Working Papers

No. 22/2015 (170)








Warsaw 2015

In the search for the optimal path to establish a funded pension system


Faculty of Economic Sciences

University of Warsaw

e-mail: mbielecki@wne.uw.edu.pl


Faculty of Economic Sciences

University of Warsaw

National Bank of Poland

e-mail: jtyrowicz@uw.edu.pl


National Bank of Poland

Warsaw School of Economics

e-mail: kmakar@sgh.waw.pl


University of Warsaw

e-mail: m.waniek@mimuw.edu.pl


We propose a politically feasible instrument for a nearly optimal transition from a pay-as-you-go to

a funded scheme in a defined contribution pension system. It consists of compensating the

transition generations in the form of pension benefits indexation more generous than would have

prevailed in a clean DC system. Thus, this instrument allows to smoothen the welfare costs of

transition over future generations via some small implicit debt. Our instrument proves robust to a

number of parametric and modeling choices.


defined contribution, pay-as-you-go, pre-funded, pension system


H55, E17, C60


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 motivation

Ever since the discussion between Samuelson (1958) and Aaron (1966) it has been a concern whether

pension system reforms – especially the privatization of the social security – can be implemented

without inflicting welfare loss on one or few cohorts. With the onset of the overlapping generations

general equilibrium models after the seminal contribution of Auerbach and Kotlikoff (1987) the

debate 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 the

early 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


While there seems to be a consensus in the literature on whether privatization can be in general

efficient – it has not been unequivocal on whether or not it can be implemented in a way that does

not inflict on utility for any of the cohorts. These disparities come from two types of reasons: model

setup and understanding of the very term ‘’privatization”. When discussing the pension system

reform, one needs to separate the systemic reform – for instance from a defined benefit to a defined

contribution system – from the method of financing. In fact, the distinction between pay-as-yougo

systems and pre-funded systems is orthogonal to the way the pensions and contributions are

computed, 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 obligatory

pension system at all.

The problem itself, is far from trivial. First, if a pay-as-you-go system is to be replaced by a

funded one there are two policy choices: either old pensions obligations are not being met or the

phase out is gradual but during the transition the pension expenditure double (the ‘old’ generations

receive benefits, but the ‘young’ generations need to save for their own pension). For the welfare

comparisons to make sense, the final and the initial fiscal stance are identical – a property that

Kotlikoff (1996) names a zero-sum generational accounting. This implies that either the transition

generations cover two pension benefits (theirs and their ancestors) or the government temporarily

raises debt, putting part of the transition costs on the future generations. Is larger distortion over

a shorter horizon preferable to a smaller one but over a longer horizon? Answer to this question

permits to decide which policy mix is (more) efficient. But is there a method for phasing in the

funding and phasing out the pay-as-you-go financing so that the welfare of no generation was


In a classic paper, Breyer (1989) concludes that in general there is no transition path from a

pay-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 well

as for the small open economy with exogenously given interest rate. The effects is smaller with

incomplete insurance, as argued by Nishiyama and Smetters (2007). 1 The loosing cohorts may be

compensated via a lump-sum transfer, which is sometimes dubbed as a Lump Sum Redistribution

Authority. 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 a

private savings.


generational government optimization problem that allows fully remunerating the cohorts that

would lose otherwise. For example, Kotlikoff (1996), Hirte and Weber (1997) search for “politically

feasible” Pareto improving transition paths by remunerating the losing cohorts with cohort-specific

lump-sum transfers. They find that mixes of debt and income taxation and debt and consumption

taxation used to finance the remuneration scheme allow for a Pareto improvement. Although

the study sheds insight on optimal taxation schemes, it still relies on a variant of the lump-sum

redistributive authority mechanism. The LSRA, however, is politically impractical, due to both

lump sum nature and the cohort-specificity of these transfers.

Given the doubts concerning the feasibility of the lump sum redistribution, a strand of the

literature focused on seeking a way to introduce a funded pillar in a way that would leave welfare of

each cohort with no loss. First, in an economy with positive externalities to capital accumulation

a la Lucas (1988), full abolition of the pension system jointly with subsidizing private savings is

Pareto improving. This has been shown formally by Belan and Pestieau (1999) and extended by

Gyárfás and Marquardt (2001), when agents live for two periods. In this approach, the final steady

state 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 to

welfare. 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 economy

standpoint, since once the subsidy scheme is created, some cohorts shall lose from the second step

of 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, Kotlikoff

et al. (1999), McGrattan and Prescott (2013) extend the earlier results of Kotlikoff (1996) showing

that abandoning PAYG scheme immediately may be universally welfare enhancing. In this setting, a

social security based on voluntary private savings is preferred to an actuarially equivalent obligatory

