sports do they do

 3

	?

Table 4 : Reasons why students do not participate in more sports and sports participation Male Female All

 4

	Not: Cost	Yes 2.556 2.889 2.722 No 3.051 2 2.621
	Not: Time	Yes 2.971 2.444 2.738 No 3 1.556 2.435
	Not: TV	Yes No 2.930 2.156 3 2.75	2.9 2.6
	Yes 2.5 Not: Other Interests No 3.242 2.368 2.896 2.059 2.297

Table 5 : Average 'Activeorder' by Gender and Hours Worked Male Female All

 5

	No hours work	1.5	3	2
	1-5 hours work	3.5	2	2.6
	6-12 hours work	3.2	2.111 2.792
	13-20 hours work 2.667	2	2.429
	21-24 hours work 2.8	2.444 2.632
	35+ hours work	5	2.5	3.333
	Total	2.939 2.222 2.635

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Clearly, each sport requires different quantities of money spent on participation and different amounts of time for participation, but universities still require information on the sports that students will participate in and therefore which facilities they need to supply.


Quantitative Methods, O47 -Measurement of Economic Growth|Aggregate Productivity < O4 -Economic Growth and Aggregate Productivity < O -Economic Development, Technological Change, and Growth, O33 -Technological Change: Choices and Consequences|Diffusion Processes < O3 -Technological Change|Research and Development < O -Economic Development, Technological Change, and Growth, F40 -General < F4 -Macroeconomic Aspects of International Trade and Finance < F -International Economics

Keywords: convergence, technology, specialization, input-output tables convergence, technology, specialization, input-output tables JEL codes: C67, F40, O33, O47

This paper analyzes whether international material input structures have converged or diverged over time. Pooled variances for 25 industries were obtained from OECD inputoutput tables in constant prices for nine countries over the period [1971][1972][1973][1974][1975][1976][1977][1978][1979][1980][1981][1982][1983][1984][1985][1986][1987][1988][1989][1990]. It is found that high-tech industries were mainly characterized by divergence of material input structures, whereas convergence was found for many low-tech, more mature industries. In line with studies on (labor) productivity growth rates, convergence of material input structures was prevalent in the 1970s, while divergence dominated in the 1980s.

Introduction

Over the past couple of decades, globalization has become a keyword to describe the increased interrelatedness of nations with respect to trade and knowledge diffusion. The question we address in this note is how production processes of industrialized countries have changed as a consequence of international technological linkages. Using information from input-output tables, we analyze for each industry whether the structures of material inputs have converged, or not. In our definition, convergence takes place if the diversity (or, variability) of production technologies used across countries decreases. 1 In this respect, our convergence concept resembles the -convergence concept that is well known in studies of labor productivity and total factor productivity convergence (see, e.g. Bernard and Jones, 1996a).

In this paper, we will view changes in material input structures in the light of so-called "technology gap" models of growth (see, e.g., [START_REF] Fagerberg | A Technology Gap Approach to Why Growth Rates Differ[END_REF]. In these models, two sources of change are highlighted: innovation and technological catch-up. Suppose a situation in which countries use identical material input structures in an industry. Any innovation will then induce a tendency to divergence, since one or a few national industries start adopting a different production technology that is perceived as an improvement over current practice.

Later on, technology flows from technological leaders to followers (see, for instance, [START_REF] Coe | International R&D Spillovers[END_REF][START_REF] Coe | International R&D Spillovers[END_REF][START_REF] Verspagen | Estimating International Technology Spillovers Using Technology Flow Matrices[END_REF] innovator(s). Ceteris paribus, material input structures will become more alike, or, in other words, convergence will prevail. 2 In the "real world", many of such processes are simultaneously at work. Convergence (divergence) occurs in a given time period if the effects of innovations by the leader are smaller (larger) than the effects of catch-up through assimilation of diffused technology by follower countries. Empirical analyses should provide answers to the question whether convergence or divergence prevails. 3 In this note, we use data from the OECD input-output tables in constant prices for nine countries over the period 1971-1990, to

Technology gap models and convergence

In traditional neoclassical economics [START_REF] Solow | A Contribution to the Theory of Economic Growth[END_REF], international differences in levels of labor productivity were seen as transitory phenomena. Due to more attractive rates of return to capital in low-productivity countries, investment rates were thought to be higher than in

2

The issue of convergence of technologies has been studied quite extensively, but almost exclusively on the basis of trends in (labor or multifactor) productivity. [START_REF] Baumol | Productivity Growth, Convergence, and Welfare: What the Long-Run Data Show[END_REF] is a classic contribution in this respect. high-productivity countries. Ultimately, all countries will tend towards the same productivity level. If (exogenous) technological progress is taken into account (countries increase their productivity at a constant pace), convergence towards a steady-state gap of productivity levels will result, and growth rates will tend to equalize. Characteristic of these models is that technology is considered to be common to all countries. Differences in savings rates are considered the main cause of productivity growth differentials.

In the technology gap literature [START_REF] Fagerberg | A Technology Gap Approach to Why Growth Rates Differ[END_REF][START_REF] Verspagen | A New Empirical Approach to Catching Up or Falling Behind[END_REF], are seminal studies, see e.g. [START_REF] Lim | Convergence and Catching Up in ASEAN: A Comparative Analysis[END_REF], for a recent contribution), a different approach is proposed. High-productivity countries attain productivity growth by means of innovation.

Since the knowledge pertaining to these innovations is assumed to disseminate slowly (or, at least, not instantaneously), low-productivity countries will initially lose ground, both in terms of productivity levels and growth rates. As soon as low-productivity countries start to "benefit" from their technological backwardness (for instance, by imitating high-productivity processes and products) catch-up can occur. Thus, the dynamics of the productivity gap between high-productivity countries and low-productivity countries is basically viewed as the outcome of two opposing forces: innovation by the leaders and assimilation by followers.

Let us for simplicity assume that there is just one productivity leader, country 0. The productivity followers are denoted by i (i=1, ..., n). Then, the basic technology gap model is given by . 4 Countries that are unable to assimilate any technology will be characterized by i =0. The better country i is at assimilating (for instance due to a relatively well-educated workforce), the more negative i will be.

+ = INI INI i i i i y y y y 0 0 0 ln ) ( & & (
For reasons of exposition, we assume that the leader's ability to innovate leads to a stable, continuous flow of innovations that yields productivity growth at rate 0 . Furthermore, let us assume that follower countries cannot innovate at all by themselves ( i =0, i=1,...,n).

The equilibrium gaps for the productivity levels can now easily be found by setting the left hand side of (1) equal to zero, that is -in equilibrium-the leader and the followers experience identical productivity growth rates. Solving for the right hand side yields:

( ) i e y y i / * 0 * 0 = (2)
Thus, the equilibrium gaps for productivity levels are larger the faster innovations arrive in the leader country and the smaller the rates of assimilation in follower countries are.

In many cases, productivity growth rates due to innovation as captured by 0 are not stable over time (see, e.g. [START_REF] Freeman | The Economics of Industrial Innovation[END_REF]. As a consequence, the equilibrium gap in

(2) and, therefore, actual productivity gaps will change over time. In the early stages of product life cycles, productivity growth is often slow ( 0 small), due to the initially limited scale at which innovative processes are used or innovated products are sold. Later on, productivity growth picks up ( 0 ), since the innovation has gained more popularity. Finally, at the time processes and products reach the stage of maturity, the rate of innovation usually goes down ( 0 ), because opportunities for further improvement get fished out. If it is 4 Note that the ratio 1) is smaller than one. assumed that the abilities to assimilate remain constant over time, the product life cycle can be held responsible for changes in the distribution of gaps, as reflected in its variance.

