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May 19, 2006
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In the coming decades, the demographic changes looming ahead will hit Germany with an impact never felt before. This applies not only to the pension system. It holds equally for the labour market, and will entail repercussions for wages and interest rates and thus growth potential and international capital flows. DB Research has analysed the complex interplay of these factors by using an overlapping generations (OLG) model. [more]
The demographic challenge: Simulations with an overlapping generations model Demography Special Current Issues Authors Bernhard Gräf +49 69 910-31738 bernhard.graef@db.com Marc Schattenberg Universität Halle +49 345 5523-323 marc.schattenberg@wiwi.uni- halle.de Editor Stefan Schneider Technical Assistant Pia Johnson Deutsche Bank Research Frankfurt am Main Germany Internet: www.dbresearch.com E-mail: marketing.dbr@db.com Fax: +49 69 910-31877 Managing Director Norbert Walter May 19, 2006 In the coming decades, the demographic changes looming ahead will hit Germany with an impact never felt before. This applies not only to the pension system. It holds equally for the labour market, and will entail repercussions for wages and interest rates and thus growth potential and international capital flows. DB Research has analysed the complex interplay of these factors by using an overlapping generations (OLG) model. The main results of our simulations: — The growth potential of the German economy will shrink from about 1 ¼% p.a. at present to a mere ¼% p.a. by about 2060. — The annual increase in real income per capita will be dampened by up to 0.3 of a percentage point up to 2050, falling to just below 1% p.a. This comes to only one-third of the annual increases in prosperity from 1955 to 2005. — Under “Status quo” conditions, the return on capital will decline by around 100 basis points by 2060. — A change of pensions policy towards “More personal provision” would drive down returns by a further 35 basis points. The findings of this study are generally in line with those of our past analyses pertaining to the demographic challenge, but they have to be interpreted with caution as they are based in some cases on very restrictive assumptions. The demographic challenge Simulations with an overlapping generations model 0.00 0.25 0.50 0.75 1.00 1.25 1.50 2000 2020 2040 2060 2080 GDP per capita Total GDP Source: DB Research Growth potential declines to 1/4% p.a. Potential GDP, % yoy 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 2000 2020 2040 2060 2080 Source: DB Research Status quo "More personal provision" (starting level standardised at 7.5%) Rate of return decreases faster in "More personal provision" scenario Return on capital Current Issues 2 May 19, 2006 The demographic challenge May 19, 2006 3 Contents 1. Introduction: Demographic consequences are unavoidable ........................................................................................... 4 1.1 Full “broadside“ to hit economy, society and politics ............................................................................... 4 1.2 OLG – several generations living side by side ........................................................................................ 5 1.3 A note to readers ..................................................................................................................................... 6 2. The simulation results: Growth potential and economic returns will decline ................................................................................. 7 2.1 The demographic change – population to steady again after 2150 ........................................................ 7 2.2 Two scenarios – “Status quo” and “More personal provision” ................................................................. 8 2.3 „Status quo” simulations .......................................................................................................................... 8 2.3.1 Contribution rate and pension replacement rate: The usual results............................................... 8 2.3.2 Shoring up the pension level – deficits of up to 3% of GDP........................................................... 9 2.4 Growth potential plummets in both scenarios.......................................................................................... 9 2.5 Overall savings ratio declines also in “More personal provision” scenario............................................ 11 2.6 Lower rate of return – asset meltdown unlikely ..................................................................................... 11 2.7 Results have to be interpreted with caution........................................................................................... 13 3. Overlapping generations models: Attempting a simple explanation................................................................................................................ 14 3.1 Why OLGs? ........................................................................................................................................... 14 3.2 The DB Research OLG model – 3 sectors, 18 generations and two life phases .................................. 15 3.3 The life-cycle hypothesis – intertemporal utility maximisation ............................................................... 17 3.4 Individuals have a time preference ........................................................................................................ 19 3.5 Perfect foresight – who can have that? ................................................................................................. 20 3.6 OLG models are dynamic equilibrium models....................................................................................... 21 3.7 The transitional stage of the adjustment process .................................................................................. 22 3.8 OLGs and farmers – what do they have in common? ........................................................................... 23 3.9 The solution of an OLG model – going the wrong way about it............................................................. 24 3.10 OLG model requires calibration ............................................................................................................. 24 4. Notes on design: The mathematical formulae for our OLG model............................................................................................. 26 4.1 Population – labour force and pensioners ............................................................................................. 26 4.2 The business sector – given by a Cobb-Douglas production function .................................................. 27 4.3 The household sector – intertemporal utility maximisation.................................................................... 28 4.4 The state sector – administrator of the pension system ........................................................................ 32 Bibliography............................................................................................................................................................ 33 (The authors would like to express their thanks to Professor Gunter Steinmann (Martin-Luther-University Halle-Wittenberg) and PD Dr. Manfred Jäger (Cologne Institute of Business Research and Martin-Luther-University Halle-Wittenberg) for their critical review of this report and their constructive comments. Special thanks go in particular to PD Dr. Jäger for his assistance in getting the programme up and running.) Current Issues 4 May 19, 2006 1. Introduction: Demographic consequences are unavoidable 1.1 Full “broadside” to hit economy, society and politics The demographic challenge is no figment of the imagination. In the coming decades, demographic developments will hit Germany with an impact never felt before. For more than 30 years the German fertility rate has fallen roughly one-third short of the replacement level that ensures a constant population. Thus, for a long time no more than two-thirds of the parent generation has been replaced. Combined with a rise in life expectancy the result will be a noticeable ageing of German society in the years ahead and, depending on the level of future immigration, the population will shrink. The demographic effects will remain fairly limited until about 2010-15. Subsequently, though, they will gain momentum drastically when the baby-boom generations start to reach retirement age, leading to a dramatic drop in the size of the potential labour force. Labour supply to shrink twice as fast as population Germany’s Federal Statistical Office shows in the medium variant of its 10th coordinated population projection that the population will remain relatively stable until 2010-15, but subsequently decline by 7.7 million, or 9%, to 75 million by 2050 1 . The average age will rise accordingly, from around 42 at present to 49. Estimates put forward by the Institute for Employment Research (IAB) indicate that, at 20%, the potential labour force (labour supply) will shrink more than twice as fast as the overall population from 2010 to 2050 because of the baby-boom effect 2 . At the same time, the net immigration of 200,000 persons per year assumed by the IAB substantially curbs the decline triggered by the demographic development. Without migration, the potential labour force would shrink by 18.2 million, or roughly 40%, by 2050. However, analysis from the Halle Institute for Economic Research (IWH) suggests that the positive effects of net immigration will feed through differently in east and west Germany 3 . Echo effects will reverberate for a long time Even if the fertility rate rises rapidly to the replacement level it will still not be able to stop the trend that began in the 1970s. At best it might be able to slow it down in the longer run. This is partly be- cause of a phenomenon referred to as demographic echo effects. Since relatively few children have been born since the 1970s, there is now a relative paucity of potential mothers. The number of child- ren born in west Germany declined by nearly half between 1965 and 2004, from 1.04 million to 577,000. The medium variant says that 1 See Statistisches Bundesamt: Bevölkerung Deutschlands bis 2050, 10. koordi- nierte Bevölkerungsvorausberechnung, Wiesbaden 2003. The population project- ion covers 9 variants which differ on the basis of various assumptions in respect of life expectancy and annual net immigration. All variants are predicated on an unchanged fertility rate of 1.4 children per woman. The medium variant (variant 5) is based on an increase in life expectancy at birth of 6.3 years for boys, to 81.1, until 2050, and of 5.8 years for girls, to 86.6, as well as on net immigration of 200,000 persons per year. 2 See Fuchs, J. and K. Dörfler (2005). Projektion des Erwerbspersonenpotenzials bis 2050. Annahmen und Datengrundlage. IAB Forschungsbericht, No. 25/2005. 3 See Steinmann, G. and S. Tagge (2002). Determinanten der Bevölkerungsent- wicklung in West- und Ostdeutschland. In Wirtschaft im Wandel (4), 2002. 75 80 85 90 95 100 105 2004 2014 2024 2034 2044 Total population Potential labour force 2004 = 100 Source: Federal Statistical Office, IAB Potential labour force to shrink twice as fast as population 1 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 1955 1965 1975 1985 1995 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Births (left) Fertility rate (right) Source: Federal Statistical Office Echo effects will reverberate for a long time m (left), children per woman (right) 2 The demographic challenge May 19, 2006 5 the number of women of child-bearing age will fall from nearly 20 million to just over 14 million in 2050. Demographic change will not only hit social-security systems The problems of the state pension system are often cited in order to illustrate the effects of the demographic challenge. The Federal Statistical Office calculates that the old-age dependency ratio, i.e. the number of persons over 65 in relation to 100 persons aged 15 to 64, will rise in Germany – in a best-case scenario 4 – from just over 28 now to over 45 by 2050. In a worst-case scenario 5 it will in fact double to 57 ½. Correspondingly, the ratio of pensioners to con- tributors to the state pension system, i.e. the central factor for our pay-as-you-go pension system, will jump noticeably. Therefore, radical reforms of the social-security system are virtually unavoid- able. The demographic effects on the economy will go much further than this, though. They range from profound changes in the labour market with related effects on the relative prices of labour (wages) and capital (interest rates) and a country’s growth potential and extend to changes in consumer demand and thus in sectoral structures, through to effects on international capital flows. Partial analyses are of limited help All of these effects are, in addition, interdependent. Partial analyses may be helpful to get an idea about the effect of demographic pro- cesses on individual parts of our economy. However, they are un- able to capture a complete picture of what awaits Germany in the years ahead. 