PAYG scheme because individual life-cycle savings path departs from a flat contribution rate. In

order to smoothen consumption in the presence of the social security contributions individuals need

to borrow with interest which in principle should be equivalent to alternative life-cycle savings path

as long as the interest rate is equal to the time preference. However, this equivalence may be broken

for two main reasons. First, if mortality is not strictly zero during the life-cycle, individuals discount

with both the time preference and probability of survival, which breaks the equivalence between

borrowing and saving in the life-cycle. Second, in many countries instruments designed for oldage

savings benefit from liquidity limitations coupled with tax redemptions, alternative investment

schemes, etc., which breaks the equivalence between the net interest earned in the pension system

as opposed to outside the pension system. For these reasons, the solutions when pension system is

abolished cannot be extended to the reforms which consist of replacing a PAYG obligatory pension

system with a funded pension system.

To the best of our knowledge, Roberts (2013) proposes a notable exception, i.e. a model seeking

an actual optimal transition path from PAYG to a funded system with no lump-sum transfers to

compensate the loosing cohorts. In his framework, replacing the PAYG security with an unfunded

one, raises temporarily capital accumulation, which lowers the interest rate, thus promoting the

adoption of a new technology. Increased total factor productivity generates – for some parameter

values – a series of supply side general equilibrium effects that make the transition cohorts better

off, relative to PAYG scenario.

In this paper we build on the rich earlier literature to propose a way to find a an optimal

transition path from a PAYG pension system to a funded system. We develop a fairly standard


OLG model with multiple cohorts of decreasing mortality and exogenous technological progress. In

the baseline scenario the economy is in transition from a pay-as-you-go defined benefit system to

a pay-as-you-go defined contribution system. In the reform scenario on the transition from DB to

DC, the pay-as-you-go pillar is complemented by a funded pillar. The contribution rates are kept

constant, but the private voluntary savings are subject to capital income tax, whereas the obligatory

pension savings are not, which replicates the feature of many pension systems across the globe, see

Gruber and Wise (2008). We contribute to the literature by proposing an instrument of phasing

in the reform, which provides fiscal pressure relief to the living cohorts. This instrument consists

of compensating the transition cohorts in the PAYG DC system. The compensation occurs via a

higher than balanced indexation schedule. Thus – unlike for example Lump Sum Redistribution

Authority – the instrument is feasible. This instrument is at the same time a state variable in the

process of identifying the optimal transition path to a (partially) pre-funded pension system. We

thus 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 standard

in the literature, featuring the demographics of an aging economy about to decelerate the rate of

technological progress due to catching up. In fact, there are a few countries which underwent a

similar reform (transition from PAYG DB to partially funded DC), for example Chile, the Baltic

States, Sweden as well as Poland, whose data we use. The details of the calibration are described

in section 4 the data for a country which actually underwent such a reform in 1999 – Poland. The

results as well as analysis of their stability are discussed in 5, while in the concluding paragraphs

we derive the policy implications of this study.

2 The model

There is a constant returns to scale Cobb-Douglas aggregate production function and constant

labor-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, and

are 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 consumption

and leisure 1 − l j,t , where l j,t denotes fraction of time supplied to the labor market by agent aged

j in year t.

2.1 Production

Using 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:


(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 augmenting

exogenous economic growth γ t = z t /z t−1 . 2 Note that if the return rate on capital is r t then the rental

2 Bouzahzah et al. (2002) and Romaniuk (2009) discuss the sensitivity of OLG models to the assumptions

concerning growth in the light of policy reforms and show that when analyzing pension systems, there is little


ate must be r t +d, where d denotes capital depreciation. Firm optimization naturally implies w t =

(1 − α)Kt α zt

1−α L −α



r t + d = αKt

α−1 (z t L t ) 1−α

2.2 Consumers’ choice

Consumers optimize lifetime utility derived from leisure and consumption:


U j,t = u(c j,t , l j,t ) + β s π j+s,t+s

u (c j+s,t+s , l j+s,t+s ) , with (2)

π j,t


u (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 and

j + 1. Discounting takes into account time preference β and probability of survival π. Consumers

know the age and time specific mortality rates ex ante. At all points in time, consumers who survive

until the age of j = 79 die with certitude when reaching the age of J = 80. Changes in longevity

and fertility rates are operationalized in our model by decreasing across time the size of the 20-year

old cohort as well as decreasing the mortality rates.