INI INI i y y 0 / in (
In the traditional technology gap literature described so far, technologies are indicated by productivity levels. In this study, we adopt a different perspective and use material input structures as indicators of technology. If assimilation of leading technologies is predominant, the industry-specific input structures will become more similar across countries. If innovation is a more important factor, however, the contrary is true. The innovating industry will adopt processes characterized by an input structure that deviates from what its foreign counterparts do. As a consequence, input structures will become more heterogeneous.

In the literature on productivity convergence, scholars often investigate trends in measures of dispersion, like the distribution's variance. If the dispersion in productivity or income levels is decreasing, Bernard andJones (1996a), Sala-i-Martin (1996) and [START_REF] Proietti | Convergence in Italian Regional Per-Capita GDP[END_REF], among others, denote this as -convergence. Several measures of dispersion have been used in the literature, ranging from variances, standard deviations, and standard deviations of transformed variables to weighted or unweighted coefficients of variation. In this paper, we will focus on intertemporal comparisons of variances, because statistical theory provides some natural yardsticks for assessing -convergence and -divergence for material input structures.

In the technology gap model represented by (1), the variance of equilibrium gaps is likely to increase ( -divergence) during the transition from the early stage to the stage of growth. High-productivity innovators will leap forward, while more backward countries will stay further behind, which implies increased dispersion of productivity levels. During the subsequent transition from the stage of growth to the maturity stage, however, the variance of equilibrium gaps could well decrease ( -convergence). Since adaptation to the equilibrium In the next section, we will outline how we adapt a measure of -convergence in the literature on productivity convergence to study convergence of material input structures at the industry level.

Methodology

We base our analysis on input-output tables in constant prices. From these, we obtain the elements ij z with the (domestic plus imported) intermediate deliveries from industry i to industry j (i, j = 1,…,n) and domestic gross outputs j x in industry j. The elements

j ij ij x z a / =
denote the input coefficients of material input i per unit of output in industry j.

Convergence (respectively divergence) of material input structures in industry j would imply that the jth columns of the matrices A A A A of input coefficients for the various countries become more (respectively less) similar. In line with the concept of -convergence for productivity levels outlined in the previous section, we analyze the changes in the similarity of the columns over time. 5 Let the countries be denoted by the index r = 1,…, t m and note that the number of countries included in the sample differs over time. Then for each industry j (= 1,…,n) at time t, we have This expression reflects the extent to which the input coefficient a ij differs from the average value computed for the full set of countries. If the material input structures for industry j would be exactly identical across countries, v ij (t) would be zero. The use of any input I as a proportion of gross output is the same for each country.

= = t m r ij r ij t ij t a t a m t v 1 2 )] ( ) ( [ 1 1 ) ( , with = = t m r r ij t ij t a m t a 1 ) ( 1 ) ( (3)
If some elements a ij vary across countries for industry j, a summary measure is needed to indicate the dissimilarity of international production technologies. The pooled variance, obtained by taking the average of the variances v ij (t), is the most straightforward candidate summary measure, i.e.

= = n i ij j t v n t v 1 ) ( 1 ) ( (4)
If ) 1 ( + t v j
is "substantially" smaller than ) (t v j , we will speak of convergence between period t and t+1. This resembles the conventional notion of -convergence, introduced in the previous section: from period t to period t+1, technologies have become more alike, most probably as a consequence of technology assimilation. Analogously, we will take a ) 1 ( + t v j that is "substantially" larger than ) (t v j as an indication of divergence between t and t+1. As a yardstick, we use the corresponding F-statistic. For example (also other percentiles will be used),

• convergence took place if ) 05 . 0 ( ) ( / ) 1 ( ) 1 ( ) 1 ( 1 + < + t t m n m n j j F t v t v • divergence took place if ) 95 . 0 ( ) ( / ) 1 ( ) 1 ( ) 1 ( 1 + > + t t m n m n j j F t v t v 5 F o r P e e r R e v i e w 9
It should be stressed that we use the critical values of F-distributions only as a yardstick to distinguish between "large" and "small" differences. The application of an F-test to decide whether the differences are significantly different (in a statistical sense), would require unrealistically strong assumptions. That is, we would have to assume that the observations

) (t a r ij are normally distributed with mean ) (t i µ
and a common variance ) ( 2 t . Moreover, the observations would need to be independent across countries r and supplier industries i. In particular the assumption regarding independency seems to be violated in reality, because country-specific substitution effects, for example, may play a role. Nevertheless, comparisons to critical values of F-distributions give sensible indicators of the strength of convergence or divergence processes.

Data Data Data Data

We study the changes in material input structures in the way outlined in the previous section on the basis of a set of national input-output tables compiled by the OECD (OECD, 1995). It contains input-output tables for ten developed countries, using a 35-industry classification. 6

For each country, three to five tables are available, roughly for the period 1968-1990.

Unfortunately, the years for which tables are compiled do not exactly coincide. We decided to 6 These countries are Australia (AU), Canada (CA), Denmark (DK), France (FR), Germany (GE), Japan (JP),

The Netherlands (NL), the United Kingdom (UK) and the United States of America (US). The OECD database also contains a single 1 presents this grouping of tables. 7

INSERT TABLE 1

Our analyses are based on the tables that contain all intermediate inputs, both domestically produced and imported (in the OECD database these tables are encoded as "TIOK"). This choice is in accordance with the idea that material input structures should resemble technologies of industries. Mere changes in the mix of domestically produced and imported inputs should not affect these representations, unless domestically produced inputs from supplying industry i would be different from imported inputs from industry i. Since the OECD constructed the tables according to an internationally harmonized industry classification, risks that substantial systematic differences affect the results are strongly reduced.

The interindustry transactions recorded in the tables for any country are denominated in the national currency. Further, the tables are expressed in constant prices. As a consequence, information on (physical) quantities of inputs (which represent the material input structures we are interested in) is thus approximated as well as possible, because inflation and changes in exchange rates do not contaminate the coefficients. 8

Since national statistical agencies did not construct their national input-output tables in exactly the same way, the OECD tables are not fully comparable. Some industries are not contained as separate entities in tables for some countries, whereas they do for others (see

7
We decided to adopt the grouping suggested by OECD (1995, p. 7), except for one table. That is, we included UK(1979) in the third subperiod, whereas OECD (1995) assigned it to the second subperiod. Our grouping yields less variance within groups with respect to timing. , 1995, p. 12). To make the tables as comparable as possible we had to aggregate a limited number of industries. The aggregation scheme is included in Appendix A. We finally computed the pooled variances for 25 industries, each of them based on material input coefficients vectors that consist of 25 elements.

Results

Applying equations ( 3) and ( 4) to the OECD (1995) tables and using the specified criteria for convergence and divergence, yields the results documented in Table 2. The rightmost columns refer to trends over the entire period of analysis, i.e. [1971][1972][1973][1974][1975][1976][1977][1978][1979][1980][1981][1982][1983][1984][1985][1986][1987][1988][1989][1990]. An overwhelming majority of industries (i.e., 19 out of 25) has experienced either convergence or divergence of material input structures, as measured by the yardstick of the 10 th , respectively the 90 th , percentiles of the appropriate F-distributions.