1.2. OLG – several generations living side by side To obtain an overview our report uses a global model to analyse the demographic effects on key macroeconomic variables such as consumption, savings, interest rates, wages and public finances. The model investigates their joint impact and also takes consider- ation of their interdependencies. This requires an approach that extrapolates not only historical data and behavioural patterns into the future, but also factors in complex interactions between the generations. To this end, we used an Overlapping Generations (OLG) model. Academic researchers regard this type of model as a very suitable instrument to analyse demographic processes 6 . The OLG approach is based on the fact that at any given time in an economy there are several generations of different ages living side by side and interacting. With every period observed, the generations advance in age – with the oldest generation dying and a new one being born. 4 Variant 3 of the 10th coordinated population projection. It looks for a low life expectancy (increase in life expectancy of boys at birth of 4.1 years by 2050 and of girls of 4.9 years) and for net immigration of foreigners totalling 300,000 per year. 5 Variant 7 of the 10th coordinated population projection. It assumes a high life expectancy (increase in life expectancy of boys at birth of 7.8 years by 2050 and of girls of 7.3 years) and for net immigration of foreigners totalling 100,000 per year. 6 See Auerbach, A.J. and L.J. Kotlikoff (1987). Dynamic Fiscal Policy. Börsch- Supan, A., A. Ludwig and J. Winter (2003/2004). Aging, pension reforms, and capital flows: A multi-country simulation model. MEA Discussion Paper No. 28, April 2003 and update as Discussion Paper No. 64, August 2004. Schmidt, S. (2004). Computerbasierte Anwendungen von Modellen sich überlappender Generationen. In ZEW Konjunkturreport No. 3, 2004. 25 30 35 40 45 50 55 60 2002 2012 2022 2032 2042 Population variant 3 Variant 5 Variant 7 Source: Federal Statistical Office Old-age dependency ratio jumps in all variants 65+ per 100 persons aged 15-64 3 Overlapping generations G1 G2 G3 G4 G1 G2 G3 G4 G1 G2 G3 G4 G1 G2 G3 G4 G1 G2 G3 G4 2000 2020 2040 2060 2080 4 Current Issues 6 May 19, 2006 OLG based on life-cycle theory The basis of an OLG model is the microeconomic level. Each generation is represented by an economic agent who, according to the life-cycle model, has specific age-related consumption and saving patterns and who maximises his utility over his entire lifetime. For one thing, this makes it possible to depict the population in its generational structure. For another, it is possible to perform a dynamic analysis of the intergenerational feedback effects of demographic developments and/or economic policy measures. Within the model, demographic shocks lead to changes in the consumption and saving patterns of individual generations. How- ever, the decisions of the current generations also influence the behaviour of subsequent generations. Equally, anticipated decisions of future generations have effects on the actions taken by today’s generations 7 . Results have to be interpreted with caution As with all models the findings of an OLG model depend on how the underlying theory is formulated. An OLG model is based on neo- classical theory although, granted, the assumptions are very restrictive. Despite the assumption-related restrictions, e.g. that we have a closed economy, such a model is constructive when analys- ing demographic processes. For instance, it takes account of inter- dependencies and interactions, providing indications of the magnitude of changes in the observed variables, such as the interest rate. 1.3 A note to readers Readers with little time to spare will find a brief description and explanation of our model simulations in chapter 2. Those who would like to spend more time on the theoretical foundations of OLG models in all their complexity and debate the economic inter- pretation of how they function are recommended to peruse chapter 3. Anyone who also enjoys mathematical formulae will find sufficient food for thought in chapter 4. 7 See chapters 3 and 4 for a description, economic interpretation and mathematical formulation of the OLG model. The demographic challenge May 19, 2006 7 2. The simulation results: Growth potential and economic returns will decline 2.1 The demographic change – population to steady again after 2150 DB Research’s OLG model: A short description Our OLG model is based on a closed economy and comprises three sectors – households, businesses and the state. Households have perfect foresight and maximise the utility of their consumption over their entire lifetime. As described in the life-cycle hypothesis they spread their consumption over time by varying their saving patterns. The distribution of lifetime consumption over the different periods is thus only predicated on the households’ time preferences, their willingness and ability to forgo consumption in the present, and the prevailing interest rate 8 . Our model covers a total of 18 generations. Eleven generations are in their working phase, seven in their retirement phase. We say that children are supported financially by their parents so they are not taken explicitly into consideration. The retired households draw a state pension and use up their savings for consumption purposes. The households in their working phase provide businesses with their labour, which according to a Cobb- Douglas production function produces a good which can be both consumed as well as invested. Interest rates and wages are determined by the marginal productivity of labour and capital. The state is reduced to its function as the administrator of the pension system. Population development is key variable The variable at the core of an OLG model is population develop- ment. It is the cause of the adjustment processes and it determines the economic interactions that an OLG model reflects. An OLG model is based on equilibrium solutions. If the model is initially in equilibrium but then disturbed by a demographic shock, such as a decrease in the size or an increase in the age of the generations, it will gradually find a new steady state in the long run (see point 3.6). The transition path is of particular interest here. It describes the effects of the demographic change on our model economy. Since a new equilibrium emerges again only in the long term, our OLG model requires a very protracted survey period – up to about 2150 – to produce a solution. For this reason we will start by discussing the assumptions on population development for the coming 150 years. However, we will subsequently only depict the results of our simulations up to 2080 since the strongest effects of the demographic change will have materialised by then. Demographic change will have repercussions for a long time Until 2050 the model simulations are based on the demographic development charted in the medium variant of the Federal Statistical Office’s latest population projection. It says that the population will remain relatively constant up to 2010-15, but then decline by 9% to around 75 million by 2050. Because of the baby-boom effects the working-age population will in fact shrink by 20% during this period. We projected the population trend after 2050 on the assumption that the fertility rate will gradually converge with the replacement level by 8 See Deaton, A. (1992). Understanding consumption. Demography Households Businesses State Pensio- ners Labour force Cobb-Douglas- production function Life-cycle hypothesis: Perfect foresight, maximum utility Limited to pension system Model economy 5 40 50 60 70 80 90 100 110 120 130 2000 2040 2080 2120 2160 Source: DB Research Population Labour force Pensioners Population steady again after 2150 2000 = 100 6 Current Issues 8 May 19, 2006 the end of the millennium. As a result, the shrinking process slows down with a noticeable time lag and the population steadies again roughly around the middle of the next millennium. Our projections suggest that the population in our “model world” will decline by a further 15% from 2050 to 2080 and then remain stable from about 2150. Old-age dependency ratio: Set to nearly double by 2060 Because of the ageing pattern, substantial shifts emerge in the population structure of our “model world” during the transition to a new steady state. The old-age dependency ratio, i.e. the number of individuals in our model belonging to the retired generations for every 100 individuals in the labour force, increases from 60 to 110 by about 2060. It then gradually retraces to its starting level by 2150. 2.2 Two scenarios – “Status quo” and “More personal provision” Taking the discussed population developments as our basis, we simulate two scenarios in our OLG model. The first scenario is called “Status quo”. In it we set out the current framework of Germany’s pension system and extrapolate it into the future. The pay-as-you-go financing method with the budget constraint of the statutory pension system predetermines either the rate increase necessary to secure the current level of pension replacement or the potential replacement rate if the contribution rate remains constant. If the state steps into the breach and holds both policy variables constant, our model can determine the resultant budget deficits and increases in public debt besides arriving at conclusions about the long-term sustainability of such policy. We called the second scenario “More personal provision”. In it, the contribution rate paid by labour into the state pension system is held constant, so the given population trend results in the pension level falling accordingly. This opens a demographic pension gap that households close by increasing their personal provision for retirement. This scenario would be in keeping with the ongoing development in Germany of the reform initiated with the state- subsidised “Riester” pension. 2.3 “Status quo” simulations 2.3.1 Contribution rate and pension replacement rate: The usual results The findings of our model simulation for the two policy variables “contribution rate” and “replacement rate” are in line with the usual results of model calculations for the statutory pension system and the well-known “rules of thumb” 9 . Because of the budget constraint of the statutory pension system it is held that when the federal subsidy is disregarded the contribution rate is equal to the old-age dependency ratio times the replacement rate. So when the pension level is constant the contribution rate has to develop like the old-age dependency ratio. In our model, the contribution rate rises from just over 26% (which, taking consideration of the federal subsidy, is in line with the current contribution rate of 19.5%) to almost 45%, if the demographic adjustment rates have to be shouldered fully by the labour force. Conversely, if the pensioners bear the full cost of the 9 See Bräuninger, D and B. Gräf. (2005). Spürbare Rentenlücken trotz Reformen. Deutsche Bank Research. Aktuelle Themen, Demografie Spezial Nr. 312. January 12, 2005. Frankfurt am Main. 50 60 70 80 90 100 110 120 2000 2040 2080 2120 2160 Source: DB Research Number of pensioners per 100 workers Old-age dependency ratio: Back at former level by about 2150 7 20 25 30 35 40 45 50 2000 2020 2040 2060 2080 Source: DB Research Status quo: Contribution rate nearly doubles % of labour income 8 The demographic challenge May 19, 2006 9 adjustments, the net replacement rate from the state pension system drops back from 70% to almost 40% by 2060. 2.3.2 Shoring up the pension level – deficits of up to 3% of GDP In our model the state is reduced to its function as the administrator of old-age security. If in the “Status quo” scenario the budget constraint of the statutory pension system is abolished and the state offsets the reduction in the pension level when the contribution rate remains constant by granting a federal subsidy, public finances come under substantial pressure due to demographic forces. The government budget, which is initially balanced, then slides into a deficit which it continues to show over the entire survey period up to 2060, when the figure reaches over 3% of GDP per year. Govern- ment debt (currently just over 60%) thus comes to exceed GDP from 2050. But debt is then still far from its peak. While our simulation suggests that the deficits will gradually start to ease again after 2080 and that the budget will return to equilibrium by about 2150, the debt burden will continue to increase to nearly 350% of GDP by 2150 because of the growing interest burden. Such a policy cannot be regarded as sustainable in the long run 10 . The sustainability gap is equal to the annual deficits. Thus to shore up the long-term sustainability of the public finances it will be necessary to enact consolidation measures of over 3% of GDP per year at times. Neither the two extreme solutions (increase in the contribution rate or reduction of the replacement rate) nor the deficit solution are feasible alternatives. For this reason we have employed our OLG model to analyse the effects of the reform scenario “More personal provision”, setting it in the following discussion against the various aspects of the “Status quo” scenario. 