Consumers are free to chose their labor supply during the working period, but once they reach

the age of ¯J they retire and are unable to supply any labor. In exchange for supplying age-specific

amount of labor l j,t consumers receive earned income of w t , which is the market clearing marginal

productivity of labor. The labor income tax τ l and social security contributions τ s are deducted

from 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 unintentional

bequests, which are then redistributed equally within their cohort as beq j,t . Thus, for working age

population (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 provides

the analytic solution of this consumer problem.

2.3 Pension system and the government

The 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. Pension

systems are indexed by ι, which corresponds to either defined contribution PAYG (frequently

referred 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 initial

steady state). The baseline scenario consists of a transition to a defined contribution pay-as-you-go

system, whereas the reform scenario consists of a defined contribution system which is partially


or no effect of endogenizing productivity growth. Thus, for clarity, we model this economy as governed by exogenous

productivity growth.


Defined contribution PAYG (NDC) The DC pension system collects contributions and uses

them to finance the contemporaneous benefits, but records them as individual stock of pension

savings and at retirement converts this stock to an annuity. For simplicity we denote by τ NDC the

obligatory contribution that goes into the PAYG system, whereas b NDC denote benefits from this

component of the pension system. The benefit at agent’s age ¯J is given by:

∑ [



b NDC s=1

Π s i=1 (1 + rNDC




¯J+i−1 ) τ NDC

t− ¯J+s−1 w t− ¯J+s−1 l s,t− ¯J+s−1

∏ J

s= ¯J π (6)


where 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 a

stock of actual savings, which are converted to an annuity at retirement. The benefit at agent’s

age ¯J is given by:

∑ [



b F ¯J,t DC s=1

Π s i=1 (1 + rF




¯J+i−1 ) τ F DC

t− ¯J+s−1 w t− ¯J+s−1 l s,t− ¯J+s−1

∏ J

s= ¯J π ,


which implies that after abandoning the defined benefit system, the pension benefits are given by

b DC

j,t = b NDC

j,t + b F j,t

DC and also τ DC = τ NDC + τ F DC with τ DC = τ DB

The DC systems are by construction balanced, i.e. the benefits paid out to an agent equal the

contributions 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 is

possible that the system has revenues in excess/short of current needs. We thus specify, that this

is the government that collects social security contributions and pays out pensions.

subsidy t = τt

NDC w t L t −


j= ¯J

N j,t b NDC

j,t (7)

We also assume that in the initial steady state the pension system runs a deficit to replicate the

empirical regularity that pension systems usually get reformed only as they gradually become

fiscally unsustainable. 3

The interest rate r F DC which accrues the individual obligatory savings in the capital pillar of the

pension system is a market interest rate r > r NDC . Thus, there are two main differences between

the obligatory and voluntary savings in our model. First, obligatory savings are determined at

the rate τ F DC and thus do not follow the life-cycle, whereas the private savings s j,t can even be

negative at any given point in life as long as the agent is not indebted at J. Indeed, the life-time

variability of private savings is one of the sources of the welfare gain in earlier studies which abolish

the pension system at all, e.g. Kotlikoff (1996), McGrattan and Prescott (2013). Second, there

is 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 system

dies. Our setting is identical to replacing a DB system with a DC one somewhere in the past and forming the funded

pillar when the last cohort with DB PAYG pensions reaches J.


which replicates the feature in many pension systems, including the US, Switzerland, UK, France

and with no exceptions all the countries which in the 1990s wave reformed their pension systems

following the guidelines by The World Bank, see Holzman and Stiglitz (2001), Gruber and Wise


The government Naturally, in addition to balancing the social security, the government collects

taxes 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 the

outstanding debt.

T t =





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 around

45% of GDP, which was the actual value of debt to GDP ratio in 1999. For the purpose of our

exercise, we hold this debt to GDP ratio fixed over time.

2.4 The reform: forming the funded pillar

In the baseline scenario the economy runs a defined contribution system funded on the pay-as-yougo

basis. Thus, agents expect gradual decrease in pension benefits due to longevity (we keep the

retirement 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 debt

overhang from the deficit in the pension system is reduced (the initial subsidy is positive). the

economy is in transition between DB and DC systems, with gradually decreasing pension benefits

due 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 the

size of the funded pillar is increased in a linear fashion by one percentage point, until it reaches 1/3

of the entire contribution (i.e. the share set by the legislation). This diversion of funds from the