INSERT TABLE 2

In particular, many manufacturing industries that are widely considered as "low-tech" -such as textiles (4), wood products (5), paper (6), petroleum (8), and basic metals (11) -appear to have converged. Convergence is also found for primary industries -agriculture (1) and mining (2) -and for a limited number of services industries. This is in line with the prediction from the technology gap literature briefly discussed in Section 2, based on the argument that innovation by technological leaders has slowed down in these "mature" industries. The technology gap will thus be narrowed if the ability of follower countries to assimilate diffused The results for most high-tech and medium-tech manufacturing industries support this explanation. Most industries that belong to this group -such as chemicals (7), plastics (9), machinery (13), ships ( 14), and transport equipment, (15) -show either divergence or no discernible tendency. An increase of the leaders' innovation rates will widen the technology gap inducing divergence, if abilities to assimilate knowledge remain unaltered. A similar argument seems to hold for business services ( 24), an industry characterized by a high degree of organizational innovation.

It should be noted, however, that we are not able to explain the findings for every industry. For example, the low-tech industries glass and stone (10) and metal products ( 12)

show divergence, whereas the high-tech industry instruments ( 16) is found to have experienced convergent tendencies. Specialization might play a role in this (for example, metal products is characterized by a high degree of product differentiation which may be taken as an indication of specialization).

Several generations of trade theories predict that countries will specialize in different production processes [START_REF] Jones | Factor Proportions and the Heckscher-Ohlin Theorem[END_REF][START_REF] Krugman | Intraindustry Specialization and the Gains from Trade[END_REF][START_REF] Grossman | Imperfect Competition and International Trade[END_REF]. The OECD inputoutput tables, however, are rather aggregated, whereas specialization typically takes place at the more detailed level of "subindustries". Each of these can be characterized by its own material input structure. Once such subindustries are aggregated into an industry, the subindustries in which a country is specialized will most strongly influence the structure of the (aggregate) industry. Changes in the input structure of an industry may therefore also be partly due to changes in the mix of its subindustries, as caused by changes in specialization patterns. The problem, however, is that an increase in specialization may lead to convergence affect our results. in one case and to divergence in an other. Therefore, although specialization may play some role, it cannot be used as a factor to explain the observed patterns of convergence/divergence, because the direction of its effect cannot be ascertained.

Looking at the results for shorter subperiods, it is not surprising that a far more heterogeneous pattern is found than for the results over the entire period. The bottom row

shows that an initial tendency towards convergence was gradually overturned. In the late 1980s, divergence rather than convergence of material input structures was found most often.

This result is in line with studies that focus on convergence and divergence of labor productivity levels (see, e.g., Bernard & Jones, 1996a, 1996b[START_REF] Los | The 'Appropriate Technology' Explanation of Productivity Growth Differentials: An Empirical Approach[END_REF].

From the perspective of single industries, only few show a more or less persistent development, such as agriculture (1), ships ( 14), other manufacturing (17), utilities (18) and government services (25). For the other industries, it is remarkable that subperiods with convergence are often followed by subperiods of divergence (and vice versa). These counteracting short-run effects certainly call for further analysis at a more detailed industry level. At the present level, they seem to have blurred the long-run tendencies indicated by the results for the entire period.

Conclusions

This note explored opportunities to incorporate changing patterns of material input structures into analyses of international convergence and divergence. For the period 1971-1990, the analysis based on pooled variances of input coefficients revealed that high-tech industries were mainly characterized by divergence of material input structures, whereas convergence was found for many low-tech, more mature industries. The above-mentioned cell-specific variances can now be used to compute the pooled variances expressed in equation ( 4): v 1 (0) = (0.005+0+0.005)/3 = 0.0033 and v 1 (1) = (0+0+0.005)/3 = 0.0017. In this specific case, the pooled variance for industry 1's material inputs has decreased. To get insight into the strength of this process, we compute the ration between v 1 (1) and v 1 (0), which is 0.50 and compare this value to ) 0 5 . 0 ( 86) DK( 85) FR( 85) GE( 86) JP( 85) NL( 86) UK( 84) US( 85)

"1990" AU(89) CA( 90) DK( 90) FR( 90) GE( 90) JP( 90) UK( 90) US(90)

Note: First column contains labels for subperiods. Values between parentheses refer to years. Notes: The numbers represent the ratios of pooled variances between the final year and the initial year of the periods indicated in the column headings, for the industries in the corresponding rows (see equation ( 4)).

The letter C indicates 'significant' convergence. In this case *, **, and *** show that the observed ratio is smaller than the 10th, 5th, or 1st percentile, respectively, of the corresponding F-distribution. The letter D indicates 'significant' divergence. In this case, *, **, and *** show that the observed ratio is larger than the 90th, 95th, or 99th percentile, respectively, of the corresponding F-distribution.

  may induce technological catch-up, since latecomer countries might learn how to produce according to the technologies operated by the original 1 To be more exact, we propose the reduction in the pooled variance of the material input coefficients of an industry as an indicator of convergence.

  compute our convergence indicator for 25 industries. The rest of the paper is organized as follows. The next section gives a more detailed presentation of the basic technology gap model. The interactions between innovation and technological catch-up are formalized and hypotheses about the typical industries for which convergence resp. divergence of material input structures should prevail are formed. Section 3 presents the convergence indicator we propose. Section 4 discusses the data we use and Section 5 deals with the results. Section 6 concludes.

3

  Hoen (2002, Chapter 7) contains a first study, approaching the question from the perspective of trade theory, however.

  is generally considered as a slow process due to relatively modest abilities to assimilate, actual convergence and divergence processes are long-run phenomena.

8

  It should be mentioned that the base years used for deflation are not identical for all countries, which may decreased. This would yield clearcut tendencies towards convergence of the material input structures.

  input coefficients for the first column and the input coefficients themselves yield v 11 (0) = (0.2-0.25) 2 + (0.3-0.25) 2 = 0.005 and v 11 (1) = (0.2-0.2) 2 + (0.2-0.2) 2 = 0. Similarly, we find v 21 (0) = 0, v 21 (1) = 0, v 31 (0) = 0.005 and v 21 (1) = 0.005.

Industry classification for analyses based on OECD (1995) data

  literature on technology gap models. As is in line with studies that focus on convergence and divergence of (labor) productivity growth rates, convergence of material input structures was prevalent in the 1970s, while divergence dominated in the 1980s.

	F o r Agriculture, forestry and fishery Appendix A: No. Description 1. 2. Mining and quarrying 3. Food, beverage and tobacco 4. Textiles, apparel and leather 5. Wood products and furniture 6. Paper, paper products and printing 7. Chemicals, including drugs and medicines 8. Petroleum and coal products 9. Rubber and plastic products derived from the F o r 10. Non-metallic mineral products 11. Basic metals F 12. Metal products o 13. Machinery, including electronics r 14. Shipbuilding and repairing	OECD IO code ISIC Rev. 2 code 1 1 2 2 3 31 4 32 5 33 6 34 7+8 351+352 9 353+354 10 355+356 11 36 12+13 37 14 381 15+16+17+18 382+383 19 3841
	P e e r P e e r 15. Other transport equipment 16. Professional goods P 17. Other manufacturing e 18. Electricity, gas and water 19. Construction e 20. Wholesale and retail trade r 21. Restaurants and hotels	20+21+22 23 24 25 26 27 28	384-3841 385 39 4 5 61+62 63
	R e v R e v 29 30 R 31+32 e v 25. Community, social and government services 33+34+35 22. Transport and storage services 23. Communication services 24. Financial and business services	71 72 8 9
		i e i e i e
		w w w
		These results confirm hypotheses
	14 16

Table 2 .

