In the search for the optimal path to establish a funded pension systemMARCIN BIELECKIFaculty of Economic SciencesUniversity of Warsawe-mail: mbielecki@wne.uw.edu.plJOANNA TYROWICZFaculty of Economic SciencesUniversity of WarsawNational Bank of Polande-mail: jtyrowicz@uw.edu.plKRZYSZTOF MAKARSKINational Bank of PolandWarsaw School of Economicse-mail: kmakar@sgh.waw.plMARCIN WANIEKUniversity of Warsawe-mail: m.waniek@mimuw.edu.plAbstractWe propose a politically feasible instrument for a nearly optimal transition from a pay-as-you-go toa funded scheme in a defined contribution pension system. It consists of compensating thetransition generations in the form of pension benefits indexation more generous than would haveprevailed in a clean DC system. Thus, this instrument allows to smoothen the welfare costs oftransition over future generations via some small implicit debt. Our instrument proves robust to anumber of parametric and modeling choices.Keywords:defined contribution, pay-as-you-go, pre-funded, pension systemJEL:H55, E17, C60Acknowledgements The support of National Science Centre (grant UMO-­‐2012/01/D/HS4/04039) is gratefully acknowledged. All opinions expressed are those of the authors and have not been endorsed by NSC. The remaining errors are ours.Working Papers contain preliminary research results.Please consider this when citing the paper.Please contact the authors to give comments or to obtain revised version.Any mistakes and the views expressed herein are solely those of the authors.

1 Introduction and motivationEver since the discussion between Samuelson (1958) and Aaron (1966) it has been a concern whetherpension system reforms – especially the privatization of the social security – can be implementedwithout inflicting welfare loss on one or few cohorts. With the onset of the overlapping generationsgeneral equilibrium models after the seminal contribution of Auerbach and Kotlikoff (1987) thedebate on Pareto-optimal pension reforms has gained momentum, see among others Kotlikoff et al.(1999), Diamond (2004), Joines (2005), Krueger and Kubler (2006), McGrattan and Prescott (2013),Roberts (2013). Indeed, the static comparisons of initial and final steady states – frequent in theearly literature – were be replaced by full cohort-by-cohort comparisons of welfare gains/losses,which is also key from the political economy and politics perspective, cfr. Holzman and Stiglitz(2001).While there seems to be a consensus in the literature on whether privatization can be in generalefficient – it has not been unequivocal on whether or not it can be implemented in a way that doesnot inflict on utility for any of the cohorts. These disparities come from two types of reasons: modelsetup and understanding of the very term ‘’privatization”. When discussing the pension systemreform, one needs to separate the systemic reform – for instance from a defined benefit to a definedcontribution system – from the method of financing. In fact, the distinction between pay-as-yougosystems and pre-funded systems is orthogonal to the way the pensions and contributions arecomputed, but the literature has often combined the two dimensions. For example Breyer (1989),Hirte and Weber (1997) or Gyárfás and Marquardt (2001) only change the method of financing,keeping other features of the balanced pay-as-you-go pension system at status quo whereas Kotlikoff(1996) and McGrattan and Prescott (2013) refer to privatization as eliminating any obligatorypension system at all.The problem itself, is far from trivial. First, if a pay-as-you-go system is to be replaced by afunded one there are two policy choices: either old pensions obligations are not being met or thephase out is gradual but during the transition the pension expenditure double (the ‘old’ generationsreceive benefits, but the ‘young’ generations need to save for their own pension). For the welfarecomparisons to make sense, the final and the initial fiscal stance are identical – a property thatKotlikoff (1996) names a zero-sum generational accounting. This implies that either the transitiongenerations cover two pension benefits (theirs and their ancestors) or the government temporarilyraises debt, putting part of the transition costs on the future generations. Is larger distortion overa shorter horizon preferable to a smaller one but over a longer horizon? Answer to this questionpermits to decide which policy mix is (more) efficient. But is there a method for phasing in thefunding and phasing out the pay-as-you-go financing so that the welfare of no generation wasdeteriorated?In a classic paper, Breyer (1989) concludes that in general there is no transition path from apay-as-you-go system to a funded system that does not incur welfare loss on at least one cohort.This result holds both for the closed economy, where the interest rate arises endogenously, as wellas for the small open economy with exogenously given interest rate. The effects is smaller withincomplete insurance, as argued by Nishiyama and Smetters (2007). 1 The loosing cohorts may becompensated via a lump-sum transfer, which is sometimes dubbed as a Lump Sum RedistributionAuthority. Indeed, if overall the reform is welfare enhancing, there exists a solution to the inter-1 In a similar spirit Krueger and Kubler (2006) analyze the introduction of a PAYG social security instead of aprivate savings.1

generational government optimization problem that allows fully remunerating the cohorts thatwould lose otherwise. For example, Kotlikoff (1996), Hirte and Weber (1997) search for “politicallyfeasible” Pareto improving transition paths by remunerating the losing cohorts with cohort-specificlump-sum transfers. They find that mixes of debt and income taxation and debt and consumptiontaxation used to finance the remuneration scheme allow for a Pareto improvement. Althoughthe study sheds insight on optimal taxation schemes, it still relies on a variant of the lump-sumredistributive authority mechanism. The LSRA, however, is politically impractical, due to bothlump sum nature and the cohort-specificity of these transfers.Given the doubts concerning the feasibility of the lump sum redistribution, a strand of theliterature focused on seeking a way to introduce a funded pillar in a way that would leave welfare ofeach cohort with no loss. First, in an economy with positive externalities to capital accumulationa la Lucas (1988), full abolition of the pension system jointly with subsidizing private savings isPareto improving. This has been shown formally by Belan and Pestieau (1999) and extended byGyárfás and Marquardt (2001), when agents live for two periods. In this approach, the final steadystate has no obligatory pension system, but rather voluntary – and subsidized – savings. Moreover,there exists a transition path along which the subsidy scheme can be phased out with no harm towelfare. This effectively leaves an economy with no pension system. Also, Gyárfás and Marquardt(2001) argue that such two-step reform may not be implementable from the political economystandpoint, since once the subsidy scheme is created, some cohorts shall lose from the second stepof the transition, even if they gain relative to the PAYG scheme.The results are similar with ex ante and ex post within cohort heterogeneity. In fact, Kotlikoffet al. (1999), McGrattan and Prescott (2013) extend the earlier results of Kotlikoff (1996) showingthat abandoning PAYG scheme immediately may be universally welfare enhancing. In this setting, asocial security based on voluntary private savings is preferred to an actuarially equivalent obligatoryPAYG scheme because individual life-cycle savings path departs from a flat contribution rate. Inorder to smoothen consumption in the presence of the social security contributions individuals needto borrow with interest which in principle should be equivalent to alternative life-cycle savings pathas long as the interest rate is equal to the time preference. However, this equivalence may be brokenfor two main reasons. First, if mortality is not strictly zero during the life-cycle, individuals discountwith both the time preference and probability of survival, which breaks the equivalence betweenborrowing and saving in the life-cycle. Second, in many countries instruments designed for oldagesavings benefit from liquidity limitations coupled with tax redemptions, alternative investmentschemes, etc., which breaks the equivalence between the net interest earned in the pension systemas opposed to outside the pension system. For these reasons, the solutions when pension system isabolished cannot be extended to the reforms which consist of replacing a PAYG obligatory pensionsystem with a funded pension system.To the best of our knowledge, Roberts (2013) proposes a notable exception, i.e. a model seekingan actual optimal transition path from PAYG to a funded system with no lump-sum transfers tocompensate the loosing cohorts. In his framework, replacing the PAYG security with an unfundedone, raises temporarily capital accumulation, which lowers the interest rate, thus promoting theadoption of a new technology. Increased total factor productivity generates – for some parametervalues – a series of supply side general equilibrium effects that make the transition cohorts betteroff, relative to PAYG scenario.In this paper we build on the rich earlier literature to propose a way to find a an optimaltransition path from a PAYG pension system to a funded system. We develop a fairly standard2