Social 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 pressure

implies either of the two: higher taxes which hurt all the living cohorts or higher debt which hurts

all the cohorts repaying it. While it is not realistic to assume the debt can increase for as long as it

takes to finance the transition, the actual burden on the future generations comprises not only debt

per se, but also the servicing costs. Since the interest rate is typically in excess of the contributions

growth, 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 the

transition cohorts to receive an indexation of pension benefits in excess of r NDC . In fact, we index

both contributions and post-retirement benefits in the notional pillar with

r NDC′


= r NDC


+ g(r F DC


− r NDC

t ),

where g denotes the generosity in excess of the rules specified in (6) i.e. defined contribution

system. This way the transition cohorts receive from the pension system slightly more than they

contributed, but the fiscal burden occurs only once these cohorts actually retire, i.e. is costly


postponed to the future. The indexation factor g can be either year specific or cohort specific. In

the case of the former, any given cohort may receive a different lifetime compensation scheme than

either 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 somehow

analogous 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 (only

to net out the welfare loss) to the selected cohorts. Thus, it is possibly second-best from a welfare

perspective (with LSRA offering superior welfare outcomes due to the lack of general equilibrium

effects), but a politically more feasible option.

2.5 Welfare accounting

The 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 and

welfare 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 of

the 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 signifies

welfare improvement for cohort born in period t.

While this method of welfare accounting is standard in the literature, it also has the advantage

of directly measuring the welfare of each single cohort. Since our interest in this paper lies in

seeking 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 a

pre-funded pillar in an optimal way.

3 The algorithm for an optimal transition path

This section describes the algorithm of seeking an optimal path of introducing the funded pillar

in the pension system. It provides characteristics of program modules and presents the general

algorithm. The model is solved twice: for the baseline and for the reform scenario. First, using

the Gauss-Seidel algorithm we compute the initial and the final steady state. In the first steady

state the share of pension savings that goes into the funded pillar is equal to strictly 0. In the final

steady state it amounts to a rate of 33%. 4 The initial and the final steady states do not depend on

the way the pre-funded pillar is established and thus the path of the instrument has no bearing on

these 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 about

220 periods. Yet, to allow the path seeking algorithm more flexibility, we allow for the instrument

to be at play for the first 200 periods. While the impact of actual generosity is negligible after

4 The actual value of this parameter has been provided by the Polish legislation, but remains irrelevant for the

proper functioning of the algorithm.


150 periods (all cohorts already participate in the funded pillar), computationally it allows more

flexibility. Thus, We set the length of the transition path to 300 periods (to cover at least 200

periods 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 pension

system which was initially pay-as-you-go, both being defined contribution systems. We thus seek

for such a schedule of generosity g t so that no cohort was worse off due to introducing a funded


At the beginning the path of generosity g t is set to 0. We seek a vector of T = 200 values of

g t , which allows for the reform scenario to be at least weakly preferred to the baseline scenario

for all cohorts. 5 Throughout the exercise we employ the minimax criterion, which in our context

translates to the minimization of the loss of the biggest loser of the reform. In that way, even if the

outcome of the reform for some of the cohorts is negative, we at least ensure that the maximum

loss is not too severe. To be specific, the adopted threshold of .001 translates to a loss of a dollar

out of each thousand dollars of the lifetime income.

The year-specific version of the algorithm iteratively identifies those cohorts whose welfare

equivalent is lower than the threshold and increases the generosity for the last 20 years of life of

the 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 experience

an increase in welfare. We do so either until the threshold is hit or until the iterations limit of 1000

is reached.

The cohort-specific version of the algorithm relies on allocating ‘excessive’ indexation of pension

benefits to the cohorts who bear the cost of the reform. The procedure first identifies the most hurt

cohort and then slightly increases the generosity for this cohort. In this case there is no reason to

suspect there are spillovers to the neighbors of the cohort in terms of welfare, what is more likely is

that the welfare of its neighbors is decreased via general equilibrium effects. Thus, the procedure

usually requires more iterations to reach a solution. We do so either until the threshold is hit or

until the iterations limit of 1000 is reached.

4 Calibration

The calibration procedure aims to replicate certain micro- and macroeconomic features of the Polish

economy in 1999. The demographic structure is retrieved from the demographic projection by EU’s

Economic Policy Committee Working Group on Aging Populations and Sustainability (henceforth

AWG) 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 fertility

and 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 assumed

to 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 depreciation

rate is assumed at 5%. The discounting factor β and depreciation rate d are calibrated to yield the

5 A clear corollary of this research is that it is perfectly possible that some cohorts are worse off from having their

DB pensions being replaced by DC pensions. However, our focus in this study is on seeking an optimal path of

introducing a funded pillar – not reforming the way the pensions are computed.