 2 Convergence and divergence of sectoral input structures in the OECD, 1971-1990. Industry 71-76 C/D 76-80 C/D 80-85 C/D 85-90 C/D 71-90 C/D

	1	0.695 C*** 1.072	0.714 C** 0.823 C*	0.438 C***
	2	0.970		0.769 C** 0.746 C** 1.408 D*** 0.783 C*
	3	0.791 C*	1.219	0.730 C** 1.190	0.838
	4	1.072		0.774 C*	1.283 D*	0.709 C*** 0.755 C**
	5	0.587 C*** 1.263 D** 0.686 C*** 1.243 D*	0.632 C***
	6	1.042		1.113	0.655 C*** 1.084	0.824 C*
	7	1.406 D** 0.910	0.696 C*** 1.147	1.020
	8	0.918		0.219 C*** 4.328 D*** 0.422 C*** 0.367 C***
	9	0.747 C** 1.023	0.988	1.504 D*** 1.136
	10	0.969		1.119	0.938	1.203	1.223 D*
	11	0.969		0.362 C*** 1.194	1.033	0.433 C***
	12	1.074		0.918	1.257 D*	1.093	1.354 D**
	13	1.033		0.940	0.759 C** 2.180 D*** 1.606 D***
	14	1.109		1.337 D** 1.265 D*	1.563 D*** 2.934 D***
	15	1.928 D*** 0.778 C*	1.086	1.363 D** 2.220 D***
	16	0.965		0.792 C*	0.946	0.984	0.711 C**
	17	0.845 C*	0.771 C*	0.849	0.807 C*	0.446 C***
	18	0.761 C** 0.770 C** 0.991	1.052	0.611 C***
	19	0.882		1.862 D*** 0.796 C*	1.104	1.444 D***
	20	1.174		0.548 C*** 1.187	1.094	0.836
	21	0.738 C** 0.574 C*** 1.457 D*** 0.979	0.604 C***
	22	0.810 C*	0.826	1.157	1.152	0.892
	23	0.523 C*** 0.985	1.094	1.483 D*** 0.837
	24	0.559 C*** 1.362 D** 0.616 C*** 2.847 D*** 1.336 D**
	25	0.537 C*** 0.469 C*** 0.806 C*	1.181	0.240 C***
	Total	C:	11	11	10	4	12
		D:	2	4	5	8	7

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Appendix B: Illustration of computations

Consider the following simplified tables with input coefficients, for the hypothetical countries A and B (m t =2, see equation ( 3)), for t=0 and t=1.

Country A, t=0

Country B, t=0


Peter Kesting
email: kesting@microec.hhl.de.

Simultaneously in an Aging and Stagnating Economy A theoretical investigation and a simulation of the German case

Keywords: Aging, pension, demographic distribution mass, intergenerational income distribution. JEL-Classification: H 00, H 55

This paper investigates the possibilities and restrictions of intergenerational income development for aging and stagnating economies.

In a first step, the basic logic of aging will be investigated in a simple dynamic model. In particular, the investigation points out the existence of the demographic distribution mass and its meaning for intergenerational income development. It proves that the demographic distribution mass makes it possible for wages and pensions to grow simultaneously in an aging and stagnating economy.

In a second step, the development of wages and pensions is simulated for the German case through to the year 2050. It is demonstrated that (under normal circumstances) it can be expected that the burden from aging can almost always be overcompensated by the economic growth of the respective year. Against this background, aging appears to be rather a problem of acceptance and income distribution than of real income reductions.

3 an intensive research on pensions. One important stream of this research was a comparison of the existing public pay-as-you-go system with different types of private, capital-based pension systems (Homburg 2000, Thøgersen 2001, Karayel 2006). In particular, there were (comparative) investigations about the meaning of different pension regimes for capital markets (Lavi/Spivak 1999, Winter 2002), domestic investment, domestic demand and economic growth (Börsch-Supan 1995, Börsch-Supan 1999) as well as on individual returns (Schnabel 1998). In addition to this, there was an intensive discussion about the consequences of a transformation from one system to the other (Börsch-Supan 2000b). It soon became clear that a transformation from a pay-as-you-go system to a capital-based pension system would burden the respective generation with both the duties of the old system and the provision for its own pension. In connection with this, the phrase 'the sandwich generation' was coined (Raphael/Schlesinger 1994). 6 As an alternative, the Swiss model and the meaning of corporate pensions also received some attention (Bütler 1999).

However, in all these discussions, one effect has been completely overlooked:

This effect is fairly well known in statistics, but its meaning has never been investigated for aging societies. The effect is of importance for aging, since it produces an 'unbounded' income, ready to be distributed among the generations. In this paper, we refer to this unbounded income as demographic distribution mass. This paper departs from the origin of all current problems: The income development of the generations as a whole. The particular aim of this paper is to The first step is focused on the theoretical elaboration of the effect. For practical reasons, this analysis is not based on an OLG model, which currently dominates research on pensions, but instead on a continuous dynamic model. The main result of the theoretical part is to proof that it is indeed possible for both pensions and wages to grow simultaneously in an aging and stagnating society.

However, this result is based on the assumption of a dynamic pension system, in that the development of pensions is tied to the development of net wages. Following this, selected alternative pension formulas are investigated. It is shown that these alternative formulas always lead to absolute income reductions for one of the respective generations in the course of aging.

As a result, it becomes clear from the theoretical part of this paper that the possibilities and restrictions of income development in an aging society are mainly determined by two opposing forces: Technical progress on the one hand and aging on the other. Subsequently, the question arises; how do these two opposing forces relate in the real world and, as a consequence, how relevant is the statement that bothpensions and wages -can grow simultaneously in an aging and stagnating society?

What meaning does the demographic distribution mass have for real aging processes? These questions will be answered in a second step exemplarily for the case of the intergenerational income development in Germany from the present through to 2050 (including the 'aging wave' of the 2030s). As a result of the simulation, it can be shown that the burden from increasing pensions alone is of conservative assumptions, it is not to be expected that aging alone will ever make significant income reductions for one of the generations (or both) necessary -not even at the peak of the aging wave. On the contrary, the simulation shows that incomes can remain stable, even in a contracting economy, as long as the GDP does not decrease by more than 0.45% on average.

Note that the analysis of this paper only concerns different possibilities of income development for the two respective groups, the active population (meaning people actively contributing to the pension system) and pensioners as a whole. This means: The analysis does not take any income differences within the generations into consideration and it does not investigate the income development of other groups, such as the unemployed or children. Additionally, this paper does not contain information on how to reach identified income paths, but rather which income paths are possible for a given development of GDP and a given aging process. Although it might appear so at the first glance, this paper does not introduce any production or growth theory and takes no account of the capital side. As a result, it assumes increases in labour productivity as externally given, coming out of a 'black box.' Finally, the paper does not refer to how the different intergenerational income developments are realised, but only which different intergenerational income developments are possible. Hence, there is nothing to find about particular pension systems, about public or private, pay-as-you-go or capital-based pensions. This procedure follows the insight that ultimately the pensions paid in period t are always and completely part of the GDP of this very period. With respect to the question of how much pension is to be paid in t, this is always a matter of distribution of this

THE MODEL

The following model is designed to represent the 'stylised' characteristics of an aging and stagnating economy. For simplification, we assume that the total population of the modelled society, B t , is constant over time (in fact, the population in Germany and a couple of other OECD-countries is actually decreasing):

(1) B B t = .