2.4 Growth potential plummets in both scenarios GDP growth rates are nearly equal in the two scenarios… The two scenarios produce nearly identical results for the rate of potential output growth. This is not very surprising considering that according to the Solow growth model which is central to neo- classical theory – also in its augmented Ramsey form 11 – economic growth depends exclusively, in a long-term steady state, on population growth and the rate of technological progress. In the transition from the initial equilibrium to the new steady state our model shows that growth potential drops off noticeably at first and reaches its nadir at about ¼% p.a. between 2060 and 2065. This is assuming that potential growth is initially close to 1 ¼% 12 , a rate reached in the new steady state by about 2150. The OLG simulations show that potential growth declines more strongly during 10 In its most recent calculations on the long-term sustainability of Germany’s public finances for the statutory pension system the ifo Institute reports that the shortfalls will come to merely 0.9% (baseline variant) or 1.2% of GDP (risk variant). How- ever, it assumes an increase in the contribution rate from 19.5% to 23.6% (base- line variant) or 24.5% (risk variant) respectively by 2050 besides a simultaneous decline in the gross replacement rate from close to 48% to 37.8% (36.5%). See Werding, M. and A. Kaltschütz (2005). Modellrechnungen zur langfristigen Trag- fähigkeit der öffentlichen Finanzen. ifo Beiträge zur Wirtschaftsforschung, 2005. 11 The Solow model works on the assumption that the savings ratio is constant, whereas the augmented Ramsey version allows intertemporal utility maximisation to determine the ratio endogenously. It can be proved that the results are identical in both models. See Barro, R.J. and X. Sala-i-Martin (2004). Economic Growth. 12 According to recent research, this is the value estimated to reflect current growth potential. See Kamps, C., C.-P. Meier and F. Oskamp (2004). Wachstum des Produktionspotenzials in Deutschland bleibt schwach. Kieler Diskussionsbeiträge (Kiel Institute for World Economics), No. 414. September 2004. 35 45 55 65 75 2000 2020 2040 2060 2080 Source: DB Research Status quo: Net pension replacement rate declines by 30 pp Net pension from statutory pension system, % of labour income 9 0 50 100 150 200 250 300 350 400 2000 2040 2080 2120 2160 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Budget- balance (right) Debt (left) Continual deficits of up to 3% of GDP % GDP 10 Source : DB Research Current Issues 10 May 19, 2006 the transition than determined in an earlier analysis by Deutsche Bank Research 13 . The main reason for the difference is that at the time it was assumed that the capital stock would deliver unchanged contributions to growth. However, in the OLG model the households adapt to the demographic development by varying their savings ratio, which leads in a closed economy to a decrease in the capital stock and thus to falling growth contributions from capital as a factor of production. The increase in GDP per capita hardly differs in the two scenarios either. Because of the downturn in the population, it is higher than the growth of potential output. Our model finds that the per-capita growth of GDP is curbed for demographic reasons by up to 0.3 of a percentage point per year until 2050 14 , falling to less than 1% p.a. This represents just one-third of the annual prosperity growth over the past five decades (real GDP per capita from 1955 to 2005: +2.7% p.a.). … but GDP levels still differ Even though the rates of potential output growth as a whole and per capita are roughly equal in the two scenarios, there are still notice- able differences between the GDP levels and thus per-capita in- come in the initial steady state. To calculate the initial steady state we solved our model for the past 60 years. Since the households have perfect foresight, they anticipate the demographic change and thus the decline in the level of their state pension. Accordingly, even before the population starts to age and shrink they save a larger amount in the “More personal provision” scenario than in the “Status quo” scenario so that they can maintain their consumption level during retirement. This increases the capital stock, enabling a roughly 10% higher level of GDP on the given production factors labour and capital than in the “Status quo” scenario. In a nutshell: We will have to get used to substantially lower prosperity gains going forward. 13 See Gräf, B. (2003). German growth potential: facing the demographic challenge. Deutsche Bank Research. Current Issues / Demography Special. December 11, 2003. Frankfurt am Main. 14 A. Ludwig (2005) finds fairly strong damping effects of up to 0.5 of a percentage point. Aging and Economic Growth: The Role of Factor Markets and of Fund- amental Pension Reforms. MEA Discussion Paper No. 94, February 2005, while an OECD analysis suggests the damping effects on per-capita income come to 0.2-0.3 of a percentage point: see Martins, J.O., F. Gonand, P. Antolin, C. de la Maisonneuve, and K.-Y. Yoo (2005). The Impact of Ageing on Demand, Factor Markets and Growth. OECD Economic Department Working Papers No. 420. March 2005. 0.00 0.25 0.50 0.75 1.00 1.25 1.50 2000 2020 2040 2060 2080 GDP per capita Total GDP Source: DB Research Growth potential declines to 1/4% p.a. Potential GDP, % yoy 11 The demographic challenge May 19, 2006 11 2.5 Overall savings ratio declines also in “More personal provision” scenario The way the individuals in our OLG model interact can be seen best by looking at the savings ratio. It is the variable that reflects how households react to changes in the general demographic conditions and pensions policy measures. For these purposes we assume that our “OLG households” only save for their retirement. “Status quo”: The purely demographic effect Chart 12 shows the development of the household savings ratio for the “Status quo” scenario when the contribution rate remains un- changed and the state ensures a constant pension replacement rate. In this case, the households have no reason to change their saving pattern. The age-specific savings ratio shows the ideal curve according to the life-cycle hypothesis (see 3.3). It is high for the working-age generations and strongly negative for the pensioner generations. The overall savings ratio corresponds to the weighted average of the age- (generation-)specific ratios. In the “Status quo” scenario the pattern of the savings ratio is only shaped by how the demographics develop, i.e. by how many households there are in the given generation. The chart thus shows the purely demographic effect when the number of older households with a low or even negative savings ratio increases. Until 2020 the overall ratio in our “model world” is still relatively constant. During this period the baby boomers are still in their income phase, and thus the phase in which they save the most. This is followed by an accelerated decline during which the baby boomers go into retirement and start to tap their savings to maintain their consumption level. According to our model findings the savings ratio will more than halve by 2060-65, from 11% to about 5%. “More personal provision”: Overall decline in savings ratio, but higher than in “Status quo” If – as in our “More personal provision” scenario – the households have to dip into their own pockets to secure the level of their retirement provision, the overall savings ratio also declines but annually it remains 1 to 2 percentage points higher than in our “Status quo” scenario. This shows that the additional savings account for somewhat less than half of the purely demographic effect, and that the demographic effect is thus dominant. 2.6 Lower rate of return – asset meltdown unlikely Status quo: Rate of return falls by over 100 basis points In our OLG model, which is based on a closed economy, the interest rate is determined by the marginal productivity of capital, that is, by the contribution to production generated by adding one unit of capital. The return on capital falls if labour is scarcer and capital more abundant. For this reason, the rate of return in countries with a shrinking population tends to decline. Assuming the population develops as in our “Status quo” scenario, our OLG model suggests that the rate of return will decline by just over 100 basis points. Note that the rate of return modelled is not the market return on capital, but the return on the entire capital stock. “More personal provision” accompanied by 135 basis point decline The decline in the rate of return is actually amplified by increased personal provision. If the current pension reform aimed at increasing -100 -80 -60 -40 -20 0 20 40 1 3 5 7 9 11 13 15 17 Source: DB Research Generations Status quo: Generation- specific savings ratio follows life-cycle hypothesis Savings ratio, % 13 0.0 0.5 1.0 1.5 2.0 2.5 2000 2020 2040 2060 2080 Source: DB Research "More personal provision": Additional savings %-points Additional savings vs status quo 14 2 4 6 8 10 12 2000 2020 2040 2060 2080 Source: DB Research Status quo: Savings ratio halves by 2060 Savings ratio, % 12 Current Issues 12 May 19, 2006 the level of funded private provision is continued rigorously, the savings ratio will not fall off as much as otherwise, so – compared to labour – capital will be more abundant than in the “Status quo” scenario. Under these circumstances, the rate of return will fall by a further 35 bp or so 15 . Since – as with the GDP level – the rate of return in the “More personal provision” scenario has a different starting level than in the “Status quo” scenario, the starting value was standardised in the chart at the “Status quo” value of 7.5%. This was done to demonstrate the demographic effect more clearly. The solution offered by our model is roughly one percentage point below that at about 6.5%. The reason is that in the “More personal provision" scenario higher savings lead to a larger capital stock and thus to a lower interest rate. International diversification can curb decline in rate of return The findings discussed apply on the premise of a closed economy. If this assumption is abandoned and capital is allowed to move freely, Börsch-Supan shows that the decrease in the rate of return in the “More personal provision” scenario can be limited to slightly less than one percentage point 16 if investment is diversified within the OECD states. But since ageing processes will appear there too, in the best of cases with a time lag, Börsch-Supan says the diversification effects will be strongest when capital is invested in “young” economies that generate high returns. This offers the chance of the reform-related decline in the rate of return being at least partly offset by international diversification. Empirical studies show, however, that investment portfolios still tend to have a “home bias”, i.e. people tend to invest their money in the domestic market 17 . Asset meltdown: Fears are unfounded The asset meltdown hypothesis assumes that the baby-boom generations start to retire as of 2010-15 and then sell their assets in order to finance part of their consumption during their old age, i.e. they “consume” their assets. Because there will be many sellers then but fewer buyers because of the demographic development, the prices of equities, fixed-income securities and real estate will – it is presumed – all plummet. The dreaded asset crash would drastically reduce the fruits of personal provision. The findings of our OLG simulations offer no indication of a potential asset meltdown. Rather, they show a slow, moderate decline in the overall rate of return on capital. However, this is where the model’s limitations become evident. Disregarding the costs of capital adjustment 18 , the interest rate in neoclassical OLG models is determined by 15 Using OLG model simulations of its own, the OECD arrives at more moderate changes in the rate of return. In its “No reform” scenario the decline comes to only about 30 bp up to 2050, while in its “Pension saving” scenario, which roughly corresponds to our “More personal provision” scenario, the decrease exceeds 100 bp. Note, though, that this is from a starting level of 4 ½%. See Martins, J.O et al. (2005). The Impact of Ageing on Demand, Factor Markets and Growth. OECD Economic Department Working Papers No. 420. March 2005. 