Figure 1: AWG’s projections

Population aged 20−99 (in millions)

20 25 30

2000 2050 2100 2150 2200


Total Factor Productivity growth rate (in %)

1.5 2 2.5 3 3.5 4

2000 2020 2040 2060


(a) Total population size

(b) Total Factor Productivity

share of investment in GDP at the level of 21% 6 and the interest rate of approximately 7%.While

this value may seem high, note that it applies to the economy still in transition with high TFP

growth. Nishiyama and Smetters (2007) find the corresponding interest rate of 6.2% for the US. 7

The leisure preference parameter φ was calibrated to replicate the labor force participation rate of


The share of government expenditure in GDP is assumed to be constant both in the steady

states as well as along the transition at the level of 20%. The initial debt to GDP ratio is set at

the level of 45%, corresponding to the data. We set the capital income tax rate at the de iure level

of 19% as there are no exemptions. Note however, that the return on assets in the funded pension

funds is tax-free. The consumption tax τ c was set at 11%, which matches the rate of revenues from

this tax in aggregate consumption in 1999. The social security contribution rate was calibrated to

replicate the resulting pensions to GDP ratio of 5%. We allow the income tax rate to vary freely

in order to close the government budget constraint. The resulting 17.4% is slightly higher than

the 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.2

sums up the above considerations.

5 Results

In 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 clear

benefit 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 bar

high for the path seeking algorithm. For example, Makarski et al. (2015) show that financing the

6 Depending on the period over which the average is taken, it ranges from 20.8% for a 10 year average (5 years

before and 5 years after the reform), 23.1% for 2 year window and 24.1% in the year of the reform. The average for

the 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 pension

funds with a balanced portfolio strategy in the period 1999-2009.


Table 1: Calibrated parameters


α capital share of income 0.33

d depreciation rate 0.05

β discounting factor 0.9735

φ preference for leisure 0.825

γ 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 market

interest rate, whereas ? allows for the unconstrained growth of debt. In our setting the debt is not

allowed to increase (all fiscal adjustment needs to occur instantaneously via taxes). The shadow

price of the implicit debt is exactly the indexation of pensions 8 – pure in the case of the baseline

scenario 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. We

discuss two separate sets of results – for the cohort-specific generosity and for the time-specific

generosity, as described above. This analysis is done with the assumption that the funded pillar

is introduced immediately but at a slow rate, which is our preferred calibration. The intuition

behind such a modeling choice is that a slowly introduced reform should have relatively low welfare

effects on each cohort for two reasons – the amount of funds diverted from the PAYG pillar to the

pre-funded one in each year (pension reform costs) is low, whereas the share of pension benefits

coming from the funded pillar (pension reform benefits) increases very slowly. Subsequently, the

welfare effects and the performance of the algorithm/policy instrument depending on the rate and

the timing of the reform. We explicitly compare delayed reforms to the immediate ones and the

slow 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 the

demography, technological progress as well as preference parameters.

5.1 Baseline calibration

In the conservative parametrization we find the generosity path to compensate mostly the future

cohorts, who start bearing the costs associated paying ‘excessively’ high pensions to the initial

8 See Kuhle (2010) for an extensive comparison of implicit and explicit pension debt with idiosyncratic shocks.


transition generations. Thus, generosity is relatively low at first, because at this stage the majority

of 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, the

welfare costs appear. We report the adjustment in taxes, which signifies the timing of periods in

which 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 the

timing is clearly related, there is no clear translation between the the behavior of the indexation

generosity 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 .015

1900 2000 2100 2200 2300

Year of birth

Consumption equivalents from baseline of no generosity

Consumption equivalents from best found year−specific path

Consumption equivalents from best found cohort−specific path

Year−specific generosity path

0 .1 .2 .3 .4

Cohort−specific generosity path

0 .05 .1 .15

2000 2100 2200 2300


1900 2000 2100 2200 2300

Year of birth

(b) Path of year-specific generosity t

(c) The path of cohort-specific generosity t

Comparing the two welfare paths, clearly the cohort-specific generosity has a larger potential

to smoothen consumption, which is also signified by the smaller and more monotonic per-cohort

transfers, as expressed by the ‘excessive’ indexation. Larger scope for fine tuning the welfare effects

does not translate, however, to a substantially lower number of ‘losing’ cohorts, when compared to

the per-year generosity.

In the above results a number of exogenous assumptions may be affecting the results. First, the

reform is really slow, but starts immediately, which may interact with the demographic transition


and the relative situation of the initial old. To address this point in the subsequent section we

explicitly analyze the effects of timing and rate of the reform on the welfare and the performance

of 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 rate

is relatively high, which favors the implicit debt over the explicit one. Thus, in the next section

we test if the algorithm is able to find a similarly welfare enhancing path if these calibrations are

relaxed towards more conservative ones.