To focus the analysis on the development of wages, pensions and pension efforts in the course of aging, we will additionally assume that the total population of the modelled society consists of only two different income groups: Firstly, a certain number of W t identical workers and secondly, a certain number of P t identical pensioners at every point in time:8 model world. As explained above, this simplification serves the purpose to focus the investigation on the development of the two remaining income groups -workers and pensioners -in the aging process.

However, it should be mentioned that the Statistisches Bundesamt forecasts a decrease in the number

t t P W B + = .
The gross output of the economy is entirely produced by the workers and will be completely (and equally) distributed among them, so that their gross income percapita in t is Y t /W t . All pensioners receive an equal per-capita pension of p t . These pensions consist entirely of transfers, which are equally subtracted from the gross income of the workers. Correspondingly, the (gross) output of the economy Y t is distributed as:

(3)

t t t t t P p W w Y + = .
In this equation, w t represents the workers' net wages per capita in t, consisting of the gross income minus the pension transfers. Consequently, the per capita pension transfers r t are:

(4)

t t t t W P p r = .
Wages, as well as pensions (per capita), are not assumed to be constant, but rather to change in the course of the aging process. To begin with, we will assume that both incomes do not develop independently of each other, but in the fixed ratio:

(5) p w t t = .

of young people -a development, which, of course, only causes the entire process of aging. This development, again, has a relieving effect on the financial burden of the active population. 8 This is the case of a dynamic pension system in that (corresponding to the public pension system in Germany) the development of pensions is linked to the development of (net) wages.

We will model the aging process of the society as a decrease in the number of workers with a constant rate (but decreasing absolute number) over time:

(6)

t t W t W = .
The number of workers in t is then:

(6a) t t e W W = 0 .
Consequently (still holding the assumption of a constant population), the number of pensioners increases relatively to that of workers in the course of time, but with a decreasing rate. In t, the ratio between pensioners and workers is:

(6b) 1 0 = t t t e W B W P .
Figure 1 schematically depicts the aging process of the modelled society: This technical progress is modelled as a constant increase of the number of efficiency units per worker l t in the course of time with a rate of (it is assumed that the working hours per capita remain constant over time):

(7)

t t l t l = .
The number of efficiency units per worker in t is then:

(7a) l l e t t = 0 .
The efficiency units are nominated so that with one efficiency unit of labour exactly one unit of output can be produced: l t = t ( t denotes the potential per capita output in t). Hence, the development of the gross production potential of the economy is determined by two opposing forces: The increasing labour productivity and the decreasing number of workers:

(8)

t t t t t W l W y Y = = ( ( .
As a result of ( 6) and ( 7), the development of the production potential follows the path: It follows that also the development of the economy's gross product is determined by the interplay of technical progress -the increase of labour productivity -on the one hand, and the aging process -the decrease of the numbers of workers -on the other.

F
According to (9), three cases can be distinguished: For the time being, we want to turn to case (a) and to investigate the scenario of a 10 This assumption might appear a little heroic for Germany and many other OECD-countries and requires some justification. Primarily it is just another simplification, which spares the necessity of an independent explanation for the development of the economy's gross product (beyond the neoclassical equilibrium framework). However, most of all, this assumption can be justified because it isolates the income effect of aging from effects that result from a change of unemployment in the economy. 2050. However, it must be emphasised that the basic results of this theoretical part of the paper -the existence and the meaning of the demographic distribution massdo not depend on this assumption. Additionally, this assumption supports the analysis, as it isolates aging from growth and focuses the analysis on the relevant processes. However, this assumption will be relaxed at the end of the theoretical analysis and in particular later in the simulation of the German case.

Analysis of the Income Development in an Aging, Stagnating Economy,

Retaining the Dynamic Pension

Let us first turn to the development of (net) wages in our model. This development is obviously a result of the interplay of two different factors: Firstly, an increase of the gross income in the result of increasing labour productivity and secondly, an increase of pension transfers in the result of aging. The question is, how these two factors combine and how -as a consequence -the per capita (net) wages of the workers will develop in the course of time. According to (3), the per capita net wages in t are:

(11)

t t t t W p P Y w = .
Because of ( 2) and ( 5), ( 11) can be transformed to:

F o r P e e r R e v i e w 12 ( 12
) t t t t W B W Y w + = .
The derivation of the wage with respect to t is:

(13) 2 ) ( ) 1 ( t t t t t W B W W Y t w + = .
Because of ( 2) and ( 12), ( 13) can finally be transformed to:

(14)

t t t t t w P W W t w + = ) 1 ( .
Equation ( 14) describes the development of the per capita (net) wages in the course of time. It is obvious that under normal conditions (i.e. when , W t , w t > 0 and 0 < < 1) this term is positive, which means nothing more than that the net wages will increase in the course of time. However, because of ( 5) this means nothing more than that pensions will also increase. Consequently, it is indeed possible that in an aging and stagnating economy with a constant population, all incomes -wages as well as pensions -can simultaneously increase over the course of time.

How do we come to this result; how is it possible that wages and pensions can both increase simultaneously, although the gross product as well as the population of this economy remains constant? The reason for this is an effect, which is fairly well known in statistics, but which, so far, has never been applied to the investigation of aging. In fact, even in our model, one group of the economy has to put up with a decreasing income. This group consist of all workers, who additionally become

F o r P e e r R e v i e w
13 pensioners in the course of aging, i.e. W t persons per period. As a consequence, a residual income of W t (w t -p t ) emerges, which can be distributed among the different income groups of the economy -in our model for the time being according to the algorithm of the dynamic pension. In the following, we will label this residual income as demographic distribution mass. However, it must be emphasised that this effect only holds for an aging society. In a rejuvenating society we would observe the opposite effect: The demographic distribution mass would become negative and incomes would decrease in the case of stagnation.

However, this mechanism should not belie the fact that aging always brings about financial burdens for at least one generation. In the assumed scenario of a dynamic pension, this burden materialises in a continuous increase of the pension ratio. The crucial thing is that this rate lies below the increase of the (per capita) gross income and provides room for net income growth. Hence, 0 <

P + W W ) - (1 t t t < 1 if 0 < < 1 and
so both wages and pensions can increase simultaneously. As a matter of fact, this net income growth is below the productivity growth: A growth, which could be realised for the wages, if there were no aging (of course, in this case the entire economy would grow).

One important implication of this result is that it is possible that both wages and pensions can increase simultaneously even in the third of the above cases -the contracting economy, in that | | < | | and the gross product of the economy decreases.

The critical borderline for income growth in this economy is at the point where wages and pensions are constant over time. In this case, the gross product is:

F o r P e e r R e v i e w 14 (15) P p + W w = Y t t t
and the growth rate of the economy's gross output is:

(16) t t t Y W w p Y t Y Y ) ( = = & & .
Given that , W t > 0, this growth rate is negative, as long as wages exceed pensions (per capita) and the more negative, the larger the difference between these two incomes is. 11 This means nothing more than that even in a contracting economy with a constant population it is possible that wages and pensions increase simultaneously in the long run -of course only when the economy's population is aging, i.e. when the number of workers decreases relative to that of pensioners.