OLG model with multiple cohorts of decreasing mortality and exogenous technological progress. Inthe baseline scenario the economy is in transition from a pay-as-you-go defined benefit system toa pay-as-you-go defined contribution system. In the reform scenario on the transition from DB toDC, the pay-as-you-go pillar is complemented by a funded pillar. The contribution rates are keptconstant, but the private voluntary savings are subject to capital income tax, whereas the obligatorypension savings are not, which replicates the feature of many pension systems across the globe, seeGruber and Wise (2008). We contribute to the literature by proposing an instrument of phasingin the reform, which provides fiscal pressure relief to the living cohorts. This instrument consistsof compensating the transition cohorts in the PAYG DC system. The compensation occurs via ahigher than balanced indexation schedule. Thus – unlike for example Lump Sum RedistributionAuthority – the instrument is feasible. This instrument is at the same time a state variable in theprocess of identifying the optimal transition path to a (partially) pre-funded pension system. Wethus offer at the same time an algorithm for seeking an optimal pension system reform.The paper is structured as follows. In section 2 we present the model and the reform scenario.Subsequently, section 3 we describe the algorithm. The model is calibrated to values standardin the literature, featuring the demographics of an aging economy about to decelerate the rate oftechnological progress due to catching up. In fact, there are a few countries which underwent asimilar reform (transition from PAYG DB to partially funded DC), for example Chile, the BalticStates, Sweden as well as Poland, whose data we use. The details of the calibration are describedin section 4 the data for a country which actually underwent such a reform in 1999 – Poland. Theresults as well as analysis of their stability are discussed in 5, while in the concluding paragraphswe derive the policy implications of this study.2 The modelThere is a constant returns to scale Cobb-Douglas aggregate production function and constantlabor-augmenting technical progress. Equilibrium factor prices are time-varying but deterministic.Agents live for up to 80 periods, i.e. j = 1, . . . , 80, with j = 1 representing the actual age of 20, andare subject to mortality risk with survival probabilities known ex ante. Agents have two choices:inter-temporal between consumption c j,t and savings s j,t , and intra-temporal between consumptionand leisure 1 − l j,t , where l j,t denotes fraction of time supplied to the labor market by agent agedj in year t.2.1 ProductionUsing capital and labor the producers provide a composite consumption good with the Cobb-Douglas production function Y t = K α t (z t L t ) 1−α . Firms solve the following problem:max(Y t,K t,L t)Y t − w t L t − (r t + d) K t (1)subject to Y t = K α t (z t L t ) 1−α ,where L t and K t represent aggregate labor supply and capital stock. We allow for labor augmentingexogenous economic growth γ t = z t /z t−1 . 2 Note that if the return rate on capital is r t then the rental2 Bouzahzah et al. (2002) and Romaniuk (2009) discuss the sensitivity of OLG models to the assumptionsconcerning growth in the light of policy reforms and show that when analyzing pension systems, there is little3

ate must be r t +d, where d denotes capital depreciation. Firm optimization naturally implies w t =(1 − α)Kt α zt1−α L −αtandr t + d = αKtα−1 (z t L t ) 1−α2.2 Consumers’ choiceConsumers optimize lifetime utility derived from leisure and consumption:J−j∑U j,t = u(c j,t , l j,t ) + β s π j+s,t+su (c j+s,t+s , l j+s,t+s ) , with (2)π j,ts=1u (c j,t , l j,t ) = c j,t (1 − l j,t ) φ . (3)The notation of π in equation (3) conveys the non-zero probability of dying between the age of j andj + 1. Discounting takes into account time preference β and probability of survival π. Consumersknow the age and time specific mortality rates ex ante. At all points in time, consumers who surviveuntil the age of j = 79 die with certitude when reaching the age of J = 80. Changes in longevityand fertility rates are operationalized in our model by decreasing across time the size of the 20-yearold cohort as well as decreasing the mortality rates.Consumers are free to chose their labor supply during the working period, but once they reachthe age of ¯J they retire and are unable to supply any labor. In exchange for supplying age-specificamount of labor l j,t consumers receive earned income of w t , which is the market clearing marginalproductivity of labor. The labor income tax τ l and social security contributions τ s are deductedfrom earned income to yield disposable labor income. Interest earned on savings r t is taxed with τ k .In addition, there is a consumption tax τ c . The agents hit with mortality shock leave unintentionalbequests, which are then redistributed equally within their cohort as beq j,t . Thus, for working agepopulation (j < ¯J) the budget constraint at time t is given by:(1 + τ c t )c j,t + s j,t = (1 − τ s t )(1 − τ l t)w t l j,t + (1 + (1 − τ k t )r t )s j−1,t−1 + beq j,t (4)whereas for the retired population (j ≥ ¯J) it takes the form of:(1 + τ c t )c j,t + s j,t = (1 − τ l t)b ι j,t + (1 + (1 − τ k t )r t )s j−1,t−1 + beq j,t , (5)where b ι j,t denotes pension benefit from system ι for person of age j in time t. Appendix A providesthe analytic solution of this consumer problem.2.3 Pension system and the governmentThe government taxes consumption and income from capital and labor, issues and services debt,purchases goods (fixed exogenous share of GDP), and pays (a part of) retirement benefits. Pensionsystems are indexed by ι, which corresponds to either defined contribution PAYG (frequentlyreferred to as Notionally Defined Contribution - NDC) or funded defined contribution system, i.e.ι ∈ {NDC, F DC}. The economy starts from the defined benefit pay-as-you-go system (the initialsteady state). The baseline scenario consists of a transition to a defined contribution pay-as-you-gosystem, whereas the reform scenario consists of a defined contribution system which is partiallyfunded.or no effect of endogenizing productivity growth. Thus, for clarity, we model this economy as governed by exogenousproductivity growth.4