5.2 The timing and the rate of the reform

We consider the following modeling choices for the timing and the rate of the reform. First, we

want to analyze what is the role for the timing of the reform. To address this point we allow

two specifications: immediate and delayed phasing in of the reform. With the delayed phasing in

of the reform, the initial pension benefit recipients do not bear the costs of forming the capital

pillar 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 when

the 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 causes

larger fiscal adjustment to finance the gap in the PAYG pillar of the pension system. Yet, it also

contributes to higher capital accumulation. Because we keep public debt constant, there is no

crowding out of the private capital, thus the obligatory pension savings in the capital pillar fully

translate 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 benefits

for the cohorts with negative consumption equivalents – allows to smoothen consumption despite

increased taxation. All analyzed paths for the phasing in of the capital pillar are displayed in

Figures E.8 and E.10 in the Appendix. We show the results for the year-specific generosity paths. 9

The basic intuition for the results is as follows. For the reforms paths which burden a larger

number of adjacent cohorts may, the instrument / algorithm may be unable to provide significant

improvements 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 require

higher compensation. The two effects are likely to propagate each other. Thus, one should expect

that neither for the very fast nor for the very slow reforms an optimal path can be found. As far

as delaying is concerned, for the reforms which are enacted later, finding an optimal path should

be more feasible, because all affected cohorts are able to fully internalize the change in the pension

system. 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 the

instrument is able to compensate the oldest cohorts, but as a consequence cohorts born between

2000 and 2030 loose as much consumption equivalent as 1% of their lifetime income. Similar results

are 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 capital


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.


Figure 3: Welfare effects of reform with generosity t for the alternative phasing in of the reform

−.02 −.01 0 .01 .02 .03

1900 2000 2100 2200 2300

Year of birth

−.02 −.01 0 .01 .02 .03

1900 2000 2100 2200 2300

Year of birth

Reform immediate, no generosity

Reform immediate, with generosity

Reform delayed, no generosity

Reform delayed, with generosity

Reform immediate, no generosity

Reform immediate, with generosity

Reform delayed, no generosity

Reform delayed, with generosity

(a) Immediate reform

(b) 1 pp per year

−.01 0 .01 .02 .03

1900 2000 2100 2200 2300

Year of birth

−.005 0 .005 .01 .015

1900 2000 2100 2200 2300

Year of birth

Reform immediate, no generosity

Reform immediate, with generosity

Reform delayed, no generosity

Reform delayed, with generosity

Reform immediate, no generosity

Reform immediate, with generosity

Reform delayed, no generosity

Reform delayed, with generosity

(c) 0.5 pp per year

(d) 0.25 pp per year

66 years and 132 years to complete. The extremely slow reform in Figure 3d is similar to the

baseline results in Section 5.1. The intermediate solutions show the specificity of the analyzed

policy instrument. First, it concentrates the welfare costs of the reform in fewer cohorts – black

lines in Figures 3b and 3c. Second, in the case of the relatively slow reforms (66 years to reach

1/3 of contributions going to the capital pillar), delaying the reform combined with the policy

instrument arrives at nearly Pareto enhancing welfare path. In fact, the welfare costs fall short of


5.3 Testing the robustness to alternative exogenous assumptions

For the specification presented in Section 5.1 we develop a series of alternative calibrations which

permit to test if the algorithm proposed provides stable results. This specification encompasses

a slow and immediate reform, which implies relatively low welfare costs, but also little room for

the policy instrument / algorithm to provide welfare enhancement. Thus, it is a conservative

environment to test the sensitivity of the instrument / algorithm to the alternative assumptions

about the exogenous parameters. In each of the analysis we change one of the exogenous parameters


(or their paths), keeping all other calibration parameters unchanged. We also show the results form

the previous section as baseline scenario in each case.

The robustness checks concern: utility function parameters, exogenous fertility and technological

progress paths. For the utility function parameters we check two cases: lower patience

δ and lower preference for consumption 1 − φ. With lower patience agents are likely to prefer

contemporaneous consumption to future pension benefits. Thus, instantaneous tax adjustments will

require higher generosity to maintain the same welfare, which sets the bar for optimum transition

path 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 increased

contemporaneous 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 assumes

that it will be gradually decreasing. Yet, the fertility rate is already extremely low in the initial

steady state – at 1.4. Moreover, decreasing size of the population implies that initially there is a

much higher tax base, whereas in the future generosity needs to be provided for by a much smaller

population. To see of the algorithm is robust to this parametrization, we also include a scenario with

a constant rather than decreasing fertility. In practice, this implies that the population stabilizes

earlier in the simulation. Please note that this scenario does not affect the initial steady state


Finally, we also consider two alternative paths for the catching up. While in the final steady

state the economy is always projected to experience technological progress of 1.7% per annum, the

starting point is actually at 4% per annum. The speed of convergence may matter for the ability

to find an optimum path in the interaction with the speed of implementing the reform. Thus, we

include 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 the

actual values of ‘excessive generosity’ differ between the calibrations, the time variation is largely

the same. This is indicative of the property that the obtained ‘optimal’ reform paths are fairly

stable for a given schedule of introducing the funded pillar.