16 In his analyses Börsch-Supan uses a multi-country OLG model based explicitly on Germany, France, Italy and individual blocs representing the other EU members, North Armerica and the rest of the world. See Börsch-Supan et al. (2004). Aging, pension reforms, and capital flows: A multi-country simulation model. Discussion Paper No. 64, August 2004. 17 See French, K.R. and J.M Poterba (1991): Investor diversification and international equity markets. In American Economic Review (81) 1991. 18 See A. B. Abel for more on the effect of the costs of capital adjustment (2002). The effects of a baby boom on stock prices and capital accumulation in the presence of social security. July 2002. 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 2000 2020 2040 2060 2080 Source: DB Research Status quo "More personal provision" (starting level standardised at 7.5%) Rate of return decreases faster in "more personal provision" scenario Return on capital 15 The demographic challenge May 19, 2006 13 productivity and not by the interplay of supply and demand for capital. However, some plausibility considerations suggest that the fear of an asset meltdown is exaggerated. Assets will be cashed in gradually Since the baby boomers will enter retirement over several decades, any potential loss of wealth will also ensue over a long period. More- over, it is unclear how many of the baby boomers will actually dis- pose of their assets. Besides, they will not do it all at once. The life- cycle hypothesis assumes that pensioners use up their savings only gradually as they grow older. And at present even this is not yet the case. So far, German households have continued to accumulate savings in their old age and build up assets – albeit not as markedly as in midlife (see 3.3 for further details). Additionally, an asset melt- down becomes increasingly unlikely when the investment portfolio is diversified internationally since potential disposals are spread among many countries with differing demographic developments. 2.7 Results have to be interpreted with caution The results of our OLG simulations sound plausible and are of a magnitude that has already been established in partial analyses. Nevertheless, they should be interpreted with caution since they are based in some cases on very restrictive assumptions. The main assumptions that could limit the significance of the findings are: — The model is based on a closed economy. — The households have perfect foresight. They maximise their consumption over their entire lifetime and can decide their optimum consumption or savings plan from the outset. — The time preferences of the households and their risk aversion levels are constant across the entire survey period. — The labour participation rate is constant and equal to 1, i.e. in our model all households of working age “work”. — Wages are completely flexible. — The labour market clears, i.e. there is no unemployment. — There is no transfer of assets by inheritance. — Companies produce goods according to a Cobb-Douglas production function. — Only one good is produced and it can be both consumed as well as invested. — The state is merely the administrator of the pension scheme. Besides, population development – the central variable – is fraught with considerable uncertainty over the long period needed for an OLG model to produce a solution. This applies for example to the assumption that the fertility rate will return towards the replacement rate, which facilitates the solution of an OLG model. But if the fertility rate remains below the replacement rate, the population will continue to shrink unabatedly. Neoclassical theory does allow the emergence of a new steady state even when the size of the population steadily decreases. However, the stability of the system and thus the attainment of a new steady state is in this case not always guaranteed. Current Issues 14 May 19, 2006 3. Overlapping generations models: Attempting a simple explanation Overlapping generations models are highly complex. We shall therefore outline the theory on which they are based, describe how they are structured as well as how they operate and we shall detail their shortcomings. So as not to intimidate our readers we have omitted the mathematical formulae for our OLG model and will attempt to exemplify the model’s extremely complex concepts 19 . 3.1. Why OLGs? Why is there currently more use being made of OLG models, in contrast to the conventional forecasting models that are, for example, regression-based? After all, OLG models are difficult to understand – even for those experienced in econometrics – and can only be solved using complex iterative processes performed on a computer running special software. There are two main factors in their favour. OLG models explicitly factor in demographic developments… Firstly, OLG models enable demographic developments to be taken into account explicitly. They can thus be said to be the “most natural” models for depicting demographic transition. After all, the basic idea of OLG models – as the name says – is that at any one time there are several generations living side by side with differing age-specific consumption and spending patterns. With each period observed, the generations advance in age – with a new generation being born and the oldest one dying. The generations thus overlap as exemplified in chart 16. Demographic contraction and ageing processes can be illustrated realistically. … as well as changes in behaviour… Secondly, OLG models focus on behavioural changes and interactions between the individual generations. Unprecedented demographic developments lie ahead of us in the coming decades. Although population ageing has occurred in the past the resulting economic burdens have been limited in scope. One measure of this burden is the total dependency ratio, which states how many “young people” and “old people” have to be supported by those in the workforce and is expressed by the number of people under 15 and over 65 per 100 people of working age between 15 and 64. In 2005 the total dependency ratio was not much higher than in 1950. By 2050, in contrast, it will have risen from just under 50 to more than 70 based on the medium variant of the Federal Statistical Office’s population estimate. The reforms required to mitigate these demographic challenges are bound to cause tangible changes in the behaviour of economic agents. One example of this is that the ageing of society in a pay-as-you-go state pension system results in either the pension level falling if the contribution rate is kept constant or the contribution rate having to rise for the existing pension level to be maintained. If, however, the state pension level falls, households will attempt to compensate for this by making greater personal provision and saving more. This means that using conventional forecasting methodology based on projecting historical data into the future could produce highly 19 The most important equations that form the mathematical basis for our OLG model are shown in chapter 4. Overlapping generations G1 G2 G3 G4 G1 G2 G3 G4 G1 G2 G3 G4 G1 G2 G3 G4 G1 G2 G3 G4 2000 2020 2040 2060 2080 16 30 35 40 45 50 55 60 65 70 75 80 50 60 70 80 90 00 10 20 30 40 50 From 2005 on the basis of variant 5 Source: Federal Statistical Office No. of under-15s and over-65s per 100 people between the ages of 15 and 64 Total dependency ratio to rise to unprecedented levels 17 The demographic challenge May 19, 2006 15 inaccurate forecasts. This is because regression models by their very nature are based on the assumption of structural constancy, which has to be called into question in the light of demographic developments. … and interactions The behaviour of economic agents has a microeconomic basis in an OLG model. Each generation is represented by an economic agent who, according to the life-cycle model, has specific age-related consumption and saving patterns and who maximises his utility over his entire lifetime. Demographic shifts within this model thus lead to changes in the consumption and saving patterns of individual generations. However, these changes not only influence the behaviour of existing generations but also the actions of generations to come. Current generations also anticipate the decisions that will be made by future generations, so the prospective choices made by the latter do also impact the actions of today’s generations. The origins of OLGs emerged in 1958 OLG models have a long tradition. Their genesis came in a suggestion by Samuelson 20 back in 1958, which was latched onto and developed by Diamond in 1965 21 and was then refined by Auerbach and Kotlikoff 22 in 1987. In their original form, however, their main focus was not yet on demographic change. The primary interest of Auerbach and Kotlikoff was the dynamic analysis of the impact of differing policy measures. Since then OLG models have been refined and used extensively to simulate demographic processes and potential policy measures in order to lessen their negative impact. A more recent example of this is the OLG model developed at the Mannheim Research Institute for the Economics of Aging (MEA) 23 . 3.2 The DB Research OLG model – 3 sectors, 18 generations and two life phases For reasons of simplicity our OLG model is focused on a closed economy and comprises three sectors – households, businesses and government. The state merely administers the pension system In order to focus solely on the impact of demographics on the pension system and thus on the public purse our model reduces the state to its function as the administrator of the pension system. This implies that the state’s other revenues and other expenditures are always equal. One policy variable in our model is the net pension level, which is currently about 70% for the so-called “benchmark pensioner” 24 . The balanced budget assumption enables the direct calculation of the contribution (tax) rate required to finance the state pension. By the same token, it is of course also possible to establish the net pension level that can be achieved with a constant 20 See Samuelson, P.A.: An exact consumption-loan model of interest with or without social contrivance of money in: Journal of Political Economy, 66 (1958). 21 See Diamond, P.A.: National debt in a neoclassical growth model in: American Economic Review, 55 (1965). 22 See Auerbach, A.J. and L.J. Kotlikoff: Dynamic Fiscal Policy (1987). 23 See Börsch-Supan, A. et al.: Aging, pension reform, and capital flows: A multi- country simulation model, Discussion Paper No. 28, April 2003 and updated as Discussion Paper No. 64, August 2004. 24 The so-called “benchmark pensioner“ receives a gross annual income equal to the average income of all members of the statutory pension system and pays contributions into the SPS for 45 years. Current Issues 16 May 19, 2006 contribution rate. If the adjustment to demographic changes is not implemented via the net pension level and/or the contribution rate, the state continuously runs a deficit. This enables deductions to be made regarding the sustainability of public finances. Business output in accordance with a neoclassical production function The enterprise sector is characterised by a representative company with a Cobb-Douglas neoclassical production function, which describes output as a function of the input factors labour and capital as well as technological advancement. The capital stock (= the savings of all generations), which is written down at a constant annual rate, and the available labour supply, which is the number of working persons alive at a given juncture, constitute the factor inputs that businesses deploy for production. The labour supply and the microeconomically driven consumption and savings patterns thus determine macroeconomic output. The product is a homogeneous good that can be both consumed and saved (invested). The special properties of the neoclassical Cobb-Douglas production function make it easy to model. It is based on the assumptions of perfect competition in the factor markets and constant economies of scale in production. Perfect competition means that both input factors labour and capital are remunerated according to their marginal product, i.e. according to their contribution of one additional unit to output. If it is assumed that companies expect to sell everything that they produce at the prevailing prices, that wages and prices are completely flexible and that no costs are incurred either by workers seeking new jobs or by businesses wishing to expand or downsize their workforce, then companies operate according to the following regime. If real wages are lower than the marginal product of labour, it is worth businesses demanding additional labour, as they thereby boost production