Defined contribution PAYG (NDC) The DC pension system collects contributions and usesthem to finance the contemporaneous benefits, but records them as individual stock of pensionsavings and at retirement converts this stock to an annuity. For simplicity we denote by τ NDC theobligatory contribution that goes into the PAYG system, whereas b NDC denote benefits from thiscomponent of the pension system. The benefit at agent’s age ¯J is given by:∑ []¯J−1b NDC s=1Π s i=1 (1 + rNDCt−¯J,t=¯J+i−1 ) τ NDCt− ¯J+s−1 w t− ¯J+s−1 l s,t− ¯J+s−1∏ Js= ¯J π (6)s,twhere r NDC denotes the indexation rate in the obligatory notional defined contribution pillar,which is defined by legislation as the payroll growth rate.Defined contribution funded (FDC) In the funded pillar the collected contributions form astock of actual savings, which are converted to an annuity at retirement. The benefit at agent’sage ¯J is given by:∑ []¯J−1b F ¯J,t DC s=1Π s i=1 (1 + rFDCt−=¯J+i−1 ) τ F DCt− ¯J+s−1 w t− ¯J+s−1 l s,t− ¯J+s−1∏ Js= ¯J π ,s,twhich implies that after abandoning the defined benefit system, the pension benefits are given byb DCj,t = b NDCj,t + b F j,tDC and also τ DC = τ NDC + τ F DC with τ DC = τ DBThe DC systems are by construction balanced, i.e. the benefits paid out to an agent equal thecontributions she has made to the system, whether the benefits are funded or financed contemporaneously.However, depending on the size and life expectancy of all cohorts living at time t, is ispossible that the system has revenues in excess/short of current needs. We thus specify, that thisis the government that collects social security contributions and pays out pensions.subsidy t = τtNDC w t L t −J∑j= ¯JN j,t b NDCj,t (7)We also assume that in the initial steady state the pension system runs a deficit to replicate theempirical regularity that pension systems usually get reformed only as they gradually becomefiscally unsustainable. 3The interest rate r F DC which accrues the individual obligatory savings in the capital pillar of thepension system is a market interest rate r > r NDC . Thus, there are two main differences betweenthe obligatory and voluntary savings in our model. First, obligatory savings are determined atthe rate τ F DC and thus do not follow the life-cycle, whereas the private savings s j,t can even benegative at any given point in life as long as the agent is not indebted at J. Indeed, the life-timevariability of private savings is one of the sources of the welfare gain in earlier studies which abolishthe pension system at all, e.g. Kotlikoff (1996), McGrattan and Prescott (2013). Second, thereis no capital income tax on obligatory savings, whereas the voluntary savings are subject to τ k ,3 In fact, many countries begin with an unbalanced defined benefit system financed on a pay-as-you-go basis.Typically, these are the obligations that need to be honored untill the last cohort which receives PAYG DB systemdies. Our setting is identical to replacing a DB system with a DC one somewhere in the past and forming the fundedpillar when the last cohort with DB PAYG pensions reaches J.5

which replicates the feature in many pension systems, including the US, Switzerland, UK, Franceand with no exceptions all the countries which in the 1990s wave reformed their pension systemsfollowing the guidelines by The World Bank, see Holzman and Stiglitz (2001), Gruber and Wise(2008).The government Naturally, in addition to balancing the social security, the government collectstaxes on earnings, interest and consumption and spends a fixed share of GDP on unproductive(but necessary) consumption. Given that the government is indebted, it naturally also services theoutstanding debt.T t =J∑j=1[]τt c c j,t + τt k r t s j,t + τt(1 l − τt s )w t l j,t + τtb l ι j,t N j,t (8)G t + subsidy t + r t D t−1 = T t + (D t − D t−1 ) (9)with G t = γ g Y t . We set initial steady state debt D t at the initial data level steady state at around45% of GDP, which was the actual value of debt to GDP ratio in 1999. For the purpose of ourexercise, we hold this debt to GDP ratio fixed over time.2.4 The reform: forming the funded pillarIn the baseline scenario the economy runs a defined contribution system funded on the pay-as-yougobasis. Thus, agents expect gradual decrease in pension benefits due to longevity (we keep theretirement age constant). On the other hand, the pension system is gradually getting balanced,which reduces the fiscal tension. In fact, taxes will gradually decrease because the initial debtoverhang from the deficit in the pension system is reduced (the initial subsidy is positive). theeconomy is in transition between DB and DC systems, with gradually decreasing pension benefitsdue to the increasing longevity of the agents while holding the retirement age fixed.In the reform scenario gradually a funded pillar is formed. Over the course of each 5 years thesize of the funded pillar is increased in a linear fashion by one percentage point, until it reaches 1/3of the entire contribution (i.e. the share set by the legislation). This diversion of funds from theSocial Insurance Fund generates a deficit, which needs to be subsidized, according to the equation(7). This in turn, leads to higher fiscal pressure, following equation (9). Higher fiscal pressureimplies either of the two: higher taxes which hurt all the living cohorts or higher debt which hurtsall the cohorts repaying it. While it is not realistic to assume the debt can increase for as long as ittakes to finance the transition, the actual burden on the future generations comprises not only debtper se, but also the servicing costs. Since the interest rate is typically in excess of the contributionsgrowth, repaying the debt implies undeniably increasing the taxes somewhere in the future.In order to avoid this type of adjustment, we introduce a different approach. We allow for thetransition cohorts to receive an indexation of pension benefits in excess of r NDC . In fact, we indexboth contributions and post-retirement benefits in the notional pillar withr NDC′t= r NDCt+ g(r F DCt− r NDCt ),where g denotes the generosity in excess of the rules specified in (6) i.e. defined contributionsystem. This way the transition cohorts receive from the pension system slightly more than theycontributed, but the fiscal burden occurs only once these cohorts actually retire, i.e. is costly6