However, the dynamic pension is rather a German specific and even in this case it is fundamentally questioned in the current discussion. Hence, in the following we investigate the effect of three alternative pension formulas: (I) a determination of the pension ratio, (II) a determination of the (absolute) amount of per capita pension transfers, and (III) a linking of the pension to the gross income. 15 to the pension system -constant and to opt out of the dynamic pension system for good. In fact, every increase in the contribution ratio leads to an intensive mediadiscussion, which puts serious pressure on the Federal Government. 12 In the following we will first investigate how the fixing of a particular contribution ratio will affect the income development in our stagnating model economy with a constant population. 13 Therefore, we assume that:

Discussion of the Effect of a Change in the Pension Formula

(17) 17), the wages are now:

12 For instance, an article in the Frankfurter Allgemeinen Zeitung from November 6, 2002 reads: "The President of the Confederation of German Employer's Association (BDA), Dieter Hundt characterized the plans of the governing coalition to increase pension duties as an additional blow against growth and employment policy. 'These plans are a fatal consequences of a misguided economic and social policy and of insufficient reforms of the pension system.'…" (own translation). 13 We still assume that the 'contribution ratio' represents all transfers from the active to the inactive , which means that -still holding the assumption of a stagnating economy -wages increase at exactly the same rate as labour productivity. However, this means nothing more than that in the case of a constant contribution ratio, the workers can enjoy the fruits of technical progress without any restriction. Due to the fact that the number of pensioners is increasing (while the gross pension revenue remains constant over time), there is a decrease in per capita pensions in the course of aging. Consequently, fixing the contribution ratio results in loading the burden of aging on to the pensioners.

F
(II) The result is even more distinctive in the case where the per capita pension transfer of the workers (in absolute terms) is fixed instead of the contribution ratio. This is the case, when:

(21) , However, of particular interest are the wages: These do not only increase with the rate of the technical progress, but even more so than with the rate:

(24) ) ( r w t w t t + =
As a result of a determination of the per capita pension transfer, the workers do not only enjoy the increase in labour productivity completely; moreover, the entire demographic distribution mass will also be distributed among them.

(III) It is easy to comprehend that, in contrast to (I) and (II), a dynamic linking of the pension to the development of the (per capita) gross income leads to an increase in pensions with the rate of the technical progress. In this case, wages will decrease, as long as W t /B t > p t /w t .

Finally, it becomes clear that only one of the investigated pension formulas results in both a wage and pension increase simultaneously in an aging and stagnating economy (although at the price of an increasing pension ratio): The dynamic pension, which is linked to the development of the (net) wages. All other investigated formulas cause one of the respective income groups to be burdened with absolute income losses in the course of time.

So far, we have concentrated on the theoretical framework. In the following, we now turn to the German case and investigate the expectations for the income development in this particular country. Up until now, only a theoretical possibility has been expounded. But how relevant is this possibility to understand the real development in aging societies? What meaning does the demographic distribution mass have there? The answer to these questions obviously depends mainly on two developments: Technical progress on the one hand and aging on the other. In the end, whether the theoretical possibility of increasing wages and pensions in an aging and stagnating economy becomes real, merely depends on which of the developments actually predominates in the country.

Consequently, the following simulation is based on assumptions about the development of labour productivity on the one hand and about the demographic development in Germany through to the year 2050 on the other.

However, by now it is impossible to give a reliable prediction for the development of the labour productivity for longer periods. Thus, we have to take a look at the past. Figure 2 19 constantly increased by rates between 0.5% and 5.2%. The average increase was 2.8%, although it slightly reduced over the course of time -in Germany (1991-2001) the average increase of labour productivity was only about 2.0%.

In contrast to the development of labour productivity, the demographic development in Germany can be predicted very accurately. In its forecast, the Statistisches Bundesamt distinguishes between two different scenarios: The development of a long-term net immigration of +100,000 people per year and that of a long term net immigration of +200,000 people per year. Since the retirement age is also relevant for the analysis of the development of the different income groups, we will additionally distinguish between the following two scenarios: An average retirement age of 60 and one of 65.

In all these different scenarios, the total population and the number of workers tend to decrease and that of the pensioners to increase: Consequently, in all different scenarios, the German society is aging. The extent and course of this aging process becomes particularly clear from a view of the age-ratio: The age-ratio records how many pensioners have to be supported on average by one worker and thus gives a clue as to the burden of the active part of a society (i.e. a pension ratio of 0.4 means that on average, one worker has to support 0.4 pensioners). It can also be recognised in figure 3 that a high immigration number and a late retirement age mostly just delay aging and only marginally diminish it: For both scenarios, the Statistisches Bundesamt forecasts a doubling of the age-ratio until the year 2050, although on a significantly higher level with an annual immigration of +100,000 per year and a retirement age of 60: In this case, the age-ratio increases from about 0.4 in 1999 to about 0.8 in 2050, whereas it only increases from about 0.25 in 1999 to about 0.5 in 2050 with an annual immigration of +200,000 per year and a retirement age of 65. 14 First Scenario: Annual Net Immigration of +100,000 People Per Year and a

Retirement Age of 60

In the first scenario, we are going to simulate the most 'pessimistic' forecast, where the aging of the German society progresses most steeply and on the highest level. For the time being, we will additionally stick to the assumption of a dynamic pension system, where the pension development is tied to that of net wages. The question is, if and to what extent it is possible for wages and pensions to increase simultaneously in the German case. The continuous line in figure 4 shows the (identical) 14 To isolate the intergenerational income development and that of the demographic distribution mass, we will exclude all people below the age of twenty from the simulation: As in the theoretical part, the total population consists only of those people aged twenty or over. However, all changes in this 'total population' through to 2050 will be completely taken into account, i.e. the above assumption of a It is clearly recognisable that even in this most pessimistic of all scenarios, wages and pensions almost always (both) increase over the entire time period (on average by about 0.62% per annum). The pattern of this increase is hardly surprising: It pretty well mirrors the demographic development. More surprising, however, is the extent of this growth: Even the increase in the aging-ratio by about 3.5% in 2024 leads to almost no decrease in incomes. In the year 1999, an increase in the agingratio by 3.7% is even accompanied by an increase in incomes by 0.2 % -and all this based on a (constant) increase in the labour productivity of just 1 %! One important reason for this development is the existence of the demographic distribution mass, which emerges as a result of the aging of the German society and considerably reduces the burden of the additional pension effort. Figure 5

FIGURE 6: SET HERE

It is this demographic distribution mass in particular, which brings about that an average increase in the labour productivity by about 0.38% is sufficient to keep incomes constant. The gross product would than decrease by about 0.45% on average. In other words: Productivity growth of even less than half a percent on average and of about 1% at the aging peak is sufficient to carry the burden resulting from the aging of the German society. As a result of a decrease in population, the GDP also decreases in this case, meaning that the requirement of a constant GDP is completely compatible with increasing per capita incomes. All of this means: The theoretical analysis of chapter 2 describes a relevant mechanism, at least for the If we assume even an average productivity growth of 2% p.a., both wages and pensions can simultaneously increase over the entire time period through to 2050 and the increase in these incomes is not seriously affected by the burdens of the aging for most of the time. 15 This means: Under normal conditions, nobody in Germany necessarily has to fear real income reductions from aging alone! So far, the good news.