and increase their income by more than the additional labour costs. So they will demand labour until this profit is exhausted. The labour market is thus always in equilibrium. The assumption of constant economies of scale leads to a proportional increase in production with total factor variation, i.e. an increase in the inputs of labour and capital by 10% each also boosts output by 10%. The sum of factor incomes from labour and capital thus equals national income. Households depicted via generation structure: Workforce & pensioners In our model there are 18 generations living at the same time. They are divided into two types of household: labour force (11 generations) and pensioners (7 generations). Each generation is characterised by a representative individual and comprises 4 annual birth cohorts. Our model world is thus quite a close approximation of reality: according to our OLG model the average household starts its working life at 18, retires at 62 and lives to be 90 on average. Generations younger than 18 are not modelled explicitly. The ratio of pensioners to workers is 0.64 at the starting point for our model, which corresponds roughly to the actual value (2003: 19.5 m pensioners compared with around 30 m people contributing to the state pension system). Since pension payments in Germany are not financed solely via contributions but also to the tune of roughly 25% by federal subsidies not included in our model the current contribut- α α- = 1 t t t L K A Y t Y A t K t L α = Output = Technological advancement = Labour supply = Capital income share of national income = Capital stock each at time t Example: At a level of technological advance- ment of 2, a capital income share of 30%, a capital stock of 500 units and a labour force of 100, the potential output is 324 units. Cobb-Douglas production function 18 The demographic challenge May 19, 2006 17 ion rate of 19.5% is insufficient to ensure our “OLG pensioners” receive a net pension level that is close to the real level of around 70%. To achieve that pension level our model requires a contribut- ion rate of over 26%, which would also be the real rate if the federal subsidy were to be removed. To simplify matters, all the working- age households in our model are in employment. The participation rate, the number of people in gainful employment as a proportion of all people of working age, is therefore 1. 25 Household economic activity is divided into two life phases, the work phase and the pension phase. During the work phase households put their labour at the disposal of companies, receive income in return and pay taxes. They use their net incomes for consumption and saving. In the pension phase households receive a state pension and spend their savings on consumption. For simplicity’s sake we have chosen not to model inheritances. 3.3 The life-cycle hypothesis – intertemporal utility maximisation Household consumption and savings decisions are made according to the life-cycle hypothesis developed by Ando und Modigliani 26 in the early 1960s. It separates the life cycle into roughly three phases: education at the beginning, work in the middle and then the retire- ment or pension phase. In the first and final phases income is lower than in the work phase. If households wish to maximise the utility of their consumption for their entire lives, they will seek to maintain a constant consumption level over time. Households smooth consumption by varying their savings In order to achieve this, households take out loans during their education phase and go into debt, in their working phase they start by paying off their debts and then accumulate assets which they use for consumption during their retirement phase alongside their pension. This is illustrated schematically in chart 20. The life-cycle hypothesis is consequently based on the assumption that households maximise their utility or consumption over a long planning horizon (intertemporally) and act in a forward-looking manner 27 . This means at the start of their economic activity the households are able to find a utility-maximising consumption or savings plan in which utility is gained by boosting consumption in the education phase and in later years corresponds precisely with the utility lost by forgoing consumption and putting aside savings during the work phase. In our model we assume that education is financed by the parent generation. As a result, our model comprises – as stated above – only the active work phase of the generations and the retirement period. Life-cycle hypothesis cannot (yet) be proven empirically for Germany However insightful the life-cycle hypothesis may sound, it cannot (yet) be proven empirically for Germany (or other countries such as France and Italy). According to the life-cycle hypothesis the savings 25 In reality not all people of working age are also gainfully employed. According to the Federal Statistical Office there were more than 73% of people aged 15 to 65 in employment in 2004, with the employment ratio for men at around 80% and for women at about 65%. 26 See Ando, A. and F. Modigliani (1963). The life-cycle hypothesis of saving: Aggregate implications and tests. In American Economic Review, 89, 1963. 27 See Kappler, M. (2002). Consumption: The life-cycle hypothesis. In ZEW Konjunkturreport No. 1, 2002. The household sector: 7 generations of pensioners 11 generations of working age 18 generations in all Each generation comprises 4 birth cohorts Age of entry into work phase: 18 years Age of entry into retirement phase: 62 years Lifetime of each generation: 90 years Old-age dependency ratio: 0.64 19 -100 -50 0 50 100 150 200 250 300 15 35 55 75 Income Con- sump- tion Savings Accumulated assets Age (years) Phase I Phase II Phase III Education Work Retirement Stilised illustration of life-cycle hypothesis Monetary units 20 Current Issues 18 May 19, 2006 ratio of young and elderly individuals should be negative and at its highest in the middle of the active phase of the life cycle. The results of the 2003 Income and Consumption Survey 28 (EVS) back up the latter assertion, as the 35 to 45 age group has the highest savings ratio of 14.8%. Under-25s 29 and over-65s are not dissavers, however, but have a positive savings ratio that actually rises among the over-80s. For the sake of accuracy the savings ratios of the individual birth cohorts derived in a longitudinal section ought to be used to test Modigliani’s life-cycle hypothesis, so as not to mix up age and birth cohort influences. However, Börsch-Supan shows that there is no appreciable difference between the age-specific savings ratios of the birth cohorts from 1909 to 1964. In a survey into the motives for saving he found that over 40% of German households “wanted to regularly save a fixed amount”, whereas only slightly over 23% of households save when income is high or consumption expenditure is low 30 . Börsch-Supan concludes that savings possess a permanent component 31 . Still, it may well be difficult to differentiate between the cohort and the time effects. According to the data from the Survey of Income and Expenditure, household financial assets are also not consistent with the life-cycle hypothesis. Although these assets are highest among the 55 to 65s they drop only slightly as age increases and actually rise again among the over-80s. Therefore, this empirical evidence also contra- dicts the life-cycle hypothesis. In its defence, however, it should be pointed out that inheritance motives may be the reason for this contradiction. Leaving an inheritance and safeguarding the next generation may certainly generate some “utility” for the bequeather. Dissaving in retirement has not been necessary to date When looking at the age-specific savings ratio it needs to be remembered that the net pension replacement rate in Germany is very high at about 70% 32 . This shows the comprehensive level of benefits that the German social security system seeks to provide. Hitherto pensioners were thus well provided for, which meant they did not have to dip into their personal financial assets. This will probably change in future. The net replacement rate of the state pension will drop to less than 50% by 2035 due to shifting demo- graphics, if neither the statutory maximum contribution rate of 22% nor the retirement age is raised 33 . Increased personal provision and dissaving during retirement will therefore be required in future to plug the pension gap. The example of the Netherlands demonstrates that high replace- ment rates, i.e. the ratio of state pension to previous earnings, are probably the main reason why the age-dependent savings profiles are much flatter than the life-cycle hypothesis would suggest. There the public pension replacement rate of 50% is much lower than in Germany (70%), France (80%) and Italy (90%). Accordingly, the 28 See Statistisches Bundesamt, Wirtschaftsrechnungen, Einkommens- und Verbrauchsstichprobe – Ausgewählte Ergebnisse zu den Einkommen und Ausgaben privater Haushalte, 1. Halbjahr 2003, September 2004. 29 The Federal Statistical Office logs only those under-25s that have their own income. 30 See Börsch-Supan, A. and L. Essig (2002). Sparen in Deutschland, Ergebnisse der ersten SAVE-Studie. 2002. 31 See Börsch-Supan, A. (2005). Risiken im Lebenszyklus, Theorie und Evidenz. January 2005 32 The pension level of the so-called benchmark pensioner. 33 See for example Bräuninger, D. and B. Gräf (2005). Spürbare Rentenlücken trotz Reformen. Deutsche Bank Research. Aktuelle Themen, Demografie Spezial No. 312, January 2005. Frankfurt am Main. 0 2 4 6 8 10 12 14 16 <25 25- 35 35- 45 45- 55 55- 65 65- 70 70- 80 >80 Age group in years Source: Federal Statistical Office Savings ratio (still) not negative among the retired Savings ratio (EVS H1 2003), % 21 0 10 20 30 40 50 60 70 <25 25- 35 35- 45 45- 55 55- 65 65- 70 70- 80 >80 Age group in years Source: Federal Statistical Office 55-65 age group has the largest financial assets Financial assets per household (EVS 2003), EUR '000 22 The demographic challenge May 19, 2006 19 age-specific savings ratios display almost typical life-cycle characteristics 34 . For example, the savings ratio of the over-70s in the Netherlands is negative. Against this background, it is certainly justified to use a model based on the life-cycle hypothesis to simulate future developments that occur over an extended period of five decades. 3.4 Individuals have a time preference If it is assumed that a household has no preferences regarding the temporal distribution of its consumption, this means that the consumption of the same amount of goods in each period generates the same utility, regardless of whether the consumption occurs today or at some later date. In other words, a uniform distribution over time would maximise the utility for an individual. However, as a rule it is assumed that there is a greater preference for consumption in the present than in the future, i.e. consumption is valued more highly the sooner it occurs. This is a fundamental assumption in neoclassical economic theory. Temporal preference is determined by the individual rate of time preference. It states the rate at which the volume of goods must grow so a household is indifferent to whether consumption occurs in the present or during a certain period in the future. Consumption thus rises over time when the time preference rate is positive. The resulting utility is, however, the same in all periods. It is possible to mathematically determine consumption in future periods by simple compounding of current consumption or discounting of future consumption to the present (determining the net present value) using a compound interest formula in which the individual time preference rate is substituted for the interest rate 35 . Chart 24 displays the optimum consumption plan including and excluding time preference. The example is based on the assumption that households consume for 10 years and can afford to consume 100 units. Excluding time preference households would divide their consumption into 10 equal units. By contrast, the optimum con- sumption of households with a positive time preference is different, if the households wish to stabilise their utility over their entire lifetime. Such households prefer consumption in the present rather than in the future. With a time preference rate of 2.5 they would only be prepared to postpone consumption from the first year to the second if they could then consume 2.5% more. In the third year it would be slightly over 5% and in the 10th year nearly 25%. Their optimum consumption plan would thus in the first year allow consumption of just under 9 units, while in the 10th year it would be a good 11 units. 34 See Börsch-Supan, A. (2000). Global Aging: Issues, Answers, More Questions. MEA Working Paper No. 55, July 2004 and Börsch-Supan, A. et al. (2000).The German Savings Puzzle. September 2000. 