postponed to the future. The indexation factor g can be either year specific or cohort specific. Inthe case of the former, any given cohort may receive a different lifetime compensation scheme thaneither of the adjacent cohorts. This approach seems especially attractive from the policy standpoint,since year-specific changes are typically easily enacted. However, they are usually fairly random,i.e. do not form a consistent pattern. In the case of the latter, the compensation is somehowanalogous to the earlier literature, because it offers a cohort specific transfer, similar to the LSRA.Yet, it is not a lump-sum transfer of the welfare surplus but rather a directed compensation (onlyto net out the welfare loss) to the selected cohorts. Thus, it is possibly second-best from a welfareperspective (with LSRA offering superior welfare outcomes due to the lack of general equilibriumeffects), but a politically more feasible option.2.5 Welfare accountingThe market clearing conditions and general equilibrium definition can be found in Appendix D.With the utility of j aged agent in period t is defined as in Equation (3), we denote allocation andwelfare in the baseline scenario (no funded pillar) with superscript B and in the reform scenario(gradually established funded pillar) with the superscript R. Then the consumption equivalent ofthe reform is computed according to the following formula:U 1,t ((1 + µ t )˜c B t , ˜l B t ) = U 1,t (˜c R t , ˜l R t ) (10)where ˜c t = (c 1,t , c 2,t+1 , ..., c J,t+J−1 ) and ˜l t = (l 1,t , l 2,t+1 , ..., l J,t+J−1 ). A positive value of µ t signifieswelfare improvement for cohort born in period t.While this method of welfare accounting is standard in the literature, it also has the advantageof directly measuring the welfare of each single cohort. Since our interest in this paper lies inseeking the Pareto-efficient way of introducing a pre-funded pillar, this is particularly adequate.In the following section we explain in detail the nature of the instrument used to implement apre-funded pillar in an optimal way.3 The algorithm for an optimal transition pathThis section describes the algorithm of seeking an optimal path of introducing the funded pillarin the pension system. It provides characteristics of program modules and presents the generalalgorithm. The model is solved twice: for the baseline and for the reform scenario. First, usingthe Gauss-Seidel algorithm we compute the initial and the final steady state. In the first steadystate the share of pension savings that goes into the funded pillar is equal to strictly 0. In the finalsteady state it amounts to a rate of 33%. 4 The initial and the final steady states do not depend onthe way the pre-funded pillar is established and thus the path of the instrument has no bearing onthese values.The transition path is found iteratively as a path between the initial and the final steady state,using also the Gaus-Seidel algorithm. The length of this path is such that the population stabilizes,which in our setting commences after 140 periods. Thus, the minimum length of the path is about220 periods. Yet, to allow the path seeking algorithm more flexibility, we allow for the instrumentto be at play for the first 200 periods. While the impact of actual generosity is negligible after4 The actual value of this parameter has been provided by the Polish legislation, but remains irrelevant for theproper functioning of the algorithm.7

150 periods (all cohorts already participate in the funded pillar), computationally it allows moreflexibility. Thus, We set the length of the transition path to 300 periods (to cover at least 200periods of instrument + 80 periods until the last cohort dies).For each path of generosity – i.e. given reform parametrization – once the path is computed,consumption equivalents are evaluated for each cohort. The goal of the system is to find optimal(in terms of relation described above) generosity path for introducing the funded pillar in a pensionsystem which was initially pay-as-you-go, both being defined contribution systems. We thus seekfor such a schedule of generosity g t so that no cohort was worse off due to introducing a fundedpillar.At the beginning the path of generosity g t is set to 0. We seek a vector of T = 200 values ofg t , which allows for the reform scenario to be at least weakly preferred to the baseline scenariofor all cohorts. 5 Throughout the exercise we employ the minimax criterion, which in our contexttranslates to the minimization of the loss of the biggest loser of the reform. In that way, even if theoutcome of the reform for some of the cohorts is negative, we at least ensure that the maximumloss is not too severe. To be specific, the adopted threshold of .001 translates to a loss of a dollarout of each thousand dollars of the lifetime income.The year-specific version of the algorithm iteratively identifies those cohorts whose welfareequivalent is lower than the threshold and increases the generosity for the last 20 years of life ofthe oldest of them. The logic behind this procedure is that generosity increases welfare ‘forward’,and so the cohorts in the close neighborhood ‘to the right’ of the selected one shall also experiencean increase in welfare. We do so either until the threshold is hit or until the iterations limit of 1000is reached.The cohort-specific version of the algorithm relies on allocating ‘excessive’ indexation of pensionbenefits to the cohorts who bear the cost of the reform. The procedure first identifies the most hurtcohort and then slightly increases the generosity for this cohort. In this case there is no reason tosuspect there are spillovers to the neighbors of the cohort in terms of welfare, what is more likely isthat the welfare of its neighbors is decreased via general equilibrium effects. Thus, the procedureusually requires more iterations to reach a solution. We do so either until the threshold is hit oruntil the iterations limit of 1000 is reached.4 CalibrationThe calibration procedure aims to replicate certain micro- and macroeconomic features of the Polisheconomy in 1999. The demographic structure is retrieved from the demographic projection by EU’sEconomic Policy Committee Working Group on Aging Populations and Sustainability (henceforthAWG) together with the projected path of the labor-augmenting total factor productivity growth.The demographic projection covers years 1999-2050, and we assume that after 2050 both fertilityand mortality stays constant, meaning that in our model the demography stabilizes after year 2130,as can be seen in Figure 1a. The projected TFP growth rate starts at 3.8% in 1999 and is assumedto slowly converge to the value of 1.7% by the year 2040, see Figure 1b.The capital share of income is assumed at the standard 0.33 level, and the annual depreciationrate is assumed at 5%. The discounting factor β and depreciation rate d are calibrated to yield the5 A clear corollary of this research is that it is perfectly possible that some cohorts are worse off from having theirDB pensions being replaced by DC pensions. However, our focus in this study is on seeking an optimal path ofintroducing a funded pillar – not reforming the way the pensions are computed.8

Figure 1: AWG’s projectionsPopulation aged 20−99 (in millions)20 25 302000 2050 2100 2150 2200YearTotal Factor Productivity growth rate (in %)1.5 2 2.5 3 3.5 42000 2020 2040 2060Year(a) Total population size(b) Total Factor Productivityshare of investment in GDP at the level of 21% 6 and the interest rate of approximately 7%.Whilethis value may seem high, note that it applies to the economy still in transition with high TFPgrowth. Nishiyama and Smetters (2007) find the corresponding interest rate of 6.2% for the US. 7The leisure preference parameter φ was calibrated to replicate the labor force participation rate of56.8%.The share of government expenditure in GDP is assumed to be constant both in the steadystates as well as along the transition at the level of 20%. The initial debt to GDP ratio is set atthe level of 45%, corresponding to the data. We set the capital income tax rate at the de iure levelof 19% as there are no exemptions. Note however, that the return on assets in the funded pensionfunds is tax-free. The consumption tax τ c was set at 11%, which matches the rate of revenues fromthis tax in aggregate consumption in 1999. The social security contribution rate was calibrated toreplicate the resulting pensions to GDP ratio of 5%. We allow the income tax rate to vary freelyin order to close the government budget constraint. The resulting 17.4% is slightly higher thanthe 16-17% ratio of income tax revenues to total taxable labor income in the late 1990s in Poland,and is understood to capture the revenues from other non-modeled taxes. The following Table C.2sums up the above considerations.5 ResultsIn the process of seeking the optimal transition path we take a set of conservative assumptions.First, we set that the government services debt at the prevailing interest rate. Thus, there is no clearbenefit for the government to borrow at a discount relative to private investment opportunities.Also we do not allow the debt to adjust, keeping it fixed at 45% of GDP. This is setting the barhigh for the path seeking algorithm. For example, Makarski et al. (2015) show that financing the6 Depending on the period over which the average is taken, it ranges from 20.8% for a 10 year average (5 yearsbefore and 5 years after the reform), 23.1% for 2 year window and 24.1% in the year of the reform. The average forthe period between 1995 (first reliable post-transition data) and 2010 amounts to 20.7%.7 In the Polish case, the average real annual rate of return at the level of 7.5% was achieved by the open pensionfunds with a balanced portfolio strategy in the period 1999-2009.9