The bad news is that a retention of the dynamic pension would considerably increase the pension ratio, according to the simulation, from 20% to over 35%. Again, this increase in pension transfers would remain below the increase in gross wages (per capita), so that the net wages would still increase. However, experience shows that such an increase in burdens always has problematic consequences: In particular, it reduces the motivation of the active part of the population and forces it into the black economy or abroad (Tanzi 1980, Frey/Weck 1983, Schneider/Ernste 2000).

Nevertheless, it must be emphasised that (due to the fact that net wages are still increasing) these are not problems of diminishing wealth, but rather of perception, of attitude towards public burdens, and possibly of international competitiveness.

The relative meaning of aging and pension transfers for the German income development may be highlighted by figure 7: 15 Due to the higher level of income, the demographic distribution mass is larger in this scenario than in the case of an assumed productivity growth of 1%: It is now about € 6.6 billion per year, on average. 7 compares the effect of an exit from the dynamic pension system on the one hand and that of an isolated productivity increase by 2% p.a. (maintaining the dynamic pension system) on the other, with the scenario in which nothing changes, i.e. the productivity growth remains at 1% p.a. and the dynamic pension is maintained. As demonstrated above, even if nothing is changed, per capita (net) wages will almost always increase, during the entire time period by about 38 %.

However, the simulation shows that the income effect of an exit from the dynamic pension system is far beyond that of 2% productivity growth: In the first scenario, the income will increase only by about 68%, whereas in the second scenario, it will increase by more than 130%. (This result might not be too surprising if one takes into account that an average productivity growth of about 0.38% is sufficient to maintain wages and pensions during this time period.) However, fixing the pension ratio at the value of 20% would lead to a considerable decrease in pensions by an average of about 0.36 % p.a., reducing the pensions over the entire period by about 20% (and even by an average of more than 0.8% p.a. over the time period until 2031, the year in which pensions reach their base). At the peak of the 'aging wave' around 2024, the annual decrease in pensions would be even larger than -2% for five and larger than -1.5% for nine years in a row.

In this case, the pensions would indeed 'no longer be secure' (as prophesied many times in the media). For a comparison, we will take a brief look at the most optimistic scenario with a net immigration of +200,000 people per year and a retirement age of 65. Figure 8 As is to be expected from the demographic development, the development of wages and pensions in scenario 2 is delayed compared with scenario 1, but follows a similar pattern. Nevertheless, in scenario 2, wages and pensions increase a little more, by roughly 0.73% p.a. on average (compared to 0.62% p.a. in the first scenario) and remain positive even in the aging peak during the 2030s. With an amount of € 4.3 billion, the average demographic distribution mass is lower in scenario 2 (compared to € 4.8 billion in scenario 1), but higher at the aging peak (€ 10.5 billion compared to € 10.2 billion in scenario 1).

One important difference, however, is that the development in scenario 2 takes place at a significantly lower level overall than that of scenario 1: In particular, the pension ratio departs only at roughly 15% in 1998 and approaches only 26% in 2050 in scenario 2 (compared to 20 % in 1998 and 35 % in 2050 in scenario 1). This

CONCLUSIONS

The results of this paper are somewhat confusing and should be handled with some care: It should not entrap us to play down the problems of aging, assuming that since it does not lead to absolute income reductions, aging does not significantly burden the active population, for at least two reasons: Firstly, increasing pension transfers are not the only consequence of aging, but aging will also affect the costs of health care and perhaps the mental condition of the society. Additionally, aging is not the only burden, the active population has to shoulder in many OECD countries.

Burdens also result from unemployment, domestic and foreign debt, inflating social systems and so on. Hence, the analysis of this paper has to be set within the overall context of economic development. Secondly, even if no real income reductions are necessary, aging always leads to an increased burden for at least one generation. This increasing burden is particularly materialised in an increasing pension rate: If we already have such resistances against an increase in the pension ratio above 20%, how should the reactions be to 30%? The harmful effects of increasing pension rates are well known. They especially affect the general dynamics of an economy. In addition to this, increasing burdens are usually unequally distributed within the 16 For instance, in a speech before the LVU-Unternehmertag on June 3, 2004, BDA-President, Dieter Hundt comes to the conclusion: "Due to the demographic development, we have no choice but to extend working life." Source: http://www.lvu.de/04/0603_8.html (own translation). In their coalition talks, SPD and CDU both explicitly expressed the extension of working life as being one of their political targets. See. Frankfurter Allgemeine Zeitung, October 28, 2005. However, one thing might become clear from the analysis of this paper: The financial burdens of aging alone do not have to ruin us. On the contrary, it is theoretically possible that the entire population of Germany will be much better off in 2050 than at the present. The background of this insight is that technical progress is a strong force, which allows us to produce an ever-growing wealth with a continuously reduced working force. Even at the peak of aging, it is not to be expected that the remaining, reduced working force will not be capable of increasing the wealth of the population, as long as technical progress does not stop and we do not face different restrictions. On the contrary: Obviously, most OECD countries are not even capable of employing the remaining workforce. As a result, unemployment grows continuously or remains at a high level. In this connection, Horst Siebert even asks, if the Germans are 'running out of work' (Siebert 1994).

Against this background, the aim of this paper is to shift the perception of the problem setting: The main problem of aging is not that it affects the economy's capability to produce an increasing wealth, but that increasing burdens are not accepted by the active population and negatively affect economic dynamics. Hence, aging first and foremost is a problem of psychology and incentives. Nevertheless, in discussions, we must always take the fact into consideration that we cannot stop aging and that there is a social responsibility to provide for this part of the population, which created the economy state from which we are moving away.

Moreover, in an aging society, pensioners constitute an ever growing part of the voting force. We are not too far away from a time when half of the voting population will consist of pensioners or people close to pension age. Faced with this One crucial factor is certainly the visibility of pension duties and their perception as burdens. In this connection, it does not appear to be very supportive when workers find their pension duties explicitly displayed on the pay roll (as it currently is the case in Germany). In this regard, a private, capital-based pension system appears much less problematic, since it gives the active generation the impression of caring for its own old-age provision and not of paying into an anonymous pension system.

F

  of the demographic distribution mass for the income development in an aging and stagnating economy. The argumentation of this paper proceeds in two steps:

  for German income development. Even under very

  independent of how this distribution is arranged.7

FIGURE

  FIGURE 1: SET HERE

  (a) In the case | | = | |, both determinants cancel each other out exactly so that the total output remains constant over time: The economic development of the country 'stagnates'. (b) In the case | | > | |, the decrease of the number of workers is overcompensated by the increase of productivity and the total output increases in the course of time. (c) On the contrary, in the case | | < | |, the total output of the economy decreases.

  It becomes clear from the simulation below that this case comes quite close to the expectations for the development of Germany through to the year

  (I) The current German discussion is dominated by the demand to keep the contribution ratio -i.e. the part of the gross income, which every worker contributes 11 However, | t | (ceteris paribus) decreases in the course of time.

  the (fixed) per capita transfer. Since the numerator decreases in equation (21) while it is constant in equation (17), it follows (not very surprisingly) (II) p t decreases even stronger in the course of time than in (I).