35 The compound interest rate formula is: K n = K 0  (1+i) n , where K n = capital after n years, K 0 = starting capital, i = interest rate in %, n = number of years. Rearranging the formula gives us: K o = K n / (1+i) n . Projected onto time preference, K 0 and K n correspond respectively to consumption at the beginning of and in period n, while i would then be the time preference rate. -20 0 20 40 60 80 100 120 140 25 30 35 40 45 50 55 60 65 70 75 Italy France Germany Netherlands Age (years) * Cohort-adjusted, indexed at age 40 = 100 Source: Börsch-Supan (2004) Savings ratio* of over-70s is negative in the Netherlands 23 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 1234567891 0 Without time preference With time preference Time Ideal consumption schedule with/without time preference Assumptions: Lifetime consumption amounts to 100 units in each cast, rate of time preference 2.5 24 Current Issues 20 May 19, 2006 Time preference not refuted by empirical evidence The theoretical construct of time preference, for which something can be said at a purely intuitive level, seems to be backed up by empirical evidence. Time preference cannot be measured directly. It can however be estimated using the real interest rate 36 . Falling real interest rates are accompanied by a decline in the appeal of postponing consumption, or rather the appeal of bringing forward credit-financed consumption increases on account of the lower costs. Thus, where a positive time preference exists the savings ratio ought to decline in a corresponding manner during such phases. Chart 25 does not show this correlation to hold in each year, which would be a surprise as the savings ratio (especially at the margin) is determined by many different factors such as expectations regarding the future employment situation and the income and wealth developments. Nevertheless, the trend since 1980 appears to confirm the existence of time preference, even though cohort effects need to be borne in mind with regard to the savings ratio. 3.5. Perfect foresight – who can have that? In our model, households have perfect foresight. They know their future income stream and can therefore draw up their optimum consumption/savings plan at the beginning of their working lives. Although this assumption simplifies the solution of the utility maximisation problem, it is very restrictive. Individuals’ lives are subject to a variety of imponderables related to biometric, economic, familial and political circumstances 37 . Examples of biometric imponderables are the chances of becoming unfit for work and longevity risk, that is the risk of living longer than one’s accumulated resources last. Economic risks lie particularly in the possibility of losing one’s job. Furthermore, financial assets and human capital can suffer value impairment. Political risks include, for example, legislative changes that call into question previously obtained entitlements, and family-related risks consist of separation, divorce and childlessness. In view of these uncertainties there seems – at first glance – to be little logic behind the assumption that individuals already know their entire future income streams at the start of their working lives and also exactly when they will die and factor this information into their planning. More sophisticated OLG models factor in mortality probabilities in order to control the latter problem 38 . Yet, in order to keep our model tractable we have decided not to incorporate them. 36 Theoretically it can be shown in a simple two-period model that with the optimum time structure of consumption the time preference rate tallies with the interest rate. However it cannot be ruled out that there are also impatient households (whose time preference rate exceeds the market rate) that consume too much initially, whereas patient households (whose time preference rate is lower than the market rate) save more initially, with the result that the time preference rate can vary from the market interest rate. In our model we have allowed this possibility but at the same time limited it (see consumption equation 3.0 in chapter 4 (Model design) in this report) by incorporating intertemporal risk aversion, which states how households react to differences between the time preference rate and the market rate. 37 See Börsch-Supan, A. (2005). Risiken im Lebenszyklus, Theorie und Evidenz. January 2005 38 See Börsch-Supan, A. et al. (2004). Aging, pension reforms, and capital flows: A multi-country simulation model. MEA (Mannheim Research Institute for the Economics of Aging) Discussion Paper No. 64. August 2004. 8 9 10 11 12 13 14 15 16 80 85 90 95 00 1 2 3 4 5 6 7 8 Savings ratio, % (left) Real 10Y bond yield, % (right) Consumers have time preference Sources: Deutsche Bundesbank, DB Research 25 0 2 4 6 8 10 80 84 88 92 96 00 04 Actual development Sources: Federal Statistical Office, DB Research With constant savings ratio since 1980 Households smooth their consumption Private consumption, % yoy 26 The demographic challenge May 19, 2006 21 At least signs of consumption smoothing are detectable If households had perfect foresight, they would smooth their consumption over time. Although there is no empirical evidence of “smooth” consumption, households are known to not only gear their consumption to current disposable income but also react to fluctuations in income by varying their savings patterns. This supports the permanent income hypothesis. With an unchanged savings ratio some of the income-driven blips in private consumption would have been far more pronounced than they actually turned out to be. In periods of weaker income growth, households have curtailed their saving in favour of consumption (particularly in 1982/83 and from 1992 to 2000). When income has grown faster than expected (for example in 1991/92) there has been a corresponding increase in saving. The data of the Income and Expenditure Survey does reveal a pronounced age-specific consumption pattern, with consumption rising initially with age and peaking among 45 to 55-year-olds. Nevertheless, consumption develops considerably “more smoothly” than income. The assumption of a certain degree of foresight consequently does not seem quite as far-fetched as it appears at first glance. Demographic change does not come as a surprise Factoring perfect foresight into an OLG model is also supported by the fact that demographic developments do not catch economic agents unaware. Demographic change has been with us for generations and will merely creep along for the time being. It is also a constant long-run process, i.e. major fluctuations in population size or structure are unlikely. It thus is a long-term trend to which economic agents can adapt and take the appropriate action over the long term. All the same, this frequently used justification should not be too comforting in view of the looming demographic pension gap. Results based on perfect foresight form “benchmark” Auerbach and Kotlikoff 39 introduce another argument in favour of the assumption. In their opinion the assumption of perfect foresight is a good instrument for analysing the behaviour of economic agents. They argue that in reality the future development, particularly of incomes, is both overestimated and underestimated unsystematic- ally. The findings obtained under the assumption of perfect foresight could thus be used as a benchmark. 3.6 OLG models are dynamic equilibrium models The solution to an OLG model is based on the neoclassical equilibrium, or steady state. This determines the long-term trend. In the steady state of a closed economy all aggregate measures (i.e. national income, savings, capital stock etc.) grow at the same rate as the population and all per-capita readings are constant. Accord- ing to the neoclassical production function, the input factors labour and capital are remunerated in accordance with their marginal productivity. Labour and capital as well as wages and interest rates are perfectly balanced due to production technology and are in a constant steady state. If this equilibrium is disturbed by a demo- graphic shock, i.e. by the ageing and shrinking of the population, in the long run the system will return to the steady state with the 39 See Auerbach, A.J. and L.J. Kotlikoff (1987). Dynamic Fiscal Policy. 1000 1500 2000 2500 3000 3500 4000 < 25 25- 35 35- 45 45- 55 55- 65 65- 70 70- 80 > 80 Income Consumption Age in years Source: Federal Statistical Office Consumption smoothing: Evidence from the EVS Per household and month (EUR) 27 40 50 60 70 80 90 100 110 120 130 2000 2040 2080 2120 2160 Source: DB Research Population Labour force Pensioners Population steady again after 2150 2000 = 100 28 50 60 70 80 90 100 110 120 2000 2040 2080 2120 2160 Source: DB Research Old-age dependency ratio: At roughly its old level again Number of pensioners per 100 workers 29 Current Issues 22 May 19, 2006 labour-to-capital ratio reverting to its original equilibrium and real wages and interest rates moving back to their original levels. New steady state to be regained in around 2150 In our model we have assumed that the population starts to decline around 2010 and even out after 2050. This will result from a gradual convergence of the fertility rate to the replacement rate, thus ensuring a stable population size. According to our modelling, a new steady state with a stationary population will be reached in about 2150. At this point, the old-age dependency ratio will return to its pre-demographic-shock level. 3.7 The transitional stage of the adjustment process The adjustment process with respect to the transition from original equilibrium to the new steady state can be described as follows: we assume that our starting position is a steady state. The size of the labour force is perfectly matched with the capital stock. If the population now starts to shrink, the labour supply falls. Capital will now become more plentiful relative to labour, so the labour to capital ratio declines. Wage rates will thus rise and interest rates fall, and the output (= income) attainable with the new factor resources will be lower than at the start. If the savings ratio were constant, savings would decline to the same degree as the population, as would consequently investment and the capital stock. A new equilibrium would be achieved. Since the savings ratio is, however, endogenous, i.e. it is determined by the decisions made by households within the model, it will decline due to the lower real interest rates and the dissaving by baby-boomers entering retirement. The combined effect of these three factors (decrease in population, in GDP, and in the savings rate) will lead to an acceleration in the rate of decline of the capital stock. As a result, in our model the capital stock will shrink faster than the population from 2060 onwards. Then, the labour force will be decreasing relative to the capital stock and real interests start to rise, while real wage levels begin to fall. Due to the higher real interest rates the savings ratio will increase, which slows the decline in the capital stock. This process will take place until the labour to capital ratio returns to that of the starting equilibrium and the original real wage and interest rate levels are reached once again. A new steady state is entered – from around 2150 in our model. 