Table 1: Calibrated parametersParametersα capital share of income 0.33d depreciation rate 0.05β discounting factor 0.9735φ preference for leisure 0.825γ g share of govt expenditure in GDP 20%D/Y share of public debt to GDP 45%τ k capital income tax 19%τ c consumption tax 11%τ ι effective social security contribution 6.2%Outcome values (initial steady state)(dk)/y share of investment in GDP 21%b/y share of pensions in GDP 5.0%r interest rate 7.2%labor force participation rate 56.9%τ l labor income tax 17.4%reform via public debt can yield higher efficiency and more equal welfare distribution across cohorts.However, in the case of ?Makarski et al. (2015) government services debt at a third of the marketinterest rate, whereas ? allows for the unconstrained growth of debt. In our setting the debt is notallowed to increase (all fiscal adjustment needs to occur instantaneously via taxes). The shadowprice of the implicit debt is exactly the indexation of pensions 8 – pure in the case of the baselinescenario and augmented for cohorts or years in the case of the reform.We first present the results of the estimation from the calibration described in Section 4. Wediscuss two separate sets of results – for the cohort-specific generosity and for the time-specificgenerosity, as described above. This analysis is done with the assumption that the funded pillaris introduced immediately but at a slow rate, which is our preferred calibration. The intuitionbehind such a modeling choice is that a slowly introduced reform should have relatively low welfareeffects on each cohort for two reasons – the amount of funds diverted from the PAYG pillar to thepre-funded one in each year (pension reform costs) is low, whereas the share of pension benefitscoming from the funded pillar (pension reform benefits) increases very slowly. Subsequently, thewelfare effects and the performance of the algorithm/policy instrument depending on the rate andthe timing of the reform. We explicitly compare delayed reforms to the immediate ones and theslow pace of introducing the funded pillar to faster ones. Finally, for the preferred specification,we demonstrate the robustness of our algorithm to the alternative assumptions concerning thedemography, technological progress as well as preference parameters.5.1 Baseline calibrationIn the conservative parametrization we find the generosity path to compensate mostly the futurecohorts, who start bearing the costs associated paying ‘excessively’ high pensions to the initial8 See Kuhle (2010) for an extensive comparison of implicit and explicit pension debt with idiosyncratic shocks.10

transition generations. Thus, generosity is relatively low at first, because at this stage the majorityof the shift in the contributions is compensated by taxing a relatively large base, see Figure D.7a.However, when the future pension benefits needs to be adjusted for the additional indexation, thewelfare costs appear. We report the adjustment in taxes, which signifies the timing of periods inwhich the rules for higher pension benefits get activated in Figure D.7 in the Appendix. Generally,both the taxes and the deficit in the pension system adjust contemporaneously. Yet, although thetiming is clearly related, there is no clear translation between the the behavior of the indexationgenerosity and the strength of the fiscal adjustment.Figure 2: Welfare effects and indexation generosity in the pension system(a) Consumption equivalents−.005 0 .005 .01 .0151900 2000 2100 2200 2300Year of birthConsumption equivalents from baseline of no generosityConsumption equivalents from best found year−specific pathConsumption equivalents from best found cohort−specific pathYear−specific generosity path0 .1 .2 .3 .4Cohort−specific generosity path0 .05 .1 .152000 2100 2200 2300Year1900 2000 2100 2200 2300Year of birth(b) Path of year-specific generosity t(c) The path of cohort-specific generosity tComparing the two welfare paths, clearly the cohort-specific generosity has a larger potentialto smoothen consumption, which is also signified by the smaller and more monotonic per-cohorttransfers, as expressed by the ‘excessive’ indexation. Larger scope for fine tuning the welfare effectsdoes not translate, however, to a substantially lower number of ‘losing’ cohorts, when compared tothe per-year generosity.In the above results a number of exogenous assumptions may be affecting the results. First, thereform is really slow, but starts immediately, which may interact with the demographic transition11

and the relative situation of the initial old. To address this point in the subsequent section weexplicitly analyze the effects of timing and rate of the reform on the welfare and the performanceof our policy instrument. Second, initially the population is not old yet, so the tax base is wider,thus allowing for a smaller scope of the tax adjustment. Also, the initial technological progress rateis relatively high, which favors the implicit debt over the explicit one. Thus, in the next sectionwe test if the algorithm is able to find a similarly welfare enhancing path if these calibrations arerelaxed towards more conservative ones.5.2 The timing and the rate of the reformWe consider the following modeling choices for the timing and the rate of the reform. First, wewant to analyze what is the role for the timing of the reform. To address this point we allowtwo specifications: immediate and delayed phasing in of the reform. With the delayed phasing inof the reform, the initial pension benefit recipients do not bear the costs of forming the capitalpillar at all, whereas working age cohorts fully internalize the effects of having the funded pillar.Please note that even with the delayed phasing in of the reform the reform is introduced whenthe demographic transition is still ongoing, because our becomes stable only after 160 periods.Second, we analyze the role of the speed at which the capital pillar is formed. Faster reform causeslarger fiscal adjustment to finance the gap in the PAYG pillar of the pension system. Yet, it alsocontributes to higher capital accumulation. Because we keep public debt constant, there is nocrowding out of the private capital, thus the obligatory pension savings in the capital pillar fullytranslate to the accumulated capital stock in the economy. The two effects counteract each other,whereas the policy instrument we employ – ‘excessively’ generous indexation of pension benefitsfor the cohorts with negative consumption equivalents – allows to smoothen consumption despiteincreased taxation. All analyzed paths for the phasing in of the capital pillar are displayed inFigures E.8 and E.10 in the Appendix. We show the results for the year-specific generosity paths. 9The basic intuition for the results is as follows. For the reforms paths which burden a largernumber of adjacent cohorts may, the instrument / algorithm may be unable to provide significantimprovements because the periods of original fiscal adjustment and the consequences of ‘excessive’indexation begin to overlap. In addition, faster reforms generate larger costs, which in turn requirehigher compensation. The two effects are likely to propagate each other. Thus, one should expectthat neither for the very fast nor for the very slow reforms an optimal path can be found. As faras delaying is concerned, for the reforms which are enacted later, finding an optimal path shouldbe more feasible, because all affected cohorts are able to fully internalize the change in the pensionsystem. The results are presented in Figure 3.In Figure 3a we show the results for the fastest possible phasing in of the reform – immediate(black lines) and delayed by 40 years (gray lines). 10 In the case of the immediate reform theinstrument is able to compensate the oldest cohorts, but as a consequence cohorts born between2000 and 2030 loose as much consumption equivalent as 1% of their lifetime income. Similar resultsare obtained for the delayed reform, which suggests that for a one-off reform there year-specific‘excessive’ indexation is not able to account for the welfare costs associated with forming the capitalpillar.In the subsequent panels of Figure 3 we show the results for the reforms which take 33 years,9 The results for the cohort-specific generosity are available upon request.10 The paths of the generosity parameter are shown in Figures E.9 and E.11.12