The results are fairly intuitive. For the agents with higher preference for leisure, generosity

is lower in both cases, as they need to be compensated for decreased consumption less. In the

scenarios with faster TFP growth, part of the necessary increase in taxation can be compensated

for by the faster accumulation of wealth. Thus, the compensations needed are lower. The opposite

holds for the paths with slower TFP growth. There are also no significant differences in the cohort

distribution of welfare – if preferences exhibit higher impatience welfare ‘break-even’ is experienced

already by older cohorts, which is consistent with the expectations. As expected, the role of fertility

scenario is negligible.

6 Conclusions

Our objective in this paper has been to find a transition path from a pay-as-you-go scheme to

a pre-funded scheme that leaves no cohort worse off after the reform in the defined contribution

setting. While the introduction of the funded pillar may well be gradual, the costs still need to be

distributed over subsequent cohorts in the search for Pareto-optimality. Thus, forming a funded

pillar when PAYG scheme exists requires either of the three: increase in tax-rate, an increase in

public debt or a decrease in other public spending. It is customary in this literature to treat non-


Figure 4: Welfare effects of reform with generosity t for the alternative calibrations

0 .005 .01 .015 .02

0 .005 .01 .015 .02

1900 2000 2100 2200 2300

Year of birth

Constant fertility

Faster TFP growth convergence

Slower TFP growth convergence

Lower patience

Higher weight of leisure in utility

Baseline scenario

1900 2000 2100 2200 2300

Year of birth

Constant fertility

Faster TFP growth convergence

Slower TFP growth convergence

Lower patience

Higher weight of leisure in utility

Baseline scenario

(a) year-specific generosity t

(b) cohort-specific generosity t

Figure 5: Paths of generosity t for the alternative calibrations

0 .1 .2 .3 .4

0 .05 .1 .15

2000 2100 2200 2300


Constant fertility

Faster TFP growth convergence

Slower TFP growth convergence

Lower patience

Higher weight of leisure in utility

Baseline scenario

1900 2000 2100 2200 2300

Year of birth

Constant fertility

Faster TFP growth convergence

Slower TFP growth convergence

Lower patience

Higher weight of leisure in utility

Baseline scenario

(a) year-specific generosity t

(b) cohort-specific generosity t

pension public expenditure as a fixed variable, because typically it is modeled as pure waste and

thus changing its value directly alternates welfare accounting. Increase in taxes concentrates the

costs 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 often

national laws and international treaties, whereas systematic increase in public debt can translate

to 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-sum

taxation to propose Pareto-optimal reforms in the pension systems. Alternatively, previous studies

designed a redistribution scheme between cohorts (e.g. a governmental Lump Sum Redistribution

Authority). The assumption of balanced pension system is not likely to hold in reality, whereas

lump sum taxation proves to be politically challenging. We propose an alternative approach, by

shifting the costs of the reform to the future generations by the means of implicit rather than

explicit debt. We develop a model of an OLG economy, which starts from a PAYG DC system

and reforms to a (partially) pre-funded DC. The funded pillar is introduced gradually over one


generation. We keep the debt share in GDP and non-pension government expenditure fixed from

the baseline scenario, but introduce for the transition cohorts additional indexation scheme. This

indexation scheme compensates the cohorts which experience the costs of the reform. We thus rely

on implicit debt as an instrument to find an optimal transition path. The advantage of relying on

implicit 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 a

series of simulations searching the indexation scheme that leaves no cohort worse off. Our analysis

is closely calibrated to the case of Poland, which implemented such reform in 1999. However, we

perform a variety of sensitivity analyses, altering the set of assumptions concerning the demographic

scenarios, preferences and exogenous technological progress path. In each of these cases our policy

instrument – 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 between

the contributions to the pension system and the subsequent pension benefits. While they maximize

lifetime utility and have perfect foresight, there is no derivative of the pension benefits with respect

to labor in the consumer problem solution. Providing this extension is likely to result in a behavioral

response in terms of labor supply decisions. This brings about another premise, i.e. the shape of

the utility function. Namely, with the Cobb-Douglas utility function, there is only minor reaction of

labor to changes in the pension system. Relying on a different utility function – such as Greenwood

et al. (1988) – could also matter for the ability of the algorithm to find the Pareto-optimal transition