2 4 6 8 10 12 14 16 2000 2040 2080 2120 2160 Source: DB Research Savings ratio in OLG model not constant Savings ratio, % 30 The demographic challenge May 19, 2006 23 3.8 OLGs and farmers – what do they have in common? Intertemporal utility maximisation of households including their time preference (that is the optimum consumption/saving plan of the generations in our OLG model) can be illustrated using a simple ”farmer model”. The starting point is a farmer 40 who at the start of his working life has 100 kg of wheat which he can either consume or sow. The possibility of stockpiling the wheat is ruled out for reasons of simplicity. Thanks to successful husbandry the farmer’s harvest is 40% higher than the annual amount that he sows 41 . He knows that he still has 10 years to live, he is interested in consuming at a constant rate over the years and does not want to have any wheat remaining at the end of his life. He now considers what amount of wheat he can consume – or rather has to sow – each year in order to maximise his consumption (utility) over his entire life. Let us assume that at the beginning of his working life he decides to consume 32 kg of wheat each year. This means that in the first year he has 68 kg for sowing, which would yield him a harvest of 95.2 kg in the second year. If he consumes 32 kg again, then he has 63.2 kg to sow in the third year. The amounts sown and harvested therefore decline every year. After the sixth year he would find that he had just 15.5 kg for sowing, which would enable him to harvest 21.6 kg of wheat. This would be too little, as he would like to consume 32 kg of wheat every year, so he would have to revise his plan and consume less from the very beginning. He would solve his maximisation problem at an annual consumption of about 29.6 kg and have found the ideal consumption/sowing plan for the next ten years. Farmers also get to retire… This illustrative example can now be expanded to include a retire- ment phase for the farmer. The game remains the same. The work phase is simply reduced, for example by three years to seven, and a pension phase is added that lasts three years, during which the farmer neither sows nor harvests, but consumes. Under these conditions at the start of the eighth year the farmer requires enough wheat to last for three years of consumption. In this case he would have decided on annual consumption of slightly more than 28 kg of wheat at the start of his work phase. … and have a time preference Farmers can also have a time preference for their consumption decision, i.e they prefer to consume in the present rather than in the future. With a time preference rate of 2.5 our farmer would only be prepared to postpone consumption until next year if he could then consume 2.5% more. In the following year his consumption would have to be 5.1% higher than in the first year and 28% higher in the 10th year than in the first to generate the same utility as in the starting period. With the farmer consuming 29.6 kg in the first year according to the optimum consumption/sowing schedule excluding time preference his consumption would then be 30.3 kg in the second year and 37 kg in the 10th year. With such a consumption/ 40 The farmer’s problem is based on concepts developed by Ramsey way back in 1928. He sought to answer the question of how much an economy should save on a long-term horizon. See Ramsey, F. (1928). A mathematical theory of saving. In Economic Journal (38). 41 To simplify matters it is assumed that sowing, harvest and consumption occur simultaneously. The farmer problem* Poor planning: Annual consumption not optimised Year Cons. Seed Yield kg/year kg/year kg/year 1 32.0 68.0 - 2 32.0 63.2 95.2 3 32.0 56.5 88.5 4 32.0 47.1 79.1 5 32.0 33.9 65.9 6 32.0 15.5 47.5 7 32.0 - 21.6 8 32.0 - - 9 32.0 - - 10 32.0 - - The optimum cons./sowing schedule Year Cons. Seed Yield 1 29.6 70.4 - 2 29.6 69.0 98.6 3 29.6 67.0 96.6 4 29.6 64.2 93.8 5 29.6 60.2 89.8 6 29.6 54.7 84.3 7 29.6 47.0 76.6 8 29.6 36.2 65.8 9 29.6 21.1 50.7 10 29.6 0.0 29.6 Retired farmers** Year Cons. Seed Yield 1 28.2 71.8 - 2 28.2 72.3 100.5 3 28.2 73.1 101.3 4 28.2 74.2 102.3 5 28.2 75.6 103.8 6 28.2 77.7 105.9 7 28.2 80.6 108.8 8 28.2 - 112.8 9 28.2 - - 10 28.2 - - ** no sowing from eighth year onwards Farmers prefer current consumption*** Year Cons. Seed Yield 1 28.0 72.0 - 2 28.7 72.0 100.8 3 29.4 71.4 100.9 4 30.2 69.8 100.0 5 30.9 66.8 97.7 6 31.7 61.8 93.5 7 32.5 54.0 86.5 8 33.3 42.2 75.6 9 34.1 25.0 59.1 10 35.0 0.0 35.0 *** Rate of time preference 2.5 * Assumptions: farmer lives 10 years and starts with 100 kg of wheat. He can consume or sow. Sowing yields a harvest of 140%. Question: what is his optimum consumption schedule if he does not want to have any wheat left over at the end? 31 Current Issues 24 May 19, 2006 savings schedule, however, he cannot achieve his aim of consuming and sowing for 10 years and not having any wheat left over because his consumption already exceeds the harvest in year 9. Such a consumption/sowing schedule would therefore not be the ideal one for the farmer. To achieve his target with a time preference rate of 2.5 he would have to reduce his consumption in the first year to about 28 kg, thus consuming 1.6 kg less than in his optimum sowing schedule without time preference. Until the 10th year his consumption then climbs to around 35 kg. Although he consumes 5.4 kg more wheat than in his optimum consumption/sowing schedule without time preference, the resulting utility would be lower for him since consumption of 35 kg in the 10th year corresponds to a present value of a mere 28 kg, compared with 29.6 kg excluding time preference. OLG – lots of farmers! If (1) the farmer is given the chance to accumulate some savings, (2) there are many farmers of differing ages, each preferring current consumption, (3) farmers are prompted to interact (i.e. their sowing and consumption plans are not independent of each other), and (4) the assumption is made that during each period of observation one young generation of farmers joins and the eldest dies, one would essentially have constructed an OLG model. 3.9 The solution of an OLG model – going the wrong way about it Just like a farmer, who knows his original and his final position, the neoclassical equilibrium condition means that the solution for the model in its original state and final state (new steady state following a demographic shock) is known. As already described, the new equlibrium must have the same optimum labour-capital ratio as the original equilibrium. Factoring in the demographic shock, the appropriate capital stock can thus be established. The adjustment path between old and new equilibria can then be determined mathematically with the aid of an iterative recursive process. 3.10 OLG model requires calibration In order for an OLG model to be a good approximation of the economy, it has to be calibrated using past readings for several variables used in the model 42 . There is the time preference rate, which impacts the distribution of consumption over time, the capital income share of national income, which is required for specifying the neoclassical production function, as well as the depreciation rate for the capital stock. In addition, there are the political variables, like the contribution rate to the state pension system and the net pension replacement level. Some of these calibration factors, for example the political variables, are easy to determine. With the others, such as the time preference rate, we have relied on the standard 42 An overview of other calibration methods and problems is provided by Ludwig, A . (2005). Moment estimation in Auerbach-Kotlikoff models: How well do they match the data? MEA (Mannheim Research Institute for the Economics of Aging) Discussion Paper No. 93. March 2005. 27 28 29 30 31 32 33 34 35 36 37 1234567891 0 Without time preference With time preference rate of 2.5 Years The farmer problem: Time preference alters optimum cons./sowing schedule Optimum consumption plan, kg per year 32 Assumptions & calibration factors in OLG model Population 2005-2050 -10% Starting values Contribution rate for SPS 26.3% (equals 19.5% plus federal subsidy) SPS net replacement rate 70% Public sector debt/GDP 60% Scenarios 1. "Status quo" 1.1 Adjustment via replacement rate Contribution rate for SPS 26 .3% 1.2 Adjustment via contribution rate SPS net replacement rate 70% 1.3 Deficit solution Contribution rate for SPS 26 .3% SPS net replacement rate 70% 2. "More personal provision" Contribution rate for SPS 26 .3% State pension system net replacement rate drops to 40%. Households take out personal provision to cover the demographic gap Calibration parameters Calibration income share of national income 30% Rate of time preference 2.5% Risk aversion 2 Capital stock depreciation rate 5% Starting position of "Status quo" model in 2005 Savings ratio 11% Real interest rate 7.5% 33 The demographic challenge May 19, 2006 25 estimates used in many OLG models 43 , partly based on the findings of Auerbach and Kotlikoff. With the calibration factors we selected, our OLG model delivers real national income growth of around 7 ½% and a household savings ratio of over 11% in its starting position. Since the calibration factors (excluding the political variables) for the entire simulation period, that is for over 50 years, are kept constant, it has to be remembered that the demographic development is uncertain. This means demographic shocks that also result in changes to these calibration factors cannot be ruled out. For example, the time preference of individuals could change in a rapidly ageing society 44 . Bernhard Gräf, +49 69 910-31738 (bernhard.graef@db.com) Marc Schattenberg, Universität Halle (+49 345 5523323, marc.schattenberg@wiwi.uni-halle.de) 43 See Auerbach, A.J. and L.J. Kotlikoff (1987). Dynamic Fiscal Policy. Börsch- Supan, A. et al. (2004). Aging, pension reforms, and capital flows: A multi-country simulation model. MEA (Mannheim Research Institute for the Economics of Aging) Discussion Paper No. 64, August 2004. Hviding, K. and M. Merette (1998). Macroeconomic effects of pension reforms in the context of ageing populations: overlapping generations model simulations for seven OECD countries, OECD Economic Department Working Papers No. 2001, July 1998. Martins, J.O. et al. (2005). The Impact of Ageing on Demand, Factor Markets and Growth. OECD Economic Department Working Papers No. 420. March 2005. 44 See Bishai, D.M.(2004). Does time preference change with age? In Journal of Population Economics, Vol. 17. Current Issues 26 May 19, 2006 4. Notes on design: The mathematical formulae for our OLG model Basically, our model was constructed using mathematical formulae that correspond to those found in most OLG models 45 . In the following we shall list and discuss the main equations. 4.1. Population – labour force and pensioners At any given time, the population is made up of the members of the labour force on the one hand and pensioners on the other. There are 18 generations in our model, with 11 in employment and 7 in retirement. Each generation is characterised by one representative household. (1.0) t t t R L Bv += where t Bv = Size of the population at time t t L = Number of workers (labour supply) at time t t R = Number of retired people at time t t = Time index Hence: (1.1) * = = 11 1 i i t t NE L where i t NE = Number of employed in generation i at time t i = Generations index (i = 1,2,…,11) and: (1.2) * = = 18 12 j j t t NR R where j t NR = Number of retired in generation j at time t j = Generations index (j = 12,13,…,18) As a result: (1.3) * * = = + = 18 12 11 1 j j t i i t t NR NE Bv 45 Supplementary reading can be found in: Auerbach, A.J. and L.J. Kotlikoff (1987). Dynamic Fiscal Policy; Börsch-Supan, A. et al. (2004). Aging, pension reforms, and capital flows: A multi-country simulation model. MEA (Mannheim Research Institute for the Economics of Aging) Discussion Paper No. 64, August 2004; Dietrich, K. (2000). OLG-Modelle; Heer, B. and M. Jäger. Lecture Notes on Mathematics for Economists, available at www. finomica.de; Maußner, A. (2004). Dynamic General Equilibrium Modelling – Computational Methods and Applications. The demographic challenge May 19, 2006 27 4.2 The business sector – given by a Cobb-Douglas production function The business sector is modelled on a representative company whose output is given by a Cobb-Douglas production function: (2.0) α α- = 1 t t t L K A Y where t Y = Output at time t A = Level of technological advancement t K = Capital stock at time t t L = Labour supply at time t α = Capital income as a share of national income Output is a homogeneous good that can be both consumed and invested. The capital stock consists of the total assets (cumulative savings including interest) of all the generations at the time of the respective survey (see Household sector equation 3.14). The labour supply is determined by the number of individuals of working age. The inputs of the production factors capital and labour are remunerated on the basis of their marginal productivity. The following equation describes the return on capital which is achieved by differentiating the production function with respect to capital when the rate of depreciation is factored in: (2.1) δ α δ α α - = - ∂ ∂ = - - 1 1 ) /( t t t t t L K A K Y r where t r = Real interest rate at time t δ = Depreciation rate for the capital stock By analogy to the remuneration of capital input, the real wage works out at: (2.2) ( ) α α α - - = ∂ ∂ = t t t t t L K A L Y w 1/ where t w = Real wage at time t Current Issues 28 May 19, 2006 4.3 The household sector – intertemporal utility maximisation Each generation is characterised by one representative household. The decision-making process is geared towards the maximisation of utility, while taking into account the life-time budget constraint. Given their perfect foresight, the individuals optimise their lifetime consumption and savings plans from the start of the model. The consumption of a representative household of generation i at time t can be formally described as such: (3.0) θ ρ 1 1 1 1 * * * * * * * * * + + = - t i t i t r C C where i t C = Consumption of generation i at time t i t C 1 - = Consumption of generation i at- time t-1 t r = Real interest rate at time t ρ = Individual rate of time preference θ 1 = Intertemporal elasticity of substitution This means that the consumption decision in period t depends on the relation between the market interest rate and the individual rate of time preference, weighted