Figure 3: Welfare effects of reform with generosity t for the alternative phasing in of the reform−.02 −.01 0 .01 .02 .031900 2000 2100 2200 2300Year of birth−.02 −.01 0 .01 .02 .031900 2000 2100 2200 2300Year of birthReform immediate, no generosityReform immediate, with generosityReform delayed, no generosityReform delayed, with generosityReform immediate, no generosityReform immediate, with generosityReform delayed, no generosityReform delayed, with generosity(a) Immediate reform(b) 1 pp per year−.01 0 .01 .02 .031900 2000 2100 2200 2300Year of birth−.005 0 .005 .01 .0151900 2000 2100 2200 2300Year of birthReform immediate, no generosityReform immediate, with generosityReform delayed, no generosityReform delayed, with generosityReform immediate, no generosityReform immediate, with generosityReform delayed, no generosityReform delayed, with generosity(c) 0.5 pp per year(d) 0.25 pp per year66 years and 132 years to complete. The extremely slow reform in Figure 3d is similar to thebaseline results in Section 5.1. The intermediate solutions show the specificity of the analyzedpolicy instrument. First, it concentrates the welfare costs of the reform in fewer cohorts – blacklines in Figures 3b and 3c. Second, in the case of the relatively slow reforms (66 years to reach1/3 of contributions going to the capital pillar), delaying the reform combined with the policyinstrument arrives at nearly Pareto enhancing welfare path. In fact, the welfare costs fall short of0.01%.5.3 Testing the robustness to alternative exogenous assumptionsFor the specification presented in Section 5.1 we develop a series of alternative calibrations whichpermit to test if the algorithm proposed provides stable results. This specification encompassesa slow and immediate reform, which implies relatively low welfare costs, but also little room forthe policy instrument / algorithm to provide welfare enhancement. Thus, it is a conservativeenvironment to test the sensitivity of the instrument / algorithm to the alternative assumptionsabout the exogenous parameters. In each of the analysis we change one of the exogenous parameters13

(or their paths), keeping all other calibration parameters unchanged. We also show the results formthe previous section as baseline scenario in each case.The robustness checks concern: utility function parameters, exogenous fertility and technologicalprogress paths. For the utility function parameters we check two cases: lower patienceδ and lower preference for consumption 1 − φ. With lower patience agents are likely to prefercontemporaneous consumption to future pension benefits. Thus, instantaneous tax adjustments willrequire higher generosity to maintain the same welfare, which sets the bar for optimum transitionpath higher than in the baseline scenario. Similar reasoning holds for higher preference for leisure –agents will be less likely to adjust upwards labor supply and downward consumption, thus increasedcontemporaneous taxation will be more costly than in the baseline calibration in terms of welfare.See Table C.2 in the Appendix for the details of these alternative calibrations.For the exogenous fertility, the demographic projection used in the baseline calibration assumesthat it will be gradually decreasing. Yet, the fertility rate is already extremely low in the initialsteady state – at 1.4. Moreover, decreasing size of the population implies that initially there is amuch higher tax base, whereas in the future generosity needs to be provided for by a much smallerpopulation. To see of the algorithm is robust to this parametrization, we also include a scenario witha constant rather than decreasing fertility. In practice, this implies that the population stabilizesearlier in the simulation. Please note that this scenario does not affect the initial steady statecalibration.Finally, we also consider two alternative paths for the catching up. While in the final steadystate the economy is always projected to experience technological progress of 1.7% per annum, thestarting point is actually at 4% per annum. The speed of convergence may matter for the abilityto find an optimum path in the interaction with the speed of implementing the reform. Thus, weinclude a faster convergence and a slower convergence, see Figure C.6 in the Appendix.The results of these sensitivity checks are reported in Figures 4 and 5 and suggest that while theactual values of ‘excessive generosity’ differ between the calibrations, the time variation is largelythe same. This is indicative of the property that the obtained ‘optimal’ reform paths are fairlystable for a given schedule of introducing the funded pillar.The results are fairly intuitive. For the agents with higher preference for leisure, generosityis lower in both cases, as they need to be compensated for decreased consumption less. In thescenarios with faster TFP growth, part of the necessary increase in taxation can be compensatedfor by the faster accumulation of wealth. Thus, the compensations needed are lower. The oppositeholds for the paths with slower TFP growth. There are also no significant differences in the cohortdistribution of welfare – if preferences exhibit higher impatience welfare ‘break-even’ is experiencedalready by older cohorts, which is consistent with the expectations. As expected, the role of fertilityscenario is negligible.6 ConclusionsOur objective in this paper has been to find a transition path from a pay-as-you-go scheme toa pre-funded scheme that leaves no cohort worse off after the reform in the defined contributionsetting. While the introduction of the funded pillar may well be gradual, the costs still need to bedistributed over subsequent cohorts in the search for Pareto-optimality. Thus, forming a fundedpillar when PAYG scheme exists requires either of the three: increase in tax-rate, an increase inpublic debt or a decrease in other public spending. It is customary in this literature to treat non-14