  FIGURE 2: SET HERE

FIGURE

  FIGURE 3: SET HERE

  FIGURE 4: SET HERE

  FIGURE 5: SET HERE

  holding the assumption of a dynamic linking of pensions to net wages), with a maximum of about € 10 billion. How much this demographic distribution mass in fact contributes to the simulated income development can be recognised, if we turn back to figure 4: The dotted line exhibits the size of the demographic distribution mass in relation to the (absolute) amount of wages paid in the German economy in the previous year. It shows that the meaning of the demographic distribution mass is considerable, particularly during those years, in which the agingratio increases very steeply: For instance, in the year 2024, it causes an increase in wages and pensions of almost 0.4% alone. In other words: Without the demographic distribution mass, incomes would not only stagnate during the peak of the aging process, but actually decrease by about 0.4 %.

  FIGURE 7: SET HERE

  FIGURE 8: SET HERE

  demand for an extension of the retirement age in Germany.16

  in cases of hardship.

  should also face the question of how to prepare the active population to accept increasing burdens from aging.

  , the gross pension revenue is constant due to the fact that the decrease in the number of workers exactly equals the per capita increase in the gross income. The derivation of p t with respect to t is:

	=	t P Y t p	p	t	=	t P Y	,
	where denotes the (fixed) contribution ratio. In the stagnating economy (with | | =
	| |)(18)	t p = t	2 YW t t P	,
	which is negative under normal conditions (i.e. when , , Y, W t , P t >0). According
	to (3) and (

Page 35 of 36 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  CHANGE IN THE AGE RATIO IN GERMANY (IN PERCENT, COMPARED TO THE PREVIOUS YEAR).CONTINUOS LINE: WITH AN ANNUAL NET IMMIGRATION OF +100,000 PEOPLE AND A RETIREMENT AGE OF 60; DOTTED LINE: WITH AN ANNUAL NET IMMIGRATION OF +200,000 PEOPLE AND A RETIREMENT AGE OF 65. SIZE OF THE DEMOGRAPHIC DISTRIBUTION MASS (IN BILLION EUROS) PER ANNUM AT AN ASSUMED ANNUAL INCREASE OF LABOUT PRODUCTIVITY OF 1 % AND A DYNAMIC PENSION SYSTEM. SOURCE: DEVELOPMENT OF ANNUAL PER CAPITA NET WAGES (IN EUROS), MAINTAINING A DYNAMIC PENSION SYSTEM WITH AN ASSUMED PRODUCTIVITY GROWTH OF 1 % P.A. (CONTINUOUS LINE), WITH A FIXED PENSION RATIO OF 20 % AND AN ASSUMED PRODUCTIVITY GROWTH OF 1 % (DOTTED LINE), AND MAINTAINING THE DYNAMIC PENSION SYSTEM WITH AN ASSUMED PRODUCTIVITY GROWTH OF 2 % (CROSSED

	Page 33 of 36								Submitted Manuscript Submitted Manuscript Submitted Manuscript Submitted Manuscript	Page 32 of 36 Page 34 of 36 Page
	FIGURE 1: AGING PROCESS OF THE MODELLED SOCIETY B, W t , P t B FIGURE 3: SOURCE: STATISTISCHES BUNDESAMT FIGURE 5: STATISTISCHES BUNDESAMT, OWN CALCULATIONS 12 FIGURE 7: LINE). SOURCE: STATISTISCHES BUNDESAMT, OWN CALCULATIONS
	8 10 4%									P
	3% 100000 6				W t
	2%	2 80000 4 90000												t
	1%	0 70000
	0% -2 60000
	-1% -4 40000 1999 50000	2003		2007	2011	2015	2019	2023	2027	2031	2035	2039	2043	2047
	-2%
	o r F o r F o r FIGURE 2: DEVELOPMENT OF LABOUR PRODUCTIVITY PER WORKING HOUR IN THE ALTE LÄNDER AND IN GERMANY SINCE 1971. SOURCE: SACHVERSTAENDIGENRAT ZUR BEGUTACHTUNG DER GESAMTWIRTSCHAFTLICHEN ENTWICKLUNG, OWN CALCULATIONS. F o r 1999 2003 2007 2011 2015 2019 2023 2027 2031 2035 2039 2043 2047 F o r F o r r o 30000 F 1998 2002 2006 2010 2014 2018 2022 2026 2030 2034 2038 2042 2046 2050
	P e P e P e P P P FIGURE 6: REQUIRED PRODUCTIVITY INCREASE (CONTINUOUS LINE) AND REQUIRED CHANGE IN THE GDPIN 4% 6% P 5% Alte Länder e e e e
	2% 3% 1,0% 1,2%					e r e r e r e r e r e r required productivity grow th e r	Germany
	0% 1% -0,2% 1971 0,0% 0,2% 0,4% 0,6% 0,8% 1,0% 1,2% -1,0% -0,6% -0,4% -0,2% 0,0% 0,2% 0,4% 0,6% 0,8% 1,2% 1,4% 0,8% 1,0% 0,6% 0,4% -0,8% 0,2% 0,0% -0,2%	1998 1998	1974 2002 2002	R e v i e w R e v i e w R e v i e w 1989 1991 1994 1997 2000 R e v i e w BUNDESAMT, OWN CALCULATIONS 1977 1980 1983 1986 R e v i e w R e v i e w required change of GDP R e v 2006 2010 2014 2018 2022 2026 2030 2034 2038 2042 2046 i e w 2006 2010 2014 2018 2022 2026 2030 2034 2038 2042 2046
			1998	2002	2006	2010	2014	2018	2022	2026	2030	2034	2038	2042	2046	2050
															31 32 33 34 35

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK FIGURE 4: CONTINUOS LINE: CHANGE OF WAGES AND PENSIONS COMPARED WITH THE PREVIOUS YEAR (IN %); DOTTED LINE: SIZE OF THE DEMOGRAPHIC DISTRIBUTION MASS (IN RELATION TO THE WAGES OF THE PREVIOUS YEAR) WITH AN ASSUMED PRODUCTIVITY INCREASE OF 1 % P.A.. SOURCE: STATISTISCHES Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK ORDER TO KEEP WAGES AND PENSIONS CONSTANT OVER TIME (DOTTED LINE). SOURCE: STATISTISCHES BUNDESAMT, OWN CALCULATIONS FIGURE 8: CHANGE IN WAGES AND PENSIONS COMPARED WITH THE PREVIOUS YEAR (IN %);DOTTED LINE: SCENARIO 1; CONTINUOUS LINE: SCENARIO 2. SOURCE: STATISTISCHES BUNDESAMT, OWN CALCULATIONS

36 of 36 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

This insight has already been recognised by[START_REF] Mackenroth | Die Reformpolitik durch einen deutschen Sozialplan, Schriften des Vereins für Socialpolitik[END_REF]. Of course, this does not mean that we can neglect the fact that the institutional arrangement of pensions has various effects -not only on income distribution, but also on economic growth and, thus, the size of the cake to distribute. For the discussion of the Mackenroth hypothesis see also[START_REF] Feldstein | Social Security and Private Savings: International Evidence in an Extended Life-Cycle Model[END_REF],[START_REF] Schmähl | Über den Satz "Aller Sozialaufwand muß immer aus dem Volkseinkommen der laufenden Periode gedeckt werden[END_REF] [START_REF] Homburg | Ein Schnellkurs in Sachen Rentenreform[END_REF].

This implies that all other income groups, in particular unemployed and children, do not exist in our

Note that technical progress is a precondition for the stagnation of an aging economy. Without technical progress, any aging economy would be contracting. I will address this point later.