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Appendix A. Consumer problem solution

For j < ¯J:


c 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−1

s=0 (1 + φ) β s π j+s,t+s

π j,t

+ ∑ ( )] (A.11)


s= ¯J−j β s π j+s,t+s

π j,t

φ(1 + τt c )c j,t

l j,t = 1 −

(1 − τt s)(1 − τ t l)w 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 − (1 + τ c t )c j,t ,

Ω j,t =

Γ j,t =




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

∏ s

i=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

∏ s

i=1 (1 + (1 − τ t+i k )r .


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+s

π j,t

φ(1 + τt c )c j,t

l j,t = 1 −

(1 − τt s)(1 − τ t l)w t

s 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 ,


Γ j,t =


s= ¯J−j

(1 − τt+s)b l ι j+s,t+s + beq j+s,t+s

∏ s

i=1 (1 + (1 − τ t+i k )r .


The numerators in equations (A.12) and (A.13) represent the present discounted value of the

remaining lifetime income.

Appendix B. Market clearing conditions and general equilibrium

The goods market clearing condition is standardly defined as


N j,t c j,t + G t + K t+1 = Y t + (1 − d)K t ,



where we denote the size of the j-aged cohort at period t as N j,t . This equation is equivalent

to stating that at each point in time the price for capital and labor would be set such, that the


demand for the goods from the consumers, the government and the producers would be met. This

necessitates clearing in the labor and in the capital markets. Thus labor is supplied and capital

accumulates according to:


L t = N j,t l j,t and K t+1 = (1 − d)K t +




N j,t ŝ j,t ,


where ŝ j,t denotes private savings s j,t as well as accrued obligatory contributions in the funded

pillar 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 the

problem 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.


Appendix C. Alternative calibrations

Table C.2: Alternative calibrations

Parameters Baseline Impatience Lazy

α capital share of income 0.33 0.33 0.33

d depreciation rate 0.05 0.05 0.05

β discounting factor 0.9735 0.9700 0.9735

φ preference for leisure 0.825 0.825 1

γ 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 4

2000 2020 2040 2060 2080 2100


Standard TFP growth path

Faster TFP growth convergence

Slower TFP growth convergence


Appendix D. Fiscal adjustments on the transition path

Figure D.7: Changing pension system balance and tax adjustments

0 .01 .02 .03

2000 2100 2200 2300


SIF subsidy −− no generosity

SIF subsidy −− with year−specific generosity

SIF subsidy −− with cohort−specific generosity

(a) Deficit in the pension system as a share of GDP in the

baseline scenario and in the reform scenarios

.13 .14 .15 .16 .17

2000 2100 2200 2300


Labor income tax −− no generosity

Labor income tax −− with year−specific generosity

Labor income tax −− with cohort−specific generosity

(b) Taxes in baseline scenario of no generosity and the

reform scenarios with cohort and year specific generosities


Appendix E. Alternative paths of the reform

Figure E.8: Immediate reform paths

.6 .7 .8 .9 1

2000 2050 2100 2150 2200


Reform immediate, funded pillar increases by 0.25 pp. per year

Reform immediate, funded pillar increases by 0.5 pp. per year

Reform immediate, funded pillar increases by 1 pp. per year

Reform immediate, funded pillar created at once

Figure E.9: Immediate reform paths

0 .5 1 1.5

2000 2050 2100 2150 2200


Reform immediate, funded pillar increases by 0.25 pp. per year

Reform immediate, funded pillar increases by 0.5 pp. per year

Reform immediate, funded pillar increases by 1 pp. per year

Reform immediate, funded pillar created at once


Figure E.10: Immediate reform paths

.6 .7 .8 .9 1

2000 2050 2100 2150 2200


Reform delayed, funded pillar increases by 0.25 pp. per year

Reform delayed, funded pillar increases by 0.5 pp. per year

Reform delayed, funded pillar increases by 1 pp. per year

Reform delayed, funded pillar created at once

Figure E.11: Immediate reform paths

0 .2 .4 .6 .8 1

2000 2050 2100 2150 2200


Reform delayed, funded pillar increases by 0.25 pp. per year

Reform delayed, funded pillar increases by 0.5 pp. per year

Reform delayed, funded pillar increases by 1 pp. per year

Reform delayed, funded pillar created at once


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