Figure 4: Welfare effects of reform with generosity t for the alternative calibrations0 .005 .01 .015 .020 .005 .01 .015 .021900 2000 2100 2200 2300Year of birthConstant fertilityFaster TFP growth convergenceSlower TFP growth convergenceLower patienceHigher weight of leisure in utilityBaseline scenario1900 2000 2100 2200 2300Year of birthConstant fertilityFaster TFP growth convergenceSlower TFP growth convergenceLower patienceHigher weight of leisure in utilityBaseline scenario(a) year-specific generosity t(b) cohort-specific generosity tFigure 5: Paths of generosity t for the alternative calibrations0 .1 .2 .3 .40 .05 .1 .152000 2100 2200 2300YearConstant fertilityFaster TFP growth convergenceSlower TFP growth convergenceLower patienceHigher weight of leisure in utilityBaseline scenario1900 2000 2100 2200 2300Year of birthConstant fertilityFaster TFP growth convergenceSlower TFP growth convergenceLower patienceHigher weight of leisure in utilityBaseline scenario(a) year-specific generosity t(b) cohort-specific generosity tpension public expenditure as a fixed variable, because typically it is modeled as pure waste andthus changing its value directly alternates welfare accounting. Increase in taxes concentrates thecosts of the reform on transition cohorts, so it cannot produce a Pareto-optimal reform. Thus,one is left with public debt. However, in real world the public debt levels are regulated by oftennational laws and international treaties, whereas systematic increase in public debt can translateto financial and/or currency crisis before the economy reaches the new steady state.Earlier literature in the field typically relied on balanced pension systems or on lump-sumtaxation to propose Pareto-optimal reforms in the pension systems. Alternatively, previous studiesdesigned a redistribution scheme between cohorts (e.g. a governmental Lump Sum RedistributionAuthority). The assumption of balanced pension system is not likely to hold in reality, whereaslump sum taxation proves to be politically challenging. We propose an alternative approach, byshifting the costs of the reform to the future generations by the means of implicit rather thanexplicit debt. We develop a model of an OLG economy, which starts from a PAYG DC systemand reforms to a (partially) pre-funded DC. The funded pillar is introduced gradually over one15

generation. We keep the debt share in GDP and non-pension government expenditure fixed fromthe baseline scenario, but introduce for the transition cohorts additional indexation scheme. Thisindexation scheme compensates the cohorts which experience the costs of the reform. We thus relyon implicit debt as an instrument to find an optimal transition path. The advantage of relying onimplicit debt is that it reduces slightly the costs of servicing the debt for the future generations.The proposed indexation scheme is at the same time a path seeking algorithm. We run aseries of simulations searching the indexation scheme that leaves no cohort worse off. Our analysisis closely calibrated to the case of Poland, which implemented such reform in 1999. However, weperform a variety of sensitivity analyses, altering the set of assumptions concerning the demographicscenarios, preferences and exogenous technological progress path. In each of these cases our policyinstrument – and at the same time algorithm – performs satisfactorily.This study is based on a number of premises. First, agents in the model see no link betweenthe contributions to the pension system and the subsequent pension benefits. While they maximizelifetime utility and have perfect foresight, there is no derivative of the pension benefits with respectto labor in the consumer problem solution. Providing this extension is likely to result in a behavioralresponse in terms of labor supply decisions. This brings about another premise, i.e. the shape ofthe utility function. Namely, with the Cobb-Douglas utility function, there is only minor reaction oflabor to changes in the pension system. Relying on a different utility function – such as Greenwoodet al. (1988) – could also matter for the ability of the algorithm to find the Pareto-optimal transitionpath.16

ReferencesAaron, H.: 1966, The social insurance paradox, Canadian Journal of Economics and PoliticalScience/Revue canadienne de economiques et science politique 32(03), 371–374.Auerbach, A. J. and Kotlikoff, L. J.: 1987, Dynamic fiscal policy, Cambridge University Press.Belan, P. and Pestieau, P.: 1999, Privatizing Social Security: A Critical Assessment, The GenevaPapers on Risk and Insurance. Issues and Practice 24(1), 114–130.Bouzahzah, M., De la Croix, D. and Docquier, F.: 2002, Policy reforms and growth in computableOLG economies, Journal of Economic Dynamics and Control 26(12), 2093–2113.Breyer, F.: 1989, On the Intergenerational Pareto Efficiency of Pay-as-you-go Financed PensionSystems, Journal of Institutional and Theoretical Economics (JITE) / Zeitschrift fur die gesamteStaatswissenschaft 145(4), 643–658.Diamond, P.: 2004, Social Security, American Economic Review 94(1), 1–24.Greenwood, J., Hercowitz, Z. and Huffman, G. W.: 1988, Investment, capacity utilization, and thereal business cycle, American Economic Review pp. 402–417.Gruber, J. and Wise, D. A.: 2008, Social security and retirement around the world, University ofChicago Press.Gyárfás, G. and Marquardt, M.: 2001, Pareto improving transition from a pay-as-you-go to afully funded pension system in a model of endogenous growth, Journal of Population Economics14(3), 445–453.Hirte, G. and Weber, R.: 1997, Pareto improving transition from a pay-as-you-go to a fully fundedsystem - is it politically feasible?, FinanzArchiv/Public Finance Analysis pp. 303–330.Holzman, R. and Stiglitz, J. E.: 2001, New ideas about old age security, Washington, DC: WorldBank.Joines, D. H.: 2005, Pareto-Improving Social Security Reform, 2005 Meeting Paper 396, Societyfor Economic Dynamics.Kotlikoff, L. J.: 1996, Privatizing Social Security: How It Works and Why It Matters, Tax Policyand the Economy pp. 1–32.Kotlikoff, L. J., Smetters, K. and Walliser, J.: 1999, Privatizing social security in the united states– comparing the options, Review of Economic Dynamics 2(3), 532–574.Krueger, D. and Kubler, F.: 2006, Pareto-Improving Social Security Reform when FinancialMarkets Are Incomplete, American Economic Review 96(3), 737–755.Kuhle, W.: 2010, The optimum structure for government debt, Technical report, Munich Centerfor the Economics of Aging (MEA) at the Max Planck Institute for Social Law and Social Policy.Lucas, R. E.: 1988, On the mechanics of economic development, Journal of Monetary Economics22(1), 3 – 42.17

Makarski, K., Tyrowicz, J. and Hagemejer, J.: 2015, Analyzing the efficiency of the pension reform:the role for the welfare effects of the fiscal closure, Macroeconomic Dynamics forthcoming.McGrattan, E. R. and Prescott, E. C.: 2013, On financing retirement with an aging population,NBER Working Paper 18760, National Bureau of Economic Research.Nishiyama, S. and Smetters, K.: 2007, Does Social Security Privatization Produce EfficiencyGains?, Quarterly Journal of Economics 122, 1677–1719.Roberts, M. A.: 2013, Pareto-improving Pension Reform Through Technological Implementation,Scottish Journal of Political Economy 60(3), 317–342.Romaniuk, K.: 2009, Pareto Improving Transition from a Pay-as-You-Go to a Fully Funded PensionSystem in a Model of Endogenous Growth: The Impact of Uncertainty, SSRN Scholarly PaperID 1613251, Social Science Research Network.Samuelson, P. A.: 1958, An exact consumption-loan model of interest with or without the socialcontrivance of money, The Journal of Political Economy pp. 467–482.18