by θ . A higher interest rate means that future goods become cheaper, which leads to higher consumption in the future. A high rate of time preference means that the households attach less importance to future consumption and prefer to consume today. θ describes the relative risk aversion of a household as being equal to the elasticity of marginal utility (accordingly, θ 1 is the intertemporal elasticity of substitution for consumption). It says how strongly the households react to deviations between their individual rate of time preference and the market interest rate and how willing they are to distance themselves from their optimum intertemporal level of allocation to consumption. If θ is high, consumption is smoothed substantially. The consumption of all generations at time t is then: (3.1) i t i t C C * = = 18 1 where t C = Consumption of all generations at- time t and the consumption of a generation across its entire lifetime is determined by: (3.2) i t t i H C C * = = = 18 1 where H C = Lifetime consumption of a household “born” in t=1 The demographic challenge May 19, 2006 29 Households maximise the utility of their consumption over their entire lifetime. For the intertemporal maximisation of utility we assume a common utility function for the constant relative risk aversion (CRRA) of the form: (3.3) ( ) θ θ - - = 1 1 1 i t i t C u where i t u = Utility of generation i at time t Following: (3.4) () max! 1 1 ) 1 ( 1 1 18 1 → - - = - = * θ θ ρ i t t t C U where U = Utility of generation i across its entire lifetime with the terms t ) 1 ( 1 ρ - corresponding to the discount factors of the utility function. The utility function is chosen in such a way that marginal utility is positive, but declining, and the elasticity of marginal utility is constant 46 . The household sector offers the business sector its labour, draws income, pays taxes (= contributions to the state pension system) and divides its income into consumption and savings. The consumption equation (3.0) thus also implicitly describes the households’ decision on savings. (3.5) i t i t i t C E S - - = ) 1 ( τ where i t S = Savings of generation i at time t τ = Tax rate (contribution rate) paid to state pension system i t E = Labour income of generation i a time t Where: (3.6) i t t i t NE w E = and: 46 In related literature, the utility function is frequently supplemented by the term -1 as such: ( ) { } 1 1 1 ) ( 1 - - = - ⊗ θ θ i t i t C C U . This merely serves to close the gap in the utility function discussed in 3.4 in case 1 = θ . ) ( i t C U ⊗ converges for 1 → θ towards ( ) i t C ln ; for further information see also i.a. Barro, R.J. and X. Sala-i-Martin (1995). Economic Growth. Current Issues 30 May 19, 2006 (3.7) i t i t E Ev ) 1 ( τ - = where i t Ev = Disposable income of generation i at time t The households’ savings ratio is then: (3.8) i t i t i t Ev S SQ = where i t SQ = Savings ratio of generation i at time t The cumulative savings including interest are equal to the assets of a household at a given point in time: (3.9) i t i t t i t S V r V + + = - 1 ) 1 ( where i t V = Assets of generation i at time t It is assumed that households are born without assets ) 0 ( 1 = t K . In the employment phase the households acquire assets but leave no assets in the model world when they pass away ) 0 ( 18 = t K . This results in the budget constraint a) for a household in the employment phase: (3.10) i t i t i t t i t C Ev V r V - + + = + + ) 1 ( 1 1 where 1 1 + + i t V = Assets of generation i+1 at time t+1 i = Generations index (i = 1,2,…,11) in the employment phase b) for a household in the retirement phase: (3.11) j t j t j t t j t C P V r V - + + = + + ) 1 ( 1 1 where j t P = Pension of generation j at time t j = Generations index (j = 12,13,…,18) The demographic challenge May 19, 2006 31 A household in the pension phase draws no labour income, but receives income from the state pension system. Assuming that pensioner households do not save money, but instead dissave, and their assets are exhausted when they die, then: (3.12) j t j t C P < Thus: (3.13) j t t j t V r V ) 1 ( 1 1 + < + + and: (3.14) 0 18 = t V The capital stock used by the companies for production equals the assets of all the households at a given point in time: (3.15) i t i i t i t t S V V K * * = = = = = 18 1 18 1 where t K = Capital stock at time t and i t S = Cumulative savings including interest of generation i at time t And the change in the capital stock (factoring in depreciation) is equal to net investment: (3.16) t t t K K net I ) 1 ( ) ( 1 δ - - = + As a consequence, investments are equal to savings at all times: (3.17) t t t t S gross I K net I = = + ) ( ) ( δ Current Issues 32 May 19, 2006 4.4 The state sector – administrator of the pension system The state is limited to its function as the administrator of the pension system. Its budget equation is: (4.0) t t t P T B - = where t B = The budget balance at time t t T = Tax revenues = pension contributions at time t t P = Pension disbursements at time t If the budget is in equilibrium, i.e. adjustments are made via the contribution rate and/or pension level, it holds that: (4.1) t t t t t t T L w R p P = = =τ where t P = Total pension disbursements at time t t p = Average pension disbursement per household in the pension phase at time t and as a consequence the usual contribution rate or replacement rate equation of a pay-as-you-go pension system applies: (4.2) * * * * * * * * * * * * * * * * * = t t t t w p L R τ where t t L R = Old-age dependency ratio at time t (Number of retired to number of employed) t t w p = Pension level at time t (pension disbursement per retired person in relation to labour income per person of working age) If the budget is not in equilibrium, the state debt works out at: (4.3) t t t t B D r D ++ = - 1 ) 1 ( where t D = Debt level at time t The demographic challenge May 19, 2006 33 Bibliography Abel, A.B. (2002). The effects of a baby boom on stock prices and capital accumulation in the presence of social security. July 2002. Ando, A. and F. Modigliani (1963). The life-cycle hypothesis of saving: Aggregate implications and tests. In American Economic Review (89) 1963. Auerbach, A.J. and L.J. Kotlikoff (1987). Dynamic Fiscal Policy. Barro, R.J. and X. Sala-i-Martin (2004). Economic Growth. Bishai, D.M. (2004). Does time preference change with age? In Journal of Population Economics. Vol. 17. 2004. Börsch-Supan, A. (2004). Global Aging: Issues, Answers, More Questions. MEA Working Paper No. 55. July 2004. Börsch-Supan, A. (2005). Risiken im Lebenszyklus, Theorie und Evidenz. January 2005. Börsch-Supan and L. Essig (2002). Sparen in Deutschland, Ergebnisse der ersten SAVE-Studie. Börsch-Supan, A., A. Ludwig and J. Winter (2003 and 2004). Aging, pension reforms, and capital flows: A multi-country simulation model, MEA (Mannheim Research Institute for the Economics of Aging) Discussion Paper No. 28, April 2003 and update as Discussion Paper No. 64, August 2004. Börsch-Supan, A., A. Reil-Held, R. Rodepeter, R. Schnabel and J. Winter (2000). The German Savings Puzzle. September 2000. Bräuninger, D. and B. Gräf (2005). Spürbare Rentenlücken trotz Reformen. Deutsche Bank Research. Aktuelle Themen/ Demografie Spezial No. 312. Frankfurt am Main. Deaton, A. (1992). Understanding consumption. Diamond, P.A. (1965). National debt in a neoclassical growth model. In American Economic Review (55) 1965. Dietrich, K. (2000). OLG-Modelle. French, K.R. and J.M. Poterba (1991). Investor diversification and international equity markets. In American Economic Review (81) 1991. Fuchs, J. and K. Dörfler (2005). Projektion des Arbeitsangebots bis 2050, Demografische Effekte sind nicht mehr zu bremsen. In IAB Kurzbericht No. 11/26.7.2005. Fuchs, J. and K. Dörfler (2005). Projektion des Erwerbspersonen- potenzials bis 2050, Annahmen und Datengrundlage. IAB Forschungsbericht No. 25/2005. Gräf, B. (2003). German growth potential: facing the demographic challenge. Deutsche Bank Research. Current Issues/Demography Special. December 11, 2003. Frankfurt am Main. Heer, B. and A. Maußner (2004). Dynamic General Equilibirum Modelling – Computational Methods and Applications. Hviding, K. and M. Merette (1998). Macroeconomic effects of pension reforms in the context of ageing populations: overlapping generations model simulations for seven OECD Current Issues 34 May 19, 2006 countries. OECD Economic Department Working Papers No. 2001. July 1998. Jäger, M. Lecture Notes on Mathematics for Economists, www.finomica.de. Kamps, C., C.-P. Meier and F. Oskamp (2004). Wachstum des Produktionspotenzials in Deutschland bleibt schwach, Kieler Diskussionsbeiträge (Institut für Weltwirtschaft Kiel) No. 414. September 2004. Kappler, M. (2002). Konsum. Die Lebenzyklus-Hypothese, In ZEW Konjunkturreport No. 1. 2002. Ludwig, A. (2005). Aging and Economic Growth: The Role of Factor Markets and of Fundamental Pension Reforms. MEA Discussion Paper No. 94. February 2005. Ludwig, A. (2005). Moment estimation in Auerbach-Kotlikoff models: How well do they match the data? MEA (Mannheim Research Institute for the Economics of Aging) Discussion Paper No. 93. March 2005. Martins, J.O., F. Gonand, P. Antolin, C. de la Maisonneuve and K.- Y. Yoo (2005). The Impact of Ageing on Demand, Factor Markets and Growth. OECD Economic Department Working Papers No. 420. March 2005. Ramsey, F. (1928). A mathematical theory of saving. In Economic Journal (38) 1928. Samuelson, P.A. (1958). An exact consumption-loan model of interest with or without social contrivance of money. In Journal of Political Economy (66) 1958. Schmidt, S. (2004). Computerbasierte Anwendungen von Modellen sich überlappender Generationen. In ZEW Kon- junkturreport No. 3. 2004. Solow, R. (1956). A Contribution to the Theory of Economic Growth. In Quarterly Journal of Economics, February 1956. Statistisches Bundesamt (2003). Bevölkerung Deutschlands bis 2050, 10. koordinierte Bevölkerungsvorausberechnung. Wiesbaden 2003. Statistisches Bundesamt (2004). Wirtschaftsrechnungen, Einkommens- und Verbrauchsstichprobe – Ausgewählte Ergebnisse zu den Einkommen und Ausgaben privater Haushalte. 1. Halbjahr 2003. September 2004. Steinmann, G. and S. Tagge. (2002). Determinanten der Bevöl- kerungsentwicklung in West- und Ostdeutschland. In Wirtschaft im Wandel (4) 2002. Werding, M. and A. Kaltschütz (2005). Modellrechnungen zur langfristigen Tragfähigkeit der öffentlichen Finanzen. ifo Beiträge zur Wirtschaftsforschung 2005. Current Issues Demography Special All our publications can be accessed, free of charge, on our website www.dbresearch.com You can also register there to receive our publications regularly by e-mail. 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In Australia, retail clients should obtain a copy of a Product Disclosure Statement (PDS) relating to any financial product referred to in this report and consider the PDS before making any decision about whether to acquire the product. Printed by: HST Offsetdruck Schadt & Tetzlaff GbR, Dieburg ISSN Print: 1612-314X / ISSN Internet and e-mail: 1612-3158 Demography lies at the heart of our socio-economic system. There is virtually no area that is not influenced by it, and the related trends around the world are not all pointing the same way. Especially the developing and emerging nations will see further population growth, while Europe and Japan will witness noticeable declines and an ageing of society. This is causing enormous economic, political and social challenges in the "old world". Without radical reforms the pay-as-you-go pension systems will be unsustainable within a few decades when the number of workers comes to equal the number of pensioners. The new age patterns will substantially change the structure of demand, financial market yields will probably fall, and growth potential will decrease. Reform of the US pension system Political controversies defeat demographic and financial realities ...............................................................July 19, 2005 Demographic development will not spare the public infrastructure .................................................... June 7, 2004 Major US health care reform – demographic and budgetary dimensions ....................................February 9, 2004 German growth potential: facing the demographic challenge ................................................. December 11, 2003 Ageing, the German rate of return and global capital markets.................................................. December 4, 2003 Ageing calls for further internationalisation in banking ................................................................ October 28, 2003 The silver test Ageing customers challenge companies at every level ....................................................................... October 10, 2003 Demography sends tremor through German property market ............................................... September 26, 2003
1.3.6