Appendix A. Consumer problem solutionFor j < ¯J:withc j,t = (1 + (1 − τ t k )r t )s j−1,t−1 + (1 − τt s )(1 − τt)w l t + beq j,t + Ω j,t + Γ[ j,t∑ ()(1 + τt c) ¯J−j−1s=0 (1 + φ) β s π j+s,t+sπ j,t+ ∑ ( )] (A.11)J−js= ¯J−j β s π j+s,t+sπ j,tφ(1 + τt c )c j,tl j,t = 1 −(1 − τt s)(1 − τ t l)w ts j,t = (1 − τ s t )(1 − τ l t)w t l j,t + (1 + (1 − τ k t )r t )s j−1,t−1 − (1 + τ c t )c j,t ,Ω j,t =Γ j,t =¯J−j−1∑s=1J−j∑s= ¯J−j(1 − τ s t+s)(1 − τ l t+s)w t+s l j+s,t+s + beq j+s,t+s∏ si=1 (1 + (1 − τ k t+i )r t+i)(1 − τt+s)b l ι j+s,t+s + beq j+s,t+s∏ si=1 (1 + (1 − τ t+i k )r .t+i)For j ≥ ¯J:c j,t = (1 + (1 − τ t k )r t )s j−1,t−1 + (1 − τt)b l ι j,t + beq j,t + Γ j,t[(1 + τt c) ∑J−j( )] (A.12)s= ¯J−j β s π j+s,t+sπ j,tφ(1 + τt c )c j,tl j,t = 1 −(1 − τt s)(1 − τ t l)w ts j,t = (1 − τ l t)b ι j,t + (1 + (1 − τ k t )r t )s j−1,t−1 − (1 + τ c t )c j,t ,withΓ j,t =J−j∑s= ¯J−j(1 − τt+s)b l ι j+s,t+s + beq j+s,t+s∏ si=1 (1 + (1 − τ t+i k )r .t+i)The numerators in equations (A.12) and (A.13) represent the present discounted value of theremaining lifetime income.Appendix B. Market clearing conditions and general equilibriumThe goods market clearing condition is standardly defined asJ∑N j,t c j,t + G t + K t+1 = Y t + (1 − d)K t ,j=1(B.13)where we denote the size of the j-aged cohort at period t as N j,t . This equation is equivalentto stating that at each point in time the price for capital and labor would be set such, that the19

demand for the goods from the consumers, the government and the producers would be met. Thisnecessitates clearing in the labor and in the capital markets. Thus labor is supplied and capitalaccumulates according to:¯J−1∑L t = N j,t l j,t and K t+1 = (1 − d)K t +j=1j=1J∑N j,t ŝ j,t ,(B.14)where ŝ j,t denotes private savings s j,t as well as accrued obligatory contributions in the fundedpillar of the pension system.An equilibrium is an allocation {(c 1,t , ..., c J,t ), (s 1,t , ..., s J,t ), (l 1,t , ..., l J,t ), K t , Y t , L t } ∞ t=0 and prices{w t , r t } ∞ t=0 such that• for all t ≥ 0, for all j ∈ [1, J] ((c j,t , ..., c J,t+J−j ), (s j,t , ..., s J,t+J−j ), (l 1,t , ..., l J,t+J−j ) solves theproblem of an agent j for all t, given prices;• (K t , Y t , L t ) solves the firm’s problem (1);• government sector is balanced, i.e. (8) - (9) are satisfied;• markets clear, i.e. (B.13)-(B.14) are satisfied.20

Appendix C. Alternative calibrationsTable C.2: Alternative calibrationsParameters Baseline Impatience Lazyα capital share of income 0.33 0.33 0.33d depreciation rate 0.05 0.05 0.05β discounting factor 0.9735 0.9700 0.9735φ preference for leisure 0.825 0.825 1γ g share of govt expenditure in GDP 20% 20% 20%D/Y share of public debt to GDP 45% 45% 45%τ k capital income tax 19% 19% 19%τ c consumption tax 11% 11% 11%τ ι effective social security contribution 6.2% 6.2% 6.2%Outcome values (first steady state)(dk)/y share of investment in GDP 21% 20.3% 21%b/y share of pensions in GDP 5.0% 5.0% 4.6%r interest rate 7.2% 7.5% 7.1%labor force participation rate 56.9% 56.5% 52%τ l labor income tax 17.4% 16.4% 16.4%Figure C.6: Alternative technological progress rates (γ)Total Factor Productivity growth rate (in %)1.5 2 2.5 3 3.5 42000 2020 2040 2060 2080 2100YearStandard TFP growth pathFaster TFP growth convergenceSlower TFP growth convergence21

Appendix D. Fiscal adjustments on the transition pathFigure D.7: Changing pension system balance and tax adjustments0 .01 .02 .032000 2100 2200 2300YearSIF subsidy −− no generositySIF subsidy −− with year−specific generositySIF subsidy −− with cohort−specific generosity(a) Deficit in the pension system as a share of GDP in thebaseline scenario and in the reform scenarios.13 .14 .15 .16 .172000 2100 2200 2300YearLabor income tax −− no generosityLabor income tax −− with year−specific generosityLabor income tax −− with cohort−specific generosity(b) Taxes in baseline scenario of no generosity and thereform scenarios with cohort and year specific generosities22

Appendix E. Alternative paths of the reformFigure E.8: Immediate reform paths.6 .7 .8 .9 12000 2050 2100 2150 2200YearReform immediate, funded pillar increases by 0.25 pp. per yearReform immediate, funded pillar increases by 0.5 pp. per yearReform immediate, funded pillar increases by 1 pp. per yearReform immediate, funded pillar created at onceFigure E.9: Immediate reform paths0 .5 1 1.52000 2050 2100 2150 2200YearReform immediate, funded pillar increases by 0.25 pp. per yearReform immediate, funded pillar increases by 0.5 pp. per yearReform immediate, funded pillar increases by 1 pp. per yearReform immediate, funded pillar created at once23

Figure E.10: Immediate reform paths.6 .7 .8 .9 12000 2050 2100 2150 2200YearReform delayed, funded pillar increases by 0.25 pp. per yearReform delayed, funded pillar increases by 0.5 pp. per yearReform delayed, funded pillar increases by 1 pp. per yearReform delayed, funded pillar created at onceFigure E.11: Immediate reform paths0 .2 .4 .6 .8 12000 2050 2100 2150 2200YearReform delayed, funded pillar increases by 0.25 pp. per yearReform delayed, funded pillar increases by 0.5 pp. per yearReform delayed, funded pillar increases by 1 pp. per yearReform delayed, funded pillar created at once24

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